Attitude control of acrobot by gain scheduling control based on sum of squares

The acrobot is a two-link underactuated system and its dynamics has strong nonlinearity that gives rise to difficulty of controller design. Gain scheduling (GS) control is an alternative method for stabilizing such system in a wide range of operation with desired performance specifications. In this paper, problems of GS controller design for the attitude control of the acrobot are reduced to solving polynomially parameter-dependent linear matrix inequalities (PDLMIs). The polynomially PDLMIs are relaxed by matrix sum of squares (SOS) polynomials to find feasible solutions. Several reduction techniques for the GS controller design of the acrobot are developed to decrease the amount of computation for solving the SOS problems while direct formulations are too cost to solve. For two attitude modes of the acrobot, the effectiveness of GS control is shown by experiments as well as simulations.