Preprints of the 21st IFAC World Congress (Virtual) , , July 12-17, 2020

Stability analysis by averaging: a time-delay approach ?

Emilia Fridman ∗ Jin Zhang ∗

∗ School of , University, Tel Aviv 69978, (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 |·| and the induced matrix robust sampled-data and networked control (see e.g. Frid- norm | · |, 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 ∈ 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) ∈ 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 ≤ |f(θ)|dθ |f(s)|xT (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 ∀τ ≥ T T ε1 Y RY ≤ |f(θ)|θdθ |f(θ)|x (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, ∀τ ≥ . t i i av ε x˙(t) = A( )x(t), t ≥ 0, (2.1) i=1 0 1 ε n n×n where x(t) ∈ R , A : [0, ∞) → R is piecewise- We will further integrate (2.1) on [t − ε, t] for t ≥ t1. continuous and ε > 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 − ε) d x˙(s)ds = = [x(t) − G], (2.6) A1 There exist ε1 > 0 and t1 ≥ ε1 such that ε t−ε ε dt 1 Z t s where A( )ds = Aav + ∆A(t, ε), ∀t ≥ t1, ε ∈ (0, ε1], Z t ε ε ∆ 1 t−ε G = (s − t + ε)x ˙(s)ds. (2.7) |∆A(t, ε)| ≤ σ(ε), ε t−ε (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 ε t−ε ε t−s t A1. Changing the variable s in (2.2) to θ = ε , we can 1 Z s rewrite the first equation in (2.2) as + A( )[x(s) − x(t)]ds, t ≥ t1. ε t−ε ε Z 1 t A( − θ)dθ = Aav + ∆A(t, ε), ∀t ≥ t1, ε ∈ (0, ε1] For shortness we will omit arguments of ∆A. By changing 0 ε variable εθ = t − s in the last integral, we have t t or, in terms of the fast time τ = ε , 1 Z s Z 1 A( )[x(s) − x(t)]ds t1 ε ε A(τ − θ)dθ = A + ∆A(ετ, ε), ∀τ ≥ . (2.3) t−ε av ε 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 ε ∆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 ε 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¯ ε t¯ and employing (2.2), we transform (2.1) to a time-delay with 1-periodic A(T τ¯), whereτ ¯ = ε . In general we can ∗ consider almost periodic A (in the sense of (2.3) with non- system for ε ∈ (0, ε ] 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 ε A(τ) = A cos(τ) + A sin2(3τ) + A e−τ , τ = 0 t−εθ 1 2 3 ε (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( ε )¯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→∞ delay system via direct Lyapunov-Krasovskii method. Additionally we assume the following: ∗ ∗ Given ε ∈ (0, ε1], 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) ∈ t ∗ ∗ t ε|f ( − θ)|θdθ ≤ f , t ≥ t , ε ∈ (0, ε ]. (2.10) [am , aM ] for τ ≥ 1 . i i 1 kj kj ε1 0 ε

4908 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

∗ Note that since fi ∈ [0, 1] and ε ∈ (0, ε ], we can always Then ∗ ε∗ d 4 choose fi ≤ . T −2αε T 2 VG + 2αVG ≤ εx˙ (t)Rx˙(t) − e G RG. (2.18) Theorem 2.1. Let A1 and A2 hold. Given matrices Aav, dt ε ∗ Ai (i = 1,...,N) and constants α > 0 and ε ∈ (0, ε1], To compensate the Yi-terms (distributed delay) let there exist n × n matrices P > 0, R > 0, Hi > 0 (i = 1, ··· ,N) and a scalar λ > 0 that satisfy the following Z 1  t  Z t Yi = fi − θ x˙(s)dsdθ (2.19) LMIs: 0 ε t−εθ  √ N  ∗ T X in (2.15), we employ as in Solomon and Fridman (2013) ε Ai (R + Hj)  Φ  N  j=1  X   V = V ,  0(N+2)n,n  < 0, i = 1,...,N. (2.11) H Hi  N  i=1  X  Z 1 Z t  ∗ ∗ −R − Hj  −2α(t−s) T VHi = 2 e (s − t + εθ)x ˙ (s)Hix˙(s)dsdθ j=1 0 t−εθ Here (2.20)  Φ11 Φ12 Φ13 P  with Hi > 0. Differentiating VHi , we have ∗ Φ22 Φ23 −P d Φ =   (2.12) V + 2αV = εx˙ T (t)H x˙(t)  ∗ ∗ Φ33 0Nn,n  dt Hi Hi i Z 1 Z t ∗ ∗ ∗ −λIn −2α(t−s) T −2 e x˙ (s)Hix˙(s)dsdθ with 0 t−εθ T 2 Z 1 Z t Φ11 = PAav + AavP + 2αP + λσ In, T −2αε T T ≤ εx˙ (t)Hix˙(t) − 2e x˙ (s)Hix˙(s)dsdθ. Φ12 = −AavP − 2αP, 0 t−εθ 4 ∗ Φ = − e−2αε R + 2αP, Applying further Jensen’s inequality (1.2) 22 ε∗ 1 Φ13 = −Φ23 = −P [A1, ··· ,AN ], Z t Y T H Y ≤ ε|f ( − θ)|θdθ −2αε∗ 1 1 i i i i Φ = −2e diag{ H , ··· , H }, 0 ε 33 f ∗ 1 f ∗ N Z 1 Z t 1 N t T Φ14 = −Φ24 = P, Φ44 = −λIn. × |fi( − θ)| x˙ (s)Hix˙(s)dsdθ 0 ε t−εθ Then system (2.1) is exponentially stable with a decay rate Z 1 Z t ∗ ∗ T α for all ε ∈ (0, ε ], meaning that there exists M0 > 0 such ≤ fi x˙ (s)Hix˙(s)dsdθ, that for all ε ∈ (0, ε∗] the solutions of (2.1) initialized by 0 t−εθ n we arrive at x(0) ∈ R satisfy the following inequality: 2 −2αt 2 d T 2 −2αε T |x(t)| ≤ M0e |x(0)| , ∀t ≥ 0. (2.13) VHi + 2αVHi ≤ εx˙ (t)Hix˙(t) − ∗ e Yi HiYi. dt fi Moreover, if the LMIs (2.11) hold with α = 0, then system (2.21) (2.1) is exponentially stable with a small enough decay ∗ rate α = α0 > 0 for all ε ∈ (0, ε ]. Define a Lyapunov functional as V = V (x(t), x˙ t, ε) = VP + VG + VH , (2.22) Proof : Choose T wherex ˙ t =x ˙(t + θ), θ ∈ [−ε, 0]. By Jensen’s inequality VP = z (t)P z(t), P > 0. (2.14) (1.1), for all ε ∈ (0, ε∗] Then T hx(t)i hP −P i hx(t)i 2 d h V ≥ Vp + VG ≥ −2αε ≥ c1|x(t)| V = 2[x(t) − G]T P (A + ∆A)x(t) G(t) ∗ P + e R G(t) dt P av (2.23) N Z 1   Z t (2.15) X t i with ε-independent c1 > 0. Thus, V is positive-definite. − A f − θ x˙(s)dsdθ . i i ε i=1 0 t−εθ To compensate ∆Ax in (2.15) we apply S-procedure: we add to V˙ the left-hand part of To compensate G-term we will use as in Fridman and 2 2 2 Shaikhet (2016) λ(σ |x(t)| − |∆Ax| ) ≥ 0 (2.24) with some λ > 0. Then from (2.14)-(2.24), we have 1 Z t V = e−2α(t−s)(s − t + ε)2x˙ T (s)Rx˙(s)ds, R > 0. G d d 2 2 2 ε t−ε V + 2αV ≤ V + 2αV + λ(σ |x(t)| − |∆Ax| ) dt dt (2.16) N X We have ≤ ξT Φξ + ε∗x˙ T (t)(R + H )x ˙(t), d i V + 2αV = εx˙ T (t)Rx˙(t) i=1 dt G G (2.25) 2 Z t (2.17) − e−2α(t−s)(s − t + ε)x ˙ T (s)Rx˙(s)ds. where ε t−ε T T T T T T T ξ = [x (t),G ,Y1 , ··· ,YN , x (t)∆A ] (2.26) By Jensen’s inequality (1.1) and Φ is given by (2.12). Substituting into (2.25)x ˙ = t Z PN f A x and applying Schur complements, we conclude 2GT RG ≤ (s − t + ε)x ˙ T (s)Rx˙(s)ds. i=1 i i t−ε that if

4909 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

 N N  Therefore, (2.30) can be presented as a system with √ X X ε∗ f AT (R + H ) polytopic type uncertainty, where A ,...,A correspond  Φ i i j  1 4  i=1 j=1  to the four vertices:        0(N+2)n,n  < 0 (2.27) −1 1 −1 1   A = 2π ,A = 2π ,  N  1 γ2 −γβ + 1 2 γ2 − 1 −γβ + 1  X  ∗ ∗ −R − Hj  1 1   1 1  A = 2π ,A = 2π . j=1 3 γ2 −γβ − 1 4 γ2 − 1 −γβ − 1 d we have dt V + 2αV ≤ 0, ∀t ≥ t1, implying (2.32) 2 −2α(t−t1) By verifying the feasibility of LMIs (2.11) in the four c1|x(t)| ≤ V (xt) ≤ e V (t1), t ≥ t1. (2.28) ∗ vertices, where for simplicity we take α = 0 and f1 = PN T ∗ ∗ LMIs (2.11) imply (2.27) since (2.27) is affine in i=1 fiAi . ... = f4 = 0.5, we find an upper bound ε = 0.0031 that preserves the stability of (2.30) for all ε ∈ (0, ε∗]. For all ε ∈ (0, ε∗], V defined by (2.22) is upper bounded Numerical simulations under an arbitrary initial condition as Z t1 show that the system (2.30) is stable for a bigger upper h 2 2 i ∗ V (t1) ≤ c2 |x(t1)| + |x˙(s)| ds bound ε = 0.4755, which may illustrate the conservatism t1−ε of the proposed method. with ε-independent c2 > 0. For t ∈ [0, t1], x(t) satisfies Example 2.2. (Hetel and Fridman, 2013): stabilization by (2.1), where under A2 we have |A( t )| ≤ a for some a > 0 ε fast switching. Consider a switched system ∗ d 2 2 and all t ≥ 0, ε ∈ (0, ε ]. Hence, dt |x(t)| ≤ 2a|x(t)| for  at A1x(t), t ∈ [kε, kε + βε), t ∈ [0, t1] yielding |x(t)| ≤ e |x(0)| and |x˙(t)| ≤ a|x(t)| ≤  aeat|x(0)| for t ∈ [0, t ]. Therefore, V (t ) can be further x˙(t) = (2.33) 1 1  upper bounded as A2x(t), t ∈ [kε + βε, (k + 1)ε), h Z t1 i where ε > 0, k = 0, 1, ... and β ∈ (0, 1), with unstable 2at1 2 2 2 V (t1) ≤ c2 e |x(0)| + a |x(s)| ds modes t1−ε h i (2.29) 0.1 0.3  −0.13 −0.16 ≤ c e2at1 |x(0)|2 + ε a2e2at1 |x(0)|2 A = ,A = . (2.34) 2 1 1 0.6 −0.2 2 −0.33 0.03 ≤ c e−2αt1 |x(0)|2 3 Then (2.33) can be presented as (2.1) with for some ε-independent c3 > 0. Then (2.13) follows from t (2.28) and (2.29). A(τ) = f (τ)A + f (τ)A , τ = ∈ [k, k + 1), k = 0, 1, ... 1 1 2 2 ε The feasibility of the strict LMIs (2.11) with α = 0 implies where f1(τ) = χ[k,k+β)(τ) is the indicator function of the feasibility with the same decision variables and with a [k, k + β), f2(τ) = 1 − f1(τ). Choose β = 0.4 that leads to small enough positive α = α0, and thus guarantees a small Hurwitz enough decay rate. 2 Aav = βA1 + (1 − β)A2. Here A(τ) is periodic implying ∆A = 0 and σ = 0. Example 2.1. (Khalil (2002), Example 10.10): vibrational control. Consider the suspended pendulum with the sus- The bounds (2.10) in this example can be found as follows: pension point that is subject to vertical vibrations of small Z 1 Z β t ∗ 2 ∆ ∗ amplitude and high frequency. The linearized at the upper εf1( − θ)θdθ = εθdθ ≤ 0.5ε β = f1 , 0 ε 0 equilibrium position model is given by Z 1 Z 1 t ∗ 2 ∆ ∗  t  εf2( − θ)θdθ = εθdθ ≤ 0.5ε (1 − β ) = f2 . cos 1 0 ε β x˙(t) = ε x(t) (2.30)  t t  By verifying the feasibility of LMIs (2.11) with α = 0 γ2 − cos2 −γβ − cos ε ε in the two vertices, we find an upper bound ε∗ = 0.1871 with γ > 0, β > 0. Note that we linearized f given above that preserves the stability of (2.33) for all ε ∈ (0, ε∗]. ∗ (10.32) on p. 410 of Khalil (2002) at x1 = π, x2 = 0 Compared with ε = 0.1871 that is obtained in the theory, to derive (2.30). Similar to Remark 2.1, we change the numerical simulations show that the system (2.33) with time variable t = 2πt¯ and definex ¯(t¯) = x(2πt¯) = x(t), β = 0.4 is stable for a much bigger upper bound ε∗ = 37.8. therefore, Remark 2.2. The presented approach can be applied to  2πt¯  persistently excited systems: cos 1 T x¯˙(t¯) = 2π  ε  x¯(t¯) (2.31) x¯˙(t¯) = −εp(t¯)p (t¯)¯x(t¯), t¯≥ 0, (2.35)  2πt¯ 2πt¯ γ2 − cos2 −γβ − cos wherex ¯(t¯) ∈ n, p : [0, ∞) → n is measurable and ε ε R R ε > 0 is a small parameter. Here, similar to Pogromsky Then we obtain and Matveev (2017), it is assumed that function p has the  0 1  A = 2π . following properties: av γ2 − 0.5 −γβ Boundedness: there exists a constant M such that for It follows from Theorem 10.4 of Khalil (2002) that for almost all τ ≥ 0 γ2 < 0.5 and small enough ε, (2.30) is exponentially stable. T 2 We choose γ = 0.2 and β = 1. p(τ)p (τ) ≤ M In. Since A in (2.31) is ε-periodic, we have ∆A = 0 and Persistency of excitation: there is a constant ρ > 0 such σ = 0. Note that cos ∈ [−1, 1] and cos2 ∈ [0, 1]. that

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Z 1 T Liu, K., Selivanov, A., and Fridman, E. (2019). Survey p(τ − θ)p (τ − θ)dθ ≥ ρIn, ∀τ ≥ 1. on time-delay approach to networked control. Annual 0 Reviews in Control, 48, 57–79. The system (2.35) has been studied in Pogromsky and Mikheev, Y., Sobolev, V., and Fridman, E. (1988). Asymp- Matveev (2017); Zhang et al. (2019), where sufficient con- totic analysis of digital control systems. Automation and ditions are provided to guarantee the stability. In Pogrom- Remote Control, 49, 1175–1180. sky and Matveev (2017), a bound on the decay rate has Moreau, L. and Aeyels, D. (2000). Practical stability and been derived by introducing a novel non-quadratic Lya- stabilization. IEEE Transactions on Automatic Control, punov functional. Time-varying Lyapunov functions for 45(8), 1554–1558. PE were considered in Efimov and Fradkov (2015); Verrelli Pogromsky, A.Y. and Matveev, A. (2017). Stability anal- and Tomei (2019). Note that our time-delay approach to ysis of PE systems via Steklov’s averaging technique. averaging should lead to a time-independent quadratic International Journal of Adaptive Control and Signal Lyapunov functional and simple conditions in terms of Processing, 31(1), 138–144. LMIs. Solomon, O. and Fridman, E. (2013). New stability condi- tions for systems with distributed delays. Automatica, 3. CONCLUSION 49(11), 3467–3475. Teel, A.R., Moreau, L., and Nesic, D. (2003). A unified framework for input-to-state stability in systems with The presented time-delay approach to the averaging al- two time scales. IEEE Transactions on Automatic lows, for the first time, to derive efficient constructive Control, 48(9), 1526–1544. conditions on the upper bound of the small parameter Tikhonov, A.N. (1952). Systems of differential equations that preserves the stability. This method provides a di- containing small parameters in the derivatives. Matem- rect Lyapunov approach to linear fast-varying systems. aticheskii sbornik, 73(3), 575–586. It can be extended to input-to-state stability, to linear Vasilieva, A. and Butuzov, V. (1973). Asymptotic expan- fast-varying systems with state-delay and to persistently sions of solutions of singularly perturbed equations. excited systems. Verrelli, C. and Tomei, P. (2019). Non-anticipating Lya- punov functions for persistently excited nonlinear sys- REFERENCES tems. IEEE Transactions on Automatic Control, doi: 10.1109/TAC.2019.2940139. Bogoliubov, N. and Mitropolsky, Y. (1961). Asymptotic Zhang, B., Jia, Y., and Du, J. (2019). Uniform stability of methods in the theory of non-linear oscillations. CRC homogeneous time-varying systems. International Jour- Press. nal of Control, doi: 10.1080/00207179.2019.1585954. Cheng, X., Tan, Y., and Mareels, I. (2018). On robustness analysis of linear vibrational control systems. Automat- ica, 87, 202–209. Efimov, D. and Fradkov, A. (2015). Design of impulsive adaptive observers for improvement of persistency of excitation. International Journal of Adaptive Control and Signal Processing, 29(6), 765–782. Fridman, E. (1992). Use of models with aftereffect in the problem of the design of optimal digital-control systems. Automation and remote control, 53(10), 1523–1528. Fridman, E. (2002). Effects of small delays on stability of singularly perturbed systems. Automatica, 38(5), 897– 902. Fridman, E. (2014). Introduction to time-delay systems: analysis and control. Birkhauser, Systems and Control: Foundations and Applications. Fridman, E., Seuret, A., and Richard, J.P. (2004). Robust sampled-data stabilization of linear systems: an input delay approach. Automatica, 40(8), 1441–1446. Fridman, E. and Shaikhet, L. (2016). Delay-induced stabil- ity of vector second-order systems via simple Lyapunov functionals. Automatica, 74, 288–296. Hale, J.K. and Lunel, S.M.V. (1993). Introduction to Func- tional Differential Equations. Springer-Verlag, New- York. Hetel, L. and Fridman, E. (2013). Robust sampled–data control of switched affine systems. IEEE Transactions on Automatic Control, 58(11), 2922–2928. Khalil, H.K. (2002). Nonlinear Systems. Prentice Hall, 3rd edition. Kokotovic, P.V. and Khalil, H.K. (1986). Singular pertur- bations in systems and control. IEEE press.

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