Robust Stability for Nonlinear Control: Analysis and Synthesis

鲁棒性是现代控制理论的关键概念：它涉及控制系统不仅对被控制系统的数学模型起作用的能力，而且还涉及在真实系统总是偏离模型的“现实世界”中工作的能力。对于线性系统，控制工程师已经开发了重要的方法来解决这个问题（例如h∞控制）。在过去的十年中，非线性系统鲁棒控制理论的发展也取得了实质性进展，但这一主题还处于起步阶段。在这里，我们考虑的是基于1997年乔治欧和史密斯提出的“非线性间隙度量”的投入产出理论。现有的非线性输入输出理论有一个明显的局限性，那就是它只直接适用于从静止开始的系统。本建议的核心部分是将这一理论推广到非从平衡位置开始的系统的情况。这种方法可能部分涉及到Sontag（处理非零初始条件，但鲁棒性不是框架的明确部分）与输入输出理论（鲁棒性是中心问题，但不处理非零初始条件）的非线性“状态空间”的统一。然后将研究基础理论的两个实质性应用。非线性分离原理。这些原理描述了基于状态信息的控制器可以被仅基于测量输出反馈的控制器所取代的条件。在这里，我们正在寻求使用输入输出工具来大大扩展文献中的现有结果；明确地处理对系统输入和输出的未建模动态和干扰的鲁棒性。非线性复杂性降低。在非线性控制理论中发展的许多控制设计的一个实质性障碍是，对于中等阶的系统，设计结构的递归性质意味着控制器的显式公式在项的数量上有一个非常快速的增长（随着系统的阶数）。这意味着对于中等秩序的系统来说，这些表达式太大，太笨拙。复杂性降低旨在以系统的方式降低这些表达式的复杂性，同时保持对控制系统的鲁棒性和性能的控制。问题1。和2。都很重要，我们将在相关文献中引入一个全新的工具库。同样重要的是，这些应用程序将用于说明核心输入-输出方法的强大功能。

Robustness is the key concept for modern control theory: it concerns the ability of a control system to work not only for the mathematical model of the system to be controlled, but also to work in the `real world' where the true system always deviates from the model. For linear systems, control engineers have developed significant approaches to addresses this issue (for example H-infinity control).In the past decade substantial progress in the development of a robust control theory for nonlinear systems has also been made, but this subject is very much in its infancy. Here we consider the input-output theory based on the 1997 development of the `nonlinear gap metric' due to Georgiou and Smith. A notable limitation this existing nonlinear input output theory is that it only directly applies to systems which start at rest. The core part of this proposal is to undertake the substantial generalisation of this theory to the case of systems which do not start at equilibrium positions. This approach may in part involve a unification of the nonlinear 'state-space' due to Sontag (which handles non-zero initial conditions, but robustness is not an explicit part of the framework) with the input-output theory (in which robustness is the central issue, but non zero initial conditions are not handled).Two substantial applications of the underlying theory will then be investigated.1. Nonlinear seperation principles. Such principles describe conditions under which a controller based on information about the state can be replaced by a controller based only on the feedback of the measured output. Here we are seeking to use the input-output tools to substantially extend the existing results in the literature; to explicitly handle robustness to unmodelled dynamics and disturbances at both the input and the output of the system.2. Nonlinear complexity reduction. A substantial obstacle to many of the control designs developed in nonlinear control theory, is that for systems of moderate order, the recursive nature of the design constructions means that the explicit formulae for the controller have a very rapid growth (with the order of the system) in the number of terms. This means the expressions are too large and unwieldy for practical purposes for systems of moderate order. Complexity reduction aims to reduce the complexity of these expressions in a systematic way, whilst keeping control of the consequent altered robustness and performance of the control system.Problems 1. and 2. are important in their own right, and we will be introducing a completely new armoury of tools into the relevant literature. Importantly also, these applications will serve to illustrate the power of the core input-output approach.