Discrete-time VSS
This chapter reviews some basic results obtained in the study of discrete-time (DT) variable structure control systems (DVSCS) theory during its twenty-five year history. For this purpose, the chapter is organised as follows: in Section 5.1 basic defini tions, assumptions and remarks are introduced that are necessary for the connection with continuous-time (CT) variable structure control systems (CVSCS) and form an introduction of terminologies for DVSCS. Section 5.2 is a brief overview of the more notable works in DVSCS. Section 5.3 gives the definition of a quasi-sliding mode (QSM) and a DT sliding mode (DSM). In Section 5.4, the Lyapunov stability concept is used to define invariant sets in DVSCS. Section 5.5 gives DSM existence conditions as a new motion phenomenon that is not possible in CVSCS. In Section 5.6, a basic concept of DVSCS, which is founded on the DT equivalent control method and a boundary layer concept for the system with nominal parameters, is presented, while Section 5.7 introduces some disturbance estimation methods. Section 5.8 describes two methods of DVSCS with sliding sectors. In Section 5.9 basic properties of DVSCS are given. Design methods to establish sliding surfaces are summarised in Section 5.10. Section 5.11 gives numerical examples that illustrate the properties of some DVSCS algorithms. Some issues in the practical realisation of DVSCS are given in Section 5.12 and Section 5.13 contains a list of published papers and other work that has been used for the preparation of this chapter.