论文标题
不规则LQ控制和标准LQ控制及其解决方案的差异和统一性
The Difference and Unity of Irregular LQ Control and Standard LQ Control and Its Solution
论文作者
论文摘要
自1970年代以来，不规则的线性二次控制（LQ，称为单数LQ）一直是一个长期存在的问题。本文将表明，当且仅当可以通过更改终端成本$ x'（t）hx（t）hx（t）$ x'（t）[h+p_1（t）x（t）x（t）x（t）x（t）$ p _1（t）$ p_1（t）时，不规则的LQ控制（确定性）是可以解决（用于任意初始值）的（用于任意初始值）的（用于任意初始值），而$ x'（t）$ x'（t）$ p_1（t）$ p_1（t）换句话说，不规则的控制器（如果存在）需要同时做两件事，一件事是最小化成本，另一件事是实现终端约束$ p_1（t）x（t）x（t）= 0 $，这阐明了与标准LQ控制器的不规则LQ的本质差异，以最大程度地降低成本。 通过这一突破，我们进一步研究了具有乘法噪声的随机系统的不规则LQ控制。基于Riccati方程式提供了足够的解决条件和最佳控制器。
Irregular linear quadratic control (LQ, was called Singular LQ) has been a long-standing problem since 1970s. This paper will show that an irregular LQ control (deterministic) is solvable (for arbitrary initial value) if and only if the LQ cost can be rewritten as a regular one by changing the terminal cost $x'(T)Hx(T)$ to $x'(T)[H+P_1(T)]x(T)$, while the optimal controller can achieve $P_1(T)x(T)=0$ at the same time. In other words, the irregular controller (if exists) needs to do two things at the same time, one thing is to minimize the cost and the other is to achieve the terminal constraint $P_1(T)x(T)=0$, which clarifies the essential difference of irregular LQ from the standard LQ control where the controller is to minimize the cost only. With this breakthrough, we further study the irregular LQ control for stochastic systems with multiplicative noise. A sufficient solving condition and the optimal controller is presented based on Riccati equations.
