Stability of Switched Dynamical Systems

控制的目的是影响动态系统的行为。在不同的环境条件下实现快速、精确的控制，甚至在保证稳定性和鲁棒性的同时，是所有控制系统设计的目标。当环境变化很大时，（例如故障或子系统故障）该设计可能需要使用多个控制器，每一个都适合于特定的环境。切换系统是指具有协调不同控制器之间切换的决策机制的混合动力系统。随着计算机技术的巨大进步，以及伴随而来的动力系统复杂性的增加，开关系统正变得越来越普遍。因此，这类系统的分析和综合就显得非常重要。稳定性是任何控制系统的基本要求。一旦这一点得到保证，整个系统的性能就可以在稳定性的范围内进行优化。其次，控制律中的修改可以是不确定的。本文提出了一种新的控制系统设计方法，使系统在不同的扰动下具有鲁棒性，这是传统控制系统设计和本方案中所采用的基本原理。开关系统已经发展了很多年，在该领域做出了许多贡献，但基本问题仍然没有得到解答。例如，许多类型的线性、非线性和自适应开关系统的稳定性仍然没有得到解答，而在形成由开关动作引起的瞬态响应方面进展甚微。这两个重要问题都包括在提案中。提案由于稳定无疑是最重要的悬而未决的问题，本文的第一部分讨论了一般线性定常系统与正线性系统之间的切换问题。特别注意具有对角李雅普诺夫函数的系统，因为它们对分散系统中的切换很重要。还讨论了实际上很重要的分段线性系统，理论上是容易处理的。接着，在建议的第二部分中，考虑了通过切换可以实现的性能。建议的第三部分和最后一部分致力于自适应系统中的切换。智力优点：PI认为，拟议的研究解决了基金会-这些问题的解决方案对于更好地理解开关系统至关重要。这反过来又对未来控制系统的设计至关重要。更广泛的影响：该计划将加强PI与其他国际研究人员（特别是爱尔兰汉密尔顿研究所的Robert Shorten博士和戴姆勒-克莱斯勒研究所的Jens Kalkuhl博士）之间的合作，增加与工业界的互动，最重要的是，为研究生和本科生提供先进控制技术的培训。

The objective of control is to in .uence the behavior of dynamical systems.Achiev-ing fast and accurate control under di .erent environmental conditions,even whileassuring stability and robustness,is the aim of all control system design.Whenenvironmental changes are signi .cant (e.g.faults o subsystem failures)the designmay equire the use of multiple controllers,each of which is appropriate in a speci .ccontext.By a switching system we mean a hybrid dynamical system with a decisionmechanism which orchestrates the switching between di .erent controllers.With greatadvances in computer technology,and the concomitant increase in the complexity ofdynamical systems,switching system are becoming ubiquitous.The analysis andsynthesis of such systems is consequently assuming major importance.Stability is the primary equirement of any control system.Once this is assured,optimization of the performance of the overall system can be undertaken within aframework of stability.Following this,modi .cations in the control laws can be un-dertaken to make the system robust under di .erent perturbations.This is the basicphilosophy used in conventional control system design as well as in this proposal.Even though research on switching systems has been in progress for many years andnumerous contributions have been made to the .eld,basic questions still remain unan-swered.For example the stability of many classes of linear,nonlinear,and adaptiveswitching systems still remains to be answered,and very little progress has been madein shaping the transient response resulting from the switching action.Both of theseimportant issues are included in the proposal.The proposal consists of three parts.Since stability is undoubtedly the most impor-tant unresolved issue,the .rst part of the proposal is devoted to it.Switching betweengeneral linear time-invariant systems and positive linear systems is treated in this con-text.Special attention is given to systems with diagonal Lyapunov functions in viewof their importance to switching in decentralized systems.Also treated are piecewiselinear systems which are practically important and theoretically tractable.Followingthis,in the second part of the proposal,the performance that can be achieved byswitching is considered.The third and last part of the proposal is devoted to switch-ing in adaptive systems.Intellectual Merit: The PI believes that the research proposed addresses funda-mental questions whose solutions are essential for a better understanding of switchingsystems.This in turn is vital for the design of future control systems.Broader Impacts: The esearch proposed will enhance collaboration between thePI and other international esearchers (particularly Dr.Robert Shorten at the Hamil-ton Institute,Ireland and Dr.Jens Kalkuhl of Daimler-Chrysler Research),lead toincreased interaction with industry,and most important of all,provide training toboth graduate and undergraduate students on advanced control techniques.1