On the complexity and dynamical properties of mixed logical dynamical systems via an automaton‐based realization of discrete‐time hybrid automaton

Abstract

Summary Modeling of hybrid systems using mixed logical dynamical (MLD) systems is an art. The MLD framework often introduces numerous constraints and auxiliary binary and continuous variables, which, in turn, increase the computational complexity of the optimization problems. This paper presents an automaton‐based realization for discrete‐time hybrid automaton (DHA) with both controlled and uncontrolled switching phenomena by which it is attempted to develop efficient translation techniques to MLD systems and reduce the total number of decision variables in the MLD model. Based on this DHA model, a modified version of MLD systems, which is called extended MLD (EMLD) is formally defined and represented. EMLD is derived based on the concept of forward evolution in which the one‐step delay between the change of discrete states and continuous dynamics of the existing evolution in the conventional MLD systems is eliminated. The result is that the size of EMLD model in terms of the number of variables, which determines the complexity of the synthesis problems on MLD systems, is reduced. The dynamical properties such as the nonblocking, determinism, and well‐posedness of the proposed DHA and its equivalent MLD models are investigated, and the necessary and sufficient conditions are derived. The effectiveness of the proposed ideas is shown by the numerical examples.

On the well-posedness, equivalency and low-complexity translation techniques of discrete-time hybrid automaton and piecewise affine systems

“…In [21], the equivalency relations among some of discrete-time hybrid modeling frameworks such as Mixed-Logical Dynamical (MLD) systems [22,23], Linear Complementarity (LC) systems [24], Extended Linear Complementarity (ELC) systems [25], Min-Max-Plus-Scale (MMPS) systems [26] and Piecewise Affine (PWA) systems [27,28], have been presented. The existence of many equivalent modeling frameworks for a hybrid system confirms the importance of the equivalency concept and translation techniques among these modeling frameworks.…”

“…A reason behind this importance is the fact that each modeling framework is suitable for a specific problem at hand. For instance, finding stability criteria and controller synthesis for piecewise affine systems is made easily in this modeling framework [29,30,31], while the existence and uniqueness of the solutions can be investigated easier in linear complementarity systems [24,32] and hybrid automata [33,23]. Therefore, the study on the equivalency relations among different classes of hybrid systems is of particular importance, because it provides the possibility of transferring the theoretical properties and tools from one modeling framework to another.…”

“…One reason for such a high complexity is that when a hybrid system is formulated in the decomposed DHA, the information associated with the structure of the hybrid system becomes hidden and cannot be exploited in the translation techniques. However, in automaton-based DHA [23], these structural information is explicitly available from the graphical net of the DHA, and can be efficiently utilized to solve DHA to PWA translation problem in a fast and low-complexity manner without any need to feasibility tests and applying mixed integer programming.…”

Global Practical Stabilization of Discrete-time Switched Affine Systems via Switched Lyapunov Functions and State-dependent Switching Functions

“…Although the literature on switched systems is rich and there are several books [2]- [6] and survey papers [7]- [10] in this context, however, most of the works in this domain assume a common equilibrium point for all isolated subsystems. On the other hand, one of the most important subclasses of switched systems is the switched affine systems that are very common in practice, especially in power electronics area [11]- [17].…”

On the global practical stabilization of discrete-time switched affine systems: application to switching power converters

“…The general classes of hybrid dynamical systems contain both controlled and uncontrolled switching phenomena that make their analysis and synthesis to be complex and challenging [1,2,3,4,5]. In this regard, the switched systems are the special form of the general hybrid dynamical systems which contain only controlled switching phenomena.…”