Technical CommuniqueOn stabilization of min–max systems☆
Introduction
Discrete event systems in which the operations min, max and plus appear simultaneously are known as min–max systems or min–max–plus systems. A variety of problems arising in digital circuits, computer networks and automated manufacturing plants can be described using these models (see Gunawardena, 1994b; Cochet-Terrasson, Gaubert, & Gunawardena, 1997 and references therein for more detailed discussions on the applications of min–max systems). Theoretically, such systems are natural extensions of timed discrete event systems which contain only maximum timing constraints (or only minimum timing constraints) and can be studied as so called linear models in terms of max–plus algebra (see e.g. Cuninghame-Green, 1979; Cohen, Moller, Quadrat, & Viot, 1989; Baccelli, Cohen, Olsder, & Quadrat, 1992; Gunawardena, 1998 for recent advances in this direction). But min–max systems containing both maximum and minimum constraints are non-linear in this sense and only until recently, some basic results were obtained (see Olsder, 1991; Gunawardena, 1994b; Gaubert & Gunawardena, 1998; Baccelli & Mairesse, 1998; Cochet-Terrasson, Gaubert, & Gunawardena, 1999 and references therein). The main results on min–max systems were concerned with the analysis of the properties of the systems. Among them, the Duality Theorem obtained in Gaubert and Gunawardena (1998) established the existence of cycle time vectors for min–max systems. Cycle time vector is a real vector η satisfying , where x(k) is the state vector of the system. The cycle times are also known as Lyapunov exponents (Baccelli et al. 1992; Baccelli & Hong, 1998). Let the system be given by , where Asx(k) are products of the matrices As and the state vector x(k) in max–plus algebra, the cycle time vector can then be determined as the minimum of the cycle time vectors χ(As) of the max–plus matrices, namely, according to Gunawardena (1994b). A constructive fixed point theorem was established by Cochet-Terrasson et al. (1999). The existence of a fixed point for a min–max system, is equivalent to the existence of a global cycle time, or a uniform asymptotic growing rate of the system state vector, namely, a real λ such that the cycle time vector η=(λ,…,λ)′. Similar to the linear cases (Cohen, Moller, Quadrat, & Viot, 1984; Baccelli et al., 1992), min–max systems with this property will be referred to as stable systems. As an application, stability of min–max systems plays an important role in solving the clock schedule verification (Cochet-Terrasson et al., 1999) problem.
To our best knowledge, only little work has been done on control of min–max systems. Among them, model predictive control (MPC) problem was studied in De Schutter and van den Boom (2000). In this note, we will study another interesting control problem—stabilization of min–max systems. In many applications, stability is an extremely desirable system property: for automated manufacturing systems, stability guarantees no indefinite increasing of internal buffer (storage) capacity and for digital circuits it guarantees the asymptotic synchronization of all signals. The purpose of this note is to extend the results on stabilization of max–plus systems in Cohen et al. (1984) and Baccelli et al. (1992) to min–max systems. Our results are based on upper bounding the cycle time vector via the structural property: inseparability proposed in Zhao, Zheng, and Zhu (2001).
Section snippets
Preliminaries
We shall follow the notation used in Gunawardena (1994b). The operations a∨b and a∧b are used to stand for maximum and minimum, respectively: a∨b=max(a,b) and a∧b=min(a,b). It can be seen that the operation plus distributes over both maximum and minimum. Definition 1 A min–max expressionf, is a term in the grammar: f≔x1,x2,…,xn,|f+a|f∧f|f∨f, where , is the set of real numbers. Note that expressions like 1∧x,1∨x are not treated as min–max expressions according to this definition. Definition 2 A min–max map of dimension
Main results
The general stabilization problem can be stated as to find a feedback map K such that by applyingthe closed-loop system given bywhere , is stable, i.e. has a global cycle time. d is a nonnegative integer to be determined as part of the feedback policy. As in Baccelli et al. (1992), the parameters of the feedback map K are required to be nonnegative because only non-negative delay of events can be
Conclusions
In this paper, we presented some conditions under which unstable min–max systems can be stabilized using feedbacks without causing loss of performance. Our results are based on the structural properties of min–max systems. We also extended the results for linear cases in Gaubert (1995) to obtain sub-optimal configuration of the resources needed by feedback maps. The structural stabilization by forcing the inseparability of the closed-loop systems is not always necessary for stabilizing system
Acknowledgements
This work was supported in part by NSFC (Grant No. 60074012 and Grant No. 60274011), Fundamental Research Funds from Tsinghua University.
The authors would like to thank Prof. Wende Chen, Mr. Yuegang Tao and Dr. David Pepyne for helpful discussions. They also wish to thank the anonymous reviewers for their extremely helpful suggestions and comments. The first author is grateful to Prof. Yu-Chi Ho for his hospitality during the first author's stay at Harvard University, he also acknowledges Mr.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Geir E. Dullerud under the direction of Editor Paul Van den Hof.