Optimal Stochastic Nonlinear Control of Spacecraft Angular Velocity

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This paper analytically and numerically optimizes the design of a single -gain controller in order to minimize the eff ects of random disturbances on satellite angular velocity . For a satellite the governing laws of attitude dynamics are well -known. In this paper, t hese laws are used in conjunction with both the linear Lyapunov equation and stochastic Lyapunov theory in order t o derive the optimal controller gain for minimizing the resultant error on the spacecraft angular velocity . This paper demonstrates that the same optimized gain and RMS error predictions can be calculated using stochastic Lyapunov theory as with the Lyapu nov equation. This proves a more serious underlying point, which is that the nonlinear terms in the governing equation of satellite attitude dynamics have no effect on the RMS error produced by random disturbances. First, the controller is proven to be sta ble both in the linear case, when the angular velocity can be assumed to be small, and in the nonlinear case, when it cannot. The RMS error is then calculated under both assumptions, using the Lyapunov equation for the linear case of small angular velocit y, and the Stochastic Lyapunov function for the more general nonlinear case . From this RMS error calculation the optimal gain is found by optimization. Both the linear and nonlin ear cases are studied and prove to lead to the same resulting optimal control ler. F inally, a numerical model is generated in Simulink and Matlab and is utilized to numerically prove that the derived gain leads to the minimal RMS error on the angular velocity.