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.
Top2.1 Introduction
In this chapter, we design and analyse a class of adaptive control schemes based on fuzzy logic and high order neural networks referred to as fuzzy-recurrent high order neural networks (F-RHONN’s).
Nonlinear dynamical systems can be represented by general nonlinear dynamical equations of the form
(1)The mathematical description of the system is required, so that we are able to control it. Unfortunately, the exact mathematical model of the plant, especially when this is highly nonlinear and complex, is rarely known and thus appropriate identification schemes have to be applied which will provide us with an approximate model of the plant.
It has been established that neural networks and fuzzy inference systems are universal approximators (Hornik, Stinchcombe & White, 1989), (Wang, 1994), (Passino & Yurkovich, 1998), that is, they can approximate any nonlinear function to any prescribed accuracy provided that sufficient hidden neurons and training data or fuzzy rules are available. Recently, the combination of these two different technologies has given rise to fuzzy neural or neuro fuzzy approaches, that are intended to capture the advantages of both fuzzy logic and neural networks. Numerous works have shown the viability of this approach for system modelling (Jang, 1993; Lin, 1995; Cho & Wang, 1996; Juang & Lin, 1998; Li & Mukaidono, 1995; Chiu, 1994; Lin & Cunningham, 1995; Jang & Lin, 1998; Mitra & Hayashi, 2000).
The neural and fuzzy approaches are most of the time equivalent, differing between each other mainly in the structure of the approximator chosen. Indeed, in order to bridge the gap between the neural and fuzzy approaches several researchers introduce adaptive schemes using a class of parameterized functions that include both neural networks and fuzzy systems (Cho & Wang, 1996; Juang & Lin, 1998; Li & Mukaidono, 1995; Chiu, 1994; Lin & Cunningham, 1995; Jang & Lin, 1998; Mitra & Hayashi, 2000). Regarding the approximator structure, linear in the parameters approximators are used in (Lin & Cunningham, 1995), (Chen, Lee & Chang, 1996), and nonlinear in (Spooner & Passino, 1996), (Narendra & Parthasarathy, 1990), (Polycarpou & Mears, 1998).