Fuzzy Optimal Control for Robot Manipulators

This chapter deals with the design of a Fuzzy Logic Controller based Optimal Linear Quadratic Regulator (FC-LQR) for the control of a robotic system. The main idea is to design a supervisory fuzzy controller capable to adjust the controller parameters in order to obtain the desired axes positions under variations of the robot parameters and payload variations. In the advanced control of robotic manipulators, it is important for manipulators to track trajectories in a wide range of work place. If speed and accuracy is required, the control using conventional methods is difficult to realize because of the high nonlinearity of the robot system. In control design, it is often of interest to design a controller to fulfil, in an optimal form, certain performance criteria and constraints in addition to stability. The theme of optimal control addresses this aspect of control system design. For linear systems, the problem of designing optimal controllers reduces to solving algebraic Riccati equations , which are usually easy to solve and detailed literature of their solutions can be found in many references . Nevertheless, for nonlinear systems, the optimization problem reduces to the so-called Hamilton-Jacobi (HJ) equations, which are nonlinear partial differential equations. Different from their counterparts for linear systems, HJ equations are usually difficult to solve both numerically and analytically. Improvements have also been carried out on the numerical solution of the approximated solution of HJ equations. But few results so far can provide an effective way of designing optimal controllers for general nonlinear systems. In the past, the design of controllers based on a linearized model of real control systems. In many cases a good response of complex and highly non-linear real process is difficult to obtain by applying conventional control techniques which often employ linear mathematical models of the process. One reason for this lack of a satisfactory performance is the fact that linearization of a non-linear systemmight be valid only as an approximation to the real system around a determined operating point. However, fuzzy controllers are basically non-linear, and effective enough to provide the desired non-linear control actions by carefully adjusting their parameters. In this chapter, we propose an effective method to nonlinear optimal control based on fuzzy control. The optimal fuzzy controller is designed by solving a minimization problem that minimizes a given quadratic performance function. Both the controlled system and the fuzzy controller are represented by the affine TakagiSugeno (T-S) fuzzy model taking into consideration the effect of the constant term. Most of the research works analyzed the T-S model assuming that the non-linear system is linearized 4