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212 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-22, O. 2, PRIL 1977 . . , . - . Roger N . (S’66M70) was in Chicago, IL, on January 25, 1946. He received the B.S. degree in control systems engineering from the University of Illinois, Chica go, and the M.S .E. E. degree from Marquette University, Milwaukee, WI, in 1968 and 1970, respectively. From 1968 to 1971 he was a Project Engineer at General Motor s’ Delco Elect ronic s Division, Milwaukee, WI, and General Motors’ Technical Center, Warren, MI. At Delco Electronics, he was nvolved in analysis o f inert ially stabilized fireantrol systems. At General Motors’ Technical Center, he par- ticipated in the design and mechanization o f experi menta l automotive electronic systems. Since October 1971, he has been a member o f the technical staff at The Analytic Sciences Corpo ratio n, Reading, Mass . where his activities have ranged from missile error analysis, pilot modeling, and analysis of strapdown nertial ransfer alignment algorithms, to a broad spectrum of studies in the radio navigation field. These studies include: navigation algorithm analyses; airborne receiver performance specification; light test planning, data reduction and an al ysis; and developmen t o f operational software us ed in computing pub- lished tables of OMEGA propagation correc tions , and operational software used for synchronizing the OMEGA transmitter network. Survey Paper Variable Structure Systems with Sliding Modes VADIM I. UTKIN Abstmct-Variable stmctme systems consist o f a set o f continuous subsystems together with snitable switching Advantageom properties rdt rom changing struetnres according to this switching logic. Design and analysis for this clafs o f systems are surveyed in this paper. I. INTRODUCTION T E basic philosophy of the variable structure ap- proach is simp ly explained by contrasting it with the linear state regulator design for the single-input syst em i=Ax+bu. In the line ar state regulator design, the structure o f the state feedback is fixed as u=kTx where the constant parameters are chosen according to various design procedures, such as eigenval ue place ment or quadra tic minimization. In variable structure systems the control is allowed to change its structure, that is, to swit ch at any instan t from one to another member o f a set of possible continu ous functi ons o f the state. The variable structure design problem is then to select the parameters of eac h o f the struct ures an d to define the switching logic. recommended by P. R. B e lk er, Chairman of the IEEE S-CS Optimal received April 7, 1976; revised December 21, 1976. Paper Systems Committee. This wort was supported by the Coordinated Sci- ence Laboratory and the Department f Electrical Engineering, Univer- sitv of Illinois. The idea o f changing a structure is a natural one, and early utilization of this approach can be ound in the papers published about 20 years ago [18] , [19] , [26], [48], [50], [59], 61], [65], [66], [75], [88], [90]. A reward for introducing this additional complexity is the possibility to combi ne useful properties o f each of the structur es. More- over, a variable structure system can possess new proper- ties not present in any o f the structures use d For instance, an asymptotically stable system may consist of two struc- tures neither o f whic h is asymptotically stable. This possi- bility is illustrated by some early examples, which stimu- lated interest in variable structure systems VSS). Although very simple, two such examples are quoted here because they present the advantages o f changing struc- tures during a control phase. In the first example we consider a second-order system x= -*x having two structures defined by * = a: and *= where a:> a:. Th e phase por trai t cons ists of families of ellipsesFig. l(a), (b)] and hence, neither structure is asymptotically stable. However, asymptotic stability is achieved if the struct ure o f t he system is change d on the coordinate axes, that is, i f the switching logic is ifxi.0 if xi <o. ~~ -~ ~~ - - The author is with the Variable Structure Laboratory, Institute for Contro l Sciences, Moscow, SSR. The resulting phase portrait is hown n ig. l(c).
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