IEEE Trans Cybern. 2017 Oct;47(10):3393-3403. doi: 10.1109/TCYB.2016.2574747. Epub 2016 Jun 20.
In this paper, backstepping for a class of block strict-feedback nonlinear systems is considered. Since the input function could be zero for each backstepping step, the backstepping technique cannot be applied directly. Based on the assumption that nonlinear systems are polynomials, for each backstepping step, Lypunov function can be constructed in a polynomial form by sum of square (SOS) technique. The virtual control can be obtained by the Sontag feedback formula, which is equivalent to an optimal control-the solution of a Hamilton-Jacobi-Bellman equation. Thus, approximate dynamic programming (ADP) could be used to estimate value functions (Lyapunov functions) instead of SOS. Through backstepping technique, the control Lyapunov function (CLF) of the full system is constructed finally making use of the strict-feedback structure and a stabilizable controller can be obtained through the constructed CLF. The contributions of the proposed method are twofold. On one hand, introducing ADP into backstepping can broaden the application of the backstepping technique. A class of block strict-feedback systems can be dealt by the proposed method and the requirement of nonzero input function for each backstepping step can be relaxed. On the other hand, backstepping with surface dynamic control actually reduces the computation complexity of ADP through constructing one part of the CLF by solving semidefinite programming using SOS. Simulation results verify contributions of the proposed method.
本文研究了一类块严格反馈非线性系统的反推控制。由于在每步反推过程中输入函数可能为零，因此不能直接应用反推技术。基于非线性系统是多项式的假设，对于每步反推，可以通过和的平方（SOS）技术将李雅普诺夫函数构造为多项式形式。虚拟控制可以通过 Sontag 反馈公式获得，它相当于最优控制——哈密顿-雅可比-贝尔曼方程的解。因此，可以使用近似动态规划（ADP）代替 SOS 来估计值函数（李雅普诺夫函数）。通过反推技术，最终利用严格反馈结构构建了整个系统的控制李雅普诺夫函数（CLF），并通过构建的 CLF 得到了一个可镇定控制器。所提出方法的贡献有两个方面。一方面，将 ADP 引入反推可以拓宽反推技术的应用范围。可以用所提出的方法来处理一类块严格反馈系统，并且可以放宽对每步反推过程中输入函数不为零的要求。另一方面，通过使用 SOS 求解半定规划来构建 CLF 的一部分，表面动态控制的反推实际上降低了 ADP 的计算复杂度。仿真结果验证了所提出方法的贡献。
