Dynamic output-feedback control for continuous-time interval positive systems under L1 performance
Introduction
Positive systems are systems whose state variables and outputs take only non-negative values for non-negative inputs and initial conditions. Such systems are encountered in almost all branches of science and technology for example, ecology [1], industrial engineering [2]. Since positive systems are defined on cones rather than linear spaces, well-established methods to system analysis and controller design may fail to give satisfactory results. This motivates the development of positive system theory.
After the development of a system-theoretic approach to positive systems in [3], a large number of theoretical contributions have appeared in the literature [4], [5], [6], [7], [8], [9], [10], [11], [42], [43], [44]. To name a few, a positive state-space representation of a given transfer function has been characterized in [12]. Necessary and sufficient conditions for positive realizability by means of convex analysis have been derived in [13]. Reachability and controllability for positive systems have been investigated thoroughly in [14], [15]. The synthesis problem of state-feedback controllers guaranteeing the closed-loop system to be positive and asymptotically stable has been investigated by the linear matrix inequality (LMI) approach and the linear programming approach in [16], [17], respectively. Stability theory for non-negative and compartmental dynamic systems with time delay has been investigated in [18], [19], [20], [21]. Some results on 2-D positive systems can be found in [7], [22]. As for the results on model reduction problem for positive systems, we refer readers to [23], [24].
Most existing results on positive systems are based on quadratic Lyapunov functions [25], [26], for which reliable and effective numerical approaches are available. The results are often formulated under the LMI framework. Recently, some new results based on linear Lyapunov functions have emerged [17], [27], [28], [29], [30], [31], [32]. The motivation is that the state of positive systems is non-negative, making a linear Lyapunov function a valid candidate. Compared with quadratic Lyapunov function based results, the new results in terms of linear programming are more amenable to analysis and computation. Moreover, the applications of the so-called linear Lyapunov functions in the analysis of positive linear systems naturally lead to a variety of results based on a linear setting, which stimulates the use of L1-gain as a performance index for positive linear systems. It is noted that some frequently used costs such as H∞ norm are based on the L2 signal space [33] and these costs are not very natural to describe some of the features of practical physical systems. By contrast, 1-norm can provide a more useful description for positive systems because 1-norm gives the sum of the values of the components, which is more appropriate, for instance, if the values represent the amount of material or the number of animals in a species.
On the other hand, it is worth noting that the system parameters are usually assumed to be exactly known in the literature [34], [35]. However, practical systems are often affected by environmental changes, variations, perturbations or disturbances, and consequently it is inevitable that uncertainties enter the system parameters [36]. Owing to the complexity caused by parameter uncertainties, the synthesis problems for uncertain positive systems have not been fully investigated. Moreover, it is difficult to design output-feedback controllers compared with state-feedback ones because the problem is not convex [37], [45], [46], [47]. Although considerable efforts have been made to tackle this nonconvex problem, for positive systems with interval uncertainties, the output-feedback controller synthesis is challenging and deserves to be tackled.
Motivated by the aforementioned discussions, in this paper, we consider the dynamic output-feedback stabilization problem for continuous-time interval positive systems under L1-induced performance. For a given positive system, the objective is to construct a stabilizing dynamic output-feedback controller such that the closed-loop system is robustly stable and satisfies a prescribed L1-induced performance. The main contributions of the paper are: (1) an L1-induced performance index is presented for continuous time positive systems with interval uncertainties, which fits well into a linear Lyapunov function framework; (2) Desired dynamic output-feedback controllers are derived with which the stability of the closed-loop system is guaranteed and the proposed performance is satisfied; (3) An iterative convex optimization approach is developed to solve the conditions.
The remaining parts of this article are organized as follows. In Section 2, preliminary results are presented, and L1-induced performance are introduced. In Section 3, the exact value of L1-induced norm is computed and a characterization is developed under which the positive linear system is robustly stable and satisfies the performance. In Section 4, the dynamic output-feedback controller design method for interval positive systems is put forward based on the analysis conditions. One example is provided in Section 5 to show the effectiveness and applicability of the theoretical results. Conclusions are given in Section 6.
Section snippets
Notation and preliminaries
In this section, we introduce notations and several results concerning continuous-time linear positive systems.
Let be the set of real numbers; denotes the set of n-column real vectors; is the set of all real matrices of dimension n × m. Let denote the non-negative orthants of ; that is, if then is equivalent to x ≥ ≥0. is the set of natural numbers. For a matrix aij denotes the element located at the ith row and the jth column. A ≥ ≥0 (respectively, A >
Performance analysis
In this section, we compute the exact value of L1-induced norm for positive system (1). Then, we develop a novel characterization under which system (1) is stable and satisfies the performance in (8).
First, we give the following theorem through which the value of L1-induced norm of system (1) can be computed directly. A similar result has also appeared in [31] and we have provided an independent proof.
Theorem 1 For a stable positive with the exact value of the L1-induced norm from w to z is
Dynamic output-feedback controller design
In this section, we consider the dynamic output-feedback stabilization problem for interval positive systems. The general controller structure under consideration is of the form where and are the controller matrices to be designed.
Define and then it follows from (23) and (27) that the augmented system can be described by where
Illustrative example
In this section, we present one illustrative example to demonstrate the applicability of the proposed results.
Among different population models, the Lotka–Volterra population model is the most classical and widely used in population dynamics and control. In this example, we investigate the population dynamics of pests described by a Lotka–Volterra model. An external disturbance is brought into consideration and our aim is to annihilate the pests in a certain area.
Here, we consider the following
Conclusions
In this paper, we have studied the synthesis problem of L1-induced dynamic output-feedback controllers for interval continuous positive systems. A method has been derived to compute the exact value of the L1-induced norm for positive systems and a characterization has been proposed to ensure the asymptotic stability of the controlled system with a prescribed L1-induced performance level. Based on the performance characterization, conditions for the existence of a desired dynamic output-feedback
Acknowledgments
This work is supported by NSFC 61503184, 61573184 and 61503037.
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