Application of Exponential Splitting Methods to Fast Switching Theory Maurizio Porfiri D. Gray Roberson Daniel J. Stilwell The Bradley Department of Electrical and Computer Engineering Virginia Polytechnic Institute and State University Blacksburg, VA 24061 {mporfiri, grayr, stilwell}@vt.edu Abstract— Stability of periodic linear switched systems is studied. By combining the method of averaging with exponential splitting, less conservative bounds on stabilizing switching rates are obtained than with other approximation techniques. Such bounds are useful for analysis and design of switching control laws. The stability analysis is generalized to a wider class of time-varying systems. I. I NTRODUCTION A switched dynamic system is characterized by a family of subsystems and a policy for switching among them. Let Θ= {A 1 ,...,A M } be a family of matrix-valued coefficients and ρ : R + →{1,...,M } a switching policy. Then ˙ x = A ρ(t) x represents a linear switched system with time-invariant sub- systems. Switched systems find application in diverse areas of engineering, such as power electronics, communication networks, hybrid control, traffic flow, and biosystem mod- eling. The books [1] and [2] contain excellent reviews of switched system research. A well-studied switching protocol is periodic switching. Under this scheme, ρ is T -periodic and switching occurs in a state-independent manner. The resulting periodic linear system may be analyzed using a number of techniques. For example, [3] uses Floquet analysis to study the periodic system. The main drawback of the Floquet approach is its inherent dependence on matrix logarithms, which are difficult to compute and hinder qualitative understanding. Alternatives to Floquet analysis are explored for example in [4], [5], [6]. A basic result is that periodic switched system stability is related to the stability of an auxiliary time-invariant average system. Stability of the average system is inherited by the switched system if the switching rate is sufficiently fast. Determining the slowest allowable switching rate (or the maximum switching period) is important for stability analysis and control design. An estimate is obtained by specializing the method of averaging proposed in [7] for general linear time-varying systems, e.g. [8]. This approach relies on a decomposition of the state transition matrix. For the time-varying system ˙ x = A(t)x, where A(t)≤ α for all t, the transition matrix satisfies Φ(s + τ,s)= e ¯ A τ (s)τ + E τ (s) where ¯ A τ (s)= 1 τ  s+τ s A(σ)dσ and E τ (s)≤ α 2 τ 2 e ατ This decomposition is well known and appears in several textbooks, e.g. [9]. Henceforth this approach to estimating the maximum switching period is referred to as the standard approach, and the estimate itself as the standard estimate. A different approach is described in [6], where the estimate is based on perturbation analysis of system spectral properties. The estimate of [6] is difficult to compute, as it involves quantifying the sensitivity of matrices to perturbations. The standard estimate, on the other hand, requires knowledge of elementary quantities and is relatively easy to compute. However, it generally leads to conservative results that may be a few orders of magnitude less than the actual maximum switching period. The present work characterizes the slowest stabilizing switching rate with a computationally tractable expression that yields less conservative bounds than the standard ap- proach. Loosely speaking, improved bounds are due to ac- counting for commutation relationships among the elements of Θ. The general idea involves combining the standard approach with established numerical techniques. By exploit- ing the concept of matrix exponential splitting, widely used in numerical analysis of partial differential equations, e.g. [10], a novel switching rate estimate is obtained. The results are extended to a more general class of linear time-varying systems, and new results on the method of averaging are established. In particular, the effects of state perturbations on the stability of periodic linear switched systems are analyzed. The paper is organized as follows. Section II describes mathematical tools that are used for stability analysis. Sec- tion III contains the principle contribution, an improved es- timate of the maximum stabilizing switching period. Section IV addresses the stability of more general time-varying sys- tems and presents stability conditions for perturbed switched systems. II. STABILITY ANALYSIS TOOLS We consider the linear time-varying system ˙ x = A(t)x (1) Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006 FrC16.2 1-4244-0210-7/06/$20.00 ©2006 IEEE 5935