IEEE Trans Cybern. 2018 Feb;48(2):500-509. doi: 10.1109/TCYB.2016.2643687. Epub 2017 Jan 10.
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.
平方和（SOS）多项式为处理许多控制问题中出现的不等式约束提供了一种计算上可行的方法。它也可以在自适应动态规划的框架中作为逼近器。本文提出了一种多项式非线性系统最优控制的近似解。在给定的衰减系数下，将哈密顿-雅可比-伊萨亚斯方程松弛为具有一组不等式的优化问题。应用策略迭代技术并将约束不等式限制在 SOS 之后，将优化问题划分为一系列可行的半定规划问题。随着收敛解的获得，进一步将衰减系数最小化到更低的值。经过迭代，得到了最小增益的近似解和相关的最优控制器。四个例子被用来验证所提出算法的有效性。
