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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 shows that this problem can be solved by utilizing the dynamic programming approach and the inverse optimal approach. The author employs the dynamic programming approach to derive the Hamilton-Jacobi-Bellman (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. 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 and provides good convergence rates of state trajectories to an equilibrium point. The author considers the three-axis attitude stabilization problem of a rigid body to illustrate the optimal control design method for TS-type fuzzy systems. The author designs the optimal three-axis attitude stabilizing control law for a rigid body based on this optimal control design method and analyzes its control performance by numerical simulations.

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