Selected Contributions on Design, Convergence, and Stability of Robust Nonlinear Controllers and Observers

The design and analysis of robust controllers and observers is a highly relevant topic in control and dynamical systems theory. The sliding mode approach, in particular, offers a variety of techniques and tools for the design of nonlinear robust control schemes. This habilitation thesis studies selected aspects of the design and analysis of sliding mode controllers and observers.

The thesis is divided into three main parts. The first part deals with design aspects. Therein, a design methodology for higher order sliding mode controllers is first shown. Starting from a nominal state-feedback controller, the methodology yields a sliding mode controller by adding a discontinuous integral term, thus essentially robustifying it. A notable feature of the approach is that certain perturbations are fully rejected while, at the same time, nominal performance is preserved in the unperturbed case. Next, a design methodology for differentiators is shown, i.e., for observers for integrator chains. The methodology allows to generate a whole family of differentiators. At the same time, it allows to prescribe their convergence time, i.e., the time it takes for the differentiator outputs to converge to the true derivatives. Finally, the design of a sliding mode controller in the presence of actuator constraints is discussed.

The second part deals with the estimation of convergence time of sliding mode control schemes. For the well-known super-twisting algorithm, which may be used as either a sliding mode controller or a sliding mode observer, the exact analytic computation of its convergence time for the unperturbed case is shown. For the perturbed case, the derivation of optimal upper bounds of the convergence time is discussed. Moreover, the convergence time of a nonlinear homogeneous state-feedback controller is estimated, which may also be used as a starting point for the higher order sliding mode controller design from the first part of the thesis.

The third part analyzes the stability of various sliding mode control schemes. Most prominently, a necessary and sufficient stability criterion for the super-twisting algorithm is presented. It allows to fully characterize the stability of all sliding mode controllers and observers that are based on it. Furthermore, the stability of some existing higher order sliding mode controllers is analyzed. Their stability or instability is proven, respectively, by means of sum-of-squares based Lyapunov functions or by counterexamples.