On Hyper Exponential Stabilization of Linear State-Delay Systems Andrey Polyakov, Denis Efimov, Emilia Fridman, Wilfrid Perruquetti, Jean-Pierre Richard
To cite this version:
Andrey Polyakov, Denis Efimov, Emilia Fridman, Wilfrid Perruquetti, Jean-Pierre Richard. OnHyper Exponential Stabilization of Linear State-Delay Systems. Conference on Decision and Control 2017, Dec 2017, Melbourne, Australia. hal-01587780
HAL Id: hal-01587780 https://hal.inria.fr/hal-01587780 Submitted on 14 Sep 2017
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On Hyper Exponential Stabilization of Linear State-Delay Systems
Andrey Polyakov, Denis Efimov, Emilia Fridman, Wilfrid Perruquetti and Jean-Pierre Richard
Abstract— A control design algorithm for hyper exponential extends the results of [2] and relaxes the restrictions to stabilization of multi-input multi-output linear control system the plant model. The key differences are as follows: the with state-delays is presented based on method of Implicit problem of global hyper exponential stabilization is studied; Lyapunov-Krasovskii Functional (ILKF). The procedure of con- trol parameters tuning is formalized by means of Linear Matrix the control design procedure is presented for multi input Inequalities (LMIs). The theoretical results are supported with system; the restriction to the matrix of the state-delayed numerical simulations. term is relaxed; the issues of practical implementation of the obtained implicit control law are also studied. In particular, I.INTRODUCTION we present a sampled approach to ILKF-based control real- Convergence rate is one of control performance indexes, ization and we prove that it is robust with respect to variation which quantitatively characterizes the transient speed of a of sampling period. controlled process. Its tuning is always required in order Notation: R is the field of real numbers, R+ = {x ∈ R : to fulfill some time constraints of control system. From x > 0}; N is the set of natural numbers; k·k is the Euclidian mathematical point of view convergence rate is well specified norm in Rn; C(X,Y ) is the space of continuous maps X → in the context of the stability theory [1]. Indeed, exponential Y , where X,Y are some subsets of finite dimensional spaces; stability determine the convergence rate of linear ordinary n 0 Ch = C([−h, 0], R ) and Ch := {ϕ ∈ Ch : ϕ(0) = 0} is differential equations. To define ”fast” control this paper the linear subspace of Ch; k · kh is the uniform norm in Ch; n×n uses linear system as the reference point for comparison In ∈ R - the identity matrix; 0 denotes zero element, n of convergence rate. Namely, a nonlinear system is said to e.g. 0 ∈ R is the zero vector but 0∈Ch is the zero-valued be fast if it demonstrates transients motions faster than any function; diag{λ1,.., λn} - diagonal matrix; positive definite linear one, i.e. if the convergence rate of the nonlinear system continuous function σ : R → R belongs to the class K if it is faster than any exponential. In [2] such systems were called is strictly increasing on R+ and σ(0) = 0; if additionally it hyper exponential. The so-called finite-time and fixed-time is radially unbounded then σ belongs to K∞. stable nonlinear systems (see, e.g. [3], [4], [5], [6], [7], [8], [9], [10]) are also locally/globally hyper exponentially stable. II.PROBLEM STATEMENT Modern control theory basically deals with control of A. System Description and Basic Assumptions mathematical models, which are just an approximation of a real (physical) plant under some assumptions on its behavior. Let us consider the linear state-delay control system Linear models are still the most popular way to describe x˙(t) = Ax(t) + Lx(t − h) + Bu(t), (1) behavior of a plant. Time delays are usual effects in control systems [11], [12], [13], [14]. They appear in the form x(τ) = ϕ0(τ), τ ∈ [−h, 0], (2) of transport delays (due to a finite speed of propagation n of information) as well as models of actuators or due to where x(t) ∈ R is the vector of the current state, u(t) ∈ m n×n approximation of models of mathematical physics described R is the control input, A, L ∈ R are system matrices, n×m by partial differential equations. B ∈ R is the matrix of control gains, h > 0 is the state This paper presents a control design algorithm for hyper delay, ϕ ∈ Ch is the vector-valued function, which defines exponential stabilization of a multi-input multi-output lin- the initial state of the time-delay system. ear control system with state-delays. The ILKF method is Assumption 1: We assume that all parameters of the sys- utilized for this purpose. To design a control law, which tem (1) are known, the pair {A, B} is controllable and the realizes global hyper exponential stabilization of the state- matrix B is of full column rank, i.e. rank(B) = m ≤ n. delay system we use two properly adapted ILKF. This paper An additional assumption concerning the matrix L is be given below (see Assumption 2). The paper is partially supported by ANR Finite4SoS and Russian Science Foundation under grant 17-19-01422 (in Section V). B. Control aim Andrey Polyakov, Denis Efimov, Wilfrid Perruquetti and Jean-Pierre Richard are with Inria Lille, 40. av Halley, Villenueve d’Ascq, France, (e- We restrict a class of admissible feedback control laws to mail: andrey.polyakov(denis.efimov, wilfrid.perruquetti)@inria.fr). m continuous operators u˜ : Ch → R such that Emilia Fridman is with Department of Electrical Engineering and Sys- tems, Tel Aviv University, 69978, Israel (e-mail: [email protected]) u(t) =u ˜(x (t)), u˜(0) = 0, (3) Andrey Polyakov and Denis Efimov are also with ITMO University, 49 h Kronverkskiy av., 197101, Saint-Petersburg, Russia Wilfrid Perruquetti and Jean-Pierre Richard are also with Ecole Centrale where xh(t) ∈ Ch, xh(t)(θ) = x(t + θ), −h ≤ θ ≤ 0 is the de Lille, Villenueve d’Ascq, France(e-mail: [email protected]). distributed state of the time delay system (1). 0 10 where k · kZ and k · kY are norms in the Banach spaces Z and Y, respectively. i =0 Theorem 1: [2] If there exists a continuous functional Q : R+ × Ch → R such that: i =1 C1) Q is continuously F-differentiable on R+ × Ch; C2) for any y ∈Ch there exists V ∈R+ such that Q(V, y)=0; )
t ∂Q(V,y) ( α −1 C3) < 0 for all V ∈ + and y ∈ Ch; i, ∂V R
ρ 10
− C4) there exist qi ∈ IK∞, i = 1, 2 such that for all V ∈ +
e R 0 q1(V, ky(0)k) ≤ Q(V, y), ∀y ∈ Ch\Ch, Q(V, y) ≤ q2(V, kykh), ∀y ∈ Ch\{0}, i =2
C5) for all (V, xh(t)) ∈ Ω with x(t) satisfying (1) we have
∂Q(V,xh(t)) ≤ −σ(kx(t)k), ∀t ∈ +, −2 ∂t R 10 0 0.2 0.4 0.6 0.8 1 t where σ : R → R and Fig. 1. Hyper exponential rate of convergence Ω = {(V, y) ∈ R+ × Ch : Q(V, y) = 0} . (6) The goal of the paper is to design a control law u˜, which Then the origin of the system (1) is Lyapunov stable if σ is stabilizes the origin of the system with a hyper exponential a nonnegative function and asymptotically stable if σ ∈ K. convergence rate in the sense of Definition 1. It is worth stressing that V is treated as a scalar parameter > r+1 Given vector α = (α0, α1, ..., αr) ∈ R+ with r ≥ 0 in all conditions of Theorem 1. However, the conditions C1)- let us define recursively the following family of functions C4) allow the Implicit Function Theorem [15] to be applica- ρi−1,α(s) Q(V, ϕ) = 0 ρ0,α(s) = α0s, ρi,α(s) = αi(e − 1), (4) ble to the equation that guarantees the existence of a positive definite functional V : C([−h, 0], Rn) → where i = 1, 2, ..., r. Obviously ρ (0) = 0. The Fig. 1 i,α R : Q(V (ϕ), ϕ) = 0 satisfying all standard conditions depicts e−ρi,α(t) with t > 0 for i = 0, 1, 2 and α = 1 in the i σ1(kϕ(0)k) ≤ V (ϕ) ≤ σ2(kϕkh) required for a Lyapunov- logarithmic scale in order to show a difference of decay rates. Krasovskii functional. The exponential decay rate corresponds to the straight line Definition 3: A functional Q satisfying the conditions (i = 0) in the logarithmic scale. The considered functions C1)-C4) is called an ILKF candidate. decrease faster than any exponential one if i ≥ 1. Implicit Lyapunov-Krasovskii functional-based analysis of Definition 1 (Rated Hyper Exponential Stability, [2]): exponential and hyper exponential stability requires some The origin of the closed-loop system (1) is said to be additional restrictions to the class of functionals. globally hyper exponentially stable of degree r ∈ N with Corollary 1: Let Q , Q be ILKF candidates and r+1 1 2 the convergence rate α∈ if there exists β ∈K∞: ∗ R+ C4 ) ∃c > 0 such that q11(cs, s) ≥ 0 if s < 1 and −ρr,α(t) kxh(t)kh ≤ β(kϕ0kh)e , t > 0. (5) q12(cs, s) ≥ 0 if s > 1, where q1i is defined by the condition r = 0 For the latter definition gives the exponential stability. C4) of Theorem 1 for Qi, i = 1, 2; ∗ III.PRELIMINARIES C5 ) for (V, xh(t)) ∈ Ω such that x(t) satisfies (1) one has i A. Method of Implicit Lyapunov-Krasovskii Functional ∂Qi(V,xh(t)) (−1) ∂Qi(V,xh(t)) ∂t ≤2α0V ln eV ∂V , ∀t∈R+, Below we utilize a special class of functions introduced in the following definition. Denote Γ=(σ,s)∈ 2 :q(σ,s)=0 . where α0 > 0 and i = 1, 2; R+ ∗ 2 C6 ) for y ∈ Ch one has Q1(1, y) = Q2(1, y) , Definition 2: [2] The function q : R+ → R, (σ, s) → then the origin of system (1) is globally hyper exponentially q(σ, s) is said to be of the class IK∞ iff 2 stable with degree r = 1 and convergence rate α = (α0, 1). 1) q is continuous on R+; Proof. Let us consider the functional V : C → defined 2) ∀s∈R+ ∃σ ∈R+ :q(σ, s)=0; h R+ implicitly as a solution to Q (V, ϕ) = 0, ϕ ∈ C where i = 1 3) the function q(·, s) is strictly decreasing if s ∈ R+ is fixed; i h 4) the function q(σ, ·) is strictly increasing if σ ∈ is fixed; if Q1(1, ϕ) < 0 and i = 2 otherwise. Due to monotonicity R+ ∗ 5) lim σ =0, lim s=0 and lim σ =+∞. condition C3) (see Theorem 1) and the condition C6 ) we s→0+ σ→0+ s→+∞ derive that {ϕ ∈ Ch : Q1(1, ϕ) < 0} = {ϕ ∈ Ch : (σ,s)∈Γ (σ,s)∈Γ (σ,s)∈Γ Q (1, ϕ) < 0}. From C3) and C∗6) one has Q (V (ϕ), ϕ)=0 If q ∈ IK∞ then (see, [2]) there exists a unique function 2 1 if V (ϕ)≤1 and Q (V (ϕ), ϕ)=0 if V (ϕ)≥1. σ ∈ K∞, such that q(σ(s), s) = 0 for all s ∈ +. 2 R ∗ Recall that the operator g : Z → Y, where Z and Taking into account Condition C5 ) and the Implicit are Banach spaces, is called F-differentiable (Frechet` Function [15] we derive Y h i−1 differentiable) at z ∈ if there exists a linear bounded ˙ ∂Qi(V,xh(t)) ∂Qi(V,xh(t)) 0 Z V (xh(t)) = − ∂V ∂V = (V,x (t))∈Ω operator Dgz0 : Z → Y such that h −2α V (x (t)) 1 + (−1)i ln(V (x (t)) , kg(z)−g(z )−Dg (z−z )k 0 h h 0 z0 0 Y kz−z k → 0 as kz − z0k → 0 where i = 1 if V (xh(t)) < 1 and i = 2 if V (xh(t)) > 1. The 0 Z Z function V (xh(·)) : R+ → R+ has no classical derivative at Assumption 2: Let us assume that Φ transforms L to a t1 ∈ R+ : V (xh(t1)) = 1, but it has negative upper right- lower triangular block form, i.e. hand derivative implying isolation of such a time instant t1. L11 0 0 ... 0 −ρ1,α(t) −1 L21 L22 0 ... 0 The obtained estimate yields V (xh(t)) ≤ V (ϕ0)e ΦLΦ = ...... , ∗ L L L ... L with α = (α0, 1). Finally, the condition C4 ) implies k1 k2 n3 kk −1 −ρ1,α(t) ni×nj ckϕ(0)k ≤ V (ϕ), i.e. kxh(t)k ≤ c V (ϕ0)e . where the blocks Lij ∈ R have the same dimensions as the corresponding blocks in the matrix ΦAΦ−1. B. ILKF Candidates for Linear State-Delay System B. Main Result Let us consider the linear dilations (see, [16]) in Rn For i = 1, 2 let us introduce the linear dilations (see, [17]) d(λ) = eGd ln λ with λ > 0, (7) Gd ln λ di(λ) = e i with λ > 0, (10) n×n where the matrix Gd ∈ R is known as the generator of d satisfying, obviously, the following identity where Gd = diag{rijIn }, d 1 i j d(λ) = Gdd(λ), λ > 0. dλ λ i+1 −1 nj ×nj rij = 1 + (−1) (k − j)µ, µ ∈ (0, k ) and Inj ∈ R Lemma 1: Let di be two linear dilations (i = 1, 2) of - the identity matrices for j = 1, 2, ..., k. In [17] the dilations the form (7) and the nonlinear functionals Qi : R+ × (10) were utilized for finite/fixed-time control design of C([−h, 0], Rn) → R be defined as follows delay-free control systems. > 1 1 Let us define the control law as follows Qi(V,ϕ)=−1+ϕ(0) di( V )P di( V )ϕ(0)+ µ(2τ+h) 0 i (8) −1 R (−1) h > 1 1 u(xh(t))= B˜ u0(Φxh(t)) +u ˜(V (t), xh(t)), (11) eV ϕ (τ)di( V )R di( V )ϕ(τ)dτ, −h where where i = 1, 2 and P,R ∈ n×n are symmetric matrices. If R u0(ϕ)=− (Ak1 Ak2 ... Akk)ϕ(0) − (Lk1 Lk2 ... Lkk)ϕ(−h) ∃γ > 0 such that and > 1+(−1)iµ 1 G P +PGd >γP, P > 0, u˜(V, ϕ) =V Kdi Φϕ(0), ϕ ∈ Ch, di i (9) V G> R+RG − µR >γR, R > 0, with di di n i = 1 if Q1(1,yh(t))≤1, ∂Qi 2 otherwise, then ∂V < 0 for ϕ ∈ Ch such that ϕ 6= 0 ∈ Ch. The proof immediately follows from the formula and V is a solution to Qi(V, ϕ) = 0, where the functionals Q are defined by (8), (9), (10). ∂Qi 1 > −1 −1 i ∂V = − V ϕ (0)di(V )(Hri P + PHri )di(V )ϕ(0)+ ∗ µ(2τ+h) It is worth stressing that due to conditions C4) and C6 ) 0 i 1 1 (−1)iµ R eV (−1) h 2τ+h ϕ>(τ)d Rd ϕ(τ)dτ h V V we have i=1 for V ≤1 and i=2 for V >1. −h + V − If we denote y =Φx then the cosed-loop system becomes 0 µ(2τ+h) R (−1)i h > 1 1 i eV ϕ (τ)d( V )(Hr R+RHr )d( V )ϕ(τ)dτdθ 1+(−1) µ 1 −h y˙(t) = A0 + BKV di V y(t) − L0y(t − h), − V Qi(V,ϕ)+1 with ≤ −γ V n 1 if Q1(1,yh(t))≤1, Therefore, the functionals (8) are ILKF candidates and they Qi(V, yh(t)) = 0, i = 2 otherwise, satisfy Conditions C4∗) and C6∗) of Corollary 1. −1 −1 where A0 and L0 are the block matrices Φ AΦ and Φ LΦ IV. HYPER EXPONENTIAL CONTROL DESIGNFOR with zero blocks in the last row (due to u0). LINEAR SYSTEM Let us calculate the time derivatives of the ILKF (8) along A. System Decomposition the trajectories of the latter system ∂Q (V, y (t)) 1 1 dI (t) It is well-known (see, for example, [17]) that under i h = 2y>(t)d P d y˙(t) + i , Assumption 1 there exists a matrix Φ allowing the decom- ∂t i V i V dt position where yh(t) ∈ Ch : yh(t)(θ) = y(t + θ), θ ∈ [−h, 0] and 0 A12 0 ... 0 0 ! t 0 0 A23 ... 0 0 µ(2(τ−t)+h) −1 −1 Z i h ΦAΦ = ...... ,B0 =Φ B = ... . (−1) > 1 1 0 0 0 ... A(k-1)k 0 Ii(t) = eV y (τ)di Rdi y(τ)dτ. ˜ V V Ak1 Ak2 An3 ... Akk B t−h where k is the number of blocks (in both rows and columns) Let us denote z(t) = d 1 y(t), s(t) = d 1 y(t − and A ∈ ni×nj are blocks of the transformed matrix A, i V i V ij R h),L (V ) = d 1 L d (V ). Taking into account n + n + .. + n = n n ≤ n (A ) = i i V 0 i such that 1 2 k , i−1 i, rank i i+1 1 (−1)iµ ˜ m×m di A0di(V )=V A0 we derive ni, B ∈ R , nk = m. By analogy with single-input V control systems the corresponding block form can be called > 1 1 2y(t) di V P di V y˙(t) = canonical. The simple recursive algorithm for construction > (−1)iµ > z(t) V (P (A+BK)+(A+BK) P ) −PLi(V ) z(t) of the matrix Φ can be found in [17]. s(t) > s(t) −Li (V )P 0 d On the other hand, one has Ξ(λ) for λ ∈ [0, 1] it is sufficient to show dλ Ξ(λ) ≥ 0 if (−1)i λ > 0. For this purpose let us rewrite Ξ as follows > 2µ ln eV Ii(t) > dIi(t) z (t)Rz(t) s (t)Rs(t) k−1 = − − µ . dt (−1)i −µ h (−1)i P (2q−2) ˜ −1 ˜> (eV ) (eV ) Ξ(λ) = λ LqR Lq + q=1 −1 µ d(1+ln(ρ)) d(µ ρ ) i µ(−1)i k−1 ln eV (−1) ≤ V Since dρ ≤ dρ ,∀ρ≥1 then µ P P j+p−2 ˜ −1 ˜> ˜ −1˜> λ LjR Lp+LpR Lj . (−1)i > 2µ ln eV (Qi(V,yh(t))+1) j=1p>j ∂Qi(V,yh(t)) z(t) z(t) ∂t ≤ s(t) Θi s(t) − h , The latter identity immediately yields the next proposition. where Proposition 1: If for 1 ≤ j < p ≤ k − 1 the following system of matrix inequalities (−1)iµ > µ 2 ! V (P (A+BK)+(A+BK) P +e R+ h P ) PLi(V ) > −R −1 > −1 > Θi = L (V )P . 0≤L˜j R L˜ + L˜pR L˜ if j+p is odd, i (−1)i µ p j (eV ) ˜ −1˜> ˜ −1˜> ˜ −1˜> (16) 0≤Lj+p R Lj+p +Lj R Lp+LpR Lj if j+p is even, 2 2 Therefore, if Θi < 0 and P,R satisfy (9) then all conditions µ ∂Qi Qi(V,yh)+1 holds then (15) is fulfilled. of Corollary 1 hold for α0 = since ≤ −γ γh ∂V V Remark 1: If we denote Z = R−1, β > 0, X = P −1 and due to Lemma 1. To guarantee Θi < 0 it is sufficient to ask Y = KP −1 then the system of matrix inequalities (9), (12), > µ 2 P (A+BK)+(A+BK) P +e R+ h PPL0 > −R ≤ 0 (12) (16) can be rewriten as the system of LMIs: L0 P eµ > > > µ > 2 −1 > −1 > AX+BY +XA +Y B +e L0ZL0 + h XX together with Li(V )R L (V ) ≤ L0R L for V ≤ 1 if −Z ≤ 0 i 0 X µ i = 1 and for V ≥ 1 if i = 2. e Hri X +XHri − µX >γX, X > 0, i = 1, 2 Under Assumption 2 we can represent L as follows: > > i 0≤L˜jZL˜ + L˜pZL˜ if j+p is odd, (17) Pk−1 (−1)i+1µ(j−1) p j Li(V ) = j=1 V ΠjL0, ˜ ˜> ˜ ˜> ˜ ˜> 0≤Lj+p ZLj+p+LjZLp+LpZLj if j+p is even, where Πj is the projector to the block diagonal matrix with 2 2 non-zero blocks only in j-th lower diagonal, namely, 1 ≤ j < p ≤ k − 1,
L11 0 0 ... 0 0 0 which immidiately follows from Schur complement (e.g. [18]) 0 L22 0 ... 0 0 0 Note that the last two LMIs disappear if k = 2 or if all 0 0 L33 ... 0 0 0 L˜ := Π L = ...... , 1 1 0 non-diagonal blocks in L0 are zeros (i.e. Lq p =0 for q 6=p). 0 0 0 ... Lk-2 k-2 0 0 0 0 ... 0 Lk−1 k−1 0 0 0 0 ... 0 0 0 0 0 ... 0 0 0 V. ASPECTSOF PRACTICAL IMPLEMENTATION L21 0 ... 0 0 0 ˜ 0 L32 ... 0 0 0 (13) A. Sampled-Time ILF Control L2 := Π2L0 = ...... , 0 0 ... Lk-1 k-2 0 0 0 0 ... 0 0 0 In order to realize the control algorithm (11) in practice ... we need to know Lyapunov-Krasovskii functional V , but it 0 0 0 ... 0 ! is defined implicitly. The ILKF control can be implemented ˜ 0 0 0 ... 0 Lk−1 := Πk−1L0 = ...... for linear control systems that admit the on-line variation Lk-1 1 0 0 ... 0 0 0 0 ... 0 of the feedback gains. Indeed, for any fixed V0 the control Let us consider the positive definite matrix valued-function u˜(V0, s) defined by (11) becomes a static linear feedback. k−1 k−1 Let us denote X (j−1) ˜ −1 X (p−1) ˜> Ξ(λ) = λ LjR λ L , (14) {ϕ ∈ C : Q (V, ϕ) ≤ 1} if V < 1, p Π(V ):= h 1 (18) j=1 p=1 {ϕ ∈ Ch : Q2(V, ϕ) ≤ 1} if V ≥ 1. where λ ∈ [0, 1]. Obviously, for i = 1, 2 one has Lemma 2: Let the conditions of Theorem 2 hold and i+1 L (V )R−1L>(V )=Ξ V (−1) µ and L R−1L> =Ξ (1) , −1 i i 0 0 u(t) = B0 u0(yh(t)) +u ˜(V0, yh(t)), (19)
i+1 where V (−1) µ ∈ (0, 1). Therefore, we have proven the where u˜ is defined by (11) with an arbitrary fixed positive next result. number V0 ∈ R+. Then the ellipsoid Π(V0) ⊂ Ch is Theorem 2: Let Assumptions 1-2 hold. If the system of positively invariant set of the closed-loop system (1), (2), matrix inequalities (9), (12) is feasible for some P,R ∈ (19) and the origin is globally exponentially stable. n×n m×n R , K ∈ R and Proof. Let us consider the functional V : Ch → R+ defined as V = Qi(V0, ϕ) + 1, where i = 1 if V0 < 1 Ξ(1) ≥ Ξ(λ) for all λ ∈ [0, 1] (15) and i = 2 if V0 ≥ 1. Due to Proposition 1 it satisfy the then the origin of the closed-loop system (1), (11) is globally condition σ1(kϕ(0)k) ≤ V (ϕ) ≤ σ2(kϕkh) for some K∞ ˙ ∂Q(V0,yh(t)) hyper exponentially stable with degree r = 1 and conver- functions σ1 and σ2. Since V (yh(t)) = ∂t then ˙ gence rate α = µ , 1 . repeating the proof of Theorem 2 we derive V (yh(t)) ≤ γh i −1 (−1) The parametric inequality (15) can be checked numerically −2h ln eV0 V (yh(t)) along the trajectories of the using a sufficiently dense grid on the segment [0, 1].A closed-loop linear system (1), (2), (19). more conservative (but much more constructive) sufficient In the next corollary we study the case where the value V condition can also be provided. Indeed, to guarantee Ξ(1) ≥ in (11) can be changed only in some sampled time instances ti > 0. In this case, the control law (11) becomes linear Since according to Lemma (2) u(t) = K∗y(t) is linear switched feedback. static stabilizing feedback then for sufficiently small ε ∈ R+ ∗ Corollary 2: If 1) the conditions of Theorem 2 hold, 2) we have yh(t) → 0 as t → ∞, i.e. there exists t ≥ tN +∞ ˜ ˜ {tj}i=0 is an arbitrary sequence of time instances: such that V∗(t∗) = V∗ and V∗(t) < V∗ for t > t∗. This contradicts with our assumption and means limi→∞ Vi = 0 0 = t0 < t1 < t2 < ... and lim tj = +∞, j→+∞ implying that the closed-loop system (1) with sampled-time implementation of the ILKF control algorithm is globally 3) the control u is defined as u(t) = B−1u (y (t)) + 0 0 h asymptotically stable. u˜(V , y (t)) on each time interval [t , t ), where j h j j+1 The proven corollary guarantees that the ILFK control u˜(V, y (t)) is defined by and V ∈ : h (11) j R+ with sampled variation of V guarantees asymptotic stabi- Q (V , y (t )) = 0 with i = 1 if Q (1, y (t )) < 1 and i j h j 1 h j lization of the closed-loop system (1) independently on the i = 2 otherwise, then the closed-loop system is globally (1) sampling period. asymptotically stable. Proof. Let V be a Lyapunov-Krasovksii functional implicitly B. Digital Implementation defined by the equation Qi(V, ϕ) = 0, where i = 1 if The sampled ILKF control (11) implementation requires Q1(1, ϕ) < 0 and i = 2 otherwise. Note that Q1(V (ϕ), ϕ) = solving the equation Q(V, ϕ) = 0 numerically and on-line 0 if V (ϕ) ≤ 1 and Q2(V (ϕ), ϕ) = 0 if V (ϕ) ≥ 1 (see in order to find an appropriate value of Vi at the time instant Corollary 1). Let us denote Vj = V (yh(tj)). +∞ ti. Fortunately, for practical reasons rather simple numerical I. To prove that the sequence {Vj}j=1 is monotone de- procedures can be utilized. creasing let us consider the time interval [tj, tj+1) and the Denote as before V := V (t ) and y := y (t ) and ˜ ˜ j j j h j functional Vj : Ch → R+ defined Vj(ϕ) := Qi(Vj, ϕ) + 1 suppose that the implicit part control is realized in the (with i = 1 if Vj < 1 and i = 2 if Vj ≥ 1), which (according sampled way, i.e. u˜(Vj, s) on the time interval [tj, tj+1), to Lemma 2) is Lyapunov-Krasovski functional for the −1 where 0 = t0 < t1 < t2 < ... and lim tj = +∞. system (1), (2) with u(t) = B0 u0(yh(t)) +u ˜(Vj, yh(t)). Algorithm 1: d ˜ Repeating the proof of Lemma 2 we derive dt Vj(yh(t)) ≤ INITIALIZATION: V =1; a=V ; b=1; i 0 min −1 (−1) ˜ −2h ln eVj Vj(yh(t)) for t ∈ [tj, tj+1). Hence, STEP : If Q (1, y ) + 1 > 0 then i = 2 else i = 1; V˜j(yh(t)) < V˜j(yh(tj)) for all (tj, tj+1] and i j endif; ˜ ˜ Qi(Vj, yh(t)) = Vj(yh(t)) − 1 < Vi(yh(tj)) − 1 = If Qi(a, yj) + 1 > 0 then a = b; b = 2b; Qi(Vj, yh(tj)) = 0 = Qi(V (yh(t)), yh(t)), a elseif Qi(a, yj)+1<0 then b=a;a=max{ 2 ,Vmin} t ∈ (tj, tj+1], i = 1, 2. a+b else c = 2 ; If Q (c, y )+1 then b = c; For any fixed ϕ ∈ Ch\{0} the functional Qi(·, ϕ): R+ → R i j is monotone decreasing (see Proposition 1). Then the latter else a=max{Vmin, c}; endif; chain of inequalities implies V (yh(t)) < V (yh(tj)) if t ∈ +∞ endif; (tj, tj+1], i.e. the sequence {Vj}j=1 is monotone decreasing V = b; and yh(t) ∈ Π(Vj) for t ≥ tj. Moreover, V (yh(t)) ≤ i If y ∈ C is fixed and STEP of the presented algorithm V (yh(0)) for all t ≥ 0, i.e. the origin of the system (1) j h is Lyapunov stable. is applied recurrently many times to the same yj then II. Since the functional V is positive definite then Algorithm 1 realizes: ∞ 1) a localization of the unique positive root of the equation the monotone decreasing sequence {Vj}j=1 (with Vj = Q (V, y ) = 0, i.e. V ∈ [a, b], where i = 1 if Q (1, y ) < 1 V (yh(tj))) converge to some limit. Let us show now that this i j j 1 j and i = 2 otherwise; limit is zero. Suppose the contrary, i.e. lim Vj = V∗ > 0 or j→∞ 2) improvement of the obtained localization by means of ∀ε>0, ∃N =N(ε): V ≤V
x1 TABLE I x2 x3 x4 COMPARISON OF LINEAR AND ILKF CONTROLS 2 x5 Linear Control ILKF Control −6 10x0 kx(3)k = 0.0115 kx(3)k = 1.5134 · 10 1 −6 30x0 kx(3)k = 0.0344 kx(3)k = 1.6028 · 10 −6 100x0 kx(3)k = 0.1146 kx(3)k = 1.5787 · 10 −6 300x0 kx(3)k = 0.3439 kx(3)k = 1.5238 · 10 ) t
( 0 x
done also for linear stabilizing control u = u0+Kx, which is −1 obtained from (11) for the fixed V = 1. For linear control the simulations show that stabilization error essentially depended
−2 of the norm of initial condition, but in the ILKF case the stabilization error is uniformly (with respect to initial
0 0.5 1 1.5 2 2.5 3 t condition) bounded for t ≥ 3. Such a behavior is similar to the fixed-time time stability that is well-known (see, e.g. [5], Fig. 2. Simulation results for ILFK control [19]) for delay-free systems. due to finite numerical precision of digital devices. REFERENCES
VI.NUMERICAL EXAMPLE [1] A. M. Lyapunov, The general problem of the stability of motion. Taylor & Francis, 1992. Let us consider the system (1) with h = 0.5 and [2] A. Polyakov, D. Efimov, W. Perruquetti, and J.-P. Richard, “Implicit Lyapunov-Krasovski Functionals for Stability Analysis and Control 0 0 1 0 0 ! 0 0 0 0 0 ! 0 0 0 1 0 0 0 −1 0 0 Design of Time-Delay Systems,” IEEE Transactions on Automatic A= 9 0 −0.1 0 2 ,Ad = 0.5 0 0.1 0 0 , Control, vol. 60, no. 12, pp. 3344–3349, 2015. 0 19.6 0 −0.3 0 0 −0.3 0 1 0 [3] S. P. Bhat and D. S. Bernstein, “Geometric homogeneity with appli- 0 0.1 0 0 −2 0 0 0 0 0.1 cations to finite-time stability,” Mathematics of Control, Signals and 0 0 ! 1 0 0 0 0 ! Systems, vol. 17, pp. 101–127, 2005. 0 0 0 0 1 0 0 B = 0 0 , Φ= 0 −1 0 0 0 , [4] W. Perruquetti, T. Floquet, and E. Moulay, “Finite-time observers: ap- 1 0 0 0 0 1 0 plication to secure communication,” IEEE Transactions on Automatic 0 1 4.5 0 0 −0.05 1 Control, vol. 53, no. 1, pp. 356–360, 2008. where the matrix Φ transforms the original system to the [5] A. Polyakov, “Nonlinear feedback design for fixed-time stabilization block forms with n = 1, n = n = 2 and of linear control systems,” IEEE Transactions on Automatic Control, 1 2 3 vol. 57(8), pp. 2106–2110, 2012. A = 1,A = 0 2 ,A = ( 0 ) ,A = 0 19.6 , [6] J. Moreno and M. Osorio, “Strict Lyapunov functions for the super- 1 2 2 3 −1 0 31 9 32 4.4 −0.1 twisting algorithm,” IEEE Transactions on Automatic Control, vol. 57, −0.3 0 1 0 pp. 1035–1040, 2012. A33 = 0 −2.1 ,B0 = ( 0 1 ) [7] E. Cruz-Zavala, J. Moreno, and L. Fridman, “Uniform robust exact 0.5 0.1 0 differentiator,” IEEE Transactions on Automatic Control, vol. 56, L11 = 0,L21 = ( 0 ) ,L22 = ( 1 0 ) , no. 11, pp. 2727–2733, 2011. 0 0 0.3 1 0 [8] V. Andrieu, L. Praly, and A. Astolfi, “Homogeneous Approximation, L31 = −0.475 ,L32 = ( 0 0 ) ,L33 = ( 0 1 ) . Recursive Observer Design, and Output Feedback,” SIAM Journal of Control and Optimization, vol. 47, no. 4, pp. 1814–1850, 2008. Using the system of LMIs (17) we derive [9] N. Nakamura, H. Nakamura, and Y. Yamashita, “Homogeneous sta- 38.7075 13.7738 −0.2037 0.0300 2.3676 ! bilization for input-affine homogeneous systems,” in Conference on 13.7738 6.0479 −0.1354 0.0201 1.1062 Decision and Control, 2007, pp. 80–85. P = −0.2037 −0.1354 4.8892 −0.8901 −0.0330 [10] A. Levant, “Homogeneity approach to high-order sliding mode de- 0.0300 0.0201 −0.8901 0.2188 0.0049 2.3676 1.1062 −0.0330 0.0049 0.2810 sign,” Automatica, vol. 41, no. 5, pp. 823–830, 2005. 0.1331 0.0164 0.0000 0.0000 0.0011 ! [11] J.-P. Richard, “Time-delay systems: an overview of some recent 0.0164 0.6516 −0.0006 −0.0000 0.0433 R = 0.0000 −0.0006 0.0770 −0.0076 −0.0002 advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667– 0.0000 −0.0000 −0.0076 0.0477 −0.0001 1694, 2003. 0.0011 0.0433 −0.0002 −0.0001 0.0655 [12] E. Fridman, Introduction to Time-Delay Systems: Analysis and Con- −4.3153 −2.3893 52.9390 −12.6925 −0.5232 K = −188.0978 −89.2779 2.2053 −0.3252 −19.9263 trol. Springer, 2014. [13] S. Choi and J. Hedrick, “Robust throttle control of automotive The Figure 2 depicts simulation results for the system (1) engines,” ASME Journal of Dynamic Systems, Measurement, and with the control (11) and the constant initial condition ϕ(t) = Control, vol. 118, pp. 92–98, 1996. x , where x = (0.2, 0.05, 0, 0, 0). The numerical simulation [14] I. Karafyllis and Z.-P. Jiang, Stability and Stabilization of Nonlinear 0 0 Systems. Springer, 2011. has been done using the explicit Euler method with the [15] B. Driver, Analysis Tools with Applications. Springer, 2003. sampling period 10−3. The implicit Lyapunov function has [16] A. Polyakov, J.-M. Coron, and L. Rosier, “On finite-time stabilization been calculated using Algorithm 1. The integral term in of evolution equations: A homogeneous approach,” in Conference on Decision and Control, 2016, pp. (https://hal.inria.fr/hal–01 371 089). Qi has been approximated by the Simpson rule. As it was [17] A. Polyakov, D. Efimov, and W. Perruquetti, “Robust stabilization of expected the simulation results show that realization of fast mimo systems in finite/fixed time,” International Journal of Robust stabilization need large declinations (overshooting) of some and Nonlinear Control, vol. 26, no. 1, pp. 69–90, 2016. [18] A. Poznyak, Advanced Mathematical Tools for Automatic Control variables during the transients (see, behavior of the variable Engineers. Volume 1: Deterministic Technique. Elsevier, 2008. x5 on the Figure 2). [19] A. Polyakov, D. Efimov, and W. Perruquetti, “Finite-time and Fixed- In order to compare the hyper exponential control with time Stabilization: Implicit Lyapunov Function Approach,” Automat- ica, vol. 51, no. 1, pp. 332–340, 2015. exponential (linear) one the numerical simulations have been