Design and Discretization of Adaptive Sliding-Mode Controllers

滑模控制（SMC）是一种具有良好鲁棒性的成熟控制方法。SMC的一个优点是它衰减非结构化不确定性的能力，除了上界之外不需要任何信息。缺点是，许多SMC方法不能处理显示显式状态依赖的不确定性。互补的，现代自适应控制器能够处理结构不确定性，保证李雅普诺夫稳定性，假设参数变化缓慢。然而，获得良好的性能和瞬态行为可能是困难的。我们建议将这两种方法结合起来，利用它们各自的优势来提高控制系统的整体鲁棒性和性能，同时保持执行器的动作适度低。这是通过使用间接自适应控制器实现的，该控制器通过利用最大可用信息对结构化不确定性产生“估计”。为了获得最佳的鲁棒性，我们对剩余的非结构化不确定性使用SMC方法。这样，大部分控制信号被给予自适应部分，而控制器的smc部分的控制作用较少。基于确定性等价原理，我们探索了自适应SMC方法的新组合，并分析了它们的优点和要求。用于高阶SMC方法的Lyapunov函数将产生全新的自适应控制律，使闭环系统在面对大量不确定性时具有鲁棒稳定性。为了实现这些控制方案，我们将把它们转换成离散时间版本。然而，SMC的数字化实现可能会导致系统变量出现不理想的振荡。这种振荡，通常被称为离散抖振，可能导致机械部件过度磨损和控制精度降低。这些抖振效应的幅值和频率特性在很大程度上取决于控制算法的性质和所采用的离散化方案。所提出的结构化和非结构化部分的不确定性分离为减少不连续控制增益提供了很大的潜力。因此，也可以减少离散抖振。由于自适应律部分继承了标称SMC的连续性，因此可能会产生额外的离散抖振源。因此，我们将深入分析基于滑模的自适应控制器的离散化过程和数字化实现。分析的目的是找出这些控制器的哪些特性在离散化后会占上风。在这方面，特别关注闭环系统的稳定性和所达到的控制精度。我们的方法将应用于最先进的纳米定位阶段。该系统的模型结构显示了特定的不确定性，适合于评估所提出的概念并与现有方法进行比较。

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.