Optimal Control of Fuzzy Systems with Application to Rigid Body Attitude Control DOI: http://dx.doi.org/10.5772/intechopen.82181

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

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. 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

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

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

Fuzzy controllers can perform the nonlinear control actions because fuzzy logic

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

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

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

artificial neural network and the genetic algorithm.

and modify their rules.

points [5].

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table.

Aerospace Engineering

subject to a multistage fuzzy system. Esfahani and Sichani [28] studied the problem of optimal fuzzy H∞-tracking control design for nonlinear systems that are represented by using the TS fuzzy modeling scheme. Through these studies, the optimal control for fuzzy systems has quite been progressed.

with a design-model obtained by conventional linear approximation techniques, the TS-type fuzzy model can be seen as a good design-model for approximating a nonlinear dynamic system because it retains the essential features of a nonlinear dynamic system with a linguistic description in terms of fuzzy IF-THEN rules, by which the TS-type fuzzy system is valid over a range of operating points within fuzzy sets. Thus, when we design the optimal controller for a nonlinear dynamic system, it is adequate to approximate a nonlinear dynamic system with a TS-type fuzzy model and to use a TS-type fuzzy model rather than a linearized designmodel. In addition, one can expect that the optimal controller designed for a TS-type fuzzy system has a wider range of optimality than the optimal controller designed for a linearized design-model because the former guarantees the optimal-

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

As a control design example in this chapter, the three-axis attitude stabilization problem of a rigid body is considered to illustrate the optimal control design method for TS-type fuzzy systems presented in this chapter. The attitude motion of a rigid body is basically represented by a set of two Equations [36]: (i) Euler's dynamic equation, which describes the time derivative of the angular velocity vector and (ii) the kinematic equation, which relates the time derivatives of the orientation angles to the angular velocity vector. For representing the orientation angles of a rigid body, there exist several kinematic parameterizations such as Euler angles, Gibbs vector, Cayley-Rodrigues parameters, and modified Rodrigues parameters [37, 38], which are singular three-dimensional parameter representations, and quaternion (also called Euler parameters), which is a nonsingular four-dimensional parameter representation. Note that three-dimensional parameter representations exhibit singular orientations because the Jacobian matrix is singular for some orientations. On the other hand, the quaternion consists of four parameters subject to the unit length constraint and is a globally nonsingular parameter for describing

In this chapter, the equations of motion of a rigid body including dynamics and kinematics are considered. The kinematic equation of a rigid body considered in this chapter is described by the quaternion. The equations of motion of a rigid body considered in this chapter describe a system in cascade interconnection, and the backstepping method of [39] can be efficiently utilized to apply the optimal control design method presented in this chapter to the three-axis attitude stabilization

The optimal attitude stabilization problem of a rigid body has been addressed by several researchers [40–42]. Also, there have been many studies which consider performance indices such as time and/or fuel in the formulation of the optimal attitude stabilization problem of a rigid body [43–49], in which the optimal regulation problems for angular velocity subsystem of a rigid body and for some quadratic

The optimal attitude control problem of the complete attitude motion of a rigid body, which includes dynamics as well as kinematics, has been investigated by many researchers: Carrington and Junkins [50] used a polynomial expansion approach to approximate the solution of H-J-B equation. Rotea et al. [51] showed that Lyapunov functions including a logarithmic term in the kinematic parameters result in linear controllers with a finite quadratic performance index. For the general quadratic performance index, they also presented sufficient conditions which guarantee the existence of a linear and suboptimal stabilizing controller. Tsiotras [52] derived a new class of globally asymptotically stabilizing feedback control laws as well as a family of exponentially stabilizing optimal control laws for the complete attitude motion of a nonsymmetric rigid body. Later, Tsiotras [53] presented a

ity over a range of operating points within fuzzy sets.

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

the body orientation [36].

problem of a rigid body.

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performance indexes have mainly been addressed.

Since TS-type fuzzy systems essentially have a nonlinear nature due to the IF-THEN fuzzy implication and fuzzy inference, we see that a nonlinear optimal control method is suitable for designing an optimal control law for TS-type fuzzy systems. In addition, when we design an optimal control law for TS-type fuzzy systems, it is often required that the control design should allow us to control the convergence rates of state trajectories to an equilibrium point. The decay rate of the closed-loop dynamics may be used to achieve this requirement in the control design. These observations motivate the author to study an optimal control of TS-type fuzzy systems that can provide good convergence rates of state trajectories to an equilibrium point in this chapter.

More specifically, in this chapter, the author presents a theoretical result on the optimal control of nonlinear dynamic systems. In this theoretical result, the author presents the optimal control problem for nonlinear dynamic systems and solves this problem by utilizing the dynamic programming approach [29] and the inverse optimal approach [30]. Note that Kalman [31] first proposed the inverse optimal approach to establish some gain and phase margins of a linear quadratic regulator. Also, note that the conventional direct optimal approach is based on seeking a stabilizing controller that minimizes a given performance index. On the other hand, the inverse optimal approach avoids the task of solving the Hamilton-Jacobi-Bellman (H-J-B) equation numerically but finds a stabilizing controller first and then shows its optimality with respect to a posteriorly determined performance index. In this chapter, the author employs the dynamic programming approach to derive the H-J-B equation associated with the optimal control problem for nonlinear dynamic systems. Then, the author presents an analytic way to solve the H-J-B equation with the help of the inverse optimal approach, by which the author establishes a systematic approach for designing the optimal controller for nonlinear dynamic systems. The resulting optimal controller takes the form of state feedback LgV controller and has a relaxed control gain structure.

Then, based on the theoretical result presented in this chapter, the author establishes an optimal control design for TS-type fuzzy systems that guarantees the global asymptotic stability of an equilibrium point and the optimality with respect to a cost function, which incorporates a penalty on the state and control input vectors, and provides good convergence rates of state trajectories to an equilibrium point. The problem appearing in this optimal control design for TS-type fuzzy systems is given as a linear matrix inequality (LMI)-based problem. From the results, the optimal controller can be found by a simple controller design procedure, which is essentially given as LMIs. The control design involving LMIs is particularly useful in practice because LMIs can be efficiently solved by recently developed interior-point methods (e.g., [32, 33]). One of the algorithms belonging to the interior-point methods can be found in [34], and an implementation of the algorithm in [34] is included in the LMI Control Toolbox of MATLAB [35], which will be used as the solver for the LMIbased problem appearing in the optimal control design.

Note that the optimal controller for a nonlinear dynamic system is in general designed for a linearized design-model, which is obtained by conventional linear approximation techniques, because it is sometimes difficult to solve the nonlinear optimal control problem associated with a nonlinear dynamic system. Clearly, the optimal controller designed for a linearized design-model guarantees its optimality only at an equilibrium point used to design a linearized design-model. Compared
