Gain Scheduling Controller Synthesis for Control Moment Gyroscope Using Improved Approximation

Abstract

This paper presents gain scheduling (GS) control of a variable speed control moment gyroscope (VSCMG) based on sum of squares (SOS). Nonlinear motion equations of the VSCMG are complicated because they contain many trigonometric functions of angles of gimbals. The dynamics varies depending on the angles of the gimbals. In this study, the difficulty of control design of the VSCMG is solved by two methods. First, the complicated nonlinear model is transformed to the linear parameter varying model, such that the linear control method can be applied, to make control design easy by using a proposed approximation method. The sine function and the cosine function are generally approximated by the first-order Taylor series expansion in ordinary controller synthesis. The model obtained by the first-order Taylor series expansion ignores the nonlinear dynamics. But, in this study, those nonlinear functions are highly accurately approximated by using the proposed approximation method. The proposed approximation method is based on a high-order Taylor series expansion and a high-order Padé approximation. By using redundant representations, the synthesis condition can be reduced to polynomially parameter-dependent linear matrix inequalities (PDLMIs). Second, GS controller whose gains depend on the angles is applied. The polynomially PDLMIs can be relaxed to finite design conditions based on matrix SOS polynomials. The GS controller is designed by solving the finite SOS conditions. By using those methods, GS controller depending on the nonlinearities is designed. The effectiveness of the proposed controller is illustrated by simulations and experiments.

Keywords

• linear parameter-varying model
• linear matrix inequalities
• sum of squares polynomials
• robust semidefinite programming
• variable speed control moment gyroscope