Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback

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Finite-time stabilization and H _ ∞ control of nonlinear delay systems via output feedback

Ta T. H. Trang, Vu N. Phat, Samir Adly

To cite this version:

Ta T. H. Trang, Vu N. Phat, Samir Adly. Finite-time stabilization and H_∞ control of nonlinear delay systems via output feedback. Journal of Industrial and Management Optimization, AIMS, 2016, 12 (1), pp.303-315. �10.3934/jimo.2016.12.303�. �hal-01313201�

FINITE-TIME STABILIZATION AND H∞ CONTROL OF NONLINEAR DELAY SYSTEMS VIA OUTPUT FEEDBACK

Ta T.H. Trang and Vu N. Phat

Institute of Mathematics, VAST 18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam

Adly Samir

Universit´e de Limoges, Laboratoire XLIM

123, avenue Albert Thomas, 87060 Limoges CEDEX, France

Abstract. This paper studies the robust finite-timeH∞control for a class of nonlinear systems with time-varying delay and disturbances via output feed- back. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel delay-dependent conditions for the existence of out- put feedback controllers are established in terms of linear matrix inequalities (LMIs). The proposed conditions allow us to design the output feedback con- trollers which robustly stabilize the closed-loop system in the finite-time sense.

An application toH∞ control of uncertain linear systems with interval time- varying delay is also given. A numerical example is given to illustrate the efficiency of the proposed method.

1. Introduction. The concept of finite-time stability (FTS) (or short-time sta- bility) introduced by Dorato [5] plays an important role in stability theory of dif- ferential equations. A system is said to be finite-time stable if its state does not exceed a certain threshold during a specified time interval. Compared with the Lyapunov stability, finite-time stability concerns the boundedness of system during a fixed finite-time interval. It is noted that, a system may be finite-time stable but not Lyapunov asymptoticaly stable, and vice versa. A lot of interesting results on finite-time stability and stabilization in the context of linear delay systems have been obtained (see, e.g. [2,8,11,14] and the references therein).

On the other hand, one of the most important problems is theH∞control of time- delay systems via output feedback controllers. The main principle of the output feedback control is to utilize the measured output to excite the plant. Since the controller can be easily implemented in practice, the output feedback control has attracted a lot of attention over the past few decades and has been applied to many areas for example, in motor engine control, constrained robotics, networked control systems, communication and biological systems, etc. [1, 3, 6, 9, 15, 18, 21, 22].

So far, however, compared with numerous research results on Lyapunov stability withH_∞ control, few results on finite-timeH_∞ control have been obtained in the literature.

2010Mathematics Subject Classification. Primary: 93D20, 34D20; Secondary: 37C75.

Key words and phrases. Finite-time stabilization,H∞control, output feedback, time-varying delay, Lyapunov function, linear matrix inequality.

The finite-timeH_∞ control for switched linear systems with time-varying delay has been studied in [19, 20], but the results were limited either to discrete-time systems or to the systems with constant delays. In [12,16] some delay-dependent conditions for finite-time H∞ control are extended to linear systems with time- varying delays, but the delay function is differentiable and the stabilizing control is designed via state feedback.

In this paper, we propose a new design tool to solve the robust finite-timeH_∞ control for nonlinear systems with interval time-varying delays via output feedback controls. The novel features here are that the interval time-varying delay is present in the observation output, and the output feedback controllers to be designed must satisfy some robust finite-time stability constraints on the closed-loop poles. Using new generalized Jensen integral inequality we select a new simpler set of Lyapunov- Krasovskii functionals to derive delay-dependent sufficient conditions for solving robust finite-time H_∞ control via output feedback controls. The conditions are obtained in terms of LMIs, which can be determined by utilizing MATLABs LMI Control Toolbox. The approach allows us to apply toH_∞control of uncertain linear systems with interval non-differentiable time-varying delay.

The paper is organized as follows. The definition of FTS for nonlinear systems with interval time-varying delays and the problem statement are given in Section 2. Sufficient conditions for designing output feedback controllers of finite-timeH∞

control problem, an application to H_∞ control of uncertain linear systems with interval time-varying delay with numerical examples are given in Section 3.

2. Preliminaries. The following notations will be used throughout this paper, R⁺denotes the set of all nonnegative real numbers;Rⁿ denotes then−dimensional space with the scalar productx^>yand the vector normk·k;R^n×rdenotes the space of all matrices of (n×r)− dimension. A^> denotes the transpose of A; a matrix A is symmetric if A=A^>;I denotes the identity matrix;λ(A) denotes the set of all eigenvalues ofA; λmax(A) = max{Re(λ) : λ∈λ(A)};λmin(A) = min{Re(λ) : λ∈λ(A)};C¹([−τ,0],Rⁿ) denotes the set of all Rⁿ−valued continuously differen- tiable functions on [−τ,0];L2([0, T],R^r) stands for the set of all square-integrable R^r−valued functions on [0, T]. The symmetric terms in a matrix are denoted by

∗. Matrix A is positive definite (A >0) if (Ax, x)>0 for all x6= 0. The segment of the trajectory x(t) is denoted by xt = {x(t+s) : s ∈ [−τ,0]} with its norm kxtk= sup

s∈[−τ,0]

kx(t+s)k.

Consider the following nonlinear control systems with time-varying delay and disturbances











˙

x(t) =A1x(t) +A2x(t−h(t)) +Bu(t) +Gw(t) +f(t, x(t), x(t−h(t)), u(t), w(t)) z(t) =C1x(t) +C2x(t−h(t)), t≥0, x(t) =ϕ(t), t∈[−h2,0],

(1)

where x(t)∈Rⁿ, u(t)∈R^m, z(t)∈R^p are, respectively, the state, the control, the observation vector,A1, A2∈R^n×n, B∈R^n×m, G∈R^n×r, C1, C2 ∈R^p×n are given constant matrices. The delay functionh(t) is continuous and satisfies

0≤h₁≤h(t)≤h₂, ∀t≥0. (2)

The initial functionϕ∈C¹([−h2,0],Rⁿ) and the disturbancew(t) is a continuous function satisfying

Z T 0

w(t)^>w(t)dt≤d. (3)

The nonlinear function f(t, x, y, u, w) is globally Lipschitzian in (x, y, u, w) such that

∃a1, a2, a3, a4>0 :kfk²≤a1kxk²+a2kyk²+a3kuk²+a4kwk² (4) for allx, y∈Rⁿ, u∈R^m, w∈R^r.

Under the above assumptions onh(·), f(·) and the initial functionϕ(t),the sys- tem (1) has a unique solutionx(t, φ) on [0,+∞) (see [10], Theorem 1.2).

To study the robust finite-time H_∞ control of the system (1), the following definitions will be used later.

Definition 2.1(Robust finite-time stabilization). For given positive numbersT, c1, c2, c2 > c1,and a positive definite matrix R, the system (1) is said to be robustly finite-time stabilizable w.r.t. (c1, c2, T, R) if there exists an output feedback con- troller u(t) = F z(t) such that the following condition holds for all disturbances satisfying (3) and∀t∈[0, T]

max

sup

−h2≤s≤0

ϕ(s)^>Rϕ(s), sup

−h2≤s≤0

˙

ϕ(s)^>Rϕ(s)˙

≤c1⇒x(t)^>Rx(t)≤c2. Definition 2.2 (Robust finite-timeH_∞ control). Given γ >0, the robust finite- timeH∞ control problem for the systems (1) has a solution if

1. The system (1) is robustly finite-time stabilizable w.r.t. (c1, c2, T, R).

2. There is a numberc0>0 such that sup

RT

0 kz(t)k²dt c₀kϕk²+RT

0 kw(t)k²dt

≤γ, (5)

where the supremum is taken over all ϕ ∈ C¹([−h2,0],Rⁿ) and non-zero disturbancesw(.) satisfying (3).

We introduce the following technical well-known propositions for the proof of the main result.

Proposition 1 (Schur complement lemma [4]). Given constant matrices X, Y, Z with appropriate dimensions satisfying X = X^> and Y = Y^> > 0. Then X + Z^>Y⁻¹Z <0if and only if

X Z^>

Z −Y

<0.

Proposition 2 (Generalized Jensen integral inequality [17]). For a given matrix R > 0, any differentiable function ϕ : [a, b] → Rⁿ, then the following inequality holds

Z b a

˙

ϕ(u)Rϕ(u)du˙ ≥ 1

b−a(ϕ(b)−ϕ(a))^>R(ϕ(b)−ϕ(a)) + 12

b−aΩ^>RΩ, whereΩ = ϕ(b) +ϕ(a)

2 − 1

b−a Z b

a

ϕ(u)du.

3. Output feedback finite-time H_∞ control. Before stating the main result, the following notations of several matrices variables are defined for simplicity.

P =R^1/2P R^1/2, U1=R^1/2U1R^1/2, U2=R^1/2U2R^1/2, X1=R^1/2X1R^1/2, X2=R^1/2X2R^1/2, S=R^1/2SR^1/2,

α1=λmin(P), α2=λmax(P) +h1λmax(U1) +h2λmax(U2) + 0.5h³₁λmax(X1) + 0.5h³₂λmax(X2) + 0.5(h2−h1)²(h2+h1)λmax(S),

α3=λmax(P) +h1λmax(U1) +h2λmax(U2) + 0.5h³₁λmax(X1) + 0.5h³₂λmax(X2) + 0.5(h₂−h₁)²(h₂+h₁)λ_max(S₂), Ψ¹= (Ψ¹_ij)_11×11, Ψ²= (Ψ²_ij)_6×11, Ψ³=diag

−0.5N,−0.5N,−1

a₃N+ 1 2a₃I,−1

a₃N+ 1

2a₃I,−0.5N,−0.5N

, Ψ¹₁₁=P A1+A^>₁P+U1+U2+a1I+ηC₁^>C1−4X1−4X2+BKC1

+C₁^>K^>B^>−0.5BN B^>, Ψ¹₂₂=−U1−4X₁−4S, Ψ¹₃₃=−U2−4X2−4S,Ψ¹₄₄=−8S+a2I+ηC₂^>C2,

Ψ¹₅₅=h²₁X₁+h²₂X₂+ (h₂−h₁)²S−2Q,Ψ¹₆₆=a₄I−γηI,Ψ¹₇₇=−I, Ψ¹₈₈=−12X1,Ψ¹₉₉=−12X2,Ψ¹_10,10= Ψ¹_11,11=−12S,Ψ¹₁₂=−2X1, Ψ¹₁₃=−2X2,Ψ¹₁₄=P A2+ηC₁^>C2,Ψ¹₁₅=A^>₁Q,Ψ¹₁₆=P G,Ψ¹₁₇=P

Ψ¹₁₈= Ψ¹₂₈= 6X1,Ψ¹₁₉= Ψ¹₃₉= 6X2,Ψ¹₂₄= Ψ¹₃₄=−2S,Ψ¹₄₅=A^>₂Q,Ψ¹₅₆=QG, Ψ¹_2,11= Ψ¹_3,10= Ψ¹_4,10= Ψ¹_4,11= 6S,Ψ¹₅₇=Q, and Ψ¹_ij= 0 for the others, Ψ²₁₁=P B, Ψ²₁₂=C₁^>K^>−0.5BN, Ψ²₁₃=C₁^>K^>, Ψ²₄₄= Ψ²₄₅=C₂^>K^>, Ψ²₅₆=QB, Ψ²_ij = 0 for the others.

The following theorem gives a sufficient condition for robust finite-timeH_∞control via output feedback of the system (1).

Theorem 3.1. For given positive constantsT, c₁, c₂, γand a positive definite matrix R,the robust finite-time H_∞ control of the system (1) has a solution if there exist a positive scalar η,symmetric positive definite matricesP, U₁, U₂, X₁, X₂, S, N and matricesQ, K such that the following conditions hold

Ψ =

Ψ¹ Ψ²

∗ Ψ³

<0, (6)

α2c1+γηd≤α1c2e^−ηT. (7) The output feedback controller is given byu(t) =N⁻¹Kz(t), t≥0.

Proof. Consider the following Lyapunov-Krasovskii functional associated to the sys- tem (1): V(t, x_t) =

4

X

i=1

V_i(t, x_t), where

V₁(t, x_t) =e^ηtx(t)^>P x(t), V₂(t, x_t) =

2

X

i=1

e^ηt Z t

t−h_i

x(s)^>U_ix(s)ds,

V₃(t, x_t) =

2

X

i=1

h_ie^ηt Z 0

−h_i

Z t t+s

˙

x(τ)^>X_ix(τ)dτds,˙

V4(t, xt) =(h2−h1)e^ηt Z −h1

−h2

Z t t+s

˙

x(τ)^>Sx(τ)dτds.˙ It is not difficult to verify that

α₁x(t)^>Rx(t)≤V(t, x_t), ∀t: 0≤t≤T, (8) V(0, x₀)≤α₂ sup

−h2≤s≤0

x(s)^>Rx(s),x(s)˙ ^>Rx(s)˙ ≤α₂c₁, (9)

V(0, x0)≤α3kϕk². (10)

Taking the derivative ofV_i(t, x), i= 1, . . . ,4, along the solution of the system, we get the following estimations

V˙1(t, xt)≤ηV1+e^ηt

x(t)^> P A1+A^>₁P+ 2P BN⁻¹B^>P+C₁^>F^>N F C1 x(t) + 2x(t)^>P Gw(t) +x(t−h(t))^>C₂^>F^>N F C₂x(t−h(t))

+ 2x(t)^>P A2x(t−h(t)) + 2x(t)^>P f(t, x, xh, u, w) , V˙₂(t, x_t) =ηV₂+e^ηt

x(t)^>(U₁+U₂)x(t)−x(t−h₁)^>U₁x(t−h₁)

−x(t−h₂)^>U₂x(t−h₂) , V˙3(t, xt) =ηV3+e^ηt

˙

x(t)^>(h²₁X1+h²₂X2) ˙x(t)−

2

X

i=1

hi

Z t t−hi

˙

x(s)^>Xix(s)ds˙ . Applying Proposition2, we have

−hi

Z t t−hi

˙

x(s)^>Xix(s)ds˙ ≤ −4x(t)^>Xix(t)−4x(t−hi)^>Xix(t−hi)

−4x(t)^>X_ix(t−h_i)s+12

h_ix(t−h_i)^>X_i Z t

t−hi

x(s)ds +12

hi

x(t)^>X_i Z t

t−hi

x(s)d−12 h²_i

Z t t−hi

x(s)^>dsX_i Z t

t−hi

x(s)ds.

Hence,

V˙₃(t,x_t)≤ηV₃+e^ηt

˙

x(t)^>(h²₁X₁+h²₂X₂) ˙x(t) +x(t)^>(−4X1−4X₂)x(t)

−4x(t−h1)^>X1x(t−h1)−4x(t−h2)^>X2x(t−h2)−4x(t)^>X1x(t−h1)

−4x(t)^>X2x(t−h2) +

2

X

i=1

12 hi

x(t)^>Xi

Z t t−hi

x(s)ds

+

2

X

i=1

12 hi

x(t−hi)^>Xi

Z t t−hi

x(s)ds−

2

X

i=1

12 h²_i

Z t t−hi

x(s)^>dsXi

Z t t−hi

x(s)ds . Using the same calculation as in ˙V₃(t, x_t), we get

V˙₄(t, x_t) =ηV₄+e^ηt

(h₂−h₁)²x(t)˙ ^>Sx(t)˙ −(h₂−h₁)

Z t−h(t) t−h₂

˙

x(s)^>Sx(s)ds˙

−(h₂−h₁) Z t−h1

t−h(t)

˙

x(s)^>Sx(s)ds˙

≤ηV4+e^ηt

(h2−h1)²x(t)˙ ^>Sx(t)˙ −8x(t−h(t))^>Sx(t−h(t))

−4x(t−h2)^>Sx(t−h2)−4x(t−h1)^>Sx(t−h1)−4x(t−h(t))^>Sx(t−h2)

−4x(t−h(t))^>Sx(t−h₁)− 12 (h2−h(t))²

Z t−h(t) t−h2

x(s)^>dsS

Z t−h(t) t−h2

x(s)ds

− 12 (h(t)−h1)²

Z t−h1

t−h(t)

x(s)^>dsS Z t−h1

t−h(t)

x(s)ds

+ 12

h₂−h(t)x(t−h(t))^>S

Z t−h(t) t−h2

x(s)ds+ 12

h₂−h(t)x(t−h2)^>S

Z t−h(t) t−h2

x(s)ds

+ 12

h(t)−h₁x(t−h1)^>S Z t−h1

t−h(t)

x(s)ds+ 12

h(t)−h₁x(t−h(t))^>S Z t−h1

t−h(t)

x(s)ds . Multiplying both sides of (1) bye^ηt2 ˙x(t)^>Qfrom the right, we obtain

e^ηt

−2 ˙x(t)^>Qx(t)+2 ˙˙ x(t)^>Q(A1+BF C1)x(t) + 2 ˙x(t)^>Q(A2+BF C2)x(t−h(t)) + 2 ˙x(t)^>QGw(t) + 2 ˙x(t)^>Qf(t, x, xh, u, w)

= 0.

Consequently, 0≤e^ηt

˙

x(t)^>(−2Q+ 2QBN⁻¹B^>Q) ˙x(t) +x(t)^>C₁^>F^>N F C₁x(t) + 2 ˙x(t)^>QA₁x(t) +x(t−h(t))^>C₂^>F^>N F C₂x(t−h(t))

+ 2 ˙x(t)^>QA2x(t−h(t)) + 2 ˙x(t)^>QGx(t) + 2 ˙x(t)^>Qf(t, x, xh, u, w) . (11) Adding the inequality (11) and the zero term

e^ηt

f(t, x, x_h, u, w)^>f(t, x, x_h, u, w)−f(t, x, x_h, u, w)^>f(t, x, x_h, u, w) +γηw(t)^>w(t)−γηw(t)^>w(t) +ηz(t)^>z(t)−ηz(t)^>z(t)

= 0 to ˙V(t, x_t), and using the estimation (4) forf(t, x, x_h, u, w), and noting that f(t, x, xh, u, w)^>f(t, x, xh, u, w)≤x(t)^> a1I+ 2a3C₁^>F^>F C1

x(t) +a3w(t)^>w(t) +x(t−h(t))^> a2I+ 2a3C₂^>F^>F C2

x(t−h(t)), ηz(t)^>z(t) =ηx(t)^>C₁^>C₁x(t) + 2ηx(t)^>C₁^>C₂x(t−h(t))

+ηx(t−h(t))^>C₂^>C2x(t−h(t)), we have

V˙(t, x_t)≤ηV(t, x_t) +e^ηt

x(t)^>(P A₁+A^>₁P+U₁+U₂+a₁I+ηC₁^>C₁

−4X₁−4X₂+ 2P BN⁻¹B^>P+ 2C₁^>F^>N F C₁+ 2a₃C₁^>F^>F C₁B x(t) +w(t)^>(a₄I−γηI)w(t) +x(t−h₁)^>(−U1−4X₁−4S)x(t−h₁) +x(t−h2)^>(−U2−4X2−4S)x(t−h2) +x(t−h(t))^>

−8S+a2I+ηC₂^>C2

+ 2C₂^>F^>N F C2+ 2a3C₂^>F^>F C2

x(t−h(t))

+ ˙x(t)^> h²₁X1+h²₂X2+ (h2−h1)²S−2Q+ 2QBN⁻¹B^>Q

˙ x(t)

−f(.)^>f(.) + 2x(t)^>(P A₂+ηC₁^>C₂)x(t−h(t)) + 2x(t)^>P Gw(t) + 2x(t)^>P f(.)

−4x(t)^>X1x(t−h1)−4x(t)^>X2x(t−h2)−4x(t−h(t))^>Sx(t−h2) + 2 ˙x(t)^>QGw(t) + 2 ˙x(t)^>Qf(.)−

2

X

i=1

12 h²_i

Z t t−hi

x(s)^>dsXi

Z t t−hi

x(s)ds

+

2

X

i=1

12 hi

x(t)^>Xi

Z t t−hi

x(s)ds+

2

X

i=1

12 hi

x(t−hi)^>Xi

Z t t−hi

x(s)dsr

− 12 (h2−h(t))²

Z t−h(t) t−h2

x(s)^>dsS

Z t−h(t) t−h2

x(s)ds−4x(t−h(t))^>Sx(t−h₁)

− 12 (h(t)−h1)²

Z t−h1

t−h(t)

x(s)^>dsS Z t−h1

t−h(t)

x(s)ds+ 2 ˙x(t)^>QA₁x(t)

+ 12

h₂−h(t)x(t−h(t))^>S

Z t−h(t) t−h2

x(s)ds+ 2 ˙x(t)^>QA2x(t−h(t))

+ 12

h2−h(t)x(t−h2)^>S

Z t−h(t) t−h2

x(s)ds+ 12 h(t)−h1

x(t−h1)^>S Z t−h1

t−h(t)

x(s)ds

+ 12

h(t)−h1

x(t−h(t))^>S Z t−h1

t−h(t)

x(s)ds

+e^ηtγηw(t)^>w(t)−e^ηtηz(t)^>z(t).

We obtain

V˙(t, xt)−ηV(t, xt)≤e^ηtξ(t)^>W ξ(t) +e^ηtγηw(t)^>w(t)−e^ηtηz(t)^>z(t), (12) where

ξ(t)^>=h

x(t)^>, x(t−h₁)^>, x(t−h₂)^>, x(t−h(t))^>,x(t)˙ ^>, w(t)^>, f(.)^>, 1

h1

Z t t−h1

x(s)^>ds, 1 h2

Z t t−h2

x(s)^>ds, 1 h2−h(t)

Z t−h(t) t−h2

x(s)^>ds, 1

h(t)−h1

Z t−h₁ t−h(t)

x(s)^>dsi , W = (Wij)_11×11,

W11=P A1+A^>₁P+U1+U2+a1I+ηC₁^>C1−4X1−4X2

+ 2P BN⁻¹B^>P+ 2C₁^>F^>N F C1+ 2a3C₁^>F^>F C1, W₂₂=−U1−4X₁−4S, W₃₃=−U2−4X₂−4S,

W44=−8S+a2I+ηC₂^>C2+ 2C₂^>F^>N F C2+ 2a3C₂^>F^>F C2,

W₅₅=h²₁X₁+h²₂X₂+ (h₂−h₁)²S−2Q+ 2QBN⁻¹B^>Q, W₆₆=a₄I−γηI, W77=−I, W88=−12X1, W99=−12X2, W10,10=W11,11=−12S,

W₁₂=−2X₁, W₁₃=−2X₂, W₁₄=P A₂+ηC₁^>C₂, W₁₅=A^>₁Q, W₁₆=P G, W17=P , W18=W28= 6X1, W19=W39= 6X2, W24=W34=−2S, W₄₅=A^>₂Q, W₅₆=QG, W_2,11=W_3,10=W_4,10=W_4,11= 6S, W57=Q, andWij = 0 for the others.

Therefore, from (12) it follows that d

dt e^−ηtV(t, x_t)

≤ξ(t)^>W ξ(t) +γηw(t)^>w(t)−ηz(t)^>z(t). (13)