线性约束反馈综合的子流形上的策略优化
Policy Optimization over Submanifolds for Linearly Constrained Feedback Synthesis
AI总结：
本文研究了Schur稳定控制器流形上的线性约束策略优化问题，提出了一种不依赖指数映射或回缩的Newton型算法，并证明了其局部收敛性，最后通过数值实验验证了算法性能。
摘要：
本文研究在Schur稳定控制器流形上的线性约束策略优化问题，该流形配备了在最优控制问题背景下自然产生的黎曼度量。我们对一般的约束光滑代价函数进行了外在分析，从而便于将任何此类约束问题纳入该框架。通过研究该流形的二阶几何，我们提出了一种Newton型算法，该算法不依赖指数映射或回缩，同时确保了局部收敛性保证。相反，该算法依赖于所开发的稳定性证书和约束的线性结构。随后，我们将该方法应用于两个著名的约束最优控制问题。最后，几个数值示例展示了所提出算法的性能。
英文摘要：
In this paper, we study linearly constrained policy optimization over the manifold of Schur stabilizing controllers, equipped with a Riemannian metric that emerges naturally in the context of optimal control problems. We provide extrinsic analysis of a generic constrained smooth cost function, that subsequently facilitates subsuming any such constrained problem into this framework. By studying the second order geometry of this manifold, we provide a Newton-type algorithm that does not rely on the exponential mapping nor a retraction, while ensuring local convergence guarantees. The algorithm hinges instead upon the developed stability certificate and the linear structure of the constraints. We then apply our methodology to two well-known constrained optimal control problems. Finally, several numerical examples showcase the performance of the proposed algorithm.
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