The Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications

MAX PLANCK INSTITUTE Elgersburg Workshop Elgersburg February 11-14, 2013 The Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications Timo Reis 1 Matthias Voigt 2 1 Department of Mathematics University of Hamburg Hamburg, Germany 2 Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Germany FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Max Planck Institute Magdeburg Matthias Voigt, 1/26

1 Introduction 2 Kalman-Yakubovich-Popov Lemma 3 Characterization via Even Matrix Pencils 4 Generalized Lur e Equations and Singular Optimal Control Max Planck Institute Magdeburg Matthias Voigt, 2/26

Introduction & Preliminaries Differential-algebraic equations Consider the differential-algebraic equation Eẋ(t) = Ax(t) + Bu(t), Ex(0) = Ex 0 with E, A K n n, B K n m, descriptor vector x(t) K n, and input vector u(t) K m. Spectral density functions Define the spectral density function by [ ] ( se A) Φ(s) = 1 [ ] [ ] B Q S (se A) 1 B I m S, R I m with Q = Q K n n, S K n m, R = R K m m. Max Planck Institute Magdeburg Matthias Voigt, 4/26

Objective and Applications Spectral density function [ ( se A) Φ(s) = 1 B Question: I m S R ] [ Q S Φ(iω) 0 iω Λ(E, A)? ] [ ] (se A) 1 B I m Max Planck Institute Magdeburg Matthias Voigt, 5/26

Objective and Applications Spectral density function [ ( se A) Φ(s) = 1 B Question: I m S R ] [ Q S Φ(iω) 0 iω Λ(E, A)? ] [ ] (se A) 1 B I m Goal of this Talk: Present Equivalent Algebraic Conditions via linear matrix inequalities, algebraic matrix equations (Riccati, Lur e), structured matrices and pencils. Why is this important? Characterization of system properties such as dissipativity, structure-preserving model order reduction, feasibility and solution of optimal control problems. Max Planck Institute Magdeburg Matthias Voigt, 5/26

The ODE Case (E = I n ) with Nonsingular R Spectral density function [ ( sin A) Φ(s) = 1 B I m S R ] [ Q S ] [ ] (sin A) 1 B I m Max Planck Institute Magdeburg Matthias Voigt, 6/26

The ODE Case (E = I n ) with Nonsingular R Spectral density function [ ( sin A) Φ(s) = 1 B I m S R ] [ Q S Kalman-Yakubovich-Popov lemma ] [ ] (sin A) 1 B I m Let [A, B] be controllable. Then Φ(iω) 0 iω Λ(A) if and only if there exists X = X K n n with [ ] A X + XA + Q XB + S B X + S 0. R Max Planck Institute Magdeburg Matthias Voigt, 6/26

The ODE Case (E = I n ) with Nonsingular R Kalman-Yakubovich-Popov lemma Let [A, B] be controllable. Then Φ(iω) 0 iω Λ(A) if and only if there exists X = X K n n with [ ] A X + XA + Q XB + S B X + S 0. R The algebraic Riccati equation Solvability of the LMI is equivalent to solvability of A X + XA + Q (XB + S)R 1 (B X + S ) = 0, X = X. Max Planck Institute Magdeburg Matthias Voigt, 6/26

The ODE Case (E = I n ) with Nonsingular R Kalman-Yakubovich-Popov lemma Let [A, B] be controllable. Then Φ(iω) 0 iω Λ(A) if and only if there exists X = X K n n with [ ] A X + XA + Q XB + S B X + S 0. R The algebraic Riccati equation Solvability of the LMI is equivalent to solvability of A X + XA + Q (XB + S)R 1 (B X + S ) = 0, X = X. The Hamiltonian matrix Solvability criteria can be given in terms of the spectrum of [ A BR 1 S BR 1 B ] S R 1 S Q ( A BR 1 S ). Max Planck Institute Magdeburg Matthias Voigt, 6/26

The ODE Case with Singular R [Reis 11] The Kalman-Yakubovich-Popov lemma still holds, but neither the algebraic Riccati equation nor the Hamiltonian matrix can be formulated! Max Planck Institute Magdeburg Matthias Voigt, 7/26

The ODE Case with Singular R [Reis 11] The Kalman-Yakubovich-Popov lemma still holds, but neither the algebraic Riccati equation nor the Hamiltonian matrix can be formulated! The Lur e equation Solvability of the LMI is equivalent to solvability of A X + XA + Q = K K, XB + S = K L, R = L L, X = X for (X, K, L) K n n K p n K p m and minimal p. Max Planck Institute Magdeburg Matthias Voigt, 7/26

The ODE Case with Singular R [Reis 11] The Kalman-Yakubovich-Popov lemma still holds, but neither the algebraic Riccati equation nor the Hamiltonian matrix can be formulated! The Lur e equation Solvability of the LMI is equivalent to solvability of A X + XA + Q = K K, XB + S = K L, R = L L, X = X for (X, K, L) K n n K p n K p m and minimal p. The even matrix pencil Solvability criteria can be given in terms of the spectrum of 0 si n + A B si n + A Q S. B S R Max Planck Institute Magdeburg Matthias Voigt, 7/26

Controllability Concepts for DAEs Controllability of DAEs [Rosenbrock 74] Further, let rank E = r and S K n,n r be a matrix with im S = ker E. Then, [E, A, B] is called (a) R-controllable, if rank [ se A B ] = n for all s C; (b) impulse controllable, if rank [ E AS B ] = n; (c) strongly controllable, if it is R-controllable and impulse controllable. Why impulse controllability? [Bunse-Gerstner, Mehrmann, Nichols 94] Regularization under feedback transformations: Impulse controllability It exists F K m n such that se (A + BF ) is regular and of index at most 1. Max Planck Institute Magdeburg Matthias Voigt, 9/26

Kalman-Yakubovich-Popov Lemma Kalman-Yakubovich-Popov lemma Assume that [E, A, B] is strongly controllable. Furthermore, let {( } x V = K u) n+m : Ax + Bu im E. Then the following two statements are equivalent: (i) Φ(iω) 0 for all iω Λ(E, A). (ii) There exists some X K n n such that ( ) [ ] ( x A X + X A + Q X B + S x u B X + S 0, E R u) X = X E. ( ) x for all V. u Max Planck Institute Magdeburg Matthias Voigt, 10/26

Kalman-Yakubovich-Popov Lemma Kalman-Yakubovich-Popov lemma Assume that [E, A, B] is strongly controllable. Furthermore, let {( ) } x V = K n+m : Ax + Bu im E. u Then the following two statements are equivalent: (i) Φ(iω) 0 for all iω Λ(E, A). (ii) There exists some X K n n such that [ ] A X + X A + Q X B + S B X + S R V 0, E X = X E. Max Planck Institute Magdeburg Matthias Voigt, 10/26

Characterization via Even Matrix Pencils Even matrix pencils A matrix pencil se A is called even, if E = E and A = A. Spectral Property: Hamiltonian eigensymmetry, i.e., Relation to even matrix pencils λ Λ(E, A) = λ Λ(E, A). We consider the matrix pencil 0 se + A B se A = se + A Q S. B S R Max Planck Institute Magdeburg Matthias Voigt, 12/26

Even Kronecker Canonical Form Definition: even Kronecker canonical form [Thompson 76] Let se A be an even pencil. Then there exists a nonsingular U C n such that U H (se A)U = diag (D 1,..., D k ) where each D j, j = 1,..., k is of one of the following structures: Type 1 (finite non-imaginary eigenvalues): s + µ j 1......... 1 s + µ j s + µ j 1......... 1 s + µ j C[s] 2l j 2l j, with finite non-imaginary eigenvalues µ j C +, µ j C. Max Planck Institute Magdeburg Matthias Voigt, 13/26

Even Kronecker Canonical Form Definition: even Kronecker canonical form [Thompson 76] Let se A be an even pencil. Then there exists a nonsingular U C n such that U H (se A)U = diag (D 1,..., D k ) where each D j, j = 1,..., k is of one of the following structures: Type 2 (finite imaginary eigenvalues): 1 si µ j ε j 1 C[s]l j l j, si µ j......... with finite imaginary eigenvalues iµ j ir and block signature ε j { 1, 1}. Max Planck Institute Magdeburg Matthias Voigt, 13/26

Even Kronecker Canonical Form Definition: even Kronecker canonical form [Thompson 76] Let se A be an even pencil. Then there exists a nonsingular U C n such that U H (se A)U = diag (D 1,..., D k ) where each D j, j = 1,..., k is of one of the following structures: Type 3 (infinite eigenvalues): si 1 ε j si C[s]l j l j, 1 with block signature ε j { 1, 1}.......... Max Planck Institute Magdeburg Matthias Voigt, 13/26

Even Kronecker Canonical Form Definition: even Kronecker canonical form [Thompson 76] Let se A be an even pencil. Then there exists a nonsingular U C n such that U H (se A)U = diag (D 1,..., D k ) where each D j, j = 1,..., k is of one of the following structures: Type 4 (singular structure): 1 s...... 1 s 1. s..... 1 s C[s] (2l j +1) (2l j +1). Max Planck Institute Magdeburg Matthias Voigt, 13/26

Spectral conditions Theorem Let rank E = r and normalrank Φ = p. Then the following statements are equivalent: (i) Φ(iω) 0 for all iω Λ(E, A). (ii) In the EKCF of se A, the blocks have the following structure: All blocks of Type 2 (imaginary evs) have even size and negative signature. There exist exactly 2(n r) + p blocks of Type 3 (infinite evs). There exist p blocks of Type 3 with positive signature. The remaing 2(n r) blocks of Type 3 are either of even size; or the number of odd-sized blocks with positive and negative signature is equal. Max Planck Institute Magdeburg Matthias Voigt, 14/26

Spectral conditions Theorem Let rank E = r and normalrank Φ = p. Then the following statements are equivalent: (i) Φ(iω) 0 for all iω Λ(E, A). (ii) In the EKCF of se A, the blocks have the following structure: All blocks of Type 2 (imaginary evs) have even size and negative signature. There exist exactly 2(n r) + p blocks of Type 3 (infinite evs). There exist p blocks of Type 3 with positive signature. The remaing 2(n r) blocks of Type 3 are either of even size; or the number of odd-sized blocks with positive and negative signature is equal. There exist exactly m p blocks of Type 4 (singular structure). Max Planck Institute Magdeburg Matthias Voigt, 14/26

The Optimal Control Problem (Regular ODE Case) Cost functionals J (x, u) = Optimization problem 0 ( x(t) u(t) ) [ Q S S R { V + (x 0 ) = inf J (x, u) : ẋ(t) = Ax(t) + Bu(t) Questions ] ( ) x(t) dt. u(t) with x(0) = x 0 and } lim x(t) = 0. t Under which conditions is the optimization problem feasible, i.e., V + (x 0 ) > for all x 0 K n? If yes, how to compute the optimal solution? Max Planck Institute Magdeburg Matthias Voigt, 16/26

Boundedness Theorem [Willems 71] Let [A, B] be controllable. Then the following statements are equivalent: 1 The optimization problem is feasible. 2 It exists a solution X = X K n n that solves the LMI [ ] A X + XA + Q XB + S B X + S 0. R Max Planck Institute Magdeburg Matthias Voigt, 17/26

Optimal Solution Theorem [Lancaster, Rodman 95] The optimal costs are given by V (x 0 ) = x 0 X + x 0, where X + is the unique maximal and semi-stabilizing solution of the algebraic Riccati equation A X + XA + Q (XB + S)R 1 (B X + S ) = 0, X = X. Maximalility: X X + for all other solutions X of the ARE; Semi-Stabilization: Closed-loop matrix A BR 1 (B X + + S ) has only eigenvalues in the closed left half-plane. Optimal control signal û(t) = R 1 (B X + + S )x(t). Max Planck Institute Magdeburg Matthias Voigt, 18/26

Construction of X + Construction of X + [Lancaster, Rodman 95] Let [A, B] be controllable and [ A BR 1 S BR 1 B S R 1 S Q ( A BR 1 S ) ] [ ] [ ] Y1 Y1 = T, Y 2 Y 2 with Y := [ ] Y1 Y2 K n+m,n and T K n n. If Y spans a semi-stable Lagrangian invariant subspace, then 1 X + = Y 2 Y 1 1 ; 2 T = A BR 1 (B X + + S ) with Λ(T ) C ir. Max Planck Institute Magdeburg Matthias Voigt, 19/26

The Singular Optimal Control Problem for DAEs Cost functionals J (x, u) = Optimization problem 0 ( x(t) u(t) ) [ Q S S R ] ( ) x(t) dt. u(t) { V + (Ex 0 ) = inf J (x, u) : Eẋ(t) = Ax(t) + Bu(t) Questions with Ex(0) = Ex 0 and } lim Ex(t) = 0. t Under which conditions is the optimization problem feasible, i.e., V + (Ex 0 ) > for all Ex 0 K n? If yes, how to compute the optimal solution? Max Planck Institute Magdeburg Matthias Voigt, 20/26

Boundedness Theorem [Reis, V. 13] Let [E, A, B] be strongly controllable. Then the following statements are equivalent: 1 V + (Ex 0 ) > for all Ex 0 K n. 2 It exists a solution X K n n that solves the LMI [ ] A X + X A + Q X B + S B X + S R V 0, E X = X E with V = {( ) x K n+m u } : Ax + Bu im E. Max Planck Institute Magdeburg Matthias Voigt, 21/26

Optimal Solution Theorem The optimal costs are given by [Reis, V. 13] V (Ex 0 ) = x 0 E X + x 0, where (X +, K +, L +, V +, Σ + ) K n n K p n K p m K n n r K n r n r with minimal p is a maximal and semi-stabilizing solution of the generalized Lur e equation A X + X A + Q = K K + V ΣV, X B + S = K L + V ΣV B, R = L L + B V ΣV B, E X = X E with im V = Def (E, A) and a signature matrix Σ. Max Planck Institute Magdeburg Matthias Voigt, 22/26

Optimal Solution Theorem [Reis, V. 13] A X + X A + Q = K K + V ΣV, X B + S = K L + V ΣV B, R = L L + B V ΣV B, E X = X E Maximalility: E X E X + for all other solutions (X, K, L, V, Σ) of the generalized Lur e equation; Uniqueness: When X 1 + and X 2 + are parts of two maximal solutions then E X 1 + = E X 2 +. Semi-Stabilization: Closed-loop matrix pencil [ ] se + A B K + L + has only eigenvalues in the closed left half-plane. Max Planck Institute Magdeburg Matthias Voigt, 22/26

Optimal Control Signal [Reis, V. 13] Optimal control signal The optimal control signal û( ) fulfills the DAE Eẋ(t) = Ax(t) + Bû(t) + δ 0 Ex 0, 0 = K + x(t) + L + û(t) (in the distributional sense). Max Planck Institute Magdeburg Matthias Voigt, 23/26

Construction of X + [Reis, V. 13] Let [E, A, B] be strongly controllable and 0 se + A B se + A Q S Y 1 Y 2 = Z 1 Z 2 ( sẽ Ã ) B S R Y 3 Z 3 with Y := [ ] Y1 Y2 Y3 K 2n+m n+m and Z := [ ] Z1 Z2 Z3 K 2n+m n+p and sẽ Ã K[s] n+p n+m. If Y spans a semi-stable E-neutral right deflating subspace of dimension n + m, then 1 X + = Y 2 Y 1 with an arbitrary right inverse Y 1 of Y 1 ; [ ] 2 se + A B sẽ Ã = K + L + with Λ ( Ẽ, Ã ) C ir { }. Max Planck Institute Magdeburg Matthias Voigt, 24/26

Conclusions What have we shown in this talk Kalman-Yakubovich-Popov lemma for DAEs; relation to even matrix pencils; linear-quadratic optimal control using generalized Lur e equations. What have we not shown in this talk Factorization of spectral density functions (spectral factorization); application to dissipative systems. Max Planck Institute Magdeburg Matthias Voigt, 25/26

Conclusions What have we shown in this talk Kalman-Yakubovich-Popov lemma for DAEs; relation to even matrix pencils; linear-quadratic optimal control using generalized Lur e equations. What have we not shown in this talk Factorization of spectral density functions (spectral factorization); application to dissipative systems. Open problems How to solve generalized Lur e equations; generalization to J -spectral factorization. Max Planck Institute Magdeburg Matthias Voigt, 25/26

Thanks for your Attention! References Reis 11: Lur e equations and even matrix pencils, Linear Algebra Appl., 434(1), Jan. 2011, pp. 152 173. Rosenbrock 74: Structural properties of linear dynamical systems, Int. J. Control, 20, 1974, pp. 191 202. Bunse-Gerstner, Mehrmann, Nichols 94: Regularization of descriptor systems by output feedback, IEEE Trans. Automat. Control, 39, 1994. pp. 1742 1748. Thompson 76: The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil, Linear Algebra Appl., 14(2), 1976, pp. 135 177. Willems 71: Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automat. Control, AC-16(6), Dec. 1971, pp. 621 634. Lancaster, Rodman 95: The Algebraic Riccati Equation, Oxford University Press, 1995. Reis, Voigt 13: Spectral factorization and linear-quadratic optimal control of differential-algebraic systems, 2013, in preparation. Max Planck Institute Magdeburg Matthias Voigt, 26/26