# Robust Exponential Stabilization of Uncertain Impulsive Bilinear Time-Delay Systems with

Authors: Yang Shujie; Shi Bao; Zhang Qiang; Pan Tetie
Journal: Journal of Control Science and Engineering
 (2010)

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## Abstract

This paper investigates the problem of robust exponential stabilization
for uncertain impulsive bilinear time-delay systems with saturating actuators. By using the Lyapunov function and Razumikhin-type techniques, two classes of impulsive systems are considered: the systems with unstable
discrete-time dynamics and the ones with stable discrete-time dynamics. Sufficient conditions for robust stabilization are obtained in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of the
theoretical results.


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## Body

## 1. Introduction

In practical control systems, impulsive dynamical systems are very important and have attracted considerable interest in science and engineering during the past decades. Two classical monographs are Lakshmikantham et al. [1] and Bainov and Simeonov [2]. In general, as reported in [1], impulsive systems provide a natural framework for mathematical modeling of many real-world evolutionary processes where the states undergo abrupt changes at certain instants or at variable instants, or autonomous systems with impulsive effects. Stability properties of impulsive systems have been extensively studied in the literatures. We refer to Bainov and Simeonov [2], Li, Soh and Xu [3, 4], Li, Wen and Soh [5], Yang [6], Chen and Zheng [7], Barreiro and Baños [8], and the references therein. The impulsive control method based on stability theory of impulsive dynamical systems has been widely used, see [7], Xu and Teo [9], and Liu, Eberhard and Teo [10].Saturating nonlinear actuators is one of the common nonlinearities and exists in many practical systems. If a control system is designed to be stable without considering the saturation, the stability of the closed-loop-controlled system cannot be guaranteed, see Ma and Zhang [11]. Two important papers on saturation control are Full [12] and Sontag and Sussmann [13]. However, as we have known, no papers have considered the saturation control in impulsive systems.This work is inspired by Wu and Wei [7], in which the authors considered the problems of robust stability and stabilization for uncertain impulsive time-delay systems. Unfortunately, they need all the impulsive time sequences to satisfy some strict conditions. That is, the length of the intervals between two adjacent time instants must have upper bound or lower bound. But in practical systems, it is always impossible or difficult to obtain it. In this paper, robust stabilization for uncertain impulsive time-delay systems with saturating actuators is considered. With saturation control, new conditions based on Lapunov-Razumikhin function and LMIs are established which can easily be used for systems with any time sequences.


## 2. Problem Formulation and Preliminaries

Throughout this paper, if not explicitly given, matrices are assumed to have compatible dimensions. For symmetric matricesA and B, the notation A≥B (A>B, A≤B, A<B) means A-B is positive semidefinite (positive definite, negative semidefinite, negative definite) matrix. λmax(·) and λmin(·) represent the maximum and minimum eigenvalues of the corresponding matrix, respectively. ∥·∥ denotes Euclidean norm for vectors or the spectral norm of matrices. PC([-τ,0],ℝn) denotes the set of piecewise right continuous function ϕ:[-τ,0]→ℝn with the norm defined by ∥ϕ∥τ=sup-τ≤θ≤0∥ϕ(θ)∥, where τ is a known positive constant.Consider uncertain bilinear impulsive systems with time-delay and saturating actuators(1)ẋ(t)=A(t)x(t)+A1(t)x(t-τ(t))+B(t)sat⁡(u(t)),t≠tk,Δx(t)=(C(t)-I)x(t-)+∑i=1msat⁡(ui(t))Bi(t)x(t-),t=tk,x(t0+θ)=ϕ(θ),t0=0,θ∈[-τ,0],
where x(t)∈ℝn is the system state, u(t)∈ℝm is the continuous control input, 0≤τ(t)≤τ, Δx(tk)=x(tk+)-x(tk-) describes the state jumping at impulsive time instant t=tk with x(tk+)=x(tk)=limh→0+x(tk+h), x(tk-)=limh→0-x(tk+h), k∈ℕ={1,2,…} and 0<t1<t2<⋯<tk<⋯ (tk→∞ as k→∞).  A(t),A1(t),B(t),Bi(t)(i=1,2,…,m), and C(t) are matrix functions with time-varying uncertainties ΔA(t), ΔA1(t), ΔB(t), ΔBi(t) (i=1,2,…,m) and ΔC(t). In this paper, we impose the following hypothesis on the above uncertainties:(2)(ΔA(t),ΔA1(t),ΔB(t))=D1F1(t)(NA,NA1,NB),(ΔC(t),ΔBi(t))=D2F2(t)(NC,NBi),(i=1,2,…,m),
where D1,D2,NA,NA1,NB,NC,NBi(i=1,2,…,m) are known real constant matrices and Fi(t) are unknown matrix functions satisfying ∥Fi(t)∥≤1 with the elements of Fi(t) is lebesgue measurable, (i=1,2). The saturating function Sat(u(t)) is defined as(3)Sat(u(t))=(sat⁡(u1(t),…,sat⁡(um(t))))T,(4)sat⁡(uj(t))={-Uj,uj(t)≤-Uj,uj(t),-Uj<uj(t)<Uj,Uj,uj(t)≥Uj,
where Uj>0(j=1,2,…,m) are known real constants representing the upper bound of the control.In the following, we will divide two cases to consider the robust stabilization of system (1). We denote by 𝒩min(β) the class of impulsive time sequences that satisfy infk{tk-tk-1}≥β and denote by 𝒩max(β) the class of impulsive time sequences that satisfy supk{tk-tk-1}≤β.We need the following lemmas.Lemma 1 (see Su, Chu, Lu and Ji [14]).
For any vectorsx,y∈ℝn and real matrices A,D,E,F and P>0 with FTF≤I, the following inequalities hold for any scalar ɛ>0:(1)
2xTDFEy≤(1/ɛ)xTDDTx+ɛyTETEy,(2)
2xTy≤xTP-1x+yTPy and for P-ɛDDT>0,(3)
(A+DFE)TP-1(A+DFE)≤(1/ɛ)ETE+AT(P-ɛDDT)-1A.Lemma 2 (Schur complement [14]).
For given symmetric matrixS=(S11S12S21S22) with S11∈ℝr×r, the following three conditions are equivalent:(1)
S<0,(2)
S11<0,S22-S12TS11-1S12<0,(3)
S22<0,S11-S12S22-1S12T<0.


## 3. Robust Stabilization

In this section, we consider the robust stabilization of system (1).Definition 3.
System (1) with Fi(t)=0 (i=1,2) and u(t)=0 is said to be exponentially stable over a given class 𝒩 if there are scalars ɛ>0,δ>0 and v>0 such that ϕ∈PC([-τ,0],ℝn) with ∥ϕ∥τ≤δ implying ∥x(t;t0,ϕ)∥<ɛexp(-v(t-t0)) for t≥t0.Definition 4.
System (1) is said to be robustly exponentially stabilizable over a given class 𝒩 with respect to static state feedback if there exists a memoryless linear state feedback law, such that the corresponding closed-loop system is robustly exponentially stable for all uncertainties satisfying (2).For anyϕ∈PC([-τ,0],ℝn), we denote x(t;t0,ϕ) by x(t). Let us now consider designing a memoryless state feedback control law of the form(5)u(t)=2Kx(t)
to stabilize system (1), where 2K is a constant gain to be designed.Now we decompose the saturation nonlinear functionSat(u(t)) as the sum of a linear part and another nonlinear part (dead zone nonlinearity) Δu(t) as Sat(u(t))=u(t)+Δu(t), then by [14], (5) implies that(6)Δu(t)TΔu(t)≤xT(t)KTKx(t).Substituting (5) into system (1), we have(7)ẋ(t)=(A(t)+2B(t)K)x(t)+A1(t)x(t-τ(t))+B(t)Δu(t),t≠tk,Δx(t)=(C(t)-I)x(t-)+∑i=1msat⁡(ui(t))Bi(t)x(t-),t=tk,x(t0+θ)=ϕ(θ),t0=0,θ∈[-τ,0],Lemma 5.
Assume that there exist matrixX>0 and positive scalars ɛ1, ɛ2, μ such that(8)(-μXXNCTXCTX0⋯0*-ɛ1I00U1NB1⋯UmNBm**ɛ1D2D2T-X0U1B1⋯UmBm***-(ɛ2∑i=1mUi)-1I0⋯0****-ɛ2I⋯0⋯⋯⋱*****⋯-ɛ2I)≤0,and set V(t)=xT(t)Px(t), where P=X-1. Then
(9)V(tk)≤μV(tk-).Proof.
By the second equation of (1), we have
(10)x(tk)=(C(tk)+∑i=1msat⁡(ui(tk))Bi(tk))x(tk-)V(tk)-μV(tk-)=xT(tk-)Ξ1x(tk-),
where Ξ1=(C(t)+∑i=1msat(ui(t))Bi(t))TP(C(t)+∑i=1msat(ui(t))Bi(t)). By Lemmas 1 and 2, (2), (4), and (8) imply that Ξ1≤0, hence (9) holds.Theorem 6.
Assume that there exist matricesX>0, Y and positive scalars α, d, ɛ1, ɛ2, ɛ3 and 0<μ<1 such that (8) and
(11)(Ψ1A1XBYT2YTNB1T+XNATD1*-αX00XNA1T0**-d-1I0NBT0***-dI00****-ɛ3I0*****-ɛ3-1I)<0
hold, then for any bounded time-delay τ(t)≤τ, system (1) can be robustly stabilized by the control law (5) with K=YX-1 for any impulsive time sequence {tk}, where Ψ1=αμ-1X+AX+XAT+2BY+2YTBT.Proof.
LetP=X-1, λ0=λmin(P), λ1=λmax(P). Substituting Y=KX-1 in (11) and pre- and postmultiplying it by diag{P,P,I,I,I,I} yield
(12)(Ψ1′PA1PBKT(NA+2NBK)TPD1*-αP00NA1T0**-d-1I0NBT0***-dI00****-ɛ3I0*****-ɛ3-1I)<0,
where Ψ1′=αμ-1P+(A+2BK)TP+P(A+2BK). Hence, there exists small positive scalar μ̃, such that(13)(Ψ1′′PA1PBKT(NA+2NBK)TPD1*-αexp⁡(-μ̃τ)P00NA1T0**-d-1I0NBT0***-dI00****-ɛ3I0*****-ɛ3-1I)<0,where Ψ1′′=(αμ-1+μ̃)P+(A+2BK)TP+P(A+2BK).For givenρ>0, choose 0<δ≤μλ0/λ1ρ. We assume that the initial function ϕ∈PC([-τ,0],ℝn) satisfying ∥ϕ∥τ≤δ. Set W(t)=exp(μ̃t)V(t). In the following, we will prove that for t≥t0-τ,(14)W(t)<λ0ρ2.First, we prove that (14) holds for t∈[t0-τ,t1). It is noticed that for θ∈[-τ,0], W(t0+θ)≤λ1∥ϕ∥τ2≤μλ0ρ2<λ0ρ2. So it needs only to prove that, for t∈(t0,t1), (14) holds. Otherwise, there exists t∈(t0,t1), such that W(t)≥λ0ρ2. Set t*=inf{t∈[t0,t1):W(t)≥λ0ρ2}. Then t*∈(t0,t1), W(t*)=λ0ρ2. Set t¯=sup{t∈[t0,t*):W(t)≤μλ0ρ2}. Then t¯∈[t0,t*) and W(t¯)=μλ0ρ2. Hence, for t∈[t¯,t*] and θ∈[-τ,0], we have(15)μW(t+θ)≤μλ0ρ2≤W(t)≤λ0ρ2.
It follows with (6) that(16)D+W(t)≤exp⁡(μ̃t)(μ̃V(t)+D+V(t))+α(μ-1W(t)-W(t-τ(t)))+d-1(xT(t)KTKx(t)-Δu(t)TΔu(t))exp⁡(μ̃t)≤exp⁡(μ̃t)ηT(t)Ξ2η(t),
where (17)ηT(t)=(xT(t),x(t-τ(t))T,Δu(t)T),Ξ2=(Ψ2′PA1PB*-αexp⁡(-μ̃τ)P0**-d-1I)+d-1(KT00)(K00)+ɛ3-1((NA+2NBK)TNA1TNBT)(NA+2NBKNA1NB)+ɛ3(PD100)(D1TP00),
By Lemma 2, (13) implies that Ξ2<0, which yields by (16) that for t∈[t¯,t*],(18)D+W(t)<0.
It follows that W(t*)≤W(t¯). This is a contradiction with the fact W(t*)=λ0ρ2>W(t¯)=μλ0ρ2. Thus, (14) holds for t∈[t0-τ,t1).Now we assume that (14) holds for t∈[t0-τ,tm); we will prove that (14) still holds for t∈[tm,tm+1). On the contrary, there exists some t∈[tm,tm+1), such that W(t)>λ0ρ2. Set t*=inf{t∈[tm,tm+1):W(t)≥λ0ρ2}. Then because of (9), we have W(tm)≤μλ0ρ2<λ0ρ2, which implies that t*∈(tm,tm+1) and W(t*)=λ0ρ2. Set t¯=sup{t∈[tm,t*):W(t)≤μλ0ρ2}. Then t¯∈[tm,t*) and W(t¯)=μλ0ρ2. Hence, for t∈[t¯,t*], we obtain that (15) holds, which leads that (16) and (18) hold. It follows that W(t*)≤W(t¯), which contradicts the fact W(t*)=λ0ρ2>W(t¯)=μλ0ρ2. Thus, (14) holds for t∈[t0-τ,tm+1). By mathematical induction, (14) holds for t≥t0-τ. That is, system (1) can be robustly stabilized.Remark 7.
By Theorem6, in the case μ<1, different from Chen and Zheng [7], we can stabilize impulsive systems with any time sequences by saturation control.Remark 8.
In Chen and Zheng [7], the authors need continuous control input uc(t) and impulsive control input ud(t). But in this paper, we only need continuous control input u(t).Forμ=1, we have the following result.Theorem 9.
Assume that there exist matricesX>0, Y and positive scalars α, d, ɛ1, ɛ2, and ɛ3 such that (8) and (11) hold, then for any bounded time-delay τ(t)≤τ, system (1) can be robustly stabilized by the control law (5) with K=YX-1 for any impulsive time sequence {tk}, where Ψ2=αX+AX+XAT+2BY+2YTBT.Proof.
Choose a sufficiently small scalarh>0, and set μ¯=1-h. Then for ∥ϕ∥τ≤δ≤μ¯λ0/λ1ρ with ϕ∈PC([-τ,0],ℝn), similar to the proof of Theorem 6, we can obtain the result of Theorem 9. The proof is complete.Forμ>1, we have the following result.Theorem 10.
Assume that for a prescribed scalarβ>0, there exist matrices X>0, Y and positive scalars α, d, ɛ1, ɛ2, ɛ3, and μ>1 such that (8) and
(19)(Ψ3A1XBYT2YTNBT+XNATD1*-αX00XNA1T0**-d-1I0NBT0***-dI00****-ɛ3I0*****-ɛ3-1I)<0
hold, then for any bounded time-delay τ(t)≤τ, there exists a positive scalar h, such that system (1) can be robustly stabilized by the control law (5) with K=YX-1 over time sequence 𝒩min(β-h), where Ψ3=(β-1lnμ+αμ)X+AX+XAT+2BY+2YTBT.Proof.
LetP=X-1, λ0=λmin(P), λ1=λmax(P), and V(t)=xT(t)Px(t). Substituting Y=KX-1 in (19) and pre- and postmultiplying it by diag{P,P,I,I,I,I} yield
(20)(Ψ3′PA1PBKT(NA+2NBK)TPD1*-αP00NA1T0**-d-1I0NBT0***-dI00****-ɛ3I0*****-ɛ3-1I)<0,
where Ψ3′=(β-1lnμ+αμ)P+(A+2BK)TP+P(A+2BK). Hence, there exist small positive scalars μ̃ and h, such that(21)(Ψ3′′PA1PBKT(NA+2NBK)TPD1*-αexp⁡(-μ̃τ)P00NA1T0**-d-1I0NBT0***-dI00****-ɛ3I0*****-ɛ3-1I)<0,where Ψ3′′=(μ1+αμ+μ̃)P+(A+2BK)TP+P(A+2BK), μ1=(β-h)-1lnμ.For givenρ>0, choose 0<δ≤λ0/μλ1ρ. We assume that the initial function ϕ∈PC([-τ,0],ℝn) satisfying ∥ϕ∥τ≤δ. Set W(t)=exp(μ̃t)V(t). In the following, we will prove that for t≥t0-τ, (14) holds. First, we prove that (14) holds for t∈[t0-τ,t1). It is noticed that for θ∈[-τ,0], W(t0+θ)≤λ1∥ϕ∥τ2≤μ-1λ0ρ2<λ0ρ2. Similar to the proof of Theorem 6, it needs only to prove that, for t∈(t0,t1), (14) holds. Otherwise, there exists t∈(t0,t1), such that W(t)≥λ0ρ2. Set t*=inf{t∈[t0,t1):W(t)≥λ0ρ2}. Then t*∈(t0,t1), W(t*)=λ0ρ2. Set t¯=sup{t∈[t0,t*):W(t)≤μ-1λ0ρ2}. Then t¯∈[t0,t*) and W(t¯)=μ-1λ0ρ2. Hence, for t∈[t¯,t*] and θ∈[-τ,0], we have(22)W(t+θ)≤λ0ρ2≤μW(t).
By (6), (7), and (22), we have (23)D+W(t)≤exp⁡(μ̃t)(μ̃V(t)+D+V(t))+α(μW(t)-W(t-τ(t)))+d-1(xT(t)KTKx(t)-Δu(t)TΔu(t))exp⁡(μ̃t)≤exp⁡(μ̃t)ηT(t)Ξ3η(t)-μ1W(t),
where (24)Ξ3=(Ψ3′′PA1PB*-αexp⁡(-μ̃τ)P0**-d-1I)+d-1(KT00)(K00)+ɛ3-1((NA+2NBK)TNA1TNBT)(NA+2NBKNA1NB)+ɛ3(PD100)(D1TP00),
By Lemma 2, (21) implies that Ξ3<0, which yields by (23) that for t∈[t¯,t*],(25)D+W(t)<-μ1W(t).
It follows that W(t*)≤W(t¯). This is a contradiction with the fact W(t*)=λ0ρ2>W(t¯)=μ-1λ0ρ2. Thus, (14) holds for t∈[t0-τ,t1).Now we assume that (14) holds for t∈[t0-τ,tm); we will prove that (14) still holds for t∈[tm,tm+1). First, we claim that W(tm-)≤μ-1λ0ρ2. Otherwise, we have W(tm-)>μ-1λ0ρ2. Here we must consider two cases.Case 1.
W(t)>μ-1λ0ρ2 for all t∈[tm-1,tm). It follows that (22) holds according to the fact that (14) holds for t∈[t0-τ,tm). Hence (23) and (25) hold for t∈[tm-1,tm). They lead that
(26)W(tm-)≤W(tm-1)exp⁡(-μ1(tm-tm-1))≤λ0ρ2exp⁡(-μ1(β-h))=μ-1λ0ρ2.
This is also a contradiction.Case 2.
There exists somet∈[tm-1,tm), such that W(t)≤μ-1λ0ρ2. Set t¯=sup{t∈[tm-1,tm):W(t)≤μ-1λ0ρ2}. Then t¯∈[tm-1,tm), W(t¯)=μ-1λ0ρ2 and (22) for t∈[t¯,tm]. It follows that (23) and (25) hold, which yields that W(tm-)≤W(t¯)=μ-1λ0ρ2. This is a contradiction which implies that W(tm-)≤μ-1λ0ρ2, and hence by (9), we obtain that W(tm)≤μW(tm-)≤λ0ρ2.Now we assume that (14) is not true for t∈[tm,tm+1). Set t*=inf{t∈[tm,tm+1):W(t)≥λ0ρ2}. Then t*∈[tm,tm+1) and W(t*)=λ0ρ2. If there exists some t∈[tm,t*), such that W(t)≤μ-1λ0ρ2, then set t¯=sup{t∈[tm,t*):W(t)≤μ-1λ0ρ2}, otherwise, we set t¯=tm. Then t¯∈[tm,t*). W(t¯)<λ0ρ2. It follows that (22), (23), and (25) hold, which yields that W(t*)≤W(t¯)<λ0ρ2. This is a contradiction, which implies that (14) is true for t∈[tm,tm+1). By mathematical induction, (14) holds for t≥t0-τ. That is, system (1) can be robustly stabilized over time sequence 𝒩min(β-h).Remark 11.
On the conditions in Theorem10, with the prescribed scalar β>0, we can stabilize not only the impulsive systems over time sequence 𝒩min(β), but some ones over other time sequences.


## 4. Numerical Examples

In this section, two illustrative examples will be presented to show the effectiveness of the results obtained.Example 12.
Consider the uncertain impulsive time-delay systems (1) with the following specifications:  μ=0.8, d=1, α=2, ɛ1=0.3034, ɛ2=0.5, ɛ3=0.5, U1=U2=1,0≤τ(t)≤γ=:const, A=(-40.20.1-5),A1=(100.21),B=(10.10.21), C=(-0.10-0.2-0.1),NA=(0.1010.1), NA1=(0.100.10.1), NB=(0.0100.10.1), NC=(0.10.100.1), B1=(0.10-10.1), B2=(0.10.200.1), NB1=(0.1-0.100.1),NB2=(0.10.100.1), D1=(10.101), D2=(10.101). Then by solving LMIs (8) and (11), we obtain the following results: X=(0.5330-0.0297-0.02970.6117),  Y=(-1.2292-0.1268-0.2300-0.8236). By Theorem 6, the controller u(t)=YX-1x(t)=(-2.3241-0.3201-0.5079-1.3710)x(t) is obtained. It exponentially stabilizes the impulsive system over any impulsive time sequence according to Theorem 6.Example 13.
Consider the uncertain impulsive time-delay systems (1) with the following specifications: μ=1,d=1, α=2, ɛ1=0.4049, ɛ2=0.5, ɛ3=0.5, U1=U2=1, 0≤τ(t)≤γ=:const, A=(-30.20.1-5), A1=(100.21), B=(10.20.20.7), C=(0.10-0.2-0.1), NA=(0.10.210.1), NA1=(0.1010.1), NB=(0.100.20.1), NC=(0.10.100.1), B1=(0.10-10.1), B2=(0.10.200.1), NB1=(0.1-0.100.1), NB2=(0.10.100.1), D1=(10.111), D2=(10.10.21). Then by solving LMIs (8) and (11), we obtain the following results: X=(0.7144-0.0180-0.01800.8380),  Y=(-2.4545-0.8386-0.6829-1.5880). By Theorem 6, the controller u(t)=YX-1x(t)=(-3.4628-1.0751-1.0042-1.9166)x(t) is obtained. It exponentially stabilizes the impulsive system over any impulsive time sequence according to Theorem 6.Example 14.
Consider the uncertain impulsive time-delay systems (1) with the following specifications: μ=1.5, d=1, α=2, ɛ1=0.1100, ɛ2=0.5, ɛ3=0.5, U1=U2=1, 0≤τ(t)≤γ=:const, A=-10I, A1=B=D1=I, C=-0.1I, NA=NA1=NC=0.1I, NB=B1=NB1=NB2=0.1I, B2=diag{1,0.1}, D2=diag{3,1}. Then by solving LMIs (8) and (19), we obtain that β=0.4055, X=diag{1.2150,1.0491}, Y=diag{0.7386,0.4180}. By Theorem 10, the controller u(t)=YX-1x(t)=(0.6079000.3984)x(t) is obtained. It exponentially stabilizes the impulsive system over impulsive time sequence 𝒩min(0.4055-h) according to Theorem 10, where h>0 is some small scalar. In fact, we can obtain some h≥0.002.


## 5. Conclusion

This paper studied a class of uncertain impulsive time-delay systems. Based on Lyapunov function and Razumikhin-type techniques, we obtain conditions which can guarantee the above systems robustly exponentially stabiliable for any impulsive time sequences and improves some related results.



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