Analysis Tools and Control Algorithms for Nonlinear Dynamical Systems

9988813尽管线性、时不变、有限维系统的控制设计领域已经非常成熟，但非线性、时不变、（甚至是有限维）系统的控制仍有许多有待发现的地方。由于非线性在工程系统中无处不在，因此对该主题缺乏了解会减缓技术进步的情况并不罕见。值得注意的是，例如，在过去三年中开发的现代抗上卷控制综合技术最近需要解决执行器饱和不可忽略的工业高性能振动衰减控制问题。本研究的目的是为非线性连续时间动力系统提供额外的稳定性分析工具和控制算法。这些工具和算法将是解决工程问题的关键。为了通用性，将强调作为差分和前移包涵体建模的系统。这类模型能够解决依赖于不连续控制的系统和那些使用离散或混合动态实现控制算法的系统。本文将着重讨论集渐近稳定性的lyapunov型特征。该项目的一些具体目标是：1)对最近关于集合的强渐近稳定性的逆Lyapunov定理提供重要的扩展；2)修正这些逆Lyapunov函数结构，使它们适用于集合的弱渐近稳定性，并生成局部Lipschitz弱Lyapunov函数；3)证明渐近可控非线性系统的局部Lipschitz弱Lyapunov函数如何产生局部Lipschitz控制Lyapunov函数；4)说明与逆Lyapunov函数构造相关的优化问题如何产生有意义的后退地平线控制策略；5)用离散时间模型逼近分析了这些水平渐退控制策略的效果；6)给出一致全局渐近稳定性的新的、积分的刻画；7)利用这些新的特征生成新的非线性控制算法。这些目标补充了首席研究员的其他工作，旨在将最近发展的非线性控制理论转化为工业。此外，这些目标与更广泛的目标是一致的，即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当今的控制工程问题。＊＊＊

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. ***