# State Estimation for Switched Time-Varying Systems with Delay and Nonlinear Disturbance: An Integral Inequality Method

Authors: Fanwei Meng; Yuangong Sun
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper focuses on the problem of state estimation for certain switched time-varying systems with time-varying delay and nonlinear disturbance. By using an integral inequality technique and a method used in positive systems, we have established several explicit criteria for state estimation of the system, which reduce to stability criteria for some particular cases. The involved nonlinear disturbance of the system takes more general form including both the internal disturbance and the external disturbance. Three numerical examples are also given to verify the validity of the obtained theoretical results.


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## Body

## 1. Introduction

Switched system is a particular hybrid system containing a number of subsystems and a switching signal. Each subsystem is usually described by a definite differential equation or difference equation. For switched systems, the issue of stability plays a key role in system analysis. As a result, stability of switched systems has received considerable attention during the past several decades owing to its extensive applications in automotive engine control system [1], chemical process control system [2], multiagent systems [3, 4], and so on.Some basic problems in stability and design of switching systems were put forward by Liberzon and Morse [5]. Later, there are several very important monographs devoted to the stability analysis and design of switched systems; e.g., see Liberzon [6] and Sun and Ge [7]. There are also many interesting results for stability of switched systems in [8–15]. In most of the existing references, the Lyapunov-Krasovskii functional method was most commonly used for switched time-invariant systems. It seems to us that little attention has been paid to the stability of switched time-varying systems. Recently, by using a positive system method, exponential stability of switched time-varying systems with delay and nonlinear disturbance was investigated in [16, 17].Integral inequality plays an important role in qualitative analysis of delay systems [18–22, 22–25]. For example, stabilization of switched systems with impulsive effects and disturbances was studied in [18] by using the Gronwall integral inequality. By introducing a generalized Gronwall-Bellman inequality, the authors established stability criteria under arbitrary switching for switched systems with general nonlinear disturbances in [21]. Later, the main results in [21] were extended to switched delay systems with nonlinear disturbances in [25]. The same method was also applied to study a class of switched delay systems in [23], where global exponential stability criteria for the system were established.Note that time delay has attracted much attention in the theory analysis of switched systems due to its detrimental effects on system performance such as oscillation [24, 26–30] and stability [31–36]. Inspired by the work in [18, 25], we will use a delay integral inequality technique and a method developed in positive systems [37, 38] to study the problem of state estimation for a class of switched time-varying systems with time-varying delay and nonlinear disturbance. The main contributions of this paper are as follows: (1) unlike most existing results in the literature, all the subsystems considered in this paper are time-varying; (2) explicit global (local) state estimation criteria will be established for the cases when the nonlinear disturbance satisfies linear and nonlinear growth conditions, respectively; (3) the nonlinear disturbance of the system has more general form which contains both the internal state disturbance and the external input disturbance, and hence it contains some cases in the literature.The rest of this paper is organized as follows. In Section2, we introduce some notations and preliminaries that are essential for deriving the main results of this paper. Section 3 then focuses on establishing explicit state estimation criteria for the system. Simulations are given to illustrate the main results in Section 4. Finally, conclusions are drawn in Section 5.


## 2. Problem Statements and Preliminaries

In the sequel, denote byRn×n the set of n×n-dimensional real matrices, and denote by Rn the n-dimensional Euclidean space with the vector norm x=∑i=1nxi, where x=(xi)∈Rn for i∈n={1,2,…,n}. Set x=(xi) and xp=(xip). For two vectors x=(xi)∈Rn and y=(yi)∈Rn, we write x≻(≺,⪯)y if xi>(<,≤)yi for i∈n. For a matrix A=[aij]∈Rn×n, we write A⪰0 if aij≥0 for i,j∈n.Now, we consider the following switched time-varying system of the form(1)x˙t=Aσttxt+Bσttxt-τt+Fσtt,xt,xt-τt,t≥0,where x(t)∈Rn is the state, σ(t):[0,∞)→m={1,2,…,m} is the switching signal which is a piecewise constant function, Ak(t)=[aij(k)(t)]∈Rn×n and Bk(t)=[bij(k)(t)]∈Rn×n are continuous matrix functions for k∈m and i,j∈n, τ(t):[0,∞)→[0,∞) is the time-varying delay satisfying 0≤τ(t)≤h, h>0 is a constant, Fk(t,x,y):[0,∞)×Rn×Rn→Rn is a continuous vector function such that system (1) has a unique solution for each initial condition x(t)=ϕ(t), t∈[-h,0], and ϕ(t):[-h,0]→Rn is a continuous vector function.We need the following assumptions for establishing the main results of this paper.(H1) There exist continuous functions λ(t)≥0 and η(t)≥0 for t≥0, constant matrices A¯k=[a¯ij(k)] and B¯k=[b¯ij(k)], k∈m, such that aijkt≤a¯ij(k)λ(t) if i≠j, aij(k)(t)≤a¯ij(k)λ(t) if i=j, and bijkt≤b¯ij(k)η(t) for i,j∈n.(H2) There exist continuous functions λ~(t)≥0 and η~(t)≥0 for t≥0, constant matrices Mk=[mij(k)]⪰0 and Nk=[nij(k)]⪰0, k∈m, an n-dimensional vector function α(t)=(αi(t))⪰0, and a constant p>0 such that (2)Fkt,xt,xt-τt⪯λ~tMkxtp+η~tNkxt-τtp+αt,where the first two parts in the right-hand side of above inequality are interpreted as the internal state disturbances, and the third part α(t) is defined as the external input disturbance.The following two lemmas play a crucial role in the state estimation of system (1).Lemma 1 (see [39]).
Assume thatc≥0 is a constant, u(t) and f(t) are nonnegative continuous functions defined on [0,∞), and u(t) satisfies the following integral inequality (3)ut≤c+∫0tfsusds,t≥0.Then we have (4)ut≤ce∫0tfsds,t≥0.Lemma 2 (see [21]).
Assume thatu(t), f(t), and g(t) are nonnegative continuous functions defined on [0,∞), and u(t) satisfies the following integral inequality (5)ut≤k+∫0tfsus+gsupsds,t≥0,where k≥0 and p>0(p=1) are constants. If (6)k1-p-p-1∫0tgse∫0sp-1fτdτds>0,t≥0then we have (7)ut≤e∫0tfsdsk1-p-p-1∫0tgse∫0sp-1fvdvds1/1-p,t≥0.


## 3. Main Results

We first study the case ofp=1 in Assumption (H2).Theorem 3.
Assume that (H1) and (H2) with p=1 hold. If there exists an n-dimensional vector ξ=(ξi)≻0 such that(8)ξTA¯k≺0,k∈m,and (9)∫0∞e∫0tγ1λs-γ3λ~sdsξTαtdt<∞,then all solutions of system (1) satisfy(10)xt≤k~1mini∈n⁡ξie∫0tγ2ηs+γ4η~se∫s-hsγ1λv-γ3λ~vdv-γ1λs+γ3λ~sds,t≥0.where(11)k~1=ϕhmaxi∈n⁡ξi+∫0∞e∫0tγ1λs-γ3λ~sdsξTαsds,γ1=mink∈m,i∈n⁡ηikξi,γ2=maxk∈m,i∈n⁡θikξi,γ3=maxk∈m,i∈n⁡η~ikξi,γ4=maxk∈m,i∈n⁡θ~ikξi,ϕh=max-h≤t≤0ϕ(t), -ηi(k), θi(k), η~i(k), θ~i(k) are the ith entry of vectors ξTA¯k, ξTB¯k, ξTMk, and ξTNk, respectively.Proof.
Let(12)Vxt=ξTxt.Without loss of generality, assume that σ(t)=k. Denote by D+V(x(t)) the right derivative of V(x(t)) along the trajectory of system (1). We get from assumptions (H1) and (H2) that (13)D+xit≤x˙itsign⁡xit≤∑j=1nλta¯ijkxjt+ηtb¯ijkxjt-τt+∑j=1nλ~tmijkxjt+η~tnijkxjt-τt+αit,i∈n,t≥0.Therefore, (14)D+Vxt≤ξTλtA¯k+λ~tMkxt+ξTηtB¯k+η~tNkxt-τt+ξTαt,t≥0.According to definitions of γi for i=1,2,3,4, we derive from the above inequality that (15)D+Vxt≤-γ1λt+γ3λ~tVxt+γ2ηt+γ4η~tVxt-τt+ξTαt,t≥0.Multiply the above inequality by e∫0tγ1λs-γ3λ~sds and let (16)ut=e∫0tγ1λs-γ3λ~sdsVxt.Then we have (17)D+ut≤γ2ηt+γ4η~te∫t-τttγ1λs-γ3λ~sdsut-τt+e∫0tγ1λs-γ3λ~sdsξTαt,t≥0.Integrating it from 0 to t, we obtain (18)ut≤u0+∫0∞e∫0tγ1λs-γ3λ~sdsξTαtdt+∫0tγ2ηs+γ4η~se∫s-hsγ1λv-γ3λ~vdvus-τsds,t≥0.Set(19)ωt=k~1+∫0tγ2ηs+γ4η~se∫s-hsγ1λv-γ3λ~vdvus-τsds,t≥0,(20)ωt=k~1,-h≤t≤0,where k~1 is defined as in Theorem 3. Since ω(t) is a monotone nondecreasing function on [-h,∞), we get (21)ut-τt≤ωt-τt≤ωt,t≥0.Consequently, (22)ut≤ωt≤k~1+∫0tγ2ηs+γ4η~se∫s-hsγ1λv-γ3λ~vdvωsds,t≥0.Combining this and Lemma 1, it implies that (23)ut≤ωt≤k~1e∫0tγ2ηs+γ4η~se∫s-hsγ1λv-γ3λ~vdvds,t≥0.By using the definition of u(t), we have (24)Vxt≤k~1e∫0tγ2ηs+γ4η~se∫s-hsγ1λv-γ3λ~vdv-γ1λs+γ3λ~sds,t≥0.Therefore, (10) holds. This completes the proof of Theorem 3.Remark 4.
For the particular case whenλ(t)≡λ~(t)≡η(t)≡η~(t)≡1, we get from Theorem 3 that all solutions of system (1) satisfy (25)xt≤k~1mini∈n⁡ξieγ2+γ4eγ1-γ3h-γ1+γ3t,t≥0.It implies that system (1) is globally exponentially stable if (γ2+γ4)e(γ1-γ3)h-γ1+γ3<0.Next, we consider the case ofp>1 in Assumption (H2).Theorem 5.
Assume that (H1) and (H2) with p>1 hold. If there exists an n-dimensional vector ξ=(ξi)≻0 such that (8) holds,(26)∫0∞eγ1∫0tλsdsξTαtdt<∞,
and(27)k~21-p-p-1∫0tgse∫0sp-1fvdvds>0,t≥0,then the corresponding solutions of system (1) satisfy(28)xt≤e∫0tfs-γ1λsdsmini∈n⁡ξik~21-p-p-1∫0tgse∫0sp-1fvdvds1/1-p,t≥0,where(29)fs=γ2ηseγ1∫s-hsλvdv,gs=eγ11-p∫0sλvdvγ5λ~s+γ6η~seγ1p∫s-hsλvdv,k~2=ϕhmaxi∈n⁡ξi+∫0∞eγ1∫0tλsdsξTαtdt,γ5=maxk∈m,i∈n⁡η~ikξip,γ6=maxk∈m,i∈n⁡θ~ikξip,γ1, γ2, η~i(k), and θ~i(k) are defined as in Theorem 3.Proof.
LetV(x(t))=ξTxt. Following the same discussion in Theorem 3, we have (30)D+Vxt≤λtξTA¯kxt+ηtξTB¯kxt-τt+λ~tξTMkxtp+η~tξTNkxt-τtp+ξTαt,t≥0.By using the definitions of γi for i=5,6,7,8 and the basic inequality (a+b)p≥ap+bp for a,b≥0 and p>1, we can derive from the above inequality that (31)D+Vxt≤-γ1λtVxt+γ2ηtVxt-τt+γ5λ~tVpxt+γ6η~tVpxt-τt+ξTαt,t≥0.By multiplying the above inequality by eγ1∫0tλ(s)ds and letting u(t)=eγ1∫0tλ(s)dsV(x(t)), we obtain (32)D+uxt≤eγ1∫0tλsdsξTαt+γ2ηteγ1∫t-τttλsdsut-τt+γ5λ~teγ11-p∫0tλsdsupt+γ6η~teγ11-p∫0tλsdseγ1p∫t-τttλsdsupt-τt,t≥0.Integrating it from 0 to t, we get (33)ut≤ξTx0+∫0∞eγ1∫0tλsdsξTαtdt+∫0tγ2ηseγ1∫s-hsλvdvus-τsds+∫0tγ5λ~seγ11-p∫0sλvdvupsds+∫0tγ6η~seγ11-p∫0sλvdveγ1p∫s-hsλvdvups-τsds,t≥0.Set (34)ωt=k~2+∫0tγ2ηseγ1∫s-hsλvdvus-τsds+∫0tγ5λ~seγ11-p∫0sλvdvupsds+∫0tγ6η~seγ11-p∫0sλvdveγ1p∫s-hsλvdvups-τsds,t≥0,(35)ωt=k~2,-h≤t≤0,where k~2 is defined as in Theorem 5. It can be seen that ω(t) is a monotone nondecreasing function on [-h,∞) and u(t-τ(t))≤ω(t-τ(t))≤ω(t) for t≥0. Therefore, (36)ωt≤k~2+∫0tγ2ηseγ1∫s-hsλvdvωsds+∫0teγ11-p∫0sλvdvγ5λ~s+γ6η~seγ1p∫s-hsλvdvωpsds,t≥0.That is, (37)ωt≤k~2+∫0tfsωsds+∫0tgsωpsds,t≥0,where f(s) and g(s) are defined as in Theorem 5. Note that (27) holds. We conclude from Lemma 2 that (38)ut≤wt≤e∫0tfsdsk~21-p-p-1∫0tgse∫0sp-1fvdvds1/1-p,t≥0.It implies that (28) holds true. This completes the proof of Theorem 5.Remark 6.
Note that Theorem5 gives a local state estimation result of system (1). That is, the state estimation (28) only holds for the solutions of system (1) satisfying (27). Let λ(t)≡λ~(t)≡η(t)≡η~(t)≡1. If γ1-γ2eγ1h>0 and k~2<γ5+γ6eγ1ph/γ1-γ2eγ1h1/1-p, we have that condition (27) holds and the corresponding solutions of system (1) satisfy (39)xt≤k~21/1-pmini∈n⁡ξie-γ1-γ2eγ1ht,t≥0.That is, system (1) is locally exponentially stable.Finally, we introduce the following assumption on the nonlinear disturbance(H3) There exist continuous functions λ~(t)≥0 and η~(t)≥0 for t≥0, an n-dimensional vector function α(t)⪰0, and constants γ~5≥0, γ~6≥0, p>0 such that (40)ξTFkt,xt,xt-τt⪯γ~5λ~tξTxtp+γ~6η~tξTxt-τtp+αt,where ξ≻0 is an n-dimensional vector.Similar to the proof of Theorem5, the following result is immediate. Hence, we omit its proof.Theorem 7.
Assume that (H1) holds and p>0(p=1). If there exists an n-dimensional vector ξ=(ξi)≻0 such that (8), (26), (H3), and (27) hold, where γ5 and γ6 are replaced by γ~5 and γ~6, respectively, then the corresponding solutions of system (1) satisfy (28).Remark 8.
Note that condition (27) is always valid for 0<p<1. Therefore, Theorem 7 gives a global state estimation result for system (1). For the particular case when λ(t)≡λ~(t)≡η(t)≡η~(t)≡1 and 0<p<1, a straightforward computation yields that all solutions of system (1) satisfy (41)xt≤k~21-pe-1-pγ1-γ2eγ1ht+γ5+γ6eγ1ph/γ1-γ2eγ1h1/1-pmini∈n⁡ξi,t≥0,where k~2, γ1, γ2, γ5, and γ6 are defined as in Theorems 3 and 7. Consequently, all solutions of system (1) are bounded if γ1-γ2eγ1h>0.


## 4. Numerical Examples

In this section, three examples are given to illustrate the main results.Example 1.
Consider system (1) with m=2, τ(t)=0.2+0.1sin⁡t, (42)A1t=0.5cos⁡t-1.50.2sin⁡t0.2cos⁡t0.5sin⁡t-1.5,B1t=0.2t1+t0.1sin⁡t0.1cos⁡t0.2sin⁡t,A2t=0.5sin⁡t-20.7sin⁡t0.7cos⁡t0.5cos⁡t-2,B2t=0.1sin⁡t0.1cos⁡t0.1t1+t0.1t1+t,F1t,xt,xt-τt=0.1sin⁡x1t+0.1sin⁡x2t-τt+e-t0.2sin⁡x2t+0.1sin⁡x1t-τt+e-t,F2t,xt,xt-τt=0.2sin⁡x2t+0.1sin⁡x1t-τt+e-t0.2sin⁡x1t+0.2sin⁡x2t-τt+e-t.A straightforward computation yields(43)A¯1=-10.20.2-1,B¯1=0.20.10.10.2,A¯2=-1.50.70.7-1.5,B¯2=0.10.10.10.1,M1=0.1000.2,N1=00.10.10,M2=00.20.20,N2=0.1000.2.Choosing ξ=(1,1)T, then ξTA¯i≺0 for i=1,2, γ1=0.8, γ2=0.3, γ3=γ4=0.2, h=0.3. Note that (γ2+γ4)e(γ1-γ3)h-γ1+γ3<0, and ∫0∞eγ1te-tdt<∞. By Theorem 3 and Remark 4, system (1) is exponentially stable. Figure 1 shows the state trajectories of the system, where λ=1 and the switching signal is defined as in Figure 2.Figure 1
The state trajectories of system (1) with τ(t)=0.2+0.1sin⁡t.Figure 2
The given switching signal.Example 2.
Consider system (1) with m=2, τ(t)=0.1+0.05sin⁡t, (44)A1t=-3+sin⁡t-cos⁡tsin⁡t-3+cos⁡t,B1t=-0.3cos⁡t0.2sin⁡t-0.1sin⁡t-0.3cos⁡t,A2t=-4+cos⁡t2sin⁡t-2cos⁡t-4+sin⁡t,B2t=-0.4sin⁡t-0.1cos⁡t0.2cos⁡t-0.3sin⁡t,F1t,xt,xt-τt=0.1x12t+0.05x22t-τt+0.5e-2t0.1x22t+0.05x12t-τt+0.5e-2t,F2t,xt,xt-τt=0.1x22t+0.0x12t-τt+0.5e-2t0.1x12t+0.05x22t-τt+0.5e-2t.A straightforward computation yields that(45)A¯1=-211-2,B¯1=0.30.20.10.3,A¯2=-322-3,B¯2=0.40.20.10.3,M1=0.1000.1,N1=00.050.050,M2=00.10.10,N2=0.05000.05.Choosing ξ=(1,1)T, it can be seen that ξTA¯i≺0 for i=1,2, γ1=1, γ2=0.5, γ5=0.1, γ6=0.05, h=0.15. Note that γ1>γ2eγ1h and ∫0∞eγ1te-2tdt<∞. By Theorem 5 and Remark 6, system (1) is exponentially stable if ϕh<γ5+γ6e2γ1h/γ1-γ2eγ1h1/1-2-1≅0.5. Figure 3 shows the state trajectories of the system, where the switching signal is defined as in Figure 2.Figure 3
The state trajectories of system (1) with τ(t)=0.1+0.05sin⁡t.Example 3.
Consider system (1) with m=2, τ(t)=0.5+0.5sin⁡t, (46)A1t=0.5sin⁡t-2sin⁡tcos⁡t0.5cos⁡t-2,B1t=-0.2sin⁡t0.1cos⁡t0.1cos⁡t-0.2sin⁡t,A2t=cos⁡t-20.5t1+t0.5sin⁡tsin⁡t-2,B2t=0.1sin⁡t0.1cos⁡t0.2sin⁡t0.1cos⁡t,F1t,xt,xt-τt=0.2x11/3t+0.1x21/3t+0.1x21/3t-τt+e-t0.1x11/3t+0.1x21/3t+0.2x11/3t-τt+e-t,F2t,xt,xt-τt=0.3x11/3t+0.2x21/3t+0.2x21/3t-τt+e-t0.2x11/3t+0.3x21/3t+0.1x11/3t-τt+e-t.Based on a direct computation, we get (47)A¯1=-1.511-1.5,B¯1=0.20.10.10.2,A¯2=-10.50.5-1,B¯2=0.10.10.20.1,M1=0.20.10.10.1,N1=00.10.20,M2=0.30.20.20.3,N2=00.20.10.If we choose ξ=(1,1)T, we have that ξTA¯i≺0 for i=1,2, γ1=0.5, γ2=0.3, h=1. Note that γ1>γ2eγ1h, ∫0∞eγ1te-tdt<∞, and (H3) holds for appropriate constants γ~5 and γ~6. By Theorem 7 and Remark 8, we conclude that each solution of system (1) is bounded. Figure 4 shows the state trajectories of the system, where the switching signal is defined as in Figure 2.Figure 4
The state trajectories of system (1) with τ(t)=0.5+0.5sin⁡t.


## 5. Conclusions

The problem of state estimation for switched time-varying systems with time-varying delay and nonlinear disturbance has been discussed in this paper. When the nonlinear disturbance satisfies linear and nonlinear growth conditions, explicit global (local) state estimation criteria have been established. For some particular cases, exponential stability and boundness of the system are taken into consideration. The method used in this paper is mainly based on the integral inequality technique. Finally, three numerical examples demonstrate the effectiveness of our main results.



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