From 8c985a102247ef3c1e537923238ab70461e7ebf4 Mon Sep 17 00:00:00 2001 From: Cafolkes Date: Sat, 10 Aug 2019 11:19:46 -0700 Subject: [PATCH 1/9] Initial commit of implementation of Laplace averaging based Koopman eigenfunction generation for prediction and control of inverted pendulum. Core functionality implemented but predicition still not working reliably. --- scripts/Koopman-ID/.DS_Store | Bin 0 -> 6148 bytes scripts/Koopman-ID/DMD/CompanionMatrix_DMD.m | 54 + scripts/Koopman-ID/DMD/Exact_DMD.m | 71 + scripts/Koopman-ID/DMD/Hankel_DMD.m | 65 + scripts/Koopman-ID/DMD/HarmonicAverage.m | 22 + scripts/Koopman-ID/DMD/SVDenhanced_DMD.m | 74 ++ scripts/Koopman-ID/Resources/.DS_Store | Bin 0 -> 6148 bytes .../interfaces/matlab/qpOASES.mexmaci64 | Bin 315512 -> 398620 bytes .../matlab/qpOASES_sequence.mexmaci64 | Bin 324192 -> 403220 bytes scripts/Koopman-ID/XY_DDMD_R4.m | 1171 +++++++++++------ scripts/Koopman-ID/construct_eigfuncs.m | 107 ++ scripts/Koopman-ID/koopman_eigen_id_laplace.m | 140 ++ scripts/Koopman-ID/main_comparison_pendulum.m | 37 +- scripts/Koopman-ID/normalize_data.m | 11 + scripts/Koopman-ID/obsvk.m | 6 + scripts/Koopman-ID/phifun_mat.m | 7 + scripts/Koopman-ID/rbf_grad.m | 53 + .../Koopman-ID/test_eigfunction_generation.m | 244 ++++ scripts/Koopman-ID/utils/collect_data.m | 8 +- .../utils/evaluate_model_performance.m | 27 +- scripts/Koopman-ID/utils/extendedDMD.m | 22 +- scripts/Koopman-ID/utils/koopman_eigen_id.m | 109 +- scripts/Koopman-ID/utils/learn_B_koop.m | 101 +- .../Koopman-ID/utils/sim_closed_loop_mpc.m | 23 +- scripts/Koopman-ID/utils/sim_prediction.m | 17 +- 25 files changed, 1854 insertions(+), 515 deletions(-) create mode 100644 scripts/Koopman-ID/.DS_Store create mode 100755 scripts/Koopman-ID/DMD/CompanionMatrix_DMD.m create mode 100755 scripts/Koopman-ID/DMD/Exact_DMD.m create mode 100755 scripts/Koopman-ID/DMD/Hankel_DMD.m create mode 100755 scripts/Koopman-ID/DMD/HarmonicAverage.m create mode 100755 scripts/Koopman-ID/DMD/SVDenhanced_DMD.m create mode 100644 scripts/Koopman-ID/Resources/.DS_Store mode change 100644 => 100755 scripts/Koopman-ID/Resources/qpOASES-3.1.0/interfaces/matlab/qpOASES.mexmaci64 mode change 100644 => 100755 scripts/Koopman-ID/Resources/qpOASES-3.1.0/interfaces/matlab/qpOASES_sequence.mexmaci64 create mode 100644 scripts/Koopman-ID/construct_eigfuncs.m create mode 100644 scripts/Koopman-ID/koopman_eigen_id_laplace.m create mode 100644 scripts/Koopman-ID/normalize_data.m create mode 100755 scripts/Koopman-ID/obsvk.m create mode 100644 scripts/Koopman-ID/phifun_mat.m create mode 100644 scripts/Koopman-ID/rbf_grad.m create mode 100644 scripts/Koopman-ID/test_eigfunction_generation.m diff --git a/scripts/Koopman-ID/.DS_Store b/scripts/Koopman-ID/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..da67d24bc6df9f8baf526f268c3f11900c0e9e51 GIT binary patch literal 6148 zcmeHK%}(1u5Z=uXT?Zj@Kos?~m)?Su03x{{nGg=WRdUrHC<-=?5=)LZiXE&r4f1=G zm*LC<@H(CSNt6&MryxKx((E@oJG1NkZ0&joA+$O2YlM^uAq-Sv&VuFgUABxgKp zfK<&9ducrILg6jNqT_WkAm3e)j7UNP5|a7%cjpblAWo{)dz#A^ifbPjE3wi>>D=jw z)JfcA6gS=Bne2Ualb&bnZ$gAK+x5N&owQxvIuTLg22p3Ik^{d3AwPcvkuTDw7)5@p z=5h1@V-~a8({AY6gS943@ zBUG|!a0)kI%xyC4)|^YxA54cANBg4HGM_NBwkuhBd3*dY!ha@W(EHyCSI#gBMbw2^ zr%ychC}<)B$N)01dIrql!`4^76s~{_AOmlV0eL=fP>Hs|OrttFpiw0NU;*4pz?QXy z%;5%YgPBGc0iik-P^VH;Vo;q9es1DygPBI1PN*q9s9Bks3WciGVScW{3AHt1iwqzG ziwxv-w=D1f)2HkI#U$(@1IWO;Vt^J}POAx9GI#6J=Hy+=K`%h1$hb`7ECmd86+^7N dig!SjfS*eP&^DN91P=)Q2q+q`K?Yuwfq$w+Y)$|G literal 0 HcmV?d00001 diff --git a/scripts/Koopman-ID/DMD/CompanionMatrix_DMD.m b/scripts/Koopman-ID/DMD/CompanionMatrix_DMD.m new file mode 100755 index 0000000..9de28e4 --- /dev/null +++ b/scripts/Koopman-ID/DMD/CompanionMatrix_DMD.m @@ -0,0 +1,54 @@ +function [ KModes,KEv,Norms ] = CompanionMatrix_DMD( Data ) +% Dynamic Mode Decomposition as presented by +% "Spectral analysis of nonlinear flows" by Rowley et al., Journal of FLuid +% Mechanics, 2009 + + +% inputs : +% Data - the data matrix: each column is a a set of measurements done at +% each instant - the sampling frequency is assumed to be constant + +% outputs: +% 1 - KModes- Koopman or Dynamic Modes: structures that scale exponentially +% with time - the exponent is given by Koopman or Dynamic Eigenvalues and +% could be imaginary as well + +% 2- KEv - Koopman or Dynamic Eigenvalues: the exponents and frequencies +% that make up the time-varaiation of data in time + +% 3- Norms - Euclidean (vector) norm of each mode, used to sort the data + + + + +X=Data(:,1:end-1); + +c = pinv(X)*Data(:,end); + +m = size(Data,2)-1; +C = spdiags(ones(m,1),-1,m,m); +C(:,end)=c; %% companion matrix + +[P,D]=eig(full(C)); %% Ritz evalue and evectors + +KEunsrtd = diag(D); %% Koopman Evalues +KMunsrtd = X*P; %% Koopman Modes + +Cnorm = sqrt(sum(abs(KMunsrtd).^2,1)); %% Euclidean norm of Koopman Modes +[Norms, ind]=sort(Cnorm,'descend'); %% sorting based on the norm + +KEv= KEunsrtd(ind); %% Koopman Eigenvalues + +KModes = KMunsrtd(:,ind); %% Koopman Modes + + + +end + + +%=========================================================================% +% Hassan Arbabi - 08-17-2015 +% Mezic research group +% UC Santa Barbara +% arbabiha@gmail.com +%=========================================================================% \ No newline at end of file diff --git a/scripts/Koopman-ID/DMD/Exact_DMD.m b/scripts/Koopman-ID/DMD/Exact_DMD.m new file mode 100755 index 0000000..3c5c7c4 --- /dev/null +++ b/scripts/Koopman-ID/DMD/Exact_DMD.m @@ -0,0 +1,71 @@ +function [ ProjectedModes,DEv,ExactModes,Norm ] = Exact_DMD( X,Y,varargin ) +% Dynamic Mode Decomposition as presented by +% "On Dynamic Mode Decomposition: theory and applications" by Tu et al., +% arXiv, 2013 + + + +% inputs : +% Data Sets X and Y- should have the same size +% each column of X is a set of measurements done at an instant +% each column of Y is the image of the corresponding columns from X + +% or + +% ( X,Y,Tol ) - with Tol (optional) being the threshold for filtering thru SVD - the +% default value is 1e-10 + + + +% outputs: +% 1 - Projected Dynamic Modes +% 2 - Exact Dynamic Modes +% 3 - Dynamic Eigenvalues +% 4 - Norms - Euclidean (vector) norm of each mode, used to sort the data + +% setting SVD hard threshold +if isempty(varargin) + Tol=1e-10; +else + Tol=varargin{1}; +end + + +disp(['Tolerance used for filtering in DMD:',num2str(Tol)]) + +[U,S,V]=svd(X,'econ'); + +k = find(diag(S)>Tol,1,'last'); +disp(['DMD subspace dimension:',num2str(k)]) + +U = U(:,1:k); V=V(:,1:k); S=S(1:k,1:k); + +Atilde = ((U'*Y) *V )* diag((1./diag(S))); + + +[W,DEv]=eig(Atilde); +DEv = diag(DEv); + +ProjectedModes = U*W; + +ExactModes = bsxfun(@times,1./DEv.',((Y*V)*S^(-1))*W); + + + +if nargout>3 + b = pinv(ProjectedModes)*X(:,end); % terminal coordinates in the Koopman subspace + [Norm,Index]=sort(abs(b),'descend'); + DEv = DEv(Index); + ProjectedModes = ProjectedModes(:,Index); + disp('modes sorted based on energy contribution to the last snapshot') +end +NORM=Norm +end + + +%=========================================================================% +% Hassan Arbabi - 08-17-2015 +% Mezic research group +% UC Santa Barbara +% arbabiha@gmail.com +%=========================================================================% \ No newline at end of file diff --git a/scripts/Koopman-ID/DMD/Hankel_DMD.m b/scripts/Koopman-ID/DMD/Hankel_DMD.m new file mode 100755 index 0000000..710f54e --- /dev/null +++ b/scripts/Koopman-ID/DMD/Hankel_DMD.m @@ -0,0 +1,65 @@ +function [ HModes, HEvalues, Norms ] = Hankel_DMD( Data,n,m,varargin ) +%HANKEL-Dynamic Mode Decompsition as presented in +% "Ergodic theory, dynamic mode decomposition and computation of Koopman +% spectral properties" by H. Arbabi and I. Mezic, arXiv:1611.06664 + +% inputs : +% Data - the data matrix: each row is a time series data on an observable +% m,n: number of columns and rows, respectively, in the data Hankel +% matrices: size(Data,2)>m+n and preferably m>>n + +% or + +% ( Data,n,m,tol )- with Tol (optional) being the threshold for filtering thru SVD - the +% default value is 1e-10 + + + + +% outputs: +% 1 - HModes- Dynamic modes which approximate the Koopman eigenfunctions + +% 2- HEvalues - Koopman or Dynamic Eigenvalues: the exponents and frequencies +% that make up the time-varaiation of data in time + +% 3- Norms - The L2-norm of Koopman eigenfunction contribution to the +% observable + + +% setting the SVD hard threshold for DMD +if isempty(varargin) + Tol=1e-10; +else + Tol=varargin{1}; +end + +index1 = 1:n; +index2 = n:n+m-1; + +X = []; Y=[]; + +for ir = 1:size(Data,1) + + % Hankel blocks () + c = Data(ir,index1).'; r = Data(ir,index2); + H = hankel(c,r).'; + c = Data(ir,index1+1).'; r = Data(ir,index2+1); + UH= hankel(c,r).'; + + % scaling of Hankel blocks + if ir>1 + alpha = norm(H(:,1))/norm(X(:,1)); + H = alpha*H; + UH= alpha* UH; + end + + % the data matrices fed to exact DMD + X=[X,H]; Y=[Y,UH]; + +end + % the DMD + [HModes,HEvalues,~,Norms ] = DMD.Exact_DMD( X,Y,Tol ); + + +end + diff --git a/scripts/Koopman-ID/DMD/HarmonicAverage.m b/scripts/Koopman-ID/DMD/HarmonicAverage.m new file mode 100755 index 0000000..5c059d9 --- /dev/null +++ b/scripts/Koopman-ID/DMD/HarmonicAverage.m @@ -0,0 +1,22 @@ +function [ Average ] = HarmonicAverage( Data,w,t,Filter) +% computes the harmonic average using the Filter + +HarmonicWeight = 2*exp(-1i*w'*t); + +% using the specified filter +if exist('Filter','var') + switch Filter + case 'Hamming' + HarmonicWeight=bsxfun(@times,HarmonicWeight,hamming(length(t))'); + case 'Hann' + HarmonicWeight=bsxfun(@times,HarmonicWeight,hann(length(t))'); + case 'exponential' % the exponential weighting defined by Das & Yorke 2015 + HarmonicWeight=bsxfun(@times,HarmonicWeight,ExpWeight(length(t))*length(t)); + otherwise + disp('filter not defined ... no filter used') + end +end + +Average = HarmonicWeight*Data'./length(t); + +end \ No newline at end of file diff --git a/scripts/Koopman-ID/DMD/SVDenhanced_DMD.m b/scripts/Koopman-ID/DMD/SVDenhanced_DMD.m new file mode 100755 index 0000000..13ac713 --- /dev/null +++ b/scripts/Koopman-ID/DMD/SVDenhanced_DMD.m @@ -0,0 +1,74 @@ +function [ DModes,DEv,Norm ] = SVDenhanced_DMD( Data,varargin ) +% Dynamic Mode Decomposition as presented by +% "Dynamic Mode Decomposition of numerical and experimental data" by Peter +% J. Schmid, Journal of Fluid Mechanics, 2010 + +% inputs : +% Data - the data matrix: each column is a a set of measurements done at +% each instant - the sampling frequency is assumed to be constant + +% or + +% ( Data,Tol ) - with Tol (optional) being the threshold for filtering thru SVD - the +% default value is 1e-10 + +% outputs: +% 1 - DModes- Koopman or Dynamic Modes: structures that scale exponentially +% with time - the exponent is given by Koopman or Dynamic Eigenvalues and +% could be imaginary as well + +% 2- DEv - Koopman or Dynamic Eigenvalues: the exponents and frequencies +% that make up the time-varaiation of data in time + +% 3- Norm - The norm here is set to the energy contribution of each mode to +% the last snapshot of data and then used to sort the modes + + + + +tic +if isempty(varargin) + Tol=1e-10; +else + Tol=varargin{1}; +end + +disp(['Tolerance used for filtering in DMD:',num2str(Tol)]) + +X=Data(:,1:end-1); +Y=Data(:,2:end); + + +[U,S,V]=svd(X,0); +r = find(diag(S)>Tol,1,'last'); +disp(['DMD subspace dimension:',num2str(r)]) + +U = U(:,1:r); +S = S(1:r,1:r); +V = V(:,1:r); + +Atilde = (U'*(Y*V)) / S; + +[w,lambda]=eig(Atilde); + +DModes = U*w; +DEv = diag(lambda); + +if nargout>2 + b = pinv(DModes)*Data(:,end); % terminal coordinates in the Koopman subspace + [Norm,Index]=sort(abs(b),'descend'); + DEv = DEv(Index); + DModes = DModes(:,Index); + disp('modes sorted based on energy contribution to the last snapshot') +end + +disp(['time to compute DMD:',num2str(toc)]) + +end + +%=========================================================================% +% Hassan Arbabi - 08-17-2015 +% Mezic research group +% UC Santa Barbara +% arbabiha@gmail.com +%=========================================================================% diff --git a/scripts/Koopman-ID/Resources/.DS_Store b/scripts/Koopman-ID/Resources/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..22ca61209c049a30e639b745054c4b2aca92acf9 GIT binary 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Drmac, I. Mezic, R. Mohr: Data driven modal decompositions: analysis -% and enhancements, SIAM Journal on Scientific Computing. -% (arXiv preprint, https://arxiv.org/abs/1708.02685v1) -% The code is prepared as a suplement to [1], only to facilitate easier -% understanding of algorithmic details and experimenting. -% Released under the 3-Clause BSD License, see below. -% =====......................................................................... -% Input -% ===== -% -% X (real/complex 2D-array) :: Data matrix. Assumed with more rows than columns. -% -% Y (real/complex 2D-array) :: Data matrix, Y = A*X for some otherwise -% inaccessible matrix A. Both X and Y should -% be tall matrices -% Separate code (another function, more memory -% efficient) is used if Y is just shifted X -% -% to_scale (string) :: Specifies whether snapshot normalization is -% needed. Equilibrating the snapshots in ell_2 -% is a good idea in this setting. It does not -% apply if a forgetting factor is incorporated. -% --> If to_scale == 'Scale' -% then X = X*D, Y = Y*D where D is diagonal -% matrix such that the new X has unit columns. -% -% to_center (string) :: Optional data centering. For experiments only! -% --> If to_center == 'Center' -% then X = X-(1/m)*X*ones(m,1)*ones(1,m) and -% Y = Y-(1/m)*Y*ones(m,1)*ones(1,m). Note that -% after centering, again, Y = A*X. -% --> Set to_center = 'Ignore', except for R&D -% experimenting. -% -% k_mode (integer) :: Truncation mode for setting numerical rank k of -% X, based on the singular values sigma(i). -% --> If k_mode == -1, k is selected as -% k = max( i : sigma(i) >= sigma(1)*tol. ) -% --> If k_mode == -2, k is selected as -% k = max( i>1 : sigma(i) > sigma(i-1)*tol -% --> If k > 0 and k <= min(m,n) -% the value of k_mode is then understood -% as the caller's request to use k=k_mode -% dominant left singular vectors of X. -% -% tol (real, >0 ) :: Tolerance threshold for truncating the -% singular values in defining numerical -% rank of scaled data. (Only for the values -% k_mode = -1, k_mode = -2.) -% [!] the recommended value is tol=n*eps, -% where eps is the round-off unit. -% -% nobal (string) :: Specifies whether to switch of the balancing -% in the function eig() that is used to -% compute the eigenvalues and eigenvectors -% of the Rayleigh quotient. -% --> If nobal == 'NoBalance', the balancing is -% not used. -% --> Set nobal = 'Ignore' if not sure what is -% this about. -% -% n_ref (integer) :: Specifies how many of the Ritz pairs will be -% refined. If n_ref exceds k, then n_ref is -% set to k (=the total number of Ritz pairs). -% [!!] In case of large dimensions, refining -% too many Ritz pairs is time consuming. -% -% ref_select (integer) :: Specifies how to select n_ref Ritz pairs -% that will be refined. -% --> If ref_select == 1, -% refine n_ref pairs with smallest residuals -% --> If ref_select == 2, -% refine n_ref pairs closest to a given target -% (see the input parameter target) -% --> If ref_select == 3, -% refine n_ref pairs closest to the imaginary axis -% --> If ref_select == 4, -% refine n_ref pairs closest to the unit -% circle -% --> If ref_select == 5, -% refine n_ref pairs closest to the real axis -% -% target (real/complex) :: Specifies a point of interest when selecting -% which Ritz pairs to refine (see ref_select) -% Only the Ritz values closest to target will -% be refined (and their Ritz vectors). -% -% overwrite (string) :: Specifes whether the refined Ritz vectors will -% overwrite the original ones, or returned in a -% separate array -% --> If overwrite == 'OverWrite' then the selected -% and improved Ritz vectors overwite the -% original ones. Otherwise, they are returned -% in a separate array. -% -% file_save (string) :: Specifies whether the selected variables -% shold be saved in a file. -% --> If file_save == 'NoSave', then no data is saved. -% --> Otherwise, selected variables are saved in -% the file file_save (given on input). -% ======........................................................................ -% Output -% ======........................................................................ -% Z (real/complex 2D-array) :: Ritz vectors, normalized so that -% ||Z(:,i)||_2 = 1. If the refinement is -% requested with overwrite option, then some -% of the columns of Z are refined Ritz -% vectors. Their indices are then listed in -% the integer array RSI. -% -% Lambda (real/complex 1D-array) :: Ritz values -% -% rez (real 1D-array) :: 2-norms of the reziduals -% rez(i) = ||A*Z(:,i)-Lambda(:,i)*Z(:,i)||_2 -% -% --> The following four arrays are void if the refiement is not requested. -% When calling this function, use tildas if they are not needed. -% -% RQ_ref (real/complex 1D-array) :: Rayleigh quotients with refined Ritz -% vectors. -% -% RSI (integer array) :: If n_ref > 0, RSI contains the indices -% of those Ritz pairs that have been refined. -% -% Z_ref (real/complex 2D-array) :: Refined Ritz vectors, in the case when -% they do not overwrite the orginal ones. -% -% rez_ref (real 1D-array) :: The residuals of the refined Ritz pairs. -% -% --> The following two array arguments are returned for post-procesing such as -% e.g. refinement. Use tildas if they are not needed. -% -% U (real/complex 2-D array) :: The left singular vectors of the matrix X. -% -% AxU_k (real/complex 2-D array) :: The underlying operator A applied to the -% k leading left singular vectors of X. -%............................................................................... -%............................................................................... -% AIMdyn Inc. -% Coded by Zlatko Drmac, Department of Mathematics, University of Zagreb. -% drmac@math.hr -% Version 1.0 , January 3, 2017. -% Version 1.1 , November 11, 2017. -%............................................................................... -%............................................................................... -% -% The data matrix is n-by-m and it is assumed that the number of its rows -% is much larger than the number of its columns. -% -[ n, m ] = size( X ) ; -if m > n - warning('XY_DMD_R4 :(o -> the number of snapshots exceeds space dimension'); -end -% Optionally, the data can be centered. This is a delicate issue. Only for R&D. -if ( strcmp(to_center,'Center') ) - X = X - ((1/m)*(X*ones(m,1))) * ones(1,m) ; - Y = Y - ((1/m)*(Y*ones(m,1))) * ones(1,m) ; -end -% The intermediate and the final results can be saved for post festum analysis. -save_to_file = ~strcmp(file_save,'NoSave') ; -% -if ( strcmp(to_scale,'Scale') ) - % Column norms of the X-data matrix - D = ones(m,1) ; - for i = 1 : m - D(i) = norm(X(:,i)) ; - end - % Remove zero columns (unlikely event, but ..) - JX = find( D > 0 ) ; - if ( length(JX) < m ) - X = X(:,JX) ; D = D(JX) ; Y = Y(:,JX) ; - m = size(X,2) ; - end - % - % The data samples are scaled to the unit ell_2 sphere and then the SVD is - % computed. The numerical rank is determined for the scaled data (uness the - % caller has hardcoded the number of left singular vectors). - % One has to be careful here, e.g. if SNR is small. Aditional scaling may - % be necessary with carefully chosen weights. If weighting is an issue, - % use the weighted version of DMD, see [1]. - % - [ U, Sigma, V ] = svd( X*diag(1./D), 'econ' ) ; S = diag(Sigma) ; -else - [ U, Sigma, V ] = svd( X, 'econ' ) ; S = diag(Sigma) ; -end -% -% Determine the numerical rank k of X, based on the k_mode and tol -if ( k_mode == -1 ) - % classics - if ( tol < eps ) - warning('XY_DMD_R4 :( -> parameter is too small') - end - k = find( S/S(1) > tol, 1, 'last' ) ; -elseif ( k_mode == -2 ) - % more conservative, useful for R&D purposes - if ( tol <= eps ) - warning('XY_DMD_R4 :( -> parameter is too small') - end - %k = find( S(2:m)./S(1:m-1) < tol, 1, 'first' ) ;must be changed, for now use - k = find( S(2:m)./S(1:m-1) > tol, 1, 'last' ) ; -elseif ( (k_mode > 0) && (k_mode <= min(m,n)) ) - % the caller inputs k as the required number of the left singular - % vectors of X - k = k_mode ; -else - error('XY_DMD_R4 :( -> parameter had an illegal value') -end -% -% The image of U(:,1:k) under the action of A -if ( strcmp(to_scale,'Scale') ) - AxU_k = Y*(diag(1./D)*( (V(:,1:k).*(ones(m,1)*(1./S(1:k))')))) ; -else - AxU_k = Y*(((V(:,1:k).*(ones(m,1)*(1./S(1:k))')))) ; -end -% -% -if ( n_ref > 0 ) - % refinement of Ritz pairs requested - % QR factorization of the matrix [ U(:,1:k), A*U(:,1:k)] - % - [ ~, R ] = qr( [ U(:,1:k), AxU_k ], 0 ) ; - % - % The Rayleigh quotient Sk = U_k' * A * U_k, expressed without using A - % Sk = (((U(:,1:k)' * Y)*diag(1./D))*V(:,1:k) ).*(ones(k,1)*(1./S(1:k))') ; - % - Sk = R(1:k,k+1:2*k) ; % Recall that Sk = U(:,1:k)'*A*U(:,1:k) ; - % - % Caveat: Here we assume that the QR is computed so that even for complex - % matrices the diagonal of R is real, as is the Matlab's function qr(). - % Otherwise, use linear algebra to determine proper scaling (easy). - for i = 1 : k - if ( R(i,i) < 0 ) - Sk(i,:) = -Sk(i,:) ; - end - end - if ( n_ref > k ) - warning('XY_DMD_R4 :( -> exceeds the total number of pairs.') - n_ref = k ; - end -else - % .. no refinement - % The Rayleigh quotient Sk = U_k' * A * U_k, expressed without using A - % Sk=(((U(:,1:k)' * Y)*diag(1./D))*V(:,1:k)).*(ones(k,1)*(1./S(1:k))') - Sk = U(:,1:k)'*AxU_k ; -end -% -%............................................................................... -% -% Ritz values -% -if ( strcmp(nobal,'NoBalance')) - [W, Lambda] = eig( Sk, 'vector', 'nobalance' ) ; -else - [W, Lambda] = eig( Sk, 'vector' ) ; -end -% -% -Z = U(:,1:k) * W ; % Ritz vectors, normalized in l_2 norm. -AxZ = AxU_k * W ; -% Next, for each Ritz pair compute its residual. -rez = zeros(k,1) ; -for i = 1 : k - rez(i) = norm( AxZ(:,i) - Lambda(i)*Z(:,i) ) ; -end -% -if save_to_file - save( file_save, 'U','Sigma','V','k','AxU_k','Sk','Z','Lambda','rez' ) ; -end -%............................................................................... -% -if ( n_ref > 0 ) - % refinement requested ; select the specified pairs - switch ref_select - case 1, - % nref pairs with smallest residuals - [~, idummy] = sort(rez,'ascend') ; - RSI = idummy(1:n_ref) ; - case 2, - % nref pairs closest to target (provided on input) - [~,idummy] = sort(abs(Lambda-target) ,'ascend') ; - RSI = idummy(1:n_ref) ; - case 3, - % nref pairs closest to the imaginary axis - [~,idummy] = sort(abs(real(Lambda)) ,'ascend') ; - RSI = idummy(1:n_ref) ; - case 4, - % nref pairs closest to the unit circle - [~,idummy] = sort(abs(abs(Lambda)-1) ,'ascend') ; - RSI = idummy(1:n_ref) ; - case 5, - % nref pairs closest to the real axis - [~,idummy] = sort(abs(imag(Lambda)) ,'ascend') ; - RSI = idummy(1:n_ref) ; - % more cases: define property and write a selection procedure - % that extracts the corresponding indicies - otherwise - error('XY_DMD_R4 :( -> parameter had an illegal value') - end -%... -% -VR = zeros(k,n_ref) ; RQ_ref = zeros(n_ref,1) ; rez_ref = zeros(n_ref,1) ; -% -% refinement of the selected pairs -for ii = 1 : n_ref - i = RSI(ii) ; - RL = [ R(1:k,k+1:2*k) - Lambda(i)*R(1:k,1:k) ; ... - R(k+1:2*k,k+1:2*k) ] ; - [~,~,VL] = svd( RL, 'econ') ; -% The singular vector of the smallest singular value is the representation -% of the refined eigenvector in the basis of U(:,1:k). - VR(:,ii) = VL(:,k) ; -% - rez_ref(ii) = norm(RL*VL(:,k)) ; % = k-th (smallest) singular value of RL -% More direct way is to use rez_ref(ii) = SL(k,k) with -% [~,SL,VL] = svd( RL, 'econ') instead of [~,~,VL] = svd( RL, 'econ') above; -% This formula is intended for the case of using less accurate SVD method, -% when the smallest singular value may be computed inaccurately, but the -% corresponding singular vector is computed reasonably accurately. In an -% optimal implemenation (not considered here) a specially tailored method -% would comput just VL(:,k), without computig the whole SVD of RL. -% -% Use the refined Ritz vectors to compute Rayleigh quotients as better -% eigenvalue approximations, with smaller residuals than with Lambda(1:k). -% In most cases, the improvement is mild, but it is an improvement. - RQ_ref(ii) = VL(:,k)' * Sk * VL(:,k) ; -end -% -% The matrix of the refined Ritz vectors -if ( strcmp(overwrite,'OverWrite') ) - Z(:,RSI) = U(:,1:k)*VR ; Z_ref = [] ; - Lambda(RSI) = RQ_ref ; - rez(RSI) = rez_ref ; -else - Z_ref = U(:,1:k)*VR ; -end -% -else - RSI = [] ; RQ_ref = [] ; rez_ref = [] ; Z_ref = [] ; -end -U = U(:,1:k) ; -end -% -%=============================================================================== -% Copyright (c) 2016-2017 AIMdyn Inc. -% All right reserved. -% -% 3-Clause BSD License -% -% Additional copyrights may follow. -% -% -% Redistribution and use in source and binary forms, with or without -% modification, are permitted provided that the following conditions are met: -% -% 1. Redistributions of source code must retain the above copyright notice, -% this list of conditions and the following disclaimer. -% 2. Redistributions in binary form must reproduce the above copyright notice, -% this list of conditions and the following disclaimer in the documentation -% and/or other materials provided with the distribution. -% 3. Neither the name of the copyright holder nor the names of its contributors -% may be used to endorse or promote products derived from this software -% without specific prior written permission. -% -% The copyright holder provides no reassurances that the source code -% provided does not infringe any patent, copyright, or any other -% intellectual property rights of third parties. The copyright holder -% disclaims any liability to any recipient for claims brought against -% recipient by any third party for infringement of that parties -% intellectual property rights. -% -% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDER AND CONTRIBUTORS "AS IS" AND -% ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED -% WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE -% DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE -% FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL -% DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR -% SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER -% CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, -% OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE -% OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. -% -%=============================================================================== - +function [ Z, Lambda, rez, RQ_ref, RSI, Z_ref, rez_ref, U, AxU_k ] = ... + XY_DDMD_R4( X, Y, to_scale, to_center, k_mode, tol, nobal, ... + n_ref, ref_select, target, overwrite, file_save) + +% + +% Refined Rayleigh-Ritz procedure for Data Driven Dynamic Mode Decomposition + +% with data driven Residual bounds(DD-DMD-RRRR). + +% This is a sample code that provides algorthmic details of the method + +% described in + +% [1] Z. Drmac, I. Mezic, R. Mohr: Data driven modal decompositions: analysis + +% and enhancements, SIAM Journal on Scientific Computing. + +% (arXiv preprint, https://arxiv.org/abs/1708.02685v1) + +% The code is prepared as a suplement to [1], only to facilitate easier + +% understanding of algorithmic details and experimenting. + +% Released under the 3-Clause BSD License, see below. + +% =====......................................................................... + +% Input + +% ===== + +% + +% X (real/complex 2D-array) :: Data matrix. Assumed with more rows than columns. + +% + +% Y (real/complex 2D-array) :: Data matrix, Y = A*X for some otherwise + +% inaccessible matrix A. Both X and Y should + +% be tall matrices + +% Separate code (another function, more memory + +% efficient) is used if Y is just shifted X + +% + +% to_scale (string) :: Specifies whether snapshot normalization is + +% needed. Equilibrating the snapshots in ell_2 + +% is a good idea in this setting. It does not + +% apply if a forgetting factor is incorporated. + +% --> If to_scale == 'Scale' + +% then X = X*D, Y = Y*D where D is diagonal + +% matrix such that the new X has unit columns. + +% + +% to_center (string) :: Optional data centering. For experiments only! + +% --> If to_center == 'Center' + +% then X = X-(1/m)*X*ones(m,1)*ones(1,m) and + +% Y = Y-(1/m)*Y*ones(m,1)*ones(1,m). Note that + +% after centering, again, Y = A*X. + +% --> Set to_center = 'Ignore', except for R&D + +% experimenting. + +% + +% k_mode (integer) :: Truncation mode for setting numerical rank k of + +% X, based on the singular values sigma(i). + +% --> If k_mode == -1, k is selected as + +% k = max( i : sigma(i) >= sigma(1)*tol. ) + +% --> If k_mode == -2, k is selected as + +% k = max( i>1 : sigma(i) > sigma(i-1)*tol + +% --> If k > 0 and k <= min(m,n) + +% the value of k_mode is then understood + +% as the caller's request to use k=k_mode + +% dominant left singular vectors of X. + +% + +% tol (real, >0 ) :: Tolerance threshold for truncating the + +% singular values in defining numerical + +% rank of scaled data. (Only for the values + +% k_mode = -1, k_mode = -2.) + +% [!] the recommended value is tol=n*eps, + +% where eps is the round-off unit. + +% + +% nobal (string) :: Specifies whether to switch of the balancing + +% in the function eig() that is used to + +% compute the eigenvalues and eigenvectors + +% of the Rayleigh quotient. + +% --> If nobal == 'NoBalance', the balancing is + +% not used. + +% --> Set nobal = 'Ignore' if not sure what is + +% this about. + +% + +% n_ref (integer) :: Specifies how many of the Ritz pairs will be + +% refined. If n_ref exceds k, then n_ref is + +% set to k (=the total number of Ritz pairs). + +% [!!] In case of large dimensions, refining + +% too many Ritz pairs is time consuming. + +% + +% ref_select (integer) :: Specifies how to select n_ref Ritz pairs + +% that will be refined. + +% --> If ref_select == 1, + +% refine n_ref pairs with smallest residuals + +% --> If ref_select == 2, + +% refine n_ref pairs closest to a given target + +% (see the input parameter target) + +% --> If ref_select == 3, + +% refine n_ref pairs closest to the imaginary axis + +% --> If ref_select == 4, + +% refine n_ref pairs closest to the unit + +% circle + +% --> If ref_select == 5, + +% refine n_ref pairs closest to the real axis + +% + +% target (real/complex) :: Specifies a point of interest when selecting + +% which Ritz pairs to refine (see ref_select) + +% Only the Ritz values closest to target will + +% be refined (and their Ritz vectors). + +% + +% overwrite (string) :: Specifes whether the refined Ritz vectors will + +% overwrite the original ones, or returned in a + +% separate array + +% --> If overwrite == 'OverWrite' then the selected + +% and improved Ritz vectors overwite the + +% original ones. Otherwise, they are returned + +% in a separate array. + +% + +% file_save (string) :: Specifies whether the selected variables + +% shold be saved in a file. + +% --> If file_save == 'NoSave', then no data is saved. + +% --> Otherwise, selected variables are saved in + +% the file file_save (given on input). + +% ======........................................................................ + +% Output + +% ======........................................................................ + +% Z (real/complex 2D-array) :: Ritz vectors, normalized so that + +% ||Z(:,i)||_2 = 1. If the refinement is + +% requested with overwrite option, then some + +% of the columns of Z are refined Ritz + +% vectors. Their indices are then listed in + +% the integer array RSI. + +% + +% Lambda (real/complex 1D-array) :: Ritz values + +% + +% rez (real 1D-array) :: 2-norms of the reziduals + +% rez(i) = ||A*Z(:,i)-Lambda(:,i)*Z(:,i)||_2 + +% + +% --> The following four arrays are void if the refiement is not requested. + +% When calling this function, use tildas if they are not needed. + +% + +% RQ_ref (real/complex 1D-array) :: Rayleigh quotients with refined Ritz + +% vectors. + +% + +% RSI (integer array) :: If n_ref > 0, RSI contains the indices + +% of those Ritz pairs that have been refined. + +% + +% Z_ref (real/complex 2D-array) :: Refined Ritz vectors, in the case when + +% they do not overwrite the orginal ones. + +% + +% rez_ref (real 1D-array) :: The residuals of the refined Ritz pairs. + +% + +% --> The following two array arguments are returned for post-procesing such as + +% e.g. refinement. Use tildas if they are not needed. + +% + +% U (real/complex 2-D array) :: The left singular vectors of the matrix X. + +% + +% AxU_k (real/complex 2-D array) :: The underlying operator A applied to the + +% k leading left singular vectors of X. + +%............................................................................... + +%............................................................................... + +% AIMdyn Inc. + +% Coded by Zlatko Drmac, Department of Mathematics, University of Zagreb. + +% drmac@math.hr + +% Version 1.0 , January 3, 2017. + +% Version 1.1 , November 11, 2017. + +%............................................................................... + +%............................................................................... + +% + +% The data matrix is n-by-m and it is assumed that the number of its rows + +% is much larger than the number of its columns. + +% + +[ n, m ] = size( X ) ; + +if m > n + + warning('XY_DMD_R4 :(o -> the number of snapshots exceeds space dimension'); + +end + +% Optionally, the data can be centered. This is a delicate issue. Only for R&D. + +if ( strcmp(to_center,'Center') ) + + X = X - ((1/m)*(X*ones(m,1))) * ones(1,m) ; + + Y = Y - ((1/m)*(Y*ones(m,1))) * ones(1,m) ; + +end + +% The intermediate and the final results can be saved for post festum analysis. + +save_to_file = ~strcmp(file_save,'NoSave') ; + +% + +if ( strcmp(to_scale,'Scale') ) + + % Column norms of the X-data matrix + + D = ones(m,1) ; + + for i = 1 : m + + D(i) = norm(X(:,i)) ; + + end + + % Remove zero columns (unlikely event, but ..) + + JX = find( D > 0 ) ; + + if ( length(JX) < m ) + + X = X(:,JX) ; D = D(JX) ; Y = Y(:,JX) ; + + m = size(X,2) ; + + end + + % + + % The data samples are scaled to the unit ell_2 sphere and then the SVD is + + % computed. The numerical rank is determined for the scaled data (uness the + + % caller has hardcoded the number of left singular vectors). + + % One has to be careful here, e.g. if SNR is small. Aditional scaling may + + % be necessary with carefully chosen weights. If weighting is an issue, + + % use the weighted version of DMD, see [1]. + + % + + [ U, Sigma, V ] = svd( X*diag(1./D), 'econ' ) ; S = diag(Sigma) ; + +else + + [ U, Sigma, V ] = svd( X, 'econ' ) ; S = diag(Sigma) ; + +end + +% + +% Determine the numerical rank k of X, based on the k_mode and tol + +if ( k_mode == -1 ) + + % classics + + if ( tol < eps ) + + warning('XY_DMD_R4 :( -> parameter is too small') + + end + + k = find( S/S(1) > tol, 1, 'last' ) ; + +elseif ( k_mode == -2 ) + + % more conservative, useful for R&D purposes + + if ( tol <= eps ) + + warning('XY_DMD_R4 :( -> parameter is too small') + + end + + %k = find( S(2:m)./S(1:m-1) < tol, 1, 'first' ) ;must be changed, for now use + + k = find( S(2:m)./S(1:m-1) > tol, 1, 'last' ) ; + +elseif ( (k_mode > 0) && (k_mode <= min(m,n)) ) + + % the caller inputs k as the required number of the left singular + + % vectors of X + + k = k_mode ; + +else + + error('XY_DMD_R4 :( -> parameter had an illegal value') + +end + +% + +% The image of U(:,1:k) under the action of A + +if ( strcmp(to_scale,'Scale') ) + + AxU_k = Y*(diag(1./D)*( (V(:,1:k).*(ones(m,1)*(1./S(1:k))')))) ; + +else + + AxU_k = Y*(((V(:,1:k).*(ones(m,1)*(1./S(1:k))')))) ; + +end + +% + +% + +if ( n_ref > 0 ) + + % refinement of Ritz pairs requested + + % QR factorization of the matrix [ U(:,1:k), A*U(:,1:k)] + + % + + [ ~, R ] = qr( [ U(:,1:k), AxU_k ], 0 ) ; + + % + + % The Rayleigh quotient Sk = U_k' * A * U_k, expressed without using A + + % Sk = (((U(:,1:k)' * Y)*diag(1./D))*V(:,1:k) ).*(ones(k,1)*(1./S(1:k))') ; + + % + + Sk = R(1:k,k+1:2*k) ; % Recall that Sk = U(:,1:k)'*A*U(:,1:k) ; + + % + + % Caveat: Here we assume that the QR is computed so that even for complex + + % matrices the diagonal of R is real, as is the Matlab's function qr(). + + % Otherwise, use linear algebra to determine proper scaling (easy). + + for i = 1 : k + + if ( R(i,i) < 0 ) + + Sk(i,:) = -Sk(i,:) ; + + end + + end + + if ( n_ref > k ) + + warning('XY_DMD_R4 :( -> exceeds the total number of pairs.') + + n_ref = k ; + + end + +else + + % .. no refinement + + % The Rayleigh quotient Sk = U_k' * A * U_k, expressed without using A + + % Sk=(((U(:,1:k)' * Y)*diag(1./D))*V(:,1:k)).*(ones(k,1)*(1./S(1:k))') + + Sk = U(:,1:k)'*AxU_k ; + +end + +% + +%............................................................................... + +% + +% Ritz values + +% + +if ( strcmp(nobal,'NoBalance')) + + [W, Lambda] = eig( Sk, 'vector', 'nobalance' ) ; + +else + + [W, Lambda] = eig( Sk, 'vector' ) ; + +end + +% + +% + +Z = U(:,1:k) * W ; % Ritz vectors, normalized in l_2 norm. + +AxZ = AxU_k * W ; + +% Next, for each Ritz pair compute its residual. + +rez = zeros(k,1) ; + +for i = 1 : k + + rez(i) = norm( AxZ(:,i) - Lambda(i)*Z(:,i) ) ; + +end + +% + +if save_to_file + + save( file_save, 'U','Sigma','V','k','AxU_k','Sk','Z','Lambda','rez' ) ; + +end + +%............................................................................... + +% + +if ( n_ref > 0 ) + + % refinement requested ; select the specified pairs + + switch ref_select + + case 1, + + % nref pairs with smallest residuals + + [~, idummy] = sort(rez,'ascend') ; + + RSI = idummy(1:n_ref) ; + + case 2, + + % nref pairs closest to target (provided on input) + + [~,idummy] = sort(abs(Lambda-target) ,'ascend') ; + + RSI = idummy(1:n_ref) ; + + case 3, + + % nref pairs closest to the imaginary axis + + [~,idummy] = sort(abs(real(Lambda)) ,'ascend') ; + + RSI = idummy(1:n_ref) ; + + case 4, + + % nref pairs closest to the unit circle + + [~,idummy] = sort(abs(abs(Lambda)-1) ,'ascend') ; + + RSI = idummy(1:n_ref) ; + + case 5, + + % nref pairs closest to the real axis + + [~,idummy] = sort(abs(imag(Lambda)) ,'ascend') ; + + RSI = idummy(1:n_ref) ; + + % more cases: define property and write a selection procedure + + % that extracts the corresponding indicies + + otherwise + + error('XY_DMD_R4 :( -> parameter had an illegal value') + + end + +%... + +% + +VR = zeros(k,n_ref) ; RQ_ref = zeros(n_ref,1) ; rez_ref = zeros(n_ref,1) ; + +% + +% refinement of the selected pairs + +for ii = 1 : n_ref + + i = RSI(ii) ; + + RL = [ R(1:k,k+1:2*k) - Lambda(i)*R(1:k,1:k) ; ... + + R(k+1:2*k,k+1:2*k) ] ; + + [~,~,VL] = svd( RL, 'econ') ; + +% The singular vector of the smallest singular value is the representation + +% of the refined eigenvector in the basis of U(:,1:k). + + VR(:,ii) = VL(:,k) ; + +% + + rez_ref(ii) = norm(RL*VL(:,k)) ; % = k-th (smallest) singular value of RL + +% More direct way is to use rez_ref(ii) = SL(k,k) with + +% [~,SL,VL] = svd( RL, 'econ') instead of [~,~,VL] = svd( RL, 'econ') above; + +% This formula is intended for the case of using less accurate SVD method, + +% when the smallest singular value may be computed inaccurately, but the + +% corresponding singular vector is computed reasonably accurately. In an + +% optimal implemenation (not considered here) a specially tailored method + +% would comput just VL(:,k), without computig the whole SVD of RL. + +% + +% Use the refined Ritz vectors to compute Rayleigh quotients as better + +% eigenvalue approximations, with smaller residuals than with Lambda(1:k). + +% In most cases, the improvement is mild, but it is an improvement. + + RQ_ref(ii) = VL(:,k)' * Sk * VL(:,k) ; + +end + +% + +% The matrix of the refined Ritz vectors + +if ( strcmp(overwrite,'OverWrite') ) + + Z(:,RSI) = U(:,1:k)*VR ; Z_ref = [] ; + + Lambda(RSI) = RQ_ref ; + + rez(RSI) = rez_ref ; + +else + + Z_ref = U(:,1:k)*VR ; + +end + +% + +else + + RSI = [] ; RQ_ref = [] ; rez_ref = [] ; Z_ref = [] ; + +end + +U = U(:,1:k) ; + +end + +% + +%=============================================================================== + +% Copyright (c) 2016-2017 AIMdyn Inc. + +% All right reserved. + +% + +% 3-Clause BSD License + +% + +% Additional copyrights may follow. + +% + +% + +% Redistribution and use in source and binary forms, with or without + +% modification, are permitted provided that the following conditions are met: + +% + +% 1. Redistributions of source code must retain the above copyright notice, + +% this list of conditions and the following disclaimer. + +% 2. Redistributions in binary form must reproduce the above copyright notice, + +% this list of conditions and the following disclaimer in the documentation + +% and/or other materials provided with the distribution. + +% 3. Neither the name of the copyright holder nor the names of its contributors + +% may be used to endorse or promote products derived from this software + +% without specific prior written permission. + +% + +% The copyright holder provides no reassurances that the source code + +% provided does not infringe any patent, copyright, or any other + +% intellectual property rights of third parties. The copyright holder + +% disclaims any liability to any recipient for claims brought against + +% recipient by any third party for infringement of that parties + +% intellectual property rights. + +% + +% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDER AND CONTRIBUTORS "AS IS" AND + +% ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED + +% WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + +% DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE + +% FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL + +% DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR + +% SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER + +% CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, + +% OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + +% OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +% + +%=============================================================================== + + + diff --git a/scripts/Koopman-ID/construct_eigfuncs.m b/scripts/Koopman-ID/construct_eigfuncs.m new file mode 100644 index 0000000..0901348 --- /dev/null +++ b/scripts/Koopman-ID/construct_eigfuncs.m @@ -0,0 +1,107 @@ +function [A_koop, z_eigfun] = construct_eigfuncs(n, Ntraj, Ntime, ... + N_basis, A_nom, B_nom, K_nom, Xstr, deltaT) + + %Identify lifted state space model using approximate Koopman invariant + %subspace + + %Inputs: + % n - Number of states + % m - Number of control inputs + % N_lambda - Number of eigenvalue candidates to use + % N_basis - Number of boundary functions to use + % basis_function - Type of basis functions (only rbfs implemented) + % rbf-type - Type of rbf function + % center_type - What centers to use for rbfs ('data'/'random') + % eps - Width of rbf if rbf type has width parameter + % plot_basis - (true/false) determines to plot basis functions + % xlim - Plot limits x-axis + % ylim - Plot limits y-axis + % A_edmd - Passive dynamics matrix from EDMD + % A_nom - Nominal state space model passive dynamics + % B_nom - Nominal state space model actuated dynamics + % K_nom - Nominal linear feedback control law gains + % Xacc - Trajectory data + % Yacc - Trajectory data at next time step + % Uacc - Control input for trajectory data + % Xacc_c - Trajectory data, perturbed control + % Yacc_c - Trajectory data at next time step, perturbed control + % Uacc_c - Control input for trajectory data, perturbed control + + %Outputs: + % A_edmd - Passive dynamics matrix in lifted space + % B_edmd - Actuation matrix in lifted space + % C_edmd - Projection matrix from lifted space to output + + + % Generate eigenfunctions and learn autonomous dynamics: + + disp('Starting autonomous dynamics learning...'); tic + + A_c = A_nom + B_nom*K_nom; + A_c_d = expm(A_c*deltaT); %Discrete time dynamics matrix + cent = rand(n,N_basis)*2*pi - pi; + rbf_type = 'gauss'; + eps_rbf = 1; + + % Set up nonlinear transformations and their gradients + zfun = @(xx) [xx'; rbf(xx',cent,rbf_type,eps_rbf)]; + zfun_grad = @(xx) [eye(2); rbf_grad(xx',cent,rbf_type,eps_rbf)]; + + % Prepare data matrices + Y = []; + Z_mplus1 = []; + Z_m = []; + for i = 1 : Ntraj + X_mplus1 = reshape(Xstr(:,i,2:end),n,Ntime); + X_m = reshape(Xstr(:,i,1:end-1),n,Ntime); + + % Set up Y matrix (targets), Y(:,i) = x_(i+1) - A_nom*x_i + Y = [Y X_mplus1-A_c_d*X_m]; + + % Set up Z_mplus1 matrix (inputs), phi(x_(i+1)) + Z_mplus1 = [Z_mplus1 zfun(X_mplus1')]; + + % Set up Z_m matrix (inputs), phi(x_i) + Z_m = [Z_m zfun(X_m')]; + end + + %Set up constraint matrix + con1 = zfun_grad([0 0]); + + disp('Solving optimization problem...') + N = size(Z_m,1); + cvx_begin + variable C(n,N); + minimize (norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro') + 0.01*norm(C,'fro')) + subject to + {C*con1 == zeros(n)}; + cvx_end + fprintf('Solved, optimal value excluding regularization: %.10f, MSE: %.10f\n', norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro'),immse(Y, A_c_d*C*Z_m - C*Z_mplus1)) + %fprintf('Constraint violation: %.4f \n', sum(sum(abs(C*con1)))) + + yfun = @(xx) xx + C*zfun(xx'); % Full learned diffeomorphism + + % Calculate eigenfunctions for linearized system + [~,D] = eig(A_c_d); + [V_a,~] = eig(A_c_d'); + + %Define powers (only implemented for n=2): + max_power = 5; + a = 0 : max_power; + [P,Q] = meshgrid(a,a); + c=cat(2,P',Q'); + powers=reshape(c,[],2); + + linfunc = @(xx) (xx'*V_a)'; + phifun = @(xx) (prod(linfunc(xx).^(powers')))'; + lambd = prod(diag(D).^(powers'))'; + + % Construct scaling function + gfun = @(xx) xx./pi; %Scale state space into unit cube + + % Construct eigenfunctions for nonlinear system + z_eigfun = @(xx) phifun_mat(phifun, gfun(yfun(xx))); + A_koop = diag(lambd); +end + + diff --git a/scripts/Koopman-ID/koopman_eigen_id_laplace.m b/scripts/Koopman-ID/koopman_eigen_id_laplace.m new file mode 100644 index 0000000..b81538c --- /dev/null +++ b/scripts/Koopman-ID/koopman_eigen_id_laplace.m @@ -0,0 +1,140 @@ +function [A_koop, B_koop, C_koop, phi_fun_v] = koopman_eigen_id_laplace(n,m, Ntraj, Ntime, ... + N_basis,basis_function, rbf_type, center_type, eps_rbf, N_lambda, ... + lambda_type, A_edmd, A_nom, B_nom, K_nom, Xacc, Yacc, Xstr, Xacc_c,... + Yacc_c, Uacc_c, Xstr_c, Ustr_c, timestr, deltaT, learn_type) + + %Identify lifted state space model using approximate Koopman invariant + %subspace + + %Inputs: + % n - Number of states + % m - Number of control inputs + % N_lambda - Number of eigenvalue candidates to use + % N_basis - Number of boundary functions to use + % basis_function - Type of basis functions (only rbfs implemented) + % rbf-type - Type of rbf function + % center_type - What centers to use for rbfs ('data'/'random') + % eps - Width of rbf if rbf type has width parameter + % plot_basis - (true/false) determines to plot basis functions + % xlim - Plot limits x-axis + % ylim - Plot limits y-axis + % A_edmd - Passive dynamics matrix from EDMD + % A_nom - Nominal state space model passive dynamics + % B_nom - Nominal state space model actuated dynamics + % K_nom - Nominal linear feedback control law gains + % Xacc - Trajectory data + % Yacc - Trajectory data at next time step + % Uacc - Control input for trajectory data + % Xacc_c - Trajectory data, perturbed control + % Yacc_c - Trajectory data at next time step, perturbed control + % Uacc_c - Control input for trajectory data, perturbed control + + %Outputs: + % A_edmd - Passive dynamics matrix in lifted space + % B_edmd - Actuation matrix in lifted space + % C_edmd - Projection matrix from lifted space to output + + + % Generate eigenfunctions and learn autonomous dynamics: + + disp('Starting autonomous dynamics learning...'); tic + + A_c = A_nom + B_nom*K_nom; + A_c_d = expm(A_c*deltaT); %Discrete time dynamics matrix + cent = rand(n,50)*2*pi - pi; + rbf_type = 'gauss'; + eps_rbf = 1; + + % Set up nonlinear transformations and their gradients + zfun = @(xx) [xx'; rbf(xx',cent,rbf_type,eps_rbf)]; + zfun_grad = @(xx) [eye(2); rbf_grad(xx',cent,rbf_type,eps_rbf)]; + + % Prepare data matrices + Y = []; + Z_mplus1 = []; + Z_m = []; + for i = 1 : Ntraj + X_mplus1 = reshape(Xstr(:,i,2:end),n,Ntime); + X_m = reshape(Xstr(:,i,1:end-1),n,Ntime); + + % Set up Y matrix (targets), Y(:,i) = x_(i+1) - A_nom*x_i + Y = [Y X_mplus1-A_c_d*X_m]; + + % Set up Z_mplus1 matrix (inputs), phi(x_(i+1)) + Z_mplus1 = [Z_mplus1 zfun(X_mplus1')]; + + % Set up Z_m matrix (inputs), phi(x_i) + Z_m = [Z_m zfun(X_m')]; + end + + %Set up constraint matrix + con1 = zfun_grad([0 0]); + + disp('Solving optimization problem...') + N = size(Z_m,1); + cvx_begin + variable C(n,N); + minimize (norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro') + 1*norm(C,'fro')) + subject to + {C*con1 == zeros(n)}; + cvx_end + fprintf('Solved, optimal value excluding regularization: %.10f, MSE: %.10f\n', norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro'),immse(Y, A_c_d*C*Z_m - C*Z_mplus1)) + %fprintf('Constraint violation: %.4f \n', sum(sum(abs(C*con1)))) + + yfun = @(xx) xx + C*zfun(xx'); % Full learned diffeomorphism + + % Calculate eigenfunctions for linearized system + [~,D] = eig(A_c_d); + [V_a,~] = eig(A_c_d'); + + %Define powers (only implemented for n=2): + max_power = 5; + a = 0 : max_power; + [P,Q] = meshgrid(a,a); + c=cat(2,P',Q'); + powers=reshape(c,[],2); + + linfunc = @(xx) (xx'*V_a)'; + phifun = @(xx) (prod(linfunc(xx).^(powers')))'; + lambd = prod(diag(D).^(powers'))'; + + % Construct scaling function + gfun = @(xx) xx./pi; %Scale state space into unit cube + + % Construct eigenfunctions for nonlinear system + z_eigfun = @(xx) [xx; phifun_mat(phifun, gfun(yfun(xx)))]; + %z_eigfun = @(xx) phifun_mat(phifun, gfun(yfun(xx))); + + %A_koop = diag(lambd); + Z_x = z_eigfun(Xacc); + Z_y = z_eigfun(Yacc); + k=1; + A_koop = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(Z_x,1)-n)]; + %A_koop = [A_koop; (Z_x'\Z_y(n/2+1:n,:)')']; + A_state = lasso(Z_x',Z_y(n/2+1:n,:)','Lambda',1e-5)'; + A_koop = [A_koop; A_state]; + A_koop = [A_koop; [zeros(length(lambd),n) diag(lambd)]]; + + % Calculate projection of states onto eigenfunctions + %C_koop = (Zacc'\Xacc')'; + C_koop = zeros(n,size(A_koop,1)); + C_koop(:,1:n) = eye(n); + fprintf('Learning autonomous dynamics done, training error %.2f, execution time: %1.2f s \n', immse(Yacc,C_koop*A_koop*Z_x), toc); + + % Learn actuated dynamics: + disp('Starting actuated dynamics learning...'); tic + [B_koop, phi_fun_v] = learn_B_koop(n, m, Ntraj, Ntime, z_eigfun, A_koop,... + C_koop, Xstr_c, Ustr_c, Xacc_c, Yacc_c, Uacc_c, deltaT, learn_type); + fprintf('Learning actuated dynamics done, execution time: %1.2f s \n', toc); + + % Subtract effect of nominal controller from A_koop: + A_koop(:,1:n) = A_koop(:,1:n)-B_koop*K_nom; + + % Evaluate training error: +% Ypred = zeros(size(Yacc_c)); +% +% C_koop*(A_koop*phi_fun_v(Xacc_c)+B_koop*Uacc_c); +% disp(immse(Ypred,Yacc_c)); +end + + diff --git a/scripts/Koopman-ID/main_comparison_pendulum.m b/scripts/Koopman-ID/main_comparison_pendulum.m index a65692b..8309a6b 100644 --- a/scripts/Koopman-ID/main_comparison_pendulum.m +++ b/scripts/Koopman-ID/main_comparison_pendulum.m @@ -14,7 +14,7 @@ %implementation. clear all; close all; clc; -addpath('./Resources'); addpath('./utils') +addpath('./Resources'); addpath('./utils'); addpath('Resources/qpOASES-3.1.0/interfaces/matlab/'); rng(2141444) %% **************** Model and simulation parameters *********************** @@ -22,20 +22,19 @@ m = 1; g = 9.81; l = 1; f_u = @(t,x,u) [x(2,:); g/l*sin(x(1,:))+ u]; n = 2; m = 1; %Number of states and control inputs -Q = eye(2); R = 1; %LQR penalty matrices for states and control inputs +Q = eye(2); R = 0.1; %LQR penalty matrices for states and control inputs %Simulation parameters: Ntime = 200; %Length of each trajectory (# of time steps) Ntraj = 20; %Number of trajectories deltaT = 0.01; %Time step length of simulation -X0_A = 2*rand(n,Ntraj)-1; %Sample initial points for each trajectory for learning A -X0_A = 0.95*pi*X0_A./vecnorm(X0_A,2,1); %Normalize so initial points lie on unit circle +X0_A = (2*pi*rand(n,Ntraj)-pi); Xf_A = zeros(n,Ntraj); %Terminal points for each trajectory for learning A -X0_B = 0.6*(2*pi*rand(n,Ntraj)-pi); %Sample initial points for each trajectory for learning B +X0_B = (2*pi*rand(n,Ntraj)-pi); %Sample initial points for each trajectory for learning B Xf_B = 0.6*(2*pi*rand(n,Ntraj)-pi); %Terminal points for each trajectory for learning B %E-DMD parameters: -N_basis_edmd = 20; %Number of basis functions +N_basis_edmd = 36; %Number of basis functions basis_function_edmd = 'rbf'; %Type of basis function rbf_type_edmd = 'thinplate'; %RBF type center_type_edmd = 'data'; %Centers of rbf ('data' - pick random points from data set, 'random' pick points uniformly at random @@ -53,11 +52,11 @@ plot_basis_koop = false; %Plots the basis functions if true xlim = [-1, 1]; %Plot limits ylim = [-1, 1]; %Plot limits -learn_type = 'single-step'; %Learn B-matrix with single step or multi-step prediction horizon +learn_type = 'multi-step'; %Learn B-matrix with single step or multi-step prediction horizon %Test simulation parameters: Nsim = 5; -Tsim = 1; +Tsim = 2; plot_results = true; X0_sim = 2*rand(n,Nsim)-1; @@ -70,39 +69,37 @@ % Collect data to learn autonomous dynamics: autonomous_learning = true; -U_perturb = 0.5*randn(Ntime,Ntraj); %Add normally distributed noise to nominal controller +U_perturb = randn(Ntime,Ntraj); %Add normally distributed noise to nominal controller [Xstr, Xacc, Yacc, Ustr, Uacc, timestr] = collect_data(n,m,Ntraj,... Ntime,deltaT,X0_A, Xf_A ,K_nom,f_u,U_perturb, autonomous_learning); % Collect data to learn controlled dynamics: autonomous_learning = false; -U_perturb_c = 0.5*randn(Ntime,Ntraj); %Add normally distributed noise to nominal controller +U_perturb_c = 4*rand(Ntime,Ntraj)-2; %Add normally distributed noise to nominal controller [Xstr_c, Xacc_c, Yacc_c, Ustr_c, Uacc_c, timestr_c] = collect_data(n,m,Ntraj,... Ntime,deltaT,X0_B,Xf_B,K_nom,f_u,U_perturb_c, autonomous_learning); fprintf('Data collection done, execution time: %1.2f s \n', toc); %% *********************** Model Identification *************************** - % Identify model using E-DMD to get eigenvalues and model to use for % comparison: disp('Starting EDMD...'); tic -[A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m,N_basis_edmd,basis_function_edmd,... +[A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m, Ntraj, Ntime, N_basis_edmd,basis_function_edmd,... rbf_type_edmd, center_type_edmd, eps_rbf_edmd, plot_basis_edmd, xlim,... - ylim, Xacc, Yacc, Uacc); + ylim, Xacc, Yacc, Uacc, Xstr, A_nom, B_nom, K_nom, deltaT); fprintf('EDMD done, execution time: %1.2f s \n', toc); -%Identify model using Koopman eigenfunctions: - -[A_koop, B_koop, C_koop, phi_fun_v] = koopman_eigen_id(n, m, Ntraj, Ntime, N_basis_koop,... +[A_koop, B_koop, C_koop, phi_fun_v] = koopman_eigen_id_laplace(n, m, Ntraj, Ntime, N_basis_koop,... basis_function_koop, rbf_type_koop, center_type_koop, eps_rbf_koop, ... - N_lambda, lambda_type, A_edmd, A_nom, B_nom, K_nom, Xacc, Xstr, Xacc_c,... + N_lambda, lambda_type, A_edmd, A_nom, B_nom, K_nom, Xacc, Yacc, Xstr, Xacc_c,... Yacc_c, Uacc_c, Xstr_c, Ustr_c, timestr, deltaT, learn_type); %% ************************ Analysis of Results *************************** [mse_edmd_avg, mse_koop_avg, mse_edmd_std, mse_koop_std,... - mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop] = ... + mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop,... + cost_nom, cost_edmd, cost_koop] = ... evaluate_model_performance(K_nom, A_edmd, B_edmd, C_edmd, A_koop, B_koop, ... C_koop, Nsim, X0_sim, Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results); @@ -110,6 +107,6 @@ fprintf('Predicition performance: \nAverage MSE: \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \nStandard Deviation MSE: \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \n', ... mse_edmd_avg, mse_koop_avg, mse_edmd_std, mse_koop_std); fprintf('-----------------------------------------------------------------\n'); - fprintf('Closed loop performance (MPC): \nAverage MSE: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \nEnergy consumption proxy: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \n', ... - mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop); + fprintf('Closed loop performance (MPC): \nAverage MSE: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \nEnergy consumption proxy: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f, \nAccumulated MPC cost: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \n', ... + mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop, cost_nom, cost_edmd, cost_koop); fprintf('-----------------------------------------------------------------\n'); \ No newline at end of file diff --git a/scripts/Koopman-ID/normalize_data.m b/scripts/Koopman-ID/normalize_data.m new file mode 100644 index 0000000..3ef4e0b --- /dev/null +++ b/scripts/Koopman-ID/normalize_data.m @@ -0,0 +1,11 @@ +function [X_norm, X_mean, X_std] = normalize_data(X) + + %Make sure each row corresponds to a time step: + if size(X,1)>size(X,2) + X = X'; + end + + X_mean = mean(X,2); + X_std = std(X,0,2); + X_norm = (X-X_mean)./X_std; +end \ No newline at end of file diff --git a/scripts/Koopman-ID/obsvk.m b/scripts/Koopman-ID/obsvk.m new file mode 100755 index 0000000..66191ee --- /dev/null +++ b/scripts/Koopman-ID/obsvk.m @@ -0,0 +1,6 @@ +function [ O ] = obsvk( A, C, k ) +O = []; +for i = 1:k + O = [O;C*A^(i-1)]; % Inefficient +end + diff --git a/scripts/Koopman-ID/phifun_mat.m b/scripts/Koopman-ID/phifun_mat.m new file mode 100644 index 0000000..5c0c83a --- /dev/null +++ b/scripts/Koopman-ID/phifun_mat.m @@ -0,0 +1,7 @@ +function res = phifun_mat(phifun,X) + [n,m] = size(X); + res = []; + for i = 1 : m + res = [res phifun(X(:,i))]; + end +end \ No newline at end of file diff --git a/scripts/Koopman-ID/rbf_grad.m b/scripts/Koopman-ID/rbf_grad.m new file mode 100644 index 0000000..85fbf1b --- /dev/null +++ b/scripts/Koopman-ID/rbf_grad.m @@ -0,0 +1,53 @@ +% X: nxN +% C: rbf center(s) - nx1 or nxK, if C is n x K, then the result is STACKED +% over the centers K, i.e., of dimension K x N +% eps: kernel width for Gaussian type rbfs (optional) +% k: polyharmonic coefficient for polyharmonic rbfs (optional) +% Note that all RBFs are symmetrical so evaluation of of a single point on +% multiple centers can by done as evaluation of multiple points (the +% centers in this case) on a single point (the center) + +function Y = rbf_grad( X,C, type, eps, k ) + type = lower(type); + if(~exist('eps','var') || isempty(eps)) + eps = 1; + end + if(~exist('k','var') || isempty(k)) + k = 1; + end + + N = size(X,2); %number of points + n = size(X,1); + Cbig = C; + Y = zeros(size(C,2),n); + for i = 1:size(Cbig,2) + C = Cbig(:,i); + C = repmat( C,1, N ); + if (n>1) + r_squared = sum( (X - C).^2 ); + else + r_squared = abs( (X - C).^2 ); + end + switch type + case 'thinplate' + error('Gradient of RBF type not implemented') + %y = r_squared.*log(sqrt(r_squared)); % + 0.5*sqrt(r_squared); % makes nonnegative, try to impose nonnegativity of K + %y(isnan(y)) = 0; + case 'gauss' + y = -2*eps^2.*(X-C)'.*exp(-eps^2*r_squared); + %y = exp(-eps^2*r_squared); + case 'invquad' + error('Gradient of RBF type not implemented') + %y = 1 ./ (1+eps^2*r_squared); + case 'invmultquad' + error('Gradient of RBF type not implemented') + %y = 1 ./ sqrt((1+eps^2*r_squared)); + case 'polyharmonic' + error('Gradient of RBF type not implemented') + %y = r_squared.^(k/2).*log(sqrt(r_squared)); + %y(isnan(y)) = 0; + otherwise + error('RBF type not recognize') + end + Y(i,:) = y; + end diff --git a/scripts/Koopman-ID/test_eigfunction_generation.m b/scripts/Koopman-ID/test_eigfunction_generation.m new file mode 100644 index 0000000..4c1676a --- /dev/null +++ b/scripts/Koopman-ID/test_eigfunction_generation.m @@ -0,0 +1,244 @@ +clear all; close all; clc; +addpath('./Resources'); addpath('./utils') +rng(2141444) + +%% **************** Model and simulation parameters *********************** +%Dynamics: +m = 1; g = 9.81; l = 1; +f_u = @(t,x,u) [x(2,:); g/l*sin(x(1,:))+ u]; +n = 2; m = 1; %Number of states and control inputs +Q = eye(2); R = 1; %LQR penalty matrices for states and control inputs + +%Simulation parameters: +Ntime = 200; %Length of each trajectory (# of time steps) +Ntraj = 10; %Number of trajectories +deltaT = 0.01; %Time step length of simulation +X0_A = 2*rand(n,Ntraj)-1; %Sample initial points for each trajectory for learning A +X0_A = 0.95*pi*X0_A./vecnorm(X0_A,2,1); %Normalize so initial points lie on unit circle +Xf_A = zeros(n,Ntraj); %Terminal points for each trajectory for learning A +X0_B = 0.6*(2*pi*rand(n,Ntraj)-pi); %Sample initial points for each trajectory for learning B +Xf_B = 0.6*(2*pi*rand(n,Ntraj)-pi); %Terminal points for each trajectory for learning B + +%% ************************** Data Collection ***************************** +disp('Starting data collection...'); tic + +% Calculate nominal model with Jacobian and find nominal control gains: + +[A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R); + +% Collect data to learn autonomous dynamics: +autonomous_learning = true; +U_perturb = 0.0*randn(Ntime,Ntraj); %Add normally distributed noise to nominal controller +[Xstr, Xacc, Yacc, Ustr, Uacc, timestr] = collect_data(n,m,Ntraj,... + Ntime,deltaT,X0_A, Xf_A ,K_nom,f_u,U_perturb, autonomous_learning); + +fprintf('Data collection done, execution time: %1.2f s \n', toc); + +%% ************** Prepare data and learn diffeomorphism h ***************** +A_c = A_nom + B_nom*K_nom; +A_c_d = expm(A_c*deltaT); %Discrete time dynamics matrix +N_basis = 100; +cent = rand(n,N_basis)*2*pi - pi; +rbf_type = 'gauss'; +eps_rbf = 1; +zfun = @(xx) [xx'; rbf(xx',cent,rbf_type,eps_rbf)]; +zfun_grad = @(xx) [eye(2); rbf_grad(xx',cent,rbf_type,eps_rbf)]; +% x = sym('x',[1,n]); +% zfun_grad = zfun(x); +Y = []; +Z_mplus1 = []; +Z_m = []; +for i = 1 : Ntraj + X_mplus1 = reshape(Xstr(:,i,2:end),n,Ntime); + X_m = reshape(Xstr(:,i,1:end-1),n,Ntime); + + % Set up Y matrix (targets), Y(:,i) = x_(i+1) - A_nom*x_i + Y = [Y X_mplus1-A_c_d*X_m]; + + % Set up Z_mplus1 matrix (inputs), phi(x_(i+1)) + Z_mplus1 = [Z_mplus1 zfun(X_mplus1')]; + + % Set up Z_m matrix (inputs), phi(x_i) + Z_m = [Z_m zfun(X_m')]; +end + +%Set up constraint matrix +con1 = zfun_grad([0 0]); + +disp('Solving optimization problem...') +N = size(Z_m,1); +cvx_begin + variable C(n,N); + minimize (norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro') + 0.01*norm(C,'fro')) + subject to + {C*con1 == zeros(n)}; +cvx_end +fprintf('Solved, optimal value excluding regularization: %.10f, MSE: %.10f\n', norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro'),immse(Y, A_c_d*C*Z_m - C*Z_mplus1)) +fprintf('Constraint violation: %.4f \n', sum(sum(abs(C*con1)))) + +yfun = @(xx) xx + C*zfun(xx'); % Full learned diffeomorphism +%% Calculate eigenfunctions for linearized system +[V,D] = eig(A_c_d); +[V_a,D_a] = eig(A_c_d'); + +%Define powers (only implemented for n=2): +max_power = 3; +a = 0 : max_power; +[P,Q] = meshgrid(a,a); +c=cat(2,P',Q'); +powers=reshape(c,[],2); + +linfunc = @(xx) (xx'*V_a)'; +phifun = @(xx) (prod(linfunc(xx).^(powers')))'; +lambd = prod(diag(D).^(powers'))'; + +%% Construct scaling function +gfun = @(xx) xx./pi; %Scale state space into unit cube + +%% Construct eigenfunctions for nonlinear system +phi_nonlin = @(xx) phifun(gfun(yfun(xx))); +A_eig = diag(lambd); + +%% Check how well the evolution of the eigenfunctions are described by the eigenvalues on linearized system: +X_test = zeros(n,Ntime+1); +X_test(:,1) = Xstr(:,1,1); +for i = 2 : Ntime+1 + X_test(:,i) = A_c_d*X_test(:,i-1); +end + +N = size(lambd,1); +Z_test = zeros(N,Ntime+1); +Z_eig = zeros(N,Ntime+1); +Z_test(:,1) = phifun(X_test(:,1)); +Z_eig(:,1) = phifun(X_test(:,1)); +for j = 2 : Ntime+1 + Z_eig(:,j) = A_eig*Z_eig(:,j-1); + Z_test(:,j) = phifun(X_test(:,j)); +end + +afigure +t = 0 : deltaT : Ntime*deltaT; +for i = 1 : N + subplot(4,4,i) + plot(t,Z_eig(i,:),t,Z_test(i,:)) + xlabel('Time (sec)'); + ylabel('Observable value') + if i == N + legend('Eigval evolution', 'True') + end +end + + +%% Check how well the evolution of the eigenfunctions are described by the eigenvalues: +X_test = reshape(Xstr(:,1,:),n,Ntime+1); +N = size(lambd,1); +Z_test = zeros(N,Ntime+1); +Z_eig = zeros(N,Ntime+1); +Z_test(:,1) = phi_nonlin(X_test(:,1)); +Z_eig(:,1) = phi_nonlin(X_test(:,1)); +for j = 2 : Ntime+1 + Z_eig(:,j) = A_eig*Z_eig(:,j-1); + Z_test(:,j) = phi_nonlin(X_test(:,j)); +end + +afigure +t = 0 : deltaT : Ntime*deltaT; +for i = 1 : N + subplot(4,4,i) + plot(t,Z_eig(i,:),t,Z_test(i,:)) + xlabel('Time (sec)'); + ylabel('Observable value') + if i == N + legend('Eigval evolution', 'True') + end +end + +%% Predict evolution of nonlinear system + + + +%% Old code: (Delete when verified not needed) +% lin_pred_acc = zeros(size(Xacc,1),Ntime*Ntraj); +% Y_res_acc = zeros(size(Xacc,1),Ntime*Ntraj); +% t = deltaT:deltaT:Ntime*deltaT; +% +% afigure +% for i = 1 : Ntraj +% +% lin_pred = A_nom*reshape(Xstr(:,i,1:end-1),size(Xacc,1),Ntime); +% lin_pred_acc(:,(i-1)*Ntime+1:i*Ntime) = lin_pred; +% Y_res = reshape(Xstr(:,i,2:end),size(Xacc,1),Ntime)-lin_pred; +% Y_res_acc(:,(i-1)*Ntime+1:i*Ntime) = Y_res; +% +% % subplot(2,Ntraj/2,i) +% % plot(t, lin_pred(1,:), t, reshape(Xstr(1,i,2:end),1,Ntime),... +% % t, lin_pred(2,:), t, reshape(Xstr(2,i,2:end),1,Ntime)); +% % xlabel('Time (sec)') +% % ylabel('State value') +% % fprintf('Error %i: %.4f\n', i, norm(lin_pred-reshape(Xstr(:,i,2:end),size(Xacc,1),Ntime))); +% end +% +% %% Test eigenfunction generation +% +% center_type = 'random'; +% N = Ntraj*Ntime; +% N_basis = 40; +% +% switch center_type +% case 'random' +% cent = rand(n,N_basis)*2*pi - pi; +% case 'data' +% cent = datasample(Xacc',N_basis)'+0.05*(rand(n,N_basis)*2-1); +% end +% +% rbf_type = 'gauss'; +% eps_rbf = 1; +% zfun = @(xx) [xx'; rbf(xx',cent,rbf_type,eps_rbf)]; +% X = zfun(Xacc'); +% Y = zfun(Yacc'); +% D_norm = diag(vecnorm(X')); +% X = D_norm*X; +% Y = D_norm*Y; +% +% to_scale = 'Scale'; +% to_center = 'Ignore'; +% k_mode = -1; +% tol = N_basis*eps; +% nobal = 'Ignore'; +% n_ref = 0; +% ref_select = 1; +% target = [0 0]'; %Not used because ref_select not equal ot 2 +% overwrite = 'OverWrite'; +% file_save = 'NoSave'; +% +% % [Z, Lambda, rez, RQ_ref, RSI, Z_ref, rez_ref, U, AxU_k] = ... +% % XY_DDMD_R4( X, Y, to_scale, to_center, k_mode, tol, nobal, n_ref,... +% % ref_select, target, overwrite, file_save); +% +% A = Y/X; +% error = norm(A*X-Y) +% [V,D] = eig(A'); +% rel_inds = find(diag(abs(D)) > 0.1); +% V_red = V(:,rel_inds); +% D_red = D(rel_inds,rel_inds); +% +% z_eigfun = @(xx) (zfun(xx)'*V_red)'; +% A_koop = D_red; %Discrete time autonomous dynamics; +% Zacc = z_eigfun(Yacc'); +% +% %Compare z with its linear evolution +% z = z_eigfun(Xacc(:,1)'); +% for i = 1 : Ntime +% z = [z A_koop*z(:,end)]; +% end +% +% %% +% t = deltaT : deltaT : Ntime*deltaT; +% n_plot = 15; +% afigure +% for i = 1 : n_plot +% subplot(n_plot,1,i) +% plot(t,Zacc(i,1:end),t,z(i,2:end)) +% legend('True', 'Linear evolution') +% end +% \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/collect_data.m b/scripts/Koopman-ID/utils/collect_data.m index d2aa418..e142df1 100644 --- a/scripts/Koopman-ID/utils/collect_data.m +++ b/scripts/Koopman-ID/utils/collect_data.m @@ -1,5 +1,5 @@ function [Xstr, Xacc, Yacc, Ustr, Uacc, time_str] = collect_data(n,m,Ntraj,... - Ntime,deltaT,X0,Xf_A,K_nom,f_u,U_perturb, autonomous_learning) + Ntime,deltaT,X0,Xf,K_nom,f_u,U_perturb, autonomous_learning) %Collect data from the "true" system with nominal controller plus %perturbation %Inputs: @@ -28,7 +28,7 @@ Xcurrent = X0; Xstr(:,:,1) = X0; for i = 1:Ntime - Ucurrent = K_nom*(Xcurrent-Xf_A)+U_perturb(i,:); + Ucurrent = K_nom*(Xcurrent-Xf)+U_perturb(i,:); Xnext = sim_timestep(deltaT, f_u, 0, Xcurrent, Ucurrent); Xstr(:,:,i+1) = Xnext; Xacc(:,Ntraj*(i-1)+1:Ntraj*(i)) = Xcurrent; @@ -40,8 +40,8 @@ Ustr(:,:,i) = Ucurrent; Uacc(:,Ntraj*(i-1)+1:Ntraj*i) = Ucurrent; else - Ustr(:,:,i) = Ucurrent - K_nom*Xcurrent; - Uacc(:,Ntraj*(i-1)+1:Ntraj*i) = Ucurrent - K_nom*Xcurrent; + Ustr(:,:,i) = Ucurrent - K_nom*(Xcurrent-Xf); + Uacc(:,Ntraj*(i-1)+1:Ntraj*i) = Ucurrent - K_nom*(Xcurrent-Xf); end end end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/evaluate_model_performance.m b/scripts/Koopman-ID/utils/evaluate_model_performance.m index b8e3e64..1461aad 100644 --- a/scripts/Koopman-ID/utils/evaluate_model_performance.m +++ b/scripts/Koopman-ID/utils/evaluate_model_performance.m @@ -1,5 +1,6 @@ function [mse_edmd_avg, mse_koop_avg, mse_edmd_std, mse_koop_std,... - mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop] = ... + mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop,... + cost_nom, cost_edmd, cost_koop] = ... evaluate_model_performance(K_nom, A_edmd, B_edmd, C_edmd, A_koop, B_koop, ... C_koop, Nsim, X0, Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results) @@ -8,19 +9,19 @@ n_edmd = size(A_edmd,1); n_koop = size(A_koop,1); Ntime = Tsim/deltaT; - A_koop_d = expm(A_koop*deltaT); %Discretize A_koop %Prediction task: [x,x_edmd,x_koop,mse_edmd_avg,mse_koop_avg,mse_edmd_std,mse_koop_std]... = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,f_u,liftFun,... - phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop_d,B_koop,C_koop); + phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop); %Closed loop task: [x_track,x_edmd_track,x_koop_track,mse_nom_track, mse_edmd_track,... - mse_koop_track,t_plot,traj_d, E_nom, E_edmd, E_koop] = ... + mse_koop_track,t_plot,traj_d, E_nom, E_edmd, E_koop,... + cost_nom, cost_edmd, cost_koop] = ... sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,... Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,... - C_edmd,A_koop_d,B_koop,C_koop); + C_edmd,A_koop,B_koop,C_koop); %Plot results if plot_results @@ -34,10 +35,10 @@ plot([0:Ntime]*deltaT,reshape(x_edmd(1,i,:),Ntime+1,1), '--r','linewidth',lw) plot([0:Ntime]*deltaT,reshape(x_koop(1,i,:),Ntime+1,1), '--g','linewidth',lw-1) end - axis([0 Tsim min(x_edmd(1,1,:))-0.15 max(x_edmd(1,1,:))+0.15]) - title('Predictor comparison, 1st trajectory - $x_1$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); + axis([0 Tsim min(x(1,1,:))-0.2 max(x(1,1,:))+0.2]) + title('Prediction comparison, 1st trajectory - $x_1$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); set(gca,'fontsize',20) - LEG = legend('True','EDMD','Koopman e-func','location','southwest'); + LEG = legend('True','EDMD','Koopman e-func','location','best'); set(LEG,'interpreter','latex') h1.Position = h1.Position + [0 0.1 0 -0.1]; %Modify size of 1st subplot @@ -48,8 +49,8 @@ plot([0:Ntime]*deltaT,reshape(x_edmd(2,i,:),Ntime+1,1), '--r','linewidth',lw) plot([0:Ntime]*deltaT,reshape(x_koop(2,i,:),Ntime+1,1), '--g','linewidth',lw-1) end - axis([0 Tsim min(x_edmd(2,1,:))-0.15 max(x_edmd(2,1,:))+0.15]) - title('Predictor comparison, 1st trajectory - $x_2$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); + axis([0 Tsim min(x(2,1,:))-0.2 max(x(2,1,:))+0.2]) + title('Prediction comparison, 1st trajectory - $x_2$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); set(gca,'fontsize',20) %LEG = legend('True','EDMD','Koopman e-func','location','southwest'); set(LEG,'interpreter','latex') @@ -60,12 +61,12 @@ hold on for i=1:Ntraj scatter(Xstr(1,i,1),Xstr(2,i,1),'+') - p(i) = plot(squeeze(Xstr(1,i,:)),squeeze(Xstr(2,i,:)),'Color',[0.2,0.2,0.2],'linewidth',1); + p(i) = plot(squeeze(Xstr(1,i,:)),squeeze(Xstr(2,i,:)),':k','linewidth',1); %alpha(p(i),0.2) end for i = 1 : Nsim - plot(reshape(x(1,i,:),Ntime+1,1),reshape(x(2,i,:),Ntime+1,1),'-r') - plot(reshape(x_edmd(1,i,:),Ntime+1,1),reshape(x_edmd(2,i,:),Ntime+1,1),'-b') + plot(reshape(x(1,i,:),Ntime+1,1),reshape(x(2,i,:),Ntime+1,1),'-k') + plot(reshape(x_edmd(1,i,:),Ntime+1,1),reshape(x_edmd(2,i,:),Ntime+1,1),'-r') plot(reshape(x_koop(1,i,:),Ntime+1,1),reshape(x_koop(2,i,:),Ntime+1,1),'-g') end %scatter(cent(1,:),cent(2,:),'o') diff --git a/scripts/Koopman-ID/utils/extendedDMD.m b/scripts/Koopman-ID/utils/extendedDMD.m index f91d7ae..ef2d376 100644 --- a/scripts/Koopman-ID/utils/extendedDMD.m +++ b/scripts/Koopman-ID/utils/extendedDMD.m @@ -1,8 +1,8 @@ -function [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m,N_basis,basis_function,... - rbf_type, center_type, eps, plot_basis, xlim, ylim, Xacc, Yacc, Uacc) +function [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m,Ntraj, Ntime, N_basis,basis_function,... + rbf_type, center_type, eps, plot_basis, xlim, ylim, Xacc, Yacc, Uacc, Xstr, A_nom, B_nom, K_nom, deltaT) %Identify lifted state space model using Extended Dynamic Mode - %Decomposition + %Inputs: % n - Number of states % m - Number of control inputs @@ -53,8 +53,15 @@ % ******************************* Lift *********************************** %disp('Starting LIFT GENERATION') - liftFun = @(xx)( [xx;rbf(xx,cent,rbf_type,eps)] ); - Nlift = N_basis + n; + switch basis_function + case 'rbf' + liftFun = @(xx)( [xx;rbf(xx,cent,rbf_type,eps)] ); + case 'koopman_efunc' + [~, liftFun] = construct_eigfuncs(n, Ntraj, Ntime, N_basis, ... + A_nom, B_nom, K_nom, Xstr, deltaT); + liftFun = @(xx) [xx; liftFun(xx)]; + end + Nlift = length(liftFun(zeros(n,1))); Xlift = liftFun(Xacc); Ylift = liftFun(Yacc); %fprintf('Lifting DONE, time = %1.2f s \n', toc); @@ -71,6 +78,11 @@ A_edmd= M(1:Nlift,1:Nlift); B_edmd = M(1:Nlift,Nlift+1:end); C_edmd = M(Nlift+1:end,1:Nlift); + + % Add known structure (overwrite parts of the learned matrices): + A_edmd(n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_edmd,2)-n)]; + C_edmd = zeros(size(C_edmd)); + C_edmd(:,1:n) = eye(n); %fprintf('Regression done, time = %1.2f s \n', toc); diff --git a/scripts/Koopman-ID/utils/koopman_eigen_id.m b/scripts/Koopman-ID/utils/koopman_eigen_id.m index 1c328ed..504a9fa 100644 --- a/scripts/Koopman-ID/utils/koopman_eigen_id.m +++ b/scripts/Koopman-ID/utils/koopman_eigen_id.m @@ -1,12 +1,11 @@ function [A_koop, B_koop, C_koop, phi_fun_v] = koopman_eigen_id(n,m, Ntraj, Ntime, ... N_basis,basis_function, rbf_type, center_type, eps_rbf, N_lambda, ... - lambda_type, A_edmd, A_nom, B_nom, K_nom, Xacc, Xstr, Xacc_c,... + lambda_type, A_edmd, A_nom, B_nom, K_nom, Xacc, Yacc, Xstr, Xacc_c,... Yacc_c, Uacc_c, Xstr_c, Ustr_c, timestr, deltaT, learn_type) %Identify lifted state space model using approximate Koopman invariant %subspace - %Decomposition %Inputs: % n - Number of states % m - Number of control inputs @@ -42,44 +41,86 @@ %N_g0 = N_basis+n; %N_gen = N_lambda*N_g0; - cent_g0 = datasample(Xacc',N_basis)'+0.05*(rand(n,N_basis)*2-1); - gfun = @(xx) [xx'; rbf(xx',cent_g0,rbf_type,eps_rbf)]; +% cent_g0 = datasample(Xacc',N_basis)'+0.05*(rand(n,N_basis)*2-1); +% gfun = @(xx) [xx'; rbf(xx',cent_g0,rbf_type,eps_rbf)]; - switch lambda_type +% switch lambda_type +% case 'random' +% lambda = 2*(rand(N_lambda,1)*2-1) + 2i*(rand(N_lambda,1)*2-1); +% case 'DMDmixing' +% lambda_dmd = dmd(Xstr,deltaT); +% lambda = zeros(N_lambda,1); +% lambda(1:length(lambda_dmd)) = lambda_dmd; % start the first lambda to dmd ones +% dlambda = N_lambda - length(lambda_dmd); % how many more to add +% nlambda = length(lambda_dmd); % starting lambda number +% +% vlambda = [0,1 +% 1,0 +% 2,0 +% 0,2 +% 2,1 +% 1,2 +% 3,1 +% 1,3 +% 3,2 +% 2,3]; +% +% while nlambda < N_lambda % stop when we have enough lambda +% nlambda = nlambda+1; +% for k =1:length(lambda_dmd) % combination of dmd lambdas +% lambda(nlambda) = lambda(nlambda) + lambda_dmd(k)*vlambda(nlambda,k); +% end +% end +% case 'eDMD' +% lambda = log(eig(A_edmd))/deltaT; +% end + center_type = 'random'; + N = Ntraj*Ntime; + N_basis = 2*N; + + switch center_type case 'random' - lambda = 2*(rand(N_lambda,1)*2-1) + 2i*(rand(N_lambda,1)*2-1); - case 'DMDmixing' - lambda_dmd = dmd(Xstr,deltaT); - lambda = zeros(N_lambda,1); - lambda(1:length(lambda_dmd)) = lambda_dmd; % start the first lambda to dmd ones - dlambda = N_lambda - length(lambda_dmd); % how many more to add - nlambda = length(lambda_dmd); % starting lambda number - - vlambda = [0,1 - 1,0 - 2,0 - 0,2 - 2,1 - 1,2 - 3,1 - 1,3 - 3,2 - 2,3]; - - while nlambda < N_lambda % stop when we have enough lambda - nlambda = nlambda+1; - for k =1:length(lambda_dmd) % combination of dmd lambdas - lambda(nlambda) = lambda(nlambda) + lambda_dmd(k)*vlambda(nlambda,k); - end - end - case 'eDMD' - lambda = log(eig(A_edmd))/deltaT; + cent = rand(n,N_basis)*2*pi - pi; + case 'data' + cent = datasample(Xacc',N_basis)'+0.05*(rand(n,N_basis)*2-1); end %N_lambda = length(lambda); - y_reg = @(x) [x(1),x(2)]; +% y_reg = @(x) [x(1),x(2)]; +% +% [phi_fun, A_koop, C_koop, phi_grid] = get_phi_A(Xstr, timestr, lambda, gfun,y_reg); + + zfun = @(xx) [xx'; rbf(xx',cent,rbf_type,eps_rbf)]; + X = zfun(Xacc'); + Y = zfun(Yacc'); + + to_scale = 'Scale'; + to_center = 'Ignore'; + k_mode = -1; + tol = N_basis*eps; + nobal = 'Ignore'; + n_ref = 0; + ref_select = 1; + target = [0 0]'; %Not used because ref_select not equal ot 2 + overwrite = 'OverWrite'; + file_save = 'NoSave'; + + [Z, Lambda, rez, RQ_ref, RSI, Z_ref, rez_ref, U, AxU_k] = ... + XY_DDMD_R4( X, Y, to_scale, to_center, k_mode, tol, nobal, n_ref,... + ref_select, target, overwrite, file_save ); + + z_eigfun = @(xx) Z'*zfun(xx); + A_koop = diag(Lambda); %Discrete time autonomous dynamics; + Zacc = z_eigfun(Xacc'); + + %Compare z with its linear evolution + z = z_eigfun(Xacc(:,1)'); + for i = 1 : 1000 + z = [z A_koop*z(:,end)]; + end + - [phi_fun, A_koop, C_koop, phi_grid] = get_phi_A(Xstr, timestr, lambda, gfun,y_reg); + C_koop = (Zacc/Xacc)'; fprintf('Learning autonomous dynamics done, execution time: %1.2f s \n', toc); diff --git a/scripts/Koopman-ID/utils/learn_B_koop.m b/scripts/Koopman-ID/utils/learn_B_koop.m index cf17c2d..0ec27c2 100644 --- a/scripts/Koopman-ID/utils/learn_B_koop.m +++ b/scripts/Koopman-ID/utils/learn_B_koop.m @@ -1,4 +1,4 @@ -function [B_koop, phi_fun_v] = learn_B_koop(n, m, Ntraj, Ntime, phi_fun, A_koop,... +function [B_koop, phi_fun_v] = learn_B_koop(n, m, Ntraj, Ntime, phi_fun_v, A_koop,... C_koop, Xstr_c, Ustr_c, Xacc_c, Yacc_c, Uacc_c, deltaT, learn_type) % Setup regression problem min ||Q * vec(Blift) - b||_2^2 corresponding to @@ -12,45 +12,88 @@ %Outputs: - A_koop_d = expm(A_koop*deltaT); %Discretize A_koop - phi_fun_v = @(x) phiFunction(phi_fun,x); + %phi_fun_v = @(x) phiFunction(phi_fun,x); switch learn_type case 'multi-step' - Obs = obsv(A_koop_d,C_koop); % Not an efficient implementation +% Obs = obsv(A_koop_d,C_koop); % Not an efficient implementation +% nc = size(C_koop,1); +% b = zeros(Ntime*nc,Ntraj); +% Q = zeros(Ntraj*Ntime*nc,size(A_koop_d,1)*m); +% X0 = Xstr_c(:,:,1); +% +% Obsx0 = Obs*phi_fun_v(X0); +% for j = 1:Ntime +% b(nc*(j-1)+1:nc*j,:) = Obsx0( j*nc + 1 : (j+1)*nc, : ) - Xstr_c(:,:,j+1); +% tmp = 0; +% for k = 0 : j-1 +% kprime = j - k -1; +% tmp = tmp + kron(Ustr_c(:,:,k+1)',Obs(kprime*nc + 1 : (kprime+1)*nc,:)); +% end +% Q(Ntraj*nc*(j-1)+1:Ntraj*nc*(j),:) = tmp; +% end +% b = reshape(-b,Ntraj*Ntime*nc,1); +% +% B_koop = pinv(Q)*b; % = vec(Blift) %Add regularization +% B_koop = reshape(B_koop,size(A_koop_d,1),m); % Unvectorize + + % Regression for B (Milan Korda implementation) + % Setup regression problem min ||Q * vec(Blift) - b||_2^2 corresponding to + % min_{Blift} \sum_{j=1}^{Nsim} ||xpred - xtrue||^2 where + % xpred = C*Alift^j*x0_lift + sum_{k=0}^{j-1} kron(u_k',C*Alift^{j-k-1})*vec(BLift) + % (And this sum over all trajectories) + Obs = obsvk(sparse(A_koop),C_koop,Ntime+1); % Not an efficient implementation + b = []; nc = size(C_koop,1); - b = zeros(Ntime*nc,Ntraj); - Q = zeros(Ntraj*Ntime*nc,size(A_koop_d,1)*m); - X0 = Xstr_c(:,:,1); - - Obsx0 = Obs*phi_fun_v(X0); - for j = 1:Ntime - b(nc*(j-1)+1:nc*j,:) = Obsx0( j*nc + 1 : (j+1)*nc, : ) - Xstr_c(:,:,j+1); - tmp = 0; - for k = 0 : j-1 - kprime = j - k -1; - tmp = tmp + kron(Ustr_c(:,:,k+1)',Obs(kprime*nc + 1 : (kprime+1)*nc,:)); - end - Q(Ntraj*nc*(j-1)+1:Ntraj*nc*(j),:) = tmp; + Q = zeros(Ntraj*Ntime*nc,size(A_koop,1)*m); + for q = 1 : Ntraj + fprintf('Building regression matrices for B: %f percent complete \n', 100*q / Ntraj) + x0 = reshape(Xstr_c(:,q,1),n,1); + Obsx0 = Obs*phi_fun_v(x0); + for j = 1:Ntime + b = [b ; Obsx0( j*nc + 1 : (j+1)*nc, : ) - reshape(Xstr_c(:,q,j+1),n,1)] ; + tmp = 0; + for k = 0 : j-1 + kprime = j - k - 1; + tmp = tmp + kron(reshape(Ustr_c(:,q,k+1),m,1)',Obs(kprime*nc + 1 : (kprime+1)*nc,:)); + end + Q((q-1)*Ntime*nc + (j-1)*nc + 1 : (q-1)*Ntime*nc + j*nc,:) = tmp; + end end - b = reshape(-b,Ntraj*Ntime*nc,1); + b = -b; + + k = 10; + %B_koop = Q\b; % = vec(Blift) + B_koop = (Q'*Q + k*eye(size(Q,2)))\(Q'*b); %Regularize B-learning + B_koop = reshape(B_koop,size(A_koop,1),m); % Unvectorize - B_koop = pinv(Q)*b; % = vec(Blift) %Add regularization - B_koop = reshape(B_koop,size(A_koop_d,1),m); % Unvectorize case 'single-step' %Calculate residual after autonomous dynamics are subtracted Z_x = phi_fun_v(Xacc_c); Z_y = phi_fun_v(Yacc_c); - unusable_data_x = find(sum(isnan(Z_x),1) > 0); - unusable_data_y = find(sum(isnan(Z_y),1) > 0); - unusable_data = unique([unusable_data_x unusable_data_y]); - Z_x(:,unusable_data) = []; %Delete unsable elements - Z_y(:,unusable_data) = []; %Delete unsable elements - Uacc_c(:,unusable_data) = []; %Delete unsable elements - - Racc = Z_y - A_koop_d*Z_x; %Calculate residual when subtracting autonomous dynamics - %Racc = phi_fun_v(Yacc_c) - A_koop*phi_fun_v(Xacc_c); + Racc = Z_y-A_koop*Z_x; %Calculate residual when subtracting autonomous dynamics + + k = 10; + %B_koop = (Uacc_c'\Racc')'; + B_koop = ((Uacc_c*Uacc_c' + k*eye(size(Uacc_c,1)))\(Uacc_c*Racc'))'; + if sum(sum(isnan(B_koop))) > 0 + disp('Warning: NAN-values in B-matrix, terminating.') + return + end + case 'multi-step_mod' + %Calculate residual after autonomous dynamics are subtracted + Racc = []; + Z_evol = zeros(length(phi_fun_v(zeros(n,1))),Ntime+1); + for i = 1 : Ntraj + Z_evol(:,1) = phi_fun_v(reshape(Xstr_c(:,i,1),n,1)); + Z_y = phi_fun_v(reshape(Xstr_c(:,i,2:end),n,Ntime)); + for j = 2 : Ntime+1 + Z_evol(:,j) = A_koop*Z_evol(:,j-1); + end + Racc = [Racc Z_evol(:,2:end)-Z_y]; + end + B_koop = (Uacc_c'\Racc')'; if sum(sum(isnan(B_koop))) > 0 disp('Warning: NAN-values in B-matrix, terminating.') diff --git a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m index bca6c13..59954bb 100644 --- a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m +++ b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m @@ -1,6 +1,7 @@ -function [x_nom,x_edmd,x_koop,mse_nom,mse_edmd,mse_koop, t_plot, traj_d, E_nom, E_edmd, E_koop]... +function [x_nom,x_edmd,x_koop,mse_nom,mse_edmd,mse_koop, t_plot, traj_d, E_nom, E_edmd, E_koop,... + cost_nom, cost_edmd, cost_koop]... = sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,Tsim,... - f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop_d,... + f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,... B_koop,C_koop) %Set up trajectory to track: @@ -35,7 +36,7 @@ z_koop(:,1) = phi_fun_v(traj_d(:,1)); % Define Koopman controller - Q = eye(2); % Weight matrices + Q = 1000*eye(2); % Weight matrices R = 0.01; Tpred = 0.1; % Prediction horizon Np = round(Tpred / deltaT); @@ -45,7 +46,7 @@ % Build Koopman MPC controller MPC_edmd = getMPC(A_edmd,B_edmd,C_edmd,0,Q,R,Q,Np,-100, 100, xlift_min, xlift_max,'qpoases'); - MPC_koop = getMPC(A_koop_d,B_koop,C_koop,0,Q,R,Q,Np,-100, 100, xlift_min, xlift_max,'qpoases'); + MPC_koop = getMPC(A_koop,B_koop,C_koop,0,Q,R,Q,Np,-100, 100, xlift_min, xlift_max,'qpoases'); % Get Jacobian of the true dynamics (for local linearization MPC) x = sym('x',[2 1]); syms u; @@ -70,7 +71,7 @@ % Local linearization MPC Aloc = double(subs(Jx,[x;u],[x_nom(:,i);u_nom(:,i)])); % Get local linearization Bloc = double(subs(Ju,[x;u],[x_nom(:,i);u_nom(:,i)])); - Cloc = double(subs(f_ud_sym,[x;u],[x_nom(:,i);u_nom(:,i)])) - Aloc*x_nom(:,i) - Bloc*u_nom(:,i); + %Cloc = double(subs(f_ud_sym,[x;u],[x_nom(:,i);u_nom(:,i)])) - Aloc*x_nom(:,i) - Bloc*u_nom(:,i); [U_nom,~,optval] = solveMPCprob(Aloc,Bloc,eye(2),[],Q,R,Q,Np,-100, 100,[],[],x_nom(:,i),yr); % Get control input u_nom(:,i) = U_nom(1:m,1); if(optval == Inf) % Detect infeasibility @@ -85,12 +86,11 @@ x_edmd(:,i+1) = sim_timestep(deltaT, f_u, 0, x_edmd(:,i), u_edmd(:,i)); % Koopman e-func MPC -% z_koop(:,i) = phi_fun_v(x_koop(:,i)); % Lift -% disp(sum(sum(isnan(z_koop(:,i))))); -% u_koop(:,i) = MPC_koop(z_koop(:,i),yr) + K_nom*x_koop(:,i); % Get control input -% x_koop(:,i+1) = sim_timestep(deltaT, f_u, 0, x_koop(:,i), u_koop(:,i)); + z_koop(:,i) = phi_fun_v(x_koop(:,i)); % Lift + u_koop(:,i) = MPC_koop(z_koop(:,i),yr);% + K_nom*x_koop(:,i); % Get control input + x_koop(:,i+1) = sim_timestep(deltaT, f_u, 0, x_koop(:,i), u_koop(:,i)); - disp([u_nom(:,i), u_edmd(:,i) u_koop(:,i)]) + %disp([u_nom(:,i), u_edmd(:,i) u_koop(:,i)]) end if(isempty(ind_inf)) @@ -105,4 +105,7 @@ E_nom = norm(u_nom); E_edmd = norm(u_edmd); E_koop = norm(u_koop); + cost_nom = sum(diag(x_nom(:,2:end)'*Q*x_nom(:,2:end))) + sum(diag(u_nom'*R*u_nom)); + cost_edmd = sum(diag(x_edmd(:,2:end)'*Q*x_edmd(:,2:end))) + sum(diag(u_edmd'*R*u_edmd)); + cost_koop = sum(diag(x_koop(:,2:end)'*Q*x_koop(:,2:end))) + sum(diag(u_koop'*R*u_koop)); end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/sim_prediction.m b/scripts/Koopman-ID/utils/sim_prediction.m index 63d0e78..7f59afa 100644 --- a/scripts/Koopman-ID/utils/sim_prediction.m +++ b/scripts/Koopman-ID/utils/sim_prediction.m @@ -1,8 +1,8 @@ function [x,x_edmd,x_koop,mse_edmd_avg,mse_koop_avg,mse_edmd_std,mse_koop_std]... = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,f_u,liftFun,... - phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop_d,B_koop,C_koop) + phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop) - u = 5*rand(m,Nsim,Ntime)-2.5; + u = zeros(m,Nsim,Ntime); x = zeros(n,Nsim,Ntime+1); z_edmd = zeros(n_edmd,Nsim,Ntime+1); @@ -11,20 +11,19 @@ z_edmd(:,:,1) = liftFun(X0); z_koop(:,:,1) = phi_fun_v(X0); - - % Simulate all systems and initial points - K_mod = K_nom + randn(size(K_nom)); for i = 1:Ntime + %Controller: + u(:,:,i) = K_nom*x(:,:,i) + 2*rand()-1; + %True dynamics: - x(:,:,i+1) = sim_timestep(deltaT,f_u,0,x(:,:,i), K_mod*x(:,:,i)+u(:,:,i)); + x(:,:,i+1) = sim_timestep(deltaT,f_u,0,x(:,:,i), u(:,:,i)); %EDMD predictor: - z_edmd(:,:,i+1) = A_edmd*z_edmd(:,:,i) + B_edmd*(K_mod*C_edmd*z_edmd(:,:,i) + u(:,:,i)); + z_edmd(:,:,i+1) = A_edmd*z_edmd(:,:,i) + B_edmd*u(:,:,i); %Koopman eigenfunction predictor: - z_koop(:,:,i+1) = A_koop_d*z_koop(:,:,i) + ... - B_koop*(K_mod*C_koop*z_koop(:,:,i)+u(:,:,i)-K_nom*C_koop*z_koop(:,:,i)); + z_koop(:,:,i+1) = A_koop*z_koop(:,:,i) + B_koop*u(:,:,i); end % Calculate corresponding predictions and MSE From b2af0bdd7c87af1f42c2a10b67abcb21b640d6ac Mon Sep 17 00:00:00 2001 From: Cafolkes Date: Sun, 11 Aug 2019 11:09:03 -0700 Subject: [PATCH 2/9] Added EDMD option, transient-case still has issues that (potentially) must be resolved --- scripts/Koopman-ID/construct_eigfuncs.m | 12 +++--- scripts/Koopman-ID/koopman_eigen_id_laplace.m | 20 ++++++---- scripts/Koopman-ID/main_comparison_pendulum.m | 11 ++++-- scripts/Koopman-ID/utils/extendedDMD.m | 38 +++++++++++++++++-- scripts/Koopman-ID/utils/learn_B_koop.m | 31 ++++++++++----- 5 files changed, 83 insertions(+), 29 deletions(-) diff --git a/scripts/Koopman-ID/construct_eigfuncs.m b/scripts/Koopman-ID/construct_eigfuncs.m index 0901348..417d3a2 100644 --- a/scripts/Koopman-ID/construct_eigfuncs.m +++ b/scripts/Koopman-ID/construct_eigfuncs.m @@ -1,5 +1,5 @@ function [A_koop, z_eigfun] = construct_eigfuncs(n, Ntraj, Ntime, ... - N_basis, A_nom, B_nom, K_nom, Xstr, deltaT) + N_basis, A_nom, B_nom, K_nom, Xstr, Xf, deltaT) %Identify lifted state space model using approximate Koopman invariant %subspace @@ -39,7 +39,7 @@ A_c = A_nom + B_nom*K_nom; A_c_d = expm(A_c*deltaT); %Discrete time dynamics matrix - cent = rand(n,N_basis)*2*pi - pi; + cent = rand(n,50)*2*pi - pi; rbf_type = 'gauss'; eps_rbf = 1; @@ -51,9 +51,11 @@ Y = []; Z_mplus1 = []; Z_m = []; + Xstr_shift = zeros(size(Xstr)); for i = 1 : Ntraj - X_mplus1 = reshape(Xstr(:,i,2:end),n,Ntime); - X_m = reshape(Xstr(:,i,1:end-1),n,Ntime); + Xstr_shift(:,i,:) = Xstr(:,i,:) - Xf(:,i); + X_mplus1 = reshape(Xstr_shift(:,i,2:end),n,Ntime); + X_m = reshape(Xstr_shift(:,i,1:end-1),n,Ntime); % Set up Y matrix (targets), Y(:,i) = x_(i+1) - A_nom*x_i Y = [Y X_mplus1-A_c_d*X_m]; @@ -72,7 +74,7 @@ N = size(Z_m,1); cvx_begin variable C(n,N); - minimize (norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro') + 0.01*norm(C,'fro')) + minimize (norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro') + 1*norm(C,'fro')) subject to {C*con1 == zeros(n)}; cvx_end diff --git a/scripts/Koopman-ID/koopman_eigen_id_laplace.m b/scripts/Koopman-ID/koopman_eigen_id_laplace.m index b81538c..c0aad29 100644 --- a/scripts/Koopman-ID/koopman_eigen_id_laplace.m +++ b/scripts/Koopman-ID/koopman_eigen_id_laplace.m @@ -110,9 +110,9 @@ Z_y = z_eigfun(Yacc); k=1; A_koop = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(Z_x,1)-n)]; - %A_koop = [A_koop; (Z_x'\Z_y(n/2+1:n,:)')']; - A_state = lasso(Z_x',Z_y(n/2+1:n,:)','Lambda',1e-5)'; - A_koop = [A_koop; A_state]; + A_koop = [A_koop; (Z_x'\Z_y(n/2+1:n,:)')']; + %A_state = lasso(Z_x',Z_y(n/2+1:n,:)','Lambda',1e-4, 'Alpha', 0.8)'; + %A_koop = [A_koop; A_state]; A_koop = [A_koop; [zeros(length(lambd),n) diag(lambd)]]; % Calculate projection of states onto eigenfunctions @@ -131,10 +131,16 @@ A_koop(:,1:n) = A_koop(:,1:n)-B_koop*K_nom; % Evaluate training error: -% Ypred = zeros(size(Yacc_c)); -% -% C_koop*(A_koop*phi_fun_v(Xacc_c)+B_koop*Uacc_c); -% disp(immse(Ypred,Yacc_c)); +% Ypred_c = zeros(size(Xstr_c(:,:,2:end))); +% Zpred = phi_fun_v(Xstr_c(:,:,1)); +% err = 0; +% for i = 1:Ntime +% Zpred = A_koop*Zpred+B_koop*Ustr_c(:,:,i); +% Ypred_c(:,:,i) = C_koop*Zpred; +% err = err + sum(sum((Ypred_c(:,:,i) - Xstr_c(:,:,i+1)).^2)); +% end +% err = err/(Ntime*Ntraj); +% disp(err); end diff --git a/scripts/Koopman-ID/main_comparison_pendulum.m b/scripts/Koopman-ID/main_comparison_pendulum.m index 8309a6b..1acf49a 100644 --- a/scripts/Koopman-ID/main_comparison_pendulum.m +++ b/scripts/Koopman-ID/main_comparison_pendulum.m @@ -52,7 +52,7 @@ plot_basis_koop = false; %Plots the basis functions if true xlim = [-1, 1]; %Plot limits ylim = [-1, 1]; %Plot limits -learn_type = 'multi-step'; %Learn B-matrix with single step or multi-step prediction horizon +learn_type = 'multi-step_mod'; %Learn B-matrix with single step or multi-step prediction horizon %Test simulation parameters: Nsim = 5; @@ -71,7 +71,7 @@ autonomous_learning = true; U_perturb = randn(Ntime,Ntraj); %Add normally distributed noise to nominal controller [Xstr, Xacc, Yacc, Ustr, Uacc, timestr] = collect_data(n,m,Ntraj,... - Ntime,deltaT,X0_A, Xf_A ,K_nom,f_u,U_perturb, autonomous_learning); + Ntime,deltaT,X0_B, Xf_B ,K_nom,f_u,U_perturb, autonomous_learning); % Collect data to learn controlled dynamics: autonomous_learning = false; @@ -87,9 +87,14 @@ disp('Starting EDMD...'); tic [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m, Ntraj, Ntime, N_basis_edmd,basis_function_edmd,... rbf_type_edmd, center_type_edmd, eps_rbf_edmd, plot_basis_edmd, xlim,... - ylim, Xacc, Yacc, Uacc, Xstr, A_nom, B_nom, K_nom, deltaT); + ylim, Xacc, Yacc, Uacc, Xstr, Xf_B, A_nom, B_nom, K_nom, deltaT); fprintf('EDMD done, execution time: %1.2f s \n', toc); +% basis_function_koop = 'koopman_efunc'; +% [A_koop, B_koop, C_koop, phi_fun_v] = extendedDMD(n,m, Ntraj, Ntime, N_basis_edmd,basis_function_koop,... +% rbf_type_edmd, center_type_edmd, eps_rbf_edmd, plot_basis_edmd, xlim,... +% ylim, Xacc, Yacc, Uacc, Xstr, Xf_B, A_nom, B_nom, K_nom, deltaT); + [A_koop, B_koop, C_koop, phi_fun_v] = koopman_eigen_id_laplace(n, m, Ntraj, Ntime, N_basis_koop,... basis_function_koop, rbf_type_koop, center_type_koop, eps_rbf_koop, ... N_lambda, lambda_type, A_edmd, A_nom, B_nom, K_nom, Xacc, Yacc, Xstr, Xacc_c,... diff --git a/scripts/Koopman-ID/utils/extendedDMD.m b/scripts/Koopman-ID/utils/extendedDMD.m index ef2d376..89c78f6 100644 --- a/scripts/Koopman-ID/utils/extendedDMD.m +++ b/scripts/Koopman-ID/utils/extendedDMD.m @@ -1,5 +1,5 @@ function [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m,Ntraj, Ntime, N_basis,basis_function,... - rbf_type, center_type, eps, plot_basis, xlim, ylim, Xacc, Yacc, Uacc, Xstr, A_nom, B_nom, K_nom, deltaT) + rbf_type, center_type, eps, plot_basis, xlim, ylim, Xacc, Yacc, Uacc, Xstr, Xf, A_nom, B_nom, K_nom, deltaT) %Identify lifted state space model using Extended Dynamic Mode %Decomposition @@ -57,8 +57,8 @@ case 'rbf' liftFun = @(xx)( [xx;rbf(xx,cent,rbf_type,eps)] ); case 'koopman_efunc' - [~, liftFun] = construct_eigfuncs(n, Ntraj, Ntime, N_basis, ... - A_nom, B_nom, K_nom, Xstr, deltaT); + [A_koop, liftFun] = construct_eigfuncs(n, Ntraj, Ntime, N_basis, ... + A_nom, B_nom, K_nom, Xstr, Xf, deltaT); liftFun = @(xx) [xx; liftFun(xx)]; end Nlift = length(liftFun(zeros(n,1))); @@ -80,7 +80,7 @@ C_edmd = M(Nlift+1:end,1:Nlift); % Add known structure (overwrite parts of the learned matrices): - A_edmd(n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_edmd,2)-n)]; + A_edmd(1:n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_edmd,2)-n)]; C_edmd = zeros(size(C_edmd)); C_edmd(:,1:n) = eye(n); @@ -93,4 +93,34 @@ % plot(eig(Alift),'*') % cplot(1,0,0) % axis equal + + % *************** Force known terms of A-matrix *********************** + if strcmp(basis_function, 'koopman_efunc') + %Set up regression for A and B: + N = length(liftFun(ones(n,1))); + X = [Xlift; Uacc]; + Y = Yacc(n/2+1:n,:); + %X_mean = mean(X,2); + %X_std = std(X,0,2); + %X_std(find(X_std<1e-6)) = 1; + %X = (X-X_mean)./X_std; + %A_state = X'\Y'; + %A_state = A_state.*X_std; + A_state = lasso(X',Y','Lambda',1e-6, 'Alpha', 0.8); + + A_edmd = zeros(N); + A_edmd(1:n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_edmd,2)-n)]; + A_edmd(n/2+1:n,:) = A_state(1:N,:)'; + A_edmd(n+1:end,n+1:end) = A_koop; + + Y_kin = Yacc(1:n/2,:) - A_edmd(1:n/2,:)*Xlift; + B_kin = Uacc'\Y_kin'; + Ylift_mod= Ylift(n+1:end,:) - A_koop*Xlift(n+1:end,:); + B_mod = Uacc'\Ylift_mod'; + B_edmd = [B_kin'; A_state(N+1:end,:); B_mod']; + + %Subtract effect of nominal controller from eigenfunctions: + A_edmd(n+1:end,1:n) = A_edmd(n+1:end,1:n)-B_mod'*K_nom; + + end end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/learn_B_koop.m b/scripts/Koopman-ID/utils/learn_B_koop.m index 0ec27c2..d2174e6 100644 --- a/scripts/Koopman-ID/utils/learn_B_koop.m +++ b/scripts/Koopman-ID/utils/learn_B_koop.m @@ -62,9 +62,10 @@ end b = -b; - k = 10; - %B_koop = Q\b; % = vec(Blift) - B_koop = (Q'*Q + k*eye(size(Q,2)))\(Q'*b); %Regularize B-learning + k = 1e-3; + B_koop = Q\b; % = vec(Blift) + %B_koop = (Q'*Q + k*eye(size(Q,2)))\(Q'*b); %Regularize B-learning + %B_koop = lasso(Q,b,'Lambda',1e-2); B_koop = reshape(B_koop,size(A_koop,1),m); % Unvectorize case 'single-step' @@ -83,21 +84,31 @@ end case 'multi-step_mod' %Calculate residual after autonomous dynamics are subtracted +% Racc = []; +% Z_evol = zeros(length(phi_fun_v(zeros(n,1))),Ntime+1); +% for i = 1 : Ntraj +% Z_evol(:,1) = phi_fun_v(reshape(Xstr_c(:,i,1),n,1)); +% Z_y = phi_fun_v(reshape(Xstr_c(:,i,2:end),n,Ntime)); +% for j = 2 : Ntime+1 +% Z_evol(:,j) = A_koop*Z_evol(:,j-1); +% end +% Racc = [Racc Z_evol(:,2:end)-Z_y]; +% end +% +% B_koop = (Uacc_c'\Racc')'; Racc = []; + U_r_acc = []; Z_evol = zeros(length(phi_fun_v(zeros(n,1))),Ntime+1); for i = 1 : Ntraj Z_evol(:,1) = phi_fun_v(reshape(Xstr_c(:,i,1),n,1)); - Z_y = phi_fun_v(reshape(Xstr_c(:,i,2:end),n,Ntime)); for j = 2 : Ntime+1 Z_evol(:,j) = A_koop*Z_evol(:,j-1); end - Racc = [Racc Z_evol(:,2:end)-Z_y]; + Yres = C_koop*Z_evol(:,2:end)-reshape(Xstr_c(:,i,2:end),n,Ntime) ; + Racc = [Racc phi_fun_v(Yres)]; + U_r_acc = [U_r_acc reshape(Ustr_c(:,i,:),m,Ntime)]; end - B_koop = (Uacc_c'\Racc')'; - if sum(sum(isnan(B_koop))) > 0 - disp('Warning: NAN-values in B-matrix, terminating.') - return - end + B_koop = -(U_r_acc'\Racc')'; end end \ No newline at end of file From a6e767424e3258ccf8de2b3a9ba92087b63de486 Mon Sep 17 00:00:00 2001 From: Cafolkes Date: Sun, 11 Aug 2019 12:04:19 -0700 Subject: [PATCH 3/9] Minor fixes --- scripts/Koopman-ID/main_comparison_pendulum.m | 18 +++++++++--------- scripts/Koopman-ID/utils/extendedDMD.m | 2 +- 2 files changed, 10 insertions(+), 10 deletions(-) diff --git a/scripts/Koopman-ID/main_comparison_pendulum.m b/scripts/Koopman-ID/main_comparison_pendulum.m index 1acf49a..e6d02b5 100644 --- a/scripts/Koopman-ID/main_comparison_pendulum.m +++ b/scripts/Koopman-ID/main_comparison_pendulum.m @@ -90,15 +90,15 @@ ylim, Xacc, Yacc, Uacc, Xstr, Xf_B, A_nom, B_nom, K_nom, deltaT); fprintf('EDMD done, execution time: %1.2f s \n', toc); -% basis_function_koop = 'koopman_efunc'; -% [A_koop, B_koop, C_koop, phi_fun_v] = extendedDMD(n,m, Ntraj, Ntime, N_basis_edmd,basis_function_koop,... -% rbf_type_edmd, center_type_edmd, eps_rbf_edmd, plot_basis_edmd, xlim,... -% ylim, Xacc, Yacc, Uacc, Xstr, Xf_B, A_nom, B_nom, K_nom, deltaT); - -[A_koop, B_koop, C_koop, phi_fun_v] = koopman_eigen_id_laplace(n, m, Ntraj, Ntime, N_basis_koop,... - basis_function_koop, rbf_type_koop, center_type_koop, eps_rbf_koop, ... - N_lambda, lambda_type, A_edmd, A_nom, B_nom, K_nom, Xacc, Yacc, Xstr, Xacc_c,... - Yacc_c, Uacc_c, Xstr_c, Ustr_c, timestr, deltaT, learn_type); +basis_function_koop = 'koopman_efunc'; +[A_koop, B_koop, C_koop, phi_fun_v] = extendedDMD(n,m, Ntraj, Ntime, N_basis_edmd,basis_function_koop,... + rbf_type_edmd, center_type_edmd, eps_rbf_edmd, plot_basis_edmd, xlim,... + ylim, Xacc, Yacc, Uacc, Xstr, Xf_B, A_nom, B_nom, K_nom, deltaT); + +% [A_koop, B_koop, C_koop, phi_fun_v] = koopman_eigen_id_laplace(n, m, Ntraj, Ntime, N_basis_koop,... +% basis_function_koop, rbf_type_koop, center_type_koop, eps_rbf_koop, ... +% N_lambda, lambda_type, A_edmd, A_nom, B_nom, K_nom, Xacc, Yacc, Xstr, Xacc_c,... +% Yacc_c, Uacc_c, Xstr_c, Ustr_c, timestr, deltaT, learn_type); %% ************************ Analysis of Results *************************** diff --git a/scripts/Koopman-ID/utils/extendedDMD.m b/scripts/Koopman-ID/utils/extendedDMD.m index 89c78f6..db23618 100644 --- a/scripts/Koopman-ID/utils/extendedDMD.m +++ b/scripts/Koopman-ID/utils/extendedDMD.m @@ -120,7 +120,7 @@ B_edmd = [B_kin'; A_state(N+1:end,:); B_mod']; %Subtract effect of nominal controller from eigenfunctions: - A_edmd(n+1:end,1:n) = A_edmd(n+1:end,1:n)-B_mod'*K_nom; + %A_edmd(n+1:end,1:n) = A_edmd(n+1:end,1:n)-B_mod'*K_nom; end end \ No newline at end of file From 61240f8041c19200b7a655f14d3d6c42c41fe0dd Mon Sep 17 00:00:00 2001 From: Cafolkes Date: Mon, 12 Aug 2019 15:53:47 -0700 Subject: [PATCH 4/9] Minor fixes and refactoring of code to improve performance and readability of KEEDMD implementation --- scripts/Koopman-ID/KEEDMD.m | 60 +++++++++++++++++ scripts/Koopman-ID/construct_eigfuncs.m | 7 +- scripts/Koopman-ID/main_comparison_pendulum.m | 65 ++++++++----------- .../utils/evaluate_model_performance.m | 29 +++++---- scripts/Koopman-ID/utils/extendedDMD.m | 51 ++------------- .../utils/find_nominal_model_ctrl.m | 6 +- .../Koopman-ID/utils/sim_closed_loop_mpc.m | 16 ++--- scripts/Koopman-ID/utils/sim_prediction.m | 21 +++++- 8 files changed, 138 insertions(+), 117 deletions(-) create mode 100644 scripts/Koopman-ID/KEEDMD.m diff --git a/scripts/Koopman-ID/KEEDMD.m b/scripts/Koopman-ID/KEEDMD.m new file mode 100644 index 0000000..d10d561 --- /dev/null +++ b/scripts/Koopman-ID/KEEDMD.m @@ -0,0 +1,60 @@ +function [A_koop, B_koop, C_koop, liftFun] = KEEDMD(n,m,Ntraj, Ntime, N_basis,... + pow_eig_pairs, Xacc, Yacc, Uacc, Xstr, Xf, A_nom, B_nom, K_nom, deltaT) + %Identify lifted state space model using Extended Dynamic Mode + %Decomposition + + %Inputs: + % n - Number of states + % m - Number of control inputs + % N_basis - Number of basis functions to use + % basis_function - Type of basis functions (only rbfs implemented) + % rbf-type - Type of rbf function + % center_type - What centers to use for rbfs ('data'/'random') + % eps - Width of rbf if rbf type has width parameter + % plot_basis - (true/false) determines to plot basis functions + % xlim - Plot limits x-axis + % ylim - Plot limits y-axis + % Xacc - Trajectory data + % Yacc - Trajectory data at next time step + % Uacc - Control input for trajectory data + + %Outputs: + % A_edmd - Passive dynamics matrix in lifted space + % B_edmd - Actuation matrix in lifted space + % C_edmd - Projection matrix from lifted space to outputs + + + [A_eigfuncs, liftFun] = construct_eigfuncs(n, Ntraj, Ntime, N_basis, ... + pow_eig_pairs, A_nom, B_nom, K_nom, Xstr, Xf, deltaT); + liftFun = @(xx) [xx; liftFun(xx)]; + + Nlift = length(liftFun(zeros(n,1))); + Xlift = liftFun(Xacc); + Ylift = liftFun(Yacc); + %fprintf('Lifting DONE, time = %1.2f s \n', toc); + + % ********************** Build predictor ********************************* + + %Set up regression for A and B: + X = [Xlift; Uacc]; + Y = Yacc(n/2+1:n,:); + A_state = lasso(X',Y','Lambda',1e-4, 'Alpha', 0.8); + + %Perform regression and enforce known structure: + A_koop = zeros(Nlift); + A_koop(1:n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_koop,2)-n)]; + A_koop(n/2+1:n,:) = A_state(1:Nlift,:)'; + A_koop(n+1:end,n+1:end) = A_eigfuncs; + + Y_kin = Yacc(1:n/2,:) - A_koop(1:n/2,:)*Xlift; + B_kin = Uacc'\Y_kin'; + Ylift_mod= Ylift(n+1:end,:) - A_eigfuncs*Xlift(n+1:end,:); + B_mod = (Uacc-K_nom*Xacc)'\Ylift_mod'; + B_koop = [B_kin'; A_state(Nlift+1:end,:); B_mod']; + + C_koop = zeros(n,size(A_koop,1)); + C_koop(1:n,1:n) = eye(n); + + %Subtract effect of nominal controller from A: + A_koop(n+1:end,1:n) = A_koop(n+1:end,1:n)-B_mod'*K_nom; +end \ No newline at end of file diff --git a/scripts/Koopman-ID/construct_eigfuncs.m b/scripts/Koopman-ID/construct_eigfuncs.m index 417d3a2..ba71c83 100644 --- a/scripts/Koopman-ID/construct_eigfuncs.m +++ b/scripts/Koopman-ID/construct_eigfuncs.m @@ -1,5 +1,5 @@ function [A_koop, z_eigfun] = construct_eigfuncs(n, Ntraj, Ntime, ... - N_basis, A_nom, B_nom, K_nom, Xstr, Xf, deltaT) + N_basis, pow_eig_pairs, A_nom, B_nom, K_nom, Xstr, Xf, deltaT) %Identify lifted state space model using approximate Koopman invariant %subspace @@ -39,7 +39,7 @@ A_c = A_nom + B_nom*K_nom; A_c_d = expm(A_c*deltaT); %Discrete time dynamics matrix - cent = rand(n,50)*2*pi - pi; + cent = rand(n,N_basis)*2*pi - pi; rbf_type = 'gauss'; eps_rbf = 1; @@ -88,8 +88,7 @@ [V_a,~] = eig(A_c_d'); %Define powers (only implemented for n=2): - max_power = 5; - a = 0 : max_power; + a = 0 : pow_eig_pairs; [P,Q] = meshgrid(a,a); c=cat(2,P',Q'); powers=reshape(c,[],2); diff --git a/scripts/Koopman-ID/main_comparison_pendulum.m b/scripts/Koopman-ID/main_comparison_pendulum.m index e6d02b5..13a740c 100644 --- a/scripts/Koopman-ID/main_comparison_pendulum.m +++ b/scripts/Koopman-ID/main_comparison_pendulum.m @@ -28,10 +28,13 @@ Ntime = 200; %Length of each trajectory (# of time steps) Ntraj = 20; %Number of trajectories deltaT = 0.01; %Time step length of simulation -X0_A = (2*pi*rand(n,Ntraj)-pi); -Xf_A = zeros(n,Ntraj); %Terminal points for each trajectory for learning A -X0_B = (2*pi*rand(n,Ntraj)-pi); %Sample initial points for each trajectory for learning B -Xf_B = 0.6*(2*pi*rand(n,Ntraj)-pi); %Terminal points for each trajectory for learning B +rng('shuffle'); %Set seed of random number generator +%X0 = 0.8*(2*pi*rand(n,Ntraj)-pi); +%Xf = 0.2*(2*pi*rand(n,Ntraj)-pi); %Terminal points for each trajectory for learning A +X0 = (2*rand(n,Ntraj)-1); +X0 = pi*X0./vecnorm(X0); +Xf = zeros(n,Ntraj); %Terminal points for each trajectory for learning A + %E-DMD parameters: N_basis_edmd = 36; %Number of basis functions @@ -42,44 +45,33 @@ plot_basis_edmd = false; %Plots the basis functions if true %Koopman eigenfunction parameters: -N_basis_koop = 20; %Number of basis functions -basis_function_koop = 'rbf'; %Type of basis function -rbf_type_koop = 'thinplate'; %RBF type -center_type_koop = 'data'; %Centers of rbf ('data' - pick random points from data set, 'random' pick points uniformly at random -eps_rbf_koop = 1; %RBF width -lambda_type = 'eDMD'; %Specifies how to choose candidate lambdas -N_lambda = 10; %Number of candidate eigenvalues (when random lambdas are used) +N_basis_diff = 50; %Number of RBFs to use for lifting when learning the diffeomorphism +pow_eig_pairs = 5; %Highest power of eigenpairs to use when generating eigenfunctions for linearized system plot_basis_koop = false; %Plots the basis functions if true xlim = [-1, 1]; %Plot limits ylim = [-1, 1]; %Plot limits -learn_type = 'multi-step_mod'; %Learn B-matrix with single step or multi-step prediction horizon %Test simulation parameters: Nsim = 5; -Tsim = 2; +Tsim = 1; plot_results = true; -X0_sim = 2*rand(n,Nsim)-1; +X0_sim = 0.8*(2*pi*rand(n,Nsim)-pi); +Xf_sim = 0.2*(2*pi*rand(n,Nsim)-pi); %% ************************** Data Collection ***************************** disp('Starting data collection...'); tic % Calculate nominal model with Jacobian and find nominal control gains: -[A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R); +[A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R,[0; 0]); %Linearization around 0. % Collect data to learn autonomous dynamics: -autonomous_learning = true; -U_perturb = randn(Ntime,Ntraj); %Add normally distributed noise to nominal controller +U_perturb = 2*randn(Ntime,Ntraj); %Add normally distributed noise to nominal controller [Xstr, Xacc, Yacc, Ustr, Uacc, timestr] = collect_data(n,m,Ntraj,... - Ntime,deltaT,X0_B, Xf_B ,K_nom,f_u,U_perturb, autonomous_learning); - -% Collect data to learn controlled dynamics: -autonomous_learning = false; -U_perturb_c = 4*rand(Ntime,Ntraj)-2; %Add normally distributed noise to nominal controller -[Xstr_c, Xacc_c, Yacc_c, Ustr_c, Uacc_c, timestr_c] = collect_data(n,m,Ntraj,... - Ntime,deltaT,X0_B,Xf_B,K_nom,f_u,U_perturb_c, autonomous_learning); + Ntime,deltaT,X0, Xf ,K_nom,f_u,U_perturb, true); fprintf('Data collection done, execution time: %1.2f s \n', toc); + %% *********************** Model Identification *************************** % Identify model using E-DMD to get eigenvalues and model to use for @@ -87,31 +79,26 @@ disp('Starting EDMD...'); tic [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m, Ntraj, Ntime, N_basis_edmd,basis_function_edmd,... rbf_type_edmd, center_type_edmd, eps_rbf_edmd, plot_basis_edmd, xlim,... - ylim, Xacc, Yacc, Uacc, Xstr, Xf_B, A_nom, B_nom, K_nom, deltaT); + ylim, Xacc, Yacc, Uacc); fprintf('EDMD done, execution time: %1.2f s \n', toc); -basis_function_koop = 'koopman_efunc'; -[A_koop, B_koop, C_koop, phi_fun_v] = extendedDMD(n,m, Ntraj, Ntime, N_basis_edmd,basis_function_koop,... - rbf_type_edmd, center_type_edmd, eps_rbf_edmd, plot_basis_edmd, xlim,... - ylim, Xacc, Yacc, Uacc, Xstr, Xf_B, A_nom, B_nom, K_nom, deltaT); - -% [A_koop, B_koop, C_koop, phi_fun_v] = koopman_eigen_id_laplace(n, m, Ntraj, Ntime, N_basis_koop,... -% basis_function_koop, rbf_type_koop, center_type_koop, eps_rbf_koop, ... -% N_lambda, lambda_type, A_edmd, A_nom, B_nom, K_nom, Xacc, Yacc, Xstr, Xacc_c,... -% Yacc_c, Uacc_c, Xstr_c, Ustr_c, timestr, deltaT, learn_type); +disp('Starting KEEDMD...'); tic +[A_koop, B_koop, C_koop, phi_fun_v] = KEEDMD(n,m,Ntraj, Ntime, N_basis_diff,... + pow_eig_pairs, Xacc, Yacc, Uacc, Xstr, Xf, A_nom, B_nom, K_nom, deltaT); +fprintf('KEEDMD done, execution time: %1.2f s \n', toc); %% ************************ Analysis of Results *************************** -[mse_edmd_avg, mse_koop_avg, mse_edmd_std, mse_koop_std,... +[mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std,... mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop,... cost_nom, cost_edmd, cost_koop] = ... evaluate_model_performance(K_nom, A_edmd, B_edmd, C_edmd, A_koop, B_koop, ... - C_koop, Nsim, X0_sim, Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results); + C_koop, Nsim, X0_sim, Xf_sim, Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results); fprintf('-----------------------------------------------------------------\n'); - fprintf('Predicition performance: \nAverage MSE: \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \nStandard Deviation MSE: \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \n', ... - mse_edmd_avg, mse_koop_avg, mse_edmd_std, mse_koop_std); + fprintf('Predicition performance: \nAverage MSE: \n -Linearization at origin: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \nStandard Deviation MSE: \n - Linearization at origin: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \n', ... + mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std); fprintf('-----------------------------------------------------------------\n'); - fprintf('Closed loop performance (MPC): \nAverage MSE: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \nEnergy consumption proxy: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f, \nAccumulated MPC cost: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \n', ... + fprintf('Closed loop performance (MPC): \nAverage MSE: \n - Linearization at origin: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \nEnergy consumption proxy: \n - Linearization at origin: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f, \nAccumulated MPC cost: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \n', ... mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop, cost_nom, cost_edmd, cost_koop); fprintf('-----------------------------------------------------------------\n'); \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/evaluate_model_performance.m b/scripts/Koopman-ID/utils/evaluate_model_performance.m index 1461aad..ee7e8f1 100644 --- a/scripts/Koopman-ID/utils/evaluate_model_performance.m +++ b/scripts/Koopman-ID/utils/evaluate_model_performance.m @@ -1,8 +1,8 @@ -function [mse_edmd_avg, mse_koop_avg, mse_edmd_std, mse_koop_std,... +function [mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std,... mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop,... cost_nom, cost_edmd, cost_koop] = ... evaluate_model_performance(K_nom, A_edmd, B_edmd, C_edmd, A_koop, B_koop, ... - C_koop, Nsim, X0, Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results) + C_koop, Nsim, X0, Xf, Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results) n = size(C_koop,1); m = size(B_koop,2); @@ -11,8 +11,9 @@ Ntime = Tsim/deltaT; %Prediction task: - [x,x_edmd,x_koop,mse_edmd_avg,mse_koop_avg,mse_edmd_std,mse_koop_std]... - = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,f_u,liftFun,... + [x,x_origin, x_edmd,x_koop, mse_origin_avg, mse_edmd_avg,mse_koop_avg,... + mse_origin_std, mse_edmd_std,mse_koop_std]... + = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,Xf,f_u,liftFun,... phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop); %Closed loop task: @@ -25,29 +26,31 @@ %Plot results if plot_results - lw=4; + lw=2; afigure(1) h1 = subplot(2,2,1); hold on for i = 1 %Only plot first trajectory - plot([0:Ntime]*deltaT,reshape(x(1,i,:),Ntime+1,1),'linewidth',lw); hold on - plot([0:Ntime]*deltaT,reshape(x_edmd(1,i,:),Ntime+1,1), '--r','linewidth',lw) - plot([0:Ntime]*deltaT,reshape(x_koop(1,i,:),Ntime+1,1), '--g','linewidth',lw-1) + plot([0:Ntime]*deltaT,reshape(x(1,i,:),Ntime+1,1),':b','linewidth',lw); hold on + plot([0:Ntime]*deltaT,reshape(x_origin(1,i,:),Ntime+1,1), 'k','linewidth',lw) + plot([0:Ntime]*deltaT,reshape(x_edmd(1,i,:),Ntime+1,1), 'r','linewidth',lw) + plot([0:Ntime]*deltaT,reshape(x_koop(1,i,:),Ntime+1,1), 'g','linewidth',lw) end axis([0 Tsim min(x(1,1,:))-0.2 max(x(1,1,:))+0.2]) title('Prediction comparison, 1st trajectory - $x_1$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); set(gca,'fontsize',20) - LEG = legend('True','EDMD','Koopman e-func','location','best'); + LEG = legend('True','Linearization at 0', 'EDMD','Koopman e-func','location','best'); set(LEG,'interpreter','latex') h1.Position = h1.Position + [0 0.1 0 -0.1]; %Modify size of 1st subplot h2 = subplot(2,2,2); hold on for i = 1 %Only plot first trajectory - plot([0:Ntime]*deltaT,reshape(x(2,i,:),Ntime+1,1),'linewidth',lw); hold on - plot([0:Ntime]*deltaT,reshape(x_edmd(2,i,:),Ntime+1,1), '--r','linewidth',lw) - plot([0:Ntime]*deltaT,reshape(x_koop(2,i,:),Ntime+1,1), '--g','linewidth',lw-1) + plot([0:Ntime]*deltaT,reshape(x(2,i,:),Ntime+1,1),':b','linewidth',lw); hold on + plot([0:Ntime]*deltaT,reshape(x_origin(2,i,:),Ntime+1,1), 'k','linewidth',lw) + plot([0:Ntime]*deltaT,reshape(x_edmd(2,i,:),Ntime+1,1), 'r','linewidth',lw) + plot([0:Ntime]*deltaT,reshape(x_koop(2,i,:),Ntime+1,1), 'g','linewidth',lw) end axis([0 Tsim min(x(2,1,:))-0.2 max(x(2,1,:))+0.2]) title('Prediction comparison, 1st trajectory - $x_2$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); @@ -89,7 +92,7 @@ %axis([0 Tsim min(x_edmd(1,1,:))-0.15 max(x_edmd(1,1,:))+0.15]) title('Closed loop trajectory tracking - $x_1$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); set(gca,'fontsize',20) - LEG = legend('Reference', 'Nominal controller','EDMD','Koopman e-func','location','southeast'); + LEG = legend('Reference', 'Linearization at 0','EDMD','Koopman e-func','location','southeast'); set(LEG,'interpreter','latex') h4.Position = h4.Position + [0 0 0 0.125]; %Modify size of 4th subplot end diff --git a/scripts/Koopman-ID/utils/extendedDMD.m b/scripts/Koopman-ID/utils/extendedDMD.m index db23618..33b59d6 100644 --- a/scripts/Koopman-ID/utils/extendedDMD.m +++ b/scripts/Koopman-ID/utils/extendedDMD.m @@ -1,5 +1,5 @@ function [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m,Ntraj, Ntime, N_basis,basis_function,... - rbf_type, center_type, eps, plot_basis, xlim, ylim, Xacc, Yacc, Uacc, Xstr, Xf, A_nom, B_nom, K_nom, deltaT) + rbf_type, center_type, eps, plot_basis, xlim, ylim, Xacc, Yacc, Uacc) %Identify lifted state space model using Extended Dynamic Mode %Decomposition @@ -56,11 +56,8 @@ switch basis_function case 'rbf' liftFun = @(xx)( [xx;rbf(xx,cent,rbf_type,eps)] ); - case 'koopman_efunc' - [A_koop, liftFun] = construct_eigfuncs(n, Ntraj, Ntime, N_basis, ... - A_nom, B_nom, K_nom, Xstr, Xf, deltaT); - liftFun = @(xx) [xx; liftFun(xx)]; end + Nlift = length(liftFun(zeros(n,1))); Xlift = liftFun(Xacc); Ylift = liftFun(Yacc); @@ -80,47 +77,9 @@ C_edmd = M(Nlift+1:end,1:Nlift); % Add known structure (overwrite parts of the learned matrices): - A_edmd(1:n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_edmd,2)-n)]; - C_edmd = zeros(size(C_edmd)); - C_edmd(:,1:n) = eye(n); + %A_edmd(1:n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_edmd,2)-n)]; + %C_edmd = zeros(size(C_edmd)); + %C_edmd(:,1:n) = eye(n); %fprintf('Regression done, time = %1.2f s \n', toc); - - % Plot the eigenvalues - % cplot = @(r,x0,y0) plot(x0 + r*cos(linspace(0,2*pi,200)),y0 + r*sin(linspace(0,2*pi,200)),'-'); - % afigure - % hold on - % plot(eig(Alift),'*') - % cplot(1,0,0) - % axis equal - - % *************** Force known terms of A-matrix *********************** - if strcmp(basis_function, 'koopman_efunc') - %Set up regression for A and B: - N = length(liftFun(ones(n,1))); - X = [Xlift; Uacc]; - Y = Yacc(n/2+1:n,:); - %X_mean = mean(X,2); - %X_std = std(X,0,2); - %X_std(find(X_std<1e-6)) = 1; - %X = (X-X_mean)./X_std; - %A_state = X'\Y'; - %A_state = A_state.*X_std; - A_state = lasso(X',Y','Lambda',1e-6, 'Alpha', 0.8); - - A_edmd = zeros(N); - A_edmd(1:n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_edmd,2)-n)]; - A_edmd(n/2+1:n,:) = A_state(1:N,:)'; - A_edmd(n+1:end,n+1:end) = A_koop; - - Y_kin = Yacc(1:n/2,:) - A_edmd(1:n/2,:)*Xlift; - B_kin = Uacc'\Y_kin'; - Ylift_mod= Ylift(n+1:end,:) - A_koop*Xlift(n+1:end,:); - B_mod = Uacc'\Ylift_mod'; - B_edmd = [B_kin'; A_state(N+1:end,:); B_mod']; - - %Subtract effect of nominal controller from eigenfunctions: - %A_edmd(n+1:end,1:n) = A_edmd(n+1:end,1:n)-B_mod'*K_nom; - - end end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/find_nominal_model_ctrl.m b/scripts/Koopman-ID/utils/find_nominal_model_ctrl.m index 295dc5d..d7a397b 100644 --- a/scripts/Koopman-ID/utils/find_nominal_model_ctrl.m +++ b/scripts/Koopman-ID/utils/find_nominal_model_ctrl.m @@ -1,4 +1,4 @@ -function [A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n, m, f_u, Q, R) +function [A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n, m, f_u, Q, R, x0) %Linearize around origin and design linear feedback control law using LQR %Inputs: @@ -13,8 +13,8 @@ x = sym('x',[n;1]); u = sym('u',[m;1]); - A_nom = double(subs(jacobian(f_u(0,x,u),x),[x;u],[0;0;0])); - B_nom = double(subs(jacobian(f_u(0,x,u),u),[x;u],[0;0;0])); + A_nom = double(subs(jacobian(f_u(0,x,u),x),[x;u],[x0;0])); + B_nom = double(subs(jacobian(f_u(0,x,u),u),[x;u],[x0;0])); K_nom = -lqr(A_nom,B_nom,Q,R); end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m index 59954bb..9e1104a 100644 --- a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m +++ b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m @@ -36,9 +36,9 @@ z_koop(:,1) = phi_fun_v(traj_d(:,1)); % Define Koopman controller - Q = 1000*eye(2); % Weight matrices + Q = 5*eye(2); % Weight matrices R = 0.01; - Tpred = 0.1; % Prediction horizon + Tpred = Tsim/2;%0.1; % Prediction horizon Np = round(Tpred / deltaT); traj_d = [traj_d traj_d(:,end).*ones(n,Np)]; %Add extra points to trajectory to allow prediction horizon xlift_min = [];%[ymin ; nan(Nlift-1,1)]; % Constraints @@ -86,11 +86,9 @@ x_edmd(:,i+1) = sim_timestep(deltaT, f_u, 0, x_edmd(:,i), u_edmd(:,i)); % Koopman e-func MPC - z_koop(:,i) = phi_fun_v(x_koop(:,i)); % Lift - u_koop(:,i) = MPC_koop(z_koop(:,i),yr);% + K_nom*x_koop(:,i); % Get control input + z_koop(:,i) = phi_fun_v(x_koop(:,i)); %Lift + u_koop(:,i) = MPC_koop(z_koop(:,i),yr);% Get control input x_koop(:,i+1) = sim_timestep(deltaT, f_u, 0, x_koop(:,i), u_koop(:,i)); - - %disp([u_nom(:,i), u_edmd(:,i) u_koop(:,i)]) end if(isempty(ind_inf)) @@ -105,7 +103,7 @@ E_nom = norm(u_nom); E_edmd = norm(u_edmd); E_koop = norm(u_koop); - cost_nom = sum(diag(x_nom(:,2:end)'*Q*x_nom(:,2:end))) + sum(diag(u_nom'*R*u_nom)); - cost_edmd = sum(diag(x_edmd(:,2:end)'*Q*x_edmd(:,2:end))) + sum(diag(u_edmd'*R*u_edmd)); - cost_koop = sum(diag(x_koop(:,2:end)'*Q*x_koop(:,2:end))) + sum(diag(u_koop'*R*u_koop)); + cost_nom = sum(diag((x_nom(:,2:end)-traj_d(:,2:end))'*Q*(x_nom(:,2:end)-traj_d(:,2:end)))) + sum(diag(u_nom'*R*u_nom)); + cost_edmd = sum(diag((x_edmd(:,2:end)-traj_d(:,2:end))'*Q*(x_edmd(:,2:end)-traj_d(:,2:end)))) + sum(diag(u_edmd'*R*u_edmd)); + cost_koop = sum(diag((x_koop(:,2:end)-traj_d(:,2:end))'*Q*(x_koop(:,2:end)-traj_d(:,2:end)))) + sum(diag(u_koop'*R*u_koop)); end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/sim_prediction.m b/scripts/Koopman-ID/utils/sim_prediction.m index 7f59afa..a462854 100644 --- a/scripts/Koopman-ID/utils/sim_prediction.m +++ b/scripts/Koopman-ID/utils/sim_prediction.m @@ -1,24 +1,35 @@ -function [x,x_edmd,x_koop,mse_edmd_avg,mse_koop_avg,mse_edmd_std,mse_koop_std]... - = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,f_u,liftFun,... +function [x,x_origin, x_edmd,x_koop, mse_origin_avg, mse_edmd_avg,mse_koop_avg,... + mse_origin_std, mse_edmd_std,mse_koop_std]... + = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,Xf,f_u,liftFun,... phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop) u = zeros(m,Nsim,Ntime); + % Find linearized model at the origin: + [A_origin, B_origin, ~] = find_nominal_model_ctrl(n,m,f_u,eye(n),eye(m),[0; 0]); + f_origin = @(t,x,u) A_origin*x + B_origin*u; + x = zeros(n,Nsim,Ntime+1); + x_origin = zeros(n,Nsim,Ntime+1); z_edmd = zeros(n_edmd,Nsim,Ntime+1); z_koop = zeros(n_koop,Nsim,Ntime+1); + x(:,:,1) = X0; + x_origin(:,:,1) = X0; z_edmd(:,:,1) = liftFun(X0); z_koop(:,:,1) = phi_fun_v(X0); % Simulate all systems and initial points for i = 1:Ntime %Controller: - u(:,:,i) = K_nom*x(:,:,i) + 2*rand()-1; + u(:,:,i) = K_nom*(x(:,:,i)-Xf) + 2*rand()-1; %True dynamics: x(:,:,i+1) = sim_timestep(deltaT,f_u,0,x(:,:,i), u(:,:,i)); + %Linearization at origin: + x_origin(:,:,i+1) = sim_timestep(deltaT,f_origin,0,x_origin(:,:,i), u(:,:,i)); + %EDMD predictor: z_edmd(:,:,i+1) = A_edmd*z_edmd(:,:,i) + B_edmd*u(:,:,i); @@ -29,17 +40,21 @@ % Calculate corresponding predictions and MSE x_edmd = zeros(n,Nsim,Ntime+1); x_koop = zeros(n,Nsim,Ntime+1); + mse_origin = zeros(Nsim,1); mse_edmd = zeros(Nsim,1); mse_koop = zeros(Nsim,1); for i = 1 : Nsim x_edmd(:,i,:) = C_edmd * reshape(z_edmd(:,i,:),n_edmd,Ntime+1); %EDMD predictions x_koop(:,i,:) = C_koop * reshape(z_koop(:,i,:),n_koop,Ntime+1); %Koopman eigenfunction predictions + mse_origin(i) = immse(reshape(x(:,i,:),n,Ntime+1), reshape(x_origin(:,i,:),n,Ntime+1)); mse_edmd(i) = immse(reshape(x(:,i,:),n,Ntime+1), reshape(x_edmd(:,i,:),n,Ntime+1)); mse_koop(i) = immse(reshape(x(:,i,:),n,Ntime+1), reshape(x_koop(:,i,:),n,Ntime+1)); end + mse_origin_avg = mean(mse_origin); mse_edmd_avg = mean(mse_edmd); mse_koop_avg = mean(mse_koop); + mse_origin_std = std(mse_origin); mse_edmd_std = std(mse_edmd); mse_koop_std = std(mse_koop); end \ No newline at end of file From 108a4ff0edbead3bdb05a5a2b61cf83f88e6cedb Mon Sep 17 00:00:00 2001 From: Cafolkes Date: Thu, 15 Aug 2019 20:31:36 -0700 Subject: [PATCH 5/9] Modified model evaluation and improved plotting and error reporting --- .../utils/evaluate_model_performance.m | 147 ++++++++++++------ .../Koopman-ID/utils/sim_closed_loop_mpc.m | 10 +- 2 files changed, 104 insertions(+), 53 deletions(-) diff --git a/scripts/Koopman-ID/utils/evaluate_model_performance.m b/scripts/Koopman-ID/utils/evaluate_model_performance.m index ee7e8f1..83f0a54 100644 --- a/scripts/Koopman-ID/utils/evaluate_model_performance.m +++ b/scripts/Koopman-ID/utils/evaluate_model_performance.m @@ -17,19 +17,28 @@ phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop); %Closed loop task: - [x_track,x_edmd_track,x_koop_track,mse_nom_track, mse_edmd_track,... - mse_koop_track,t_plot,traj_d, E_nom, E_edmd, E_koop,... - cost_nom, cost_edmd, cost_koop] = ... - sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,... - Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,... - C_edmd,A_koop,B_koop,C_koop); + q = [1 5 50]; + r = [0.01 0.01 0.01]; + for i = 1 : length(q) + Q = q(i)*eye(n); + R = r(i)*eye(m); + [x_track{i},x_edmd_track{i},x_koop_track{i},mse_nom_track{i}, mse_edmd_track{i},... + mse_koop_track{i},t_plot,traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... + cost_nom{i}, cost_edmd{i}, cost_koop{i}] = ... + sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,... + Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,... + C_edmd,A_koop,B_koop,C_koop,Q,R); + end %Plot results if plot_results lw=2; + font_size = 16; + set(gcf,'units','pixel') + close all; %TODO: Remove after figure design afigure(1) - h1 = subplot(2,2,1); + subplot(3,2,1); hold on for i = 1 %Only plot first trajectory plot([0:Ntime]*deltaT,reshape(x(1,i,:),Ntime+1,1),':b','linewidth',lw); hold on @@ -38,13 +47,16 @@ plot([0:Ntime]*deltaT,reshape(x_koop(1,i,:),Ntime+1,1), 'g','linewidth',lw) end axis([0 Tsim min(x(1,1,:))-0.2 max(x(1,1,:))+0.2]) - title('Prediction comparison, 1st trajectory - $x_1$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); - set(gca,'fontsize',20) - LEG = legend('True','Linearization at 0', 'EDMD','Koopman e-func','location','best'); + title('\textbf{Prediction Performance}','interpreter','latex'); + xlabel('Time [s]','interpreter','latex'); + ylabel('$x_1$','interpreter','latex'); + set(gca,'fontsize',font_size) + LEG = legend({'True','Lin at 0', 'EDMD','KEEDMD'},'location','best', 'NumColumns',2); set(LEG,'interpreter','latex') - h1.Position = h1.Position + [0 0.1 0 -0.1]; %Modify size of 1st subplot + %h1.Position = h1.Position + [0 0.1 0 -0.1]; %Modify size of 1st subplot - h2 = subplot(2,2,2); + %h2 = subplot(2,2,2); + h2 = subplot(3,2,3) hold on for i = 1 %Only plot first trajectory plot([0:Ntime]*deltaT,reshape(x(2,i,:),Ntime+1,1),':b','linewidth',lw); hold on @@ -53,47 +65,86 @@ plot([0:Ntime]*deltaT,reshape(x_koop(2,i,:),Ntime+1,1), 'g','linewidth',lw) end axis([0 Tsim min(x(2,1,:))-0.2 max(x(2,1,:))+0.2]) - title('Prediction comparison, 1st trajectory - $x_2$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); - set(gca,'fontsize',20) + %title('Prediction comparison, 1st trajectory - $x_2$','interpreter','latex'); + xlabel('Time [s]','interpreter','latex'); + ylabel('$x_2$','interpreter','latex'); + set(gca,'fontsize',font_size) %LEG = legend('True','EDMD','Koopman e-func','location','southwest'); set(LEG,'interpreter','latex') - h2.Position = h2.Position + [0 0.1 0 -0.1]; %Modify size of 1st subplot + %h2.Position = h2.Position + [0 0.1 0 -0.1]; %Modify size of 1st subplot % Plot trajectories Learning - h3 = subplot(2,2,3); - hold on - for i=1:Ntraj - scatter(Xstr(1,i,1),Xstr(2,i,1),'+') - p(i) = plot(squeeze(Xstr(1,i,:)),squeeze(Xstr(2,i,:)),':k','linewidth',1); - %alpha(p(i),0.2) - end - for i = 1 : Nsim - plot(reshape(x(1,i,:),Ntime+1,1),reshape(x(2,i,:),Ntime+1,1),'-k') - plot(reshape(x_edmd(1,i,:),Ntime+1,1),reshape(x_edmd(2,i,:),Ntime+1,1),'-r') - plot(reshape(x_koop(1,i,:),Ntime+1,1),reshape(x_koop(2,i,:),Ntime+1,1),'-g') - end - %scatter(cent(1,:),cent(2,:),'o') - %axis equal - xaxis([-5 5]) - yaxis([-5 5]) - xlabel('x') - xlabel('y') - %legend('True','EDMD','Koopman e-func') - title('Training and simulated trajectories','interpreter','latex'); - h3.Position = h3.Position + [0 0 0 0.125]; %Modify size of 3rd subplot - set(gcf, 'Position', [0, 0, 1670, 980]) %Modify size of figure window +% h3 = subplot(2,2,3); +% hold on +% for i=1:Ntraj +% scatter(Xstr(1,i,1),Xstr(2,i,1),'+') +% p(i) = plot(squeeze(Xstr(1,i,:)),squeeze(Xstr(2,i,:)),':k','linewidth',1); +% %alpha(p(i),0.2) +% end +% for i = 1 : Nsim +% plot(reshape(x(1,i,:),Ntime+1,1),reshape(x(2,i,:),Ntime+1,1),'-k') +% plot(reshape(x_edmd(1,i,:),Ntime+1,1),reshape(x_edmd(2,i,:),Ntime+1,1),'-r') +% plot(reshape(x_koop(1,i,:),Ntime+1,1),reshape(x_koop(2,i,:),Ntime+1,1),'-g') +% end +% %scatter(cent(1,:),cent(2,:),'o') +% %axis equal +% xaxis([-5 5]) +% yaxis([-5 5]) +% xlabel('x') +% xlabel('y') +% %legend('True','EDMD','Koopman e-func') +% title('Training and simulated trajectories','interpreter','latex'); +% h3.Position = h3.Position + [0 0 0 0.125]; %Modify size of 3rd subplot + %Closed loop plot: - h4 = subplot(2,2,4); - plot(t_plot,traj_d(1,:),':b','linewidth',lw); hold on - plot(t_plot,x_track(1,:),'k','linewidth',lw); hold on - plot(t_plot,x_edmd_track(1,:),'--r','linewidth',lw); hold on - plot(t_plot,x_koop_track(1,:),'--g','linewidth',lw); hold on - %axis([0 Tsim min(x_edmd(1,1,:))-0.15 max(x_edmd(1,1,:))+0.15]) - title('Closed loop trajectory tracking - $x_1$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); - set(gca,'fontsize',20) - LEG = legend('Reference', 'Linearization at 0','EDMD','Koopman e-func','location','southeast'); - set(LEG,'interpreter','latex') - h4.Position = h4.Position + [0 0 0 0.125]; %Modify size of 4th subplot + h4 = subplot(3,2,[2, 4]); + plot(t_plot,traj_d(1,:),'-b','linewidth',lw); hold on + lin_typ = {'-', '--', '-.'}; + for i = 1 : length(q) + plot(t_plot,x_track{i}(1,:),strcat(lin_typ{i},'k'),'linewidth',lw); hold on + plot(t_plot,x_edmd_track{i}(1,:),strcat(lin_typ{i},'r'),'linewidth',lw); hold on + plot(t_plot,x_koop_track{i}(1,:),strcat(lin_typ{i},'g'),'linewidth',lw); hold on + end + %axis([0 Tsim min(x_edmd(1,1,:))-0.15 max(x_edmd(1,1,:))+0.15]) + title('\textbf{Closed Loop Tracking Performance}','interpreter','latex'); + xlabel('Time [s]','interpreter','latex'); + ylabel('$x_1$','interpreter','latex'); + set(gca,'fontsize',font_size) + LEG = legend({'Reference', ... + strcat('Lin at 0$_{q/r=',num2str(q(1)/r(1)),'}$'),strcat('EDMD$_{q/r=',num2str(q(1)/r(1)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(1)/r(1)),'}$'),... + strcat('Lin at 0$_{q/r=',num2str(q(2)/r(2)),'}$'),strcat('EDMD$_{q/r=',num2str(q(2)/r(2)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(2)/r(2)),'}$'),... + strcat('Lin at 0$_{q/r=',num2str(q(3)/r(3)),'}$'),strcat('EDMD$_{q/r=',num2str(q(3)/r(3)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(3)/r(3)),'}$')},... + 'location','southeast','NumColumns',2); + set(LEG,'Interpreter','latex') + %h4.Position = h4.Position + [0 0 0 0.125]; %Modify size of 4th subplot + set(gcf, 'Position', [0, 0, 1200, 600]) %Modify size of figure window + + %Prediction table: + set(h2,'Units','pixels'); + cnames = {'Mean MSE', 'Std MSE'}; + rnames = {'Linearization at 0', 'EDMD', 'KEEDMD'}; + data = [mse_origin_avg mse_edmd_avg mse_koop_avg;... + mse_origin_std mse_edmd_std mse_koop_std]'; + height = 75; + pos = [h2.Position(1) h2.Position(2)-height-70 h2.Position(3) height]; + col_width = {107 107}; + pos_title = [pos(1) pos(2)+pos(4) pos(3) 20]; + uitable('Data', data, 'ColumnName', cnames, 'RowName', rnames,'Position', pos, 'ColumnWidth', col_width,'FontSize',12); + txt_title = uicontrol('Style', 'text','Position', pos_title, 'String', 'Prediction performance over 5 trajectories','FontSize',14); + + %Closed loop table: + set(h4,'Units','pixels'); + cnames = {strcat('q/r=',num2str(q(1)/r(1))) strcat('q/r=',num2str(q(2)/r(2))) strcat('q/r=',num2str(q(3)/r(3)))}; + rnames = {'Linearization at 0', 'EDMD', 'KEEDMD'}; + data = [cost_nom{1} cost_edmd{1} cost_koop{1};... + cost_nom{2} cost_edmd{2} cost_koop{2};... + cost_nom{3} cost_edmd{3} cost_koop{3}]'; + height = 75; + pos = [h4.Position(1) h4.Position(2)-height-70 h4.Position(3) height]; + col_width = {71 71}; + pos_title = [pos(1) pos(2)+pos(4) pos(3) 20]; + uitable('Data', data, 'ColumnName', cnames, 'RowName', rnames,'Position', pos, 'ColumnWidth', col_width,'FontSize',12); + txt_title = uicontrol('Style', 'text','Position', pos_title, 'String', 'Closed loop trajectory tracking performance (MPC Cost)','FontSize',14); end end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m index 9e1104a..3380cc4 100644 --- a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m +++ b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m @@ -2,7 +2,7 @@ cost_nom, cost_edmd, cost_koop]... = sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,Tsim,... f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,... - B_koop,C_koop) + B_koop,C_koop,Q,R) %Set up trajectory to track: t = 0 : deltaT : Tsim; @@ -36,8 +36,6 @@ z_koop(:,1) = phi_fun_v(traj_d(:,1)); % Define Koopman controller - Q = 5*eye(2); % Weight matrices - R = 0.01; Tpred = Tsim/2;%0.1; % Prediction horizon Np = round(Tpred / deltaT); traj_d = [traj_d traj_d(:,end).*ones(n,Np)]; %Add extra points to trajectory to allow prediction horizon @@ -69,8 +67,10 @@ yr = traj_d(:,i:i+Np-1); % Local linearization MPC - Aloc = double(subs(Jx,[x;u],[x_nom(:,i);u_nom(:,i)])); % Get local linearization - Bloc = double(subs(Ju,[x;u],[x_nom(:,i);u_nom(:,i)])); + %Aloc = double(subs(Jx,[x;u],[x_nom(:,i);u_nom(:,i)])); % Get local linearization + %Bloc = double(subs(Ju,[x;u],[x_nom(:,i);u_nom(:,i)])); + Aloc = double(subs(Jx,[x;u],[[0 0]';0])); % Get local linearization + Bloc = double(subs(Ju,[x;u],[[0 0]';0])); %Cloc = double(subs(f_ud_sym,[x;u],[x_nom(:,i);u_nom(:,i)])) - Aloc*x_nom(:,i) - Bloc*u_nom(:,i); [U_nom,~,optval] = solveMPCprob(Aloc,Bloc,eye(2),[],Q,R,Q,Np,-100, 100,[],[],x_nom(:,i),yr); % Get control input u_nom(:,i) = U_nom(1:m,1); From 291923fc8d80201350b9b087b7f17e7cf47f9735 Mon Sep 17 00:00:00 2001 From: Cafolkes Date: Sat, 17 Aug 2019 19:59:52 -0700 Subject: [PATCH 6/9] First commit continuous implementation. Not yet working, must refine methodology --- scripts/Koopman-ID/KEEDMD.m | 60 +++-- scripts/Koopman-ID/construct_eigfuncs.m | 59 ++--- scripts/Koopman-ID/main_comparison_pendulum.m | 29 +-- scripts/Koopman-ID/movingslope.m | 219 ++++++++++++++++++ scripts/Koopman-ID/num_diff.m | 7 + scripts/Koopman-ID/rbf_dot.m | 53 +++++ scripts/Koopman-ID/rbf_grad.m | 4 +- .../Koopman-ID/test_eigfunction_generation.m | 162 ++++++------- scripts/Koopman-ID/utils/collect_data.m | 14 +- .../utils/evaluate_model_performance.m | 27 ++- scripts/Koopman-ID/utils/sim_prediction.m | 9 +- 11 files changed, 461 insertions(+), 182 deletions(-) create mode 100644 scripts/Koopman-ID/movingslope.m create mode 100644 scripts/Koopman-ID/num_diff.m create mode 100644 scripts/Koopman-ID/rbf_dot.m diff --git a/scripts/Koopman-ID/KEEDMD.m b/scripts/Koopman-ID/KEEDMD.m index d10d561..a06456e 100644 --- a/scripts/Koopman-ID/KEEDMD.m +++ b/scripts/Koopman-ID/KEEDMD.m @@ -1,5 +1,5 @@ function [A_koop, B_koop, C_koop, liftFun] = KEEDMD(n,m,Ntraj, Ntime, N_basis,... - pow_eig_pairs, Xacc, Yacc, Uacc, Xstr, Xf, A_nom, B_nom, K_nom, deltaT) + pow_eig_pairs, Xstr, Ustr, Xf, A_nom, B_nom, K_nom, deltaT) %Identify lifted state space model using Extended Dynamic Mode %Decomposition @@ -23,38 +23,62 @@ % B_edmd - Actuation matrix in lifted space % C_edmd - Projection matrix from lifted space to outputs - - [A_eigfuncs, liftFun] = construct_eigfuncs(n, Ntraj, Ntime, N_basis, ... - pow_eig_pairs, A_nom, B_nom, K_nom, Xstr, Xf, deltaT); + % ************************** Prepare data ***************************** + Xstr_shift = zeros(size(Xstr)); %Shift dynamics such that origin is fixed point + X = []; + X_dot = []; + U = []; + for i = 1 : Ntraj + Xstr_shift(:,i,:) = Xstr(:,i,:) - Xf(:,i); + X = [X reshape(Xstr_shift(:,i,:),n,Ntime+1)]; + X_dot = [X_dot num_diff(reshape(Xstr_shift(:,i,:),size(Xstr_shift,1),size(Xstr_shift,3)),deltaT)]; + U = [U reshape(Ustr(:,i,:),m,Ntime+1)]; + end + + % ******************** Construct eigenfunctions *********************** + + [A_eigfuncs, liftFun] = construct_eigfuncs(n, N_basis,pow_eig_pairs, ... + A_nom, B_nom, K_nom, X, X_dot); + + %Perform lifting liftFun = @(xx) [xx; liftFun(xx)]; - Nlift = length(liftFun(zeros(n,1))); - Xlift = liftFun(Xacc); - Ylift = liftFun(Yacc); + Xlift = []; + Xlift_dot = []; + for i = 1 : Ntraj + Xlift_temp = liftFun(reshape(Xstr_shift(:,i,:),size(Xstr_shift,1),size(Xstr_shift,3))); + Xlift = [Xlift Xlift_temp]; + Xlift_dot = [Xlift_dot num_diff(Xlift_temp,deltaT)]; + end + %fprintf('Lifting DONE, time = %1.2f s \n', toc); % ********************** Build predictor ********************************* %Set up regression for A and B: - X = [Xlift; Uacc]; - Y = Yacc(n/2+1:n,:); - A_state = lasso(X',Y','Lambda',1e-4, 'Alpha', 0.8); + X_vel = [Xlift; U]; + Y_vel = Xlift_dot(n/2+1:n,:); + A_vel = lasso(X_vel',Y_vel','Lambda',1e-1, 'Alpha', 0.8); %Perform regression and enforce known structure: A_koop = zeros(Nlift); - A_koop(1:n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_koop,2)-n)]; - A_koop(n/2+1:n,:) = A_state(1:Nlift,:)'; + A_koop(1:n/2,:) = [zeros(n/2) eye(n/2) zeros(n/2,size(A_koop,2)-n)]; + A_koop(n/2+1:n,:) = A_vel(1:Nlift,:)'; A_koop(n+1:end,n+1:end) = A_eigfuncs; - Y_kin = Yacc(1:n/2,:) - A_koop(1:n/2,:)*Xlift; - B_kin = Uacc'\Y_kin'; - Ylift_mod= Ylift(n+1:end,:) - A_eigfuncs*Xlift(n+1:end,:); - B_mod = (Uacc-K_nom*Xacc)'\Ylift_mod'; - B_koop = [B_kin'; A_state(Nlift+1:end,:); B_mod']; + Y_kin = X_dot(1:n/2,:) - A_koop(1:n/2,:)*Xlift; + X_kin = U; + B_kin = X_kin'\Y_kin'; + + Y_eig = Xlift_dot(n+1:end,:) - A_eigfuncs*Xlift(n+1:end,:); + X_eig = U-K_nom*X; + B_eig = X_eig'\Y_eig'; + + B_koop = [B_kin'; A_vel(Nlift+1:end,:); B_eig']; C_koop = zeros(n,size(A_koop,1)); C_koop(1:n,1:n) = eye(n); %Subtract effect of nominal controller from A: - A_koop(n+1:end,1:n) = A_koop(n+1:end,1:n)-B_mod'*K_nom; + A_koop(n+1:end,1:n) = A_koop(n+1:end,1:n)-B_eig'*K_nom; end \ No newline at end of file diff --git a/scripts/Koopman-ID/construct_eigfuncs.m b/scripts/Koopman-ID/construct_eigfuncs.m index ba71c83..48928f7 100644 --- a/scripts/Koopman-ID/construct_eigfuncs.m +++ b/scripts/Koopman-ID/construct_eigfuncs.m @@ -1,5 +1,5 @@ -function [A_koop, z_eigfun] = construct_eigfuncs(n, Ntraj, Ntime, ... - N_basis, pow_eig_pairs, A_nom, B_nom, K_nom, Xstr, Xf, deltaT) +function [A_koop, z_eigfun] = construct_eigfuncs(n, N_basis, pow_eig_pairs,... + A_nom, B_nom, K_nom, X, X_dot) %Identify lifted state space model using approximate Koopman invariant %subspace @@ -37,55 +37,39 @@ disp('Starting autonomous dynamics learning...'); tic - A_c = A_nom + B_nom*K_nom; - A_c_d = expm(A_c*deltaT); %Discrete time dynamics matrix - cent = rand(n,N_basis)*2*pi - pi; + A_cl = A_nom + B_nom*K_nom; + cent = 2*pi*rand(n,N_basis) - pi; rbf_type = 'gauss'; - eps_rbf = 1; - + eps_rbf = 2; + % Set up nonlinear transformations and their gradients - zfun = @(xx) [xx'; rbf(xx',cent,rbf_type,eps_rbf)]; - zfun_grad = @(xx) [eye(2); rbf_grad(xx',cent,rbf_type,eps_rbf)]; + zfun = @(xx) [xx; rbf(xx,cent,rbf_type,eps_rbf)]; + zfun_grad = @(xx) [eye(2); rbf_grad(xx,cent,rbf_type,eps_rbf)]; + zfun_dot = @(xx, xx_dot) [xx_dot; rbf_dot(xx,xx_dot,cent,rbf_type,eps_rbf)]; - % Prepare data matrices - Y = []; - Z_mplus1 = []; - Z_m = []; - Xstr_shift = zeros(size(Xstr)); - for i = 1 : Ntraj - Xstr_shift(:,i,:) = Xstr(:,i,:) - Xf(:,i); - X_mplus1 = reshape(Xstr_shift(:,i,2:end),n,Ntime); - X_m = reshape(Xstr_shift(:,i,1:end-1),n,Ntime); - - % Set up Y matrix (targets), Y(:,i) = x_(i+1) - A_nom*x_i - Y = [Y X_mplus1-A_c_d*X_m]; - - % Set up Z_mplus1 matrix (inputs), phi(x_(i+1)) - Z_mplus1 = [Z_mplus1 zfun(X_mplus1')]; - - % Set up Z_m matrix (inputs), phi(x_i) - Z_m = [Z_m zfun(X_m')]; - end + % Set up Z and Z_dot matrices: + Z = zfun(X); + Z_dot = zfun_dot(X,X_dot); %Set up constraint matrix - con1 = zfun_grad([0 0]); - + con1 = zfun_grad([0; 0]); + disp('Solving optimization problem...') - N = size(Z_m,1); - cvx_begin + N = size(Z,1); + cvx_begin variable C(n,N); - minimize (norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro') + 1*norm(C,'fro')) + minimize (norm(X_dot + C*Z_dot - A_cl*(X+C*Z),'fro') + 1*norm(C,'fro')) subject to {C*con1 == zeros(n)}; cvx_end - fprintf('Solved, optimal value excluding regularization: %.10f, MSE: %.10f\n', norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro'),immse(Y, A_c_d*C*Z_m - C*Z_mplus1)) + fprintf('Solved, optimal value excluding regularization: %.10f, MSE: %.10f\n', norm(X_dot + C*Z_dot - A_cl*(X+C*Z),'fro'),immse(X_dot+C*Z_dot, A_cl*(X+C*Z))) %fprintf('Constraint violation: %.4f \n', sum(sum(abs(C*con1)))) - yfun = @(xx) xx + C*zfun(xx'); % Full learned diffeomorphism + yfun = @(xx) xx + C*zfun(xx); % Full learned diffeomorphism % Calculate eigenfunctions for linearized system - [~,D] = eig(A_c_d); - [V_a,~] = eig(A_c_d'); + [~,D] = eig(A_cl); + [V_a,~] = eig(A_cl'); %Define powers (only implemented for n=2): a = 0 : pow_eig_pairs; @@ -102,6 +86,7 @@ % Construct eigenfunctions for nonlinear system z_eigfun = @(xx) phifun_mat(phifun, gfun(yfun(xx))); + %z_eigfun = @(xx) phifun(gfun(yfun(xx))); A_koop = diag(lambd); end diff --git a/scripts/Koopman-ID/main_comparison_pendulum.m b/scripts/Koopman-ID/main_comparison_pendulum.m index 13a740c..2c860dd 100644 --- a/scripts/Koopman-ID/main_comparison_pendulum.m +++ b/scripts/Koopman-ID/main_comparison_pendulum.m @@ -45,8 +45,8 @@ plot_basis_edmd = false; %Plots the basis functions if true %Koopman eigenfunction parameters: -N_basis_diff = 50; %Number of RBFs to use for lifting when learning the diffeomorphism -pow_eig_pairs = 5; %Highest power of eigenpairs to use when generating eigenfunctions for linearized system +N_basis_diff = 100; %Number of RBFs to use for lifting when learning the diffeomorphism +pow_eig_pairs = 2; %Highest power of eigenpairs to use when generating eigenfunctions for linearized system plot_basis_koop = false; %Plots the basis functions if true xlim = [-1, 1]; %Plot limits ylim = [-1, 1]; %Plot limits @@ -66,9 +66,9 @@ [A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R,[0; 0]); %Linearization around 0. % Collect data to learn autonomous dynamics: -U_perturb = 2*randn(Ntime,Ntraj); %Add normally distributed noise to nominal controller +U_perturb = 0.5*randn(Ntime+1,Ntraj); %Add normally distributed noise to nominal controller [Xstr, Xacc, Yacc, Ustr, Uacc, timestr] = collect_data(n,m,Ntraj,... - Ntime,deltaT,X0, Xf ,K_nom,f_u,U_perturb, true); + Ntime,deltaT,X0, Xf ,K_nom,f_u,U_perturb); fprintf('Data collection done, execution time: %1.2f s \n', toc); @@ -84,21 +84,14 @@ disp('Starting KEEDMD...'); tic [A_koop, B_koop, C_koop, phi_fun_v] = KEEDMD(n,m,Ntraj, Ntime, N_basis_diff,... - pow_eig_pairs, Xacc, Yacc, Uacc, Xstr, Xf, A_nom, B_nom, K_nom, deltaT); + pow_eig_pairs, Xstr, Ustr, Xf, A_nom, B_nom, K_nom, deltaT); fprintf('KEEDMD done, execution time: %1.2f s \n', toc); %% ************************ Analysis of Results *************************** - +disp('Starting analysis of results...'); tic [mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std,... - mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop,... - cost_nom, cost_edmd, cost_koop] = ... - evaluate_model_performance(K_nom, A_edmd, B_edmd, C_edmd, A_koop, B_koop, ... - C_koop, Nsim, X0_sim, Xf_sim, Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results); - - fprintf('-----------------------------------------------------------------\n'); - fprintf('Predicition performance: \nAverage MSE: \n -Linearization at origin: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \nStandard Deviation MSE: \n - Linearization at origin: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \n', ... - mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std); - fprintf('-----------------------------------------------------------------\n'); - fprintf('Closed loop performance (MPC): \nAverage MSE: \n - Linearization at origin: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \nEnergy consumption proxy: \n - Linearization at origin: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f, \nAccumulated MPC cost: \n - Nominal: %1.5f \n - EDMD: %1.5f \n - Koopman eigenfunctions: %1.5f \n', ... - mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop, cost_nom, cost_edmd, cost_koop); - fprintf('-----------------------------------------------------------------\n'); \ No newline at end of file + cost_nom, cost_edmd, cost_koop] = evaluate_model_performance(K_nom, ... + A_edmd, B_edmd, C_edmd, A_koop, B_koop, C_koop, Nsim, X0_sim, Xf_sim, ... + Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results); +fprintf('Analysis done, execution time: %1.2f s \n', toc); + \ No newline at end of file diff --git a/scripts/Koopman-ID/movingslope.m b/scripts/Koopman-ID/movingslope.m new file mode 100644 index 0000000..6c8b2e8 --- /dev/null +++ b/scripts/Koopman-ID/movingslope.m @@ -0,0 +1,219 @@ +function Dvec = movingslope(vec,supportlength,modelorder,dt) +% movingslope: estimate local slope for a sequence of points, using a sliding window +% usage: Dvec = movingslope(vec) +% usage: Dvec = movingslope(vec,supportlength) +% usage: Dvec = movingslope(vec,supportlength,modelorder) +% usage: Dvec = movingslope(vec,supportlength,modelorder,dt) +% +% +% movingslope uses filter to determine the slope of a curve stored +% as an equally (unit) spaced sequence of points. A patch is applied +% at each end where filter will have problems. A non-unit spacing +% can be supplied. +% +% Note that with a 3 point window and equally spaced data sequence, +% this code should be similar to gradient. However, with wider +% windows this tool will be more robust to noisy data sequences. +% +% +% arguments: (input) +% vec - row of column vector, to be differentiated. vec must be of +% length at least 2. +% +% supportlength - (OPTIONAL) scalar integer - defines the number of +% points used for the moving window. supportlength may be no +% more than the length of vec. +% +% supportlength must be at least 2, but no more than length(vec) +% +% If supportlength is an odd number, then the sliding window +% will be central. If it is an even number, then the window +% will be slid backwards by one element. Thus a 2 point window +% will result in a backwards differences used, except at the +% very first point, where a forward difference will be used. +% +% DEFAULT: supportlength = 3 +% +% modelorder - (OPTIONAL) - scalar - Defines the order of the windowed +% model used to estimate the slope. When model order is 1, the +% model is a linear one. If modelorder is less than supportlength-1. +% then the sliding window will be a regression one. If modelorder +% is equal to supportlength-1, then the window will result in a +% sliding Lagrange interpolant. +% +% modelorder must be at least 1, but not exceeding +% min(10,supportlength-1) +% +% DEFAULT: modelorder = 1 +% +% dt - (OPTIONAL) - scalar - spacing for sequences which do not have +% a unit spacing. +% +% DEFAULT: dt = 1 +% +% arguments: (output) +% Dvec = vector of derivative estimates, Dvec will be of the same size +% and shape as is vec. +% +% +% Example: +% Estimate the first derivative using a 7 point window with first through +% fourth order models in the sliding window. Note that the higher order +% approximations provide better accuracy on this curve with no noise. +% +% t = 0:.1:1; +% vec = exp(t); +% +% Dvec = movingslope(vec,7,1,.1) +% Dvec = +% Columns 1 through 7 +% 1.3657 1.3657 1.3657 1.3657 1.5093 1.668 1.8435 +% Columns 8 through 11 +% 2.0373 2.0373 2.0373 2.0373 +% +% Dvec = movingslope(vec,7,2,.1) +% Dvec = +% Columns 1 through 7 +% 0.95747 1.0935 1.2296 1.3657 1.5093 1.668 1.8435 +% Columns 8 through 11 +% 2.0373 2.2403 2.4433 2.6463 +% +% Dvec = movingslope(vec,7,3,.1) +% Dvec = +% Columns 1 through 7 +% 1.0027 1.1049 1.2206 1.3498 1.4918 1.6487 1.8221 +% Columns 8 through 11 +% 2.0137 2.2268 2.4602 2.7138 +% +% Dvec = movingslope(vec,7,4,.1) +% Dvec = +% Columns 1 through 7 +% 0.99988 1.1052 1.2214 1.3498 1.4918 1.6487 1.8221 +% Columns 8 through 11 +% 2.0137 2.2255 2.4597 2.7181 +% +% +% Example: +% Estimate the slope of a noisy curve, using a locally quadratic +% approximation. In this case, use a straight line so that we know +% the true slope should be 1. Use a wide window, since we have +% noisy data. +% +% t = 0:100; +% vec = t + randn(size(t)); +% Dvec = movingslope(vec,10,2,1) +% mean(Dvec) +% ans = +% 1.0013 +% std(Dvec) +% ans = +% 0.10598 +% +% By way of comparison, gradient gives a much noisier estimate +% of the slope of this curve. +% +% std(gradient(vec)) +% ans = +% 0.69847 +% +% +% Example: +% As a time test, generate random data vector of length 500000. +% Compute the slopes using a window of width 10. +% +% vec = rand(1,500000); +% tic +% Dvec = movingslope(vec,10,2); +% toc +% +% Elapsed time is 0.626021 seconds. +% +% +% See also: gradient +% +% +% Author: John D'Errico +% e-mail: woodchips@rochester.rr.com +% Release: 1.0 +% Release date: 10/19/07 + +% how long is vec? is it a vector? +if (nargin==0) + help movingslope + return +end +if ~isvector(vec) + error('vec must be a row or column vector') +end +n = length(vec); + +% supply defaults +if (nargin<4) || isempty(dt) + dt = 1; +end +if (nargin<3) || isempty(modelorder) + modelorder = 1; +end +if (nargin<2) || isempty(supportlength) + supportlength = 3; +end + +% check the parameters for problems +if (length(supportlength)~=1) || (supportlength<=1) || (supportlength>n) || (supportlength~=floor(supportlength)) + error('supportlength must be a scalar integer, >= 2, and no more than length(vec)') +end +if (length(modelorder)~=1) || (modelorder<1) || (modelorder>min(10,supportlength-1)) || (modelorder~=floor(modelorder)) + error('modelorder must be a scalar integer, >= 1, and no more than min(10,supportlength-1)') +end +if (length(dt)~=1) || (dt<0) + error('dt must be a positive scalar numeric variable') +end + +% now build the filter coefficients to estimate the slope +if mod(supportlength,2) == 1 + parity = 1; % odd parity +else + parity = 0; +end +s = (supportlength-parity)/2; +t = ((-s+1-parity):s)'; +coef = getcoef(t,supportlength,modelorder); + +% Apply the filter to the entire vector +f = filter(-coef,1,vec); +Dvec = zeros(size(vec)); +Dvec(s+(1:(n-supportlength+1))) = f(supportlength:end); + +% patch each end +vec = vec(:); +for i = 1:s + % patch the first few points + t = (1:supportlength)' - i; + coef = getcoef(t,supportlength,modelorder); + + Dvec(i) = coef*vec(1:supportlength); + + % patch the end points + if i<(s + parity) + t = (1:supportlength)' - supportlength + i - 1; + coef = getcoef(t,supportlength,modelorder); + Dvec(n - i + 1) = coef*vec(n + (0:(supportlength-1)) + 1 - supportlength); + end +end + +% scale by the supplied spacing +Dvec = Dvec/dt; +% all done + +end % mainline end + +% ========================================================= +% subfunction, used to compute the filter coefficients +function coef = getcoef(t,supportlength,modelorder) +% Note: bsxfun would have worked here as well, but some people +% might not yet have that release of matlab. +A = repmat(t,1,modelorder+1).^repmat(0:modelorder,supportlength,1); +pinvA = pinv(A); +% we only need the linear term +coef = pinvA(2,:); +end % nested function end diff --git a/scripts/Koopman-ID/num_diff.m b/scripts/Koopman-ID/num_diff.m new file mode 100644 index 0000000..22acd14 --- /dev/null +++ b/scripts/Koopman-ID/num_diff.m @@ -0,0 +1,7 @@ +function diff = num_diff(X, deltaT) + + diff = zeros(size(X)); + for k = 1 : size(X,1) + diff(k,:) = movingslope(X(k,:),4,2,deltaT); + end +end \ No newline at end of file diff --git a/scripts/Koopman-ID/rbf_dot.m b/scripts/Koopman-ID/rbf_dot.m new file mode 100644 index 0000000..d8ef5c0 --- /dev/null +++ b/scripts/Koopman-ID/rbf_dot.m @@ -0,0 +1,53 @@ +% X: nxN +% C: rbf center(s) - nx1 or nxK, if C is n x K, then the result is STACKED +% over the centers K, i.e., of dimension K x N +% eps: kernel width for Gaussian type rbfs (optional) +% k: polyharmonic coefficient for polyharmonic rbfs (optional) +% Note that all RBFs are symmetrical so evaluation of of a single point on +% multiple centers can by done as evaluation of multiple points (the +% centers in this case) on a single point (the center) + +function Y = rbf_dot(X,X_dot,C, type, eps, k ) + type = lower(type); + if(~exist('eps','var') || isempty(eps)) + eps = 1; + end + if(~exist('k','var') || isempty(k)) + k = 1; + end + + N = size(X,2); %number of points + n = size(X,1); + Cbig = C; + Y = zeros(size(C,2),N); + for i = 1:size(Cbig,2) + C = Cbig(:,i); + C = repmat( C,1, N ); + if (n>1) + r_squared = sum( (X - C).^2 ); + else + r_squared = abs( (X - C).^2 ); + end + switch type + case 'thinplate' + error('Gradient of RBF type not implemented') + %y = r_squared.*log(sqrt(r_squared)); % + 0.5*sqrt(r_squared); % makes nonnegative, try to impose nonnegativity of K + %y(isnan(y)) = 0; + case 'gauss' + y = -2*eps^2.*(X-C).*exp(-eps^2*r_squared); + %y = exp(-eps^2*r_squared); + case 'invquad' + error('Gradient of RBF type not implemented') + %y = 1 ./ (1+eps^2*r_squared); + case 'invmultquad' + error('Gradient of RBF type not implemented') + %y = 1 ./ sqrt((1+eps^2*r_squared)); + case 'polyharmonic' + error('Gradient of RBF type not implemented') + %y = r_squared.^(k/2).*log(sqrt(r_squared)); + %y(isnan(y)) = 0; + otherwise + error('RBF type not recognized') + end + Y(i,:) = sum(y.*X_dot,1); + end diff --git a/scripts/Koopman-ID/rbf_grad.m b/scripts/Koopman-ID/rbf_grad.m index 85fbf1b..37baa32 100644 --- a/scripts/Koopman-ID/rbf_grad.m +++ b/scripts/Koopman-ID/rbf_grad.m @@ -34,7 +34,7 @@ %y = r_squared.*log(sqrt(r_squared)); % + 0.5*sqrt(r_squared); % makes nonnegative, try to impose nonnegativity of K %y(isnan(y)) = 0; case 'gauss' - y = -2*eps^2.*(X-C)'.*exp(-eps^2*r_squared); + y = -2*eps^2.*(X-C).*exp(-eps^2*r_squared); %y = exp(-eps^2*r_squared); case 'invquad' error('Gradient of RBF type not implemented') @@ -49,5 +49,5 @@ otherwise error('RBF type not recognize') end - Y(i,:) = y; + Y(i,:) = y'; end diff --git a/scripts/Koopman-ID/test_eigfunction_generation.m b/scripts/Koopman-ID/test_eigfunction_generation.m index 4c1676a..76868c4 100644 --- a/scripts/Koopman-ID/test_eigfunction_generation.m +++ b/scripts/Koopman-ID/test_eigfunction_generation.m @@ -13,108 +13,100 @@ Ntime = 200; %Length of each trajectory (# of time steps) Ntraj = 10; %Number of trajectories deltaT = 0.01; %Time step length of simulation -X0_A = 2*rand(n,Ntraj)-1; %Sample initial points for each trajectory for learning A -X0_A = 0.95*pi*X0_A./vecnorm(X0_A,2,1); %Normalize so initial points lie on unit circle -Xf_A = zeros(n,Ntraj); %Terminal points for each trajectory for learning A -X0_B = 0.6*(2*pi*rand(n,Ntraj)-pi); %Sample initial points for each trajectory for learning B -Xf_B = 0.6*(2*pi*rand(n,Ntraj)-pi); %Terminal points for each trajectory for learning B +X0 = 2*rand(n,Ntraj)-1; %Sample initial points for each trajectory for learning A +X0 = 0.95*pi*X0./vecnorm(X0,2,1); %Normalize so initial points lie on unit circle +Xf = zeros(n,Ntraj); %Terminal points for each trajectory for learning A %% ************************** Data Collection ***************************** disp('Starting data collection...'); tic % Calculate nominal model with Jacobian and find nominal control gains: -[A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R); +[A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R,[0;0]); % Collect data to learn autonomous dynamics: autonomous_learning = true; -U_perturb = 0.0*randn(Ntime,Ntraj); %Add normally distributed noise to nominal controller -[Xstr, Xacc, Yacc, Ustr, Uacc, timestr] = collect_data(n,m,Ntraj,... - Ntime,deltaT,X0_A, Xf_A ,K_nom,f_u,U_perturb, autonomous_learning); +U_perturb = 0.0*randn(Ntime+1,Ntraj); %Add normally distributed noise to nominal controller +[Xstr, Xacc, Yacc, Ustr, U, timestr] = collect_data(n,m,Ntraj,... + Ntime,deltaT,X0, Xf ,K_nom,f_u,U_perturb); fprintf('Data collection done, execution time: %1.2f s \n', toc); %% ************** Prepare data and learn diffeomorphism h ***************** -A_c = A_nom + B_nom*K_nom; -A_c_d = expm(A_c*deltaT); %Discrete time dynamics matrix -N_basis = 100; -cent = rand(n,N_basis)*2*pi - pi; -rbf_type = 'gauss'; -eps_rbf = 1; -zfun = @(xx) [xx'; rbf(xx',cent,rbf_type,eps_rbf)]; -zfun_grad = @(xx) [eye(2); rbf_grad(xx',cent,rbf_type,eps_rbf)]; -% x = sym('x',[1,n]); -% zfun_grad = zfun(x); -Y = []; -Z_mplus1 = []; -Z_m = []; -for i = 1 : Ntraj - X_mplus1 = reshape(Xstr(:,i,2:end),n,Ntime); - X_m = reshape(Xstr(:,i,1:end-1),n,Ntime); - - % Set up Y matrix (targets), Y(:,i) = x_(i+1) - A_nom*x_i - Y = [Y X_mplus1-A_c_d*X_m]; - - % Set up Z_mplus1 matrix (inputs), phi(x_(i+1)) - Z_mplus1 = [Z_mplus1 zfun(X_mplus1')]; - - % Set up Z_m matrix (inputs), phi(x_i) - Z_m = [Z_m zfun(X_m')]; -end + Xstr_shift = zeros(size(Xstr)); %Shift dynamics such that origin is fixed point + X = []; + X_dot = []; + U = []; + for i = 1 : Ntraj + Xstr_shift(:,i,:) = Xstr(:,i,:) - Xf(:,i); + X = [X reshape(Xstr_shift(:,i,:),n,Ntime+1)]; + X_dot = [X_dot num_diff(reshape(Xstr_shift(:,i,:),size(Xstr_shift,1),size(Xstr_shift,3)),deltaT)]; + U = [U reshape(Ustr(:,i,:),m,Ntime+1)]; + end -%Set up constraint matrix -con1 = zfun_grad([0 0]); - -disp('Solving optimization problem...') -N = size(Z_m,1); -cvx_begin - variable C(n,N); - minimize (norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro') + 0.01*norm(C,'fro')) - subject to - {C*con1 == zeros(n)}; -cvx_end -fprintf('Solved, optimal value excluding regularization: %.10f, MSE: %.10f\n', norm(Y - (A_c_d*C*Z_m - C*Z_mplus1),'fro'),immse(Y, A_c_d*C*Z_m - C*Z_mplus1)) -fprintf('Constraint violation: %.4f \n', sum(sum(abs(C*con1)))) - -yfun = @(xx) xx + C*zfun(xx'); % Full learned diffeomorphism -%% Calculate eigenfunctions for linearized system -[V,D] = eig(A_c_d); -[V_a,D_a] = eig(A_c_d'); - -%Define powers (only implemented for n=2): -max_power = 3; -a = 0 : max_power; -[P,Q] = meshgrid(a,a); -c=cat(2,P',Q'); -powers=reshape(c,[],2); - -linfunc = @(xx) (xx'*V_a)'; -phifun = @(xx) (prod(linfunc(xx).^(powers')))'; -lambd = prod(diag(D).^(powers'))'; - -%% Construct scaling function -gfun = @(xx) xx./pi; %Scale state space into unit cube - -%% Construct eigenfunctions for nonlinear system -phi_nonlin = @(xx) phifun(gfun(yfun(xx))); -A_eig = diag(lambd); + N_basis = 100; + A_cl = A_nom + B_nom*K_nom; + cent = 2*pi*rand(n,N_basis) - pi; + rbf_type = 'gauss'; + eps_rbf = 2; + + % Set up nonlinear transformations and their gradients + zfun = @(xx) [xx; rbf(xx,cent,rbf_type,eps_rbf)]; + zfun_grad = @(xx) [eye(2); rbf_grad(xx,cent,rbf_type,eps_rbf)]; + zfun_dot = @(xx, xx_dot) [xx_dot; rbf_dot(xx,xx_dot,cent,rbf_type,eps_rbf)]; + + % Set up Z and Z_dot matrices: + Z = zfun(X); + Z_dot = zfun_dot(X,X_dot); + %Set up constraint matrix + con1 = zfun_grad([0; 0]); + + disp('Solving optimization problem...') + N = size(Z,1); + cvx_begin + variable C(n,N); + minimize (norm(X_dot + C*Z_dot - A_cl*(X+C*Z),'fro') + 1*norm(C,'fro')) + subject to + {C*con1 == zeros(n)}; + cvx_end + fprintf('Solved, optimal value excluding regularization: %.10f, MSE: %.10f\n', norm(X_dot + C*Z_dot - A_cl*(X+C*Z),'fro'),immse(X_dot+C*Z_dot, A_cl*(X+C*Z))) + %fprintf('Constraint violation: %.4f \n', sum(sum(abs(C*con1)))) + + yfun = @(xx) xx + C*zfun(xx); % Full learned diffeomorphism + + % Calculate eigenfunctions for linearized system + [~,D] = eig(A_cl); + [V_a,~] = eig(A_cl'); + + %Define powers (only implemented for n=2): + pow_eig_pairs = 3; + a = 0 : pow_eig_pairs; + [P,Q] = meshgrid(a,a); + c=cat(2,P',Q'); + powers=reshape(c,[],2); + + linfunc = @(xx) (xx'*V_a)'; + phifun = @(xx) (prod(linfunc(xx).^(powers')))'; + lambd = prod(diag(D).^(powers'))'; + + % Construct scaling function + gfun = @(xx) xx./pi; %Scale state space into unit cube + + % Construct eigenfunctions for nonlinear system + z_eigfun = @(xx) phifun_mat(phifun, gfun(yfun(xx))); + %z_eigfun = @(xx) phifun(gfun(yfun(xx))); + A_koop = diag(lambd); %% Check how well the evolution of the eigenfunctions are described by the eigenvalues on linearized system: -X_test = zeros(n,Ntime+1); -X_test(:,1) = Xstr(:,1,1); -for i = 2 : Ntime+1 - X_test(:,i) = A_c_d*X_test(:,i-1); -end +f_cl = @(t,x) A_cl*x; +[t_test, X_test] = ode45(f_cl,0:deltaT:2,Xstr(:,1,1)); +X_test = X_test'; N = size(lambd,1); -Z_test = zeros(N,Ntime+1); -Z_eig = zeros(N,Ntime+1); -Z_test(:,1) = phifun(X_test(:,1)); -Z_eig(:,1) = phifun(X_test(:,1)); -for j = 2 : Ntime+1 - Z_eig(:,j) = A_eig*Z_eig(:,j-1); - Z_test(:,j) = phifun(X_test(:,j)); -end +Z_test = z_eigfun(X_test); +f_eig = @(t,x) A_koop*x; +[~,Z_eig] = ode45(f_eig,0:deltaT:2,Z_test(:,1)); +Z_eig = Z_eig'; afigure t = 0 : deltaT : Ntime*deltaT; @@ -137,7 +129,7 @@ Z_test(:,1) = phi_nonlin(X_test(:,1)); Z_eig(:,1) = phi_nonlin(X_test(:,1)); for j = 2 : Ntime+1 - Z_eig(:,j) = A_eig*Z_eig(:,j-1); + Z_eig(:,j) = A_koop*Z_eig(:,j-1); Z_test(:,j) = phi_nonlin(X_test(:,j)); end diff --git a/scripts/Koopman-ID/utils/collect_data.m b/scripts/Koopman-ID/utils/collect_data.m index e142df1..58df8f9 100644 --- a/scripts/Koopman-ID/utils/collect_data.m +++ b/scripts/Koopman-ID/utils/collect_data.m @@ -1,5 +1,5 @@ function [Xstr, Xacc, Yacc, Ustr, Uacc, time_str] = collect_data(n,m,Ntraj,... - Ntime,deltaT,X0,Xf,K_nom,f_u,U_perturb, autonomous_learning) + Ntime,deltaT,X0,Xf,K_nom,f_u,U_perturb) %Collect data from the "true" system with nominal controller plus %perturbation %Inputs: @@ -35,13 +35,9 @@ Yacc(:,Ntraj*(i-1)+1:Ntraj*(i)) = Xnext; Xcurrent = Xnext; time_str(:,i+1) = i*deltaT*ones(Ntraj,1); - - if autonomous_learning - Ustr(:,:,i) = Ucurrent; - Uacc(:,Ntraj*(i-1)+1:Ntraj*i) = Ucurrent; - else - Ustr(:,:,i) = Ucurrent - K_nom*(Xcurrent-Xf); - Uacc(:,Ntraj*(i-1)+1:Ntraj*i) = Ucurrent - K_nom*(Xcurrent-Xf); - end + Ustr(:,:,i) = Ucurrent; + Uacc(:,Ntraj*(i-1)+1:Ntraj*i) = Ucurrent; end + Ucurrent = K_nom*(Xcurrent-Xf)+U_perturb(i+1,:); + Ustr(:,:,i+1) = Ucurrent; end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/evaluate_model_performance.m b/scripts/Koopman-ID/utils/evaluate_model_performance.m index 83f0a54..f575f7e 100644 --- a/scripts/Koopman-ID/utils/evaluate_model_performance.m +++ b/scripts/Koopman-ID/utils/evaluate_model_performance.m @@ -1,8 +1,7 @@ function [mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std,... - mse_nom_track, mse_edmd_track, mse_koop_track, E_nom, E_edmd, E_koop,... - cost_nom, cost_edmd, cost_koop] = ... - evaluate_model_performance(K_nom, A_edmd, B_edmd, C_edmd, A_koop, B_koop, ... - C_koop, Nsim, X0, Xf, Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results) + cost_nom, cost_edmd, cost_koop] = evaluate_model_performance(K_nom, ... + A_edmd, B_edmd, C_edmd, A_koop, B_koop, C_koop, Nsim, X0, Xf, Ntraj, ... + Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results) n = size(C_koop,1); m = size(B_koop,2); @@ -22,12 +21,18 @@ for i = 1 : length(q) Q = q(i)*eye(n); R = r(i)*eye(m); - [x_track{i},x_edmd_track{i},x_koop_track{i},mse_nom_track{i}, mse_edmd_track{i},... - mse_koop_track{i},t_plot,traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... - cost_nom{i}, cost_edmd{i}, cost_koop{i}] = ... - sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,... - Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,... - C_edmd,A_koop,B_koop,C_koop,Q,R); +% [x_track{i},x_edmd_track{i},x_koop_track{i},mse_nom_track{i}, mse_edmd_track{i},... +% mse_koop_track{i},t_plot,traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... +% cost_nom{i}, cost_edmd{i}, cost_koop{i}] = ... +% sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,... +% Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,... +% C_edmd,A_koop,B_koop,C_koop,Q,R); + +% [x_track{i},x_edmd_track{i},x_koop_track{i},mse_nom_track{i}, mse_edmd_track{i},... +% mse_koop_track{i},t_plot,traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... +% cost_nom{i}, cost_edmd{i}, cost_koop{i}] = ... +% [0,0,0,0,0,0,0,0,0,0,0,0,0,0]; + end %Plot results @@ -56,7 +61,7 @@ %h1.Position = h1.Position + [0 0.1 0 -0.1]; %Modify size of 1st subplot %h2 = subplot(2,2,2); - h2 = subplot(3,2,3) + h2 = subplot(3,2,3); hold on for i = 1 %Only plot first trajectory plot([0:Ntime]*deltaT,reshape(x(2,i,:),Ntime+1,1),':b','linewidth',lw); hold on diff --git a/scripts/Koopman-ID/utils/sim_prediction.m b/scripts/Koopman-ID/utils/sim_prediction.m index a462854..0b90e84 100644 --- a/scripts/Koopman-ID/utils/sim_prediction.m +++ b/scripts/Koopman-ID/utils/sim_prediction.m @@ -9,6 +9,10 @@ [A_origin, B_origin, ~] = find_nominal_model_ctrl(n,m,f_u,eye(n),eye(m),[0; 0]); f_origin = @(t,x,u) A_origin*x + B_origin*u; + % Define KEEDMD model: + f_koop = @(t,x,u) A_koop*x + B_koop*u; + + x = zeros(n,Nsim,Ntime+1); x_origin = zeros(n,Nsim,Ntime+1); z_edmd = zeros(n_edmd,Nsim,Ntime+1); @@ -22,7 +26,7 @@ % Simulate all systems and initial points for i = 1:Ntime %Controller: - u(:,:,i) = K_nom*(x(:,:,i)-Xf) + 2*rand()-1; + u(:,:,i) = K_nom*(x(:,:,i)-Xf) + 5*rand()-2.5; %True dynamics: x(:,:,i+1) = sim_timestep(deltaT,f_u,0,x(:,:,i), u(:,:,i)); @@ -34,7 +38,8 @@ z_edmd(:,:,i+1) = A_edmd*z_edmd(:,:,i) + B_edmd*u(:,:,i); %Koopman eigenfunction predictor: - z_koop(:,:,i+1) = A_koop*z_koop(:,:,i) + B_koop*u(:,:,i); + %z_koop(:,:,i+1) = A_koop*z_koop(:,:,i) + B_koop*u(:,:,i); + z_koop(:,:,i+1) = sim_timestep(deltaT,f_koop,0,z_koop(:,:,i), u(:,:,i)); end % Calculate corresponding predictions and MSE From 47eb8cc2bf94e61c1bfeee6d5fe51db75258a6f0 Mon Sep 17 00:00:00 2001 From: Cafolkes Date: Mon, 19 Aug 2019 19:53:27 -0700 Subject: [PATCH 7/9] Continuous time version implemented. Functional, some error codes for MPC QP problem must be resolved. --- scripts/Koopman-ID/construct_eigfuncs.m | 24 ++-- scripts/Koopman-ID/main_comparison_pendulum.m | 6 +- .../Koopman-ID/test_eigfunction_generation.m | 119 ++---------------- .../utils/evaluate_model_performance.m | 36 ++++-- .../utils/find_nominal_model_ctrl.m | 4 +- .../Koopman-ID/utils/sim_closed_loop_mpc.m | 34 +---- scripts/Koopman-ID/utils/sim_prediction.m | 12 +- 7 files changed, 62 insertions(+), 173 deletions(-) diff --git a/scripts/Koopman-ID/construct_eigfuncs.m b/scripts/Koopman-ID/construct_eigfuncs.m index 48928f7..6289e80 100644 --- a/scripts/Koopman-ID/construct_eigfuncs.m +++ b/scripts/Koopman-ID/construct_eigfuncs.m @@ -40,20 +40,20 @@ A_cl = A_nom + B_nom*K_nom; cent = 2*pi*rand(n,N_basis) - pi; rbf_type = 'gauss'; - eps_rbf = 2; + eps_rbf = 1; % Set up nonlinear transformations and their gradients zfun = @(xx) [xx; rbf(xx,cent,rbf_type,eps_rbf)]; zfun_grad = @(xx) [eye(2); rbf_grad(xx,cent,rbf_type,eps_rbf)]; zfun_dot = @(xx, xx_dot) [xx_dot; rbf_dot(xx,xx_dot,cent,rbf_type,eps_rbf)]; - % Set up Z and Z_dot matrices: - Z = zfun(X); - Z_dot = zfun_dot(X,X_dot); + % Set up Z and Z_dot matrices: + Z = zfun(X); + Z_dot = zfun_dot(X,X_dot); %Set up constraint matrix con1 = zfun_grad([0; 0]); - + disp('Solving optimization problem...') N = size(Z,1); cvx_begin @@ -66,10 +66,10 @@ %fprintf('Constraint violation: %.4f \n', sum(sum(abs(C*con1)))) yfun = @(xx) xx + C*zfun(xx); % Full learned diffeomorphism - + % Calculate eigenfunctions for linearized system - [~,D] = eig(A_cl); - [V_a,~] = eig(A_cl'); + [V_a,D] = eig(A_cl); + [W_a,~] = eig(A_cl'); %Define powers (only implemented for n=2): a = 0 : pow_eig_pairs; @@ -77,16 +77,16 @@ c=cat(2,P',Q'); powers=reshape(c,[],2); - linfunc = @(xx) (xx'*V_a)'; + linfunc = @(xx) (xx'*W_a)'./diag(V_a'*W_a); phifun = @(xx) (prod(linfunc(xx).^(powers')))'; - lambd = prod(diag(D).^(powers'))'; + lambd = prod(exp(diag(D)).^(powers'))'; + lambd = log(lambd); % Construct scaling function gfun = @(xx) xx./pi; %Scale state space into unit cube % Construct eigenfunctions for nonlinear system - z_eigfun = @(xx) phifun_mat(phifun, gfun(yfun(xx))); - %z_eigfun = @(xx) phifun(gfun(yfun(xx))); + z_eigfun = @(xx) phifun_mat(phifun, gfun(yfun(xx))); A_koop = diag(lambd); end diff --git a/scripts/Koopman-ID/main_comparison_pendulum.m b/scripts/Koopman-ID/main_comparison_pendulum.m index 2c860dd..424aba1 100644 --- a/scripts/Koopman-ID/main_comparison_pendulum.m +++ b/scripts/Koopman-ID/main_comparison_pendulum.m @@ -46,7 +46,7 @@ %Koopman eigenfunction parameters: N_basis_diff = 100; %Number of RBFs to use for lifting when learning the diffeomorphism -pow_eig_pairs = 2; %Highest power of eigenpairs to use when generating eigenfunctions for linearized system +pow_eig_pairs = 5; %Highest power of eigenpairs to use when generating eigenfunctions for linearized system plot_basis_koop = false; %Plots the basis functions if true xlim = [-1, 1]; %Plot limits ylim = [-1, 1]; %Plot limits @@ -63,7 +63,7 @@ % Calculate nominal model with Jacobian and find nominal control gains: -[A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R,[0; 0]); %Linearization around 0. +[A_nom, B_nom, C_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R,[0; 0]); %Linearization around 0. % Collect data to learn autonomous dynamics: U_perturb = 0.5*randn(Ntime+1,Ntraj); %Add normally distributed noise to nominal controller @@ -90,7 +90,7 @@ %% ************************ Analysis of Results *************************** disp('Starting analysis of results...'); tic [mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std,... - cost_nom, cost_edmd, cost_koop] = evaluate_model_performance(K_nom, ... + cost_nom, cost_edmd, cost_koop] = evaluate_model_performance(A_nom, B_nom, C_nom, K_nom, ... A_edmd, B_edmd, C_edmd, A_koop, B_koop, C_koop, Nsim, X0_sim, Xf_sim, ... Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results); fprintf('Analysis done, execution time: %1.2f s \n', toc); diff --git a/scripts/Koopman-ID/test_eigfunction_generation.m b/scripts/Koopman-ID/test_eigfunction_generation.m index 76868c4..3037319 100644 --- a/scripts/Koopman-ID/test_eigfunction_generation.m +++ b/scripts/Koopman-ID/test_eigfunction_generation.m @@ -76,8 +76,8 @@ yfun = @(xx) xx + C*zfun(xx); % Full learned diffeomorphism % Calculate eigenfunctions for linearized system - [~,D] = eig(A_cl); - [V_a,~] = eig(A_cl'); + [V_a,D] = eig(A_cl); + [W_a,~] = eig(A_cl'); %Define powers (only implemented for n=2): pow_eig_pairs = 3; @@ -86,9 +86,10 @@ c=cat(2,P',Q'); powers=reshape(c,[],2); - linfunc = @(xx) (xx'*V_a)'; + linfunc = @(xx) (xx'*W_a)'./diag(V_a'*W_a); phifun = @(xx) (prod(linfunc(xx).^(powers')))'; - lambd = prod(diag(D).^(powers'))'; + lambd = prod(exp(diag(D)).^(powers'))'; + lambd = log(lambd); % Construct scaling function gfun = @(xx) xx./pi; %Scale state space into unit cube @@ -99,12 +100,13 @@ A_koop = diag(lambd); %% Check how well the evolution of the eigenfunctions are described by the eigenvalues on linearized system: f_cl = @(t,x) A_cl*x; -[t_test, X_test] = ode45(f_cl,0:deltaT:2,Xstr(:,1,1)); +x0 = [1;1]; +[t_test, X_test] = ode45(f_cl,0:deltaT:2,x0); X_test = X_test'; N = size(lambd,1); -Z_test = z_eigfun(X_test); -f_eig = @(t,x) A_koop*x; +Z_test = phifun_mat(phifun,X_test); +f_eig = @(t,z) A_koop*z; [~,Z_eig] = ode45(f_eig,0:deltaT:2,Z_test(:,1)); Z_eig = Z_eig'; @@ -124,14 +126,9 @@ %% Check how well the evolution of the eigenfunctions are described by the eigenvalues: X_test = reshape(Xstr(:,1,:),n,Ntime+1); N = size(lambd,1); -Z_test = zeros(N,Ntime+1); -Z_eig = zeros(N,Ntime+1); -Z_test(:,1) = phi_nonlin(X_test(:,1)); -Z_eig(:,1) = phi_nonlin(X_test(:,1)); -for j = 2 : Ntime+1 - Z_eig(:,j) = A_koop*Z_eig(:,j-1); - Z_test(:,j) = phi_nonlin(X_test(:,j)); -end +Z_test = z_eigfun(X_test); +[~,Z_eig] = ode45(f_eig,0:deltaT:2,Z_test(:,1)); +Z_eig = Z_eig'; afigure t = 0 : deltaT : Ntime*deltaT; @@ -143,94 +140,4 @@ if i == N legend('Eigval evolution', 'True') end -end - -%% Predict evolution of nonlinear system - - - -%% Old code: (Delete when verified not needed) -% lin_pred_acc = zeros(size(Xacc,1),Ntime*Ntraj); -% Y_res_acc = zeros(size(Xacc,1),Ntime*Ntraj); -% t = deltaT:deltaT:Ntime*deltaT; -% -% afigure -% for i = 1 : Ntraj -% -% lin_pred = A_nom*reshape(Xstr(:,i,1:end-1),size(Xacc,1),Ntime); -% lin_pred_acc(:,(i-1)*Ntime+1:i*Ntime) = lin_pred; -% Y_res = reshape(Xstr(:,i,2:end),size(Xacc,1),Ntime)-lin_pred; -% Y_res_acc(:,(i-1)*Ntime+1:i*Ntime) = Y_res; -% -% % subplot(2,Ntraj/2,i) -% % plot(t, lin_pred(1,:), t, reshape(Xstr(1,i,2:end),1,Ntime),... -% % t, lin_pred(2,:), t, reshape(Xstr(2,i,2:end),1,Ntime)); -% % xlabel('Time (sec)') -% % ylabel('State value') -% % fprintf('Error %i: %.4f\n', i, norm(lin_pred-reshape(Xstr(:,i,2:end),size(Xacc,1),Ntime))); -% end -% -% %% Test eigenfunction generation -% -% center_type = 'random'; -% N = Ntraj*Ntime; -% N_basis = 40; -% -% switch center_type -% case 'random' -% cent = rand(n,N_basis)*2*pi - pi; -% case 'data' -% cent = datasample(Xacc',N_basis)'+0.05*(rand(n,N_basis)*2-1); -% end -% -% rbf_type = 'gauss'; -% eps_rbf = 1; -% zfun = @(xx) [xx'; rbf(xx',cent,rbf_type,eps_rbf)]; -% X = zfun(Xacc'); -% Y = zfun(Yacc'); -% D_norm = diag(vecnorm(X')); -% X = D_norm*X; -% Y = D_norm*Y; -% -% to_scale = 'Scale'; -% to_center = 'Ignore'; -% k_mode = -1; -% tol = N_basis*eps; -% nobal = 'Ignore'; -% n_ref = 0; -% ref_select = 1; -% target = [0 0]'; %Not used because ref_select not equal ot 2 -% overwrite = 'OverWrite'; -% file_save = 'NoSave'; -% -% % [Z, Lambda, rez, RQ_ref, RSI, Z_ref, rez_ref, U, AxU_k] = ... -% % XY_DDMD_R4( X, Y, to_scale, to_center, k_mode, tol, nobal, n_ref,... -% % ref_select, target, overwrite, file_save); -% -% A = Y/X; -% error = norm(A*X-Y) -% [V,D] = eig(A'); -% rel_inds = find(diag(abs(D)) > 0.1); -% V_red = V(:,rel_inds); -% D_red = D(rel_inds,rel_inds); -% -% z_eigfun = @(xx) (zfun(xx)'*V_red)'; -% A_koop = D_red; %Discrete time autonomous dynamics; -% Zacc = z_eigfun(Yacc'); -% -% %Compare z with its linear evolution -% z = z_eigfun(Xacc(:,1)'); -% for i = 1 : Ntime -% z = [z A_koop*z(:,end)]; -% end -% -% %% -% t = deltaT : deltaT : Ntime*deltaT; -% n_plot = 15; -% afigure -% for i = 1 : n_plot -% subplot(n_plot,1,i) -% plot(t,Zacc(i,1:end),t,z(i,2:end)) -% legend('True', 'Linear evolution') -% end -% \ No newline at end of file +end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/evaluate_model_performance.m b/scripts/Koopman-ID/utils/evaluate_model_performance.m index f575f7e..81a42ab 100644 --- a/scripts/Koopman-ID/utils/evaluate_model_performance.m +++ b/scripts/Koopman-ID/utils/evaluate_model_performance.m @@ -1,5 +1,5 @@ function [mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std,... - cost_nom, cost_edmd, cost_koop] = evaluate_model_performance(K_nom, ... + cost_nom, cost_edmd, cost_koop] = evaluate_model_performance(A_nom, B_nom, C_nom, K_nom, ... A_edmd, B_edmd, C_edmd, A_koop, B_koop, C_koop, Nsim, X0, Xf, Ntraj, ... Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results) @@ -13,25 +13,35 @@ [x,x_origin, x_edmd,x_koop, mse_origin_avg, mse_edmd_avg,mse_koop_avg,... mse_origin_std, mse_edmd_std,mse_koop_std]... = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,Xf,f_u,liftFun,... - phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop); + phi_fun_v,A_nom,B_nom,C_nom,K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop); %Closed loop task: q = [1 5 50]; r = [0.01 0.01 0.01]; + + %Discretize nominal system: + continuous_model_nom = ss(A_nom, B_nom, C_nom, 0); + discrete_model_nom = c2d(continuous_model_nom,deltaT); + A_nom_d = discrete_model_nom.A; + B_nom_d = discrete_model_nom.B; + C_nom_d = discrete_model_nom.C; + + %Discretize KEEDMD system: + continuous_model_koop = ss(A_koop, B_koop, C_koop, 0); + discrete_model_koop = c2d(continuous_model_koop,deltaT); + A_koop_d = discrete_model_koop.A; + B_koop_d = discrete_model_koop.B; + C_koop_d = discrete_model_koop.C; + for i = 1 : length(q) Q = q(i)*eye(n); R = r(i)*eye(m); -% [x_track{i},x_edmd_track{i},x_koop_track{i},mse_nom_track{i}, mse_edmd_track{i},... -% mse_koop_track{i},t_plot,traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... -% cost_nom{i}, cost_edmd{i}, cost_koop{i}] = ... -% sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,... -% Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,... -% C_edmd,A_koop,B_koop,C_koop,Q,R); - -% [x_track{i},x_edmd_track{i},x_koop_track{i},mse_nom_track{i}, mse_edmd_track{i},... -% mse_koop_track{i},t_plot,traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... -% cost_nom{i}, cost_edmd{i}, cost_koop{i}] = ... -% [0,0,0,0,0,0,0,0,0,0,0,0,0,0]; + [x_track{i},x_edmd_track{i},x_koop_track{i},mse_nom_track{i}, mse_edmd_track{i},... + mse_koop_track{i},t_plot,traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... + cost_nom{i}, cost_edmd{i}, cost_koop{i}] = ... + sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,... + Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v,A_nom,B_nom,C_nom,K_nom,A_edmd,B_edmd,... + C_edmd,A_koop_d,B_koop_d,C_koop_d,Q,R); end diff --git a/scripts/Koopman-ID/utils/find_nominal_model_ctrl.m b/scripts/Koopman-ID/utils/find_nominal_model_ctrl.m index d7a397b..f93fd36 100644 --- a/scripts/Koopman-ID/utils/find_nominal_model_ctrl.m +++ b/scripts/Koopman-ID/utils/find_nominal_model_ctrl.m @@ -1,4 +1,4 @@ -function [A_nom, B_nom, K_nom] = find_nominal_model_ctrl(n, m, f_u, Q, R, x0) +function [A_nom, B_nom, C_nom, K_nom] = find_nominal_model_ctrl(n, m, f_u, Q, R, x0) %Linearize around origin and design linear feedback control law using LQR %Inputs: @@ -15,6 +15,6 @@ x = sym('x',[n;1]); u = sym('u',[m;1]); A_nom = double(subs(jacobian(f_u(0,x,u),x),[x;u],[x0;0])); B_nom = double(subs(jacobian(f_u(0,x,u),u),[x;u],[x0;0])); - + C_nom = eye(n); K_nom = -lqr(A_nom,B_nom,Q,R); end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m index 3380cc4..07f5f44 100644 --- a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m +++ b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m @@ -1,7 +1,7 @@ function [x_nom,x_edmd,x_koop,mse_nom,mse_edmd,mse_koop, t_plot, traj_d, E_nom, E_edmd, E_koop,... cost_nom, cost_edmd, cost_koop]... = sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,Tsim,... - f_u,liftFun,phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,... + f_u,liftFun,phi_fun_v,A_nom,B_nom,C_nom,K_nom,A_edmd,B_edmd,C_edmd,A_koop,... B_koop,C_koop,Q,R) %Set up trajectory to track: @@ -35,28 +35,18 @@ z_edmd(:,1) = liftFun(traj_d(:,1)); z_koop(:,1) = phi_fun_v(traj_d(:,1)); - % Define Koopman controller - Tpred = Tsim/2;%0.1; % Prediction horizon + % Define MPC parameters: + Tpred = Tsim;%0.1; % Prediction horizon Np = round(Tpred / deltaT); traj_d = [traj_d traj_d(:,end).*ones(n,Np)]; %Add extra points to trajectory to allow prediction horizon xlift_min = [];%[ymin ; nan(Nlift-1,1)]; % Constraints xlift_max = [];%[ymax ; nan(Nlift-1,1)]; % Build Koopman MPC controller + MPC_nom = getMPC(A_nom,B_nom,C_nom,0,Q,R,Q,Np,-100, 100, xlift_min, xlift_max,'qpoases'); MPC_edmd = getMPC(A_edmd,B_edmd,C_edmd,0,Q,R,Q,Np,-100, 100, xlift_min, xlift_max,'qpoases'); MPC_koop = getMPC(A_koop,B_koop,C_koop,0,Q,R,Q,Np,-100, 100, xlift_min, xlift_max,'qpoases'); - % Get Jacobian of the true dynamics (for local linearization MPC) - x = sym('x',[2 1]); syms u; - f_ud_sym = sim_timestep(deltaT, f_u, 0, x, u); - Jx = jacobian(f_ud_sym,x); - Ju = jacobian(f_ud_sym,u); - - %Save indices of infeasible MPC-problems (Should never happen with no - %state constraints) - wasinfeas= 0; - ind_inf = []; - % Closed-loop simultion start for i = 1:Ntime_track if(mod(i,10) == 0) @@ -67,17 +57,7 @@ yr = traj_d(:,i:i+Np-1); % Local linearization MPC - %Aloc = double(subs(Jx,[x;u],[x_nom(:,i);u_nom(:,i)])); % Get local linearization - %Bloc = double(subs(Ju,[x;u],[x_nom(:,i);u_nom(:,i)])); - Aloc = double(subs(Jx,[x;u],[[0 0]';0])); % Get local linearization - Bloc = double(subs(Ju,[x;u],[[0 0]';0])); - %Cloc = double(subs(f_ud_sym,[x;u],[x_nom(:,i);u_nom(:,i)])) - Aloc*x_nom(:,i) - Bloc*u_nom(:,i); - [U_nom,~,optval] = solveMPCprob(Aloc,Bloc,eye(2),[],Q,R,Q,Np,-100, 100,[],[],x_nom(:,i),yr); % Get control input - u_nom(:,i) = U_nom(1:m,1); - if(optval == Inf) % Detect infeasibility - ind_inf = [ind_inf i]; - wasinfeas = 1; - end + u_nom(:,i) = MPC_nom(x_nom(:,i),yr); x_nom(:,i+1) = sim_timestep(deltaT, f_u, 0, x_nom(:,i), u_nom(:,i)); % Update true state % EDMD MPC @@ -90,10 +70,6 @@ u_koop(:,i) = MPC_koop(z_koop(:,i),yr);% Get control input x_koop(:,i+1) = sim_timestep(deltaT, f_u, 0, x_koop(:,i), u_koop(:,i)); end - - if(isempty(ind_inf)) - ind_inf = Ntime_track; - end % Calculate corresponding predictions and MSE traj_d = traj_d(:,1:end-Np); %Remove added elements from trajectory diff --git a/scripts/Koopman-ID/utils/sim_prediction.m b/scripts/Koopman-ID/utils/sim_prediction.m index 0b90e84..952eb43 100644 --- a/scripts/Koopman-ID/utils/sim_prediction.m +++ b/scripts/Koopman-ID/utils/sim_prediction.m @@ -1,18 +1,14 @@ function [x,x_origin, x_edmd,x_koop, mse_origin_avg, mse_edmd_avg,mse_koop_avg,... mse_origin_std, mse_edmd_std,mse_koop_std]... = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,Xf,f_u,liftFun,... - phi_fun_v, K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop) + phi_fun_v,A_nom,B_nom,C_nom,K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop) u = zeros(m,Nsim,Ntime); - % Find linearized model at the origin: - [A_origin, B_origin, ~] = find_nominal_model_ctrl(n,m,f_u,eye(n),eye(m),[0; 0]); - f_origin = @(t,x,u) A_origin*x + B_origin*u; - - % Define KEEDMD model: + % Define nominal and KEEDMD model: + f_nom = @(t,x,u) A_nom*x + B_nom*u; f_koop = @(t,x,u) A_koop*x + B_koop*u; - x = zeros(n,Nsim,Ntime+1); x_origin = zeros(n,Nsim,Ntime+1); z_edmd = zeros(n_edmd,Nsim,Ntime+1); @@ -32,7 +28,7 @@ x(:,:,i+1) = sim_timestep(deltaT,f_u,0,x(:,:,i), u(:,:,i)); %Linearization at origin: - x_origin(:,:,i+1) = sim_timestep(deltaT,f_origin,0,x_origin(:,:,i), u(:,:,i)); + x_origin(:,:,i+1) = sim_timestep(deltaT,f_nom,0,x_origin(:,:,i), u(:,:,i)); %EDMD predictor: z_edmd(:,:,i+1) = A_edmd*z_edmd(:,:,i) + B_edmd*u(:,:,i); From a062c22b1c659dbc568a75d5d73a0677845868a0 Mon Sep 17 00:00:00 2001 From: Cafolkes Date: Mon, 19 Aug 2019 20:44:42 -0700 Subject: [PATCH 8/9] Fixed MPC issues, implemented continuous EDMD --- scripts/Koopman-ID/KEEDMD.m | 5 +- scripts/Koopman-ID/main_comparison_pendulum.m | 20 +++--- scripts/Koopman-ID/utils/collect_data.m | 20 ++---- .../utils/evaluate_model_performance.m | 12 +++- scripts/Koopman-ID/utils/extendedDMD.m | 68 ++++++++----------- scripts/Koopman-ID/utils/sim_prediction.m | 4 +- 6 files changed, 55 insertions(+), 74 deletions(-) diff --git a/scripts/Koopman-ID/KEEDMD.m b/scripts/Koopman-ID/KEEDMD.m index a06456e..9c5a503 100644 --- a/scripts/Koopman-ID/KEEDMD.m +++ b/scripts/Koopman-ID/KEEDMD.m @@ -36,7 +36,6 @@ end % ******************** Construct eigenfunctions *********************** - [A_eigfuncs, liftFun] = construct_eigfuncs(n, N_basis,pow_eig_pairs, ... A_nom, B_nom, K_nom, X, X_dot); @@ -50,15 +49,13 @@ Xlift = [Xlift Xlift_temp]; Xlift_dot = [Xlift_dot num_diff(Xlift_temp,deltaT)]; end - - %fprintf('Lifting DONE, time = %1.2f s \n', toc); % ********************** Build predictor ********************************* %Set up regression for A and B: X_vel = [Xlift; U]; Y_vel = Xlift_dot(n/2+1:n,:); - A_vel = lasso(X_vel',Y_vel','Lambda',1e-1, 'Alpha', 0.8); + A_vel = lasso(X_vel',Y_vel','Lambda',1e-3, 'Alpha', 0.8); %Perform regression and enforce known structure: A_koop = zeros(Nlift); diff --git a/scripts/Koopman-ID/main_comparison_pendulum.m b/scripts/Koopman-ID/main_comparison_pendulum.m index 424aba1..cf324c3 100644 --- a/scripts/Koopman-ID/main_comparison_pendulum.m +++ b/scripts/Koopman-ID/main_comparison_pendulum.m @@ -35,19 +35,16 @@ X0 = pi*X0./vecnorm(X0); Xf = zeros(n,Ntraj); %Terminal points for each trajectory for learning A - %E-DMD parameters: N_basis_edmd = 36; %Number of basis functions basis_function_edmd = 'rbf'; %Type of basis function rbf_type_edmd = 'thinplate'; %RBF type center_type_edmd = 'data'; %Centers of rbf ('data' - pick random points from data set, 'random' pick points uniformly at random eps_rbf_edmd = 1; %RBF width -plot_basis_edmd = false; %Plots the basis functions if true %Koopman eigenfunction parameters: N_basis_diff = 100; %Number of RBFs to use for lifting when learning the diffeomorphism pow_eig_pairs = 5; %Highest power of eigenpairs to use when generating eigenfunctions for linearized system -plot_basis_koop = false; %Plots the basis functions if true xlim = [-1, 1]; %Plot limits ylim = [-1, 1]; %Plot limits @@ -59,39 +56,42 @@ Xf_sim = 0.2*(2*pi*rand(n,Nsim)-pi); %% ************************** Data Collection ***************************** -disp('Starting data collection...'); tic - % Calculate nominal model with Jacobian and find nominal control gains: - [A_nom, B_nom, C_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R,[0; 0]); %Linearization around 0. % Collect data to learn autonomous dynamics: +disp('Starting data collection...'); tic + U_perturb = 0.5*randn(Ntime+1,Ntraj); %Add normally distributed noise to nominal controller -[Xstr, Xacc, Yacc, Ustr, Uacc, timestr] = collect_data(n,m,Ntraj,... +[Xstr, Ustr, timestr] = collect_data(n,m,Ntraj,... Ntime,deltaT,X0, Xf ,K_nom,f_u,U_perturb); fprintf('Data collection done, execution time: %1.2f s \n', toc); %% *********************** Model Identification *************************** - % Identify model using E-DMD to get eigenvalues and model to use for % comparison: disp('Starting EDMD...'); tic + [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m, Ntraj, Ntime, N_basis_edmd,basis_function_edmd,... - rbf_type_edmd, center_type_edmd, eps_rbf_edmd, plot_basis_edmd, xlim,... - ylim, Xacc, Yacc, Uacc); + rbf_type_edmd, center_type_edmd, eps_rbf_edmd, Xstr, Ustr, Xf, deltaT); + fprintf('EDMD done, execution time: %1.2f s \n', toc); disp('Starting KEEDMD...'); tic + [A_koop, B_koop, C_koop, phi_fun_v] = KEEDMD(n,m,Ntraj, Ntime, N_basis_diff,... pow_eig_pairs, Xstr, Ustr, Xf, A_nom, B_nom, K_nom, deltaT); + fprintf('KEEDMD done, execution time: %1.2f s \n', toc); %% ************************ Analysis of Results *************************** disp('Starting analysis of results...'); tic + [mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std,... cost_nom, cost_edmd, cost_koop] = evaluate_model_performance(A_nom, B_nom, C_nom, K_nom, ... A_edmd, B_edmd, C_edmd, A_koop, B_koop, C_koop, Nsim, X0_sim, Xf_sim, ... Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results); + fprintf('Analysis done, execution time: %1.2f s \n', toc); \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/collect_data.m b/scripts/Koopman-ID/utils/collect_data.m index 58df8f9..da3f332 100644 --- a/scripts/Koopman-ID/utils/collect_data.m +++ b/scripts/Koopman-ID/utils/collect_data.m @@ -1,4 +1,4 @@ -function [Xstr, Xacc, Yacc, Ustr, Uacc, time_str] = collect_data(n,m,Ntraj,... +function [Xstr, Ustr, time_str] = collect_data(n,m,Ntraj,... Ntime,deltaT,X0,Xf,K_nom,f_u,U_perturb) %Collect data from the "true" system with nominal controller plus %perturbation @@ -10,33 +10,25 @@ %Outputs: % Xstr - (n x Ntraj x Ntime) struct with data from each timestep and % trajectory for each state - % Xacc - (n x Ntraj*Ntime) matrix with all trajectories added to it - % Yacc - (n x Ntraj*Ntime) matrix with all trajectories added to it - % Yacc_k = Xacc_k+1 - % Uacc - (m x Ntraj*Ntime) matrix with all trajectories added to it - % Xstr - (1 x Ntraj x Ntime) struct with the time stamp of each data + % Ustr - (1 x Ntraj x Ntime) struct with the time stamp of each data % point from each trajectory Xstr = zeros(n,Ntraj,Ntime+1); % *str is structure Ustr = zeros(m,Ntraj,Ntime); - Xacc = zeros(n,Ntraj*(Ntime)); % *acc is accumulated vectors - Yacc = zeros(n,Ntraj*(Ntime)); - Uacc = zeros(m,Ntraj*(Ntime)); - time_str = zeros(Ntraj,Ntime+1); + Xcurrent = X0; Xstr(:,:,1) = X0; for i = 1:Ntime Ucurrent = K_nom*(Xcurrent-Xf)+U_perturb(i,:); + Ustr(:,:,i) = Ucurrent; + Xnext = sim_timestep(deltaT, f_u, 0, Xcurrent, Ucurrent); Xstr(:,:,i+1) = Xnext; - Xacc(:,Ntraj*(i-1)+1:Ntraj*(i)) = Xcurrent; - Yacc(:,Ntraj*(i-1)+1:Ntraj*(i)) = Xnext; Xcurrent = Xnext; + time_str(:,i+1) = i*deltaT*ones(Ntraj,1); - Ustr(:,:,i) = Ucurrent; - Uacc(:,Ntraj*(i-1)+1:Ntraj*i) = Ucurrent; end Ucurrent = K_nom*(Xcurrent-Xf)+U_perturb(i+1,:); Ustr(:,:,i+1) = Ucurrent; diff --git a/scripts/Koopman-ID/utils/evaluate_model_performance.m b/scripts/Koopman-ID/utils/evaluate_model_performance.m index 81a42ab..29de7d9 100644 --- a/scripts/Koopman-ID/utils/evaluate_model_performance.m +++ b/scripts/Koopman-ID/utils/evaluate_model_performance.m @@ -26,6 +26,13 @@ B_nom_d = discrete_model_nom.B; C_nom_d = discrete_model_nom.C; + %Discretize EDMD system: + continuous_model_edmd = ss(A_edmd, B_edmd, C_edmd, 0); + discrete_model_edmd = c2d(continuous_model_edmd,deltaT); + A_edmd_d = discrete_model_edmd.A; + B_edmd_d = discrete_model_edmd.B; + C_edmd_d = discrete_model_edmd.C; + %Discretize KEEDMD system: continuous_model_koop = ss(A_koop, B_koop, C_koop, 0); discrete_model_koop = c2d(continuous_model_koop,deltaT); @@ -40,9 +47,8 @@ mse_koop_track{i},t_plot,traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... cost_nom{i}, cost_edmd{i}, cost_koop{i}] = ... sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,... - Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v,A_nom,B_nom,C_nom,K_nom,A_edmd,B_edmd,... - C_edmd,A_koop_d,B_koop_d,C_koop_d,Q,R); - + Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v,A_nom_d,B_nom_d,C_nom_d,K_nom,A_edmd_d,B_edmd_d,... + C_edmd_d,A_koop_d,B_koop_d,C_koop_d,Q,R); end %Plot results diff --git a/scripts/Koopman-ID/utils/extendedDMD.m b/scripts/Koopman-ID/utils/extendedDMD.m index 33b59d6..622a136 100644 --- a/scripts/Koopman-ID/utils/extendedDMD.m +++ b/scripts/Koopman-ID/utils/extendedDMD.m @@ -1,5 +1,5 @@ function [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m,Ntraj, Ntime, N_basis,basis_function,... - rbf_type, center_type, eps, plot_basis, xlim, ylim, Xacc, Yacc, Uacc) + rbf_type, center_type, eps, Xstr, Ustr, Xf,deltaT) %Identify lifted state space model using Extended Dynamic Mode %Decomposition @@ -24,62 +24,48 @@ % C_edmd - Projection matrix from lifted space to outputs + % ************************** Prepare data ***************************** + Xstr_shift = zeros(size(Xstr)); %Shift dynamics such that origin is fixed point + X = []; + X_dot = []; + U = []; + for i = 1 : Ntraj + Xstr_shift(:,i,:) = Xstr(:,i,:) - Xf(:,i); + X = [X reshape(Xstr_shift(:,i,:),n,Ntime+1)]; + X_dot = [X_dot num_diff(reshape(Xstr_shift(:,i,:),size(Xstr_shift,1),size(Xstr_shift,3)),deltaT)]; + U = [U reshape(Ustr(:,i,:),m,Ntime+1)]; + end + + % ******************************* Lift *********************************** + % Find RBF Centers switch center_type case 'random' cent = rand(n,N_basis)*2 - 1; case 'data' - cent = datasample(Xacc',N_basis)'+0.05*(rand(n,N_basis)*2-1); + cent = datasample(X',N_basis)'+0.05*(rand(n,N_basis)*2-1); end - - % ********************** Plot basis functions ************************* - if plot_basis - [X,Y] = meshgrid(-xlim(1):0.1:xlim(2),ylim(1):0.1:ylim(2)); - x_flat = reshape(X,[1,size(X,1)*size(X,2)]); - y_flat = reshape(Y,[1,size(X,1)*size(X,2)]); - xy_flat = [x_flat;y_flat]; - - afigure - hold on - for i=1:1 - rbf_flat = rbf( xy_flat,cent(:,i), rbf_type, eps_rbf); - rbf_pack = reshape(rbf_flat,[size(X,1),size(X,2)]); - surf(X,Y,rbf_pack) - end - alpha 0.2 - xlabel('x') - ylabel('y') - end - - % ******************************* Lift *********************************** - - %disp('Starting LIFT GENERATION') + switch basis_function case 'rbf' - liftFun = @(xx)( [xx;rbf(xx,cent,rbf_type,eps)] ); + liftFun = @(xx)( [ones(1,size(xx,2)); xx;rbf(xx,cent,rbf_type,eps)]); end Nlift = length(liftFun(zeros(n,1))); - Xlift = liftFun(Xacc); - Ylift = liftFun(Yacc); - %fprintf('Lifting DONE, time = %1.2f s \n', toc); - + Xlift = []; + Xlift_dot = []; + for i = 1 : Ntraj + Xlift_temp = liftFun(reshape(Xstr_shift(:,i,:),size(Xstr_shift,1),size(Xstr_shift,3))); + Xlift = [Xlift Xlift_temp]; + Xlift_dot = [Xlift_dot num_diff(Xlift_temp,deltaT)]; + end % ********************** Build predictor ********************************* - %disp('Starting REGRESSION') - %tic - W = [Ylift ; Xacc]; - V = [Xlift; Uacc]; + W = [Xlift_dot ; X]; + V = [Xlift; U]; VVt = V*V'; WVt = W*V'; M = WVt * pinv(VVt); % Matrix [A B; C 0] A_edmd= M(1:Nlift,1:Nlift); B_edmd = M(1:Nlift,Nlift+1:end); C_edmd = M(Nlift+1:end,1:Nlift); - - % Add known structure (overwrite parts of the learned matrices): - %A_edmd(1:n/2,:) = [eye(n/2) deltaT*eye(n/2) zeros(n/2,size(A_edmd,2)-n)]; - %C_edmd = zeros(size(C_edmd)); - %C_edmd(:,1:n) = eye(n); - - %fprintf('Regression done, time = %1.2f s \n', toc); end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/sim_prediction.m b/scripts/Koopman-ID/utils/sim_prediction.m index 952eb43..35bb077 100644 --- a/scripts/Koopman-ID/utils/sim_prediction.m +++ b/scripts/Koopman-ID/utils/sim_prediction.m @@ -7,6 +7,7 @@ % Define nominal and KEEDMD model: f_nom = @(t,x,u) A_nom*x + B_nom*u; + f_edmd = @(t,x,u) A_edmd*x + B_edmd*u; f_koop = @(t,x,u) A_koop*x + B_koop*u; x = zeros(n,Nsim,Ntime+1); @@ -31,10 +32,9 @@ x_origin(:,:,i+1) = sim_timestep(deltaT,f_nom,0,x_origin(:,:,i), u(:,:,i)); %EDMD predictor: - z_edmd(:,:,i+1) = A_edmd*z_edmd(:,:,i) + B_edmd*u(:,:,i); + z_edmd(:,:,i+1) = sim_timestep(deltaT,f_edmd,0,z_edmd(:,:,i), u(:,:,i)); %Koopman eigenfunction predictor: - %z_koop(:,:,i+1) = A_koop*z_koop(:,:,i) + B_koop*u(:,:,i); z_koop(:,:,i+1) = sim_timestep(deltaT,f_koop,0,z_koop(:,:,i), u(:,:,i)); end From c8b5897edbffbc45fd9ec455f9db2395cc3c5fcb Mon Sep 17 00:00:00 2001 From: Cafolkes Date: Wed, 28 Aug 2019 14:14:50 -0700 Subject: [PATCH 9/9] Implemented cart-pendulum and added generalization to the code allowing any dimension of the dynamics --- scripts/Koopman-ID/KEEDMD.m | 15 ++-- scripts/Koopman-ID/construct_eigfuncs.m | 34 +++++--- scripts/Koopman-ID/generate_trajectory.m | 37 ++++++++ scripts/Koopman-ID/main_comparison_pendulum.m | 74 +++++++++++----- scripts/Koopman-ID/test_trajectory_gen.m | 85 +++++++++++++++++++ scripts/Koopman-ID/utils/collect_data.m | 14 +-- .../utils/evaluate_model_performance.m | 81 +++++++++--------- scripts/Koopman-ID/utils/extendedDMD.m | 8 +- .../Koopman-ID/utils/sim_closed_loop_mpc.m | 49 ++++------- scripts/Koopman-ID/utils/sim_prediction.m | 5 +- 10 files changed, 274 insertions(+), 128 deletions(-) create mode 100644 scripts/Koopman-ID/generate_trajectory.m create mode 100644 scripts/Koopman-ID/test_trajectory_gen.m diff --git a/scripts/Koopman-ID/KEEDMD.m b/scripts/Koopman-ID/KEEDMD.m index 9c5a503..df8fe2e 100644 --- a/scripts/Koopman-ID/KEEDMD.m +++ b/scripts/Koopman-ID/KEEDMD.m @@ -1,5 +1,5 @@ function [A_koop, B_koop, C_koop, liftFun] = KEEDMD(n,m,Ntraj, Ntime, N_basis,... - pow_eig_pairs, Xstr, Ustr, Xf, A_nom, B_nom, K_nom, deltaT) + pow_eig_pairs, Xstr, Ustr, qf, A_nom, B_nom, K_nom, deltaT) %Identify lifted state space model using Extended Dynamic Mode %Decomposition @@ -29,10 +29,10 @@ X_dot = []; U = []; for i = 1 : Ntraj - Xstr_shift(:,i,:) = Xstr(:,i,:) - Xf(:,i); - X = [X reshape(Xstr_shift(:,i,:),n,Ntime+1)]; + Xstr_shift(:,i,:) = Xstr(:,i,:)-qf(:,i); + X = [X reshape(Xstr_shift(:,i,:),size(Xstr_shift,1),size(Xstr_shift,3))]; X_dot = [X_dot num_diff(reshape(Xstr_shift(:,i,:),size(Xstr_shift,1),size(Xstr_shift,3)),deltaT)]; - U = [U reshape(Ustr(:,i,:),m,Ntime+1)]; + U = [U reshape(Ustr(:,i,:),size(Ustr,1),size(Ustr,3))]; end % ******************** Construct eigenfunctions *********************** @@ -55,12 +55,13 @@ %Set up regression for A and B: X_vel = [Xlift; U]; Y_vel = Xlift_dot(n/2+1:n,:); - A_vel = lasso(X_vel',Y_vel','Lambda',1e-3, 'Alpha', 0.8); + A_vel_x = lasso(X_vel',Y_vel(1,:)','Lambda',1e-3, 'Alpha', 0.5); + A_vel_th = lasso(X_vel',Y_vel(2,:)','Lambda',1e-3, 'Alpha', 0.5); %Perform regression and enforce known structure: A_koop = zeros(Nlift); A_koop(1:n/2,:) = [zeros(n/2) eye(n/2) zeros(n/2,size(A_koop,2)-n)]; - A_koop(n/2+1:n,:) = A_vel(1:Nlift,:)'; + A_koop(n/2+1:n,:) = [A_vel_x(1:Nlift,:)'; A_vel_th(1:Nlift,:)']; A_koop(n+1:end,n+1:end) = A_eigfuncs; Y_kin = X_dot(1:n/2,:) - A_koop(1:n/2,:)*Xlift; @@ -71,7 +72,7 @@ X_eig = U-K_nom*X; B_eig = X_eig'\Y_eig'; - B_koop = [B_kin'; A_vel(Nlift+1:end,:); B_eig']; + B_koop = [B_kin'; A_vel_x(Nlift+1:end,:); A_vel_th(Nlift+1:end,:); B_eig']; C_koop = zeros(n,size(A_koop,1)); C_koop(1:n,1:n) = eye(n); diff --git a/scripts/Koopman-ID/construct_eigfuncs.m b/scripts/Koopman-ID/construct_eigfuncs.m index 6289e80..db3d8ef 100644 --- a/scripts/Koopman-ID/construct_eigfuncs.m +++ b/scripts/Koopman-ID/construct_eigfuncs.m @@ -38,13 +38,16 @@ disp('Starting autonomous dynamics learning...'); tic A_cl = A_nom + B_nom*K_nom; - cent = 2*pi*rand(n,N_basis) - pi; + cent = [5*rand(1,N_basis)-2.5;... + 2*pi/3*rand(1,N_basis)-pi/3;... + 4*rand(1,N_basis)-2;... + 4*rand(1,N_basis)-2]; rbf_type = 'gauss'; eps_rbf = 1; % Set up nonlinear transformations and their gradients zfun = @(xx) [xx; rbf(xx,cent,rbf_type,eps_rbf)]; - zfun_grad = @(xx) [eye(2); rbf_grad(xx,cent,rbf_type,eps_rbf)]; + zfun_grad = @(xx) [eye(n); rbf_grad(xx,cent,rbf_type,eps_rbf)]; zfun_dot = @(xx, xx_dot) [xx_dot; rbf_dot(xx,xx_dot,cent,rbf_type,eps_rbf)]; % Set up Z and Z_dot matrices: @@ -52,38 +55,45 @@ Z_dot = zfun_dot(X,X_dot); %Set up constraint matrix - con1 = zfun_grad([0; 0]); + con1 = zfun_grad(zeros(n,1)); disp('Solving optimization problem...') N = size(Z,1); cvx_begin variable C(n,N); - minimize (norm(X_dot + C*Z_dot - A_cl*(X+C*Z),'fro') + 1*norm(C,'fro')) + minimize (norm(X_dot + C*Z_dot - A_cl*(X+C*Z),'fro') + 1e-2*norm(C,'fro')/N*n) subject to {C*con1 == zeros(n)}; cvx_end fprintf('Solved, optimal value excluding regularization: %.10f, MSE: %.10f\n', norm(X_dot + C*Z_dot - A_cl*(X+C*Z),'fro'),immse(X_dot+C*Z_dot, A_cl*(X+C*Z))) - %fprintf('Constraint violation: %.4f \n', sum(sum(abs(C*con1)))) - - yfun = @(xx) xx + C*zfun(xx); % Full learned diffeomorphism + fprintf('Constraint violation: %.4f \n', sum(sum(abs(C*con1)))) + yfun = @(xx) xx + [zeros(1,size(xx,2));... + C(2,:)*zfun(xx);... + zeros(1,size(xx,2));... + C(4,:)*zfun(xx)]; % Full learned diffeomorphism +% yfun = @(xx) xx + C*zfun(xx); + + % Calculate eigenfunctions for linearized system [V_a,D] = eig(A_cl); [W_a,~] = eig(A_cl'); %Define powers (only implemented for n=2): a = 0 : pow_eig_pairs; - [P,Q] = meshgrid(a,a); - c=cat(2,P',Q'); - powers=reshape(c,[],2); + combs = combnk(a,n); + powers = []; + for i = 1 : size(combs,1) + powers = [powers; perms(combs(i,:))]; + end - linfunc = @(xx) (xx'*W_a)'./diag(V_a'*W_a); + linfunc = @(xx) (xx'*W_a)';%./diag(V_a'*W_a); phifun = @(xx) (prod(linfunc(xx).^(powers')))'; lambd = prod(exp(diag(D)).^(powers'))'; lambd = log(lambd); % Construct scaling function - gfun = @(xx) xx./pi; %Scale state space into unit cube + gfun = @(xx) xx./[5 4 2*pi/3 4]'; %Scale state space into unit cube % Construct eigenfunctions for nonlinear system z_eigfun = @(xx) phifun_mat(phifun, gfun(yfun(xx))); diff --git a/scripts/Koopman-ID/generate_trajectory.m b/scripts/Koopman-ID/generate_trajectory.m new file mode 100644 index 0000000..11ef673 --- /dev/null +++ b/scripts/Koopman-ID/generate_trajectory.m @@ -0,0 +1,37 @@ +function [q_t, u_t] = generate_trajectory(A,B,C,Qf,q0,qf,t,deltaT,lb,ub) + +% Extract relevant parameters +Nsim = length(t); +Ntraj = size(q0,2); +N = size(A,1); +n = size(C,1); +m = size(B,2); + +% Discretize state space matrices +continuous_model_nom = ss(A, B, C, 0); +discrete_model_nom = c2d(continuous_model_nom,deltaT); +A_d = discrete_model_nom.A; +B_d = discrete_model_nom.B; +C_d = discrete_model_nom.C; + +% Initialize variables +q_t = zeros(n,Ntraj,Nsim); +u_t = zeros(m,Ntraj,Nsim-1); +% Generate trajectories for each initial-final condition pair +for i = 1 : size(q0,2) + cvx_begin quiet + variable u(m,Nsim-1) + variable z(N,Nsim) + minimize ((C_d*z(:,Nsim)-qf(:,i))'*Qf*(C_d*z(:,Nsim)-qf(:,i)) + norm(u,'fro')); + subject to + {z(:,2:Nsim) == A_d*z(:,1:Nsim-1) + B_d*u}; + {C_d*z(:,1) == q0(:,i)}; + {z <= ub.*ones(N,Nsim)}; + {z >= lb.*ones(N,Nsim)}; + cvx_end; + + q_t(:,i,:) = C*z; %Project back to state space if trajectory is based on lifted state + u_t(:,i,:) = u; +end + + diff --git a/scripts/Koopman-ID/main_comparison_pendulum.m b/scripts/Koopman-ID/main_comparison_pendulum.m index cf324c3..7200fe4 100644 --- a/scripts/Koopman-ID/main_comparison_pendulum.m +++ b/scripts/Koopman-ID/main_comparison_pendulum.m @@ -19,52 +19,76 @@ %% **************** Model and simulation parameters *********************** %Dynamics: -m = 1; g = 9.81; l = 1; -f_u = @(t,x,u) [x(2,:); g/l*sin(x(1,:))+ u]; -n = 2; m = 1; %Number of states and control inputs -Q = eye(2); R = 0.1; %LQR penalty matrices for states and control inputs +M = .5; %Cart mass +m = 0.2; %Pendulum mass +g = 9.81; %Gravity +l = 0.4; %Pole length +f_th = 0.1; %Friction coefficient between pendulum and cart + +f_u = @(t,q,u) [q(3,:);... + q(4,:);... + (-m*g*sin(q(2,:)).*cos(q(2,:))+m*l*q(4,:).^2.*sin(q(2,:))+f_th*m*q(4,:).*cos(q(2,:)) + u)./(M+(1-cos(q(2,:)).^2)*m);... + ((M+m)*(g*sin(q(2,:))-f_th*q(2,:))-(l*m*q(4,:).^2.*sin(q(2,:)) + u).*cos(q(2,:)))./(l*(M+(1-cos(q(2,:)).^2)*m))]; + +n = 4; m = 1; %Number of states and control inputs +ub = [2.5 pi/3 2 2]'; %State constraints +lb = -ub; %State constraints +Q = 50*eye(n); R = eye(m); %LQR penalty matrices for states and control inputs %Simulation parameters: -Ntime = 200; %Length of each trajectory (# of time steps) -Ntraj = 20; %Number of trajectories +tspan = [0 2]; deltaT = 0.01; %Time step length of simulation +Ntime = (tspan(2)-tspan(1))/deltaT+1; %Length of each trajectory (# of time steps) +Ntraj = 20; %Number of trajectories rng('shuffle'); %Set seed of random number generator -%X0 = 0.8*(2*pi*rand(n,Ntraj)-pi); -%Xf = 0.2*(2*pi*rand(n,Ntraj)-pi); %Terminal points for each trajectory for learning A -X0 = (2*rand(n,Ntraj)-1); -X0 = pi*X0./vecnorm(X0); -Xf = zeros(n,Ntraj); %Terminal points for each trajectory for learning A +q0 = [4*rand(1,Ntraj)-2;... + 0.2*rand(1,Ntraj)-0.1;... + zeros(1,Ntraj);... + zeros(1,Ntraj)]; +qf = zeros(n,Ntraj); %Terminal points for each trajectory for learning A %E-DMD parameters: -N_basis_edmd = 36; %Number of basis functions +N_basis_edmd = 60; %Number of basis functions basis_function_edmd = 'rbf'; %Type of basis function rbf_type_edmd = 'thinplate'; %RBF type center_type_edmd = 'data'; %Centers of rbf ('data' - pick random points from data set, 'random' pick points uniformly at random eps_rbf_edmd = 1; %RBF width %Koopman eigenfunction parameters: -N_basis_diff = 100; %Number of RBFs to use for lifting when learning the diffeomorphism -pow_eig_pairs = 5; %Highest power of eigenpairs to use when generating eigenfunctions for linearized system +N_basis_diff = 200; %Number of RBFs to use for lifting when learning the diffeomorphism +pow_eig_pairs = 4; %Highest power of eigenpairs to use when generating eigenfunctions for linearized system xlim = [-1, 1]; %Plot limits ylim = [-1, 1]; %Plot limits %Test simulation parameters: Nsim = 5; -Tsim = 1; +Tsim = 2; plot_results = true; -X0_sim = 0.8*(2*pi*rand(n,Nsim)-pi); -Xf_sim = 0.2*(2*pi*rand(n,Nsim)-pi); +q0_sim = [1+rand(1,Nsim);... + 0.2*rand(1,Nsim)-0.1;... + zeros(1,Nsim);... + zeros(1,Nsim)]; +qf_sim = zeros(n,Nsim); %% ************************** Data Collection ***************************** % Calculate nominal model with Jacobian and find nominal control gains: -[A_nom, B_nom, C_nom, K_nom] = find_nominal_model_ctrl(n,m,f_u,Q,R,[0; 0]); %Linearization around 0. +[A_nom, B_nom, C_nom, ~] = find_nominal_model_ctrl(n,m,f_u,Q,R,zeros(4,1)); %Linearization around 0. +%Pole placement controller: +p = [-10 -6 -3 -2]; +K_nom = -place(A_nom,B_nom,p); + +% Generate trajectories for all initial and terminal points +disp('Starting trajectory generation...'); tic +t = tspan(1) : deltaT : tspan(2); Qf = 1e3*eye(n); % Time vector and penalty for final state deviations +[q_traj, ~] = generate_trajectory(A_nom, B_nom, C_nom,Qf,q0,qf,t,deltaT,lb,ub); +fprintf('Trajectory generation done, execution time: %1.2f s \n', toc); % Collect data to learn autonomous dynamics: disp('Starting data collection...'); tic U_perturb = 0.5*randn(Ntime+1,Ntraj); %Add normally distributed noise to nominal controller [Xstr, Ustr, timestr] = collect_data(n,m,Ntraj,... - Ntime,deltaT,X0, Xf ,K_nom,f_u,U_perturb); + Ntime,deltaT,q0, qf ,K_nom,f_u,U_perturb,q_traj); fprintf('Data collection done, execution time: %1.2f s \n', toc); @@ -74,23 +98,29 @@ disp('Starting EDMD...'); tic [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m, Ntraj, Ntime, N_basis_edmd,basis_function_edmd,... - rbf_type_edmd, center_type_edmd, eps_rbf_edmd, Xstr, Ustr, Xf, deltaT); + rbf_type_edmd, center_type_edmd, eps_rbf_edmd, Xstr, Ustr, qf, deltaT); fprintf('EDMD done, execution time: %1.2f s \n', toc); disp('Starting KEEDMD...'); tic [A_koop, B_koop, C_koop, phi_fun_v] = KEEDMD(n,m,Ntraj, Ntime, N_basis_diff,... - pow_eig_pairs, Xstr, Ustr, Xf, A_nom, B_nom, K_nom, deltaT); + pow_eig_pairs, Xstr, Ustr, qf, A_nom, B_nom, K_nom, deltaT); fprintf('KEEDMD done, execution time: %1.2f s \n', toc); %% ************************ Analysis of Results *************************** + +t = 0 : deltaT : Tsim; Qf = 1e3*eye(n); % Time vector and penalty for final state deviations +[q_traj_sim,~] = generate_trajectory(A_nom,B_nom,C_nom,Qf,q0_sim,qf_sim,t,deltaT,lb,ub); +[q_traj_track,~] = generate_trajectory(A_nom,B_nom,C_nom,Qf,[2 0 0 0]',[0 0 0 0]',t,deltaT,lb,ub); +q_traj_track = reshape(q_traj_track,size(q_traj_track,1),size(q_traj_track,3)); + disp('Starting analysis of results...'); tic [mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std,... cost_nom, cost_edmd, cost_koop] = evaluate_model_performance(A_nom, B_nom, C_nom, K_nom, ... - A_edmd, B_edmd, C_edmd, A_koop, B_koop, C_koop, Nsim, X0_sim, Xf_sim, ... + A_edmd, B_edmd, C_edmd, A_koop, B_koop, C_koop, Nsim, q_traj_sim, q_traj_track, ... Ntraj, Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results); fprintf('Analysis done, execution time: %1.2f s \n', toc); diff --git a/scripts/Koopman-ID/test_trajectory_gen.m b/scripts/Koopman-ID/test_trajectory_gen.m new file mode 100644 index 0000000..774b11e --- /dev/null +++ b/scripts/Koopman-ID/test_trajectory_gen.m @@ -0,0 +1,85 @@ +clear all; close all; clc; +%% **************** Model and simulation parameters *********************** +%Dynamics: +M = .5; %Cart mass +m = 0.2; %Pendulum mass +g = 9.81; %Gravity +l = 0.4; %Pole length +f_th = 0.1; %Friction coefficient between pendulum and cart + +% f_u = @(t,q,u) [q(2);... +% (-m*g*sin(q(3))*cos(q(3))+m*l*q(4)^2*sin(q(3))+f_th*m*q(4)*cos(q(3)) + u)/(M+(1-cos(q(3))^2)*m);... +% q(4);... +% ((M+m)*(g*sin(q(3))-f_th*q(4))-(l*m*q(4)^2*sin(q(3)) + u)*cos(q(3)))/(l*(M+(1-cos(q(3))^2)*m))]; + +f_u = @(t,q,u) [q(3,:);... + q(4,:);... + (-m*g*sin(q(2,:)).*cos(q(2,:))+m*l*q(4,:).^2.*sin(q(2,:))+f_th*m*q(4,:).*cos(q(2,:)) + u)./(M+(1-cos(q(2,:)).^2)*m);... + ((M+m)*(g*sin(q(2,:))-f_th*q(2,:))-(l*m*q(4,:).^2.*sin(q(2,:)) + u).*cos(q(2,:)))./(l*(M+(1-cos(q(2,:)).^2)*m))]; + + +n = 4; m = 1; %Number of states and control inputs + +%Simulation parameters: +tSpan = [0 2]; +deltaT = 0.01; %Time step length of simulation +q0 = [2 0 0 0]'; +qf = [0 0 0 0]'; + +%Linearize dynamics around x0 +Q = 20*eye(n); R = eye(m); +[A_nom, B_nom, C_nom, ~] = find_nominal_model_ctrl(n, m, f_u, Q, R, qf); + +%Pole placement controller: +p = [-10 -6 -3 -2]; +K_nom = -place(A_nom,B_nom,p); + +%% Test trajectory generator based on linearized model +N = n; %Lifting dimension +t = tSpan(1) : deltaT : tSpan(end); +N_sim = length(t); +R_traj = eye(m); +Qf = 1000*eye(n); +ub = [3 pi/3 10 10]'; +lb = -ub; + +[z_t,u_t] = generate_trajectory(A_nom,B_nom,C_nom,Qf,q0,qf,t,deltaT,lb,ub); + +afigure +for i = 1 : n + subplot(2,2,i); + plot(t,z_t(i,:)); +end + +afigure +plot(t(1:end-1),u_t(1,:)) + + +%% Test LQR controller +u_lqr = zeros(m,N_sim-1); +q_lqr = zeros(n,N_sim); +q_lqr(:,1) = q0; + +u_lqr_t = zeros(m,N_sim-1); +q_lqr_t = zeros(n,N_sim); +q_lqr_t(:,1) = q0; + +for i = 1 : N_sim-1 + u_lqr_t(:,i) = K_nom*(q_lqr_t(:,i)-z_t(:,i)); + q_lqr_t(:,i+1) = sim_timestep(deltaT,f_u,0,q_lqr_t(:,i),u_lqr_t(:,i)); +% u_lqr(:,i) = K_nom*(q_lqr(:,i)-qf); +% q_lqr(:,i+1) = sim_timestep(deltaT,f_u,0,q_lqr(:,i),u_lqr(:,i)); +end + +afigure +for i = 1 : n + subplot(2,2,i); + plot(t,q_lqr_t(i,:),t,q_lqr(i,:),t,z_t(i,:)); + legend('lqr traj', 'lqr','traj') +end + +afigure +plot(t(1:end-1),u_lqr_t(1,:),t(1:end-1),u_lqr(1,:),t(1:end-1),u_t(1,:)) +legend('lqr traj', 'lqr','traj') + + \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/collect_data.m b/scripts/Koopman-ID/utils/collect_data.m index da3f332..5692e99 100644 --- a/scripts/Koopman-ID/utils/collect_data.m +++ b/scripts/Koopman-ID/utils/collect_data.m @@ -1,5 +1,5 @@ function [Xstr, Ustr, time_str] = collect_data(n,m,Ntraj,... - Ntime,deltaT,X0,Xf,K_nom,f_u,U_perturb) + Ntime,deltaT,X0,Xf,K_nom,f_u,U_perturb,qtraj) %Collect data from the "true" system with nominal controller plus %perturbation %Inputs: @@ -14,14 +14,14 @@ % point from each trajectory - Xstr = zeros(n,Ntraj,Ntime+1); % *str is structure - Ustr = zeros(m,Ntraj,Ntime); - time_str = zeros(Ntraj,Ntime+1); + Xstr = zeros(n,Ntraj,Ntime); % *str is structure + Ustr = zeros(m,Ntraj,Ntime-1); + time_str = zeros(Ntraj,Ntime); Xcurrent = X0; Xstr(:,:,1) = X0; - for i = 1:Ntime - Ucurrent = K_nom*(Xcurrent-Xf)+U_perturb(i,:); + for i = 1:Ntime-1 + Ucurrent = K_nom*(Xcurrent-qtraj(:,:,i))+U_perturb(i,:); Ustr(:,:,i) = Ucurrent; Xnext = sim_timestep(deltaT, f_u, 0, Xcurrent, Ucurrent); @@ -30,6 +30,6 @@ time_str(:,i+1) = i*deltaT*ones(Ntraj,1); end - Ucurrent = K_nom*(Xcurrent-Xf)+U_perturb(i+1,:); + Ucurrent = K_nom*(Xcurrent-qtraj(:,:,i+1))+U_perturb(i+1,:); Ustr(:,:,i+1) = Ucurrent; end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/evaluate_model_performance.m b/scripts/Koopman-ID/utils/evaluate_model_performance.m index 29de7d9..6645eaf 100644 --- a/scripts/Koopman-ID/utils/evaluate_model_performance.m +++ b/scripts/Koopman-ID/utils/evaluate_model_performance.m @@ -1,6 +1,6 @@ function [mse_origin_avg, mse_edmd_avg, mse_koop_avg, mse_origin_std, mse_edmd_std, mse_koop_std,... cost_nom, cost_edmd, cost_koop] = evaluate_model_performance(A_nom, B_nom, C_nom, K_nom, ... - A_edmd, B_edmd, C_edmd, A_koop, B_koop, C_koop, Nsim, X0, Xf, Ntraj, ... + A_edmd, B_edmd, C_edmd, A_koop, B_koop, C_koop, Nsim, q_traj_sim, q_traj_track, Ntraj, ... Xstr, Tsim, deltaT, f_u, liftFun, phi_fun_v, plot_results) n = size(C_koop,1); @@ -12,7 +12,7 @@ %Prediction task: [x,x_origin, x_edmd,x_koop, mse_origin_avg, mse_edmd_avg,mse_koop_avg,... mse_origin_std, mse_edmd_std,mse_koop_std]... - = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,Xf,f_u,liftFun,... + = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,q_traj_sim,f_u,liftFun,... phi_fun_v,A_nom,B_nom,C_nom,K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop); %Closed loop task: @@ -44,16 +44,17 @@ Q = q(i)*eye(n); R = r(i)*eye(m); [x_track{i},x_edmd_track{i},x_koop_track{i},mse_nom_track{i}, mse_edmd_track{i},... - mse_koop_track{i},t_plot,traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... + mse_koop_track{i},traj_d, E_nom{i}, E_edmd{i}, E_koop{i},... cost_nom{i}, cost_edmd{i}, cost_koop{i}] = ... sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,... Ntime,deltaT,Tsim,f_u,liftFun,phi_fun_v,A_nom_d,B_nom_d,C_nom_d,K_nom,A_edmd_d,B_edmd_d,... - C_edmd_d,A_koop_d,B_koop_d,C_koop_d,Q,R); + C_edmd_d,A_koop_d,B_koop_d,C_koop_d,Q,R,q_traj_track); end %Plot results if plot_results lw=2; + t_plot = 0 : deltaT : Tsim; font_size = 16; set(gcf,'units','pixel') @@ -80,49 +81,26 @@ h2 = subplot(3,2,3); hold on for i = 1 %Only plot first trajectory - plot([0:Ntime]*deltaT,reshape(x(2,i,:),Ntime+1,1),':b','linewidth',lw); hold on - plot([0:Ntime]*deltaT,reshape(x_origin(2,i,:),Ntime+1,1), 'k','linewidth',lw) - plot([0:Ntime]*deltaT,reshape(x_edmd(2,i,:),Ntime+1,1), 'r','linewidth',lw) - plot([0:Ntime]*deltaT,reshape(x_koop(2,i,:),Ntime+1,1), 'g','linewidth',lw) + plot([0:Ntime]*deltaT,reshape(x(3,i,:),Ntime+1,1),':b','linewidth',lw); hold on + plot([0:Ntime]*deltaT,reshape(x_origin(3,i,:),Ntime+1,1), 'k','linewidth',lw) + plot([0:Ntime]*deltaT,reshape(x_edmd(3,i,:),Ntime+1,1), 'r','linewidth',lw) + plot([0:Ntime]*deltaT,reshape(x_koop(3,i,:),Ntime+1,1), 'g','linewidth',lw) end - axis([0 Tsim min(x(2,1,:))-0.2 max(x(2,1,:))+0.2]) + axis([0 Tsim min(x(3,1,:))-0.2 max(x(3,1,:))+0.2]) %title('Prediction comparison, 1st trajectory - $x_2$','interpreter','latex'); xlabel('Time [s]','interpreter','latex'); - ylabel('$x_2$','interpreter','latex'); + ylabel('$x_3$','interpreter','latex'); set(gca,'fontsize',font_size) %LEG = legend('True','EDMD','Koopman e-func','location','southwest'); set(LEG,'interpreter','latex') %h2.Position = h2.Position + [0 0.1 0 -0.1]; %Modify size of 1st subplot - % Plot trajectories Learning -% h3 = subplot(2,2,3); -% hold on -% for i=1:Ntraj -% scatter(Xstr(1,i,1),Xstr(2,i,1),'+') -% p(i) = plot(squeeze(Xstr(1,i,:)),squeeze(Xstr(2,i,:)),':k','linewidth',1); -% %alpha(p(i),0.2) -% end -% for i = 1 : Nsim -% plot(reshape(x(1,i,:),Ntime+1,1),reshape(x(2,i,:),Ntime+1,1),'-k') -% plot(reshape(x_edmd(1,i,:),Ntime+1,1),reshape(x_edmd(2,i,:),Ntime+1,1),'-r') -% plot(reshape(x_koop(1,i,:),Ntime+1,1),reshape(x_koop(2,i,:),Ntime+1,1),'-g') -% end -% %scatter(cent(1,:),cent(2,:),'o') -% %axis equal -% xaxis([-5 5]) -% yaxis([-5 5]) -% xlabel('x') -% xlabel('y') -% %legend('True','EDMD','Koopman e-func') -% title('Training and simulated trajectories','interpreter','latex'); -% h3.Position = h3.Position + [0 0 0 0.125]; %Modify size of 3rd subplot - %Closed loop plot: - h4 = subplot(3,2,[2, 4]); + h4 = subplot(3,2,2); plot(t_plot,traj_d(1,:),'-b','linewidth',lw); hold on - lin_typ = {'-', '--', '-.'}; - for i = 1 : length(q) + lin_typ = {'--', '-', '-.'}; + for i = 2% : length(q) plot(t_plot,x_track{i}(1,:),strcat(lin_typ{i},'k'),'linewidth',lw); hold on plot(t_plot,x_edmd_track{i}(1,:),strcat(lin_typ{i},'r'),'linewidth',lw); hold on plot(t_plot,x_koop_track{i}(1,:),strcat(lin_typ{i},'g'),'linewidth',lw); hold on @@ -132,12 +110,31 @@ xlabel('Time [s]','interpreter','latex'); ylabel('$x_1$','interpreter','latex'); set(gca,'fontsize',font_size) - LEG = legend({'Reference', ... - strcat('Lin at 0$_{q/r=',num2str(q(1)/r(1)),'}$'),strcat('EDMD$_{q/r=',num2str(q(1)/r(1)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(1)/r(1)),'}$'),... - strcat('Lin at 0$_{q/r=',num2str(q(2)/r(2)),'}$'),strcat('EDMD$_{q/r=',num2str(q(2)/r(2)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(2)/r(2)),'}$'),... - strcat('Lin at 0$_{q/r=',num2str(q(3)/r(3)),'}$'),strcat('EDMD$_{q/r=',num2str(q(3)/r(3)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(3)/r(3)),'}$')},... - 'location','southeast','NumColumns',2); - set(LEG,'Interpreter','latex') +% LEG = legend({'Reference', ... +% strcat('Lin at 0$_{q/r=',num2str(q(1)/r(1)),'}$'),strcat('EDMD$_{q/r=',num2str(q(1)/r(1)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(1)/r(1)),'}$'),... +% strcat('Lin at 0$_{q/r=',num2str(q(2)/r(2)),'}$'),strcat('EDMD$_{q/r=',num2str(q(2)/r(2)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(2)/r(2)),'}$'),... +% strcat('Lin at 0$_{q/r=',num2str(q(3)/r(3)),'}$'),strcat('EDMD$_{q/r=',num2str(q(3)/r(3)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(3)/r(3)),'}$')},... +% 'location','southeast','NumColumns',2); +% set(LEG,'Interpreter','latex') + %h4.Position = h4.Position + [0 0 0 0.125]; %Modify size of 4th subplot + + h4 = subplot(3,2,4); + plot(t_plot,traj_d(3,:),'-b','linewidth',lw); hold on + for i = 2% : length(q) + plot(t_plot,x_track{i}(3,:),strcat(lin_typ{i},'k'),'linewidth',lw); hold on + plot(t_plot,x_edmd_track{i}(3,:),strcat(lin_typ{i},'r'),'linewidth',lw); hold on + plot(t_plot,x_koop_track{i}(3,:),strcat(lin_typ{i},'g'),'linewidth',lw); hold on + end + %axis([0 Tsim min(x_edmd(1,1,:))-0.15 max(x_edmd(1,1,:))+0.15]) + xlabel('Time [s]','interpreter','latex'); + ylabel('$x_3$','interpreter','latex'); + set(gca,'fontsize',font_size) +% LEG = legend({'Reference', ... +% strcat('Lin at 0$_{q/r=',num2str(q(1)/r(1)),'}$'),strcat('EDMD$_{q/r=',num2str(q(1)/r(1)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(1)/r(1)),'}$'),... +% strcat('Lin at 0$_{q/r=',num2str(q(2)/r(2)),'}$'),strcat('EDMD$_{q/r=',num2str(q(2)/r(2)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(2)/r(2)),'}$'),... +% strcat('Lin at 0$_{q/r=',num2str(q(3)/r(3)),'}$'),strcat('EDMD$_{q/r=',num2str(q(3)/r(3)),'}$'),strcat('KEEDMD$_{q/r=',num2str(q(3)/r(3)),'}$')},... +% 'location','southeast','NumColumns',2); +% set(LEG,'Interpreter','latex') %h4.Position = h4.Position + [0 0 0 0.125]; %Modify size of 4th subplot set(gcf, 'Position', [0, 0, 1200, 600]) %Modify size of figure window diff --git a/scripts/Koopman-ID/utils/extendedDMD.m b/scripts/Koopman-ID/utils/extendedDMD.m index 622a136..af916d2 100644 --- a/scripts/Koopman-ID/utils/extendedDMD.m +++ b/scripts/Koopman-ID/utils/extendedDMD.m @@ -1,5 +1,5 @@ function [A_edmd, B_edmd, C_edmd, liftFun] = extendedDMD(n,m,Ntraj, Ntime, N_basis,basis_function,... - rbf_type, center_type, eps, Xstr, Ustr, Xf,deltaT) + rbf_type, center_type, eps, Xstr, Ustr,Xf,deltaT) %Identify lifted state space model using Extended Dynamic Mode %Decomposition @@ -30,10 +30,10 @@ X_dot = []; U = []; for i = 1 : Ntraj - Xstr_shift(:,i,:) = Xstr(:,i,:) - Xf(:,i); - X = [X reshape(Xstr_shift(:,i,:),n,Ntime+1)]; + Xstr_shift(:,i,:) = Xstr(:,i,:)-Xf(:,i); + X = [X reshape(Xstr_shift(:,i,:),size(Xstr_shift,1),size(Xstr_shift,3))]; X_dot = [X_dot num_diff(reshape(Xstr_shift(:,i,:),size(Xstr_shift,1),size(Xstr_shift,3)),deltaT)]; - U = [U reshape(Ustr(:,i,:),m,Ntime+1)]; + U = [U reshape(Ustr(:,i,:),size(Ustr,1),size(Ustr,3))]; end % ******************************* Lift *********************************** diff --git a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m index 07f5f44..d592132 100644 --- a/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m +++ b/scripts/Koopman-ID/utils/sim_closed_loop_mpc.m @@ -1,26 +1,11 @@ -function [x_nom,x_edmd,x_koop,mse_nom,mse_edmd,mse_koop, t_plot, traj_d, E_nom, E_edmd, E_koop,... +function [x_nom,x_edmd,x_koop,mse_nom,mse_edmd,mse_koop, q_traj, E_nom, E_edmd, E_koop,... cost_nom, cost_edmd, cost_koop]... = sim_closed_loop_mpc(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,Tsim,... f_u,liftFun,phi_fun_v,A_nom,B_nom,C_nom,K_nom,A_edmd,B_edmd,C_edmd,A_koop,... - B_koop,C_koop,Q,R) - - %Set up trajectory to track: - t = 0 : deltaT : Tsim; - t1 = t(1:floor(Ntime/3)); - t2 = t(floor(Ntime/3):floor(2*Ntime/3)); - t3 = t(floor(2*Ntime/3):end); - t_plot = [t1 t2 t3]; - x0 = -pi/2; %Start angle - x1 = 0.2; %Waypoint angle - x2 = -0.2; %Waypoint angle - x3 = pi/3; %Final angle - traj_d = [spline([t1(1) t1(end)], [0 [x0 x1] 0], t1) ... - spline([t2(1) t2(end)], [0 [x1 x2] 0], t2) ... - spline([t3(1) t3(end)], [0 [x2 x3] 0], t3)]; - traj_d = [traj_d; diff([traj_d traj_d(end)])/deltaT]; + B_koop,C_koop,Q,R,q_traj) %Initialize variables: - Ntime_track = size(traj_d,2)-1; + Ntime_track = size(q_traj,2)-1; x_nom = zeros(n,Ntime_track+1); u_nom = zeros(m,Ntime_track); z_edmd = zeros(n_edmd,Ntime_track+1); @@ -29,16 +14,16 @@ z_koop = zeros(n_koop,Ntime_track+1); x_koop = zeros(n,Ntime_track+1); u_koop = zeros(m,Ntime_track); - x_nom(:,1) = traj_d(:,1); - x_edmd(:,1) = traj_d(:,1); - x_koop(:,1) = traj_d(:,1); - z_edmd(:,1) = liftFun(traj_d(:,1)); - z_koop(:,1) = phi_fun_v(traj_d(:,1)); + x_nom(:,1) = q_traj(:,1); + x_edmd(:,1) = q_traj(:,1); + x_koop(:,1) = q_traj(:,1); + z_edmd(:,1) = liftFun(q_traj(:,1)); + z_koop(:,1) = phi_fun_v(q_traj(:,1)); % Define MPC parameters: Tpred = Tsim;%0.1; % Prediction horizon Np = round(Tpred / deltaT); - traj_d = [traj_d traj_d(:,end).*ones(n,Np)]; %Add extra points to trajectory to allow prediction horizon + q_traj = [q_traj q_traj(:,end).*ones(n,Np)]; %Add extra points to trajectory to allow prediction horizon xlift_min = [];%[ymin ; nan(Nlift-1,1)]; % Constraints xlift_max = [];%[ymax ; nan(Nlift-1,1)]; @@ -54,7 +39,7 @@ end % Prediction horizon of reference signal - yr = traj_d(:,i:i+Np-1); + yr = q_traj(:,i:i+Np-1); % Local linearization MPC u_nom(:,i) = MPC_nom(x_nom(:,i),yr); @@ -72,14 +57,14 @@ end % Calculate corresponding predictions and MSE - traj_d = traj_d(:,1:end-Np); %Remove added elements from trajectory - mse_nom = immse(traj_d,x_nom); - mse_edmd = immse(traj_d,x_edmd); - mse_koop = immse(traj_d,x_koop); + q_traj = q_traj(:,1:end-Np); %Remove added elements from trajectory + mse_nom = immse(q_traj,x_nom); + mse_edmd = immse(q_traj,x_edmd); + mse_koop = immse(q_traj,x_koop); E_nom = norm(u_nom); E_edmd = norm(u_edmd); E_koop = norm(u_koop); - cost_nom = sum(diag((x_nom(:,2:end)-traj_d(:,2:end))'*Q*(x_nom(:,2:end)-traj_d(:,2:end)))) + sum(diag(u_nom'*R*u_nom)); - cost_edmd = sum(diag((x_edmd(:,2:end)-traj_d(:,2:end))'*Q*(x_edmd(:,2:end)-traj_d(:,2:end)))) + sum(diag(u_edmd'*R*u_edmd)); - cost_koop = sum(diag((x_koop(:,2:end)-traj_d(:,2:end))'*Q*(x_koop(:,2:end)-traj_d(:,2:end)))) + sum(diag(u_koop'*R*u_koop)); + cost_nom = sum(diag((x_nom(:,2:end)-q_traj(:,2:end))'*Q*(x_nom(:,2:end)-q_traj(:,2:end)))) + sum(diag(u_nom'*R*u_nom)); + cost_edmd = sum(diag((x_edmd(:,2:end)-q_traj(:,2:end))'*Q*(x_edmd(:,2:end)-q_traj(:,2:end)))) + sum(diag(u_edmd'*R*u_edmd)); + cost_koop = sum(diag((x_koop(:,2:end)-q_traj(:,2:end))'*Q*(x_koop(:,2:end)-q_traj(:,2:end)))) + sum(diag(u_koop'*R*u_koop)); end \ No newline at end of file diff --git a/scripts/Koopman-ID/utils/sim_prediction.m b/scripts/Koopman-ID/utils/sim_prediction.m index 35bb077..6471961 100644 --- a/scripts/Koopman-ID/utils/sim_prediction.m +++ b/scripts/Koopman-ID/utils/sim_prediction.m @@ -1,6 +1,6 @@ function [x,x_origin, x_edmd,x_koop, mse_origin_avg, mse_edmd_avg,mse_koop_avg,... mse_origin_std, mse_edmd_std,mse_koop_std]... - = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,X0,Xf,f_u,liftFun,... + = sim_prediction(n,m,n_edmd,n_koop,Nsim,Ntime,deltaT,q_traj,f_u,liftFun,... phi_fun_v,A_nom,B_nom,C_nom,K_nom,A_edmd,B_edmd,C_edmd,A_koop,B_koop,C_koop) u = zeros(m,Nsim,Ntime); @@ -15,6 +15,7 @@ z_edmd = zeros(n_edmd,Nsim,Ntime+1); z_koop = zeros(n_koop,Nsim,Ntime+1); + X0 = q_traj(:,:,1); x(:,:,1) = X0; x_origin(:,:,1) = X0; z_edmd(:,:,1) = liftFun(X0); @@ -23,7 +24,7 @@ % Simulate all systems and initial points for i = 1:Ntime %Controller: - u(:,:,i) = K_nom*(x(:,:,i)-Xf) + 5*rand()-2.5; + u(:,:,i) = K_nom*(x(:,:,i)-q_traj(:,:,i)) + 5*rand()-2.5; %True dynamics: x(:,:,i+1) = sim_timestep(deltaT,f_u,0,x(:,:,i), u(:,:,i));