Abstract
The solution to the Hamilton-Jacobi equation associated with the nonlinear ℋ∞ 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.
Highlights
The attitude control problem is critical for most satellite applications and has attracted extensive interest
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 [6,7,8,9]
Solutions have far only been obtained under certain conditions: in the case of linear systems with a quadratic performance index, the HJE reduces to the well-known algebraic Riccati equation (ARE)
Summary
The attitude control problem is critical for most satellite applications and has attracted extensive interest. The primary purpose of this paper is to develop robust nonlinear controllers based on analytical expressions for approximate solutions to the Hamilton-Jacobi equation. We shall provide analytical expressions for the second-, third-, and fourth-order terms of the approximation solution These controllers are compared through numerical simulations with existing methods from the literature for spacecraft attitude regulation [1,2,3,4]. This controller is the solution of an appropriate nonlinear H∞ problem and is taken from the work of James et al [9]. We note here that some of these results appear in past conference proceedings [30, 31] with the present paper containing improvements to the overall presentation
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