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

On output-based accelerated stabilization of a chain of integrators: Implicit Lyapunov-Krasovskii functional approach ?

Artem N. Nekhoroshikh ∗,∗∗,∗∗∗ Denis Efimov ∗,∗∗,∗∗∗ Andrey Polyakov ∗,∗∗ Wilfrid Perruquetti ∗∗ Igor B. Furtat ∗∗∗,∗∗∗∗ Emilia Fridman †

∗ Inria, Univ. Lille, CNRS, UMR 9189 - CRIStAL, F-59000 Lille, France (e-mails: [email protected]; denis.efi[email protected]; [email protected]). ∗∗ Ecole Centrale de Lille, BP 48, Cit´eScientifique, 59651 Villeneuve-d’Ascq, France (e-mail: [email protected]) ∗∗∗ ITMO University, 49 Kronverkskiy av., 197101 Saint Petersburg, Russia (e-mail: [email protected]) ∗∗∗∗ IPME RAS, 61 Bolshoy av. V.O., 199178 Saint Petersburg, Russia (e-mail: [email protected]) † School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel (e-mail: [email protected])

Abstract: The problem of output accelerated stabilization of a chain of integrators is considered. Proposed control law nonlinearly depends on the output and its delayed values, and it does not use an observer to estimate the unmeasured components of the state. It is proven that such a nonlinear delayed control law ensures practical output stabilization with rates of convergence faster than exponential. The effective way of computation of feedback gains is given. It is shown that closed-loop system stability does not depend on the value of artificial delay, but the maximum value of delay determines the width of stability zone. The efficiency of the proposed control is demonstrated in simulations.

Keywords: Implicit Lyapunov-Krasovskii Functional (ILKF), nonlinear control, delay-dependent output feedback, Linear Matrix Inequalities (LMIs), practical stability

1. INTRODUCTION It is worth to mention that inducing artificial delays has other benefits also. For instance, in Mazenc and Malisoff Output feedback stabilization of dynamical systems is one (2016); Mazenc et al. (2019) this technique allows one of the central problems in the control theory and applica- to relax the smoothness requirements imposed on the tion. For linear systems with relative degrees greater than fictitious controls during backstepping design. one, output derivatives are necessary to construct a control In addition, in many applications convergence rate and law. Since sometimes they cannot be measured directly, the settling time are the main optimization criteria. In their estimates should be obtained, and usually for this such cases the problem of non-asymptotic (finite-time and goal the controller dynamics has to be augmented by an fixed-time) stable system design arises. For example, a con- observer Khalil (2002). In some cases, if our control has vergence rate of homogeneous systems could be changed to be embedded in an actuator with a low computational significantly by modifying only their degree of homogene- capacity, any decreasing the control low complexity and ity. Another conventional solution for acceleration consists dimension is appreciated. To this end one can use finite in feedback gains increasing. Nevertheless, for the time- time approximations of velocities (Selivanov and Fridman delayed systems such a method has a limited use: for any (2018)), which can be reconstructed based on the value of given delay h sufficiently large gains make the closed-loop position in the current instant of time and a delayed one. system unstable. It has been shown that delayed-induced feedback preserves the stability for small enough values of artificial delay Non-asymptotic stability analysis could be performed by (Borne et al. (2000); Fridman and Shaikhet (2016)). using different methods. For instance, homogeneous sys- tems described by ordinary differential equation can be investigated in a simple way. The main feature of such ? This work was partially supported by the Ministry of Science systems is that it is necessary and sufficient to study their and Higher Education of Russian Federation, passport of goszadanie behavior on a sphere in state space. However, time-delayed no. 2019-0898, by Government of Russian Federation (Grant 08-08), systems in general do not inherit such properties (Efimov by the grant of Russian Federation President (No. MD-1054.2020.8, et al. (2014)). agreement No. 075-15-2020-184).

Copyright lies with the authors 6060 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

1 One of the main theoretical tools for stability analysis • Ch is the space of continuously differentiable func- n and control synthesis of nonlinear systems is the Lya- tions [−(n−1)h, 0] → R with the norm k·kh defined 1 punov function method (Khalil (2002)). In Polyakov et al. as follows kΦkh := max kΦ(τ)k for Φ ∈ Ch; (2013) the Implicit Lyapunov Function (ILF) method has τ∈[−(n−1)h,0] 1,0 1 1 been extended to non-asymptotic analysis. Theorems on • Ch = {Φ ∈ Ch : Φ(0) = 0} is a subspace of Ch; n Implicit Lyapunov-Krasovskii Functional (ILKF) are pre- • diag{λi}j=1 is the diagonal matrix with the elements sented in Polyakov et al. (2015). Both implicit meth- λj ∈ R, j = 1, n on the main diagonal; ods allow to check all stability conditions by analyzing • if P ∈ Rn×n is symmetric, then the inequalities P > 0 the algebraic equations. Therefore, a simple and effective (P < 0) and P ≥ 0 (P ≤ 0) mean that P is positive procedure for parameter tuning based on Linear Matrix (negative) definite and semidefinite, respectively; Inequalities (LMIs) can be obtained (see, for example, • λmin(P ) and λmax(P ) are the minimal and maximal Lopez-Ramirez et al. (2018)). eigenvalues of a symmetric matrix P ∈ Rn×n, respec- tively; Nevertheless, stability analysis of time-delay systems can n×n n×1 be done not only in the time domain. For example, in • In ∈ R is the identity matrix and On ∈ R is the zero column; Kharitonov et al. (2005) necessary conditions for the j
j! existence of multiple delay controllers are presented in • i := i!(j−i)! is the binomial coefficient; n terms of Hurwitz stability of some polynomials. However, • ||A := infη∈A k − ηk is the distance between  ∈ R feedback gains and admissible delays could be found only and A ⊆ Rn. by solving transcendental equations. The present work studies the problem of nonlinear output 2.2 Auxiliary lemmas delay-dependent feedback design providing a practical ac- celerated (faster than exponential) stabilization of a chain Lemma 1. (Solomon and Fridman (2013)). Let f, κ :[a, b] of integrators. Although the chain of integrators is a quite → [0, ∞) and φ :[a, b] → R be such that integration simple model, it is also a very useful benchworking tool concerned is well defined. Then: b b b since all linear controllable systems and many nonlinear h Z i2 Z Z ones can be transformed into this particular form. κ(s)φ(s)ds ≤ f −1(s)κ(s)ds f(s)κ(s)φ2(s)ds. The goal of this work is to extend the results obtained in a a a Efimov et al. (2018) for the case of a second order system Lemma 2. (Lopez-Ramirez et al. (2018)). For ∀s ∈ [0, 1], β to the chain of n ≥ 2 integrators. However, in this paper β ∈ R>0 \{1} the function gβ(s) := |s − s| admits the the closed-loop system is not homogeneous, therefore, for following estimate stability analysis Implicit Lyapunov-Krasovskii functional 1/(1−β) max gβ(s) ≤ gβ(β ). is introduced. It is proven that the proposed control law s∈[0,1] ensures practical output stabilization with the rates of convergence faster than exponential. In contrast to Efimov 3. IMPLICIT LYAPUNOV-KRASOVKSII et al. (2018) the effective way of computation of feedback FUNCTIONAL APPROACH gains is given. Moreover, it is shown that closed-loop system stability does not depend on the value of artificial Consider the system of the form delay. Nevertheless, maximum value of delay determines 1 the width of stability zone. x˙(t) = f(xt), t ∈ R>0, x0 = Φ ∈ D ⊆ Ch, (1) where x(t) ∈ n, x ∈ 1 is the state function defined by The outline of this work is as follows. The notations and R t Ch xt(τ) := x(t + τ) with τ ∈ [−(n − 1)h, 0] with h > 0 (time auxiliary lemmas are given in Section 2. The Implicit 1 n delay) and f : Ch → R is a continuous operator. Assume Lyapunov-Krasovskii approach for stability analysis of that the origin is an equilibrium point of the system (1), time-delay systems is introduced in Section 3. The finite- i.e. f(0) = 0. A solution of the system (1) with the initial differences approximation of derivatives is touched upon in function Φ ∈ D is denoted by x(t, Φ). Section 4. The problem statement and the control design with stability analysis are considered in section 5. An In applications frequently it is only required to stabilize the example is presented in Section 6. origin of the system in some zone, the width of which is determined by technical requirements. The next definition 2. PRELIMINARIES present an asymptotically stability with respect to the set. Definition 3. The system (1) is said to be asymptotically 2.1 Notations stable with respect to the set A, if it is: Through the paper the following notations will be used: a) Lyapunov stable with respect to the set A: for any ε > 0 there exists ∆(ε) > 0 such that • is the field of natural numbers; N |x(t, Φ)|A ≤ ε for all t ≥ 0 and Φ ∈ D : kΦkh ≤ ∆; • a series of integers 1, 2, . . . , n is denoted by 1, n; b) attractive with respect to the set A: • is the field of real numbers, := {x ∈ : x > 0} R R>0 R |x(t, Φ)|A → 0 as t → ∞ for any Φ ∈ D. and R≥0 := R>0 ∪ 0; • Ci is a class of i times continuously differentiable If the set could be chosen as A = 0, then the origin of the functions R>0 → R; system (1) is asymptotically stable. The set D is called the α α 1 • d·c := sign(·)| · | ; domain of attraction of the system (1). If D = Ch, then n • k · k is the Euclidean norm in R ; the corresponding stability becomes global. 

6061 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

In this paper only asymptotic stability is considered. brevity the proof is omitted. Nevertheless, it is worth to Therefore, hereafter stability relates to asymptotic one. note that condition C5) provides V˙ ≤ −θV 1+µ, since ˙ 0 0 ˙ Before formulating the theorem concerning stability anal- Q := Qt(V, xt)+QV (V, χ)V = 0 for all (V, xt) ∈ Ω. Hence, ysis of time-delay systems using ILKF, a special class of Theorem 5 implies the stability and the transition time comparison functions introduced by the following defini- estimate (4). 2 tion. 4. FINITE-DIFFERENCES APPROXIMATION OF Definition 4. (Polyakov et al. (2015)). The function q : 2 DERIVATIVES R>0 → R, (ρ, s) → q(ρ, s) is said to be of the class IK∞ if and only if: In this section the exact form of finite-differences approx- (i) 2 imation error δi(t) :=y ˜i(t) − y (t) is presented. 1) q is continuous on R>0; i 2) for any s ∈ R>0 there exists ρ ∈ R>0 such that Proposition 6. (Selivanov and Fridman (2018)). If y ∈ C q(ρ, s) = 0; and y(i) is absolutely continuous with i ∈ N, then the 3) for any fixed s ∈ R>0 the function q(·, s) is strictly following relation holds: decreasing on >0; R y˜i−1(t) − y˜i−1(t − h) (i) 4) for any fixed ρ ∈ the function q(ρ, ·) is strictly δi(t) = − y (t) R>0 h increasing on >0; Z 0 (5) R (i+1) 5) = − ϕi(−τ)y (t + τ)dτ, lim ρ = lim s = 0, lim ρ = ∞, −ih + + s→∞ s→0 ρ→0 (ρ,s)∈Γ wherey ˜0(t) := y(t) and (ρ,s)∈Γ (ρ,s)∈Γ ϕ (ξ) := 1 − ξ/h, ξ ∈ [0, h] where Γ = {(ρ, s) ∈ 2 : q(ρ, s) = 0}. 1 R>0  Z ξ  ϕi(λ) h − ξ The next theorem on Implicit Lyapunov-Krasovskii func-  dλ + , ξ ∈ [0, h],  0 h h tional presents conditions that guarantee stability of the Z ξ  ϕi(λ) (6) nonlinear time-delayed system. ϕ (ξ) := dλ, ξ ∈ (h, ih), i+1 h  ξ−h Denote partial derivatives of functional Q(V, χ) at time Z ih ϕ (λ)  i dλ, ξ ∈ [ih, ih + h]. 0 ∂Q(V,χ)   instant t as follows: Qt(V, xt) := ∂χ f(xt) and ξ−h h χ=x t 0 ∂Q(V,χ) QV (V, xt) := ∂V . For further stability analysis some characteristics of the χ=xt functions ϕi are introduced. Theorem 5. If there exists a continuous functional Q : 1 Proposition 7. The functions ϕi(ξ) defined in (6) and R>0 × Ch → R such that: R ih ψi(ξ) := ϕi(λ)dλ possess the following properties: 1 ξ C1) for all χ ∈ Ch the function (V, τ) → Q(V, χ(τ)) is 0 continuously differentiable; P1) ϕi ≤ 0; 1 C2) for any χ ∈ Ch there exists V ∈ R>0 such that P2) ϕi(ξ) + ϕi(ih − ξ) = 1; Q(V, χ) = 0; P3) ψi(0) = ih/2 and ψi(ih) = 0; 0 C3) there exist qi ∈ IK∞, i = 1, 2 such that for all P4) ψi(ξ) = −ϕi(ξ); V ∈ R>0 P5) ϕi(ξ) is concave on ξ ∈ [0, ih/2] and convex on 1 1,0 ξ ∈ [ih/2, ih] for i ≥ 2; q1(V, kχ(0)k) ≤ Q(V, χ), ∀χ ∈ Ch \ Ch , 1 P6) ψi(ξ) ≤ (ih/2)ϕi(ξ); Q(V, χ) ≤ q (V, kχk ), ∀χ ∈ \{0}; 2 2 h Ch R ih ϕi (ξ) P7) ζi := dξ does not depend on h; 0 1 0 ψi(ξ) C4) QV (V, χ) < 0 for all V ∈ R>0 and χ ∈ Ch such that (i) Q(V, χ) = 0; P8) for all xi+1(t) = y (t) it holds: 1 Z 0 C5) for all (V, xt) ∈ Ω = {(V, xt) ∈ (Vmin,Vmax) × Ch : 2 2 2 Q(V, x ) = 0} such that x satisfies (1), we have ϕi(−τ)x ˙ i+1(t + τ)dτ ≥ δi (t), (7a) t t −ih ih 0 1+µ 0 Q (V, xt) ≤ θV Q (V, xt), ∀t ∈ >0, Z 0 t V R 2 1 2 ψi(−τ)x ˙ (t + τ)dτ ≥ δ (t). (7b) where µ ∈ (−1, 0) ∪ >0, θ > 0, Vmin,Vmax ∈ ≥0. i+1 i  R R −ih ζi Then the system is stable with the domain of attraction 1 Proof of the properties P1)–P4) can be found in Selivanov D := {Ψ ∈ Ch : Q(Vmax, Ψ) ≤ 0} (2) and Fridman (2018). Other properties can be proven using (globally stable if Vmax = +∞) with respect to the set P1)–P4) and Lemma 1, but due to space limitation are not 1 presented in this paper. A := {Ψ ∈ Ch : Q(Vmin, Ψ) ≤ 0} (3) (at the origin if Vmin = 0). It is worth to note that ζi is well-defined. Indeed, function 2 1 ϕ˜i(ξ) := ϕi (ξ)/ψi(ξ) is continuous on ξ ∈ [0, ih) and The transition between sets S1 := {Ψ ∈ Ch : Q(V1, Ψ) = 1 ϕ˜i(ih−) = 0. Therefore, functionϕ ˜i(ξ) is integrable. 0} and B2 := {Ψ ∈ Ch : Q(V2, Ψ) ≤ 0} is bounded as: −µ −µ T (µ, S1, B2) ≤ (V2 − V1 )/(θµ), (4) 5. NONLINEAR DELAY-DEPENDENT CONTROL where Vmin ≤ V2 < V1 ≤ Vmax.  5.1 Problem statement Proof. This Theorem can be proven in the same way as Theorems 2 and 3 in Polyakov et al. (2015) and for Consider a chain of n ≥ 2 integrators:

6062 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

 x˙(t) = Ax(t) + Bu(t), Φ := XA> + Y >B> + AX + BY + (ζ + 2)aI , (8) 11 n y(t) = Cx(t), Z i ϕ2(ξ¯) σ := g (α1/(1−αj )), ζ := max i dξ¯ where x(t) ∈ Rn is the state vector, u(t) ∈ R is the control j αj j R i ¯ ¯ i=1,n−1 0 ¯ ϕi(λ)dλ input, y(t) ∈ R is the output available for measurements ξ and be feasible for some S, a, b > 0, X ∈ Rn×n, X = X> > 0,   1×n On−1 In−1 > Y ∈ R , with j = 1, n. A = > ,B = [0 ... 0 1] ,C = [1 0 ... 0] . 0 On−1 Then for any h > 0 the system (8), (11) with K := YX−1: The goal is to design an output-feedback control practi- cally stabilizing the system (8) with the rate of conver- a) is globally stable with respect to the set A (3), where −1/µ gence faster than exponential. Vmin := ((n − 1)hS) , if µ < 0; b) is stable at the origin with the domain of attraction −1/µ 5.2 Main result D (2), where Vmax := ((n − 1)hS) , if µ > 0; c) converges from the set S1 to the set B2 with time 2 Define nonlinear delay-dependent control algorithm as estimate (4), where 1/θ := 2ω(µ)S .  follows: The proof of Theorem 8 is given in Appendix A. u = Kdx˜cα (9) where Corollary 9. If LMIs (13) from Theorem 8 are feasible > for some µmax, then they also hold for all µ = γµmax, K = [K ...K ] , dx˜cα = [dx˜ cα1 ... dx˜ cαn ] , 1 n 1 n 0 ≤ γ ≤ 1.  x˜ (t) − x˜ (t − h) x˜ (t) := x (t), x˜ (t) := i i , i = 1, n − 1, 1 1 i+1 h Proof. Indeed, rj(µ) could be rewritten as: αj := 1/rj, rj(µ) := 1 − (n + 1 − j)µ, j = 1, n. (10) rj(µ) = (1−γ)+γ−(n+1−j)γµmax = (1−γ)+γrj(µmax). Let us rewrite control law (9) in a form suitable for Therefore, condition (13a) can be expressed as follows practical implementation: 0 < 2(1 − γ)X + γ(XHrmax + Hrmax X) ≤ 2ω(µ)X, n j−1 i   X l X (−1) j − 1 kαj where rmax = r(µmax). u(t) = Kj zi(t) , (11) hj−1 i Clearly, this condition holds for all µ = γµ , 0 ≤ γ ≤ 1, j=1 i=0 max since 0 < XHrmax + Hrmax X ≤ 2ω(µmax)X and ω(µ) = where (1 − γ) + γω(µ ). y(t − ih), t > ih, max z (t) := i y(0), t ≤ ih. And taking into account that: rj Therefore, control (11) depends on only n + 2 parameters: ∂σj 1−rj n + 1 − j 1/(1−αj ) µ = µrj ln (rj) sign(gαj (αj )) > 0, h > 0 is the artificial delay, µ ∈ (−1, 0) ∪ (0, 1/n) is ∂µ 1 − rj the degree of nonlinearity and Kj < 0, j = 1, n are one can deduce that max kσk = kσ(µmax)k. Hence, in- feedback gains. The restrictions on effective selection of µ these parameters and the conditions to check are given in equality (13c) also holds for all µ = γµmax. 2 the following theorem. Therefore, Corollary 9 postulates that sufficiently small µ, Before formulating the main result, let us introduce an for which LMIs (13) are feasible, always could be found. Implicit Lyapunov-Krasovskii functional (ILKF) by the equality: The following proposition claims that for any feedback > −r −r gains K, obtained as the solution of LMIs (13) from Q(V, χ) := −1 + χ (0)ΛV P ΛV χ(0) Theorem 8, and fixed delay h, the nonlinear closed-loop n−1 X Z 0 2ζ h i2 (12) system (8), (11) with µ 6= 0 is always converging faster +V µ ψ (−τ) V −ri+2 χ˙ (τ) dτ, i ihS i+1 than its linear analog with µ = 0 between properly selected i=1 −ih sets S1 and B2, which depend on µ and h. > −r −rj n where P = P > 0 and ΛV := diag{V }j=1. Proposition 10. Let conditions of Theorem 8 hold for some Theorem 8. Let for some µ ∈ (−1, 0) ∪(0, 1/n) the system µ ∈ (−1, 0) ∪ (0, 1/n). Then there exist two sets S1 := 1 of LMIs: {V1 ≥ 1, Ψ ∈ Ch : Q(V1, Ψ) = 0} and B2 := {V2 ≤ 1, Ψ ∈ 1 : Q(V , Ψ) ≤ 0} such that in the system (8) and the 0 < XHr + HrX ≤ 2ω(µ)X (13a) Ch 2 control (11): XY > XI  aI X  ≥ 0, n ≥ 0, n ≥ 0, Y 1 I SI XSI T (µ, S1, B2) < T (0, S1, B2) (14) n n n (13b) b S 1 for fixed K and h > 0 satisfying ≥ 0,X ≤ I , −µ S 1 2 n V /((n − 1)S) for µ < 0, h ≤ 2  >  V −µ/((n − 1)S) for µ > 0. Φ11 bkσkBBY 1  ∗ −S 0 bkσk   ≤ 0, (13c)  ∗ ∗ −4ζ(1 − a) 1  A procedure for selection of sets S1 and B2 is given in the ∗ ∗ ∗ −S/ζ proof below. where Proof. For linear system µ = 0 the time of convergence  n 1 − (n + 1)µ, µ < 0, between sets S1 and B2 is lower-bounded: Hr := diag{rj}j=1, ω(µ) := 1, µ > 0, T (0, S1, B2) ≥ T0(h)(ln V1 − ln V2),

6063 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

5 where T0(h) is the fastest time of transition between levels 10 |x(t)| V 0 = e and V 0 = 1 for µ = 0, given delay h and fixed K. Linear output feedback 1 2 Nonlinear output feedback 10-5 Linear state feedback From (4) it follows that condition (14) is verified if: Nonlinear state feedback -15 −µ −µ 10 (V2 − V1 )/(θµ) ≤ T0(h)(ln V1 − ln V2) -25 or taking into account property c) from Theorem 8 10

−µ −µ 10-35 V1 − V2 T0(h) −µ −µ < for µ < 0, 2 -45 ln V1 − ln V2 2(1 − (n + 1)µ)S 10 −µ −µ t V2 − V1 T0(h) , s < for µ > 0, 10-55 −µ −µ 2 0 25 50 75 100 125 ln V2 − ln V1 2S which for any µ ∈ (−1, 0) (µ ∈ (0, 1/n)), h > 0 and Fig. 1. Trajectories of stabilized system with µ = −0.01

V2 ∈ [Vmin, 1] (V1 ∈ [1,Vmax]) could be guaranteed by 25 10 |x(t)| decreasing V1 (increasing V2). Linear output feedback 20 10 Nonlinear output feedback Conditions on h guarantee that A ⊂ B2 for µ < 0 and Linear state feedback 1015 S1 ⊂ D for µ > 0. 2 Nonlinear state feedback Remark 11. Since the system (8), (11) keeps a constant 1010 value of µ for all t ≥ 0, then it is clear that the set S1 105 represents the set of initial conditions. 100 −µ Remark 12. Taking into account that V2 ≤ 1 for µ < 0 -5 −µ 10 and V1 ≤ 1 for µ > 0, it follows that the maximum value 10-10 of artificial delay is bounded as: t, s 10-15 hmax := 1/((n − 1)S). (15) 0 25 50 75 100 125 On the one hand, the greater values of h leads to the bigger Fig. 2. Trajectories of stabilized system with µ = 0.01 stable set for µ < 0 and smaller domain of attraction for µ > 0. On the other hand, the smaller values of 7. CONCLUSION h could make the control system (11) unimplementable. Therefore, there is the minimum value of artificial delay The paper addresses the problem of output feedback stabilization of a chain of integrators using a nonlinear hmin depending on technical realization. delay-dependent control law that achieves the rates of convergence faster than exponential. The effective way of computation of feedback gains is presented. It is shown 6. EXAMPLE that the value of artificial delay does not affect closed- loop system stability, but determines the width of stability Consider a chain of three integrators to show the main ad- zone. Simulation results show feasibility of the proposed vantages of the proposed control law. Selecting µ = −0.01 control scheme. The efficiency of the proposed approach and µ = 0.01 one can obtain, respectively, the follow- is demonstrated in simulations, and a comparison with ing feedback gains K− = [−0.9754, −2.4398, −1.9303], a linear analogue as well as nonlinear and linear state K+ = [−0.9667, −2.4473, −1.9459] and S− = 11.4824, feedback is carried out. S+ = 11.5149. According to (15), the maximum value of artificial delay REFERENCES is hmax = 0.0434 s. Therefore, let us choose the nominal Borne, P., V, K., and Shaikhet, L. (2000). Stabilization value as h = 0.025 s to enlarge the stability zone and to of inverted pendulum by control with delay. Dynamic keep the control system implementable. Systems and Applications, 9, 501–514. The norm of the state x(t) obtained in simulation of the Efimov, D., Polyakov, A., Fridman, E., Perruquetti, W., system (8) governed by the proposed nonlinear output and Richard, J.P. (2014). Comments on finite-time delay-induced feedback (11) (blue solid line) in comparison stability of time-delay systems. Automatica, 50(7), with linear one (red solid line), as well as nonlinear (blue 1944–1947. dashed line) and linear (red dashed line) state feedback, Efimov, D., Fridman, E., Perruquetti, W., and Richard, are depicted in figures 1 and 2 in logarithmic scale. For the J.P. (2018). On hyper-exponential output-feedback sake of clarity, the feedback gains K are chosen the same stabilization of a double integrator by using artificial for all control schemes as well as parameter µ for both delay. 1–5. nonlinear feedback and artificial delay for delay-dependent Fridman, E. and Shaikhet, L. (2016). Delay-induced stabil- control laws. ity of vector second-order systems via simple lyapunov functionals. Automatica, 74, 288–296. Obviously, in the nonlinear case the rate of convergence is Khalil, H.K. (2002). Nonlinear systems; 3rd ed. Prentice- faster than in the linear one close to the origin for µ < 0 Hall, Upper Saddle River, NJ. and far outside for µ > 0. Moreover, proposed control law Kharitonov, V., Niculescu, S.I., Moreno, J., and Michiels, and nonlinear state feedback show the same performance. W. (2005). Static output feedback stabilization nec-

6064 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

essary conditions for multiple delay controllers. IEEE −r µ −r −r Taking into account that ΛV A = V AΛV and ΛV BK = Transactions on Automatic Control, 50(1), 82–86. V −rn BK = V −1+µBK, the first term in (A.2) could be Lopez-Ramirez, F., Efimov, D., Polyakov, A., and Perru- rewritten as follows: quetti, W. (2018). Fixed-time output stabilization and Σ = 2V µx>Λ−rP (A + BK)Λ−rx fixed-time estimation of a chain of integrators. Interna- 1 V V (A.3) +2V µx>Λ−rP BKd + 2V µx>Λ−rP BKd , tional Journal of Robust and Nonlinear Control, 28(16), V 1 V 2 4647–4665. where −1 α −r −r α −r Mazenc, F., Burlion, L., and Malisoff, M. (2019). Stabi- d1 := V dx˜c − ΛV x˜ = dV x˜c − ΛV x,˜ lization and robustness analysis for a chain of saturating −r −r −r > d2 := Λ x˜ − Λ x = Λ [0, δ1, . . . , δn−1] . integrators with imprecise measurements. IEEE Control V V V Obviously, disturbance terms d and d represent nonlin- Systems Letters, 3(2), 428–433. 1 2 ear deviation of feedback and finite-differences approxima- Mazenc, F. and Malisoff, M. (2016). New control design for tion error respectively. bounded backstepping under input delays. Automatica, 66, 48–55. The second term in (A.2) either can be upper-bounded by > 2 Polyakov, A., Efimov, D., and Perruquetti, W. (2013). using property P6) and supposing that d1 d1 ≤ kσk : Finite-time stabilization using implicit lyapunov func- −µ > µ  V h d1 d1 > −r −r i tion technique. IFAC Proceedings Volumes (IFAC- Σ2 ≤ −V S 1 + 2 − 2x ΛV P ΛV x PapersOnline), 46(23 PART 1), 140–145. (n − 1)hS kσk Polyakov, A., Efimov, D., Perruquetti, W., and Richard, or by using (7a): J.P. (2015). Implicit lyapunov-krasovski functionals  V −µ 2 Σ ≤ −4ζV µ Sd>d . for stability analysis and control design of time-delay 2 (n − 1)hS 2 2 systems. IEEE Transactions on Automatic Control, It is easy to see that the term Σ could compensate cross- 60(12), 3344–3349. 2 terms in (A.3) for V ≥ V if µ < 0 or V ≤ V if Selivanov, A. and Fridman, E. (2018). An improved time- min max µ > 0. Therefore, only global stability with respect to the delay implementation of derivative-dependent feedback. set A (3) and stability at the origin with the domain of Automatica, 98, 269–276. attraction D (2) could be obtained for µ < 0 and µ > 0 Solomon, O. and Fridman, E. (2013). New stability con- respectively. Hence, we deduce: ditions for systems with distributed delays. Automatica −2 µ −2 > > −r −r (Journal of IFAC), 49, 3467–3475. Σ2 ≤ −S V (1 + kσk d1 d1 − 2x ΛV P ΛV x) −3 µ > −4ζ(1 − S )V Sd2 d2. Appendix A. PROOF OF THEOREM 8 Taking into account that from (13b) it follows that: > 2 2 3 2 K K ≤ P ≤ SIn ≤ aS P ⇒ 1/S ≤ a, S ≤ b, Let us show that ILKF (12) satisfies all conditions from the term Σ could be finally estimated as follows: Theorem 5. Since proof of conditions C1)–C3) can be done 2 −2 µ −2 µ > > in a straightforward way (see, for example, Polyakov et al. Σ2 ≤ −S V − S(bkσk) V d1 K Kd1 µ > −r 2 −r µ > > (A.4) (2015)), we just focus on conditions C4) and C5). +2aV x ΛV P ΛV x − 4ζ(1 − a)V d2 K Kd2. −1 2 A. Proof of condition C4) of Theorem 5 Since S In ≤ aP , the third term in (A.2) is bounded: Σ ≤ ζV µ(ax>Λ−rP 2Λ−rx + S−1[V −1x˙ ]2), (A.5) The derivative of Q(V, χ) with respect to V is: 3 V V n where 0 > −r −r −1 −r VQV (V, χ) = −χ(0) Λ [HrP + PHr]Λ χ(0) V V V x˙ n = KΛV x + Kd1 + Kd2. n−1 Z 0 2ζ h i2 Combining (A.3)–(A.5), applying the Schur complement µ X −ri+2 −V (2ri+2 − µ) ψi(−τ) V χ˙ i+1(τ) dτ. > −r −1 > ihS with respect to the vector [x ΛV , (bkσk) Kd1, Kd2] i=1 −ih 0 −2 µ and using (13c), we deduce that Qt(V, xt) ≤ −S V . Taking into account (13a), one can see that: Taking into account (A.1), we conclude the proof. 2 0 < HrP + PHr ≤ 2ω(µ)P, max 2ri+2 − µ < 2ω(µ). > 2 i=1,n−1 C. Proof of the estimate d1 d1 ≤ kσk > So we finally conclude that The nonlinear disturbance term d1 d1 can be rewritten as: −ω(µ) ≤ VQ0 (V, χ) < 0 (A.1) n h i2 V > X −rj αj −rj 1 d1 d1 = |V x˜j| − |V x˜j| . (A.6) for all V ∈ R>0 and χ ∈ Ch such that Q(V, χ) = 0. Therefore, the condition C4) of Theorem 5 hold. 2 j=1 Applying (7b) to (12) for all (V, xt) ∈ Ω, we deduce that: B. Proof of condition C5) of Theorem 5 n−1 X 2ζ V −µ h i2 1 ≥ x>Λ−rP Λ−rx + V −ri+1 δ . If x(t) is the solution of the system (8), (9), then using V V ζ ihS i properties P3) and P4), we obtain i=1 i −µ h i Since ζ ≥ ζi for any h > 0 due to P7) and V ≥ (n−1)hS, Q0 (V, x ) = 2x>Λ−rP Λ−r Ax + BKdx˜cα t t V V the following inequality holds for P ≥ 2In: n−1 0 X Z 2ζ h i2 n h i2 n−1 h i2 n h i2 µ −ri+2 X −rj X −ri+1 X −rj −V ϕi(−τ) V x˙ i+1(t + τ) dτ 1 ≥ 2 V xj + 2 V δi ≥ V x˜j . −ih ihS (A.2) i=1 j=1 i=1 j=1 n−1 2 ζ h i −rj µ X −ri+2 Hence, it follows that |V x˜j| ≤ 1 and, applying Lemma +V V x˙ i+1(t) = Σ1 + Σ2 + Σ3. S > 2 i=1 2 to (A.6), one can finally conclude that d1 d1 ≤ kσk . 2

6065