Robust Control of Non-Minimum Phase Non-Linear Systems

在过去的几年里，新的强大的思想和概念已经为非线性反馈系统的全局分析和设计的一些系统方法的发展做出了贡献。这些研究工作的一个共同目标是在存在结构不确定性（如参数变化）和/或非结构不确定性（如未建模的动力学）的情况下，合成反馈控制器，产生渐近稳定性（具有规定的吸引力区域）和/或对选定的外源命令（干扰）的渐近跟踪（拒绝）。必须实现这些健壮设计方法的实际设置是，只有一组变量（而不是完整状态）可用于反馈。在这种情况下，整个设计问题当然变得更加苛刻，考虑到需要包括；反馈体系结构中的某种“状态观察者”。这个问题在跟踪的情况下尤其明显，当最现实的情况确实是只有“跟踪误差”（而不是整个参考轨迹）可用于控制目的时。对于处理特定输出映射选择的任何设计方案来说，执行上述反馈设计方法的可能性深受系统所谓“零动力学”的渐近特性的影响（当输入和初始状态约束输出等于零时，系统内部产生的动力学）。更准确地说，这些反馈设计方法的一致数量要求零动力学是渐近稳定的：具有这种性质的系统-类比于流行的术语线性系统称为“最小相位系统”。特别是，如果系统是最小相位，则有可能在扰动和输出之间实现任意小的衰减水平，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行为。由于能够鲁棒控制非最小相位非线性系统的重要性以及该领域可用结果的稀缺性，本研究将侧重于两个方向。第一部分研究存在未建模动力学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鲁棒性通常是通过观察被控系统对两个子系统的反馈互连进行处理的，其中只有一个子系统被精确建模，然后寻求一个控制律，尽可能地降低一些适当定义的互连建模/未建模组件的“增益”。在这种设置中，研究将解决寻找问题，对于各种类型的非最小相位系统，最低可实现“增益”的估计（以减少设计的保守性）。另一个方向是研究存在参数不确定性时的鲁棒跟踪问题。与线性系统的情况不同，在线性系统中，不稳定零动力学的存在通常不会阻碍规定轨迹族的渐近跟踪，而在非最小相位非线性系统的情况下，在存在大参数不确定性和任意大初始条件的情况下，没有可用的方法来实现渐近跟踪。这部分的研究是针对具有不稳定零动力学的非线性系统的半全局鲁棒跟踪的系统方法的发展。

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.