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In this article, the  H ∞control problem is investigated for polynomial systems with uncertain parameters by utilising an adaptive dynamic programming (ADP) approach. The goal of this problem is to design a controller that stabilises the system and renders the system  L 2-gain less than some disturbance attenuation level for all possible values of the uncertain parameters. A policy iteration (PI) algorithm is proposed to approximately solve the Hamilton-Jacobi-Isaacs (HJI) equation based on the sum-of-squares (SOS) optimisation. The convergence of the iterative algorithm and robustness of the closed-loop system with respect to both external disturbances and parametric uncertainties are guaranteed. Moreover, a two-loop iterative algorithm is presented to design a parametric robust  H ∞controller with a smaller  L 2-gain. The efficiency and advantage of the proposed SOS-based algorithm are demonstrated by simulation examples.

SOS-based policy iteration for H ∞ control of polynomial systems with uncertain parameters