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## Mathematical and design aspects of sliding mode control theory

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Abstract:
In many books and survey papers the authors mentioned that the research in the area of Variable Structure Systems and Sliding Mode Control (SMC) was initiated in the former Soviet Union about 40 years ago and then the developed control methodology has been receiving much more attention of the international control community within the last two decades. So first, the events happened before these “last two decades” starting from the late fifties are surveyed to demonstrate the initial ideas and hopes of the first researchers. The retrospective view will be helpful to establish the bridges between those early steps and the scientific arsenal accumulated in the sliding mode control design methodology by now. Control actions of the systems under study are assumed to be discontinuous function of the system state. For the principle operation mode the state trajectories are in the vicinity of discontinuity points. This motion is referred to as sliding mode. The scope of sliding mode control studies embraces - mathematical methods of discontinuous differential equations, - design of manifolds in the state space and discontinuous control functions enforcing motions along the manifolds, - implementation of sliding mode controllers and their applications to control of dynamic plants. The first problem concerns development of the tools to derive the equations governing sliding modes and the conditions for this motion to exist. Formally motion equations of SMC do not satisfy the conventional uniqueness-existence theorems of the ordinary differential equations theory. The reasons of ambiguity are discussed. The regularization approach to derive sliding motion equations i demonstrated and compared with other models of sliding modes. The sliding mode existence problem is studied in terms of the stability theory. Enforcing sliding modes enables decoupling of the design procedure, since the motion preceding sliding mode and motion in sliding mode are of lower dimensions and may be designed independently. On the other hand, under so called matching conditions the sliding mode equations depend neither plant parameter variations nor external disturbances. Therefore sliding mode control algorithms are efficient when controlling nonlinear dynamic plants of high dimension operating under uncertainty conditions. The design methods are demonstrated mainly for systems in the regular form. Component-wise and vector design versions of sliding mode control are discussed. The design methodology is illustrated by sliding mode control in linear systems. The concept "sliding mode control" is generalized for discrete-time systems to make feasible its implementation for the systems with digital controllers. New mathematical and design methods are needed for sliding mode control in infinite-dimensional systems including systems governed by PDE. The recent results in this area are briefly surveyed. The problem of chattering caused by unmodeled dynamics is discussed in the context of applications. The systems with asymptotic observers are shown to be free of chattering. Sliding mode control of electric drives, mobile robots, flexible bar and plate are demonstrated as application examples.

Type of Seminar:
Public Seminar
Speaker:
Ohio State University, USA
Date/Time:
Aug 23, 2004   17:15
Location:

ETH-Zentrum, ETZ E6, Gloriastrasse 35, 8006 Zurich
Contact Person:

Prof. L. Guzzella