2 IEEE TRANSACTIONSON INDUSTRIAL ELECTRONICS,VOL. 40, NO. 1, FEBRUARY 1993 Variable Structure Control: A Survey John Y. Hung, Member, IEEE, Weibing Gao, SeniorMember, IEEE, and James C. Hung, Fellow, IEEE Abstract-A tutorial account of variable structure control with brief discussions about its historical development are pre- sliding mode is presented in this paper. The purpose is to sented. introduce in a concise manner the fundamental theory, main results, and practical applications of this powerful control sys- A. The Basic Notion of VSC tem design approach. This approach is particularly attractive The basic idea of VSC was originally illustrated by a for the control of nonlinear systems. Prominent characteristics such as invariance, robustness, order reduction, and control second-order system [431, [ 1541 similar to the following chattering are discussed in detail. Methods for coping with X=y chattering are presented. Both linear and nonlinear systems are considered. Future research areas are suggested and an exten- +U 2y -x y = (la) sive list of references is included.

-*x U = where * CONVENTIONS SYMBOLS FOR 4 when s ( x , y ) > 0 = Capital, italic letters represent matrices, e.g., A and K . (1b) when s ( x , y ) < 0 = -4 Boldface, roman lower-case letters represent vectors, e.g., a and k. and Lower-case, italic letters represent scalars, e.g., a and s(x,y) = X U , u=0.5x+y. (IC) k.

A block diagram of the system is shown in Fig. l(a). The...

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