Guaranteed performance control of switched positive systems: A switching policy design method

https://doi.org/10.1016/j.jfranklin.2022.04.019Get rights and content

Abstract

In this article, a state-dependent switching policy for handling the guaranteed performance control problem, by means of the partition of the nonnegative state space, for switched positive systems, is devised. The mixed performance, induced by the linear guaranteed cost and L1-gain performance for disturbance attenuation, is investigated. A corresponding optimization problem devoting to the mixed performance is considered, including an upper bound of linear guaranteed cost and an L1-gain performance. Moreover, sufficient conditions are also established by the joint design of switching rules and controllers to ensure the positivity and mixed performance of the closed-loop systems. Eventually, two examples are used to illustrate the validness and practicability of the developed results.

Introduction

In recent years, significant attentions of many researchers have been attracted by positive systems, whose state vectors remain nonnegative under any nonnegative initial conditions. This is mainly due to the fact that positive systems play an important role in many fields including economic modeling, engineering and life sciences, among others [1]. Indeed, the significance of the study of positive systems stems also from its theory aspect, and numerous valuable achievements about positive systems have been made in the literature [2], [3], [4]. The involved research problems range from the positivity, stability, and linear dissipativity analysis to the constrained control, positive filtering, and positive observer design. In the practical implementation, [5] mentioned that positive systems may operate under different working conditions, each of them associated with a different physical model, and hence display switching features. As known to all, switched systems have the capability of modeling the switching behavior. For a review of the relevant studies on switched systems, see [6], [7], [8], [9] and the references therein. Furthermore, switched positive systems not only possess the characteristic of switching but also provide more freedom in handling the solvability of stabilization with the positivity constraint compared with the non-switched versions [10]. Therefore, the investigation of switched positive systems is fruitful [11], [12], [13], [14], [15], [18], as a supplement to general theoretical results on non-positive switched systems and their applications for many of the existing switched positive models.

Typically, when researching a real plant, it is certainly expected to guarantee jointly stability and an adequate level of performance of the underlying systems. Indeed, it is well-known that stability is a precondition to maintain the systems well work and Lyapunov theory is widely used for the stability analysis of dynamic systems. Especially, linear copositive Lyapunov functions occupy an important position in the analyses of stability and performance for switched positive systems due to the positivity of the states. In fact, stability conditions for switched positive systems with all stable subsystems have been proposed in terms of the existence of a common linear copositive Lyapunov function in [11]. But, we know that the existence of this kind of Lyapunov function is a very strong property. Therefore, motivated by this, in [13], the stability of switched positive systems, based on multiple linear copositive Lyapunov functions, was studied. Meanwhile, many results have been obtained in [20], [21], and [22] for theoretical study and [16], [18], [23] for corresponding applications.

On the other hand, the maximization/minimization of the cost function associated with the states or controls, or both is always of importance in control system synthesize. Therefore, many significant results have been discussed in [24], [25], [26], [29]. For the infinite horizons, the guaranteed cost problem has been addressed in [19] mere with respect to the states. However, in practice, many physical systems are usually subjected to both the states and the controls restrictions. Thus, a quadratic cost function associated with both the states and the controls for non-switched positive systems is presented in [28], but leads to a conservative result, since the characteristic of positivity was not used. Recently, the conservativeness in using quadratic functions can be reduced by considering linear cost functions [17] of switched positive systems, but the apriori knowledge of the sequence of modes is required, which motivates us to relax this restriction.

In engineering practice, there always exist the inevitable exogenous perturbations which may cause poor performance. Thus, the enhancement of the robustness against exogenous perturbations remains an important problem for switched positive systems. As a topic beyond stability, L1-gain performance is employed to capture the influence of the exogenous disturbance on system performance at a specified level for switched positive systems. As is pointed out in [31], it is more suitable to take into account L1-gain as a performance index in place of the conventional L2-gain, since the 1-norm represents the sum of the values of the components. For instance, it represents the amount of vehicles in a traffic system. As a consequence, great research interests have been paid to the analysis of L1-gain performance for switched positive systems in [32], [33]. Although L1-gain performance analysis has been investigated in above reports, as far as we know, no related results on mixed the linear guaranteed cost and the L1-gain performance analysis are reported for switched positive systems, which motivates this investigation in part. The difficulties originate from the simultaneous optimization of two performance indexes, including the minimal upper bound of guaranteed cost property and the corresponding minimal L1-gain disturbance resistance property.

This paper is concerned with the guaranteed performance control problem with the simultaneous optimization of two performance indexes via devising a state-dependent switching strategy. Compared with the existing literature, the results of this paper have three distinct features.

(i) A state-dependent switching paradigm is constructed via the partition of the nonnegative state space, which permits the addressed problem for each individual subsystem to be unsolvable. While in this situation, the existing methods in [13] and [30] become inapplicable.

(ii) The mixed linear guaranteed cost and L1-gain performance analysis are taken into account for switched positive systems, and the optimal solutions of two performance indexes can be jointly calculated, which can be viewed as a multi-performance index optimization problem. While the existing results in [17], [18] are only a single performance index optimization problem, as a special case.

(iii) Unlike these previous works in [21], [27] and [33], where the solvability conditions (e.g., guaranteed cost, L1-gain performance, or stability) need to be held on the entire state space, in our paper, the one is required solely on active regions of subsystems. Consequently, the proposed theoretical framework can be extended to various analysis and design synthesis for switched positive systems without requiring the solvability of the conditions on the entire state space.

A brief summary of the paper is as follows. We introduce the model of switched positive systems and formulate the study problems in Section 2. Then, in Section 3, we provide the state-dependent switching policy design and investigate the mixed guaranteed cost performance and L1-gain performance by means of linear copositive Lyapunov functions. The results of simulation are established in Section 4. Section 5 provides some conclusions.

Notation. The nonnegative orthant of the n-dimensional Euclidean space Rn is denoted by R+n. A real matrix A=[aij] is said to be Metzler matrix, if all its components satisfy aij0 for all ij. vec refers to the columns of a matrix taken from left to right and stacked one above the other and is the well-known Kronecker product. x=(x1,x2,,xn)TRn, its 1-norm is x1=k=1n|xk|. I is an identity matrix. 1n1=[111n1]T and 1n1k=[000k1,1,000n1k]T.

Section snippets

Problem formulation and preliminaries

Consider the switched positive systems expressed byx˙(t)=Aσ(t)x(t)+Bσ(t)u(t)+Wσ(t)ω(t),z(t)=Cσ(t)x(t)+Qσ(t)ω(t), in which σ(t):[0,)I^m={1,,m} stands for the switching signal and I^m indicates the amount of positive subsystems of the underlying system. Moreover, for the switching signal σ(t), we use Σ={x0;(i0,t0),(i1,t1),,(il,tl),,|ilI^m,lN} to express the corresponding switching sequence, where x0, t0, i1 and N refer to the initial state, initial time, lth switching moment and the set of

Main results

Our key aim of this section is to determine jointly the state feedback gains and the switching rule such that the problem of aforementioned is solvable for system (3). More precisely, sufficient conditions of the closed-loop positivity and asymptotic stability for the considered system are provided. Based on this analysis, the further study results, subject to the mixed guaranteed cost performance and the L1-gain performance, are presented.

Numerical example

In this section, the validity and practicability of the proposed guaranteed performance control approach are presented with the help of two examples.

Example 1

The typically characteristic of cancer growth and development is multiple genes and pathways, thus combination therapy has been viewed as the standard of care in the treatment of cancer. However, drug toxicity becomes a major concern whenever a patient takes 2 or more drugs simultaneously at the maximum tolerable dosage. A potential method is to

Conclusions

In this paper, we have accomplished the investigation on the guaranteed performance control problem for switched positive systems. Particularly, by constructing switching rules and switching controllers, the guaranteed cost property and the L1-gain property have been simultaneously attained. Then, the optimal guaranteed cost performance and the disturbance attenuation level have also been handled by solving a multi-objective optimization problem. A solvability condition of the guaranteed

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 62003070 and the Natural Scinece Foundation of Fujian Province under Grant 2020J05113.

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