论文标题
具有性能边界的价值函数和非线性控制器设计的多项式近似
Polynomial Approximation of Value Functions and Nonlinear Controller Design with Performance Bounds
论文作者
论文摘要
对于任何合适的最佳控制问题（OCP），都存在一个值函数，该值函数被定义为汉密尔顿 - 雅各比 - 贝尔曼（HJB）局部局部 - 差异方程式（PDE）的唯一粘度解决方案，可以用来设计给定OCP的最佳反馈控制器。在本文中，我们通过提出一系列平方符号（SOS）问题来大致解决HJB-PDE，每个问题总和为HJB-PDE产生多项式订阅。我们表明，多项式子溶液的产生序列会收敛到L1规范中OCP的值函数。此外，对于此序列中的每个多项式子溶液，我们表明，级别集合的相关序列会收敛到体积度量中OCP的值函数的级别集合。接下来，对于从SOS程序或任何其他方法获得的任何近似值函数（例如离散化），我们构建了一个关联的反馈控制器，并表明该控制器应用于OCP的子次数受OCP在Sobolev Norm中的OCP的近似值和真实值函数之间的距离。最后，我们以数字方式证明，通过解决提出的SOS问题，我们能够准确地近似值功能，设计控制器和估计可及的集合。
For any suitable Optimal Control Problem (OCP) there exists a value function, defined as the unique viscosity solution to the Hamilton-Jacobi-Bellman (HJB) Partial-Differential-Equation (PDE), and which can be used to design an optimal feedback controller for the given OCP. In this paper, we approximately solve the HJB-PDE by proposing a sequence of Sum-Of-Squares (SOS) problems, each of which yields a polynomial subsolution to the HJB-PDE. We show that the resulting sequence of polynomial sub-solutions converges to the value function of the OCP in the L1 norm. Furthermore, for each polynomial sub-solution in this sequence, we show that the associated sequence of sublevel sets converge to the sublevel set of the value function of the OCP in the volume metric. Next, for any approximate value function, obtained from an SOS program or any other method (e.g. discretization), we construct an associated feedback controller, and show that sub-optimality of this controller as applied to the OCP is bounded by the distance between the approximate and true value function of the OCP in the Sobolev norm. Finally, we demonstrate numerically that by solving our proposed SOS problem we are able to accurately approximate value functions, design controllers and estimate reachable sets.
