随机H∞预演控制问题研究

The deterministic H∞ preview control and fixed-lag smoothing are stated as Open Problem 51 in Open Problems in Mathematical Systems and Control Theory. The preview control is a method which makes use of the information available in advance to design the controller so as to achieve the desired performance. Compared with the traditional methods, it always leads to better results and is suitable for vehicle suspension control, unmanned vehicles, etc. Therefore, the preview control problem has significantly theoretical and practical value. The project will consider stochastic H∞ preview control(SHPC) and its applications. It aims to provide the sufficient and necessary condition (under which the SHPC is solvable) and the adapted H∞ preview controller based on the stochastic Riccati equation with the same dimension as the system (ignoring the delay). Due to that the preview control falls into the category of the control problems with delays, the project will start with addressing the indefinite linear quadratic regulator (LQR) problem for the stochastic systems with input-delays. With the aid of the link existing between the H∞ control and twice optimization, stochastic analysis, calculus of variations and game theory, and the combination of the quantitative derivation in low-dimensional space and the qualitative analysis in the infinite-dimensional abstract space, we will investigate the relationships between the SHPC, the FBSDEs and the stochastic Riccati equation and explore the essential distinctions between the stochastic and deterministic control for systems with delays. Eventually, the project expect to propose a new idea solving the control problems for the stochastic system with delays effectively.

确定系统H∞预演控制与固定滞后平滑被合列为《数学系统与控制理论的公开问题》的第51个公开问题。预演控制(PC)是指利用可提前获取的信息来设计控制器从而达到一定目的的控制方法。与其它方法相比，PC能达到更好的控制效果，适用于汽车悬架系统、无人驾驶载器等工业、国防领域。因而,PC问题兼具理论及应用价值。本项目拟针对随机系统，研究H∞PC及其应用问题，旨在基于与原系统同维的随机Riccati 方程的解，给出问题可解的充要条件及适应控制器。由于PC属于时滞控制问题的研究范畴，本项目将从研究时滞随机不定LQR问题着手，借助H∞控制与两次优化问题之间的关系，应用随机分析、变分法、对策论等理论和低维定量研究与抽象定性分析相结合的思路，考察问题与正倒向随机系统可解性和随机Riccati 方程可解性间的关系，揭示随机与确定系统时滞控制问题的本质区别，探索出一条能够有效解决时滞随机控制问题的道路。

本项目主要研究了具有输入时滞的随机系统控制问题。首先研究了具有输入时滞的随机LQR问题，提出了一种低维定量分析与抽象定性研究相结合的新思路，建立了新的离散随机系统极大值原理和抽象随机LQR可解的充要条件，给出了问题可解的充要条件与适应控制器；其次研究了随机H∞预演控制问题，借助H∞控制与两次优化问题之间的关系，应用随机分析、变分法、对策论等理论和低维定量研究与抽象定性分析相结合的思路，给出了H∞预演控制问题可解的条件与预演控制器；最后，研究了前述控制器在实际执行时可能遇到的状态不可测、量化等问题，得到了相应的解决方案。