Analysis Tools and Control Algorithms for Nonlinear Dynamical Systems

9988813 Teel虽然线性、时不变、有限维系统的控制设计领域已经非常成熟，但关于非线性、时变（甚至有限维）系统的控制仍有许多有待发现。由于非线性在工程系统中无处不在，因此缺乏对这一主题的了解减缓了技术进步并不罕见。值得注意的是，例如，在过去的三年中，现代抗饱和控制合成最近需要解决的工业，高性能的振动衰减控制问题，其中致动器饱和是不可忽略的。本研究的目的是贡献额外的稳定性分析工具和控制算法的非线性连续时间动态系统。这些工具和算法将是关键的工程意义的问题。为了一般性，系统建模为微分和前移夹杂物将被强调。这类模型能够解决依赖于不连续控制的系统和那些实施控制算法与离散或混合dynamics.In这项工作中，李雅普诺夫型集渐近稳定的特征将被强调。该项目的一些具体目标是：1）对最近关于集合强渐近稳定性的匡威李雅普诺夫定理提供重要的扩展; 2）修改这些匡威李雅普诺夫函数的构造，使它们适用于集合的弱渐近稳定性，并生成局部Lipschitz弱李雅普诺夫函数; 3）证明了这些局部Lipschitz弱李雅普诺夫函数如何产生渐近可控非线性系统的局部Lipschitz控制李雅普诺夫函数; 4）展示了如何与匡威李雅普诺夫函数构造相关联的优化问题产生有意义的滚动时域控制策略; 5）分析了这些滚动时域控制策略在使用离散时间模型近似时的有效性; 6）提供了一致全局渐近稳定性的新的积分特征; 7）使用这些新的特征来产生新的非线性控制算法。这些目标补充了主要研究者的其他工作，旨在将最近开发的非线性控制理论转化为工业。此外，这些目标是一致的，使非线性分析和控制设计的工具更强大，更简洁，更好地理解和更广泛地适用于今天的控制工程问题的更广泛的目标。 ***

9988813TeelWhile the field of control design for linear, time-invariant, finite dimensional systems is very mature, there is still a lot remaining to be discovered about control of nonlinear, time-varying, (even finite dimensional) systems. Since nonlinearities are ubiquitous in engineering systems, it is not uncommon to see the lack of knowledge about this topic slowing down technological advances. It is noteworthy that, for example, modern anti-windup control synthesis developed in the last three years was recently needed to solve an industrial, high-performance vibrational attenuation control problem where actuator saturation was non-negligible.The objective of this research is to contribute additional stability analysis tools and control algorithms for nonlinear continuous-time dynamical systems. These tools and algorithms will be keyed to problems of engineering significance. For the sake of generality, systems that are modeled as differential and forward-shift inclusions will be stressed. This class of models is able to address systems that rely on discontinuous controls and those that implement control algorithms with discrete or hybrid dynamics.In this work Lyapunov-type characterizations of set asymptotic stability will be emphasized. Some of the specific goals of the project are to 1) provide important extensions to the most recent converse Lyapunov theorems on strong asymptotic stability of sets; 2) modify these converse Lyapunov function constructions so that they apply to weak asymptotic stability of sets and generate locally Lipschitz weak Lyapunov functions; 3) show how these locally Lipschitz weak Lyapunov functions produce locally Lipschitz control Lyapunov functions for asymptotically controllable nonlinear systems; 4) show how the optimization problem associated with the converse Lyapunov function construction yields a meaningful receding horizon control strategy; 5) analyze the efficacy of these receding horizon control strategies when using discrete-time model approximations; 6) provide new, integral characterizations of uniform global asymptotic stability; 7) use these new characterizations to generate new nonlinear control algorithms.These objectives complement other work of the principal investigator that is aimed at transitioning recently developed nonlinear control theories to industry. Moreover, the objectives are consistent with the broader goal of making the tools of nonlinear analysis and control design more powerful, more concise, better understood and more widely applicable to today's control engineering problems. ***