Approximate Optimal Control Design for a Class of Nonlinear Systems by Lifting Hamilton-Jacobi-Bellman Equation
Approximate Optimal Control Design for a Class of Nonlinear Systems by Lifting Hamilton-Jacobi-Bellman Equation
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一类非线性系统提升Hamilton-Jacobi-Bellman方程的近似最优控制设计
DOI：
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发表时间：
2020
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影响因子：
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通讯作者：
N. Motee
中科院分区：
文献类型：
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作者：
A. Amini;Qiyu Sun;N. Motee
研究了一类右端为解析函数的仿射非线性系统的最优控制设计问题。我们建立在Carleman线性化的思想基础上，Carleman线性化是一种将有限维非线性系统转换（提升）为无损失的无限维线性系统的非线性过程，并将Hamilton-Jacobi-Bellman（HJB）方程提升为类似于熟悉的代数Riccati方程的无限维二次型。然后，我们提出了一个有效的方法来计算所得到的无限维方程的解，使用一个代数Riccati方程（与原始非线性系统相同的维度）和一系列的线性矩阵方程的迭代方式。可以得到任意的近似最优解使用有限截断。结果表明，所得到的近似解是对称的。利用HJB方程的这些近似解，我们构造了近似最优控制律。我们的仿真结果表明，我们的方法具有较高的精度相比，实际的最优反馈控制律和精度增加高阶截断使用。
We consider optimal control design of a class of affine nonlinear systems whose right-hand sides are analytic functions. We build upon ideas from Carleman linearization, which is a nonlinear procedure to transform (lift) a finite-dimensional nonlinear system into an infinite-dimensional linear system with no loss, and lift a Hamilton-Jacobi-Bellman (HJB) equation into an infinite-dimensional quadratic form that resembles the familiar algebraic Riccati equation. Then, we propose an efficient method to calculate solution of the resulting infinite-dimensional equation using one algebraic Riccati equation (of the same dimension as the original nonlinear system) and a series of linear matrix equations in an iterative manner. One can obtain arbitrarily near-optimal solution using finite truncations. It is shown that the resulting approximate solutions are symmetric. Using these approximate solutions to the HJB equation, we construct approximate optimal control laws. Our simulation results assert that our method enjoys high accuracy in compared to the actual optimal feedback control laws and the accuracy increases as higher-order truncations are used.