Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020 Stability analysis by averaging: a time-delay approach ? Emilia Fridman ∗ Jin Zhang ∗ ∗ School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel (e-mails: [email protected], [email protected]). Abstract: We study stability of linear systems with fast time-varying coefficients. The classical averaging method guarantees the stability of such systems for small enough values of parameter provided the corresponding averaged system is stable. However, it is difficult to find an upper bound on the small parameter by using classical tools for asymptotic analysis. In this paper we introduce an efficient constructive method for finding an upper bound on the value of the small parameter that guarantees a desired exponential decay rate. We transform the system to a model with time-delays of the length of the small parameter. The resulting time-delay system is a perturbation of the averaged LTI system which is assumed to be exponentially stable. The stability of the time-delay system guarantees the stability of the original one. We construct an appropriate Lyapunov functional for finding sufficient stability conditions in the form of linear matrix inequalities (LMIs). The upper bound on the small parameter that preserves the exponential stability is found from LMIs. Two numerical examples (stabilization by vibrational control and by time-dependent switching) illustrate the efficiency of the method. Keywords: Averaging, linear systems, time-delay systems, Lyapunov-Krasovskii method, LMIs 1. INTRODUCTION of the both sides of the system, we present the resulting system as a perturbation of the averaged system, and Asymptotic methods for analysis and control of perturbed model it as a system with time-delays of the length of the systems depending on small parameters have led to impor- small parameter. If the transformed time-delayed system tant qualitative results (Tikhonov, 1952; Kokotovic and is stable, then the original one is also stable. We assume Khalil, 1986; Khalil, 2002; Vasilieva and Butuzov, 1973; that the averaged LTI system is exponentially stable. Bogoliubov and Mitropolsky, 1961; Moreau and Aeyels, We suggest a direct Lyapunov-Krasovskii approach, and 2000; Teel et al., 2003; Cheng et al., 2018). However, by formulate sufficient exponential stability conditions in the using these methods it is difficult to find an efficient bound form of LMIs. The upper bound on the small parameter on the small parameter that preserves the stability. For that guarantees a desired decay rate for the original system singularly perturbed systems, such a bound was presented can be found from LMIs. Two numerical examples (sta- e.g. in Fridman (2002) by using direct Lyapunov method. bilization by vibrational control and by time-dependent For the sampled-data systems with fast sampling, the switching) illustrate the efficiency of the method. time-delay approach was initiated in the framework of asymptotic methods (Mikheev et al., 1988) and aver- 1.1 Necessary notations, definitions and statements aging (Fridman, 1992). Later the time-delay approach to sampled-data control via direct Lyapunov-Krasovskii Throughout the paper Rn denotes the n-dimensional Eu- method Fridman et al. (2004) led to efficient tools for clidean space with vector norm j·j and the induced matrix robust sampled-data and networked control (see e.g. Frid- norm j · j, Rn×m is the set of all n × m real matrices. man (2014); Hetel and Fridman (2013); Liu et al. (2019)). The superscript T stands for matrix transposition, and n×n In this paper we consider linear systems with fast vary- the notation P > 0, for P 2 R means that P is ing coefficients. Our objective is to propose a construc- symmetric and positive definite. The symmetric elements tive time-delay approach with a corresponding Lyapunov- of the symmetric matrix are denoted by ∗. Krasovskii method to the averaging method for these sys- We will employ extended Jensen's inequalities (Solomon tems. Differently from the classical results (see Chapter and Fridman, 2013): 10 of Khalil (2002)), where the system coefficients are Lemma 1.1. Denote supposed to be at least continuous in time, we assume them to be piecewise-continuous. This allows to apply our Z b Z b Z t results e.g. to fast switching systems. By taking average G = f(s)x(s)ds; Y = f(θ)x(s)dsdθ; a a t−θ ? This work was supported by Israel Science Foundation (grant no. n 673/19), by C. and H. Manderman Chair at Tel Aviv University, and where a ≤ b, f :[a; b] ! R, x(s) 2 R and the integration by the Planning and Budgeting Committee (PBC) Fellowship from concerned is well defined. Then for any n×n matrix R > 0 the Council for Higher Education, Israel. the following inequalities hold: Copyright lies with the authors 4907 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020 Z b Z b Under A2, A can be presented as a convex combination GT RG ≤ jf(θ)jdθ jf(s)jxT (s)Rx(s)ds; m M of the constant matrices Ai with the entries akj or akj : a a (1.1) N X t1 Z b Z b Z t A(τ) = fi(τ)Ai 8τ ≥ T T "1 Y RY ≤ jf(θ)jθdθ jf(θ)jx (s)Rx(s)dsdθ: i=1 (2.5) a a t−θ N X n2 (1.2) fi ≥ 0; fi = 1; 1 ≤ N ≤ 2 : i=1 2. A TIME-DELAY APPROACH TO STABILITY BY Note that f 6≡ 0. For a constant a , we have am = aM . AVERAGING i kj kj kj From A1 we have N Z 1 Consider the fast varying system: X t1 A f (τ − θ) dθ = A + ∆A; 8τ ≥ : t i i av " x_(t) = A( )x(t); t ≥ 0; (2.1) i=1 0 1 " n n×n where x(t) 2 R , A : [0; 1) ! R is piecewise- We will further integrate (2.1) on [t − "; t] for t ≥ t1. continuous and " > 0 is a small parameter. Similar to the Note that similar to Fridman and Shaikhet (2016), we can case of general averaging in Sect. 10.6 of Khalil (2002), present assume the following: 1 Z t x(t) − x(t − ") d x_(s)ds = = [x(t) − G]; (2.6) A1 There exist "1 > 0 and t1 ≥ "1 such that " t−" " dt 1 Z t s where A( )ds = Aav + ∆A(t; "); 8t ≥ t1;" 2 (0;"1]; Z t " " ∆ 1 t−" G = (s − t + ")_x(s)ds: (2.7) j∆A(t; ")j ≤ σ("); " t−" (2.2) Then, integrating (2.1) and taking into account (2.6) we with Hurwitz constant matrix Aav. Here σ is a strictly arrive at increasing (of class K) scalar function with σ(0) = 0. d 1 Z t s System (2.1) has almost periodic coefficients if it satifies [x(t) − G] = A( )ds · x(t) dt " t−" " t−s t A1. Changing the variable s in (2.2) to θ = " , we can 1 Z s rewrite the first equation in (2.2) as + A( )[x(s) − x(t)]ds; t ≥ t1: " t−" " Z 1 t A( − θ)dθ = Aav + ∆A(t; "); 8t ≥ t1;" 2 (0;"1] For shortness we will omit arguments of ∆A. By changing 0 " variable εθ = t − s in the last integral, we have t t or, in terms of the fast time τ = " , 1 Z s Z 1 A( )[x(s) − x(t)]ds t1 " " A(τ − θ)dθ = A + ∆A(ετ; "); 8τ ≥ : (2.3) t−" av " Z 1 t 0 1 = A( − θ)[x(t − εθ) − x(t)]dθ Remark 2.1. If A(τ) is 1-periodic, then in (2.3) we have 0 " ∆A = 0. If A(τ) is T -periodic with T > 0, scaling the Z 1 t Z t time t = T t¯ and denotingx ¯(t¯) = x(T t¯) = x(t), we can = − A( − θ) x_(s)dsdθ: 0 " t−εθ present (2.1) as Finally, denoting d T t¯ x¯(t¯) = T · A( )¯x(t¯) (2.4) z(t) = x(t) − G (2.8) dt¯ " t¯ and employing (2.2), we transform (2.1) to a time-delay with 1-periodic A(T τ¯), whereτ ¯ = " . In general we can ∗ consider almost periodic A (in the sense of (2.3) with non- system for " 2 (0;" ] and t ≥ t1 zero ∆A). For example, let A in (2.1) have the form Z 1 t Z t z_(t) = (A + ∆A)x(t) − A( − θ) x_(s)dsdθ: t av " A(τ) = A cos(τ) + A sin2(3τ) + A e−τ ; τ = 0 t−εθ 1 2 3 " (2.9) with constant n × n-matrices A1;A2;A3 and with A2 System (2.9) is a kind of a neutral type system that Hurwitz. Then, scaling the time t = 2πt¯ and denoting depends on the past values ofx _. However, this is not ¯ ¯ _ ¯ 2πt ¯ a neutral system in Hale's form (Hale and Lunel, 1993) x¯(t) = x(t), we arrive at x¯(t) = 2πA( " )¯x(t) with Z 1 because G depends onx _ and not on x. A(2π(τ − θ))dθ = 0:5A2 + ∆A; Summarizing, if x(t) is a solution to (2.1), then it satisfies 0 the time-delay system (2.9). Therefore, the stability of the where Z 1 time-delay system guarantees the stability of the original −2π(τ−θ) ∆A = A3 e dθ −! 0: system. We will derive the stability conditions for the time- 0 τ!1 delay system via direct Lyapunov-Krasovskii method. Additionally we assume the following: ∗ ∗ Given " 2 (0;"1], denote by fi > 0 (i = 1; :::; N) the following bound: A2 All entries akj(τ) of A(τ) are uniformly bounded for Z 1 τ ≥ 0 with the values from some finite intervals akj(τ) 2 t ∗ ∗ t "jf ( − θ)jθdθ ≤ f ; t ≥ t ;" 2 (0;" ]: (2.10) [am ; aM ] for τ ≥ 1 .