# Stability and Stabilization for Polytopic LPV Systems with Parameter-Varying Time Delays

Authors: Fu Chen; Shugui Kang; Fangyuan Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, we deal with the problem of stability and stabilization for linear parameter-varying (LPV) systems with time-varying time delays. The uncertain parameters are assumed to reside in a polytope with bounded variation rates. Being main difference from the existing achievements, the representation of the time derivative of the time-varying parameter is under a polytopic structure. Based on the new representation, delay-dependent sufficient conditions of stability and stabilization are, respectively, formulated in terms of linear matrix inequalities (LMI). Simulation examples are then provided to confirm the effectiveness of the given approach.


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## Body

## 1. Introduction

Linear time-varying (LPV) systems, which depend on unknown but measurable time-varying parameters, have received much attention. This is mainly due to the context of gain scheduled control of nonlinear systems. In the framework of gain scheduled control, the design can be regarded as applicable for the underlying LPV system, which is a weighted combination of linearized systems. This type of system was first presented in [1], which was extensively studied by many researchers in [2–4] and the references therein. Many approaches used the parameter-independent Lyapunov approach and parameter-dependent Lyapunov approach; see [5–12] and the references therein. It is well known that much less conservative results can be obtained by using the parameter-dependent Lyapunov approach than using the parameter-independent one. However, the difficulty of using a parameter-dependent Lyapunov function is that if the parameter is time-varying, the rate of variation needs to be taken into account. Thus, for the parameter-dependent Lyapunov function, how to seek an appropriate method to represent the derivative of the time-varying parameter is particularly important.On the other hand, time delay is frequently encountered in control systems [13–16]. It is well recognized that the time delay is a source of instability and degrades the system performance. Therefore, it is important to investigate the time-delay effects on the stability of LPV systems. Recently, there has been increasing interest in the stability analysis of LPV systems with parameter-varying time delays. For example, in [17], the analysis and state-feedback control synthesis problem for LPV systems with parameter-dependent state delays are considered. The corresponding analysis and synthesis conditions for stabilization and induced l2 norm performance are obtained in terms of LMIs that can be solved by interior-point algorithms. In [18], by using parameter-dependent Lyapunov functionals and interior-point algorithms, the stability, H∞ gain performance, L2 gain performance, and L2-to-L∞ gain performance are explored for LPV systems with parameter-varying time delays. Moreover, polytopic LPV systems whose parameter is contained in an a priori given set also have aroused the general scholars’ attention; see [19] and the references therein. However, to the best of our knowledge, there have not been any available results on the stability and stabilization on polytopic LPV systems with parameter-varying time delays.In the light of the above, this paper investigates the stability and stabilization of polytopic LPV systems with parameter-varying time delays. Firstly, an innovative representation for the rate of variation of the parameter is stated. Secondly, based on this representation and parameter-dependent Lyapunov functionals, delay-dependent sufficient conditions for the stability and stabilization are derived in terms of LMIs. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.Notation. R stands for the set of real numbers and R+ stands for the nonnegative real numbers. Rm×n is the set of m×n matrices. For a symmetric block matrix, we use ∗ as an ellipsis for the terms that are introduced by symmetry. For a real matrix X, XT denotes its transpose. And, finally, we use the symbol He(X)=X+XT, and X>0(X≥0) means that X is symmetric and positive-definite (positive semidefinite). ⊗ denotes the Kronecker products. The space of continuous functions will be denoted by C and the corresponding norm is ϕt=supt⁡ϕt.


## 2. Problem Statement

Consider a polytopic system with time parameter-varying time delays described by state-space equations.(1a)x˙t=Aϱtxt+Bϱtxt-τϱt+Cϱtut,(1b)xθ=ϕθ,θ∈-τϱ0,0,where x(t)∈Rn is the state vector, u(t)∈Rm is the control input, the real-valued initial function ϕ(θ) in (1b) is a given function in C([-ϱ(0),0],Rn), and τ(t) is a differentiable scalar function representing the parameter-varying delay. The system matrices A(ϱ(t)),B(ϱ(t)),C1(ϱ(t)),C2(ϱ(t)) and the delay τ(ϱ(t)) are dependent on the parameters ϱi(t); that is,(2)Aϱt=∑i=1NϱitAi,Bϱt=∑i=1NϱitBi,Cϱt=∑i=1NϱitCi,τϱt=∑i=1Nϱitτi,where ϱ(t)=[ϱ1(t)ϱ2(t)…ϱN(t)]T∈RN is the time-varying parameter. We make the following assumptions:(A1)
The time-varying parameterϱ(t) varies in a polytope given byϱ(t)∈ΛN, where ΛN:=ϱt∈RN:∑i=1Nϱit=1,ϱit≥0.(A2)
The time derivativeϱi˙(t) of the parameter is such thatϱi˙t≤r, where, r≥0.(A3)
The delay is bounded and the functiont-τ(t) is monotonically increasing; that is, τ(t) lies in the set Ω=τt∈CR,R:0≤τt≤τ-<∞,τ˙t<ω≤1,∀t∈R+.In this paper, we will establish stability and stabilization conditions for system (1a) with (1b) by using Lyapunov-Krasovskii functional and a novel representation of the time derivative of the parameter (t).


## 3. New Representation of the Time Derivative of the Parameter

Lemma 1.
If parameterϱ(t) satisfies assumptions (A1) and (A2), then its time derivative can be written as(3)ϱi˙t=N.r2ζit-ηit,where(4)ζt=ζ1tζ2t…ζNtT∈ΛN,ηt=η1tη2t…ηNtT∈ΛN.Proof.
To complete the proof of Lemma1, we need to distinguish two cases: r=0 and r>0.
Case  1 (r=0). It is known that ϱi˙(t)=0,i=1,2,…,N. In this situation, Lemma 1 is obviously right.
Case  2 (r>0). From assumptions (A1) and (A2), it is known that ϱi˙(t),i=1,2,…,N satisfies the following inequality:(5)-r≤ϱi˙t≤r,∑i=1Nϱi˙t=0.ϱi˙(t),i=1,2,…,N can be written as follows:(6)ϱi˙t=N.r21N+ϱi˙tNr-1N-ϱi˙tNr,i=1,2,…,N.Let(7)ζit=1N+ϱi˙tNr,ηit=1N-ϱi˙tNr,i=1,2,…,N,and then(8)ϱi˙t=N.r2ζit-ηit.It follows from (5) that(9)0≤ζit≤1,∑i=1Nζit=1,0≤ηit≤1,∑i=1Nηit=1.It follows that ζ(t)=[ζ1(t)ζ2(t)…ζN(t)]T∈ΛN and η(t)=[η1(t)η2(t)…ηN(t)]T∈ΛN.
The proof is completed.Remark 2.
The main difference for studying the stability problems of LPV time-dependent systems is how to represent the derivative of the time-varying parameter.
We note, for instance, the results presented in [20], where the rate of variation θi˙ is well defined at all times and satisfies(10)θi˙≤υiθi,υiisconstant,i=1,2,…,N.The weakness of this representation is that, in numerous practical cases, it cannot be physically justified.
We mention in a second case the results given in [21], where the time derivative is defined in a polytope such that(11)θi˙t=∑j=1Mμjthj,i=1,2,…,Nwhere μ(t)∈ΛM={μ∈RM:∑j=1Mμj=1,μj≥0},hj∈RN for all i=1,2,…,M are given vectors. And θi˙t≤r,i=1,2,…,N. Contrary to (10), expression (11) does not impose any particular condition on the derivative of the uncertain parameter, although it does not give any idea about the dimension of the polyhedral convex set, where it evolves.
The main advantage of the new representationϱi˙(t)=N.r/2(ζi(t)-ηi(t)) is that it does not directly depend on the parameter itself and that it is simply defined as a difference between two parameters that evolve in two known and well-defined polytopes.


## 4. Stability Analysis

Now, we use Lyapunov-Krasovskii functional and Lemma1 to develop robust stability condition for the unforced system (1a) with the initial condition (1b). The following technical lemma will be employed to establish our robust stability condition.Lemma 3 (see [22]).
LetX be a full-rank matrix; then, for an arbitrary negative definite matrix Y, the following inequality is satisfied:(12)XTYX<0.Theorem 4.
Consider the unforced system (1a) with the initial condition (1b). If there exist matrices 0<Q∈Rn×n,0<Pi∈Rn×n,Ej∈Rn×n,Gj∈Rn×n,Fj∈Rn×n,i=1,2,…,N,j=1,2,…,N such that(13)N.r2Pj-Pk+Q0Pi0-1-N.r2τj-τkQ0Pi00+HeEjGjFjAiBi-I<0is satisfied for i=1,2,…,N,j=1,2,…,N,k=1,2,3,…,N, then the unforced system (1a) with the initial condition (1b) is asymptotically stable.Proof.
Consider the following Lyapunov-Krasovskii type functional:(14)Vxt,ϱ=xTPϱtx+∫t-τttxTξQxξdξ,where P(ϱ(t))=∑i=1Nϱi(t)Pi.
Let(15)λ-P=maxi∈1,N⁡λmaxPi,λ_P=mini∈1,N⁡λminPi,and then we have that, for ∀x∈Rn,(16)λ_Px2≤Vxt,ϱ≤λ-P+τ-λmaxQx2.The time derivative of V(xt,ϱ(t)) along the trajectories of unforced system (1a) is given by(17)dVdt=xTtdPϱtdtxt+2xTtPϱtx˙t+xTtQxt-1-dτϱtdtxTt-τtQxt-τt=xTtxTt-τtΞ∗BTϱtPϱt-1-dτϱtdtQxtxt-τt,where(18)Ξ=dPϱtdt+PϱtAϱt+ATϱtPϱt+Q.Notice that the time derivatives of P(ϱ(t)) and τ(ϱ(t)) are, respectively, given by(19)dPϱtdt=∑i=1Nϱi˙tPi,dτϱtdt=∑i=1Nϱi˙tτi.It follows from Lemma 1 that there exist ζi(t) and ηi(t) such that(20)ϱi˙t=N.r2ζit-ηit,i=1,2,…,N,where(21)ζt=ζ1tζ2t…ζNtT∈ΛN,ηt=η1tη2t…ηNtT∈ΛN.Multiplying condition (20), respectively, by Pi and τi, we get(22)dPϱtdt=N.r2Pζt-Pηt,dτϱtdt=N.r2τζt-τηt.Multiplying condition (13), respectively, by ϱi(t),ζj(t) and ηk(t) and summing up for i=1,2,…,N,j=1,2,…,N,k=1,2,…,N, we have(23)Ψ11+Ψ22<0,where(24)Ψ11=N.r2Pζt-Pηt+Q0Pϱt0-1-N.r2τζt-τηtQ0Pϱt00,Ψ22=HeEζtGζtFζtAϱtBϱt-I.Pre- and postmultiplying condition (23), respectively, by a full-row matrix(25)I0ATϱt0IBTϱt,as well as its transpose, which combined with Lemma 3 yields(26)N.r2Pζt-Pηt+PϱtAϱt+ATϱtPϱt+QPϱtBϱt∗-1-N.r2τζt-τηtQ<0Substituting condition (22) into the above inequality, we can easily obtain that condition (17) is negative. Thus, V(xt,ϱ) is a Lyapunov functional and the unforced system (1a) with initial condition (1b) is asymptotically stable.
The proof is completed.


## 5. Stabilization

In this paper, we consider the following control law:(27)ut=Kϱtxt=∑i=1NϱitKixt,where matrices Ki,i=1,2,…N are to be designed. Then, the closed-loop system from (1a) and (27) reads(28)x˙t=Aϱt+CϱtKϱtxt+Bϱtxt-τϱt.Before giving a solution to the robust stabilization problem of system (1a) with the initial condition (1b), the following technical lemmas will be employed to establish our main results.Lemma 5 (see [10]).
LetΦ be a symmetric matrix and N,M are matrices of appropriate dimensions. The following statements are equivalent:(1)
(29)Φ<0,Φ+NMT+MNT<0.(2)
There exists a matrixF such that(30)ΦM+NFMT+FTNT-F-FT<0.Lemma 6.
Consider the system matricesA(ϱ(t)) and B(ϱ(t)); the following two conditions are equivalent:(1)
there exist matricesP(ϱ(t))>0 and Q>0 such that(31)HedPϱt/dt+Q2+PϱtAϱt0BTϱtPϱt-1-dτt/dtQ2<0;(2)
for the above matricesP(ϱ(t))>0 and Q>0, there exist a sufficient large scalar β>0 and matrices E(σ(t)),F(σ(t)),G(σ(t)),H(σ(t)) such that(32)HedPϱt/dt+Q2-βEσt000-βFσt-1-dτt/dtQ200-ETσt+Pϱt-βGσt-FTσt-Gσt0ETσt+HTσtAϱt+βHTσtFTσt+HTσtBϱtGTσt-HTσt<0,where(33)σt=σ1tσ2t…σNtT∈ΛN,Eσt=∑i=1Nσ2tEi,Fσt=∑i=1Nσ2tFi,Gσt=∑i=1Nσ2tGi,Hσt=∑i=1Nσ2tHi.Proof.
For necessity, we perform multiplication on the left by(34)I00βI0I00000I00I0,and on the right we do it by its transpose; we have(35)ΠEσt,Fσt,Gσt,Hσt<0,where(36)ΠEσt,Fσt,Gσt,Hσt=HedPϱt/dt+Q2+βHTσtAϱt000βBTϱtHσt-1-dτt/dtQ200ETσt+HTσtAϱt-βHσtFTσt+HTσtBϱt-Hσt0-ETσt+Pϱt-FσtGσt-Gσt.If we let(37)Eσt=Pϱt,Fσt=0,Hσt=Pϱtβ,then we obtain(38)ΠPϱt,0,Gσt,Pϱtβ=HedPϱt/dt+Q2+PϱtAϱt000BTϱtPϱt-1-dτϱt/dtQ200PϱtβAϱtPϱtβBϱt-Pϱtβ000Gσt-Gσt<0.Defining matrices(39)Γ=HedPϱt/dt+Q2+PϱtAϱt00BTϱtPϱt-1-dτϱt/dtQ20PϱtβAϱtPϱtβBϱt-Pϱtβ,Λ1=000T,Λ2=00IT,condition (38) then can be rewritten as(40)ΓΛ1+Λ2GTΛ1T+GΛ2T-GT-G<0.
From conditions (40) and Lemma 5, we can easily verify that condition (38) is satisfied if and only if(41)Γ<0,Γ+Λ2Λ1T+Λ1Λ2T<0;that is(42)Γ<0.
Applying Schur complement to the above inequality, we have(43)HedPϱt/dt+Q2+PϱtAϱt0BTϱtPϱt-1-dτϱt/dtQ2<-12βATϱtBTϱtPϱtAϱtBϱt,which implies that condition (32) is satisfied for a sufficiently large scalar β>0 of condition (31).
For Sufficiency, defining(44)Ξ=HedPϱt/dt+Q2-βEϱt00-βFσt-1-dτϱt/dtQ20-ETσt+Pϱt-βGσt-FTσt-Gσt,Ψ1=ETσtFTσtGTσtT,Ψ2=Aϱt+βIBϱt0T,condition (32) can be written as(45)ΞΨ1+Ψ2HσtΨ1T+HTσtΨ2T-Hσt-HHσt<0.By Lemma 5, condition (45) is equivalent to(46)Ξ+Ψ2Ψ1T+Ψ1Ψ2T=HedPϱt/dt+Q2+EϱtAϱt00BTϱtETϱt+FϱtAϱt-1-dτϱt/dtQ2+FϱtBϱt0-ETϱt+Pϱt+GϱtAϱtGϱtBϱt-FTϱt-Gϱt<0.We perform multiplication on the left by(47)I0ATϱt0IBTϱt,and on the right we do it by its transpose; we obtain(48)HedPϱt/dt+Q2+PϱtAϱt0BTϱtPϱt-1-dτt/dtQ2<0.
The proof is completed.Theorem 7.
If there exist a sufficient large scalarβ>0 and matrices 0<Xi∈Rn×n,0<Y∈Rn×n,Li∈Rm×n,Uj∈Rn×n,Vj∈Rn×n,Wj∈Rn×n,i=1,2,…,N,j=1,2,…,N, and M∈Rn×n such that the inequalities (49)HeN.r/2Xj-Xk+Y2-βUj000-Vj-1-N.r/2τj-τk2Y00-UjT+Xi-βWj-VjT-Wj0IUj+βMT+AM+CLiIVjT+BMIWjT-IN⊗M<0,i=1,2,…,N,j=1,2,…,N,k=1,2,…,N,hold, where(50)A=A1A2⋮AN,B=B1B2⋮BN,C=C1C2⋮CN,I=InIn⋮In∈RNn×n,then the matrix gains Ki=LiM-1,i=1,2,…,N, stabilize the closed-loop system (28) with the initial condition (1b).Proof.
Let(51)Xj=MTPjM,Uj=MTEjM,Vj=MTFjM,Wj=MTGjM,,Li=KiM,Y=MTQM.
Multiplying condition (49) at the left by full-row matrix(52)M-T0000M-T0000M-T0000IN⊗M-Tand at the right by its transpose, then, using Lemma 3, we have(53)HeN.r/2Pj-Pk+Q2-βEj000-Fj-1-N.r/2τj-τk2Q00-EjT+Pi-βGj-FjT-Gj0IEj+βHT+IN⊗HTA+CKiIFjT+IN⊗HTBIGjT-IN⊗H<0,and similar to the proof of Theorem 4, there exist(54)ζt=ζ1tζ2t…ζNtT∈ΛN,ηt=η1tη2t…ηNtT∈ΛN.Therefore(55)dP(ϱtdt=N.r2Pζt-Pηt,dτ(ϱtdt=N.r2τζt-τηt.Multiplying (53), respectively, by ϱi,ζj and ηk and summing up for i=1,2,…,N,j=1,2,…,N,k=1,2,…,N, we have(56)HeN.r/2Pζt-Pηt))+Q2-βEζt000-Fζt)-(1-N.r/2τζt-τηt2Q00-EζtT+Pϱt-βGζt)-FζtT-Gζt)0IEζt+βHT+IN⊗HTA+CKϱtIFζtT+IN⊗HTBIGζtT-IN⊗H<0.Substituting (55) into (56) gives(57)HedPϱt/dt+Q2-βEζt000-Fζt)-(1-dτϱt/dt2Q00-EζtT+Pϱt-βGζt)-FζtT-Gζt)0IEζt+βHT+IN⊗HTA+CKϱtIFζtT+IN⊗HTBIGζtT-IN⊗H<0.Multiplying condition (57) at the left by full-row matrix I3n(ϱ(t)⊗I)T and at the right by its transpose, then, using Lemma 3, we obtain(58)HedPϱt/dt+Q2-βEζt000-βFζt-1-dτt/dtQ200-ETζt+Pϱt-βGζt-FTζt-Gζt0ETζt+HTAϱt+βHTFTζt+HTBϱtGTζt-HT<0,and applying Lemma 6 to (58), we have(59)HedPϱt/dt+Q2+PϱtAϱt0BTϱtPϱt-1-dτt/dtQ2<0;consider the Lyapunov functional(60)Vxt,ϱ=xTPϱtx+∫t-τttxTξQxξdξ,and its time derivative along the trajectories of system (28) is given by(61)dVdt=xTtdPϱtdt+2xTtPϱtx˙t+xTtQxt-1-dτϱtdtxTt-τtQxt-τt=xTtxTt-τtHeΞ′∗BTϱtPϱt-1-dτϱtdtQxtxt-τt,where(62)Ξ′=dPϱtdt+PϱtAϱt+CϱtKϱt+Aϱt+CϱtKϱtTPϱt+Q.It follows from (59) that, for ∀x∈Rn,(63)dVdt<0,which combined with (16) yielding that V(xt,ϱ) is a Lyapunov functional. Thus, system (28) with initial condition (1b) is asymptotically stable.
The proof is completed.


## 6. Simulation Results

In this section, we give two examples to demonstrate the effectiveness of the proposed methods.Example 1.
Let us consider a polytopic system in the form of (1a) with u(t)=0. Assume that the system data are given by(64)A1=-4-2-3-7,A2=-13-15-1-10,B1=2-211.5,B2=1-120.5,ϱ1t=12+12sin⁡2t,ϱ1t=12-12sin⁡2t,τ1=0.5,τ2=1.One can verify that assumptions (A1)–(A3) are satisfied with parameters(65)N=2,r=1,τ-=1,ω=0.5.Using Theorem 4 and MATLAB LMI Control Toolbox, we can obtain a set of feasible solutions as(66)P1=0.12260.03540.03540.3726,P2=0.06980.11880.11880.4658,Q=0.21500.07270.07271.4302,E1=0.0786-0.0281-0.05390.1857,E2=0.0616-0.0114-0.05020.1711,G1=0.00850.0175-0.06570.0953,G2=0.01080.0056-0.01850.0398,F1=0.01240.0002-0.00330.0314,F2=0.00950.0053-0.00790.0454.For an initial condition (x1(0),x2(0))=(5,-3), we simulate the open-loop behavior of the system. The states are shown in Figure 1. Note that both states x1 and x2 converge to zero.Figure 1
System response:x1 (solid) and x2(dotted).Example 2.
Let us consider a polytopic system in the form of (1a). Assume that the system data are given by(67)A1=0100,A2=0021,B1=010.50,B2=0100.7,C1=0.3-0.4,C2=-0.20.5,ϱ1t=12+12sin⁡2t,ϱ1t=12-12sin⁡2t,τ1=0.5,τ2=1.One can verify that assumptions (A1)–(A3) are satisfied with parameters(68)N=2,r=1,τ-=1,ω=0.5.Now, we choose β=3 and then, using Theorem 7 and MATLAB LMI Control Toolbox, we can obtain the control gains as(69)K1=-0.0932-0.0341,K2=-0.0861-0.0969.For an initial condition (x1(0),x2(0))=(-0.4,0.4), we simulate the closed-loop behavior of the system. The states are shown in Figure 2. Note that both states x1 and x2 converge to zero.Figure 2
System response:x1(dotted) and x2 (solid).


## 7. Conclusions

This paper develops the stability and stabilization of polytopic LPV systems with parameter-varying time delays. For this purpose, an innovative representation for the rate of variation of the parameter is addressed. By using this representation and parameter-dependent Lyapunov functionals, delay-dependent sufficient conditions for the stability and stabilization are then derived in terms of LMIs. Two examples are given to illustrate the effectiveness of the methods presented in this paper.



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