不确定负虚系统的鲁棒控制及其在柔性结构振动抑制中的应用

Passivity has precise physical meaning (i.e. energy dissipating), thus it plays an important role in robust control. However, the relative degree of passive systems can not be over one, this fact limits the development and applications of passivity theory to a great extent. In this project, an uncertain class of linear time invariant systems with maximum relative degree of two, namely, negative-imaginary systems, is investigated and the relevant robust control framework will be provided, and hence this study will complement and extend the existing passivity theory and its applications. Based on the state-space characterization and the stability theorem of negative-imaginary systems, robust stabilization a class of systems with negative-imaginary uncertainties will be studied, conditions of the existence of robust stabilizing controllers will be given, and several robust stabilization methods will be studied. Controller synthesis methods will be studied to stabilize an uncertain negative-imaginary system and at the same time to satisfy an H-infinity or H2 performance of the nominal closed-loop system. Experimental applicability studies will also be undertaken on a cantilever beam test bed with piezoelectric collocated sensor/actuator pairs and to improve the performance of vibration control. Easier controller design methods for this flexible structure will be studied as well. This project, will help tackle many physical problems that can not be addressed well by the use of small-gain theorem or passivity theory, for example, the vibration suppression of flexible structures, and produce design paradigms that are easier and less conservative to use in practice; moreover, it will facilitate the development of a class of nonlinear systems with counterclockwise dynamics. Thus this project has significant theoretical meaning and practical value.

无源性因其明确的物理意义（即能量耗散），为鲁棒控制提供了重要的理论依据。而无源系统的相对阶次不能超过1，这一限制条件很大程度上制约了无源理论的发展。本课题研究一类最高阶次可达到2的线性时不变不确定负虚系统的鲁棒控制问题，将会补充并拓展现有的无源理论和应用领域。拟利用负虚系统的判定条件和稳定性定理，研究一类含有负虚不确定性的系统的鲁棒镇定控制器存在条件和设计方法。对一类不确定负虚系统，研究设计鲁棒控制器使标称闭环系统具有H无穷或H2性能的方法。搭建压电传感器和执行器同位布置的悬臂梁平台进行验证，并对这类柔性结构的振动抑制问题，应用负虚系统鲁棒控制理论，研究更便捷的控制方法。这一课题的完成，将为一类基于小增益定理和无源理论无法很好解决的控制问题，如柔性结构的振动抑制，提供理论依据和更便捷且保守性更小的设计方法，而且有利于推动具有逆时针动态的非线性系统的研究，因此具有重要的理论意义和工程价值。

负虚系统具有广泛的工程应用，如柔性结构振动控制、大型车辆编队和纳米高精度定位控制等。本课题提出了不确定负虚系统的两种鲁棒控制方法，研究设计鲁棒控制器使整个系统鲁棒稳定并且标称闭环系统满足某种性能要求，比如极点配置或者H2性能，搭建具有积分环节的负虚系统控制与分析的仿真验证平台以验证理论算法，解决带积分环节的负虚系统的鲁棒控制理论向应用转化的一些关键性技术问题。这一课题的完成, 将在理论上补充和拓展现有的无源控制理论结果，工程上为一类基于小增益定理和无源理论无法很好解决的控制问题，如带有刚体模态的柔性结构的振动抑制，提供理论依据和更便捷且保守性更小的设计方法。