Quantized feedback stabilization of continuous time-delay systems subject to actuator saturation

Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Quantized feedback stabilization of continuous time-delay systems subject to actuator saturation Gongfei Song a, *, James Lam b , Shengyuan Xu c a

CICAEET, School of Information and Control, Nanjing University of Information Science and Technology, Nanjing 210044, Jiangsu, PR China b Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong c School of Automation, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China

article

info

Article history: Received 7 April 2017 Accepted 15 April 2018

Keywords: Actuator saturation Quantized feedback control Time-delay Delay-independent stabilization Delay-dependent stabilization

a b s t r a c t In this paper, the problem of quantized feedback stabilization is investigated for continuous time-delay systems subject to actuator saturation. By utilizing two different methods, delay-independent conditions are obtained to guarantee the existence of a region of admissible initial conditions from which all trajectories of the resulting closed-loop system converge to a neighborhood of the equilibrium. Furthermore, delay-dependent conditions are developed based on the delay partitioning idea and the Lyapunov–Razumikhin functional approach, respectively. Finally, several examples are provided to demonstrate the effectiveness of the proposed approaches. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Actuator saturation arises naturally in many engineering systems. It may cause loss in performance and even instability of the closed-loop system if it is ignored in the design process. In the last few years, many researchers have investigated control system analysis and design with actuator saturation. Moreover, general systematic methods based on rigorous theory were developed (see [1–3] and [4,5]). Stability analysis and stabilization problems of delay systems have attracted much attention during the past few years (see [6–13]). When time delay and actuator saturation are present in a control system, it is important to study the problems of stability and stabilization (for instance [14–17] and [18,19]). The design of controllers such that an estimate of the domain of attraction is as large as possible was considered in [14] and [16]. The state feedback and output feedback semi-global stabilization problems were studied in [18,20], respectively, while an algebraic Riccati equations (ARE) approach was developed in [20]. A state feedback controller was designed to guarantee that there exists a region of admissible initial conditions from which all trajectories of the resulting closed-loop system converge to some bounded region ([15] and [17]). Quantized control with delay was investigated in [15], where delay-dependent conditions were proposed. The problem of robust exponential stabilization was presented, and delay-independent conditions were reported in [17]. In [21], the delay partitioning idea was introduced to study the stability of continuous systems. Recently, a new inequality which was the modified version of free-matrix-based integral inequality was derived in [22], and then improved delay-dependent stability criteria which guaranteed the asymptotic stability of the system were presented. It is worth noting that there is not an intensive literature in the problems of quantized feedback stabilization for time-delay systems with input saturation.

*

Corresponding author. E-mail address: [email protected] (G. Song).

https://doi.org/10.1016/j.nahs.2018.04.002 1751-570X/© 2018 Elsevier Ltd. All rights reserved.

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G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

Quantized feedback has found applications in many engineering systems including digital control systems and networked systems. It rises to a challenging problem in the analysis and synthesis of control systems. Various types of quantizations have been investigated and a great number of results on quantized feedback systems have been reported in the literature (see [23–32] and the references therein). Practical compressive sensing systems were investigated in [33], where two approaches to sparse signal recovery in the face of saturation error were developed. Accordingly, the problem of quantized feedback controller design still has not been studied in [33]. The networked control system with quantization and actuator saturation was discussed in [34]. However, the more complicated situation with time-delay has not been dealt with in [34]. In [35], a framework based on a kind of Lyapunov approach and an alternative stabilization method based on the chaotic behavior of piecewise affine maps were provided, respectively. Moreover, the performances of these methods for dealing with scalar linear systems have been compared in [35]. An optimal quantized feedback has been studied in [36–38], where the corresponding optimal quantizers were designed, respectively. The input-to-state stabilization (ISS) has been extended to l2 stabilization for linear systems with quantized feedback in [39]. The problem of stabilization for discrete-time linear systems via logarithmic quantized feedback was addressed in [23,30,40], where the method based on Tsypkin-type Lyapunov functions was applied in [23]. Also, many quantized feedback stabilization results for linear systems have been extended to nonlinear systems ([41] and [42]). Very recently, the problem of ISS with respect to external disturbances of control systems using measurements from a dynamic quantizer was solved in [43]. It is observed that the quantized feedback control problems for systems with logarithmic quantizer have been studied by many authors. However, the corresponding control problems of finite-range quantizer have been proposed so little. Furthermore, one should note that the stabilization problem studied in [15] involves a linear system and a saturated quantizer. In the current paper, input saturation is present so that the considered system is a nonlinear one. Motivated by [15], we stabilize a nonlinear time-delay system by quantized feedback controls based on a finite-range quantizer. However, no input saturation is considered in [15]. In our paper, both input saturation and saturated quantizer are considered, thus the results in [15] are not applicable to the present system. Compared with [15], the delay partitioning technique is adopted in this paper, which is a less conservative method. A continuous-time linear system with input saturation and quantized control law was considered in [44]. The contribution of [44] can be viewed as complementary to the results developed in [15], where the method developed in [44] provided convex optimization procedure to characterize the domain of attraction. In particular, the uniform quantizer presented in [44] does not have a finite number of quantization levels. In practice, quantizers have a finite dynamic range due to the voltage limits of hardware devices. Hence, there is a strong desire to use a finite number of quantization levels to represent the set of quantized values. Unlike the quantizer in [44], a finite-range quantizer with saturation level is addressed in this paper. Note that neither finite level quantization nor time delays are considered in [44]. Therefore, the results in [44] cannot be applied to the system considered in the current paper. Accordingly, we need to deal with the nonlinearity issues arising from actuator saturation and saturated quantizer. To the best of our knowledge, there is so far no result on domain of attraction estimation for time-delay systems with actuator saturation and saturated quantizer. In this paper, the corresponding results are developed by using a unique approach which is confirmed in the analysis of systems subject to nested saturations. Our aim here is to initiate the study in this area by utilizing an approach for the analysis of systems with nested saturations. We highlight a number of key features:

• Sufficient conditions of quantized feedback stabilization for a class of continuous systems with both time-varying delay and actuator saturation are given;

• A finite-range quantizer with saturation level is considered; • Two different techniques on how to design the quantized feedback controllers are developed. In this paper, we study control systems whose input variables are quantized. The designed quantized feedback controllers can guarantee that all trajectories of the closed-loop system will converge to a smaller region for every initial condition from the admissible domain. The delay-independent and delay-dependent estimates of the corresponding domain are provided via the linear matrix inequality (LMI) scheme, respectively. Notation. Throughout this paper, the following notations will be used. For real symmetric matrices M and N, the notation M ≥ N (respectively, M > N) means that the matrix M − N is positive semi-definite (respectively, positive definite). In and 0m,n represent the n × n identity matrix and m × n zero matrix. 1m denotes a vector of dimension m with components equal to 1. For a real vector u, u(i) denotes the ith component of vector u. ∥u∥ denotes its Euclidean vector norm. Cn,d = C ([−d, 0], Rn ) stands for the Banach space of continuous vector functions mapping the interval [−d, 0] into Rn . Furthermore, we define Sym{A} = A + AT and L(H , U1 ) = {v ∈ Rn : ∥Hi v∥ ≤ U1(i) }, where Hi is the ith row of the matrix H. For a number ρ −1 > 0 and a matrix P ∈ Rn×n , P > 0, the ellipsoid ε (P , ρ −1 ) is defined by ε (P , ρ −1 ) = {v ∈ Rn : v T P v ≤ ρ −1 }. 2. Problem formulation Consider a continuous system with a time delay in state and input saturation, which is described by x˙ (t) = Ax(t) + Ad x(t − τ (t)) + BsatU1 (u(t)),

(1)

G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

x(t) = ϕ (t),

∀t ∈ [−d, 0],

3

(2)

where x(t) ∈ Rn is the state and u(t) ∈ Rm is the control input. In system (1), A, Ad and B are known real constant matrices with appropriate dimensions. We consider a saturation function satU1 (u(t)), with each component defined as (satU1 (u(t)))(i) = satU1 (u(i) (t)) = sign(u(i) (t)) min(U1(i) , |u(i) (t)|), where U1(i) is a given scalar with U1(i) > 0, ∀i = 1, . . . , m. τ (t) is the time-varying delay of the system that satisfies 0 < τ (t) ≤ d, τ˙ (t) ≤ µ, where d and µ are known constant scalars. ϕ (t) is a vector-valued initial continuous function, that is, ϕ (t) ∈ Cn,d . Then we denote xt (θ ) = x(t + θ ), θ ∈ [−d, 0]. Similar to [2], we suppose that γ(j) belongs to the set {1, 2, 3}, for j = 1, 2, . . . , m, that is, γ(j) = 1 or γ(j) = 2 or γ(j) = 3. Let J = {γ ∈ Rm : γ(j) is the jth component of vector γ } of cardinality 3m , for example, [1 1 1 · · · 1]T ∈ J , [1 2 1 · · · 1]T ∈ J , [2 2 2 · · · 2]T ∈ J , and so on. We also use a γ ∈ J to define a diagonal matrix Γs such that

Γs (γ ) = diag{σ (γ(1) − s), σ (γ(2) − s), . . . , σ (γ(m) − s)}, s = 1, 2, 3, where

σ (γ(j) − s) =

{

1, if γ(j) = s, 0, if γ(j) ̸ = s, ∀j = 1, 2, . . . , m.

In the present paper, a finite-range quantizer is considered with general form as in [33]. By denoting Q(i) (u) = Q (u(i) ),

∀i = 1, . . . , m, where Q(·) is defined as ⎧ ∆ ⎪ ⎪ n = 1, 2, 3, . . . , U2(i) − , for u(i) > U2(i) (≜ n∆), ⎪ ⎪ 2 ⎪ ⎪ ⎨ ∆ + k∆, for k∆ < u(i) ≤ (k + 1)∆, k = 0, 1, 2, . . . , n − 1, Q (u(i) ) = 2 ⎪ ⎪ ⎪ ⎪ 0, for u(i) = 0, ⎪ ⎪ ⎩ −Q(−u(i) ), for u(i) < 0.

(3)

The range of this quantizer is [−U2(i) , U2(i) ], for U2(i) > 0, i = 1, 2, . . . , m, and the quantization error bound is ∆ > 0. Since 2 ∆ is given by the designer, Q(·) is a static uniform quantizer. Define ψ (u) = Q(u) − satU2 (u). Then, ∥ψ (u(i) )∥ ≤ ∆2 , i =

1, 2, . . . , m is satisfied. Similar to [44], we can obtain ψ (u)T T ψ (u) − T ∈ Rm×m . We consider the following quantized state feedback controller:

∆2 4

1Tm T 1m ≤ 0 for any diagonal positive definite matrix

u(t) = Q (Kx(t)), where K ∈ R follows:

m×n

(4)

is the controller gain to be determined. Then the problems considered in this paper can be described as

Problem 1 (Delay-independent Problem). Design a stabilizing gain K for system (1)–(2) with feedback controller (4) such that for any initial condition ϕ starting from an admissible condition domain Ω , all trajectories of the closed-loop system converge to a neighborhood of the equilibrium with the neighborhood strictly within Ω . In this case, the delay-independent estimates of domain are obtained. Problem 2 (Delay-dependent Problem). Determine a quantized state feedback controller of the form (4) for system (1)–(2) such that for any initial condition ϕ starting from an admissible condition domain Ω , all trajectories of the closed-loop system converge to a neighborhood of the equilibrium with the neighborhood strictly within Ω . In this case, the delay-dependent estimates of the corresponding domain are obtained. 3. Main results Before presenting the main results, we introduce the following technical lemmas, which will be used in the proof of our main results. Lemma 1 ([16]). Suppose that u(l), v (l), w (l) and p(l) ∈ R+ → R+ are scalar, continuous and nondecreasing functions, u(l), v (l), w(l) positive for l > 0, u(0) = v (0) = 0 and p(l) > l for l > 0. If there is a continuous function V : Rn → R and a positive number ρ such that for all xt ∈ MV (ρ ) := {ϕ ∈ Cn,d : V (ϕ (θ )) ≤ ρ, ∀θ ∈ [−d, 0]}, the following conditions hold. (1) u(∥x∥) ≤ V (x) ≤ v (∥x∥). (2) V˙ (x(t)) ≤ −w (∥x∥), if V (x(t + θ )) < p(V (x(t))), ∀θ ∈ [−d, 0]. Then, the solution x(t) ≡ 0 of the corresponding system is asymptotically stable. Moreover, the set MV (ρ ) is an invariant set inside the domain of attraction.

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G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

Lemma 2 ([17,45]). Let g(t) be a continuous and differentiable function with g(t) ≥ 0 for all t ≥ t0 − d and k∗ > supt0 −d≤s≤t0 g(s). Let

˙ ≤ −α1 g(t) + α2 sup g(s) + ζ for t ≥ t0 , g(t) t −d≤s≤t

where α1 , α2 and ζ are positive constants. If 0 < α2 < α1 , then g(t) ≤ r0 + k∗ e−κ (t −t0 ) for t ≥ t0 , where r0 =

ζ α1 −α2

and κ is the unique solution to

− κ = −α1 + α2 eκ d . Firstly, a delay-independent condition is developed by employing two different methods, which guarantees that the trajectories of the closed-loop system will be uniformly bounded. Theorem 1. Given scalars α and β satisfying β > α > 0. If there exist matrices Q > 0, X > 0, Z1 , Z2 , Y , and scalars δ > 0, ρ > 0 such that the following conditions hold for γ ∈ J and i = 1, 2, . . . , m: Sym{AQ + B1 } + β Q + B2 δ B2T QATd

[ [

ρ

Z1i 2 U1(i) Q

T Z1i

]

≥ 0,

[

ρ

Z2i 2 U2(i) Q

T Z2i

2(β − α )δ − m∆2 ρ > 0,

Ad Q −X

]

]

< 0,

(5)

≥ 0,

(6)

α Q − X ≥ 0,

(7)

where B1 = B(Γ1 (γ )Z1 + Γ2 (γ )Z2 + Γ3 (γ )Y ), B2 = B(Γ2 (γ ) + Γ3 (γ )),

then for all initial conditions ϕ ∈ ε (Q −1 , ρ −1 − ellipsoid ε (Q −1 ,

m∆2 ). 4(β−α )δ

m∆2 ), 4(β−α )δ

the resulting trajectories of the closed-loop system converge to the

Moreover, a quantized state feedback controller is given by u(t) = Q (YQ −1 x(t)).

Proof. Now, let P = Q −1 , H1 = Z1 Q −1 , H2 = Z2 Q −1 , K = YQ −1 and S = Q −1 XQ −1 , V (x(t)) = x(t)T Px(t). Similar to [2], we can deduce that 2x(t)T PBsatU1 (ψ (x(t)) + satU2 (Kx(t)))

[ ≤ x(t)T

ψ (x(t))T

[ ] P B1 P + P B1T P

P B2 0

B2T P

][

]

x(t) , ψ (x(t))

where B(·,i) represents the ith column of the matrix B. From (7) and (5), it can be deduced that x(t − τ (t))T Sx(t − τ (t)) ≤ α sup V (x(s)),

(8)

t −d≤s≤t

V˙ (x(t)) ≤ −β V (x(t)) + α sup V (x(s)) + δ −1 m t −d≤s≤t

By (7), it is easy to see that ρ −1 > 2

m∆2 . 4(β−α )δ

∆2 4

.

(9)

Then, following a similar argument as in [17], if (6) is satisfied then for

∆ x(θ ) = ϕ (θ ) ∈ ε (Q −1 , ρ −1 − 4(m ), θ ∈ [−d, 0], it follows that x(t) ∈ ε (Q −1 , ρ −1 ) ⊂ L(H1 , U1 ) β−α )δ

⋂

L(H2 , U2 ) for t ∈ [−d, 0].

Suppose that at t1 > 0 the trajectory x(t) first reaches the boundary of ε (Q −1 , ρ −1 ). Then, we have V (x(t1 )) = ρ −1 and supt1 −d≤s≤t1 V (x(s)) = ρ −1 . Considering (9), one has at t = t1 , V˙ (x(t)) ≤ −βρ −1 + αρ −1 + δ −1 m

∆2 4

< 0.

Then, using Lemma 2, we obtain V (x(t)) ≤

m∆2 4(β − α )δ

+ sup V (x(s))e−κ t , − κ = −β + α eκ d . −d≤s≤0

This completes the proof. □ Then a solution to Problem 1 can be obtained by solving the following optimization problem: inf

Q ,X ,Z1 ,Z2 ,Y ,δ,γ ∈J

s. t.

ρ

(5), (6) and (7).

(10)

G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

5

Remark 1. A delay-independent approach is developed to estimate an admissible initial basin ε (Q −1 , ρ −1 − a bounded region ε (Q

−1

,

m∆2 ). The first inequality in (7) guarantees that the ellipsoid 4(β−α )δ m∆2 −1 ). Moreover, the orientation of the ellipsoids (Q −1 4(β−α )δ

than the ellipsoid ε (Q −1 , ρ are the same.

ε

−

ε (Q

) and

4(β−α )δ m∆2 ) is strictly smaller 4(β−α )δ m∆2 ∆2 ) and (Q −1 4(m ) 4(β−α )δ β−α )δ

−1

, ρ −1 −

m∆2

,

ε

,

We are in a position to present another method in the following theorem. Theorem 2. Given scalars η1 > 0, η2 > 0 and η3 > 0. If there exist matrices Q > 0, S > 0, P¯ 1 > 0, Z1 , Z2 , Y , and a diagonal matrix T > 0 such that the following conditions hold for γ ∈ J and i = 1, 2, . . . , m: Sym{AQ + B1 } + η2 P¯ 1 + (1 − η1 )Q ⎣ B2T SATd

⎡

[

I T Z1i

Z1i 2 Q U1(i)

2

T 1m

−

S − Q ≥ 0,

[

I T Z2i

≥ 0,

∆

⎡ η − η2 ⎢ 1 ⎣∆ 2

]

1Tm T 1

η3

T

Z2i 2 Q U2(i)

B2

− η3 T 0

]

⎤

Ad S 0 ⎦ < 0, −S

(11)

≥ 0,

(12)

⎤ ⎥ ⎦ ≤ 0,

(13)

P¯ 1 − Q ≥ 0,

(14)

where B1 = B(Γ1 (γ )Z1 + Γ2 (γ )Z2 + Γ3 (γ )Y ), B2 = B(Γ2 (γ ) + Γ3 (γ )),

then, for all initial conditions ϕ ∈ ε (Q −1 , 1), the resulting trajectories of the closed-loop system converge to the ellipsoid ε (Q −1 P¯ 1 Q −1 , 1). Moreover, a quantized state feedback controller can be chosen by u(t) = Q(YQ −1 x(t)). −1 −1 ¯ −1 T Proof. Let P = Q −1 , H1 = Z1 Q −1 , H2 = Z2 Q −1 , K = YQ ⋂ and P1 = Q P1 Q , V (x(t)) = x(t) Px(t). Note that the inequalities in (12) ensure the ellipsoid ε (P , 1) ⊂ L(H1 , U1 ) L(H2 , U2 ). Therefore, we can show that

V˙ (x(t)) +x(t)T Px(t) − η1 (x(t)T Px(t) − 1) − η2 (1 − x(t)T P1 x(t)) − η3 (ψ (x(t))T T ψ (x(t)) −

≤

∆2 4

1Tm T 1m )

x(t)T [Sym{PA + P B1 P } + PAd SATd P + (1 − η1 )P + η2 P1 ]x(t) + 2x(t)T P B2 ψ (x(t))

− η3 ψ (x(t))T T ψ (x(t)) + x(t − τ (t))T S −1 x(t − τ (t)) + η1 − η2 + η3

∆2 4

1Tm T 1m .

(15)

By following a similar procedure as in [16] and using (11) and (13), we can deduce that there exists a scalar ξ > 0 such that x(t)T [Sym{PA + P B1 P } + PAd SATd P + (1 + 2ξ )P ]x(t) + 2x(t)T P B2 ψ (x(t)) < 0.

(16)

Suppose that V (x(t + θ )) < (1 + ξ )V (x(t)), ∀θ ∈ [−d, 0]. Therefore, by (14) and (16) it is easy to see that V˙ (x(t)) < −ξ V (x(t)). By Lemma 1, all the trajectories of the closed-loop system starting from the ellipsoid ε (Q −1 , 1) converge to the ellipsoid ε (Q −1 P¯ 1 Q −1 , 1). This completes the proof. □ Remark 2. Theorem 2 gives a delay-independent condition for the solution to Problem 1. Note that the orientation of the ellipsoids ε (Q −1 , 1) and ε (Q −1 P¯ 1 Q −1 , 1) are not the same. The inequalities in (14) guarantee that the ellipsoid ε (Q −1 P¯ 1 Q −1 , 1) is included in the ellipsoid ε (Q −1 , 1). By solving the following optimization problem, the ellipsoids ε (Q −1 , 1) and ε (Q −1 P¯ 1 Q −1 , 1) can be optimized. Also, the controller gain can be obtained. inf

Q ,S ,P¯ 1 ,Z1 ,Z2 ,Y ,T ,R,γ ∈J

s.t.

(a) (b)

χ + trace(R)

(17)

(11), (12), (13) and (14),

[ χI I

I Q

]

≥ 0,

[

P¯ 1 −Q

−Q R

]

≥ 0.

In the following, attention will be focused on the study of delay-dependent condition to solve Problem 2. The main results of this section are given in the following theorems.

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G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

Theorem 3. Consider the time delay system (1) and let ϖ0 > 0, λ be given scalars. Then, there exists a quantized state feedback controller in the form of (4) such that all the trajectories of the closed-loop system starting from the ellipsoid ε (X −1 PX −1 , ρ −1 − m∆2 ∆2 ) converge to the ellipsoid ε (X −1 PX −1 , 4m ), if there exist matrices P > 0, W > 0, R > 0, Q > 0, M, T , X , G1 , G2 , K˜ and 4ϖ0 ϖ ϖ0 ϖ scalars ϖ > 0, ρ > 0 such that the following matrix inequalities hold for γ ∈ J and i = 1, 2, . . . , m:

⎡

Ξ

de−

⎢ ⎢ − ϖ0 d T ⎢de r M ⎣d ϖ d 0 r

e−

ϖ0 d r

r

]

ϖ0 d r

⎤

T

0 ϖ0 d

d

[

≥ 0,

e−

R

− e− r R r] G2i ≥ 0, 2

0

G1i 2 P U1(i)

GT1i

d

M

r

− de−

TT

[r ρI

ϖ0 d

ρI GT2i

⎥ ⎥ ⎥ < 0, ⎦

(18)

(19)

U2(i) P

2ϖ ϖ0 − ρ m∆2 > 0,

(20)

where T XWdot Ξ = Sym{WxT (P − X )Wdot + WxT (AX + B˜1 )Wx + WxT Ad XWt + ϖ WxT B˜2 Wf − λWdot T T T ˜ + λWdot (AX + B˜1 )Wx + λWdot Ad XWt + λϖ Wdot B2 Wf + e−

d

T + Wdot RWdot + (τ − 1)e−ϖ0 d WtT WWt r [ ] [ 0n,(r +2)n+m , Wdot = 0n,(r +2)n Wx = In [ ] [ Im , Wd = 0n,n In Wf = 0m,(r +3)n [ ] [ Irn 0rn,2n+m , Ψ = M Wrt = 0rn,n

B˜1 = B(Γ1 (γ )G1 + Γ2 (γ )G2 + Γ3 (γ )K˜ ),

ϖ0 d r

Ψ } + WxT (ϖ0 P + W )Wx

ϖ0 d

+ WrT QWr − e− r WrtT QWrt − ϖ WfT IWf , ] [ ] In 0n,m , Wt = 0n,(r +1)n In 0n,n+m , ] [ ] 0n,(r +1)n+m , Wr = Irn 0rn,3n+m , ] [ ] Wx − Wt T , Wt − Wd

B˜2 = B(Γ2 (γ ) + Γ3 (γ )).

In this case, an appropriate quantized state feedback controller can be chosen by u(t) = Q (K˜ X −1 x(t)). Proof. Construct a Lyapunov functional as follows: V (t) = x(t)T P0 x(t) +

t

∫

eϖ0 (s−t) x(s)T W0 x(s)ds +

t −τ (t)

∫

t

+ t − dr

0

∫

t

∫

− dr

eϖ0 (s−t) x˙ (s)T R0 x˙ (s)dsdθ

t +θ

eϖ0 (s−t) Υ (s)T Q0 Υ (s)ds,

(21)

where

[ Υ (s) = x(s)T

x(s −

W0 = X −1 WX −1 ,

1 r

...

T

d)

x(s −

R0 = X −1 RX −1 ,

r −1 r

]T T

d)

,

P0 = X −1 PX −1 ,

Q0 = X −1 QX −1 .

Then, we have V =

d dt

V (t) + ϖ0 V (t) −

1

ϖ

ψ (x(t))T ψ (x(t))

≤ 2x(t)T P0 x˙ (t) + ϖ0 x(t)T P0 x(t) − −e

−ϖ0 d

− e−

ϖ0 d

1

ϖ

d

ψ (x(t))T ψ (x(t)) + x(t)T W0 x(t) + x˙ (t)T R0 x˙ (t) r

(1 − µ)x(t − τ (t)) W0 x(t − τ (t)) + Υ (t) Q0 Υ (t) − e T

∫

T

−

ϖ0 d r

d

d

r

r

Υ (t − )T Q0 Υ (t − )

t

r

t − dr

x˙ (s)T R0 x˙ (s)ds.

(22)

In addition, for any appropriately dimensioned matrices M0 , T0 , L and L0 , the following equations are true: 0 = 2[x(t)T L + x˙ (t)T L0 ][−˙x(t) + Ax(t) + Ad x(t − τ (t)) + BsatU1 (ψ (x(t)) + satU2 (Kx(t)))], 0 = e−

ϖ0 d r

[

2ξ (t)T M0 x(t) − x(t − τ (t)) −

∫

t

(23)

]

x˙ (s)ds ,

(24)

t −τ (t)

0 = e−

ϖ0 d r

[ 2ξ (t)T T0 x(t − τ (t)) − x(t −

d r

∫

t −τ (t)

)− t − dr

] x˙ (s)ds ,

(25)

G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

7

with

[ ξ (t) = Υ (t)T

x(t − τ (t))T

x(t − d)T

x˙ (t)T

]T ψ (x(t))T .

The ellipsoid ε (X −1 PX −1 , ρ −1 ) is contained in L(H1 , U1 ) L(H2 , U2 ), if the inequalities in (19) are feasible, where G1 = H1 X and G2 = H2 X . Hence, together with (22) and (23)–(25), it can be deduced that for X = L−1 , K˜ = KX

⋂

V ≤ x(t)T (2P0 − 2L)x˙ (t) + x(t)T (ϖ0 P0 + W0 + 2LA + 2LB˜1 X −1 )x(t) + x(t)T 2LAd x(t − τ (t)) d + x˙ (t)T (−2L0 + R0 )x˙ (t) + x˙ (t)T (2L0 A + 2L0 B˜1 X −1 )x(t) + x˙ (t)T 2L0 Ad x(t − τ (t)) r + 2x(t)T LB˜2 ψ (x(t)) + 2x˙ (t)T L0 B˜2 ψ (x(t)) − e−ϖ0 d (1 − µ)x(t − τ (t))T W0 x(t − τ (t))

+ Υ (t)T Q0 Υ (t) − e− −

ϖ0 d

−

ϖ0 d

ϖ0 d r

d

d

Υ (t − )T Q0 Υ (t − ) + e−

ϖ0 d r

[ ξ (t)T (Sym{ M0

T0

r r d 1 1 T + ( − τ (t))T0 R− ψ (x(t))T ψ (x(t)) 0 T0 ]ξ (t) − r ϖ

[ ] ] Wx − Wt })ξ (t) Wt − Wd

ξ (t) [τ ∫ t ϖ0 d 1 T T ˙ [ξ (t)T M0 + x˙ (s)T R0 ]R− − e− r 0 [M0 ξ (t) + R0 x(s)]ds +e

−e

T

r

−1

(t)M0 R0 M0T

t −τ (t) t −τ (t)

∫

r

t − dr

1 T T ˙ [ξ (t)T T0 + x˙ (s)T R0 ]R− 0 [T0 ξ (t) + R0 x(s)]ds

≤ ξ (t)T Ξ0 ξ (t) − e−

ϖ0 d

t

∫

1 T T ˙ [ξ (t)T M0 + x˙ (s)T R0 ]R− 0 [M0 ξ (t) + R0 x(s)]ds

r

t −τ (t)

−

−e

ϖ0 d

∫

t −τ (t)

r

t − dr

1 T T ˙ [ξ (t) T0 + x˙ (s)T R0 ]R− 0 [T0 ξ (t) + R0 x(s)]ds, T

(26)

where T Ξ0 = Sym{WxT (P0 − L)Wdot + WxT L(A + B˜1 X −1 )Wx + WxT LAd Wt + WxT LB˜2 Wf + Wdot (−L0 )Wdot ] [ ϖ0 d [ ] Wx − Wt T − 1 T T − T0 + Wdot L0 (A + B˜1 X )Wx + Wdot L0 Ad Wt + Wdot L0 B˜2 Wf + e r M0 } Wt − Wd

d

T + WxT (ϖ0 P0 + W0 )Wx + Wdot R0 Wdot + (µ − 1)e−ϖ0 d WtT W0 Wt + WrT Q0 Wr − e−

r

−

1

ϖ

WfT IWf + e−

ϖ0 d r

ϖ0 d r

WrtT Q0 Wrt

d

1 T 1 T [dM0 R− T0 R− 0 M0 + 0 T0 ].

r

r +3







Define L0 = λL, Λ = diag{X −1 , X −1 , . . . , X −1 , ϖ −1 I }, ΛMX −1 = M0 and ΛTX −1 = T0 . Pre- and post-multiplying (18) by diag{ΛT , X −T , X −T } and its transpose, respectively, and then applying Schur complement equivalence, we have that Ξ0 < 0. Furthermore, we can obtain V =

d dt

V (t) + ϖ0 V (t) −

1

ϖ

ψ (x(t))T ψ (x(t)) < 0.

Then using this relation and similar techniques given in Proposition 1 in [15], we can derive x(t)T P0 x(t) < e−ϖ0 (t −t0 ) ϕ (t0 )T P0 ϕ (t0 ) + [1 − e−ϖ0 (t −t0 ) ]

1

ϖ ϖ0

∥ψ (x(t))∥2

(27)

for t ≥ t0 and t0 ∈ [−d, 0]. From (20), we can verify that x(t)T P0 x(t) < ϕ (t0 )T P0 ϕ (t0 ) + proof. □ Regarding the problem of maximizing the ellipsoid ε (X −1 PX −1 , ρ −1 − problem inf

P ,W ,R,Q ,M ,T ,X ,G1 ,G2 ,K˜ ,ϖ ∈J

s. t.

m∆2 4ϖ0 ϖ

m∆2 4ϖ0 ϖ

≤ ρ −1 . This completes the

), we can solve the following optimization

ρ

(18), (19) and (20).

(28)

Theorem 4. Consider the time delay system (1) and let η1 > 0, η2 > 0, η3 > 0 be given scalars. Then, there exists a quantized state feedback controller in the form of (4) such that all the trajectories of the closed-loop system starting from the ellipsoid ε (Q −1 , 1) converge to the ellipsoid ε (Q −1 P¯ 1 Q −1 , 1), if there exist matrices Q > 0, P2 > 0, P3 > 0, P¯ 1 > 0, Z1 , Z2 , Y and a diagonal matrix

8

G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

T > 0 such that the following matrix inequalities hold for γ ∈ J and i = 1, 2, . . . , m: Sym{(A + Ad )Q + B1 } + dAd (P2 + P3 )ATd + (2d − η1 )Q + η2 P¯ 1 B2T

[

⎡

−P2 ⎣B1T + QAT

B1 + AQ − (1 + η1 )Q + η2 P¯ 1 0

B2T

[

QATd − P3

−Q

Ad Q

∆

⎡ η − η2 ⎢ 1 ⎣∆ 2

2

T 1m

[

I T Z1i

]

Z1i 2 Q U1(i)

B2 0

− η3 T

B2

− η3 T

]

< 0,

⎤ ⎦ ≤ 0,

(30)

≤ 0, 1Tm T 1

− T η ] 3 ≥ 0,

(29)

(31)

⎤ ⎥ ⎦ ≤ 0,

(32)

[

I T Z2i

Z2i 2 Q U2(i)

]

≥ 0,

(33)

P¯ 1 − Q ≥ 0,

(34)

where B1 = B(Γ1 (γ )Z1 + Γ2 (γ )Z2 + Γ3 (γ )Y ), B2 = B(Γ2 (γ ) + Γ3 (γ )).

In this case, a suitable quantized state feedback controller can be chosen by u(t) = Q (YQ −1 x(t)). Proof. Using the Newton–Leibniz formula, we have x(t − τ (t)) = x(t) −

∫

t

x˙ (s)ds t −τ (t) t

∫

[Ax(s) + Ad x(s − τ (s)) + BsatU1 (ψ (x(s)) + satU2 (Kx(s)))]ds.

= x(t) − t −τ (t)

Now, choose a Lyapunov functional candidate as V (x(t)) = x(t)T Px(t), where Q = P −1 . Then, the time-derivative of V (t) is V˙ (x(t)) = 2x(t)T P x˙ (t) = 2x(t)T P {(A + Ad )x(t) + BsatU1 (ψ (x(t)) + satU2 (Kx(t)))

∫

t

[Ax(s) + Ad x(s − τ (s)) + BsatU1 (ψ (x(s)) + satU2 (Kx(s)))]ds}.

− Ad

(35)

t −τ (t)

Set H1 = ⋂ Z1 Q −1 , H2 = Z2 Q −1 , K = YQ −1 and P1 = Q −1 P¯ 1 Q −1 . The conditions in (33) imply the ellipsoid ε (P , 1) ⊂ L(H1 , U1 ) L(H2 , U2 ). Following a similar argument as in the proof of Theorem 1, it can be verified that 2x(t)T P(−Ad )BsatU1 (ψ (x(s)) + satU2 (Kx(s)))

≤ 2x(t)T P(−Ad )B1 Px(s) + 2x(t)T P(−Ad )B2 ψ (x(s)). Therefore, we can deduce that

∫

t

2x(t)T P(−Ad )[Ax(s) + BsatU1 (ψ (x(s)) + satU2 (Kx(s)))]ds t −τ (t)

≤ dx(t)T PAd P2 ATd Px(t) +

∫

t

[B1 Px(s) + B2 ψ (x(s)) + Ax(s)]T P2−1 t −τ (t)

×[B1 Px(s) + B2 ψ (x(s)) + Ax(s)]ds.

(36)

On the other hand, by applying Schur complement equivalence to (31), it is easy to see that ATd P3−1 Ad ≤ P .

(37)

Now, pre- and post-multiplying (30) by diag{I , Q

[

x(t)T

ψ (x(t))

] T

([

T

(B1 P + A) B2T

]

, I } and its transpose, and together with (32), we can obtain [ ]) [ ] ] −P 0 x(t) B1 P + A B2 + 0 0 ψ (x(t)) −1

[ −1

P2

− η1 (x(t)T Px(t) − 1) − η2 (1 − x(t)T P1 x(t)) − η3 (ψ (x(t))T T ψ (x(t)) −

∆2 4

1Tm T 1m ) ≤ 0,

G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

9

for x(t) such that x(t)T Px(t) ≤ 1 and x(t)T P1 x(t) ≥ 1. Hence, we have

[x(t)T (B1 P + A)T + ψ (x(t))T B2T ]P2−1 [(B1 P + A)x(t) + B2 ψ (x(t))] ≤ x(t)T Px(t).

(38)

Similarly, from (29) and (32), we can obtain x(t)T [Sym{P(A + Ad ) + P B1 P } + dPAd (P2 + P3 )ATd P ]x(t) + 2x(t)T P B2 ψ (x(t)) + x(t)T 2dPx(t) < 0. We have that there exists a scalar ϵ > 0 such that x(t)T [Sym{P(A + Ad ) + P B1 P } + dPAd (P2 + P3 )ATd P ]x(t) + 2x(t)T P B2 ψ (x(t))

+ x(t)T 2d(1 + 2ϵ )Px(t) < 0.

(39)

Here, similar to [16], we define xt (θ ) = x(t + θ ), θ ∈ [−2d, 0]. It follows from (35)–(38) that V˙ (x(t)) ≤ x(t)T [Sym{P(A + Ad ) + P B1 P } + dPAd (P2 + P3 )ATd P ]x(t) + 2x(t)T P B2 ψ (x(t)) 0

∫

∫

0

V (x(t − τ (s) + s))ds.

V (x(t + s))ds +

+ −τ (t)

(40)

−τ (t)

Suppose that V (x(t + θ )) < (1 + ϵ )V (x(t)), ∀θ ∈ [−2d, 0]. This, together with (39), implies that V˙ (x(t)) < −ϵ V (x(t)). Finally, by the relationship (34), it can be shown that the ellipsoid ε (Q −1 P¯ 1 Q −1 , 1) is included in the ellipsoid ε (Q −1 , 1). This completes the proof. □ Similar to the development made in (17), the ellipsoids ε (Q −1 , 1) and ε (Q −1 P¯ 1 Q −1 , 1) can be optimized by solving the following optimization problem inf

Q ,P2 ,P3 ,P¯ 1 ,Z1 ,Z2 ,Y ,T ,R,γ ∈J

s.t.

χ + trace(R)

(29), (30), (31), (32), (33), (34),

[ χI

I Q

I

]

[ ≥ 0 and

P¯ 1 −Q

−Q R

]

≥ 0.

(41)

4. Simulation examples Some examples are presented in this section in order to illustrate the effectiveness of the proposed approach. Example 1. Consider the time-delay system with input saturation (1) described by

[ A=

0.5 1

]

1 , −1

[ Ad =

−0.1 −0.2

0.4 , 0.1

]

[ ] B=

1 . 3

Suppose scalars U1 , U2 and ∆ are chosen such that U1 = 5, U2 = 8 and ∆ = 1. Case 1: Given scalars λ = 0.5, ϖ0 = 1, µ = 1.5 and partitioning size r = 1, by solving (28), we can get the maximum allowable delay of dmax = 1.3492 and the following controller gain is obtained for d = 0.5: K = −1.2086

] −0.9554 .

[

2

2

∆ ∆ The inner ellipsoid ε (X −1 PX −1 , 4m ) (solid) and the outer ellipsoid ε (X −1 PX −1 , ρ −1 − 4m ) (solid) are described in Fig. 1. ϖ0 ϖ ϖ0 ϖ By solving (41) with η1 = 0.001, η2 = 0.02 and η3 = 0.1, we obtain the maximum allowable delay of dmax = 0.5171 and the corresponding controller gain is obtained for d = 0.5:

K = −1.2273

] −1.0778 .

[

The inner ellipsoid ε (Q −1 P¯ 1 Q −1 , 1) (dashed) and the outer ellipsoid ε (Q −1 , 1) (dashed) are shown in Fig. 1, respec∆2 tively. We see that the ellipsoid ε (X −1 PX −1 , 4m ) is smaller than the ellipsoid ε (Q −1 P¯ 1 Q −1 , 1). In addition, the ellipsoid ϖ ϖ 0

2

ε (X −1 PX −1 , ρ −1 − 4mϖ∆ϖ ) is larger than the ellipsoid ε (Q −1 , 1). Fig. 1 illustrates that the trajectories of the closed-loop system 0

starting from the initial condition domain converge to some bounded region. In this case, the results demonstrate that the method in Theorem 3 is less conservative than the method in Theorem 4. Case 2: Given scalars λ = 1.5, ϖ0 = 1.2, µ = 1.5 and r = 1, by solving (28), we can obtain the maximum allowable delay of dmax = 0.8701 and the controller gain is obtained for d = 0.5: K = −1.8851

[

] −1.4374 .

Furthermore, we give the different maximal allowed d in Table 1 for different partitioning sizes. Then, by solving (41) with η1 = 10−5 , η2 = 0.05 and η3 = 0.1, we get the maximum allowable delay of dmax = 0.7667 and the state feedback gain is obtained for d = 0.5: K = −2.0941

[

] −1.1800 .

10

G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

Fig. 1. System state trajectories with d = 0.5 (Theorems 3 and 4).

Fig. 2. System state trajectories with d = 0.5 (Theorems 3 and 4). Table 1 Maximum allowable delay by Theorem 3 for Case 2. r

1

2

3

4

5

6

dmax

0.8701

1.3939

1.8241

2.2006

2.5404

2.8532

Fig. 2 shows the inner ellipsoid ε (Q −1 P¯ 1 Q −1 , 1) (dashed), the outer ellipsoid ε (Q −1 , 1) (dashed), the inner ellip∆2 ∆2 soid ε (X −1 PX −1 , 4m ) (solid), the outer ellipsoid ε (X −1 PX −1 , ρ −1 − 4m ) (solid), and several closed-loop trajectories. ϖ ϖ ϖ ϖ 0

We can see that the ellipsoid ε (X −1 PX −1 , 2

m∆2 4ϖ0 ϖ

0

) is larger than the ellipsoid ε (Q −1 P¯ 1 Q −1 , 1). Moreover, the ellipsoid

ε(X −1 PX −1 , ρ −1 − 4mϖ∆ϖ ) is smaller than the ellipsoid ε (Q −1 , 1). In this case, the results imply that the method in Theorem 4 0

G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

11

Fig. 3. System state trajectories with d = 0.2 (Theorem 1).

Fig. 4. System state trajectories with d = 0.2 (Theorem 2).

is less conservative than the method in Theorem 3. Therefore, there is no clean-cut way to determine which is better for the methods in Theorems 3 and 4 in the more general case. Case 3: According to Theorem 1, by setting β = 0.6 and α = 0.5, we solve the optimization problem (10) to obtain ∆2 ∆2 an admissible initial basin ε (Q −1 , ρ −1 − 4(m ) and a bounded region ε (Q −1 , 4(m ). Shown in Fig. 3 are the ellipsoids β−α )δ β−α )δ m∆2 ) 4(β−α )δ

2

∆ (larger) and ε (Q −1 , 4(m ) (smaller). Also, Fig. 3 shows that the responses of the closed-loop system β−α )δ starting from the larger ellipsoid converge to the smaller ellipsoid. To illustrate Theorem 2, the corresponding parameters are chosen as η1 = 10−5 , η2 = 0.003, η3 = 0.01. By solving the optimization problem in (17), we can obtain estimates of the corresponding domain. The ellipsoids ε (Q −1 , 1) (larger), ε (Q −1 P¯ 1 Q −1 , 1) (smaller) and the trajectories of the closed-loop ∆2 system are depicted in Fig. 4. It can be seen that the ellipsoid ε (Q −1 , ρ −1 − 4(m ) is larger than the ellipsoid ε (Q −1 , 1). β−α )δ

ε (Q −1 , ρ −1 −

However, the ellipsoid ε (Q −1 P¯ 1 Q −1 , 1) is much smaller than the ellipsoid ε (Q −1 , determine which is better for the methods in Theorems 1 and 2.

m∆2 ). 4(β−α )δ

Thus, there is no straight rule to

12

G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

Fig. 5. System state trajectories with d = 0.5 (Theorems 3 and 4).

From Figs. 1 and 3, we also see that the ellipsoids ε (X −1 PX −1 , m∆2 4ϖ0 ϖ

m∆2 4ϖ0 ϖ

) and ε (Q −1 ,

m∆2 ) 4(β−α )δ

are almost of the same size. It 2

∆ then can be seen that the ellipsoid ε (X PX , ρ − ) is larger than the ellipsoid ε (Q , ρ −1 − 4(m ). In this case, β−α )δ the method in Theorem 3 is less conservative than the method in Theorem 1. Furthermore, by Figs. 2 and 3 it can be shown ∆2 ∆2 ∆2 that the ellipsoid ε (X −1 PX −1 , 4m ) is smaller than the ellipsoid ε (Q −1 , 4(m ) and the ellipsoid ε (Q −1 , ρ −1 − 4(m ) is ϖ ϖ β−α )δ β−α )δ

−1

−1

−1

0

−1

2

∆ much larger than the ellipsoid ε (X −1 PX −1 , ρ −1 − 4m ). Therefore, it is hard to determine whether Theorem 3 is better ϖ0 ϖ than Theorem 1, or vice versa. Similarly, there is no straight rule to determine which is less conservative for the methods in Theorems 4 and 2 since some tuning parameters are included in our main results.

Example 2. Consider the time-delay system with input saturation (1) described by

[ A=

0 0.5

]

1 , −1

Ad =

[ −0.6 −0.5

0.2 , 0.6

]

[ ] B=

0 . 1

In this example, we assume U1 = 5, U2 = 10, and the quantization error bound ∆ = 1. In what follows, by solving the convex optimization problem in (28) with scalars λ = 1.2, ϖ0 = 1, µ = 1.5 and [ partitioning size r ]= 1, we can obtain that the maximum allowable delay of dmax = 0.8820 and the controller gain K = −0.6977 −2.6501 for d = 0.5. Fig. 5 illustrates that the trajectories of the closed-loop system starting from the outer ellipsoid ε (X −1 PX −1 , ρ −1 − (solid) converge to the inner ellipsoid ε (X −1 PX −1 ,

m∆2 4ϖ0 ϖ

m∆2 4ϖ0 ϖ

)

) (solid). Moreover, by solving (41) with η1 = 0.001, η2 = 0.05 and

[ ] η3 = 0.1, we obtain that the maximum allowable delay of dmax = 0.5095 and the controller gain K = −0.3708 −2.0023 for d = 0.5. The inner ellipsoid ε (Q −1 P¯ 1 Q −1 , 1) (dashed), the outer ellipsoid ε (Q −1 , 1) (dashed), and some trajectories of the closed-loop system are reported in Fig. 5, respectively. Noting Fig. 5, we can deduce that the ellipsoid ε (Q −1 , 1) is ∆2 ∆2 larger than the ellipsoid ε (X −1 PX −1 , ρ −1 − 4m ). Furthermore, the ellipsoid ε (X −1 PX −1 , 4m ) is smaller than the ellipsoid ϖ0 ϖ ϖ0 ϖ ε(Q −1 P¯ 1 Q −1 , 1). Similarly, it is also difficult to determine which is less conservative for the methods in Theorems 3 and 4 in this case. 5. Conclusions This paper has developed two different methods to solve the problem of quantized feedback stabilization for a class of continuous time-delay systems with actuator saturation. By introducing two different methods to study the quantized system, delay-independent and delay-dependent conditions are obtained, respectively, which guarantee the existence of quantized feedback controllers such that all solutions of the closed-loop system starting from a bigger region converge to a smaller domain. Through simulation examples, we cannot determine which one of the two methods is less conservative in the general case. The effectiveness and the applicability of our results have been demonstrated. Acknowledgments This work was partially supported by the following grants: GRF HKU 7137/13E, The Startup Foundation for Introducing Talent of NUIST (No. S8113107001), Natural Science Fundamental Research Project of Jiangsu Colleges and Universities (No. 15 kJB120007), National Natural Science Foundation of P.R. China (No. 61503190), Natural Science Foundation of Jiangsu Province (No. BK20150927).

G. Song et al. / Nonlinear Analysis: Hybrid Systems 30 (2018) 1–13

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References [1] T. Hu, Z. Lin, Control Systems with Actuator Saturation: Analysis and Design, Birkhäuser, Boston, 2001. [2] A. Bateman, Z. Lin, An analysis and design method for linear systems under nested saturation, Systems Control Lett. 48 (2003) 41–52. [3] R. Rakkiyappan, V. Latha, Q. Zhu, Z. Yao, Exponential synchronization of Markovian jumping chaotic neural networks with sampled-data and saturating actuators, Nonlinear Anal. Hybrid Syst. 24 (2017) 28–44. [4] S. Tarbouriech, G. Garcia, J.Gomes da Silva Jr, I. Queinnec, Stability and Stabilization of Linear Systems with Saturating Actuators, Springer-Verlag, London, 2011. [5] L. Zhang, M. Chen, Event-based global stabilization of linear systems via a saturated linear controller, Internat. J. Robust Nonlinear Control 26 (2016) 1073–1091. [6] L. Wu, J. Lam, X. Yao, J. Xiong, Robust guaranteed cost control of discrete-time networked control systems, Optim. Control Appl. Methods 32 (2011) 95–112. [7] C. Souza, D. Coutinho, Delay-dependent regional stabilization of nonlinear quadratic time-delay systems, in: Proceedings of the 19th World Congress, The International Federation of Automatic Control Cape Town, South Africa, August 24–29, 2014, pp. 10084–10089. [8] E. Verriest, Causal behavior of switched delay systems as multi-mode multi-dimensional systems, in: 8th IFAC Workshop on Time-Delay Systems, vol. 42, 2009, pp. 414–419. [9] M.S. Mahmoud, Y. Xia, Improved exponential stability analysis for delayed recurrent neural networks, J. Franklin Inst. B 348 (2011) 201–211. [10] C. Hua, G. Liu, L. Zhang, X. Guan, Output feedback tracking control for nonlinear time-delay systems with tracking errors and input constraints, Neurocomputing 173 (2016) 751–758. [11] C. Hua, C. Ge, X. Guan, Synchronization of chaotic Lur’e systems with time delays using sampled-data control, IEEE Trans. Neural Netw. Learn. Syst. 26 (2015) 1214–1221. [12] Q. Han, Y. Liu, F. Yang, Optimal communication network-based H∞ quantized control with packet dropouts for a class of discrete-time neural networks with distributed time delay, IEEE Trans. Neural Netw. Learn. Syst. 27 (2016) 426–434. [13] X. Zhang, Q. Han, A. Seuret, F. Gouaisbaut, An improved reciprocally convex inequality and an augmented LyapunovCKrasovskii functional for stability of linear systems with time-varying delay, Automatica (2017). http://dx.doi.org/10.1016/j.automatica.2017.04.048. [14] E. Fridman, A. Pila, U. Shaked, Regional stabilization and H∞ control of time-delay systems with saturating actuators, Internat. J. Robust Nonlinear Control 13 (2003) 885–907. [15] E. Fridman, M. Dambrine, Control under quantization, saturation and delay: An LMI approach, Automatica 45 (2009) 2258–2264. [16] Y. Cao, Z. Lin, T. Hu, Stability analysis of linear time-delay systems subject to input saturation, IEEE Trans. Circuits Syst. I. Regul. Pap. 49 (2002) 233–240. [17] S. Oucheriah, Robust exponential convergence of a class of linear delayed systems with bounded controllers and disturbances, Automatica 42 (2006) 1863–1867. [18] Z. Lin, H. Fang, On asymptotic stabilizability of linear systems with delayed input, IEEE Trans. Automat. Control 52 (2007) 998–1013. [19] G. Song, J. Lam, S. Xu, Robust quantized output feedback control for uncertain discrete time-delay systems with saturation nonlinearity, Internat. J. Robust Nonlinear Control 25 (2015) 3515–3527. [20] B. Zhou, J. Lam, G. Duan, An ARE approach to semi-global stabilization of discrete-time descriptor linear systems with input saturation, Systems Control Lett. 58 (2009) 609–616. [21] F. Gouaisbaut, D. Peaucelle, Delay-dependent stability analysis of linear time delay systems, in: Proceeedings of IFAC Workshop on Time Delay System, L’Aquila, Italy, 2006. [22] T. Lee, J. Park, S. Xu, Relaxed conditions for stability of time-varying delay systems, Automatica 75 (2017) 11–15. [23] B. Zhou, G. Duan, J. Lam, On the absolute stability approach to quantized feedback control, Automatica 46 (2010) 337–346. [24] W. Liu, Z. Wang, M. Ni, Quantized feedback stabilization for a class of linear systems with nonlinear disturbances, Nonlinear Anal. Hybrid Syst. 8 (2013) 48–56. [25] Y. Zou, J. Lam, Y. Niu, D. Li, Constrained predictive control synthesis for quantized systems with Markovian data loss, Automatica 55 (2015) 217–225. [26] R. Lu, H. Wu, J. Bai, Networked H∞ filtering for T-S fuzzy systems with quantization and data dropouts, J. Franklin Inst. B 351 (2014) 3126–3144. [27] Y. Tan, D. Du, S. Fei, Co-design of event generator and quantized fault detection for time-delayed networked systems with sensor saturations, J. Franklin Inst. B 354 (2017) 6914–6937. [28] M. Wakaiki, Y. Yamamoto, Stabilisation of switched systems with sampled and quantised output feedback, IET Control Theory Appl. 11 (2017) 1913–1921. [29] L. Zha, J. Fang, X. Li, J. Liu, Event-triggered output feedback H∞ control for networked Markovian jump systems with quantizations, Nonlinear Anal. Hybrid Syst. 24 (2017) 146–158. [30] M. Fu, L. Xie, The sector bound approach to quantized feedback control, IEEE Trans. Automat. Control 50 (2005) 1698–1711. [31] M.S. Mahmoud, A.Y. Al-Rayyah, Y. Xia, Quantized feedback stabilization of interconnected continuous time-delay systems, IMA J. Math. Control Inform. 28 (2011) 1–17. [32] M.S. Mahmoud, A.Y. Al-Rayyah, Y. Xia, Quantised feedback stabilisation of interconnected discrete time-delay systems, IET Control Theory Appl. 5 (2011) 795–802. [33] J. Laska, P. Boufounos, M. Davenport, R. Baraniuk, Democracy in action: Quantization, saturation, and compressive sensing, Appl. Comput. Harmon. Anal. 31 (2011) 429–443. [34] H. Yang, Y. Xia, H. Yuan, J. Yan, Quantized stabilization of networked control systems with actuator saturation, Internat. J. Robust Nonlinear Control 26 (2016) 3595–3610. [35] F. Fagnani, S. Zampieri, Stability analysis and synthesis for scalar linear systems with a quantized feedback, IEEE Trans. Automat. Control 48 (2003) 1569–1584. [36] F. Bullo, D. Liberzon, Quantized control via locational optimization, IEEE Trans. Automat. Control 51 (2006) 2–13. [37] J. Delvenne, An optimal quantized feedback strategy for scalar linear systems, IEEE Trans. Automat. Control 51 (2006) 298–303. [38] S. Azuma, T. Sugie, Optimal dynamic quantizers for discrete-valued input control, Automatica 44 (2008) 396–406. [39] T. Kameneva, D. Nešić, On l2 stabilization of linear systems with quantized control, IEEE Trans. Automat. Control 53 (2008) 399–405. [40] M. Fu, L. Xie, Quantized feedback control for linear uncertain systems, Internat. J. Robust Nonlinear Control 20 (2010) 843–857. [41] D. Liberzon, Nonlinear control with limited information, Commun. Inf. Syst. 9 (2009) 41–58. [42] A. Franci, A. Chaillet, Quantised control of nonlinear systems: Analysis of robustness to parameter uncertainty, measurement errors, and exogenous disturbances, Internat. J. Control 83 (2010) 2453–2462. [43] Y. Sharon, D. Liberzon, Input to state stabilizing controller for systems with coarse quantization, IEEE Trans. Automat. Control 57 (2012) 830–844. [44] S. Tarbouriech, F. Gouaisbaut, Stabilization of quantized linear systems with saturations, in: Proceedings of IEEE Conference on Decision and Control, Atlanta, USA, December 15–17, 2010, pp. 3361–3366. [45] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, NY, USA, 1966.