Robust Control of Non-Minimum Phase Non-Linear Systems

ECS-9707891 Isidori在过去的几年里，新的强大的思想和概念促进了一些用于全局分析和设计非线性反馈系统的系统方法的发展。这些研究工作的一个共同目标是综合反馈控制器，在存在结构性不确定性(如参数变化)和/或非结构化不确定性(如具有未建模动态)的情况下，对选定的外部命令(扰动)产生渐近稳定性(具有指定的吸引域)和/或渐近跟踪(拒绝)。必须实施这些稳健设计方法的实际设置是只有一组变量(而不是整个状态)可用于反馈。在这种情况下，考虑到在反馈体系结构中包括某种类型的“状态观测器”的需要，整个设计问题当然变得更加苛刻。这一问题在跟踪方面尤其明显，因为最现实的情况确实是只有“跟踪误差”(而不是整个参考弹道)可用于控制目的。正如任何涉及选择特定输出映射的设计方案所发生的那样，实施上述反馈设计方法的可能性深深地受到系统所谓的“零动态”(当输入和初始状态约束输出为零时系统内部出现的动态)的渐近性质的影响。更准确地说，这些反馈设计方法的一致数量要求零动态是渐近稳定的：具有这种性质的系统--类似于线性系统中流行的一个术语--“最小相位系统”。具体地说，如果系统是最小相位，则可以在干扰和输出之间实现任意小的衰减，这对解决未建模动态情况下的鲁棒镇定问题的能力具有明显的积极影响。然而，许多物理系统表现出非最小相位行为。考虑到能够同时对非最小相位非线性系统进行鲁棒控制的重要性，以及该领域现有成果的稀缺性，本研究将集中在两个方向上。第一个问题研究了存在未建模动态的非最小相位非线性系统的鲁棒镇定问题。关于这种不确定性的稳健性通常是通过观察受控系统关于两个子系统的反馈互连来处理的，其中只有一个子系统被准确建模，然后寻求一种控制律，该控制律尽可能地降低互连建模/未建模组件的某些适当定义的“增益”。在这个设置中，研究将解决为各种类型的非最小相位系统寻找可实现的最低“增益”的估计的问题(从而降低设计的保守性)。另一个方向是研究参数不确定条件下的鲁棒跟踪问题。与线性系统不同的是，不稳定零动态的存在通常不会阻碍对指定轨迹族的渐近跟踪，而对于非最小相位非线性系统，在存在大参数不确定性和任意大的初始条件的情况下，还没有方法可以实现渐近跟踪。这部分研究的目的是开发系统方法，对具有不稳定零动态的非线性系统进行半全局鲁棒跟踪。

ECS-9707891 Isidori In the last few years, new powerful ideas and concepts have contributed to the development of a number of systematic methods for global analysis and design of nonlinear feedback systems. A common goal of these research efforts has been the synthesis of feedback controllers yielding asymptotic stability (with prescribed region of attraction) and/or asymptotic tracking (rejection) of selected exogenous commands (disturbances), in the presence of structured uncertainties, such as parameter variations, and/or unstructured uncertainties, such has unmodeled dynamics. The actual setup in which these robust design methods have to be implemented is that in which only a set of variables (and not the full state) is available for feedback. In this case the entire design problem becomes of course more demanding, in view of the need of including; some kind of "state observer" in the feedback architecture. This problem is particularly felt in the case of tracking, when the most realistic situation is indeed the one in which only the "tracking error" (and not the entire reference trajectory) would be available for control purposes. As it happens for any design scheme dealing with the selection of specific output maps, the possibility of implementing the above-mentioned methods for feedback design is deeply influenced by the asymptotic properties of the so-called "zero dynamics" of the system (the dynamics which arise internally in a system when input and initial state are such as to constrain the output to be identically zero). More precisely, a consistent number of these methods for feedback design require the zero dynamics to be asymptotically stable: systems having this property are - in analogy with a terminology en vogue for linear systems called "minimum phase systems". In particular, if a system is minimum phase it is possible to achieve an arbitrarily small level of attenuation between disturbances and output, with obvious positive consequences on the ability of solving pr oblems of robust stabilization in the case of unmodeled dynamics. However, many physical systems exhibit a non-minimum phase behavior. Motivated by the importance of being able to robustly control also non-minimum phase nonlinear systems and by the scarcity of available results in this area, this research will focus on two directions. The first one deals with problems of robust stabilization of non-minimum phase nonlinear systems, in the presence of unmodeled dynamics. Robustness with respect to this kind of uncertainty is usually dealt with by looking at the controlled system as to the feedback interconnection of two subsystems, only one of which is accurately modeled, and then seeking a control law which lowers, as much as possible, some suitably defined "gain" of the interconnection modeled/unmodeled component. In this setup, the research will address the problem of finding, for various classes of non-minimum phase systems, estimates of the lowest achievable "gains" (so as to reduce the conservativeness of the design). The other direction is to study problems of robust tracking in the presence of parametric uncertainties. Unlike the case of linear systems, where the presence of unstable zero dynamics is generally not an obstruction to asymptotic tracking of prescribed families of trajectories, in the case of non-minimum phase nonlinear system no method is available yet to achieve asymptotic tracking, in the presence of large parameter uncertainties and for arbitrarily large initial conditions. This part of the proposed research is directed toward the development of systematic methods yielding semiglobal robust tracking for nonlinear systems having unstable zero dynamics.