Design and Discretization of Adaptive Sliding-Mode Controllers

滑模控制（SMC）是一种成熟的方法，具有突出的鲁棒性。SMC的一个据点是它的能力，以削弱非结构化的不确定性，不需要信息以外的上限。缺点是，许多SMC方法不能处理显示出显式状态依赖性的不确定性。互补的，现代自适应控制器能够处理结构化的不确定性，保证李雅普诺夫稳定性，参数变化缓慢。然而，获得良好的性能和瞬态behaviors.We建议联合收割机这两种方法相结合，并利用各自的优势，有利于整体的鲁棒性和性能的控制系统，同时保持执行器的行动适度低。这是通过使用间接自适应控制器，产生的“估计”的结构不确定性，利用最大的可用信息。为了尽可能好的鲁棒性，我们使用SMC方法对剩余的非结构化的不确定性。这样，控制信号的大部分被提供给自适应部分，而较少的控制动作留给控制器的SMC部分。基于确定性等价原理，我们探索了自适应SMC方法的新组合，并研究了它们的优点和要求。高阶滑模控制方法的李雅普诺夫函数将产生全新的自适应控制律，使闭环系统对大类不确定性鲁棒稳定。为了实现这些控制方案，我们将它们转化为离散时间版本。然而，SMC的数字实现可能导致系统变量的不期望的振荡。这种振荡，通常被称为离散抖动，可能会导致机械部件的过度磨损和低控制精度。这些抖振效应的幅值和频率特性强烈地依赖于控制算法和所应用的离散化方案的性质。所提出的分离结构和非结构部分的不确定性提供了一个很大的潜力，以减少不连续的控制增益。因此，也可以减少离散化抖动。由于标称SMC的连续性特性部分继承自适应法，离散抖振的额外来源可能是后果。因此，我们将深入分析离散化过程和数字实现的滑模自适应控制器。分析的目标是找出这些控制器的哪些属性将在离散化后占上风。在这方面，特别注意保持闭环系统的稳定性和实现的控制精度。我们的方法将被应用到一个国家的最先进的纳米定位阶段。该系统的模型结构显示出特定的不确定性，这是适当的评估所提出的概念和现有的方法进行比较。

Sliding-mode control (SMC) is an established approach with outstanding robustness properties. A stronghold of SMC is its capacity to attenuate unstructured uncertainties, requiring no information other than an upper bound. On the downside, many SMC approaches cannot handle uncertainties that show explicit state-dependencies. Complementary, modern adaptive controllers are capable of dealing with structured uncertainties guaranteeing Lyapunov-stability, given that parameter variations are slow. Yet, obtaining good performance and transient behavior may be difficult.We propose to combine both methods and exploit their individual strengths to the benefit of the overall robustness and performance of the control system while keeping actuator action moderately low. This is achieved by using indirect adaptive controllers that yield an “estimate” for the structured uncertainty by exploiting the maximum available information. For best possible robustness we use SMC approaches on the remaining unstructured uncertainty. This way, large portions of the control signal are given to the adaptive part leaving less control action to the SMC-part of the controller. Based on the certainty equivalence principle, we explore new combinations of adaptive SMC approaches and examine their benefits and requirements. Lyapunov functions for higher-order SMC approaches will yield entirely novel adaptive control laws that render the closed-loop system robustly stable against a large class of uncertainties.For implementing these control-schemes we will transform them into discrete-time versions. Digital realization of SMC, however, may lead to undesirable oscillations in the system variables. Such oscillations, often termed discretization chattering, may cause excessive wear of mechanical components and low control accuracy. The amplitude and frequency characteristics of these chattering effects strongly depend on the properties of the control algorithm and the applied discretization scheme. The proposed separation of uncertainty in structured and unstructured parts offers a high potential to reduce the discontinuous control gains. Thus, also discretization chattering may be reduced. Since continuity properties of the nominal SMC are partly inherited to the adaptation law, additional sources of discretization chattering may be the consequence. Therefore, we will thoroughly analyze the discretization process and the digital realization of sliding-mode based adaptive controllers. The goal of the analysis is to find out which properties of these controllers will prevail after discretization. In this regard, special attention is kept on the stability properties of the closed-loop system and the achieved control accuracy.Our methods will be applied to a state-of-the-art nano-positioning stage. The model structure of this system shows specific uncertainties which are appropriate for the assessment of the proposed concepts and comparison with existing methods.