Model Reduction for Unstable Systems

Model Reduction for Unstable Systems Klajdi Sinani Virginia Tech klajdi@vt.edu Advisor: Serkan Gugercin October 22, 2015 (VT) SIAM October 22, 2015 1 / 26

Overview 1 Introduction 2 Interpolatory Model Reduction 3 Balanced Truncation 4 Rational Krylov Methods for Optimal L 2 Model Reduction 5 Balanced Truncation for Unstable Systems 6 IRKA for Unstable Systems 7 Conclusions and Future Work (VT) SIAM October 22, 2015 2 / 26

Model Reduction for Stable Linear Dynamical Systems Consider the linear dynamical system: Eẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) with x(0) = 0 where A, E R n n, B R n m, C R p n, and D R p m are constant matrices. In this presentation we assume m = p = 1. A dynamical system is asymptotically stable or stable if all of its poles lie to the left of the imaginary axis. For large n e.g. n > 10 6, the simulation is very expensive. (VT) SIAM October 22, 2015 3 / 26

Model Reduction for Linear Dynamical Systems Use model reduction to replace the original model with a lower dimension model: E r ẋ r (t) = A r x r (t) + B r u(t) y r (t) = C r x r (t) + D r u(t) with x r (0) = 0 where A r, E r R r r, B r R r m, C r R p r, and D r R p m with r << n. The outputs of the reduced system approximate the true outputs. The simulation of the reduced order model is cheaper. (VT) SIAM October 22, 2015 4 / 26

Transfer Function in the Frequency Domain Obtain the frequency domain representation of the model by computing Laplace transforms of y(t), y r (t) and u(t). The functions H(s) = C(sE A) 1 B + D H r (s) = C r (se r A r ) 1 B r + D r are the transfer functions associated with the full and reduced model respectively. (VT) SIAM October 22, 2015 5 / 26

H 2 and H Norms The H 2 norm is defined as H(s) H2 := ( 1 ) 1/2 H(iω) 2 F 2π where F represents the Frobenius norm. The H norm is defined as H H = sup H(iω) 2 ω R where 2 denotes the 2-norm of a matrix. (VT) SIAM October 22, 2015 6 / 26

Error Measures We aim to minimize the error between the full and the reduced model: (VT) SIAM October 22, 2015 7 / 26

Error Measures We aim to minimize the error between the full and the reduced model: y y r L H H r H2 u L2 (VT) SIAM October 22, 2015 7 / 26

Error Measures We aim to minimize the error between the full and the reduced model: y y r L H H r H2 u L2 y y r L2 H H r H u L2. (VT) SIAM October 22, 2015 7 / 26

Rational Interpolation Given a set of interpolation points {s i } r i C we need to construct H r such that H r (s i ) = H(s i ) for i = 1, 2, 3,..., r. We compute H r via projection in the following manner: Compute the model reduction basis V and W: V = [(s 1 E A) 1 B (s r E A) 1 B] W = [C(s 1 E A) 1 C(s r E A) 1 ] Then, E r = W T EV, A r = W T AV, C r = CV, B r = W T B. H r satisfies Hermite interpolation conditions. (VT) SIAM October 22, 2015 8 / 26

Rational Krylov Methods We want to find H r such that H H r H2 = min dim(ĥ r )=r H Ĥ r H2. Theorem (Gugercin and Beattie, 2014) Let H r (s) be the best r th order rational approximation of H with respect to the H 2 norm. Then H( λ k ) = H r ( λ k ), H ( λ k ) = H r ( λ k ) for k = 1, 2,..., r where λ k denotes the poles of the reduced system. (VT) SIAM October 22, 2015 9 / 26

Iterative Rational Krylov Algorithm (IRKA) Sketch of IRKA Pick an r-fold initial shift set selection closed under conjugation V = [(σ 1 E A) 1 )B... (σ r E A) 1 )B] W = [(σ 1 E A) T ) 1 C T... [(σ r E A) T ) 1 C T ] while (not converged) A r = W T r AV r, E r = W T r EV r, B r = W T r B, and C r = CV r Compute a pole-residue expansion of H r (s): H r (s) = C r (se r A r ) 1 B r = r i=1 σ i λ i, for i = 1,..., r V = [(σ 1 E A) 1 )B... (σ r E A) 1 )B] W T = [(σ 1 E A) T ) 1 C T... [(σ r E A) T ) 1 C T ] φ i s λ i A r = W T r AV r, E r = W T r EV r, B r = W T r B, and C r = CV r (VT) SIAM October 22, 2015 10 / 26

Observability and Reachability Consider: ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) [ ] A B Σ : = C D Balanced Truncation eliminates states which are hard to reach and observe. The reachability gramian P is used to classify hard to reach states. The observability gramian Q is used to classify hard to observe states. P and Q are the solutions of the following Lyapunov equations: AP + PA T + BB T = 0, A T Q + QA + C T C = 0 The values σ i = λ i (PQ) are known as the Hankel singular values of the system. (VT) SIAM October 22, 2015 11 / 26

Balancing Transformation Transform the gramians in order to ensure the states which are difficult to reach are also difficult to observe. (VT) SIAM October 22, 2015 12 / 26

Balancing Transformation Transform the gramians in order to ensure the states which are difficult to reach are also difficult to observe. If A is asymptotically stable, P and Q are symmetric positive semi-definite. Compute the Cholesky factor U of P and the eigendecomposition of U T QU: P = UU T, U T QU = KG 2 K T (VT) SIAM October 22, 2015 12 / 26

Truncation We obtain a balanced system [ Â ˆB ˆΣ = Ĉ ˆD ] (VT) SIAM October 22, 2015 13 / 26

Truncation We obtain a balanced system Consider the partitions: [ ] A11 A Â = 12, ˆB = A 21 A 22 [ Â ˆB ˆΣ = Ĉ ˆD [ B1 B 2 ] ], Ĉ = [ C 1 C 2 ], G = [ G11 0 0 G 22 for G = diag(σ 1,..., σ n ) = P = Q, G 1 = (σ 1,..., σ r ), G 2 = (σ r+1,..., σ N ), where σ i denotes the i-th Hankel singular values of the system for i = 1, 2,..., n. ] (VT) SIAM October 22, 2015 13 / 26

(VT) SIAM 2(σ +... + σ. October 22, 2015 13 / 26 Truncation We obtain a balanced system Consider the partitions: [ ] A11 A Â = 12, ˆB = A 21 A 22 [ Â ˆB ˆΣ = Ĉ ˆD [ B1 B 2 ] ], Ĉ = [ C 1 C 2 ], G = [ G11 0 0 G 22 for G = diag(σ 1,..., σ n ) = P = Q, G 1 = (σ 1,..., σ r ), G 2 = (σ r+1,..., σ N ), where σ i denotes the i-th Hankel singular values of the system for i = 1, 2,..., n. The reduced system of order r is [ ] A11 B Σ r = 1 D C 1 ]

Model Reduction for Unstable Linear Dynamical Systems Consider the linear dynamical system: ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) Suppose the system is unstable. We want to find a reduced model ẋ r (t) = A r x r (t) + B r u(t) y r (t) = C r x r (t) which approximates the full order system with some error measure. (VT) SIAM October 22, 2015 14 / 26

L 2 Systems L n 2 (R) = {x(t) Rn x(t) 2 dt < } Magruder, Beattie and Gugercin, 2010 showed the unstable system can be associated with a bounded map from L 2 (R) to itself. We can reduce the model with respect to the L 2 norm defined by H L2 = ( H(iω) 2 dω) 1/2 (VT) SIAM October 22, 2015 15 / 26

L 2 IRKA Split the original systems into two systems where one is strictly stable and the other is strictly unstable. Negate the unstable system Reduce each model separately using IRKA. Negate the part corresponding to the unstable part. Combine the reduced models. The error is determined component wise. (VT) SIAM October 22, 2015 16 / 26

System Transformation and Separation Let Σ = [ A B C 0 ] be an unstable system with no poles in the imaginary axis. Suppose T is a transformation such that [ ] TAT 1 A TB 1 0 B 1 CT 1 = 0 A 2 B 2 0 C 1 C 2 0 where A 1 is stable and A 2 is antistable. (VT) SIAM October 22, 2015 17 / 26

Observability and Reachability Gramians for Unstable Systems Let P 1, P 2, Q 1, Q 2 0 be solutions to the Lyapunov equations: A 1 P 1 + P 1 A T 1 + B 1 B T 1 = 0, A T 1 Q 1 + Q 1 A 1 + C T 1 C 1 = 0, ( A 2 )P 2 + P 1 ( A 2 ) T + B 2 B T 2 = 0, ( A 2 ) T Q 2 + Q 2 ( A 2 ) + C T 2 C 2 = 0. Zhou et al., 1999 showed P and Q can be computed as [ ] P = T 1 P1 0 T T 0 P 2 Q = T 1 [ Q1 0 0 Q 2 ] T T (VT) SIAM October 22, 2015 18 / 26

Truncation Recall the Hankel singular values are defined as σ i = λ i (PQ). P = Q = diag(σ 1,..., σ n ). Eliminate the states associated with the smallest singular values. (VT) SIAM October 22, 2015 19 / 26

Reduce Unstable Systems via IRKA What if we reduced an unstable system by applying IRKA directly? (VT) SIAM October 22, 2015 20 / 26

Reduce Unstable Systems via IRKA What if we reduced an unstable system by applying IRKA directly? While the algorithm may not converge, under certain conditions, the unstable poles are captured. Often, even if the poles are not captured very accurately, the stability is preserved. (VT) SIAM October 22, 2015 20 / 26

Pole Capture Initial Shifts Final Shifts -1.0000 10 1-9.9999 10 2-1.0000 10 2-9.9999 10 3 5.5949 10 3 2.9678 10 4 5.6076 10 3 1.272 10 2 8.0280 10 3 9.9689 10 2 (VT) SIAM October 22, 2015 21 / 26

Pole Capture Initial Shifts Final Shifts -1.5321 10 3 1.8467 10 2-1.8469 10 2-1.0088 10 2 5.1433 10 2 i -1.0069 10 2-5.1435 10 2 i -1.0088 10 2 5.1433 10 2 i -1.0069 10 2-5.1435 10 2 i -1.5323 10 3 5.5737 10 3 2.8033 10 4 5.5949 10 3 1.2560 10 2 5.6077 10 3 3.9537 10 2-6.7733 10 2 i 8.0280 10 3 3.9537 10 2 + 6.7733 10 2 i (VT) SIAM October 22, 2015 22 / 26

Comparisons of Algorithms r=12 L 2 IRKA L 2 IRKA Initialization Pole reflection Bal. truncation poles H 2 Relative error 0.4885 0.0457 H Relative error 0.5160 0.0041 r=12 IRKAfUS IRKAfUS Initialization Pole reflection Bal. truncation poles H 2 Relative error 0.1449 0.1754 H Relative error 0.0198 0.0423 r=12 Bal. Truncation L 2 IRKA Initialization - IRKAfUS poles H 2 Relative error 0.0717 0.0840 H Relative error 0.0110 0.0072 (VT) SIAM October 22, 2015 23 / 26

Conclusions and Future Work For unstable systems with few unstable poles, numerical simulations show IRKA captures the unstable poles. These obtained poles appear to be good initializations for L 2 IRKA. Investigate delay systems and develop techniques to reduce such systems. Consider second-order models and compare structure preserving algorithms with optimal ones such as IRKA. (VT) SIAM October 22, 2015 24 / 26

References S. Gugercin and C. Beattie (2014) Model Reduction by Rational Interpolation Model Reduction and Approximation for Complex Systems C. magruder, S. Gugercin and C. Beattie (2010) Rational Krylov Methods for Optimal L 2 Model Reduction 49th IEEE Conference on Decision Control S. Gugercin and C. Beattie (2014) Model Reduction by Rational Interpolation Model Reduction and Approximation for Complex Systems K. Zhou, G. Salomon and E. Wu (1999) Balanced Realization and Model Reduction for Unstable Systems Int. J. Robust Nonlinear Control S. Gugercin (2003) Projection Methods for Model Reduction of Large-Scale Dynamical Systems (PhD Thesis) Rice University (VT) SIAM October 22, 2015 25 / 26

Thank you! (VT) SIAM October 22, 2015 26 / 26