On common linear/quadratic Lyapunov functions for switched linear systems M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland, USA gowda@math.umbc.edu, http://www.math.umbc.edu/ gowda On common linear/quadratic Lyapunov functions for switched linear systems p.1/30
Joint work with Melania Moldovan Based on the forthcoming article in Nonlinear analysis and variational problems, Eds: P. Pardalos, Th.M. Rassias, and A.A. Khan, Springer 2009, Volume dedicated to the memory of George Isac Objective is to study switched linear systems via complementarity and duality ideas. On common linear/quadratic Lyapunov functions for switched linear systems p.2/30
Outline Continuous/discrete/positive linear systems Complementarity ideas Z-matrices and transformations Switched systems Duality ideas Positive switched systems An extension of Mason-Shorten result Miscellaneous On common linear/quadratic Lyapunov functions for switched linear systems p.3/30
Continuous Linear System Given A R n n, consider ẋ + Ax = 0. Asymptotic stability: x(t) 0 as t. Lyapunov s Theorem (1893): The following are equivalent: (1) A is positive stable. (2) There exists a P 0 with A T P + PA 0. (3) The system ẋ + Ax = 0 is asymptotically stable. On common linear/quadratic Lyapunov functions for switched linear systems p.4/30
A is positive stable: Re λ > 0 for all eigenvalues of A. Q(x) := x T Px is a quadratic Lyapunov function. When A is positive stable, Q decreases along any trajectory of the system. On common linear/quadratic Lyapunov functions for switched linear systems p.5/30
Discrete Linear System x(k + 1) = Ax(k); k = 1, 2,.... Asymptotic stability: x(k) 0 as k. Stein s Theorem (1952): The following are equivalent: (1) A is Schur stable. (2) There exists a P 0 with P A T PA 0. (3) The system x(k + 1) = Ax(k) is asymptotically stable. On common linear/quadratic Lyapunov functions for switched linear systems p.6/30
A Schur stable: λ < 1 for all eigenvalues of A. Q(x) := x T Px is a quadratic Lyapunov function. When A is Schur stable, Q decreases along any trajectory of the system. On common linear/quadratic Lyapunov functions for switched linear systems p.7/30
Positive systems All trajectories of ẋ + Ax = 0 are constrained to R+. n This happens if and only if A is a Z-matrix: All off-diagonal entries are nonpositive. If there is d > 0 such that A T d > 0, then Q(x) = x T d is a linear Lyapunov function. In this case, the system is asymptotically stable. On common linear/quadratic Lyapunov functions for switched linear systems p.8/30
Linear complementarity problems A R n n and q R n. LCP(A,q): Find x R n such that x 0, Ax + q 0, and x T (Ax + q) = 0. A is a Q-matrix if LCP(A,q) has a solution for all q. A is a P-matrix if LCP(A,q) has a unique solution for all q or equivalently, x Ax 0 x = 0. On common linear/quadratic Lyapunov functions for switched linear systems p.9/30
Semidefinite LCPs L : S n S n linear, Q S n. SDLCP(L,Q): Find X S n such that X 0, L(X) + Q 0, and X,L(X) + Q = 0. L has the Q-property if SDLCP(L,Q) has a solution for all Q S n. L has the P-property if X and L(X) commute and X L(X) 0 implies X = 0. On common linear/quadratic Lyapunov functions for switched linear systems p.10/30
LCPs and positive systems Let A be a Z-matrix. Then the following are equivalent: There exists d > 0 such that A T d > 0. A is a Q-matrix. A is a P-matrix. On common linear/quadratic Lyapunov functions for switched linear systems p.11/30
SDLCPs and linear systems A R n n, let L A (X) := AX + XA T the Lyapunov transformation Gowda-Song (2000): The following are equivalent: There exists a P 0 such that L A (P) 0. A is positive stable. L A has the Q-property L A has the P-property. On common linear/quadratic Lyapunov functions for switched linear systems p.12/30
SDLCPs and discrete linear systems For A R n n, let S A (X) := X AXA T the Stein transformation. Gowda-Parthasarathy (2000) : The following are equivalent: There exists a P 0 such that S A (P) 0. A is Schur stable (equivalently, S A is positive stable). S A has the Q-property. S A has the P-property. Do L A and S A have some sort of Z-property? On common linear/quadratic Lyapunov functions for switched linear systems p.13/30
The Z-property V Finite dimensional real Hilbert space. K proper cone in V : K is a closed convex cone with K ( K) = {0}. A linear L : V V has the Z-property if x K, y K, x,y = 0 L(x),y 0 where K := {y V : y,x 0 x K}. L A and S A have the Z-property on the semidefinite cone. On common linear/quadratic Lyapunov functions for switched linear systems p.14/30
Properties equivalent to the Z-property Schneider-Vidyasagar (1970) exp( tl)(k) K for all t 0. ẋ + L(x) = 0, x(0) K x(t) K for all t 0. Stern (1981): For a Z-transformation, the following are equivalent: There exists d int(k) such that L(d) int(k). L 1 exists and L 1 (K) K. L is positive stable. On common linear/quadratic Lyapunov functions for switched linear systems p.15/30
Gowda-Tao (2009): The above are further equivalent to: L has the Q-property with respect to K. Open problem: Does L have the P-property? On common linear/quadratic Lyapunov functions for switched linear systems p.16/30
Switched linear systems {A 1,A 2,...,A m } R n n. ẋ + A σ x = 0, σ {1, 2,...,m} is a switched linear system. Switching Signal σ : [0, ) {1, 2,...,m} is piecewise constant. Survey article by R. Shorten et al in SIAM Review, 2007, Monograph: Switching systems by D. Liberzon. On common linear/quadratic Lyapunov functions for switched linear systems p.17/30
The (uniform exponential) asymptotic stability: There exists β > 0 such that x(t) Me βt x(0) for all t, all solutions, and all signals. On common linear/quadratic Lyapunov functions for switched linear systems p.18/30
Sufficient condition for asymptotic stability: P 0 and A T i P + PA i 0 for all i = 1, 2,...,m. Then V (x) := x T Px is a Common Quadratic Lyapunov Function. Question: When do we have a CQLF? Known conditions: Commutativity, simultaneous triangularization, Lie algebraic conditions, etc. Similar results/ideas for discrete switched systems. On common linear/quadratic Lyapunov functions for switched linear systems p.19/30
Kamenetskiy and Pyatnitskiy (1987): There exists a CQLF if and only if m i=1 (A iy i + Y i A T i ) = 0, Y i 0, i Y i = 0 i. On common linear/quadratic Lyapunov functions for switched linear systems p.20/30
L i : H H linear for i = 1, 2,...,m. L 1 is positive stable and has the Z-property on proper cone K. Moldovan-Gowda (2009) The following are equivalent: There exists d K such that L i (d) K for all i = 1,...,m. m 1 LT i (y i) = 0, y i K i y i = 0 i. Proof based on duality/theorem of alternative: On common linear/quadratic Lyapunov functions for switched linear systems p.21/30
Let K 1 H 1 and K 2 H 2. L : H 1 H 2 linear. Then exactly one of the following is consistent: L(x) (K 2 ), x (K 1 ). L T (y) K1, 0 y K 2. On common linear/quadratic Lyapunov functions for switched linear systems p.22/30
Positive switched systems ẋ + A σ x = 0, σ {1, 2,...,m} where each A i is a Z-matrix. Q(x) = x T d is a common linear Lyapunov function if d > 0 and A T i d > 0 for all i. In this case, the switched system is asymptotically stable. Question: When do we have such a d? On common linear/quadratic Lyapunov functions for switched linear systems p.23/30
Complementarity ideas again x y = min{x,y} Vertical linear complementarity problem: x (A T 1 x + q 1 ) (A T mx + q m ) = 0. If q i < 0, then x 0 and A T i x q i > 0 for all i; such an x produces a vector d > 0 such that A T i d > 0 for all i. On common linear/quadratic Lyapunov functions for switched linear systems p.24/30
Gowda-Sznajder (1994) VLCP has a unique solution for all q 1,q 2,,q m if and only if all column representatives of A = {A 1,A 2,...,A m } are P-matrices. On common linear/quadratic Lyapunov functions for switched linear systems p.25/30
Mason-Shorten (2007) A 1 and A 2 are positive stable Z-matrices. The following are equivalent: There exists d > 0 in R n such that A T 1 d > 0 and AT 2 d > 0. Every column rep. of {A 1,A 2 } is positive stable. Every column rep. of {A 1,A 2 } has positive determinant. On common linear/quadratic Lyapunov functions for switched linear systems p.26/30
Song-Gowda-Ravindran (2003) A is a compact set of positive stable Z-matrices: There exists d > 0 such that A T d > 0 for all A A. Every column rep. of A is a P-matrix. On common linear/quadratic Lyapunov functions for switched linear systems p.27/30
A = {A 1,A 2,...,A m } pos. stable Z-matrices A # := set of all column representatives of A Â := { m A i D i : D i is nonnegative and diagonal, m D i = I } 1 1 On common linear/quadratic Lyapunov functions for switched linear systems p.28/30
Moldovan- Gowda (2009) The following are equivalent: (1) There exists d > 0 such that A T i i = 1, 2,...,m. (2) Every matrix in  is a P-matrix. (3) Every matrix in A # is a P-matrix. d > 0 for all (4) Every matrix in A # has positive determinant. (5) Every matrix in  has positive determinant. (6) Every VLCP corresponding to A T has a unique solution. On common linear/quadratic Lyapunov functions for switched linear systems p.29/30
Let A be a compact set of positive stable Z-matrices in R n n. Then the following are equivalent: (1) There exists a d > 0 such that A T d > 0 for all A A. (2) Every column representative of A is a P-matrix. (3) Every column representative of A has positive determinant. On common linear/quadratic Lyapunov functions for switched linear systems p.30/30