一类非线性最优控制问题的不变容许策略迭代
Invariantly Admissible Policy Iteration for a Class of Nonlinear Optimal Control Problems
摘要：
本文提出了一种迭代求解一类输入仿射非线性最优控制问题的广义逐次逼近法（SAM），称为不变容许策略迭代法（PI）。与现有的SAM不同，所提出的方法更新了下一个策略和值函数的域，以获得容许性（和不变性）。在现有的SAM中，所生成的政策的容许性得到保证的两个隐含的假设下，关于李雅普诺夫定理和不变性，这两个提出和讨论在本文中，一般是不正确的。相反，所提出的不变容许PI以更精细的方式保证了容许性，而没有这样的假设。在特定的不变可容许更新规则下，从数学上证明了更新区域相对于相应策略的可容许性和不变性。在一定条件下，我们还证明了值函数序列的单调递减性和一致收敛性。最后，通过数值模拟验证了PI控制方法的有效性.
英文摘要：
In this paper, we propose a generalized successive approximation method (SAM), called invariantly admissible policy iteration (PI), for finding the solution to a class of input-affine nonlinear optimal control problems by iterations. Unlike the existing SAM, the proposed method updates the domain of the next policy and value function for admissibility (and invariance). In the existing SAM, the admissibility of the generated policies are guaranteed under the two implicit assumptions regarding Lyapunov's theorem and invariance, both of which are presented and discussed in this paper and are generally not true. On the contrary, the proposed invariantly admissible PI guarantees the admissibility in a more refined manner, without such assumptions. The admissibility and invariance of the updated region, with respect to the corresponding policies, are mathematically prove under the specific invariant admissible update rule. We also provide monotonic decreasing and uniform convergence properties of the sequence of value functions under certain conditions. Finally, numerical simulations are presented to illustrate the proposed PI method and its effectiveness.
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