Stability and Lp-gain characterization of positive linear systems on time scales

https://doi.org/10.1016/j.jfranklin.2021.04.021Get rights and content

Abstract

The stability and Lp-gain (p{1,}) analysis problems are investigated for positive linear systems on arbitrary time scales in this paper. Firstly, by virtue of the linear copositive Lyapunov function, the uniform stability and uniformly exponential stability conditions are presented for positive linear time-varying systems on time scales. Then, we conclude that the uniformly exponential stability condition of positive linear time-invariant systems is independent of time scales. Meanwhile, it is further proved that the Lp-gain (p{1,}) conditions of these systems are also independent of time scales and can be fully characterized by system matrices. Finally, a numerical example is given to demonstrate the validity of our conclusions.

Introduction

A dynamical system is named to be positive if its state variables and outputs take nonnegative values for nonnegative initial conditions and inputs. Positive systems are often used to model dynamical phenomena whose variables are restricted to be nonnegative. Thus, positive system models are often encountered in many areas such as economics, population dynamics and ecology, etc. In the past few decades, many unique features have been found in positive systems due to the fact that their state variables are constrained on the positive orthant rather than on the whole vector space [1], [2], [3], [4], [5], [6], [7], [8].

It is well known that Lyapunov’s second method has been widely used for stability analysis of linear and nonlinear systems in both differential and difference equations, and the quadratic function V(x)=xTPx is a frequent Lyapunov candidate for the stability analysis of general systems. For positive systems, the matrix P in V(x) can be further chosen to be a diagonal definite matrix [9], [10] and the corresponding linear matrix inequalities (LMIs) on stability guaranteed can admit the existence of diagonal positive definite solutions. Based on the diagonal quadratic storage functions, Tanaka and Langbort in [11] proved that for continuous-time positive systems the Bounded Real Lemma condition can also be characterized by LMIs with a diagonal positive definite solution and they designed a structured H optimization controller. Later, Wang in [12] generalized the Bounded Real Lemma condition for discrete-time systems. The KYP Lemmas for both continuous-time and discrete-time positive systems were further presented in terms of LMIs with a diagonal matrix variable rather than a generally symmetric one in [13].

On the other hand, the copositive linear function V(x)=λTx, where λ is a vector with positive entries, often serves as an alternative Lyapunov function for positive systems due to its unique nonnegative feature. Thus, the stability conditions of positive systems can be also characterized in terms of linear programming. Furthermore, following the idea of linear storage function, the L1(resp. l1) and L(resp. l)-gains are naturally chosen for performance analysis for continuous-time (resp. discrete-time) positive systems. These gains have been investigated in [14], where gain conditions were characterized in terms of linear programming, and were fully determined by the static matrices of systems. Shen and Lam in [15] showed that these gains of positive systems with bounded and unbounded time-delays were insensitive of the delay size and can also be stated in terms of linear programming.

From above mentioned references and other references, it is not difficult to find that there are many properties hold for both continuous-time and discrete-time positive systems, for example, the diagonal stability and the insensitivity to the delay size. Thus, it is necessary to unify the research on continuous-time and discrete-time positive systems. Hilger in [16] proposed the time scale to unify differential and difference equations. Then, some pioneering works on positive systems on time scales have emerged. In practical application, it is not necessarily continuous- and discrete-time. Time scale makes up for this vacancy and is more convenient. The positive realization, controllability and observability were investigated in[17], [18], [19]. The stability problem of positive systems on general time scales was studied in[20], [21], [22]. However, the problem of whether we can provide a unified form of stability or Lp-gain (p{1,}) condition for these systems is still few investigated.

In this paper, we focus on the analysis of stability and Lp-gain (p{1,}) for positive linear systems on general time scales. Firstly, through exploiting the linear copositive Lyapunov function, we discuss the conditions of uniformly stability and uniformly exponential stability for positive linear time-varying systems on time scales. Secondly, we aim to show that the stability conditions of positive linear time-invariant systems are independent of time scales. Finally, the Lp-gain problems are investigated and the corresponding equivalent conditions are presented. Meanwhile, we explore that the Lp-gains conditions of positive linear systems are also independent of time scales and fully determined by system matrices.

Section snippets

Preliminaries

In this section, we introduce some conceptions on time scales, which can be found in [23]. A time scale is a nonempty closed subset of the real numbers set R. For the purpose of this paper, we assume that the considered time scale T is unbounded, i.e. supT=+ and contains 0. The forward jump operator is defined byσ(t):=inf{sT:t<s}.If t=supT, then σ(supT)=supT when supT is finite. If t<supT and σ(t)=t, we say that t is right-dense, while σ(t)>t, we say that t is right-scattered. The forward

Stability analysis of positive linear time-varying systems

In this section, we will investigate the stability analysis of linear time-varying systems on time scales.

Let A(t):TkRn×n be rd-continuous and consider the following linear time-varying system on the time scale TxΔ(t)=A(t)x(t),x(t0)=x0,t[t0,+)T.

In order to investigate the stability of the time-varying system (2), we firstly introduce the following stability definitions.

Definition 3.1

[25]. The linear time-varying system (2) is uniformly stable if there exists a finite constant γ>0 such that for any t0 and x

Lp-gain characterization of positive linear time-invariant systems

In this section, we mainly investigate the Lp-gain problems of positive linear time-invariant systems on the time scale T. Now, let us consider the following linear time-invariant system on the time scale TΣ:{xΔ(t)=Ax(t)+Bω(t)z(t)=Cx(t)+Dω(t)where x(t)Rn, ω(t)Rm and z(t)Rl denote system state, system input and output, respectively. Moreover, we have ARn×n, BRn×m, CRl×n and DRl×m. In the following, we always assume that initial condition x(t0)=0.

Let us start with the definition and

Numerical example

Let us consider the special case of system Σ with the following system matricesA=[20.50.20.61.50.10.30.51],B=[0.10.10.10.20.10.1],C=[0.20.10.10.10.10.2],D=[0.20.40.10.1].

Now, let us give the following four time scales:T1=R;T2=k=1+[t2k1,t2k],t1=0,tk+1:=tk+1k+1(kN);T3={tk}k1,t1=0,tk+1:=tk+1k+1(kN);T4=Z.Denote ηi:=1sup{μ(t),tTi} (for i=1,2,3,4), then we get η1=+, η2=3, η3=2, η4=1.

It is easily checked that A+ηiIn0 (for i=1,2,3), B0, C0 and D0. From Lemma 4.3 one can deduce that

Conclusions

The stability and Lp-gain (p{1,}) problems of positive linear time-invariant systems on time scales were investigated in this paper. Firstly, sufficient conditions of uniformly stability and uniformly exponential stability were given for positive linear time-varying systems on time scales. Then, it was deduced that the uniformly exponentially stability of a positive linear time-invariant system is independent of time scale. In addition, some necessary and sufficient conditions of Lp-gain (p{1

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    The research was supported by Foundation of National Natural Science of China under Grant No. 61907027.

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