1. Introduction

Since Lotfi Aliasker Zadeh introduced the fuzzy set [1] and the fuzzy logic [2], these two concepts have been successfully applied to various kinds of fields existing in the earth (e.g., see references in [3]), and the usefulness of these two concepts have also been verified in many industries.

Fuzzy logic is basically a multi-valued logic that allows intermediate values to be defined between conventional evaluations like yes or no, true or false, and big or small. The main characteristic of fuzzy concept is that it can handle some complicated phenomena of systems with the help of the fuzzy and linguistic modeling. Thus, it is possible to design knowledge-oriented intelligent systems if we use the fuzzy logic in the design of systems. This is a very important characteristic of fuzzy logic.

Most physical systems are almost nonlinear dynamic systems. Conventional control design approaches use different approximation methods such as linear, piecewise linear, and lookup table approximations to handle their nonlinearities. The linear approximation method linearizes a nonlinear dynamic system about a single equilibrium point and provides a linearized design-model for it. Then, the controller is designed for the linearized design-model to satisfy a given control objective. The linear approximation method is relatively simple, but it tends to limit control performance. In addition, the controller designed by the linear approximation method is valid only under the assumption that the states of a nonlinear dynamic system operate closely around the considered equilibrium point, which is the basic limitation of the control design by the linear approximation method [4]. The piecewise linear approximation method requires the design of several linear controllers. Thus, it works better than the linear approximation method, although it is tedious to implement. The lookup table approximation method can improve control performance, but it is difficult to debug and tune. Moreover, in complex systems where multiple inputs exist, the lookup table approximation method may be very costly to implement because large memories are required to store a lookup table.

Subsequently, many successful applications of the fuzzy control have increased the need of theoretical analysis concerning the stability and performance of fuzzy control systems. Most of all, the stability of fuzzy control systems has often been required to be verified with theoretical arguments, and there have been several significant studies for designing the stabilizing controllers for fuzzy systems with rigorous stability proofs [8–14], in which the so-called TS-type fuzzy model proposed by Takagi and Sugeno [15] has mainly been used to represent fuzzy systems. Specifically, in [8–14], the authors made TS-type fuzzy models of dynamic systems with the IF-THEN fuzzy implication and fuzzy inference and designed the stabilizing control laws for TS-type fuzzy models. Then they applied the stabilizing control laws for TS-type fuzzy models to dynamic systems. In the fuzzy control design, the knowledge of an expert can be applied to the control design for dynamic systems by

Optimal Control of Fuzzy Systems with Application to Rigid Body Attitude Control

DOI: http://dx.doi.org/10.5772/intechopen.82181

Concerning the performance of fuzzy control systems, the optimality has often been considered as an important issue in the design of fuzzy control systems, and the conventional linear optimal control method [16] has been used to design the optimal control law for TS-type fuzzy systems. On the optimality issue of fuzzy control systems, Wang [17] developed the optimal fuzzy controller for linear timeinvariant systems by utilizing the Pontryagin minimum principal. However, the design method of [17] does not have much practical implications because it may not be a good choice to use the fuzzy controller designed for linear systems directly as the controller for nonlinear fuzzy systems. Based on the linear quadratic optimal control theory [16], Wu and Lin [18] presented a design method of the optimal controllers for both continuous- and discrete-time fuzzy systems. The main strategy of [18] is to seek the optimal controller that minimizes a given performance index by solving the matrix Riccati differential equations or the steady-state algebraic Riccati equations. Later, Wu and Lin [19] addressed a quadratic optimal control problem for continuous-time fuzzy systems, which were represented by the socalled linear-like synthetic matrix form and developed a design scheme of the optimal fuzzy controller under finite or infinite horizon by utilizing the calculusof-variation method. The study of [19] is also based on solving a steady-state algebraic Riccati-like equation, but it utilizes an efficient algorithm to design the global optimal fuzzy controller. Park et al. [20] addressed the optimal control problem for continuous-time TS-type fuzzy systems. However, the design method of [20] has less redundancy in choice of feedback gains and requires undesirable high feedback gains, which are the main drawbacks of [20] in the design of the optimal controller for fuzzy systems. Kim and Rhee [21] presented a response surface methodology, and they applied this methodology to the design of an optimal fuzzy controller for a plant. Wu and Lin [22] proposed a way to design a global optimal discrete-time fuzzy controller to control and stabilize a nonlinear discrete-time TS-type fuzzy system with finite or infinite horizon time. Chen and Liu [23] studied the problem of guaranteed cost control for TS-type fuzzy systems with a time-varying delayed state. Mirzaei et al. [24] proposed an optimized fuzzy controller for antilock braking systems to improve vehicle control during sudden braking. Lin, Wang, and Lee [25] investigated a geometric property of time-optimal control problem in the TS-type fuzzy model via Lie algebra and found the time-optimal controller as the bang-bang

type with a finite number of switching by applying the maximum principle.

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Mostefai et al. [26] presented a fuzzy observer-based optimal control design for the compensation of nonlinear friction in a robot joint structure based on a fuzzy local modeling technique. Zhu [27] studied a fuzzy optimal control problem for a multistage fuzzy system to optimize the expected value of a fuzzy objective function

a linguistic expression such as the IF-THEN fuzzy implication.

In many applications, the fuzzy control, based on the fuzzy logic, provides better control performance than linear, piecewise linear, or lookup table approximation methods because it provides an efficient framework to incorporate linguistic fuzzy information from human experts. The so-called intelligent control has emerged to use the expert information in the control community. Some tools for implementing the intelligent control can be referred to artificial neural network, genetic algorithm, and fuzzy logic. Note that the artificial neural network is a learning-based device whose design is motivated by the function of human brains and components thereof. And, the genetic algorithm is inspired by Charles Robert Darwin's theory of evolution and works by creating many stochastic selection parameters to a problem. Usually, the expert information is represented by fuzzy terms like small, large, fast, and so on. Therefore, the fuzzy control is more adequate to implement the intelligent control with the expert information than the artificial neural network and the genetic algorithm.

Fuzzy controllers can perform the nonlinear control actions because fuzzy logic systems are capable of uniformly approximating any nonlinear function over a compact set to any degree of accuracy [5]. Thus, if the parameters of a fuzzy controller are carefully chosen, it is possible to design a fuzzy controller for nonlinear dynamic systems. Then, it is well known that fuzzy controllers are robust with respect to disturbances of systems because their operations are determined by fuzzy rules. Also, fuzzy controllers are customizable because it is easy to understand and modify their rules.

As the behaviors of dynamic systems become complex, the need of fuzzy scheme increases, and the linguistic analysis suggested by Zadeh [6] allows us to analyze the qualitative behaviors of systems with the fuzzy algorithms. As motivated by the study of [6], Mamdani and Assilian [7] proposed the configuration of a fuzzy system with fuzzifier and defuzzifier, and they applied the fuzzy logic to the control of a dynamic plant, and the fuzzy control has attracted a great deal of interest among researchers. Note that fuzzy system is a name for the system which has a direct relationship with fuzzy concepts (e.g., fuzzy sets and linguistic variables) and fuzzy logic [5]. Also, note that the function of fuzzifier is to map crisp points to fuzzy sets, and the function of defuzzifier is to map fuzzy sets to crisp points [5].
