On the KalmanYakubovichPopov Lemma and its Application in Model - PowerPoint PPT Presentation

Elgersburg Workshop 2013 February 11–14, 2013 On the Kalman–Yakubovich–Popov Lemma and its Application in Model Order Reduction Peter Benner Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Germany joint work (in parts) with Matthias Voigt and Xin Du, Guanghong Yang, and Dan Ye MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 1/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Overview Linear Systems Basics 1 Dissipativity and Structural Properties 2 The Kalman-Yakubovich-Popov Lemma 3 Model Reduction for LTI Systems 4 Frequency-dependent KYP Lemma and Model Reduction 5 Numerical Examples 6 Conclusions and Future Work 7 Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 2/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Linear Systems Basics 1 Dissipativity and Structural Properties 2 Dissipative Systems Dissipativity in the Frequency Domain The Kalman-Yakubovich-Popov Lemma 3 Model Reduction for LTI Systems 4 Balanced truncation for linear systems Frequency-dependent KYP Lemma and Model Reduction 5 The Frequency-dependent KYP Lemma Frequency-dependent Balanced Truncation Numerical Examples 6 RLC ladder network Butterworth filter Conclusions and Future Work 7 Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 3/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Linear Systems LTI Systems � x ( t ) = Ax ( t ) + Bu ( t ) , ˙ x (0) = x 0 Σ : y ( t ) = Cx ( t ) + Du ( t ) , with A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , D ∈ R p × m , state vector x ( t ) ∈ R n , input vector u ( t ) ∈ R m , output vector y ( t ) ∈ R p . u ( t ) x ( t ) = Ax ( t ) + Bu ( t ) ˙ y ( t ) y ( t ) = Cx ( t ) + Du ( t ) Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 4/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Stability and Controllability Definitions The system Σ is called (asymptotically) stable if lim t →∞ x ( t ) = 0 for u ≡ 0; controllable if for all x 1 ∈ R n there exist t 1 > 0 and an input signal u ( t ) such that x ( t 1 ) = x 1 . observable if y ( t ) ≡ 0 implies x ( t ) ≡ 0 (assuming u ( t ) ≡ 0). Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 5/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Stability and Controllability Definitions The system Σ is called (asymptotically) stable if lim t →∞ x ( t ) = 0 for u ≡ 0; controllable if for all x 1 ∈ R n there exist t 1 > 0 and an input signal u ( t ) such that x ( t 1 ) = x 1 . observable if y ( t ) ≡ 0 implies x ( t ) ≡ 0 (assuming u ( t ) ≡ 0). Equivalent Conditions The system Σ is (asymptotically) stable ⇐ ⇒ all eigenvalues of A are in the open left half-plane; � λ I n − A B � controllable ⇐ ⇒ rank = n for all λ ∈ C . � C T � λ I n − A T observable ⇐ ⇒ rank = n for all λ ∈ C . minimal if it is controllable and observable. Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 5/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Frequency Domain Analysis Laplace transform � ∞ e − st f ( t ) d t L{ f } ( s ) := 0 Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Frequency Domain Analysis Laplace transform � ∞ e − st f ( t ) d t L{ f } ( s ) := 0 Transfer function Assume x (0) = 0. Then � L{ ˙ x } ( s ) = A L{ x } ( s ) + B L{ u } ( s ) , L (Σ) : L{ y } ( s ) = C L{ x } ( s ) + D L{ u } ( s ) , Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Frequency Domain Analysis Laplace transform � ∞ e − st f ( t ) d t L{ f } ( s ) := 0 Transfer function Assume x (0) = 0. Then � s ( L{ x } ( s ) − x (0)) = A L{ x } ( s ) + B L{ u } ( s ) , L (Σ) : L{ y } ( s ) = C L{ x } ( s ) + D L{ u } ( s ) , Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Frequency Domain Analysis Laplace transform � ∞ e − st f ( t ) d t L{ f } ( s ) := 0 Transfer function Assume x (0) = 0. Then � s L{ x } ( s ) = A L{ x } ( s ) + B L{ u } ( s ) , L (Σ) : L{ y } ( s ) = C L{ x } ( s ) + D L{ u } ( s ) , Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Frequency Domain Analysis Laplace transform � ∞ e − st f ( t ) d t L{ f } ( s ) := 0 Transfer function Assume x (0) = 0. Then � s L{ x } ( s ) = A L{ x } ( s ) + B L{ u } ( s ) , L (Σ) : L{ y } ( s ) = C L{ x } ( s ) + D L{ u } ( s ) , Then L{ y } ( s ) = C ( sI n − A ) − 1 B L{ u } ( s ) . � �� � =: G ( s ) Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Frequency Domain Analysis Laplace transform � ∞ e − st f ( t ) d t L{ f } ( s ) := 0 Transfer function Assume x (0) = 0. Then � s L{ x } ( s ) = A L{ x } ( s ) + B L{ u } ( s ) , L (Σ) : L{ y } ( s ) = C L{ x } ( s ) + D L{ u } ( s ) , Then L{ y } ( s ) = C ( sI n − A ) − 1 B L{ u } ( s ) . � �� � =: G ( s ) The transfer function G ( s ) maps inputs to outputs in the frequency domain. Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Linear Systems Basics 1 Dissipativity and Structural Properties 2 Dissipative Systems Dissipativity in the Frequency Domain The Kalman-Yakubovich-Popov Lemma 3 Model Reduction for LTI Systems 4 Balanced truncation for linear systems Frequency-dependent KYP Lemma and Model Reduction 5 The Frequency-dependent KYP Lemma Frequency-dependent Balanced Truncation Numerical Examples 6 RLC ladder network Butterworth filter Conclusions and Future Work 7 Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 7/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Dissipative Systems Definition [Scherer, Weiland ’05] A dynamical system Σ is called dissipative with respect to a supply function s : R p × R m − → R if there exists a storage function V : R n − → R such that the dissipation inequality � t 1 V ( x ( t 1 )) ≤ V ( x (0)) + s ( y ( t ) , u ( t )) d t 0 is fulfilled for all 0 ≤ t 1 . Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 8/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Dissipative Systems Definition [Scherer, Weiland ’05] A dynamical system Σ is called dissipative with respect to a supply function s : R p × R m − → R if there exists a storage function V : R n − → R such that the dissipation inequality � t 1 V ( x ( t 1 )) ≤ V ( x (0)) + s ( y ( t ) , u ( t )) d t 0 is fulfilled for all 0 ≤ t 1 . Interpretation � t 1 s ( y ( t ) , u ( t )) d t can be seen as the energy supplied to the system 0 in the time interval [0 , t 1 ]. s ( y ( t ) , u ( t )) is a measure for the power at time t . V ( x ( t )) is the internal energy at time t . Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 8/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Quadratic Supply Functions Often, we consider � � T � W � � � y ( t ) y ( t ) S with W = W T , R = R T s ( y ( t ) , u ( t )) = S T u ( t ) u ( t ) R Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 9/35

Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References Quadratic Supply Functions Often, we consider � � T � W � � � y ( t ) y ( t ) S with W = W T , R = R T s ( y ( t ) , u ( t )) = S T u ( t ) u ( t ) R � Cx ( t ) + Du ( t ) � T � W � � Cx ( t ) + Du ( t ) � S = S T u ( t ) R u ( t ) Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 9/35