Policy Iteration for $H_\infty $ Optimal Control of Polynomial Nonlinear Systems via Sum of Squares Programming
提出一种利用平方和多项式近似求解多项式非线性系统H∞最优控制的方法，通过策略迭代将问题转化为可计算的半定规划，并最小化衰减系数以获得近似最优控制器。
Sum of squares (SOS) polynomials have provided a computationally tractable way to deal with inequality constraints appearing in many control problems. It can also act as an approximator in the framework of adaptive dynamic programming. In this paper, an approximate solution to the optimal control of polynomial nonlinear systems is proposed. Under a given attenuation coefficient, the Hamilton-Jacobi-Isaacs equation is relaxed to an optimization problem with a set of inequalities. After applying the policy iteration technique and constraining inequalities to SOS, the optimization problem is divided into a sequence of feasible semidefinite programming problems. With the converged solution, the attenuation coefficient is further minimized to a lower value. After iterations, approximate solutions to the smallest -gain and the associated optimal controller are obtained. Four examples are employed to verify the effectiveness of the proposed algorithm.