# Stability Analysis for Nonlinear Impulsive Control System with Uncertainty Factors

Authors: Zemin Ren; Shiping Wen; Qingyu Li; Yuming Feng; Ning Tang
Journal: Computational Intelligence and Neuroscience
 (2020)

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## Abstract

Considering the limitation of machine and technology, we study the stability for nonlinear impulsive control system with some uncertainty factors, such as the bounded gain error and the parameter uncertainty. A new sufficient condition for this system is established based on the generalized Cauchy–Schwarz inequality in this paper. Compared with some existing results, the proposed method is more practically applicable. The effectiveness of the proposed method is shown by a numerical example.


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## Body

## 1. Introduction

Impulse control is based on impulsive differential equation and has many applications [1–6], such as digital communication system, artificial intelligence, and financial sector. In comparison with other methods, impulse control is more efficient in dealing with the stability of complex systems. The stability is an important property of the impulsive control system. Mathematically, its goal is to stabilize an unstable system by proper impulse. Up to now, a wide variety of achievements of impulse control theory have been developed in the literature [7–13].Generally, there are at least one “impulsively” changeable state variable appearing in a plantP, which could be described as following control system:(1)x˙t=Ax+ϕx,t≠τk,Δx=Uk,x,t=τk,k=1,2,…,xt0=x0.Here,x∈ℝn denotes the state variable and Uk,x the impulse control law. We assume that the control instance satisfies(2)t0<τ1<⋯⋯<τk<τk+1⋯,limk⟶∞τk=∞.A continuous nonlinear functionϕx:ℝn⟶ℝn stratifies ϕt,0=0 and ϕx≤Lx, where L is a Lipschitz constant. Many researchers have paid more attention on control system (1) and achieved many sufficient conditions for the stability of these systems [14–20]. Feng et al. consider single state-jumps impulsive systems with periodically time windows and give stability criteria for the new model [21]. To make the nonlinear impulse control system more reasonable, parameter uncertainty and bounded gain error are introduced into the corresponding impulsive differential equations [22–25]. Considering the limitation of machine and technology, Ma et al. investigate stabilization of impulse control systems with gain error and obtain a sufficient criterion for global exponential stability [26]. Zou et al. study impulsive systems with bounded gain error and form a sufficient criterion for the stability [27].Cauchy–Schwarz inequality is an important tool to study nonlinear systems [28–31]. Recently, Peng et al. generalize the Cauchy–Schwarz inequality, which is used to deduce asymptotic stability for a class of nonlinear control systems [30]. Under the assumption Uk,x=BCx, they study the after nonlinear system:(3)x˙t=Ax+ϕx,t≠τk,Δx=BCx,t=τk,k=1,2,…,xt0=x0,where B and C are constant matrixes. Based on the generalized Cauchy–Schwarz inequality, we consider a class of nonlinear impulsive control systems with the parameter uncertainty, which can be written as follows:(4)x˙t=A+ΔAx+ϕx,t≠τk,Δx=BCx,t=τk,k=1,2,…,xt0=x0.Generally, one can express the parameter uncertainty asΔA=GFtH with FTtFt≤I. Here, matrixes G and H are given with appropriate dimensions. In this paper, we will find some conditions for the stability of system (4). We organize the paper as follows. In Section 2, we briefly introduce some related lemmas. Then, we show sufficient conditions in Section 3. The simulation experiment is shown in Section 4, and conclusion is listed in Section 4.


## 2. Related Lemmas

First of all, we introduce some lemmas to be used later. Throughout this paper,λmaxandλmin are denoted as the largest eigenvalue and the smallest eigenvalue, respectively. · is denoted as the Euclidian norm of matric or vector.Lemma 1 (see [30]).
Suppose thatP is positive definite. If x,y∈Rn satisfy xTy≤σxTxyTy for a certain σ∈0,1, then(5)xTPy2≤λmaxP−gσλminPλmaxP+gσλminP2xTPxyTPy,where gσ=1−σ/1+σ.Lemma 2.
(see [27]). Suppose that Q is symmetric and positive definite; then, for any A,B∈Rn×n and μ>0,(6)ATQB+BTQA≤μATQA+1μBTQB.Lemma 3.
(see [32]). Suppose that H is a real symmetric matrix; then,(7)λminHxTx≤xTHx≤λmaxHxTx.


## 3. The Proposed Results

We give the main results in this section. Specifically, we will analyze the stabilization of impulsive control system (4) with bounded gain error and parameter uncertainty and then list some sufficient conditions which assure the origin of the related systems is asymptotically stable.Theorem 1.
SupposeP∈ℝn×n be a symmetric and positive definite matrix, λ1=λminP, λ2=λmaxP, I be the identity matrix, λ3 be the largest eigenvalue of P−1PA+ATP, and λ4 be the largest eigenvalue of the matrix P−1I+BCTPI+BC. If(8)xTtϕxt≤σxTtxtϕxtTϕxt,for a certain σ∈0,1 and(9)λ3+2λ2λmaxGTGλmaxHTHλ1+2Lλ2−gσλ1λ2+gσλ1λ2λ1τk+1−τk≤−lnγλ4,where(10)gσ=1−σ1+σ,γ>1.
then, we obtain that the origin of impulsive control system (4) is asymptotically stable.Proof.
We choose the Lyapunov function as follows:(11)Vxt=xTtPxt.
Whent≠τk, we obtain Dini’s derivative of Vxt for impulsive control system (4) as follows:(12)D+Vxt=2xTtPA+ΔAxt+ϕxt,=2xTtPAxt+2xTtPΔAxt+2xTtPϕxt.
Next, we will calculate the three parts of the above formula (12), respectively. The matrices P−1PA+ATP and P−0.5PA+ATPP−0.5 have the same eigenvalues. By Lemma 3, we have(13)2xTtPAxt=xTtPA+ATPxt,=xTtP0.5P−0.5PA+ATPP−0.5P0.5xt,≤λ3xTtP0.5P0.5xt,=λ3Vxt.
According to the Cauchy–Schwarz inequality, we obtain(14)xTtPΔAxt≤xTtP2xtxTtΔATΔAxt.
Since parameter uncertaintyΔA=GFtH and FTtFt≤I, inequality (14) can be rewritten as(15)2xTtPΔAxt≤2xTtP2xtxTtHTFTtGTGFtHxt,≤2xTtP1/2PP1/2xtxTtHTFTtGTGFtHxt,≤2λ2VxtλmaxGTGxTtHTIHxt,≤2λ2VxtλmaxGTGλmaxHTHxTtxt,=2λ2VxtλmaxGTGλmaxHTHxTtP1/2P−1P1/2xt,≤2λ2λmaxGTGλmaxHTHλ1Vxt.
According to Lemma1, we obtain(16)2xTtPϕxt≤2Lλ2−gσλ1λ2+gσλ1xTtPxtϕxtTPϕxt,≤2Lλ2−gσλ1λ2+gσλ1λ2xTtPxtϕxtTϕxt.
Sinceϕx≤Lx, inequality (16) can be obtained as follows:(17)2xTtPϕxt≤2Lλ2−gσλ1λ2+gσλ1λ2xTtPxtxtTxt,≤2Lλ2−gσλ1λ2+gσλ1λ2xTtPxtxtTxt,≤2Lλ2−gσλ1λ2+gσλ1λ2λ1xTtPxtxtTPxt,=2Lλ2−gσλ1λ2+gσλ1λ2λ1Vxt.
Combining inequalities (13), (15), and (17), we obtain(18)D+Vxt≤λ3+2λ2λmaxGTGλmaxHTHλ1+2Lλ2−gσλ1λ2+gσλ1λ2λ1Vxt.
Whent=τk, we compute the value of V as follows:(19)Vxt+BCxt|t=τk=xt+BCxtTPxt+BCxt|t=τk,=xtTI+BCTPI+BCxt|t=τk,=xTtP0.5P−0.5I+BCTPI+BCP−0.5P0.5xt|t=τk.
It is known that the matrixP−0.5I+BCTPI+BCP−0.5 has the same eigenvalues with the matrix P−1I+BCTPI+BC. Thus, it follows from (19) that(20)Vxt+BCxt|t=τk≤λ4xTtP0.5P0.5xt|t=τk,=λ4Vxt|t=τk.
Now, we analyze the following comparison system:(21)ω˙=λ3+2λ2λmaxGTGλmaxHTHλ1+2Lλ2−gσλ1λ2+gσλ1λ2λ1ωt,t≠τk,ωτk+=λ4ωτk,ωτ0+=ω0≥0.
According to the related conclusion (see Theorem 3 in [29]), we obtain that if(22)∫τkτk+1λ3+2λ2λmaxGTGλmaxHTHλ1+2Lλ2−gσλ1λ2+gσλ1λ2λ1dt+lnγλ4≤0,γ>1.
The origin of impulsive control system (4) is asymptotically stable.Remark 1.
If the parameter uncertaintyΔA=0, the condition of (9) became the result of Theorem 3.1 in reference [30]. Thus, the proposed method is a generalization of Peng’s method.
In many practical applications, it is inevitable to put impulses with errors due to the limitation of machine and technology. So, we integrate the bounded gain error into the impulsive system (4). For simplicity, let D=BC. We rewrite the corresponding system as(23)xt=A+ΔAxt+ϕxt,t≠τk,Δxt=D+ΔDxt,t=τk,k=1,2,…,xt0=x0,where ΔD denotes the bounded gain error and has the following form: ΔD=mFtD with m>0 and FTtFt≤I. It is easy to obtain a similar analysis from Theorem 1.Theorem 2.
LetP∈ℝn×n be a symmetric and positive definite matrix, λ1=λminP, λ2=λmaxP, I be the identity matrix, and λ3 be the largest eigenvalue of P−1PA+ATP. If(24)xTtϕxt≤σxTtxtϕxtTϕxt.for a certain σ∈0,1 and(25)λ3+2λ2λmaxGTGλmaxHTHλ1+2Lλ2−gσλ1λ2+gσλ1λ2λ1τk+1−τk≤−lnγλ4,where(26)λ4=λ2λ11+μλmaxI+DTI+D+1+1μm2λmaxDTD,(27)gσ=1−σ1+σ,γ>1.
Then, the origin of impulsive control system (23) is asymptotically stable.Proof.
We choose the following Lyapunov function as follows:(28)Vxt=xTtPxt.
According to inequality (18), Dini’s derivative of Vxt for impulsive control system (23) is acquired as follows:(29)D+Vxt≤λ3+2λ2λmaxGTGλmaxHTHλ1+2Lλ2−gσλ1λ2+gσλ1λ2λ1Vxt.
Then, we just need to computeVxt+D+ΔDxt|t=τk.
We perform some calculations onVxt+D+ΔDxt|t=τkand obtain(30)Vxt+D+ΔDxt|t=τk=xt+D+ΔDxtTPxt+D+ΔDxt|t=τk,=xtTI+D+ΔDTPI+D+ΔDxt|t=τk,≤λ2xtTI+D+ΔDTI+D+ΔDxt|t=τk,≤λ2xtTI+DTI+D+I+DTΔD+ΔDTI+D+ΔDTΔDxt|t=τk.
By using Lemma2 and ΔD=mFtD, we rewrite inequality (30) as(31)Vxt+D+ΔDxt|t=τk≤λ2xTtI+DTI+D+I+DTΔD+ΔDTΔD+ΔDTI+Dxt|t=τk,≤λ2xTt1+μI+DTI+D+1+1μΔDTΔDxt|t=τk,=λ2xTt1+μI+DTI+D+1+1μm2DTFTtFtDxt|t=τk.
It follows from (15) that(32)xTtxt=xTtP1/2P−1P1/2xt≤Vxtλ1.
Combine inequalities (31) and (32) and FTtFt≤I, we obtain(33)Vxt+D+ΔDxt|t=τk=λ2xTt1+μI+DTI+D+1+1μm2DTFTtFtDxt|t=τk,≤λ2xTt1+μI+DTI+D+1+1μm2DTDxt|t=τk,≤λ2λ11+μλmaxI+DTI+D+1+1μm2λmaxDTDVxt|t=τk,=λ4Vxt|t=τk.
Here, we emit the rest analysis process, which is similar to Theorem1. Thus, from equalities (29) and (33), we obtain that if(34)λ3+2λ2λmaxGTGλmaxHTHλ1+2Lλ2−gσλ1λ2+gσλ1λ2λ1τk+1−τk≤−lnγλ4,λ4=λ2λ11+μλmaxI+DTI+D+1+1μm2λmaxDTD,the origin of impulsive control system (23) is asymptotically stable. This completes the proof.


## 4. A Numerical Example

In this section, we perform the proposed model on a numerical example to display its effectiveness. The example is produced by Qi and Chen [33]. Let x=x1,x2,x3T, ϕx=x2x3,−x1x3,x1x2T, and(35)A=−aa0c−1000−b.The corresponding state equation can be described as(36)x˙=Ax+ϕx.According to the strategy of [33], some parameters of this system are set as a=35, b=8/3, and c=25. From Figure 1, we can see that system (36) is chaotic for the initial condition: x0=3,5,10T.Figure 1
The chaotic phenomenon of system (36) with the initial condition: x0=3,5,10T.After simple calculation, we obtain that(37)ϕx=x2x32+x1x32+x1x22,≤maxx1,x2,x3x12+x22+x32,=maxx1,x2,x3x.From Figure1, we can intuitively find maxx1,x2,x3≤45. Combining with inequality (37), the parameter L can be set as 45. Since(38)xTϕx2≤19xTxϕxTϕx,the parameter σ is chosen as σ=1/9. In this section, some matrices are chosen as follows:(39)G=H=0.50000.50000.5,P=C=100010001,B=−0.5−0.010.02−0.01−0.500.020−0.5.Thus, the parameter uncertainty can be formed as(40)ΔA=0.50000.50000.50.1sint0000.1sint0000.1sint0.50000.50000.5.According to Theorem1, we calculate λ3=32.9638 and λ4=0.2729. It follows from (8) that(41)τk+1−τk≤−lnγλ463.4638.Ifγ=1.1, it yields τk+1−τk≤0.0190. We choose τk+1−τk=0.0190 and show the simulation result in Figure 2. The impulsive control system (36) is asymptotically stable.Figure 2
Time response curves for the controlled system (36) with the parameter uncertainty.
(a)(b)(c)Next, we consider the controlled system (36) with the parameter uncertainty and the bounded gain error. The gain error is detailed as ΔD=msintD in this section. We perform some similar calculation on (25) and obtain λ3=32.9638. We choose μ=1 and then obtain λ4=0.54581+m2 from (26). Let γ=1.1 and m=0.05; then,(42)τk+1−τk≤0.0080.Thus, we chooseτk+1−τk=0.0080 and show the experimental result in Figure 3. From this figure, we can obtain that the impulsive control system (36) is asymptotically stable.Figure 3
Time response curves for the controlled system (36) with parameter uncertainty and gain error.
(a)(b)(c)


## 5. Conclusion

We study the asymptotic stability of impulsive control systems with some uncertainty factors, such as the bounded gain error and the parameter. The proposed sufficient condition is established based on the generalized Cauchy–Schwarz inequality. We think the proposed issue is more practically applicable than some existing ones.



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