FWF - DasProjekt - Entwurf und Diskretisierung Adaptiver Sliding-Mode-Regler

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.