Balanced Model Reduction

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Evelyn Johnson
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1 1 Balanced Realization Balanced Model Reduction CEE 629. System Identification Duke University, Fall 17 A balanced realization is a realization for which the controllability gramian Q and the observability gramian P both equal a diagonal matrix of the Hankel singular values, Σ. In other words, the state covariance matrix of pulse or white-noise responses equals the output covariance matrix of free output responses, and they are both diagonal matrices. To transform a realization into a balanced form, we require the n n coordinate transformation matrix T from balanced coordinates x b to the given coordinates x (x = T x b ) such that the observability and controllability gramians are diagonal and equal. Substituting T A b T 1 = A, T T A T b T T = A T, C b T 1 = C, and T B b = B into Lyapunov equations, 0 = A T P A P + C T C and 0 = AQA T Q + BB T, (1) and relating the resulting Lyapunov equations to 0 = A T b ΣA b Σ + C T b C b and 0 = A b ΣA T b Σ + B b B T b, (2) we find that the condition for the coordinate transformation into balanced form is T T P T = T 1 QT T = Σ = diag(σ 1,, σ n ), σ 1 σ 2 σ n > 0. (3) A sketch of the solution of equation (3) for T follows. (T T P T )(T 1 QT T ) = Σ 2 T T P (T T 1 )QT T = Σ 2 T T P QT T = Σ 2 T T (R 1 R)P (R T R)T T = Σ 2 T T R 1 (RP R T )RT T = Σ 2 (RT T ) 1 (RP R T )(RT T ) = Σ 2 (RT T ) 1 (UΣ 2 U T )(RT T ) = Σ 2 R T R = Q... the Cholesky decomp of Q RP R T = UΣ 2 U T... the eigenvalue decomp of RP R T V Σ 2 V 1 = Σ 2... define V V = (RT T ) 1 U... define V V = T T R 1 U V 1 = U T RT T V U T = T T R 1... solve for T in terms of V V U T R = T T R T UV = T (4) 1

2 2 CEE 629 ME System Identification Duke University Fall 17 H.P. Gavin To determine V, So, T 1 QT T = Σ (V 1 U T R T ) Q (R 1 UV 1 ) = Σ... substitue T in terms of V V 1 U T R T R T RR 1 UV 1 = Σ... substitute R T R = Q V 1 U T (R T R T )(RR 1 )UV 1 = Σ V 1 (U T U)V 1 = Σ V 1 V 1 = Σ V 2 = Σ V = Σ 1/2 To compute this coordinate transformation matrix, T = R T UΣ 1/2 (5) T 1 = Σ 1/2 U T R T (6) 1. Solve the Lyapunov equations for the controllability gramian Q and the observability gramian P. 2. Compute the Cholesky decomposition of Q, R T R = Q. 3. Compute the eigenvalue decomposition of RP R T = UΣU T. This gives the Hankel singular values, Σ. 4. Evaluate equations (5) and (6) In a balanced model the first state x 1 is the most observable and the most controllable. The last state x n is the least so. The reduced balanced realization simply retains the first r states. A r = A b (1 : r, 1 : r); B r = B b (1 : r, :); C r = C b (:, 1 : r). 2 Example system in complex modal coordinates linear dynamics: system output: parameter values: ẋ = Ax + Bu y = Cx ([ pi A = diag p i. b B = i b i. C = D = 0 ]) c j c j A C n n B C n m p i = σ i + 1ω i b i = i + i 1 C C l n c j = 1 + j 1 D C l m

3 Balanced Model Reduction Lightly Damped Dynamics σ i = ω i /100 natural frequencies, Hz damping ratio number of states frequency response magnitude frequency, f, Hz

4 4 CEE 629 ME System Identification Duke University Fall 17 H.P. Gavin 2.2 Heavily Damped Dynamics σ i = ω i /10 frequency response magnitude frequency, f, Hz natural frequencies, Hz damping ratio number of states

5 Balanced Model Reduction 5 bal real.m 1 function [ Ab, Bb, Cb, G, T, Ti ] = bal_real ( A, B, C ) 2 % [ Ab, Bb, Cb, G, T, Ti ] = b a l r e a l ( A, B, C ) 3 % balanced r e a l i z a t i o n f o r a continuous time system 4 % 5 % i n p u t A,B,C a continuous time s t a t e space r e a l i z a t i o n 6 % 7 % output Ab, Bb, Cb t h e balanced continuous time s t a t e space r e a l i z a t i o n 8 % G t h e balanced o b s e r v a b i l i t y and c o n t r o l l a b i l i t y gramians 9 % T c o o r d i n a t e t r a n s f o r m a t i o n matrix 10 % Ti i n v e r s e o f c o o r d i n a t e t r a n s f o r m a t i o n matrix P = liap ( A, C ); % s o l v e l e f t Liapunov eq n A P + P A + C C = 0 13 Q = liap ( A, B ); % s o l v e r i g h t Liapunov eq n A Q + Q A + B B = 0 14 R = chol (Q); % r i g h t Cholesky f a c t o r o f Q... R R = Q [U,G2] = eig (R*P*R ); % d i a g o n a l i z a t i o n o f R P R % t h e balanced o b s e r v a b i l i t y and c o n t r o l l a b i l i t y gramian 19 G = sqrt ( real (diag(g2 ))); 21 % t h e c o o r d i n a t e t r a n s f o r m a t i o n matrix T = R *U * diag (1./ sqrt (G )); % t h e i n v e r s e o f t h e c o o r d i n a t e t r a n s f o r m a t i o n matrix Ti = diag( sqrt (G)) * U * inv(r ); % s o r t e d grammian [G,idx ] = sort (G, descend ); % determine t h e s t a t e re s o r t i n g matrix, S X sorted = S X 31 n = length( A); 32 S = zeros (n); 33 for i =1: n 34 S(i, idx ( i)) = 1; 35 end 36 Ti = S*Ti; T = T/S; % r e o r g a n i z e t h e t r a n s f o r m a t i o n matrices Ab = Ti* A* T; % balanced dynamics matrix 39 Bb = Ti*B; % balanced i n p u t matrix 40 Cb = C* T; % balanced output matrix % H. P. Gavin

6 6 CEE 629 ME System Identification Duke University Fall 17 H.P. Gavin balreal test.m 1 % b a l r e a l t e s t.m t e s t balanced model r e d u c t i o n 2 3 % S p e c i f y an LTI system in modal c o o r d i n a t e s 4 r = 1; % number o f i n p u t s 5 m = 1; % number o f o u t p u t s 6 n = ; % number o f i n t e r n a l s t a t e s ( even )... complex c o n j u g a t e p o l e s 7 8 % g e n e r a t e a random LTI system with under damped dynamics in d i a g o n a l i z e d form 9 10 z = 0.1; % H ea v il y Damped 11 z = 0.01; % L i g h t l y Damped wn = 2* pi * logspace (0.2,1.8,10); [A,B,C,D] = modallti (wn,z,m,r, unif ); damp (A) % check t h e nat l f r e q u e n c i e s and damping r a t i o s [ Ab, Bb, Cb, G, T, Ti] = bal_real ( A, B, C ); 21 w = 2* pi * logspace (0,2,500); 22 nw = length( w); 23 mag = zeros (nw,n /4); 24 pha = zeros (nw,n /4); eig_vals = NaN(n, n /4+1); 27 for iter = 1: n / r = n -2* iter +2; % remaining number o f s t a t e s Ar = Ab (1:r,1: r); Br = Bb (1:r,:); Cr = Cb (:,1: r); eig_vals (1:r, iter ) = eig (Ar ); [ mag (:, iter ), pha (:, iter )] = mybode (Ar,Br,Cr,D,1,w); end % P l o t s figure (1) 42 c l f 43 loglog (w /2/ pi, mag ) 44 legend(, 18, 16, 14, 12, 10,3); 45 legend( location, southwest ) 46 xlabel ( frequency, f, Hz ) 47 ylabel ( frequency response magnitude ) figure (2) 50 c l f 51 subplot (211) 52 semilogy(n -[0:2: n/2], abs( eig_vals )/2/ pi, ob, MarkerSize,5); 53 axis ([ n/2-1 n e2 ]) 54 ylabel ( natural frequencies, Hz ) 55 subplot (212) 56 plot (n -[0:2: n/2], - real ( eig_vals )./abs( eig_vals +eps), ob, MarkerSize,5); 57 % a x i s ( [ n/2 1 n max(max( r e a l ( e i g v a l s ). / abs ( e i g v a l s+eps ) ) ) ] ) 58 xlabel ( number of states ) 59 ylabel ( damping ratio ) % b a l r e a l t e s t

Principal Input and Output Directions and Hankel Singular Values

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