Analysis and Synthesis of Discrete-Time Sliding Mode Controllers and Observers

Research output: ThesisDoctoral Thesis


Feedback loops designed using the ideas of sliding mode control are known to exhibit a number of appealing features. A very prominent characteristic is their insensitivity against bounded matched disturbances and model uncertainties. However, it is well-known that improper discrete-time realizations of sliding mode based algorithms cause so-called discretization chattering, i.e., undesired oscillations in the control signal. These oscillations typically deteriorate the closed-loop performance or even cause damages.

This thesis deals with the analysis and development of tools for the characterization of chattering effects as well as with the development of discretization schemes that entirely avoid this effect. The theoretical results are supported by simulations and experiments.

For the characterization of chattering effects, i.e., the determination of frequency and amplitude of oscillations, frequency domain techniques such as the describing function method and the locus of perturbed relay system approach and their extension to the sampled data configuration are studied. The so-called sampled describing function approach is revisited and a formula for the computation of the locus of perturbed relay system approach is derived. In contrast to the describing function method, the locus of perturbed relay system approach yields exact results for the oscillation frequency. Stability properties of limit cycles and the basin of attraction of periodic solutions are discussed.

Then, novel discrete-time variants of the super-twisting algorithm are presented. In contrast to the commonly employed explicit Euler discretized super-twisting dynamics, the proposed schemes are exact in the sense that in the unperturbed case the controllers ensure convergence to the origin. Discretization chattering effects are avoided whilst the robustness properties are preserved. The approach is extended to a family of homogeneous differentiators, including the well-known arbitrary-order robust exact differentiator.

Finally, exploiting the notion of homogeneous eigenvalues, a new family of continuous-time arbitrary-order homogeneous state feedback controllers is derived. A formula that allows to design controllers for all combinations of the system's relative degree and the desired homogeneity degree of the closed-loop system is presented. The structure of the resulting controllers permits realization in a discrete-time environment straightforwardly using the developed ideas.
Original languageEnglish
QualificationDoctor of Technology
Awarding Institution
  • Graz University of Technology (90000)
  • Horn, Martin, Supervisor
  • Reichhartinger, Markus, Supervisor
Award date23 Aug 2019
Publication statusPublished - 2019

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