CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XIII - Sliding Mode Control - Vadim Utkin

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1 SLIDING MODE CONTROL Vadm Utkn The Oho State Unversty, Columbus,Oho, USA Keywords: Sldng mode, State space, Dscontnuty manfold, Varable structure system, Sldng mode exstence condtons, Lyapunov functon, Egenvalue placement, Chatterng, Dscrete-tme sldng mode. Contents 1. Introducton 2. Concept Sldng Mode 3. Sldng Mode Equatons 4. Exstence Condtons 5. Desgn Prncples 6. Dscrete-Tme Sldng Mode Control 7. Chatterng Problem 8. Inducton Motor Control 9. Concluson Acknowledgement Glossary Bblography Bographcal Sketch Summary The paper presents the basc concepts, mathematcal and desgn aspects of sldng mode control.it s shown that the man advantages of sldng mode control are order reducton, decouplng desgn procedures, dsturbance rejecton, nsenstvty to parameter varatons, smple mplementaton by means of conventonal power converters. The methods of suppressng chatterng, caused by dscrete-tme mplementaton and unmodeled dynamcs, are gven. The sldng mode control s demonstrated for lnear tme-nvarant systems and for control of nducton motors. 1. Introducton The sldng mode control approach s recognzed as one of the effcent tools to desgn robust controllers for complex hgh-order nonlnear dynamc plant operatng under uncertanty condtons. The research n ths area were ntated n the former Sovet Unon about 40 years ago, and then the sldng mode control methodology has been recevng much more attenton from the nternatonal control communty wthn the last two decades. The major advantage of sldng mode s low senstvty to plant parameter varatons and dsturbances whch elmnates the necessty of exact modelng. Sldng mode control enables the decouplng of the overall system moton nto ndependent partal components of lower dmenson and, as a result, reduces the complexty of feedback

2 desgn. Sldng mode control mples that control actons are dscontnuous state functons whch may easly be mplemented by conventonal power converters wth on-off as the only admssble operaton mode. Due to these propertes the ntensty of the research at many scentfc centers of ndustry and unverstes s mantaned at hgh level, and sldng mode control has been proved to be applcable to a wde range of problems n robotcs, electrc drves and generators, process control, vehcle and moton control. 2. Concept Sldng Mode. The phenomenon Sldng Mode may appear n dynamc systems governed by ordnary dfferental equatons wth dscontnuous state functons n the rght-hand sdes. The conventonal example of sldng mode a second order relay system - can be found n any text book on nonlnear control. The control nput n the second order system x+ a2 + a1x= u, u = Msgn(), s s= cx+, a, a, M, c - const 1 2 may take only two values, M and M, and undergoes dscontnutes on the straght lne s = 0 n the state plane ( x, )(Fg.1 for the case a1= a2 = 0 ). It follows from the analyss of the state plane that, n the neghborhood segment mn on the swtchng lne s = 0, the trajectores run n opposte drectons, whch leads to the appearance of a sldng mode along ths lne. The equaton of ths lne + cx = 0 may be nterpreted as the sldng mode equaton. Note that the order of the equaton s less than that of the orgnal system the sldng mode does not depend on the plant dynamcs, and s determned by parameter c only. Fgure 1. Sldng mode n a second relay system

3 Sldng mode became the prncple operaton mode n so-called varable structures systems. A varable structure system conssts of a set of contnuous subsystems wth a proper swtchng logc and, as a result, control actons are dscontnuous functons of the system state, dsturbances (f they are accessble for measurement), and reference nputs. The prevous example of the relay system wth state dependent ampltude of the control varable may serve as an llustraton of a varable structure system: u = k x sgn(), s k s constant. Fgure 2. State planes of two unstable structures Fgure 3. State plane of varable structure system Now the system wth a 1 = 0and a 2 < 0 conssts of two unstable lnear structures (u = kx and u = kx, Fg.2) wth x = 0 and s = 0 as swtchng lnes. As t s clear from the system state plane, the state reaches the swtchng lne s = 0 for any ntal condtons. Then, the sldng mode occurs on ths lne (Fg.3) wth the moton equaton + cx = 0, whle the state vector decays exponentally. Smlarly to the relay system, after the start of the sldng mode, the moton s governed by a reduced order equaton whch does not depend on the plant parameters.

4 Now we demonstrate sldng modes n non-lnear affne systems of general form, (1) x = f( x, t) + B( x, t) u u u = u + ( x, t) ( x, t) f f s ( x) > 0 s ( x) < 0 = 1,..., m, (2) n m + where x R s a state vector, u R s a control vector, u (,t) x, u ( x, t) and + s ( x ) are contnuous functons of ther arguments, u (,t) x u (,) x t. The control s desgned as a dscontnuous functon of the state such that each component undergoes dscontnutes n some surface n the system state space. Smlar to the above example, state velocty vectors may be drected towards one of the surfaces and sldng mode arses along t (arcs ab and cb n Fg.4). It may arse also along the ntersecton of two of surfaces (arc bd ). Fgure 4. Sldng mode n dscontnuouty surface and ther ntersecton Fg.5 llustrates the sldng mode n the ntersecton even f t does not exst at each of the surfaces taken separately. Fgure 5. Sldng mode n ntersecton of dscontnuty surfaces

5 For the general case (1) sldng mode may exst n the ntersecton of all dscontnuty surfaces s = 0, or n the manfold T s( x) = 0, s ( x) = [ s ( x),..., s ( x)] of dmenson n m. 1 m (3) Let us dscuss the benefts of sldng modes, f t would be enforced n the control system. Frst, n sldng mode the nput s of the element mplementng dscontnuous control s close to zero, whle ts output (exactly speakng ts average value u av) takes fnte values (Fg.6). Fgure 6. Hgh gan mplementaton by sldng mode Hence, the element mplements hgh (theoretcally nfnte) gan, that s the conventonal tool to reject dsturbance and other uncertantes n the system behavor. Unlke to systems wth contnuous controls, ths property called nvarance s attaned usng fnte control actons. Second, snce sldng mode trajectores belong to a manfold of a dmenson lower than that of the orgnal system, the order of the system s reduced as well. Ths enables a desgner to smplfy and decouple the desgn procedure. Both order reducton and nvarance are transparent for the above two second-order systems. 3. Sldng Mode Equatons So far the arguments n favor of employng sldng modes n control systems have been dscussed at the qualtatve level. To justfy them strctly, the mathematcal methods should be developed for descrbng ths moton n the ntersecton of dscontnuty surfaces and dervng the condtons for sldng mode to exst. The frst problem means dervng dfferental equatons of sldng mode. Note that for our second-order example the equaton of the swtchng lne + cx = 0 was nterpreted as the moton equaton. But even for a tme nvarant second-order relay system 1= a11x1+ a12x2 + bu 1 = a x + a x + b u, u = Msgn( s), s = cx + x ; M, a, b, c are const the problem does not look trval snce n sldng mode s = 0 s not a moton equaton. j

6 The frst problem arses due to dscontnutes n control, snce the relevant moton equatons do not satsfy the conventonal theorems on exstence-unqueness of solutons. In stuatons when conventonal methods are not applcable, the usual approach s to employ regularzaton or replacng the ntal problem by a closely smlar one, for whch famlar methods can be used. In partcular, takng nto account delay or hysteress of a swtchng element, small tme constants n an deal model, replacng a dscontnuous functon by a contnuous approxmaton are examples of regularzaton snce dscontnuty ponts (f they exst) are solated. The unversal approach to regularzaton conssts of ntroducng a boundary layer s < Δ, Δ const around the manfold s = 0, where an deal dscontnuous control s replaced by a real one such that the state trajectores are not confned to ths manfold but run arbtrarly nsde the layer (Fg. 7). The only assumpton for ths moton s that the soluton exsts n the conventonal sense. If, wth the wth of the boundary layer Δ tendng to zero, the lmt of the soluton exsts, t s taken as a soluton to the system wth deal sldng mode. Otherwse we have to recognze that the equatons beyond dscontnuty surfaces do not derve unambguously equatons n ther ntersecton, or equatons of the sldng mode. Fgure 7. Boundary layer The boundary layer regularzaton enables substantaton of so-called Equvalent Control Method ntended for dervng sldng mode equatons n manfold s = 0 n system (1). Followng ths method the sldng mode equaton wth a unque soluton may derved for the nonsngular matrx GxBx ( ) ( ), Gx ( ) = { s/ x}, det( GB) 0. Frst, the equvalent control should be found for the system (1) as the soluton to the -1 equaton s = 0 on the system trajectores (G and ( GB ) are assumed to exst) : s = G = Gf + GBu = 0, u =-( GB) Gf. eq eq -1

7 Then the soluton should be substtuted nto (1) for the control -1. (4) x= f B( GB) Gf Equaton (4) s the sldng mode equaton wth ntal condtons s( x (0),0) = 0. Snce s( x ) = 0 n sldng mode m components of the state vector may be found as a m n-m functon of the rest ( n m) ones: x2 s0( x1); x2, s0 ; x1 correspondngly, the order of the sldng mode equaton may be reduced by m : = R R and, x = f [ x, t, s ( x )], f R n m (5) The dea of the equvalent control method may be easly explaned wth the help of geometrc consderaton. Sldng mode trajectores le n the manfold s = 0 and the equvalent control u eq beng a soluton to the equaton s = 0 mples replacng the orgnal dscontnuous control by such contnuous one that the state velocty vector les n the tangental manfold and as a result the state trajectores are n ths manfold. It wll be mportant for control desgn that sldng mode equaton Bblography Is of reduced order Does not depend on control Depends on the equaton of swtchng surfaces. TO ACCESS ALL THE 23 PAGES OF THIS CHAPTER, Clck here Determnstc Non-Lnear Control, (1990). A.S. Znober,Ed., Peter Peregrnus Lmted, UK. [The book s wrtten by the author team and covers dfferent control methods, ncludng sldng mode control, for nonlnear dynamc plants operatng under uncertanty condtons]. Edwards, C. and Spurgeon, S. (1999), Sldng Mode Control: Theory and Applcatons, Taylor and Frencs, London. [The conventonal theory s presented along wth new methods of control and observaton and new applcaton areas]. Flppov, A., (1988), Dfferental Equatons wth Dscontnuous Rght-Hand Sdes, Kluver. [Mathematcal aspects of dscontnuous dynamc systems are studed n ths book]. Itks, U., (1976), Control Systems of Varable Structure, Wley, New York. [Ths book deals wth SISO control systems n canoncal form]. J.-J. E. Slotne and W. L. (1991). Appled Nonlnear Control, Prentce Hall Englewood Clffs, New Jersey. [Text book wth a chapter on sldng mode control].

8 Utkn, V. (1978). Sldng Modes and ther Applcatons n Varable Structure Systems, Mr Publ., Moscow (Translaton of the book publshed by Nauka, Moscow, 1974 n Russan). [Ths book presents the frst results on MIMO sldng mode control systems]. Utkn, V. (1992). Sldng Modes n Control and Optmzaton, Sprnger Verlag, Berln. [The book ncludes mathematcal, desgn and applcaton aspects of sldng mode control]. Utkn, V., Guldner, J. and Sh, J.X. (1999). Sldng Mode Control n Electro-Mechancal Systems, Taylor and Frencs, London. [Theoretcal desgn methods are presented n the context of applcatons for control of electrc drves, alternators, vehcle moton, manpulators and moble robots]. Varable Structure and Lyapunov Control, (1993) A.S.Znober, Ed.,Sprnger Verlag, London. [The book s prepared based on the results of VSS 1992 workshop held n Sheffeld, UK]. Varable Structure Control for Robotcs and Aerospace Applcaton, (1993), K-K. D. Young, Ed., Elsever Scence Publshers B.V., Amsterdam. [Aplcatons of sldng mode control for electrc drves, pulse-wth modulators, manpulator, flexble mechancal structures, magnetc suspenson, flght dynamcs are studed n the book]. Varable Structure Systems, Sldng Mode and Nonlnear Control, (1999), K.D. Young and U. Ozguner (Eds), Sprnger Verlag. [The book s prepared based on the results of VSS 1998 workshop held n Longboat Key, Florda, USA]. Bographcal Sketch Prof V. Utkn graduated from Moscow Power Insttute (Dpl.Eng.) and receved a Ph.D. from the Insttute of Control Scences (Moscow, Russa). He was wth the Insttute of Control Scences snce 1960, as Head of the Dscontnuous Control Systems Laboratory n Currently he s Ford Char of Electromechancal Systems at the Oho State Unversty. Prof. Utkn s one of the orgnators of the concepts of Varable Structure Systems and Sldng Mode Control. He s an author of fve books and more than 250 techncal papers. In he was n charge of an nternatonal project between hs Insttute and "Energonvest" Sarajevo on the sldng mode control of nducton motors. D.C., nducton and synchronous drves wth sldng mode control have been appled for metal-cuttng machne tools, process control and electrc cars. Hs current research nterests are control of nfnte-dmensonal plants ncludng flexble manpulators, sldng modes n dscrete tme systems and mcroprocessor mplementaton of sldng mode control, control of electrc drves and alternators, robotcs and automotve control. He s Honorary Doctor of Unversty of Sarajevo, Yugoslava, n 1972 was awarded Lenn Prze (the hghest scentfc award n the former USSR). He held vstng postons at unverstes n the USA, Japan, Italy and Germany. Prof. V.Utkn was IPC charman of 199O IFAC Congress n Tallnn; now he s Assocate Edtor of "Internatonal Journal of Control", Charman of Techncal Commttee of IEEE on Varable Structure and Sldng Mode Control, member of Admnstratve Commttee of IEEE Industral Electroncs Socety.