Supervisory Control of Fuzzy Discrete Event Systems

In order to cope with situations in which a plant's dynamics are not precisely known, we consider the problem of supervisory control for a class of discrete event systems modelled by fuzzy automata. The behavior of such discrete event systems is described by fuzzy languages; the supervisors are event feedback and can disable only controllable events with any degree. The concept of discrete event system controllability is thus extended by incorporating fuzziness. In this new sense, we present a necessary and sufficient condition for a fuzzy language to be controllable. We also study the supremal controllable fuzzy sublanguage and the infimal controllable fuzzy superlanguage when a given pre-specified desired fuzzy language is uncontrollable. Our framework generalizes that of Ramadge-Wonham and reduces to Ramadge-Wonham framework when membership grades in all fuzzy languages must be either 0 or 1. The theoretical development is accompanied by illustrative numerical examples.Comment: 12 pages, 2 figure

Al'brekht's Method in Infinite Dimensions

In 1961 E. G. Albrekht presented a method for the optimal stabilization of smooth, nonlinear, finite dimensional, continuous time control systems. This method has been extended to similar systems in discrete time and to some stochastic systems in continuous and discrete time. In this paper we extend Albrekht's method to the optimal stabilization of some smooth, nonlinear, infinite dimensional, continuous time control systems whose nonlinearities are described by Fredholm integral operators

State-Based Control of Fuzzy Discrete Event Systems

To effectively represent possibility arising from states and dynamics of a system, fuzzy discrete event systems as a generalization of conventional discrete event systems have been introduced recently. Supervisory control theory based on event feedback has been well established for such systems. Noting that the system state description, from the viewpoint of specification, seems more convenient, we investigate the state-based control of fuzzy discrete event systems in this paper. We first present an approach to finding all fuzzy states that are reachable by controlling the system. After introducing the notion of controllability for fuzzy states, we then provide a necessary and sufficient condition for a set of fuzzy states to be controllable. We also find that event-based control and state-based control are not equivalent and further discuss the relationship between them. Finally, we examine the possibility of driving a fuzzy discrete event system under control from a given initial state to a prescribed set of fuzzy states and then keeping it there indefinitely.Comment: 14 double column pages; 4 figures; to be published in the IEEE Transactions on Systems, Man, and Cybernetics--Part B: Cybernetic

Chaotic Behaviour in Some Discrete –Time Adaptive Control Systems

It has been shown that nonlinear discrete maps can display extremely rich behaviour and under certain parameter conditions to show chaotic phenomenon. This work looks at adaptive control feedback systems which can be represented as nonlinear discrete maps and shows how model mismatch can lead to undesired complicated and chaotic behaviour. Moreover that a discrete-time adaptive control system which can display chaotic behaviour can be extended into higher order systems and the results show that under certain parameter conditions, the higher order systems also behave chaotically. A generalised equation form for the eigenvalues is also given

Discrete port-Hamiltonian systems: mixed interconnections

Either from a control theoretic viewpoint or from an analysis viewpoint it is necessary to convert smooth systems to discrete systems, which can then be implemented on computers for numerical simulations. Discrete models can be obtained either by discretizing a smooth model, or by directly modeling at the discrete level itself. The goal of this paper is to apply a previously developed discrete modeling technique to study the interconnection of continuous systems with discrete ones in such a way that passivity is preserved. Such a theory has potential applications, in the field of haptics, telemanipulation etc. It is shown that our discrete modeling theory can be used to formalize previously developed techniques for obtaining passive interconnections of continuous and discrete systems

Consistent Approximations for the Optimal Control of Constrained Switched Systems

Though switched dynamical systems have shown great utility in modeling a variety of physical phenomena, the construction of an optimal control of such systems has proven difficult since it demands some type of optimal mode scheduling. In this paper, we devise an algorithm for the computation of an optimal control of constrained nonlinear switched dynamical systems. The control parameter for such systems include a continuous-valued input and discrete-valued input, where the latter corresponds to the mode of the switched system that is active at a particular instance in time. Our approach, which we prove converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, then performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical Chattering Lemma that we prove. We extend this algorithm by formulating a computationally implementable algorithm which works by discretizing the time interval over which the switched dynamical system is defined. Importantly, we prove that this implementable algorithm constructs a sequence of points by recursive application that converge to the local minimizers of the original constrained optimal control problem. Four simulation experiments are included to validate the theoretical developments