Composite control for switched impulsive time-delay systems subject to actuator saturation and multiple disturbances

Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

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

Composite control for switched impulsive time-delay systems subject to actuator saturation and multiple disturbances✩ , ✩✩ ∗

Yunliang Wei a,b , , Guo-Ping Liu b,c a

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, China School of Engineering, University of South Wales, Cardiff, CF37 1DL, UK c CTGT Center, Harbin Institute of Technology, Harbin 150001, China b

article

info

Article history: Received 7 November 2018 Received in revised form 13 July 2019 Accepted 11 October 2019 Available online xxxx Keywords: Switched impulsive systems Actuator saturation Robust H∞ control Time-delay Disturbance compensation

a b s t r a c t This paper focuses on the composite control problem for a class of uncertain switched impulsive systems with time-varying time delay, actuator saturation and multiple disturbances which are matched and mismatched with continuous control inputs. The system under consideration involves parameter uncertainties and nonlinear uncertainties, and the actuators of both continuous control inputs and impulsive control inputs are considered to be limited by saturation. Actuator saturation brings great difficulties and challenges to implement composite control based on disturbance compensation, especially when the system is with matched disturbances. A novel switching composite controller and a switching state feedback controller are constructed respectively such that the performance requirement of anti-disturbance control for the system is achieved, the former of which can realize rejection of the matched disturbances simultaneously. Based on the treatment of the saturation as polytopic differential inclusion, the design conditions of the controllers are developed to guarantee the locally robustly asymptotic stability and robust H∞ disturbance attenuation of the closed-loop system with an estimation of the domain of attraction. Finally, a simulation example is performed to validate the feasibility of our results. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Due to the physical constraint of the control channel in practice, actuator saturation is one of the ineluctable phenomena, which may degrade performance and even destroy stability of the controlled systems [1]. Therefore, it is of great importance to consider the existence of actuator saturation in the controller design [2]. Moreover, the nonlinear essence of the saturation makes the control problem of the system with actuator saturation difficult. There already exist some fundamental methods to tackle the control problem of the system subject to actuator saturation. For instance, the anti-windup control strategy can be implemented by adding compensator into a linear controller which is already designed without regard or consideration of actuator saturation [3–5]. However, when a new controller needs to be ✩ This work was supported in part by a research grant from the National Science Foundation of China:61703233,61773144,61690212, Postdoctoral Science Foundation of China:2016M602112, the Open Fund of Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, China under Grant:MCCSE2016A04, Science and Technology Planning Project of Qufu Normal University, China:xkj201513. ✩✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.nahs.2019.100825. ∗ Corresponding author at: School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, China. E-mail addresses: [email protected] (Y. Wei), [email protected] (G.-P. Liu). https://doi.org/10.1016/j.nahs.2019.100825 1751-570X/© 2019 Elsevier Ltd. All rights reserved.

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Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

designed for meeting the performance required in practice, actuator saturation is generally taken into account at the outset of control design [6–10]. As an important class of influence factors of the performance degradation for the controlled systems, the disturbances widely exist in many actual processes and practical systems. To achieve the disturbance attenuation and rejection, many traditional and classic control methods have been proposed, such as H∞ control [11], adaptive control [12] and so on. Furthermore, recent technology enhancement has enabled the control of the system with disturbances via many advanced control schemes, such as disturbance-observer-based control (DOBC) [13], active disturbance reject control (ADRC) [14] and composite hierarchical anti-disturbance control (CHADC) [15], which are based on the techniques of disturbance compensation. More and more efforts have been devoted to investigation of composite control based on disturbance compensation for the controlled systems with disturbances [6,16–19]. In [16], the problem of anti-disturbance control for a nonlinear uncertain system with an exosystem generated disturbance is investigated, the techniques of disturbance-observer-based control (DOBC) and robust observer design are combined effectively. A CHADC technique combining the H∞ control with DOBC is studied for the system with matched and mismatched disturbances in [17]. However, if the actuator of the control input is subject to saturation, it would bring great difficulties and challenges to implement disturbance compensation. In [18], a problem of DOBC has been studied for a class of nonlinear systems with input saturation, in which two kinds of design strategies are proposed. The static anti-windup compensation control for Markovian jumping system with actuator saturation and multiple disturbances in the CHADC framework has been investigated in [20]. The above results are based on the assumption of the matched disturbances generated by an exogenous system, without which they would lose efficacy. When the actuator of the control input is limited by saturation and the assumption of the matched disturbance generated by the exogenous system is removed, it would be more difficult and challenging. As a typical class of hybrid systems, the switched system consists of a finite number of subsystems and a switching law to orchestrate switching, which can been applied to model various practice systems, such as power and electronics systems, traffic control systems, network control systems, etc. [21]. In actual systems, the switching causes the states of the systems to change abruptly, which cannot be described by traditional switched system. Therefore, the switched impulsive system has been put forward and widely studied [22–29]. [22] proves that reachability is equivalent to controllability for switched impulsive system, and a necessary and sufficient criteria for both is given. In [23], the problems of stability analysis and control synthesis are investigated, sufficient conditions of globally uniformly asymptotically stability are proposed under arbitrary and dynamical dwell time switching law. Furthermore, the controllers for each subsystem and state impulsive jumping generator at switching instant are designed. Recently, some results on the control of switched impulsive system with time delay, actuator saturation and disturbances have been presented [24,25,30]. In [24], the problem of the stabilization for switched impulsive linear systems subject to input saturation is studied, which views the impulses as control or disturbances. Based on the average dwell method, the finite-time control scheme for a interconnected impulsive switched systems with neutral delay has been proposed in [30]. However, the effects of the disturbances and saturation are not fully considered in the existing results, and some control problems are still open. For instance, when the switched impulsive time-delay system is subject to nonlinear uncertainties, actuator saturation and matched and mismatched disturbances, how to design control strategies to achieve disturbance attenuation and rejection as well as stabilization of the system? This paper pertains to the disturbance attenuation and rejection for a class of uncertain switched impulsive time-delay systems with actuator saturation and multiple disturbances via composite control. Firstly, a novel switching composite controller is designed based on the bounds of the matched disturbance for each subsystem, and a switching state feedback impulsive controller is constructed for state impulsive jumping generators at switching instants, respectively. By combining the generalized differential inclusions of saturation with assistant matrix functions, locally valid polytopic representation of the closed-loop system is proposed. Based on the switched Lyapunov function approach, the design conditions of the controller are proposed to ensure the locally robustly asymptotic stability and robust H∞ disturbance attenuation of the closed-loop system with an estimation of the domain of attraction. The remainder of the paper is organized as follows. In Section 2, some preliminaries are given and the problem under consideration is formulated. Section 3 presents the main results of the paper on construction method and design conditions of the controllers. In Section 4, one simulation example is provided to illustrate the feasibility of the proposed technique. Finally, concluding remarks are shown in Section 5. 2. Preliminaries and problem statement We consider a switched impulsive system described as follows

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

x˙ (t) = Aσ (t) (t)x(t) + Adσ (t) (t)x(t − d(t)) + f (t , xt )

+ Bσ (t) {sat(u1 (t)) + ω1 (t)} + E1,σ (t) ω2 (t), ∆x(t) = (Ck (t) − I)x(t − ) + Dk (t)sat(u2 (t − )), ⎪ ⎪ ⎪ z(t) = Cσ (t) x(t) + E2,σ (t) ω2 (t), ⎪ ⎪ ⎩ x(t + θ ) = φ (θ ),

t ̸ = tk , t = tk , t0 = 0, θ ∈ [−d, 0],

(1)

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

3

where x(t) ∈ Rn is the system state, u1 (t) ∈ Rm1 is the continuous control input, u2 (t) ∈ Rm2 is the impulsive control ˙ ≤ h < 1. input, z(t) ∈ Rq is the controlled output, d(t) denotes the time-varying delay satisfying 0 ≤ d(t) ≤ d and d(t) The piecewise constant function σ (t) is called switching law which takes its values in finite set S = I[1, N ]; N is the number of the subsystems and σ (t) = i ∈ [1, N ] implies that the ith subsystem or mode is activated. For any switching time sequence 0 = t0 < t1 < · · · < tk < · · ·, σ (t) = σ (tk ) = ik ∈ I[1, N ] when t ∈ [tk , tk+1 ). ∆x(t) = x(t + ) − x(t − ), limh→0+ x(t + h) = x(t + ), limh→0− x(t + h) = x(t − ), and the impulses are activated at the switching time points tk , k = 1, 2, . . .. Without loss of generality, it is assumed that x(tk ) = x(tk+ ). The function sat : Rm → Rm stands for the standard saturation function defined as sat(u) = [sat(u1 ) sat(u2 ) · · · sat(um )]T , where sat(ui ) = sgn(ui )min{ρ, |ui |}. The disturbances comprise two parts: a matched disturbance ω1 (t) ∈ Rn and a mismatched disturbance ω2 (t) ∈ Rp . In this paper, we assume that each component of ω1 (t) ∈ L∞ and ω2 (t) ∈ L2 ∩ L∞ satisfies

|eTl ω1 (t)| ≤ ε1 , |eTl ω2 (t)| ≤ ε2 ,

(2)

where εi ∈ R+ and εi < 1, i = 1, 2. Remark 1. When a system is subjected to input saturation, the state might go unbounded for sufficiently large disturbances even though any control input is applied [31]. In this paper, the matched disturbance ω1 (t) ∈ L∞ will be removed by the extra control part of the proposed controller, and the magnitude-bounded disturbance ω2 (t) ∈ L2 ∩ L∞ is considered. For σ (t) = i ∈ I[1, N ], Ai (t), Adi (t), Bi (t), Ck (t), Dk (t), Ci (t) are time-varying matrices and in the form of Ai (t) = Ai + △Ai (t), Ad,i (t) = Adi + △Ad,i (t), Ck (t) = Ck + △Ck (t),

(3)

Dk (t) = Di + △Dk (t).

where Ai , Ad,i , Bi , Ck , Dk , Ci are real valued constant matrices and △Ai (t), △Ad,i (t), △Bi (t), △Ck (t), △Dk (t), △Ci (t) are timevarying parameter uncertainties described as

[△Ai (t), △Adi (t)] = M1i F1i (t)[Nai Ndi ], [△Ck (t), △Dk (t)] = M2 F2 (t)[N1k N2k ],

(4)

with M1i , NAi , Ndi , Nbi , M2 , N1k , N2k are real constant matrices and F1i (t), F2 (t) are time-varying uncertainties satisfying FgT Fg ≤ I ,

g = 1i, 2.

(5)

On the other hand, E1,i and E2,i are real known constant matrices. The nonlinear function f (t , xt ) : R+ × RC ([−d, 0], Rn ) → Rn is continuous and assumed to satisfy the following condition:

∥f (t , xt )∥2 ≤ ∥G1 x(t)∥2 + ∥G2 x(t − d(t))∥2 ,

t ≥ 0.

(6)

where Gi , i = 1, 2, are real constant matrices. Remark 2. The system in [24] is different from System (1), in which the uncertainties, matched and mismatched disturbances are taken into consideration well. The results in [24] already cannot realize the control for the system under consideration in this paper, especially due to the exist of the matched disturbances. In this paper, the assumption of the matched disturbances generated by a exogenous system is abandoned, the DOBC method cannot be used to reject this disturbances. Define ball and ellipsoid as follows

Ω (P , τ ) = {x ∈ Rn : xT Px ≤ τ },

(7)

where τ ∈ R+ and P > 0. Also, define a symmetric polyhedron as L(v ) = {v ∈ Rm : |eTl v| ≤ ρ,

l ∈ I[1, m]},

(8)

where ρ is a positive scalar. Let D ⊂ Rm×m with the elements to be diagonal matrices with diagonal entries being either 1 or 0. Then, there are 2m − elements in D. Suppose that each element of D is labeled as Ds , s ∈ Q = {1, . . . , 2m }, and denote D− s = I − Ds . Clearly, Ds is also an element of D.

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Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

Lemma 1 ([1]). Let u ∈ Rm and v ∈ Rm be given. If v ∈ L(v ), then the saturation sat(u) can be expressed as m

sat(u) =

2 ∑

ηs (t)(Ds u + D− s v ),

(9)

s=1

where

∑2m

s=1

ηs (t) = 1, 0 ≤ ηs (t) ≤ 1.

In this work, for t ∈ [tk−1 , tk ), k = 1, 2, . . ., we consider the controllers u1 (t) and u2 (tk− ) as

{

u1 (t) = K1σ (t) x(t) + ω ¯ σ (t) (t , x(t)),

(10)

u2 (tk− ) = K2σ (t − ) x(tk− ). k

where

[ω¯ σ (t) ]l = −ε1 sgn(eTl φ (t , σ (t), x(t))),

(11)

φ (t , σ (t), x(t)) ≜ BTσ (t) (t)Pσ (t) x(t). and v (t) is an assistant input.

Remark 3. Here, the disturbance compensation technique is applied for construction of the controller. A novel switching composite controller is designed based on the bounds of the matched disturbance, and a switching state feedback impulsive controller is constructed for state impulsive jumping generators at switching instants, respectively. However, the interconnection of the saturation and matched disturbances brings great difficulties for the construction of the controller and the analysis of the stability of the closed-loop system. On the other hand, the following assistant matrix functions are given as follows

{

v1 (t) = Hσ (t) x(t) + ω¯ σ (t) (t , x(t)), v2 (tk− ) = Hσ (t − ) x(tk− ).

(12)

k

In the following, we denote a set for t ∈ [tk−1 , tk ) as L(Hσ (t) x(t)) = {x(t) ∈ Rn : |eTl Hσ (t) x(t)| ≤ ρ − ε1 , l ∈ I[1, m]}.

(13)

For any x(t) ∈ L(Hσ (t) x(t)), the following inequality can be obtained

ρ ≥ |eTl Hσ (t) x(t)| + ε1 ≥ |eTl Hσ (t) x(t) − ε1 sgn(eTl φ (t , σ (t), x(t)))|

(14)

= |eTl (Hσ (t) x(t) + ω¯ σ (t) (t , x(t)))| = |eTl v1 (t)|.

It is easy to see that (14) means v1 (t) ∈ L(v1 (t)) for any x(t) ∈ L(Hσ (t) x(t)). Together (12) and (13) with (14), we can obtain that v2 (tk− ) ∈ L(v2 (tk− )) for any x(tk− ) ∈ L(Hσ (t − ) x(tk− )). Using Lemma 1, k for any x(t) ∈ L(Hσ (t) x(t)), t ∈ [tk−1 , tk ), k = 1, 2, . . ., one has

⎧ 2m ∑ ⎪ ⎪ ⎪ ⎪ sat(u1 (t)) = ηs (t)(Ds K1σ (t) x(t) + D− ¯ σ (t) (t , x(t)), s Hσ (t) x(t)) + ω ⎪ ⎨ s=1

(15)

2m ⎪ ∑ ⎪ ⎪ − ⎪ − − sat(u (t )) = ηh (tk− )(Dh K2σ (t − ) x(tk− ) + D− ⎪ 2 k h Hσ (tk )x(tk ) ). ⎩ k h=1

Under (15), the closed-loop system can be described as

⎧ 2m ∑ ⎪ ⎪ ⎪ ˙ x (t) = ηs (t){[Aσ (t) (t)x(t) + Adσ (t) (t)x(t − d(t)) + ω1 (t)] + f (t , xt ) ⎪ ⎪ ⎪ ⎪ s=1 ⎪ ⎪ ⎪ ⎪ ⎪ + Bσ (t) (t)[Ds Kσ (t) x(t) + D− ¯ σ (t) (t , x(t))] s Hσ (t) x(t) + ω ⎨ + E1,σ (t) ω2 (t)}, ⎪ ⎪ 2m ⎪ ⎪ ∑ ⎪ ⎪ ⎪ ∆x(t) = ηh (t){(Ck (t) − I)x(t − ) + Dk (t)[Dh K2σ (t) x(tk− ) ⎪ ⎪ ⎪ ⎪ h = 1 ⎪ ⎩ − + D− h Hσ (t) x(tk )]},

t ̸ = tk ,

t = tk .

(16)

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

5

Lemma 2 ([32]). Given any constant λ > 0 and any matrices M , Γ , U of compatible dimensions, then 2xT M Γ Ux ≤ λxT MM T x + λ−1 xT U T Ux,

(17)

for all x ∈ Rn , where Γ is an uncertain matrix satisfying Γ T Γ ≤ I. 3. Main results 3.1. Robust stabilization In this sequel, we investigate the problem of robust stabilization of system (1) with ω2 (t) = 0 under the controller (10). Theorem 1. With ω2 (t) ≡ 0, system (1) can be stabilized by the controller (10) with the estimation of the domain of attraction Υ (φ ) ≤ 1 with Υ (φ ) = maxs∈[−d,0] ∥φ (s)∥2 [λ1 + dλ2 ], if there exist matrices Xi > 0, Yi > 0, Si , Wi , Ui , i ∈ I[1, N ] and positive scalars ε1i , ε2i , gi , such that for ∀i, j ∈ I[1, N ],

Υi < 0 ,

(18)

⎡ − Xj ⎣ ∗ ∗

φj12 −Xi + ε2i M2 M2T ∗

[

]

φj13

⎤ ⎦ < 0,

0

(19)

−ε2i I

and 1

∗

Sil Xi

≥ 0,

(20)

where λ1 = maxi∈I[1,N ] λmax (Pi ) λ2 = maxi∈I[1,N ] λmax (Qi ), Sil is the lth row of Si and

⎡ 11 ϕi ⎢ ∗ ⎢ ⎢ ∗ ⎢ Υi = ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

Adi Yi

gi I

T Xi N1i

Xi G 1

0

Xi

⎤

−(1 − h)Yi ∗ ∗ ∗ ∗ ∗

0

−gi I ∗ ∗ ∗ ∗

Yi NdiT 0 −ε1i I

0 0 0 −gi I

Yi GT2 0 0 0 −gi I

0 0 0 0 0 −Yi

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

∗ ∗ ∗

∗ ∗

∗

1 T T φj12 = Xj CkT + (Dh Uj + D− h Sj ) Dk , 1 T T T φj13 = Xj N1k + (Dh Uj + D− h Sj ) N2k ,

with T ϕi11 = He{Ai Xi + Bi [Ds Wi + D− s Si ]} + ε1i Mi Mi .

On the other hand, the stabilizing control law (10) can be designed as K1i = Wi Xi−1 , K2i = Ui Xi−1 and Hi = Si Xi−1 . Proof. For any k = 1, 2, . . . and t ∈ [tk−1 , tk ), we assume that σ (t) = σ (tk−1 ) = ik−1 ∈ I[1, N ], while σ (t) = σ (tk ) = ik ∈ I[1, N ] for t ∈ [tk , tk+1 ). When t ∈ [tk , tk+1 ), we choose the Lyapunov–Krasovskii functional candidate for system (1) as follows V (x(t)) = xT (t)Pik x(t) +

∫

t

xT (s)Qik x(s)ds.

(21)

t −d(t)

In what follows, setting Pi = Xi−1 , Si = Hi Xi , i ∈ I[1, N ] and pre- and post-multiplying (20) by diag{I , Pi } yield that Ω (Pi , 1) ⊂ L(Hi x(t)). By using Lemma 1, for any x(t) ∈ Ω (Pik , 1), the closed-loop system can be modeled locally as (16). For t ∈ [tk−1 , tk ), the time derivative of V (x(t)) along the trajectories of system (16) is given by T ˙ D+ V (x(t)) = 2xT (t)Pik x˙ (t) + xT (t)Qik x(t) − (1 − d(t))x (t − d(t))Qik x(t − d(t))

≤ maxm 2xT (t)Pik {Aik (t)x(t) + Adik (t)x(t − d(t)) + Bi (t) s∈I[1,2 ]

× [Ds Kik x(t) + D− ¯ ik (t , x(t)) + ω1 (t)] + f (t , xt )} s Hik x(t) + ω + xT (t)Qik x(t) − (1 − h)xT (t − d(t))Qik x(t − d(t)).

(22)

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Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

Noticing the assumption (6), for any scalar gik > 0, one has D+ V (x(t)) ≤ max 2xT (t)Pik {Aik (t) + Bik (t)[Ds Ki + D− s Hik ]}x(t) s∈I[1,2m ]

+ 2xT (t)Pik Adik (t)x(t − d(t)) + 2xT (t)Pik Bik [ω¯ ik (t , x(t)) + ω1 (t)] + 2xT (t)Pik f (t , xt ) − gi−k 1 [f T (t , xt )f (t , xt ) − xT (t)GT1 G1 x(t) − x(t − d(t))T GT2 G2 x(t − d(t))]

(23)

+ xT (t)Qik x(t) − (1 − h)xT (t − d(t))Qik x(t − d(t)). Based on the presentation of ω ¯ ik in (11) and the assumption of (2), we can get xT (t)Pik Bik [ω ¯ ik (t , x(t)) + ω1 (t)] = φ T (t , ik , x(t))[ω¯ ik (t , x(t)) + ω1 (t)]

≤

m ∑ {−ε1 |eTl φ (t , ik , x(t))| + ε1 |eTl φ (t , ik , x(t))|}

(24)

l=1

= 0. Together (6) and (24) with (22), it follows that D+ V (x(t)) ≤ max ξ T (t)Ω0ik ξ (t),

(25)

s∈I[1,2m ]

where ξ (t) = [xT (t) xT (t − d(t)) f (t , xt )]T and

Ω0ik

⎡ 11 Ω0i =⎣ ∗ ∗

Pik Adik (t) G2 G2 − (1 − h)Qik

gi

k

⎤

Pik 0

−1 T

−gi−k 1 I

∗

⎦,

with −1 T Ω0i11k = He{Pik Aik (t) + Pik Bik [Ds Kik + D− s Hik ]} + gik G1 G1 + Qik .

Denote

π0ik

⎡ 11 π0ik =⎣ ∗ ∗

Pik

⎤

gi−1 GT2 G2 − (1 − h)Qik

0

−gi−k 1 I

⎦,

∗

Pik Adik k

(26)

with −1 T π0i11k = He{Pik Aik + Pik Bik [Ds Kik + D− s Hik ]} + gik G1 G1 + Qik .

Noticing (3) and using Lemma 2, it is easy to get that

Ω0ik = π0ik +

[

Pik M1ik 0

≤ π0ik + ε1ik

[

]

[

F1ik (t) Ndik Pik M1ik 0

][

0

Pik M1ik 0

]

[ + Ndik

]T

0

[ + ε1ik

−1

]T

][

NaiT

k

NdiT

[ F1ik (t)

k

Pik M1ik 0

(27)

]T

NaiT

k

NdiT

]T

,

k

¯ si = Ds Ki + D− where D s Hik . k k Setting Qik = Yi−1 , Wik = K1ik Xik and multiplying diag{Pik , Qik , gik , I } on both sides of (18). Further, using Schur k complement and together with (27), it yields that Ω0ik < 0.

(28)

For any x(t) ̸ = 0, t ∈ [tk , tk+1 ), from (25), we have D+ V (x(t)) < 0.

(29)

Based on the above discussion, it is known that the switched impulsive system is asymptotically stable at any time distance t ∈ [tk , tk+1 ). In what follows, we investigate the stability of the system at the impulsive and switching points.

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

7

First of all, we assume that x(tk− ) ∈ Ω (Pik−1 , 1) ⊂ L(Hik−1 x(t)). Based on (16), we have m

−

T

−

V (x(tk )) − V (x(tk )) = x (tk )

2 ∑

T ηh {Ck (tk ) + Dk (tk )[Dh K2ik−1 + D− h Hik−1 ]} Pik

h=1 m

×

2 ∑

(30)

− ηh {{Ck (tk ) + Dk (tk )[Dh K2ik−1 + D− h Hik−1 ]}x(tk )

h=1 T − xT (tk− )[Pik−1 x(tk− ) + D− h Hik−1 ]} Pik {Ck (tk ) + Dk (tk ){[Dh K2ik−1

− +D− h Hik−1 ]} − Pik−1 }x(tk ).

Using Lemma 2 and from (4), there exists a positive scalar ε2ik such that T ψtk = {Ck (tk ) + Dk (tk )[Dh K2ik−1 + D− h Hik−1 ]} Pik {Ck (tk )

+ Dk (tk )[Dh K2ik−1 + D− h Hik−1 ]} − Pik−1 ≤ (Ck + Dk )T (Pi−k 1 − ε2ik M2 M2T )−1 (Ck + Dk )

(31)

−1 + ε2i [N1k + N2k (Dh K2ik−1 + D− h Hik−1 )]T [N1k k

+ N2k (Dh K2ik−1 + D− h Hik−1 )] − Pik−1 . One the other hand, setting Uik−1 complement, we have

= K2ik−1 Xik−1 , pre- and post-multiplying (19) by diag{Xik−1 , I } and using Schur

ψtk < 0.

(32)

Substituting (32) into (30), one has V (x(tk )) < V (x(tk− )),

(33)

which means that V (t) is decreasing at the impulsive and switching time point tk . Next, we will give an estimation of the domain of the attraction for the controlled system (16). Together (21) with (33), for t ∈ [t0 , t1 ), we have xT (t)Pi0 x(t) ≤ V (x(t)) ≤ V (x(t0 ))

≤ max ∥φ (s)∥2 [λmax (Pi0 ) + dλmax (Qi0 )] s∈[−d,0]

(34)

≤ max ∥φ (s)∥2 [λ1 + dλ2 ] s∈[−d,0]

= Υ (φ ) . If the initial condition satisfies that Υ (φ ) ≤ 1, it is easy to get that x(t) ∈ Ω (Pi0 , 1). And then, when it is at the impulsive and switching point t1 , from (33), we have xT (t1 )Pi1 x(t1 ) ≤ V (x(t1 )) ≤ V (x(t1− )) ≤ V (x(t0 )) ≤ Υ (φ ).

(35)

Combining (34) with (35), for any t ≥ t0 and under the assumption that Υ (φ ) ≤ 1, we can get that x (t)Piσ (t) x(t) ≤ 1. This completes the proof. ■ T

Remark 4. To obtain the conditions of the robust stability for the closed-loop system (16), a classical construction of Lyapunov–Krasovskii function has been used. In the future, we will do research further for more advanced Lyapunov function to reduce conservatism [33–35]. 3.2. Robust H∞ control In this section, we address the robust H∞ control problem for system (1). First of all, the definition of H∞ is presented as follows Definition 1. The state feed-back control law (10) is a local H∞ control law with L2 gain less than γ if the corresponding closed-loop system is locally asymptotically stable with an estimate ∫ T of the domain ∫ofT the attraction when ω(t) = 0 and its zero state response (φ (s) = 0, ω(s) = 0, s ∈ [−d, t0 ]) satisfies 0 ∥z(t)∥2 dt ≤ γ 2 0 ∥ω(t)∥2 dt for all ω ∈ L2 [0, T ] and T ≥ 0. Theorem 2. If for prescribed scalars γ > 0, there exist matrices Xi > 0, Yi > 0, Si , Wi , Ui , i ∈ I[1, m] and positive scalars ε1i , ε2i , gi , such that (19), (20) and (36) in the top of next page with T ϕi11 = He{Ai Xi + Bi [Ds Wi + D− s Si ]} + ε1i Mi Mi , T ϕi14 = Xi N1iT + [Ds Wi + D− s Si ] Nbi ,

8

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

⎡ 11 ϕi ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Adi Yi

gi I

ϕi14

Xi G1

0

Xi

Xi CiT E2i + E1i

Xi CiT

−(1 − h)Yi ∗ ∗ ∗ ∗ ∗ ∗ ∗

0

Yi NdiT

0

Yi GT2

0

0

0

−gi I ∗ ∗ ∗ ∗ ∗ ∗

0

0

0

0

0

0

−ε1i I ∗ ∗ ∗ ∗ ∗

0

0

0

0

0

−gi I ∗ ∗ ∗ ∗

0

0

0

0

−gi I ∗ ∗ ∗

0

0

0

−Yi ∗ ∗

0

0

−γ 2 I − E2iT E2i ∗

0

−1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(36)

−I −1

then system (1) has a local H∞ control law (10) with K1i = Wi Xi , K2i = Ui Xi and Hi = Si Xi−1 , an estimate of the domain of the attraction can be given by Υ (φ ) ≤ 1. Moreover, the closed-loop system has L2 gain less than γ . Proof. It is obvious that (36) can ensure (18). And then, based on the Theorem 1, we can know that the corresponding closed-loop system (16) can be locally asymptotically stabilized by the control law (10) with K1i = Wi Xi−1 , K2i = Ui Xi−1 with an estimate of the domain of the attraction Υ (φ ) ≤ 1 when ω(t) = 0. In what follows, we assume that φ (s) = 0, ω(s) = 0, s ∈ [−d, t0 ] and ω(s) ̸ = 0, s ∈ (t0 , +∞). Then, from the process of proof for Theorem 1, it is easy to get that

¯ 0ik ζ (t), t ∈ [tk , tk+1 ), D+ V (x(t)) ≤ max ζ T (t)Ω

(37)

s∈I[1,2m ]

where ζ (t) = ((xT (t) xT (t − d) f T (t , xt )) ωT (t))T and

¯ 0ik Ω

⎡ 11 Ω0i ⎢ ∗ =⎢ ⎣ ∗ ∗

⎤

Pik Adik (t) gi−1 GT2 G2 − (1 − h)Qik

Pik 0

∗ ∗

−gi−k 1 I ∗

k

Pik E1ik ⎥ 0 ⎥ 0 0

⎦,

11 was defined in (25). with Ω0i k Denote that

J(t) = z T (t)z(t) − r 2 ωT (t)ω(t) + D+ V (t),

(38)

where V (t) was define in (21). Let Pi = Xi−1 , Qi = Yi−1 , multiplying diag{Pik , Qik , gik , I } on the both sides of (36) and using Schur complement, we have z T (t)z(t) − r 2 ωT (t)ω(t) + D+ V (t) < 0.

(39)

Under the zero initial condition, integrating both sides of (40) from 0 to T ∈ [tk , tk+1 ), one has T

∫

(z (t)z(t) − r ω (t)ω(t))dt ≤ T

2

T

0

T

∫

D+ V (t)dt 0

= (V (0) +

k ∑

(40)

(V (tj+ ) − V (tj− )) − V (T ))

j=1

≤ 0. For any T ∈ [tk , tk+1 ), (40) yields that T

∫

z T (t)z(t) ≤ r 2 0

T

∫

(ωT (t)ω(t))dt ,

(41)

0

which means that the L2 -gain condition of disturbance attenuation is satisfied. The proof is completed.

■

Remark 5. In this paper, we assume that the state impulses are activated at the switching time points. By the implementation of control on both each subsystem and states impulsive generators at the switching instants, the design methods and conditions we proposed can achieve anti-disturbance control requirement for system (1). On the other hand, the results of this paper can be used at arbitrary switch law due to the control for the state impulsive jumping generators at the switching instants. In the future, we will try to reduce the conservatism by designing more advanced switching law, such as time driven or state depended switching law.

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

9

Fig. 1. Switching law.

4. Numerical example Consider the switched nonlinear system (1) including two subsystems, the parameters of which are listed as follows: 0.15 0.1

0 .2 0.05

[ A1 =

0.1 0

[ Na1 =

]T

0.2 0.06

0.2 0

[ Na2 =

[ Ck =

0.3 0.25

]T

]

Ad1 =

[

]

]T

0 0.1

, C1 =

[

,

Ad2 =

, Nd2 = 0.1 0.15

0.08 0.03

[

,

, Nd1 = 0.3 0.1

[ A2 =

]

[

]T

0 0.1

, Dk =

0.1 0.5

[

0.03 0.3

, C2 =

0.15 0.05

[

0.05 0.1

]

] ]T

0.05 0.01

[

0.2 0.1

, M2 =

0.1 0.1

[

]

, E11 =

[

0.02 0.02

]

, M11 =

0.1 0.1

[

]

,

, Nb1 = 0.5, E21 = 0.05, G1 = 0.05,

]

]T [

, B1 =

, B2 =

[

0.1 0.1

]

, E12 =

0.01 0.01

[

]

, M12 =

[

0 0.2

]T

,

(42)

, Nb2 = 0.2, E22 = 0.02, G2 = 0.05,

0 0.1

]

, N1k =

[

0 0.1

]T

, N2k = 0.1.

Let d = 1, h = 0.1, F11 = e−t , F12 = sin(0.1 ∗ t), Fk = sin(t), ω1 (t) = 0.02 ∗ e−2t and f (t , xt ) = x1 (t − d(t)) ∗ sin(x2 (t)), under the switching law described in Fig. 1, the example is simulated for two cases as follows: Case1 (w ith ω2 (t) ≡ 0) : The following gains of the controllers (10) can be obtained by solving the conditions in Theorem 1,

[ K11 =

−6.7307 −1.0325

]T

, K12 =

[

−8.5980 −1.7384

]T

, K21 =

[ ]T [ ]T −1.2315 −1.1256 , K22 = , −0.0544 −0.0088

(43)

with parameters ε11 = 0.0698, ε12 = 0.1215, ε21 = 0.0958, ε22 = 0.0946, g1 = 0.0014, g2 = 0.0019. The state trajectories of the closed-loop system under the controllers (10) with the gains in (44) are depicted in Fig. 2. On the other hand, the state impulse trajectories at the switching times are shown in Fig. 3. It can be seen that system (1) without external disturbance ω2 (t) can be robustly stabilized by our proposed design method, and the disturbance matched with control input can be rejected. The signals of the control inputs generated by composite controller and impulsive controller (12) are shown in Figs. 4 and 5, respectively. Case2 (w ith ω2 (t) ̸ ≡ 0) : For a given H∞ performance index γ = 0.3162, by computing the conditions in Theorem 2, we can get the gains of the controllers (12) as follows: K11 =

[ ]T [ ]T [ ]T [ ]T −7.1420 −6.6289 −0.6259 −0.5127 , K12 = , K21 = , K22 = , −1.1561 −1.8759 0.1426 0.0320

(44)

with parameters ε11 = 0.0175, ε12 = 0.1308, ε21 = 0.0891, ε22 = 0.1187, g1 = 0.0010, g2 = 0.0013. Under the influence of the disturbance ω2 (t) = 0.05 ∗ sin(t), the states of the closed-loop system can be shown in Fig. 6, and Fig. 7 shows the impulsive responds of the states of the system under the controllers (10) with (44), the input signals of which are shown in Figs. 8 and 9. From Fig. 6 and controller output signal in Fig. 10, the robust H∞ performance can be achieved based on the design conditions in Theorem 2.

10

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

Fig. 2. States trajectories of system (1) without ω2 under controllers (10) with gains (43).

Fig. 3. States impulse trajectories at switching instant under controllers (10) with gains (43).

Fig. 4. Control input u1 with gains (43).

Remark 6. From this simulation, we can see that the proposed composite controller can reject the matched disturbance effectively. The controllers (10) can robustly stabilize system (1) without regard to the external disturbance ω2 (t). When the design conditions in Theorem 2 are satisfied, the controllers (10) can achieve disturbance rejection and attenuation for system (1).

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

11

Fig. 5. Impulsive control input u2 with gains (43).

Fig. 6. States trajectories of system (1) under controllers (10) with gains (44).

Fig. 7. States impulse trajectories at switching instant under controllers (10) with gains (44).

Remark 7. To show the generality of the our proposed design method, the state responses of the closed-loop system over aforementioned-generated 200 switching signals are displayed in Figs. 11 and 12, which clearly show the effectiveness of our results. 5. Conclusion The composite control of a class of uncertain switched impulsive time-delay system with actuator saturation and multiple disturbances has been discussed in this paper. A switching composite controller based on the bound of matched disturbances has been constructed for each subsystem, and a switching state feedback controller has been designed for

12

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

Fig. 8. Control input u1 with gains (44).

Fig. 9. Impulsive control input u2 with gains (44).

Fig. 10. Control output z(t).

state impulsive jumping generators at switching instant. By treating saturation with generalized differential inclusions, the closed-loop system was equivalent to a polytopic form locally. Some design conditions of the controller have been obtained to guarantee the locally robustly asymptotic stability and robust H∞ disturbance attenuation of the closed-loop system, and the domain of attraction has been estimated. Finally, a simulation example was shown to validate the feasibility and effectiveness of the results in this paper.

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

13

Fig. 11. States trajectories of system (1) without ω2 under controllers (10) with gains (43) under 200 switching signals.

Fig. 12. States trajectories of system (1) under controllers (10) with gains (44) under 200 switching signals.

References [1] T. Hu, Z. Lin, Control Systems with Actuator Saturation: Analysis and Design, Springer Science & Business Media, 2001. [2] S. Tarbouriech, G. Garcia, J.M.G.da.Silva. Jr, et al., Stability and Stabilization of Linear Systems with Saturating Actuators, Springer Science & Business Media, 2011. [3] Y.Y. Cao, Z. Lin, D.G. Ward, An antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation, IEEE Trans. Autom. Control 47 (1) (2002) 140–145. [4] W. Qi, J.H. Park, J. Cheng, et al., Anti-windup design for stochastic Markovian switching systems with mode-dependent time-varying delays and saturation nonlinearity, Nonlinear Anal. Hybrid Syst. 26 (2017) 201–211. [5] M. Hussain, M. Rehan, C.K. Ahn, et al., Robust anti-windup for one-sided lipschitz systems subject to input saturation and applications, IEEE Trans. Ind. Electron. (2018). [6] L. Zhang, E.K. Boukas, A. Haidar, Delay-range-dependent control synthesis for time-delay systems with actuator saturation, Automatica 44 (10) (2008) 2691–2695. [7] B. Xiao, Q. Hu, Y. Zhang, Adaptive sliding mode fault tolerant attitude tracking control for flexible spacecraft under actuator saturation, IEEE Trans. Control Syst. Technol. 20 (6) (2012) 1605–1612. [8] W. Qi, Y. Kao, X. Gao, et al., Controller design for time-delay system with stochastic disturbance and actuator saturation via a new criterion, Appl. Math. Comput. 320 (2018) 535–546. [9] G. Song, J. Lam, S. Xu, Quantized feedback stabilization of continuous time-delay systems subject to actuator saturation, Nonlinear Anal. Hybrid Syst. 30 (2018) 1–13. [10] Z. Yu, Y. Yang, S. Li, et al., Observer-based adaptive finite-time quantized tracking control of nonstrict-feedback nonlinear systems with asymmetric actuator saturation, IEEE Trans. Syst. Man Cybern.: Syst. (99) (2018) 1–12. [11] P. Gahinet, P. Apkarian, A linear matrix inequality approach to H∞ control, Int. J. Robust Nonlinear Control 4 (4) (1994) 421–448. [12] S. Sastry, M. Bodson, Adaptive Control: Stability, Convergence and Robustness, Courier Corporation, 2011. [13] S. Li, J. Yang, W.H. Chen, et al., Disturbance Observer-Based Control: Methods and Applications, CRC press, 2016. [14] J. Han, Active Disturbance Rejection Control Technique-the Technique for Estimating and Compensating the Uncertainties, National Defense Industry Press, Beijing, 2008, pp. 197–270. [15] L. Guo, S. Cao, Anti-Disturbance Control for Systems with Multiple Disturbances, CRC Press, 2013. [16] L. Guo, W.H. Chen, Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach, Internat. J. Robust Nonlinear Control 15 (3) (2005) 109–125.

14

Y. Wei and G.-P. Liu / Nonlinear Analysis: Hybrid Systems 35 (2020) 100825

[17] X. Wei, L. Guo, Composite disturbance-observer-based control and H∞ control for complex continuous models, Internat. J. Robust Nonlinear Control 20 (1) (2010) 106–118. [18] Y. Wei, W.X. Zheng, S. Xu, Anti-disturbance control for nonlinear systems subject to input saturation via disturbance observer, Systems Control Lett. 85 (2015) 61–69. [19] Y. Li, H. Sun, G. Zong, et al., Composite anti-disturbance resilient control for Markovian jump nonlinear systems with partly unknown transition probabilities and multiple disturbances, Internat. J. Robust Nonlinear Control 27 (14) (2017) 2323–2337. [20] X. Yao, L. Guo, L. Wu, et al., Static anti-windup design for nonlinear Markovian jump systems with multiple disturbances, Inform. Sci. 418 (2017) 169–183. [21] Z. Li, Y. Soh, C. Wen, Switched and Impulsive Systems: Analysis, Design and Applications, Springer Science & Business Media, 2005. [22] G. Xie, L. Wang, Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Trans. Automat. Control 49 (6) (2004) 960–966. [23] W. Xiang, J. Xiao, Stability analysis and control synthesis of switched impulsive systems, Internat. J. Robust Nonlinear Control 22 (13) (2012) 1440–1459. [24] Y. Chen, S. Fei, K. Zhang, Stabilization of impulsive switched linear systems with saturated control input, Nonlinear Dynam. 69 (3) (2012) 793–804. [25] G. Zong, H. Ren, L. Hou, Finite-time stability of interconnected impulsive switched systems, IET Control Theory Appl. 10 (6) (2016) 648–654. [26] Z. Ai, L. Peng, Exponential stabilisation of impulsive switched linear systems via a periodic switching scheme, IET Control Theory Appl. 11 (16) (2017) 2921–2926. [27] I. Zamani, M. Shafiee, A. Ibeas, On singular hybrid switched and impulsive systems, Internat. J. Robust Nonlinear Control 28 (2) (2018) 437–465. [28] R. Wang, L. Hou, G. Zong, et al., Stability and stabilization of continuous-time switched systems: A multiple discontinuous convex Lyapunov function approach, Internat. J. Robust Nonlinear Control 29 (5) (2019) 1499–1514. [29] R. Wang, T. Jiao, T. Zhang, et al., Improved stability results for discrete-time switched systems: A multiple piecewise convex Lyapunov function approach, Appl. Math. Comput. 353 (2019) 54–65. [30] H. Ren, G. Zong, L. Hou, et al., Finite-time resilient decentralized control for interconnected impulsive switched systems with neutral delay, ISA Trans. 67 (2017) 19–29. [31] F. Wu, Q. Zheng, Z. Lin, Disturbance attenuation by output feedback for linear systems subject to actuator saturation, Internat. J. Robust Nonlinear Control 19 (2) (2009) 168–184. [32] I.R. Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems Control Lett. 8 (1987) 351–357. [33] Y. Chen, S. Fei, Y. Li, Stabilization of neutral time-delay systems with actuator saturation via auxiliary time-delay feedback, Automatica 52 (2015) 242–247. [34] C.de. Souza, V.J.S. Leite, E.B. Castelan, et al., ISS Robust stabilization of state-delayed discrete-time systems with bounded delay variation and saturating actuators, IEEE Trans. Automat. Control (2018). [35] M. Rehan, N. Iqbal, K.S. Hong, Delay-range-dependent control of nonlinear time-delay systems under input saturation, Internat. J. Robust Nonlinear Control 26 (8) (2016) 1647–1666.