In this paper, we establish a new sufficient condition for the stability of impulsive systems with impulse time window and bounded gain error. The proposed result is more general and more applicable than some existing results. Finally, a numerical example is given to show the effectiveness of our result

Feng, Yuming

Peng, Yang

Tu, Zhengwen

Zou, Limin

English

Nonlinear Analysis: Modelling and Control

ISSN 1392-5113https://doi.org/10.15388/NA.2018.1.4Nonlinear Analysis: Modelling and Control, 2018, Vol. 23, No. 1, 40–49Impulsive control of nonlinear systems with impulse timewindow and bounded gain error∗Limin Zoua, Yang Pengb, Yuming Fengc, Zhengwen TubaSchool of Mathematics and Statistics,Chongqing Technology and Business University,Chongqing, 400067, Chinalimin-zou@163.combSchool of Mathematics and Statistics,Chongqing Three Gorges University,Chongqing, 404100, Chinapeng_yang2011@163.com; tuzhengwen@163.comcKey Laboratory of Intelligent Information Processing and Control,Chongqing, 404100, Chinayumingfeng25928@163.comReceived: February 20, 2017 / Revised: November 11, 2017 / Published online: December 14, 2017Abstract. In this paper, we establish a new sufficient condition for the stability of impulsivesystems with impulse time window and bounded gain error. The proposed result is more generaland more applicable than some existing results. Finally, a numerical example is given to show theeffectiveness of our result.Keywords: impulse time window, bounded gain error, impulsive control.1 IntroductionIn this paper, we mainly adopt the notation and terminology in [8]. Within the last threedecades, impulsive control theory had been intensively studied because impulsive controlcan be applied in many fields, such as chaotic systems [9, 15, 24, 30, 34], HIV preventionmodles [5], complex dynamical systems [2–4,11,13,14,16,18,22,23,25,31,35–38]. Thenonlinear impulsive control systems with impulses at fixed times is given byx˙ = Ax+ φ(x), t 6= τk,∆x = U(k,x), t = τk, k = 1, 2, . . . ,x(t0) = x0.(1)∗This research was supported by the Chongqing Research Program of Basic Research and Frontier Technol-ogy (grant No. cstc2017jcyjAX0032) and the Natural Science Foundation of China (grant No. 11601047).c© Vilnius University, 2018Impulsive control of nonlinear systems 41where x ∈ Rn is the state variable, t0 < τ1 < τ2 < · · · , limk→∞ τk = ∞ denote themoments when impulsive control occurs, A is an n × n constant matrix, φ : Rn → Rnis a continuous nonlinear function satisfying φ(t, 0) = 0 and ‖φ(x)‖2 6 xTLx, whereL is a diagonal matrix, and U(k,x) is the impulsive control law. Many researchers havestudied impulsive control system (1) [1,17,32,33]. But we cannot guarantee the impulseswithout any error due to the limit of equipment and technology.Recently, Feng, Li, and Huang [8] discussed the following nonlinear impulsive controlsystems with impulse time window:x˙(t) = Ax+ φ(x), mT 6 t < mT + τm,x(t) = Jmx(t−), t = mT + τm,x˙(t) = Ax(t) + φ(x), mT + τm < t < (m+ 1)T,(2)where T > 0 denotes the control period, τm is unknown within impulse time window(mT, (m + 1)T ), and Jm ∈ Rn×n is impulsive control gain. The impulsive effects canbe stochastically occurred in a impulse time window in system (2), which is more generalthan ones impulses occurred at fixed times. Some results related to impulse time windowcan be found in [6, 7, 12, 26–29, 39].In many practical applications, the impulsive control gain Jm may also contain errors,so we should take into account the influence of impulsive control gain errors on thesystems. In this paper, we consider a class of impulsive control systems with impulsetime window and bounded gain error as follows:x˙(t) = Ax+ φ(x), mT 6 t < mT + τm,x(t) = (Jm +∆Jm)x(t−), t = mT + τm,x˙(t) = Ax(t) + φ(x), mT + τm < t < (m+ 1)T,(3)where∆Jm is gain error, which is often time-varying and bounded. As pointed out in [17],we can assume that ∆Jm = mF (t)Jm, m > 0, and FT(t)F (t) 6 I .The purpose of this paper is to find some conditions on control gain Jm and impulsetime window T such that the origin of impulsive control system (3) is asymptoticallystable. We establish a new sufficient condition for the stability of system (3). Comparedwith the results shown in [8, 17, 19], our result is more general and more applicable.Finally, a numerical example is given to show the effectiveness of our result.2 Main resultsIn this section, we will give the main results. To do this, we need the following lemmas.Lemma 1. (See [21].) Let x,y ∈ Rn and ε > 0. Then2xTy 6 εxTx+ 1εyTy.Nonlinear Anal. Model. Control, 23(1):40–4942 L. Zou et al.Lemma 2. Let A,P,B ∈ Rn×n such that P is a symmetric and positive definite matrixand µ > 0. ThenATPB +BTPA 6 µATPA+ 1µBTPB.Proof. Note that for any X ∈ Rn×n, the matrix XTX is positive semidefinite. It followsthat (√µP 1/2A− 1√µP 1/2B)T(√µP 1/2A− 1√µP 1/2B)> 0.Small calculations show that the result holds. This completes the proof.Lemma 3. (See [10].) Let H be a real symmetrical matrix, and λmax(H) > λmin(H)be the largest and the smallest eigenvalues of H , respectively. Then for any x ∈ Rn, wehaveλmin(H)xTx 6 xTHx 6 λmax(H)xTx.Theorem 1. Let P ∈ Rn×n be a symmetric and positive definite matrix. If there exitg, µ, ε > 0 such that the following hold:(i) PA+ATP + εP 2 + ε−1L− gP 6 0,(ii) gT + lnλ < 0,where λ = max{λm(1+µ+m2(1+1/µ))}, λm = λmax(P )×λmax(P−1JTmJm), thenthe origin of system (3) is asymptotically stable.Proof. Let us construct the following Lyapunov function:V(x(t))= xT(t)Px(t).If mT 6 t < mT + τm, then by Lemma 1 and condition (i) we haveD+(V(x(t)))= 2xT(t)P(Ax(t) + φ(x(t)))= 2xT(t)PAx(t) + 2xT(t)Pφ(x)= xT(t)(PA+ATP)x(t) + 2xTPφ(x)6 xT(t)(PA+ATP)x(t) + εxTP 2x+1εφ(x(t))Tφ(x(t))6 xT(t)(PA+ATP)x(t) + εxTP 2x(t) +1εxT(t)Lx(t)= xT(t)(PA+ATP + εP 2 +1εL− gP)x(t) + gxT(t)Px(t)6 gV(x(t)),which implies thatV(x(t))6 V(x(mT ))eg(t−mT ). (4)Similarly, if mT + τm < t < (m+ 1)T , we also haveD+(V(x(t)))6 gV(x(t)),https://www.mii.vu.lt/NAImpulsive control of nonlinear systems 43which implies thatV(x(t))6 V(x(mT + τm))eg(t−mT−τm). (5)If t = mT + τm, then by Lemmas 2 and 3 we haveV(x(t))=((Jm +∆Jm)x(t−))TP (Jm +∆Jm)x(t−)= x(t−)T(JTmP +∆JTmP)(Jm +∆Jm)x(t−)= x(t−)T(JTmPJm + JTmP∆Jm +∆JTmPJm +∆JTmP∆Jm)x(t−)6 x(t−)T((1 + µ)JTmPJm +(1 +1µ)∆JTmP∆Jm)x(t−)6 λmax(P )x(t−)T((1 + µ)JTmJm +(1 +1µ)∆JTm∆Jm)x(t−)6 λmax(P )x(t−)T((1 + µ)JTmJm +m2(1 +1µ)JTmFTt FtJm)x(t−)6 λmax(P )x(t−)T((1 + µ)JTmJm +m2(1 +1µ)JTmJm)x(t−)= λm(1 + µ+m2(1 +1µ))V(x(t−))6 λV(x(t−)). (6)It follows from (5) and (6) thatV(x(t))6 λV(x((mT + τm)−))eg(t−mT−τm), (7)where mT + τm 6 t < (m+ 1)T .By using inequalities (4) and (7) we can derive the following results.When m = 0, if t ∈ [0, τ0), thenV(x(t))6 V (x0)egtand soV(x(τ−0 ))6 V(x0)egτ0 .If t ∈ [τ0, T ), thenV(x(t))6 λV(x(τ−0 ))eg(t−τ0) 6 λV (x0)egtand soV(x(T ))6 λV (x0)egTWhen m = 1, if t ∈ [T, T + τ1), thenV(x(t))6 V(x(T ))eg(t−T ) 6 λV (x0)egtand soV(x((T + τ1)−)) 6 λV (x0)eg(T+τ1).Nonlinear Anal. Model. Control, 23(1):40–4944 L. Zou et al.If t ∈ [T + τ1, 2T ), thenV(x(t))6 λV(x((T + τ1)−))eg(t−T−τ1) 6 λ2V (x0)egtand soV(x(2T ))6 λ2V (x0)e2gT .In general, when m = k, k = 0, 1, . . . , if t ∈ [kT, kT + τk), then we haveV(x(t))6 λkV (x0)egt 6 λkV (x0)eg(k+1)T= V (x0)egT+kgT+k lnλ = V (x0)egT+k(gT+lnλ). (8)If t ∈ [kT + τk, (k + 1)T ), we obtainV(x(t))6 λk+1V (x0)egt 6 λk+1V (x0)eg(k+1)T= V (x0)e(k+1)gT+(k+1) lnλ = V (x0)e(k+1)(gT+lnλ). (9)It follows from (8), (9), and condition (ii) thatlimt→∞V(x(t))= 0.This completes the proof.If we choose P = I in Theorem 1, then the condition of Theorem 1 can be simplifiedas follows.Corollary 1. If there exit g, µ, ε > 0 such that the following hold:(i) A+AT + (ε− g)I + ε−1L 6 0,(ii) gT + lnλ < 0,where λ = max{λm(1 + µ +m2(1 + 1/µ))}, λm = λmax(JTmJm), then the origin ofsystem (3) is asymptotically stable.The condition of Corollary 1 is similar to Theorem 1 shown in [19]. Since impulsiveeffects can be stochastically occurred in a impulse time window in system (3), Corollary 1is more general than Theorem 1 shown in [19].Sometimes, for the sake of convenience, the impulsive control gain Jm is alwaysselected as a constant matrix J , then we have the following.Corollary 2. Let P ∈ Rn×n be a symmetric and positive definite matrix. If there exitg, µ, ε > 0 such that the following hold:(i) PA+ATP + εP 2 + ε−1L− gP 6 0,(ii) gT + lnλ < 0,where λ = λmax(P )λmax(P−1JTJ)(1 + µ+m2(1 + 1/µ)), then the origin of sys-tem (3) is asymptotically stable.https://www.mii.vu.lt/NAImpulsive control of nonlinear systems 45The condition of Corollary 2 is similar to Theorem 1 shown in [8]. Since we take intoaccount the influence of impulsive control gain errors on the systems, Corollary 2 is morepractical than Theorem 1 shown in [8].In many practical applications, the parameters of impulsive control of nonlinear sys-tems contain errors. In what follows, we will consider system (3) with parameter uncer-tainty. The corresponding system can be described asx˙(t) = (A+∆A)x+ φ(x), mT 6 t < mT + τm,x(t) = (Jm +∆Jm)x(t−), t = mT + τm,x˙(t) = Ax(t) + φ(x), mT + τm < t < (m+ 1)T,(10)where ∆A is the parametric uncertainty and has the following form: ∆A = GF (t)H ,where FT(t)F (t) 6 I , while G and H are appropriate known matrices.Theorem 2. Let P ∈ Rn×n be a symmetric and positive definite matrix. If there exitg, µ, ε > 0 such that the following hold:(i) PA+ATP + (1 + ε)P 2 + ε−1L+$I − gP 6 0,(ii) gT + lnλ < 0,where $ = λmax(GTG)λmax(HTH), λ = max{λm(1 + µ+m2(1 + 1/µ))}, λm =λmax(P )λmax(P−1JTmJm), then the origin of system (10) is asymptotically stable.Proof. Let us construct the following Lyapunov function:V(x(t))= xT(t)Px(t).If mT 6 t < mT + τm, then by Lemma 1 and condition (i) we haveD+(V(x(t)))= 2xT(t)P((A+∆A)x(t) + φ(x(t)))= 2xT(t)PAx(t) + 2xT(t)P∆Ax(t) + 2xT(t)Pφ(x)= xT(t)(PA+ATP)x(t) + 2xT(t)P∆Ax(t) + 2xT(t)Pφ(x)6 xT(t)(PA+ATP)x(t) + xTP 2x+ xT∆AT∆Ax+ εxTP 2x+1εφ(x(t))Tφ(x(t))= xT(t)(PA+ATP)x(t) + (1 + ε)xTP 2x+1εφ(x(t))Tφ(x(t))+ xTHTFT(t)GTGF (t)Hx6 xT(t)(PA+ATP)x(t) + (1 + ε)xTP 2x(t) +1εxT(t)Lx(t)+ λmax(GTG)xTHTFT(t)F (t)Hx6 xT(t)(PA+ATP)x(t) + (1 + ε)xTP 2x(t) +1εxT(t)Lx(t)+ λmax(GTG)xTHTHxNonlinear Anal. Model. Control, 23(1):40–4946 L. Zou et al.6 xT(t)(PA+ATP)x(t) + (1 + ε)xTP 2x(t) +1εxT(t)Lx(t)+ λmax(GTG)λmax(HTH)xTx= xT(t)(PA+ATP + (1 + ε)P 2 +1εL+$I − gP)x(t) + gV(x(t))6 gV(x(t)),which implies thatV(x(t))6 V(x(mT ))eg(t−mT ).The rest of proof is same as that of Theorem 1, so we omit it here for simplicity. Thiscompletes the proof.3 A numerical exampleIn this section, we will illustrate the effectiveness of our result by showing simulationresults employing the Chua’s system. Throughout this section, we assume that x =[x, y, z]T.The original and dimensionless form of a Chua’s oscillator [20] is given byx˙ = α(y − x− g(x)), y˙ = x− y + z, z˙ = −βy, (11)where α and β are parameters, and g(x) is the piecewise linear characteristics of theChua’s diode, which is defined byg(x) = bx+ 0.5(a− b)(|x+ 1| − |x− 1|),where a < b < 0 are two constants.By decomposing the linear and nonlinear parts of the system in (11), we rewrite it asx˙(t) = Ax+ φ(x),whereA =−α− αb α 01 −1 10 −β 0 , φ(x) =−0.5α(a− b)(|x+ 1| − |x− 1|)00 .Simple calculations show that∥∥φ(x)∥∥2 = 0.25α2(a− b)2[(x+ 1)2 + (x− 1)2 − 2∣∣(x+ 1)(x− 1)∣∣]= 0.5α2(a− b)2(x2 + 1− |x2 − 1|)={α2(a− b)2, x2 > 1,α2(a− b)2x2, x2 6 16 α2(a− b)2x2.Thus, we can choose L = diag(α2(a− b)2, 0, 0).https://www.mii.vu.lt/NAImpulsive control of nonlinear systems 47−10−50510−2−1012−10−50510xyzFigure 1. The chaotic phenomenon of Chua’soscillator with the initial condition x(0) =(0.5, 0.3,−0.5)T.0 0.05 0.1 0.15 0.2−101x0 0.05 0.1 0.15 0.2−101y0 0.05 0.1 0.15 0.2−101ztFigure 2. Time response curves of controlledChua’s system with the initial condition x(0) =(0.5, 0.3,−0.5)T.In this example, we set the system parameters asα = 9.2156, β = 15.9946, a = −1.24905, b = −0.75735,which make Chua’s circuit (11) chaotic. Figure 1 shows the chaotic phenomenon ofChua’s oscillator with the initial condition x(0) = (0.5, 0.3,−0.5)T.Meanwhile, we choose P = I , µ = ε = 1, andJm = J = diag(−0.5,−0.5,−0.5).For the sake of simplicity, the impulsive control gain error ∆Jm is specified as∆Jm = ∆J = 0.05 sin tJ,and so λ = 0.5012. In order to satisfy condition (i) of Theorem 1, we can choose g = 31.From the following inequalitygT + lnλ < 0we have T < 0.0233. Thus, by Theorem 1 we know that the origin of system (3) isasymptotically stable. 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Impulsive control of nonlinear systems with impulse time window and bounded gain error

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Impulsive control of nonlinear systems with impulse time window and bounded gain error

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