Stability and -gain characterization of positive linear systems on time scales☆
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 is a frequent Lyapunov candidate for the stability analysis of general systems. For positive systems, the matrix in 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 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 where 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 (resp. ) and (resp. )-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 -gain () condition for these systems is still few investigated.
In this paper, we focus on the analysis of stability and -gain () 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 -gain problems are investigated and the corresponding equivalent conditions are presented. Meanwhile, we explore that the -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 . For the purpose of this paper, we assume that the considered time scale is unbounded, i.e. and contains 0. The forward jump operator is defined byIf then when is finite. If and we say that is right-dense, while we say that 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 be rd-continuous and consider the following linear time-varying system on the time scale
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 such that for any and
-gain characterization of positive linear time-invariant systems
In this section, we mainly investigate the -gain problems of positive linear time-invariant systems on the time scale . Now, let us consider the following linear time-invariant system on the time scale where and denote system state, system input and output, respectively. Moreover, we have and . In the following, we always assume that initial condition .
Let us start with the definition and
Numerical example
Let us consider the special case of system with the following system matrices
Now, let us give the following four time scales:Denote (for ), then we get .
It is easily checked that (for ), and . From Lemma 4.3 one can deduce that
Conclusions
The stability and -gain () 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 -gain (
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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