This technical note deals with the state estimation for max-plus linear systems. This estimation is carried out following the ideas of the observer method for classical linear systems. The system matrices are assumed to be known, and the observation of the input and of the output is used to compute the estimated state. The observer design is based on the residuation theory which is suitable to deal with linear mapping inversion in idempotent semiring.

Observer Design for (max,+) linear systems

The objective of this paper is to provide a concise introduction to the max-plus algebra and to max-plus linear discrete-event systems. We present the basic concepts of the max-plus algebra and explain how it can be used to model a specific class of discrete-event systems with synchronization but no concurrency. Such systems are called max-plus linear discrete-event systems because they can be described by a model that is “linear” in the max-plus algebra. We discuss some key properties of the max-plus algebra and indicate how these properties can be used to analyze the behavior of max-plus linear discrete-event systems. Next, some control approaches for max-plus linear discrete-event systems, including residuation-based control and model predictive control, are presented briefly. Finally, we discuss some extensions of the max-plus algebra and of max-plus linear systems.

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Analysis and control of max-plus linear discrete-event systems: An introduction

This paper extends the concept of Karnaugh map from a logical function to a logical system, thereby allowing both the state transition graph and state transition matrix of a logical system, as well as a digital circuit, to be constructed directly. It is found that the Karnaugh map of a logical system is actually a two-dimensional state transition table or, equivalently, a table-like state transition graph and is also a grid-type state transition matrix. For constructing the state transition graph and state transition matrix of a logical system, the computational complexities of the methods based on the proposed Karnaugh map are exponentially lower than those of the existing methods. The Karnaugh maps of logical systems are then used in the analysis and design of clocked sequential circuits and in the simplification of multioutput gate circuits. Some illustrative examples and simulations are presented to demonstrate the results and their applications.

Karnaugh Maps of Logical Systems and Applications in Digital Circuit Design

Reconstructing the logical network from the transition matrix is benefit for learning the logical meaning of the algebraic result from the algebraic representation of a BN. And so far there has no method to convert the matrix expression back to the logic expression for a BN with an arbitrary topology structure. Based on the canonical form and Karnaugh map, we propose a method for reconstructing the logical network from the transition matrix of a Boolean network in this paper.

Reconstruct the Logical Network from the Transition Matrix.

Many problems arising in communication networks, flexible manufacturing systems, traffic managements, etc., can be modelled as discrete-event systems with maximum and minimum constraints. Such systems can be described by max-min-plus systems in which operations maximization, minimization and addition appear simultaneously. This paper considers the output feedback stabilization policy of a class of single-input and single-output max-min-plus systems. The max-plus projection representation of the closed-loop system with the output feedback is introduced. The reachable and observable single-input and single-output max-min-plus systems are stabilized by using the output feedback. It is pointed out that the closed-loop systems have the same eigenvalues as the maximal Lyapunov exponent of the open-loop systems. The method mentioned in this paper is constructive in nature. Some numerical examples are presented to illustrate how the method works in practice.

Output feedback stabilization of a class of single-input and single-output max-min-plus systems

A variety of problems arising in nonlinear systems with timing constraints such as manufacturing plants, digital circuits, scheduling managements, etc., can be modeled as min–max–plus systems described by the expressions in which the operations minimum, maximum and addition appear. This paper applies the max–plus matrix method to analyze the cycle time assignability and feedback stabilizability of min–max–plus systems with min–max–plus inputs and max–plus outputs, which are nonlinear extensions of the systems studied in recent years. The max–plus projection matrix representation of closed-loop systems is introduced to establish some structural and quantitative relationships between reachability, observability, cycle time assignability and feedback stabilizability. The necessary and sufficient conditions for the cycle time assignability with respect to a state feedback and an output feedback, respectively, and the sufficient condition for the feedback stabilizability with respect to an output feedback are derived. Furthermore, one output feedback stabilization policy is designed so that the closed-loop systems take the maximal Lyapunov exponent as an eigenvalue. The max–plus matrix method based on max–plus algebra and directed graph is constructive and intuitive, and several numerical examples are given to illustrate this method.

Xiaowu Mu

Papers

Max–plus matrix method and cycle time assignability and feedback stabilizability for min–max–plus systems