The solution to the Hamilton-Jacobi equation associated with the nonlinear control problem is approximated using a Taylor series expansion. A recently developed analytical solution method is used for the second-, third-, and fourth-order terms. The proposed controller synthesis method is applied to the problem of satellite attitude control with attitude parameterization accomplished using the modified Rodrigues parameters and their associated shadow set. This leads to kinematical relations that are polynomial in the modified Rodrigues parameters and the angular velocity components. The proposed control method is compared with existing methods from the literature through numerical simulations. Disturbance rejection properties are compared by including the gravity-gradient and geomagnetic disturbance torques. Controller robustness is also compared by including unmodeled first- and second-order actuator dynamics, as well as actuation time delays in the simulation model. Moreover, the gap metric distance induced by the unmodeled actuator dynamics is calculated for the linearized system. The results indicated that a linear controller performs almost as well as those obtained using higher-order solutions for the Hamilton-Jacobi equation and the controller dynamics. 1. Introduction The attitude control problem is critical for most satellite applications and has thus attracted extensive interest. While many control methods have been developed to address this problem, most of them are concerned primarily with the optimality of attitude maneuvers [1–4]. In the present work, we shall focus on robust nonlinear control systems. We note that, throughout this paper, by nonlinear we mean the -gain of a nonlinear system. Control laws are generally developed based on mathematical models that are, at best, a close approximation of real-world phenomena. For such control methods to have any real practical value, they must be made robust with regard to unmodeled dynamics and disturbances that may act on the system. The study of robust control is therefore an essential part of the application of control theory to physical systems. In general, the development of an optimal nonlinear state feedback control law is characterized by the solution to a Hamilton-Jacobi partial differential equation (HJE) [5], while a robust nonlinear controller is obtained from the solution of one or more Hamilton-Jacobi equations [6–9]. However, no general analytical solution has yet been obtained to solve this optimization problem. Solutions have thus far only been obtained under certain
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