Joint Spectral Radius and Path-Complete Graph Lyapunov Functions (original) (raw)
Related papers
A Characterization of Lyapunov Inequalities for Stability of Switched Systems
2016
We study stability criteria for discrete-time switched systems and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past fifteen years. We prove in this note that a family of languagetheoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies stability of a switched system. Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.
Stability Criteria via Common Non-strict Lyapunov Matrix for Discrete-time Linear Switched Systems
2011
Let S = {S 1 , S 2 } ⊂ R d×d have a common, but not necessarily strict, Lyapunov matrix (i.e. there exists a symmetric positive-definite matrix P such that P − S T k PS k ≥ 0 for k = 1, 2). Based on a splitting theorem of the state space R d (Dai, Huang and Xiao, arXiv:1107.0132v1[math.PR]), we establish several stability criteria for the discrete-time linear switched dynamics x n = S σ n · · · S σ 1 (x 0 ), x 0 ∈ R d and n ≥ 1 governed by the switching signal σ : N → {1, 2}. More specifically, let ρ(A) stand for the spectral radius of a matrix A ∈ R d×d , then the outline of results obtained in this paper are: (1) For the case d = 2, S is absolutely stable (i.e., S σ n · · · S σ 1 → 0 driven by all switching signals σ) if and only if ρ(S 1 ), ρ(S 2 ) and ρ(S 1 S 2 ) all are less than 1; (2) For the case d = 3, S is absolutely stable if and only if ρ(A) < 1 ∀A ∈ {S 1 , S 2 } ℓ for ℓ = 1, 2, 3, 4, 5, 6, and 8. This further implies that for any S = {S 1 , S 2 } ⊂ R d×d with the generalized spectral radius ρ(S) = 1 where d = 2 or 3, if S has a common, but not strict in general, Lyapunov matrix, then S possesses the spectral finiteness property.
Efficient Method for Computing Lower Bounds on the ppp-radius of Switched Linear Systems
2015
This paper proposes lower bounds on a quantity called LpL^pLp-norm joint spectral radius, or in short, ppp-radius, of a finite set of matrices. Despite its wide range of applications to, for example, stability analysis of switched linear systems and the equilibrium analysis of switched linear economical models, algorithms for computing the ppp-radius are only available in a very limited number of particular cases. The proposed lower bounds are given as the spectral radius of an average of the given matrices weighted via Kronecker products and do not place any requirements on the set of matrices. We show that the proposed lower bounds theoretically extend and also can practically improve the existing lower bounds. A Markovian extension of the proposed lower bounds is also presented.
52nd IEEE Conference on Decision and Control, 2013
We introduce the concept of sos-convex Lyapunov functions for stability analysis of discrete time switched systems. These are polynomial Lyapunov functions that have an algebraic certificate of convexity, and can be efficiently found by semidefinite programming. We show that convex polynomial Lyapunov functions are universal (i.e., necessary and sufficient) for stability analysis of switched linear systems. On the other hand, we show via an explicit example that the minimum degree of an sos-convex Lyapunov function can be arbitrarily higher than a (non-convex) polynomial Lyapunov function. (The proof is omitted.) In the second part, we show that if the switched system is defined as the convex hull of a finite number of nonlinear functions, then existence of a non-convex common Lyapunov function is not a sufficient condition for switched stability, but existence of a convex common Lyapunov function is. This shows the usefulness of the computational machinery of sos-convex Lyapunov functions which can be applied either directly to the switched nonlinear system, or to its linearization, to provide proof of local switched stability for the nonlinear system. An example is given where no polynomial of degree less than 14 can provide an estimate to the region of attraction under arbitrary switching.
On the minimal degree of a common Lyapunov function for planar switched systems
2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601), 2004
In this paper we consider a planar switched systemẋ(t) = u(t)Ax(t) + (1 − u(t))Bx(t), where x ∈ R 2 , A and B are two 2 × 2 real matrices, that is asymptotically stable for every switching function u(.) : [0, ∞[→ {0, 1}. We prove that if LA,B is a common Lyapunov function for the pair (A, B), then LA,B cannot be polynomial of degree uniformly bounded over all asymptotically stable systems of the kind above. This result answer to a question open from 1999.
Switching systems with dwell time: Computing the maximal Lyapunov exponent
Nonlinear Analysis: Hybrid Systems, 2021
We study asymptotic stability of continuous-time systems with mode-dependent guaranteed dwell time. These systems are reformulated as special cases of a general class of mixed (discrete-continuous) linear switching systems on graphs, in which some modes correspond to discrete actions and some others correspond to continuous-time evolutions. Each discrete action has its own positive weight which accounts for its time-duration. We develop a theory of stability for the mixed systems; in particular, we prove the existence of an invariant Lyapunov norm for mixed systems on graphs and study its structure in various cases, including discrete-time systems for which discrete actions have inhomogeneous time durations. This allows us to adapt recent methods for the joint spectral radius computation (Gripenberg's algorithm and the Invariant Polytope Algorithm) to compute the Lyapunov exponent of mixed systems on graphs.
Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems
Discrete & Continuous Dynamical Systems - B
To estimate the growth rate of matrix products An • • • A1 with factors from some set of matrices A, such numeric quantities as the joint spectral radius ρ(A) and the lower spectral radiusρ(A) are traditionally used. The first of these quantities characterizes the maximum growth rate of the norms of the corresponding products, while the second one characterizes the minimal growth rate. In the theory of discrete-time linear switching systems, the inequality ρ(A) < 1 serves as a criterion for the stability of a system, and the inequalityρ(A) < 1 as a criterion for stabilizability. For matrix products AnBn • • • A1B1 with factors Ai ∈ A and Bi ∈ B, where A and B are some sets of matrices, we introduce the quantities µ(A, B) and η(A, B), called the lower and upper minimax joint spectral radius of the pair {A, B}, respectively, which characterize the maximum growth rate of the matrix products AnBn • • • A1B1 over all sets of matrices Ai ∈ A and the minimal growth rate over all sets of matrices Bi ∈ B. In this sense, the minimax joint spectral radii can be considered as generalizations of both the joint and lower spectral radii. As an application of the minimax joint spectral radii, it is shown how these quantities can be used to analyze the stabilizability of discrete-time linear switching control systems in the presence of uncontrolled external disturbances of the plant.
On several composite quadratic Lyapunov functions for switched systems
2006
Three types of composite quadratic Lyapunov funtions are used for deriving conditions of stabilization and for constructing switching laws for switched systems. The three types of functions are, the max of quadratics, the min of quadratics and the convex hull of quadratics. Directional derivatives of the Lyapunov functions are used for the characterization of convergence rate. Stability results are established with careful consideration of the existence of sliding mode and the convergence rate along the sliding mode. Dual stabilization result is established with respect to the pair of conjugate Lyapunov functions: the max of quadratics and the convex hull of quadratics. It is observed that the min of quadratics, which is nondifferentiable and nonconvex, may be a more convenient tool than the other two types of functions which are convex and/or differentiable.