5. Conclusion

In this chapter, the author presented a theory on the optimal control of nonlinear dynamic systems by utilizing the dynamic programming approach and the inverse optimal approach. Specifically, the author employed the dynamic programming approach to derive the Hamilton-Jacobi-Bellman (H-J-B) equation associated with the optimal control problem for nonlinear dynamic systems and utilized the inverse optimal approach to avoid the task of solving the H-J-B equation numerically.

Then, the author established an optimal control design for TS-type fuzzy systems to achieve the global asymptotic stability of an equilibrium point, the optimality with respect to a cost function, and the good convergence rates of state trajectories to an equilibrium point. Based on this optimal control design, the author presented a systematic way for designing the optimal control law for TS-type fuzzy systems.

The author showed the usefulness of the optimal control design by considering the three-axis attitude stabilization problem of a rigid body. The optimal three-axis attitude stabilizing control law for a rigid body was designed, and its control performance was analyzed by numerical simulations. The numerical simulation results demonstrated that the optimal three-axis attitude stabilizing control law designed in this chapter provides desirable optimal control performance together with good convergence rates of state trajectories to an equilibrium point.

Figure 4.

210

Figure 3.

Aerospace Engineering

q tð Þ¼ <sup>0</sup> ½ � <sup>0</sup>:3320 0:4618 0:1915 0:<sup>7999</sup> <sup>T</sup>:

0:4618 0:1915 0:7999 �

T:

Quaternion responses of the rigid body given by (38) and (41) with control laws uað Þt of (51), ubð Þt of (52), and ucð Þ<sup>t</sup> of (53) designed in this chapter at initial conditions <sup>ω</sup>ð Þ¼ <sup>t</sup><sup>0</sup> ½ � <sup>000</sup> <sup>T</sup> rad/sec and q tð Þ¼ <sup>0</sup> <sup>½</sup>0:<sup>3320</sup>

Angular velocity responses of the rigid body given by (38) and (41) with control laws uað Þt of (51), ubð Þt of (52), and ucð Þ<sup>t</sup> of (53) designed in this chapter at initial conditions <sup>ω</sup>ð Þ¼ <sup>t</sup><sup>0</sup> ½ � <sup>000</sup> <sup>T</sup> rad/sec and
