Investigating duality on stability conditions

This paper is devoted to investigate the role played by duality in stability analysis of linear time-invariant systems. We seek for a dual statement of a recently developed method for generating stability conditions, which combines Lyapunov stability theory with Finsler’s Lemma. This method, developed in the time domain, is able to generate a set of (primal) equivalent stability tests involving extra multipliers. The resulting tests have very attractive properties. Stability is characterized via Linear Matrix Inequalities and we use optimization theory to obtain the duals. The dual problems are given a frequency domain interpretation. Notation R (C) denotes the space of real (complex) vectors of dimension n. R (C) denotes the space of real (complex) matrices of dimension m× n. S (H) denotes the space of real symmetric (complex Hermitian) matrices of dimension n × n. For real or complex scalars or matrices (·) , (·), and (·) , indicate, respectively, transposition, complex conjugate, and complex conjugate plus transposition. X 0 is used to denote that X ∈ S (H) is positive definite. Analogous definitions follow for X ≺ 0, X 0, X 0, which indicate, respectively that X ∈ S (H) is, respectively, negative definite, positive semidefinite, and negative semidefinite. ∗The author is supported by a grant from FAPESP, São Paulo, Brazil. 1 1 Motivation Consider the continuous-time linear time-invariant system described by the equation ẋ(t) = Ax(t), x(0) = x0, (1) where x(t) : [0,∞) → R and A ∈ R. Using Lyapunov stability theory, an stability test for system (1) can be obtained as follows. Define the quadratic form V : R → R as V (x) := x Px, (2) where P ∈ S. Compute the quadratic form V̇ : R × R → R V̇ (x(t), ẋ(t)) := x(t) Pẋ(t) + ẋ(t) Px(t). (3) That is, the time derivative of the quadratic form (2) expressed as a function of x(t) and ẋ(t). According to Lyapunov stability theory, system (1) is asymptotically stable if for any x(0) = x0 6= 0 there exists V (x(t)) > 0, ∀x(t) 6= 0 such that 1 V̇ (x(t), ẋ(t)) < 0, ∀ẋ(t) = Ax(t), (x(t), ẋ(t)) 6= 0, t ∈ [0,∞). (4) That is, if the quadratic form (3) is negative along all trajectories of system (1). The standard approach to verify the stability condition (4) is to explicitly substitute for ẋ(t) into V̇ (·) using the system equation (1). This provides the equivalent condition V̇ (x(t)) = x(t) ( A P + PA ) x(t) < 0, ∀x(t) 6= 0, t ∈ [0,∞), (5) and the well know stability test. Lemma 1 (Lyapunov) The time-invariant linear system is asymptotically stable if, and only if, ∃P ∈ S : P 0, A P + PA ≺ 0. Although the Lyapunov method provides only a sufficient condition for stability, one can prove necessity of Lemma 1 by showing that if (1) is stable then there is always a positive definite matrix P that makes the condition given in Lemma 1 feasible. Some authors have develop alternative stability tests from a different starting point, on the frequency domain (see [2, 3]). In this context, after taking the Laplace transform, asymptotic stability of the linear system (1) can be formulated as the existence of no zeros (nontrivial solutions) of the algebraic equation (sI − A) q = 0, q 6= 0, (6) The classic statement of Lyapunov stability requires (4) to be verified for all x(t) 6= 0. It has been shown in [1] that testing for all (x(t), ẋ(t)) 6= 0 can be done without loss of generality. 2 where s ∈ C, has positive real part. That is, asymptotic stability can be characterized as @ s ∈ C, q ∈ C : s + s ≥ 0, (sI − A) q = 0, q 6= 0. (7) Condition (7) is of little use unless the zeros of (6) are explicitly computed. In order to derive an indirect stability test, that does not require computing the zeros of a rational matrix, we use a lemma given in [2]. Lemma 2 There exists a vector p = sq 6= 0 for some s+s ≥ 0 if, and only if, pq+qp 0. Hence, introducing the vector p ∈ C, p = sq, condition (7) can be rewritten as @ p ∈ C, q ∈ C : pq + qp 0, p− Aq = 0, (p, q) 6= 0. (8) Substituting p = Aq into the first inequality in (8) we obtain @ q ∈ C : Aqq + qqA 0, q 6= 0. (9) From this point on, we proceed by defining the rank-one Hermitian matrix Q ∈ H Q = qq 0, which provides the equivalent stability condition @ Q ∈ H : Q 0, AQ + QA 0, rank(Q) = 1. (10) The above is summarized as the following stability test [2]. Lemma 3 (Ben-Tal) The time-invariant linear system is asymptotically stable if, and only if, @ Q ∈ S : Q 0, AQ + QA 0. Again, with the steps shown above, we can only conclude that the condition in Lemma 2 is sufficient for stability. Some more steps are required to show that, for a system in the form (1), the rank constraint can be relaxed and that Q can be taken to be symmetric (instead of Hermitian) with no loss of generality. See [2, 3] for details or wait until Section 3 for a rigorous proof. Comparing Lemmas 1 and 3, we notice that besides the qualitative time domain versus frequency domain duality, the given tests are also mathematical duals. In fact, by defining the Lagrangian function L(P,Q) = trace [( A P + PA )