Tsallis Entropy of Fuzzy Dynamical Systems

This article deals with the mathematical modeling of Tsallis entropy in fuzzy dynamical systems. At first, the concepts of Tsallis entropy and Tsallis conditional entropy of order where is a positive real number not equal to 1, of fuzzy partitions are introduced and their mathematical behavior is described. As an important result, we showed that the Tsallis entropy of fuzzy partitions of order satisfies the property of sub-additivity. This property permits the definition of the Tsallis entropy of order of a fuzzy dynamical system. It was shown that Tsallis entropy is an invariant under isomorphisms of fuzzy dynamical systems; thus, we acquired a tool for distinguishing some non-isomorphic fuzzy dynamical systems. Finally, we formulated a version of the Kolmogorov–Sinai theorem on generators for the case of the Tsallis entropy of a fuzzy dynamical system. The obtained results extend the results provided by Markechová and Riečan in Entropy, 2016, 18, 157, which are particularized to the case of logical entropy.

A New Approach to the Control Design of Fuzzy Dynamical Systems

A new approach to the control design for fuzzy dynamical systems is proposed. For a fuzzy dynamical system, the uncertainty lies within a fuzzy set. The desirable system performance is twofold: one deterministic and one fuzzy. While the deterministic performance assures the bottom line, the fuzzy performance enhances the cost consideration. Under this setting, a class of robust controls is proposed. The control is deterministic and is not if-then rules-based. An optimal design problem associated with the control is then formulated as a constrained optimization problem. We show that the problem can be solved and the solution exists and is unique. The closed-form solution and cost are explicitly shown. The resulting control is able to guarantee the prescribed deterministic performance and minimize the average fuzzy performance.

A NEW DIRECT ADAPTIVE REGULATOR WITH ROBUSTNESS ANALYSIS OF SYSTEMS IN BRUNOVSKY FORM

The direct adaptive regulation of unknown nonlinear dynamical systems in Brunovsky form with modeling error effects, is considered in this paper. Since the plant is considered unknown, we propose its approximation by a special form of a Brunovsky type neuro–fuzzy dynamical system (NFDS) assuming also the existence of disturbance expressed as modeling error terms depending on both input and system states plus a not-necessarily-known constant value. The development is combined with a sensitivity analysis of the closed loop and provides a comprehensive and rigorous analysis of the stability properties. The existence and boundness of the control signal is always assured by introducing a novel method of parameter hopping and incorporating it in weight updating laws. Simulations illustrate the potency of the method and its applicability is tested on well known benchmarks, as well as in a bioreactor application. It is shown that the proposed approach is superior to the case of simple recurrent high order neural networks (HONN's).

Identification of Nonlinear Systems Using a New Neuro-Fuzzy Dynamical System Definition Based on High Order Neural Network Function Approximators

A new definition of adaptive dynamic fuzzy systems (ADFS) is presented in this chapter for the identification of unknown nonlinear dynamical systems. The proposed scheme uses the concept of adaptive fuzzy systems operating in conjunction with high order neural networks (HONN’s). Since the plant is considered unknown, we first propose its approximation by a special form of an adaptive fuzzy system and in the sequel the fuzzy rules are approximated by appropriate HONN’s. Thus the identification scheme leads up to a recurrent high order neural network, which however takes into account the fuzzy output partitions of the initial ADFS. Weight updating laws for the involved HONN’s are provided, which guarantee that the identification error reaches zero exponentially fast. Simulations illustrate the potency of the method and comparisons on well known benchmarks are given.

Neuro – Fuzzy Control Schemes Based on High Order Neural Network Function Approximators

The indirect or direct adaptive regulation of unknown nonlinear dynamical systems is considered in this chapter. Since the plant is considered unknown, we first propose its approximation by a special form of a fuzzy dynamical system (FDS) and in the sequel the fuzzy rules are approximated by appropriate high order neural networks (HONN’s). The system is regulated to zero adaptively by providing weight updating laws for the involved HONN’s, which guarantee that both the identification error and the system states reach zero exponentially fast. At the same time, all signals in the closed loop are kept bounded. The existence of the control signal is always assured by introducing a novel method of parameter hopping, which is incorporated in the weight updating laws. The indirect control scheme is developed for square systems (number of inputs equal to the number of states) as well as for systems in Brunovsky canonical form. The direct control scheme is developed for systems in square form. Simulations illustrate the potency of the method and comparisons with conventional approaches on benchmarking systems are given.