Imperial College London > Talks@ee.imperial > Eric C Kerrigan's list > Mathematical and Design Aspects of Sliding Mode Control

Mathematical and Design Aspects of Sliding Mode Control

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Leverhulme Lecture, joint with IEEE Control Systems Society UK&RI Chapter

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 is 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. Frobenius theorem is applied to reduce an arbitrary affine system to 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.

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