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A Control Engineers Guide t o Sliding ModeControlK. David Young, Vadim

.4bstmct- This paper presents a guide to sliding modecontrol for practicing control engineers. It offers an accurateassessment of the so-called chattering phenom enon, catalogsimplementable sliding mode control design solutions, andprovides a frame of reference for future sliding mode controlresearch.K e y w d s - Asymptotic Observers, Discontinuous Con-trol, Discrete Time Systems, Disturbance Compensation,Disturbance Estimation, Disturbance R ejection, High GainFeedback Systems, Motion Separations, Multivariable Ser-vomechanisms, Parametric Uncertainties, Parasitic Dynam-

ics, Robust Control, Sampled Data Control Systems, Singu-lar Perturbations, Sliding Mode, Uncertain System s, Vari-able Structure Control.

I. INTRODUCTIONDuring the last two decades since the publication of the

survey paper in the IEEE Transactions of Automatic Con-trol in 1977 [l ] , ignificant interest on Variable StructureSystems (VSS) and Sliding Mode Control (SMC) have beengenerated in the control research community worldwide.One of the most intriguing aspects of sliding mode is thediscontinuous nature of the control action whose primaryfunction is to switch between two distinctively differentsystem structures (or components) such that a new typeof system motion, called Sliding Mode, exists in a mani-fold. This peculiar system characteristic is claimed to re -sult in superb system performance which includes insen-sitivity to parameter variations, and complete rejection ofdisturbances. The reportedly superb system behavior ofVSS and SMC naturally invites criticisms and scepticismsfrom within the research community, and from practicingcontrol engineers alike [a]. The sliding mode control re-search community has risen to answer to some of thesecritical challenges, while at the same time, contributed tothe confusion about the robustness of SMC with incom-plete analyses, and design fixes for the so-called chatter-ing phenomenon [3].Many analytical design methods wereproposed to reduce the effects of chattering [4], [5], [6], [7],[8]- for it remains to be the only obstacle for sliding modeto become one of the most significant discoveries in moderncontrol theory; and its potential seemingly limited by theimaginations of the control researchers [9], [ l o ] , [ l l ] .

In contrast to the previously works published since the1977 article [ l ] , hich serve as a s tatu s overview [l2] , a tu -K . D. Young is with YKK Systems, 2680 LaSalle Drive, MountainView, California 94040-4770, USA, Email : [email protected]. I. Utkin is with the Departments of Electrical Engineeringand Mechanical Engineering, Ohio State University, 2015 Neil Ave.,Columbus, Ohio 43210 USA, Em a i l utkin8ee.eng.ohic-state.eduU . Ozgiiner is with th e Department of Electrical Engineering, OhioSta te University, 2015 Neil Ave., Columbus, Ohio 43210 USA, Email :[email protected]

[. Utkin, Umit Ozgiinertorial [13]of design methods, or another more recent stateof the ar t assessment [14], or yet another survey of slidingmode research [15], he purpose of this paper is to providea comprehensive guide t o Sliding Mode Control for controlengineers. It is our goal to accomplish these objectives:

Provide an accurate assessment of the chattering phe-nomenon;offer a catalog of implementable robust sliding modecontrol design solutions for real life engineering appli-cations;initiate a dialog with practicing control engineers onsliding mode control by threading the many analyti-cal underpinnings of sliding mode analysis through aseries of design exercises on a simple, yet illustrativecontrol problem; andestablish a frame of reference for future sliding modecontrol research.

The flow of the following presentation conforms to the his-torical development of VSS and SMC: First we introduceissues within Continuous Time Sliding Mode in Section 11,then in Section 111, we progress to Discrete time SlidingMode, followed with Sampled Data SMC Design in SectionIV.

11. CONTINUOUSIM E SLIDING ODESliding mode is originally conceived as system motion for

dynamic systems whose essential open loop behavior can bemodeled adequately with ordinary differential equations.The discontinuous control action, which is often referredto as Variable Structure Control (VSC), is also defined inthe continuous time domain. The resulting feedback sys-tem, the so-called VSS, is also defined in the continuoustime domain, and it is governed by ordinary differentialequations with discontinuous right hand side. The mani-fold of the state space of the system on which sliding modeoccurs is the Sliding Mode Manifold, or in brief, SlidingManifold. For control engineers, the simplest, but vividlyperceptible example is a double integrator plant, subject totime optimal control action. Due to imperfections in theimplementations of the switching curve, derivable using thePontrayagon Maximum Principle, sliding mode may occur.Sliding mode was studied in conjunction with relay con-trol for double integrator plants, a problem motivated bythe design of attit ude control systems of missiles with jetthrusters in the 1950s [16].

The so-called chattering phenomenon is generally per-ceived as motion which oscillates about the sliding man-ifold. There are two possible mechanisms which producesuch a motion. Firs t, in the absence of switching nonide-alities such as delays, i.e., when the switching device is

1996 IEEE Workshop onVariable Structure Systems0-7803-3718-2/96 $5.00 0 996 IEEE

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ideally switching at an infinite frequency. In this case, theexistence of parasitic dynamics in series with the plant .This may account for the otherwise negligible fast actua-tor and sensor dynamics, and cause a small amplitude highfrequency oscillation to appear in the neighborhood of thesliding manifold. In control engineering practice, such fastdynamics are often neglected in the open loop model forcontrol design if the associated poles are well damped, andoutside the desired bandwidth of the feedback control sys-tem. Generally, the motion of the real system is close toth at of an ideal system in which the parasitic dynamics areneglected, and the difference between the ideal and real mo-tion, which is on the order of the neglected time constants,decays rapidly. The mathematical basis for the analysis ofdynamic systems with fast motion is the theory of singu-larly perturbed differential equations [17], and its exten-sions to control theory have been developed and appliedin practice [18]. However, the theory is not applicable forVS S since they are governed by differential equations withdiscontinuous right hand sides. The interactions betweenthe parasitic dynamics and VS C generate a non-decayingoscillatory component of finite amplitude and frequency,and this is generically referred to as chattering.

Second, the switching nonidealities alone can cause suchhigh frequency oscillations. We shall focus only on the de-lay type of switching nonidealities since it is most relevantto any electronic implementation of the switching device,including both analog and digital circuits, and micropro-cessor code executions. Since the cause of th e resultingchattering phenomenon is due to time delays, discrete timecontrol design techniques such as the design of an e xt ra polator exist to mitigate the switching delays [19]. Thesedesign approaches are perhaps more familiar to control en-gineers.

Unfortunately, in practice, both the parasitic dynamicsand switching time delays exist. Since it is necessary tocompensate for the switching delays by using a discretetime control design approach, a practical SMC design mayhave to be unavoidably approached in discrete time. Weshall retu rn to t he details of discrete time SMC after weillustrate our earlier points on continuous time SM C witha simple design example, and summarize the existing ap-proaches to avoid chattering.A . Chattering due to parasitic d yna mics - a simple example

The effects of unmodeled dynamics on sliding mode canbe illustrated with an extremely simple relay control systemexample: Let the nominal plant be an integrator,

? = U , x ( t o ) = x o # O , (1)and assume that a relay controller has been designed,

U = -sgn(z) . (2)The sliding mani fold is the origin of th e state spacex = 0. Given any nonzero initial condition 50, he statetrajectory z ( t ) s driven toward the sliding manifold. Ide-

s(t = 0 , 2 t* where t* is the first time instant thatz( t*)= 0, a.e., once the sta te trajec tory reaches the slidingmanifold, it remains on it for good. However, even withsuch idealized switching device, unmodeled dynamics caninduce oscillations about th e sliding manifold. Suppose wehave ignored the existence of a second order sensor dy-namics, and the true system dynamics are governed by,

x = -sgn(x,) , ( 3 )p2x, + 2p4, + xs = 5 , (4 )


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proposed method of piecewise linear approximation reducesthe feedback system to a system with no sliding mode. Thisproposed method has a wide acceptance by many slidingmode researchers, but unfortunately i t does not resolve thecore problem of the robustness of sliding mode as exhib-ited in chattering. Many sliding mode researchers cited thework in [3], [22] s the basis of their optimism that withboundary layer control, the implementation issues of con-tinuous time sliding mode are solved. Unfortunately, theoptimism of these researchers was not shared by practicingengineers, and this may be rightly so. The effectivenessof boundary layer control is immediately challenged whenrealistic parasitic dynamics are considered. An in-depthanalysis of the interactions of parasitic actuator and sensordynamics with the boundary layer control [24] revealed theshortcomings of this approach where parasitics dynamicsmust be carefully modeled and considered in the feedbackdesign in order to avoid instability inside the boundarylayer which leads to chattering. Without such informationof the parasitic dynamics, control engineers must opt fora worst case boundary layer control design in which thedisturbance rejection properties of SM C are severely com-promised.B.l A boundary layer controller

We shall continue with the simple relay control exam-ple, and consider the design of a boundary layer controller.We assume the same second order parasitic sensor dynam-ics as before. The behavior inside the boundary layer isdescribable by a linear closed loop system,

x = -gx, + d ( t )rzx, + 27,xy + x, = 2 , (9)(10)

where d ( t ) represents a bounded, but unknown exogenousdisturbance. Whereas discontinuous control action in VS Ccan reject bounded disturbances, by replacing the switch-ing control with a. boundary layer control, the additionalassumption that d be bounded is needed since accordingto singular perturbation analysis, the residue error is pro-portional to l d I / g . Given a finite T, , we can compute theroot locus of this system with respect to the scalar pos-itive gain g > 0. An upperbound gc exists which speci-fies the crossover point of the root locus on the imaginaryaxis. Thus, for 0 < g < gc , th e behavior of this system isasymptotically stable, i . e . , for any initial point inside theboundary layer,the sliding manifold z = 0 is reached asymptotically ast ---f oc). The transient response and disturbance rejectionof this feedback system are two competing performancemeasures to be balanced by the choice of an optimum gainvalue. If we assume p = .01,he associated root locus isplotted in Figure 1 for 0.003 5 9-l 5 0.01 with a stepsize of 0.001, The critical gain is gc = 200. Thus fromthe linear analysis, a boundary layer control with g = 100results in a stable sliding mode,whereas with g = 200, os-cillatory behavior about the sliding manifold is predicted.

IXSI 5 1/91 (11)

Figure 2 shows the simulated error responses of t he closedloop system for these two gain values which agree with theanalysis. In this simulation, a unity reference commandfor the plant s tat e and a constant disturbance d ( t ) = 0.5 isintroduced. The tradeoff between chattering reduction anddisturbance rejection can be observed from the steady statevalue of -0.005 in x ( t )whose magnitude (for the stable re-sponse), or average value of -0.0025 (for the oscillatoryresponse) is proportional to the gain g. We note that evenwith g > gc , the resulting responses are only oscillatory, butstill bounded. This is because the linear analysis is validonly inside the boundary layer, and the VSC always forcesthe state trajectory back into the boundary layer region.However, as the gain increases, the frequency of oscillation,related to the magnitude of the imaginary part of the rootlocus, also increases, hence the general description chatter-ing applies as this frequency reaches the neglected resonantfrequencies of t he physical plant.

This example illustrates the advantages of boundarylayer control which lie primarily in the availability of fa-miliar linear control design tools to reduce the potentiallydisastrous chattering. However, it should also be remindedthat if the acceptable closed loop gain has to be reducedsufficiently to avoid instability in the boundary layer, theresulting feedback system performance may be significantlyinferior to t he nominal system with ideal sliding mode. Fur-thermore, the precise details of the parasitic dynamics mustbe known and used properly in the linear design.C. Observer based Slzdzng Mode Control

Recognizing the essential triggering mechanism for chat-tering is due to the interactions of the switching action withthe parasitic dynamics, an approach which utilizes asymp-totic observers to construct a high frequency by pass loophas been proposed[4]. This design exploits a localizationof the high frequency phenomenon in the feedback loop byintroducing a discontinuous feedback control loop which isclosed through an asymptotic observer of the plant [25].Since the model imperfections of the observer are suppos-edly smaller than th at in the plant, and the control isdiscontinuous only with respect to the observer variables,chattering is localized inside a high frequency loop whichbypasses the plant . However, this approach assumes thatan asymptotic observer can indeed be designed such thatthe observation error converges to zero asymptotically. Weshall discuss the various options available in observer basedsliding mode control in the following design example.C.l Design example of observer based SM C

For the relay control example, we examine the utilityof the observed based SM C in localizing the high fre-quency phenomenon. For the nominal plant, the follow-ing asymptotic observer results from applying conventionalstate space linear control design,

k = h(s , - 2 ) + U , (12)where h > 0 is the observer feedback gain, and x, is theoutput of the parasitic sensor dynamics. The SMC and the

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associated sliding manifold defined on the observer statespace is,

U = -sgn(f) . (13)The behavior of the closed loop system can be deducedfrom the following fourth order system,

Ax = -sgn(z - ) + d(t) e = z - 2 , (14)e = -he + h(z-a,)+ d ( t )T;i?# + 2T,k, +2 , = 5 . (15)(16)

Firs t we consider the case when d(t)= 0. Using an infinitegain linear function g( z - ) to approximate the switchingfunction sgn(z- ) , and since p is finite, the above systemis a singular perturbed system with g- being the parasiticparameter. The slow dynamics which are of third order canbe extracted by formally setting 9- l = 0, and z - e = 0 ,

1 = -hx,, (17)r,X,+ Z T , ~ ,+ z, = e . (18)

It is possible to further apply a singular perturbation anal-ysis to insure that given r 8 , here exists h > 0 such thatthe asymptotic observer dynamics are of first order, andits eigenvalue is approximately -h. Clearly, the adverse ef-fects of the parasitic sensor dynamics are neutralized withan observer based SM C design. If a switching function isrealized in the SMC design, the only remaining concernswill be switching tim e delays, and if the observer is to beimplemented in discrete time, the entire feedback designincluding the compensation of switching time delays maybe best carried out in the discrete time domain. Figure 3is a block diagram of this design. Note that the switchingelement is inside a feedback loop which passes through onlyth e observer, bypassing both blocks of the plant dynamics.This is the so-called high frequency bypass effects of theobserver based SM C [4], [26].

When d ( t ) # 0, its effects on the convergence of asymp-totic observers are well known. If d ( t ) is an unknown con-stant disturbance, a multivariable servomechanism formu-lation can be adopted to estimate both t he sta te and exoge-nous disturbance in a composite asymptotic observer. Theresulting feedback system is the so-called Variable Struc-ture (VS) Servomechanism design [25], [27].

For bounded but unknown disturbances with boundedtime derivative, the only known approach to solving therobustness of the asymptotic observer is to introduce ahigh gain loop around the observer itself to reject the un-known disturbance, e.g., by increasing the gain h in theobserver such that the effects of d ( t ) are adequately atten-uated. However, the requirements for disturbance atten-uation and closed loop stability must be balanced in thedesign, and if sliding mode is to be preserved in the mani-fold 2 = 0, g must be sufficiently larger th an h. A switch-ing function implementation of the SMC would seem toensure the necessary time scale separations, however, thecondition g


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D.l An SM disturbance estimatorOnce again we return to the simple relay example with

parasitic sensor dynamics for our design of a disturbanceestimator. The plant model is

j.= U + d ( t )T Z X , + 27,xs + x,3= x .

(19)(20)

We shall design a disturbance estimator with sliding modeas follows:

From the error dynamics,f = U + sgn(x, - ) . (21)

ae = -sgn(z, - ) + d ( t )+S - k , (22)if sliding mode occurs on e = 0, since u ( t ) is continuousand differentiable, /Ss .1 = O(rS ) .By solving for theequivalent control in d = 0 ,

e = x, - ,

Thus, within this estimator, there exists a signal which,under the sliding mode condition, is O(7,) close to theunknown disturbance d ( t ) . This forms the basis of a feed-back control design which utilizes this signal to compensatethe disturbance to O ( T ~ ) .he resulting control law has aconventional linear feedback component, and a disturbancecompensating component, and for this system

U = -kf - [sgn(s, - ) I e q . (24)The extraction of the equivalent control from the slidingmodel control signal is by low pass filtering. While the-oretically there exists such a low pass filter such that theequivalent control can be found, in practice, the bandwidthof the desired closed loop system, the spectrum of the dis-turbance, are all important to the selection of the cutofffrequency of this filter.

For evaluation by simulation, we let the sensor time con-stant be once again T, = 0.01, and assume the same unityreference command, and constant disturbance d ( t ) = 0.5.After canceling the disturbance, we desire a closed loopsystem with a time constant of one seconds which can beattained with k = 2. A boundary layer of 5 x replacesthe switching function in th e estimator. The closed loopeigenvalues are { -2000. (from the boundary layer), -1 ,(thedominant closed loop pole), -96.75 i 101.83(the shiftedsensor poles). For low pass filtering, a third order butter-worth filter with a 3dB corner frequency of 50 rad./sec. isused to filter the equivalent control. Figure 6 shows theerror between the reference command and the plant statewhich exhibits the desired one seconds time constant tran-sient behavior, with the exception of initial minor distor-tions which are due to the convergence of the disturbanceestimate shown in Figure 7. Despite the constant distur-bance, the steady state error is zero. While standard PIDcontrollers can achieve the same zero steady state error inthe presence of unknown constant disturbance, the track-ing error is regulated to zero even when d ( t ) s time varyingas it is reported before [8].

E . Actuator Bandluidth ConstraintsDespite its desirable properties, VSC is mostly restricted

to control engineering problems where the control input ofthe plant is, by the nature of the control actuator , nec-essarily discontinuous. Such problems include control ofelectric drives where pulse-width-modulation is not t he ex-ception, but the rule of the game. Space vehicle att itudecontrol is another example where reaction jets operatedin an on-off mode are commonly used. The thi rd exam-ple, which is closely related to the first one, is power con-verter and inverter feedback control design. For this classof applications, the chattering phenomenon still need to beaddressed, however, the arguments against using slidingmode in the feedback design are weakened. The issue inthis case is whether VSC should be utilized to improve sys-tem performance, or standard PWM techniques are to beapplied, after a standard PID type design is completed, torealize a low bandwidth servo loop. If VSC is to be used,by adopting an observer based SMC, the high frequencycomponents of the discontinuous control can be bypassed,and consequently, adverse interactions with the unmodeleddynamics which causes chattering are avoided.

In plants where control actuators have limited band-width, e.g., hydraulic actua tors, there are two possibilities:First, the ac tuator bandwidth is outside the required closedloop bandwidth. Thus the actuator dynamics become un-modeled dynamics, and our discussions in the previous sec-tions are applicable. While in linear control design, it ispossible to ignore the actuator dynamics, doing so in VSCrequires extreme care. By ignoring actua tor dynamics in aclassical SMC design, chattering is likely to occur since theswitching frequency is limited by the actuator dynamicseven in the absence of parasitic dynamics. Strictly speak-ing, sliding mode cannot occur, since the control input tothe plant is continuous.Second, the desired closed loop bandwidth is beyond theactuator bandwidth. In this case, regardless of whetherSMC or other control designs are to be used, the actu-ator dynamics are lumped together with the plant, andthe control design model encompasses the actuator-plantin series. With the actuator dynamics no longer negligi-ble, often the matching conditions which are satisfied inthe nominal plant model are violated. This results fromhaving dominant dynamics inserted between the physicalinput , such as force, and the controller output, usually anelectrical signal. The design of SMC which incorporatesthe actuator dynamics as a prefilter for the VSC was pro-posed in [28]. This design utilizes an expansion of the orig-inal state space by including state derivatives, and formu-late an SMC design such that the matching condition isindeed satisfied in the extended space. Another alternativeapproach is to utilize sliding mode to estimate the distur-bance for compensation as discussed earlier. Since slidingmode is not introduced primarily to reject disturbances,the matching conditions are of no significance in this de-sign. Provided that a suitable sliding mode exists such th atthe disturbance can be estimated from the correspondingequivalent control, this approach resolves the limitations

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imposed by actuator bandwidth constraints on the designof sliding mode based controllers.E.l An SMC Design with Pre-Filter

plant, and actuator dynamics,We shall use the example with a nominal integrator

x = 2 , + d ( t ) ,c y 2 x , + 2aXa + 5 , = U , (25)(26)

to illustrate this design. The actua tor bandwidth limita-tion is expressed in the time constant cy. Given a discontin-uous input u(t) , he rate of change of the actuator outputx a ( t ) s limited by the finite magnitude of cy. However, inorder for the disturbance d ( t ) to be rejected, z a ( t )must bean SMC. Also if x , can be designed as a control input, thenth e matching condition is clearly satisfied. But since U isthe actual input, the matching condition does not holdsfor finite a. The design begins with an assumption thatd ( t ) has continuous first and second derivatives, and thedefinitions of new state variables,

x 1 = x , x 2 = X C , x 3 = x . (27)The control U is designed as an VSC with respect to thesliding manifold,

4 x 1 , 2 , x3 ) = ClXl + c2x2 + x3 = 0 .With the equivalent control ueqcomputed from

(28)

the resulting sliding mode dynamics are found to be com-posed of two subsystems in series:

x + c2x + c1x = 0 , (30)(31), = (c ; - Cl)* + c1c2x - i .

This design shows that although the embedded p refilte r inthe plant model destroys the matching condition, an SMCcan still be designed to reject the unknown disturbance.However, it is necessary to restrict the class of disturbancesto those which have bounded derivatives. Furthermore,derivatives of the st ate are also required in the design.E.2 A,Disturbance Estimation Solution

For the nominal integrator plant with limited bandwidthactuator dynamics given by Eqns.(25,26),we introduce thesame set of sensor dynamics as in Eqn.(20) and use a distur-bance estimator similar to Eqn.(2l ),with only x , replacingU , 2 = 5 , + sgn(x, - 2 ) . (32)With sliding mode occurs on x - x = 0, the disturbance d ( t )is estimated with the equivalent control given by Eqn.(23)to O ( T ~ ) . ith the disturbance compensated, the remain-ing task is to design a linear feedback control to achieve

the desired transient performance. T he resulting feedbackcontrol law is given by

(33)= - k l 2 - 2xa - sgn(x, - ) l e q .With 7 , = 0.01, and a = 0.2, the feedback gains IC1 =31.25, and k2 = 6.25 place the poles of third order systemdynamics, which is consists of the actuator dynamics andthe integrator plant, at (-2.5, -2.5, -5}. Again, we usethe same third order butterworth low pass filter with a 50Hz bandwidth as before to filter the equivalent control sig-nal. Figure 8 shows the effects of the constant disturbanced ( t ) = 0.5 have been neutralized since the error betweenthe reference command and the plant state is reduced tozero in steady state . The disturbance estimate is shown inFigure 9 to reach its expected value in steady state.F. Frequency Shapzng

An approach which has been advocated for attenuatingthe effectsof unmodeled parasitic dynamics in sliding modeinvolves the introduction of frequency shaping in th e de-sign of the sliding manifold [5]. In stead of treating th esliding manifold as the intersection of hyperplanes definedin the s tate space of the plant, sliding manifolds whichare defined as linear operators are introduced to suppressfrequency components of the sliding mode response in adesignated frequency band. For unmodeled high frequencydynamics, this approach implants a low pass filter eitheras a prefilte r, similar to introducing an artificial actuatordynamics, or as a post-filter, functioning like sensor dynam-ics. The premise of th is so-called Frequency Shaped Slid-ing Mode design, which was motivated by flexible roboticmanipulator control applications [32], is tha t th e effects ofparasitic dynamics are as critical on the sliding manifold.However, robustness to chattering was not addressed in thisdesign. By combining frequency shaping sliding mode andthe SMC designs introduced earlier, the effects of parasiticdynamics on switching induced oscillations, aswell as theirlong term interactions with sliding mode dynamics can behandled.F.l A frequency shaped SMC designdynamics, we introduce a frequency shaping post-filter,

For the nominal integrator plant with parasitic sensor

xp+ 2w,xp + w;x, = 2 , , (34)(35)p = Plkp + 2 z p

The sliding manifold is defined as a linear operator, whichcan be expressed as a linear transfer function,

Given an estimate of the lower bound of the bandwidthof parasitic dynamics, the post-filter parameter wp an bechosen to impose a frequency dependent weighting functionin a linear quadrat ic optimal design whose solution providesan optimal sliding manifold. The optimal feedback gains

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are implemented as p l , p 2 in Eqn.(36), and they ensuresthat the sliding mode dynamic response has adequate rolloff in th e specified frequency band.G . Robust Control Design based on Lyapunov Method

For plants whose dynamic models are uncertain, robustcontrol design which utilizes Lyapunov functions of th enominal plant has been proposed. The origin of this ap-proach can be traced to work published in the 1970's byLeitmann and Gutman [33]. The resulting feedback controllaw is of the form,

n dV,v v =- Et", z E IR" (37)BTVVllBTVVll d XU = -p (z , t )where p( . , .) is a scalar feedback gain, z E Et" is the statevector, and B(z, ) E R n x ms the inpu t matrix in an affinedynamic system,

k = f(x, )+ B ( z , )u+ h(z , ) U E Rm (38)and V(z) > 0 is a Lyapunov function of the nominal openloop plant, i.e., Eqn.(38) with u = 0 and h(. , . )= 0. Forunity feedback gain, p(.;) = 1, the norm of th e abovefeedback control is equal to unity for any (5 , t ) , hus it isalso referred to as unit control.

It is critical to point out that in any uncertain plant givenby Eqn.(38) and unit control in Eqn.(37), the feedback con-trol is in fact a sliding mode control which is discontinuouson the sliding mode manifold,

s(z) = BTVV = 0 , (39)provided that the unknown disturbance denoted by theterm h( z , ) can be rejected by the choice of the scalar feed-back functional p( . , .) . Under the matching condition [35]that there exists a vector X(z, t )E Rm uch that

h ( s , )= B ( z , )X(s , t ) (40)sliding mode on s(z) = 0 is guaranteed with

p(z , t )> Xo(zc , t ) z Il4s,t)ll. (41)Since sliding mode is the principle mechanism with whichuncertainties and disturbances are rejected in robust con-tro l of uncertain systems, the robustness of these feedbackcontrollers with respect to unmodeled dynamics are identi-cal to continuous time SMC, and the respective engineeringdesign issues can be addressed as outlined in this section.

111. DISCRETE IM E LIDINGMODEWhile it is an accepted practice for control engineers to

consider the design of feedback systems in the continuoustime domain - a practice which is based on the notionthat, with sufficiently fast sampling rate, the discrete timeimplementation of the feedback loops is merely a matterof convenience due to the increasingly affordable mic roprecessor. The essential conceptual framework of the feedbackdesign remains to be in th e continuous time domain. For

VSS and SMC, the notion of sliding mode subsumes a con-tinuous time plant, and a continuous time feedback control,albeit its discontinuous, or switching characteristics. How-ever, Sliding Mode, with its conceptually continuous timecharacteristics, is more difficult to quantify when a discretetime implementation is adopted. When control engineersapproach discrete time control, the choice of sampling rateis an immediate, and extremely critical design decision,unfortunately, in continuous time Sliding Mode, desiredclosed loop bandwidth does not provide any useful guide-lines for the selection of sampling rate . In the previoussection, we indicate that asymptotic observers or slidingmode observers can be constructed to eliminate chatter-ing. Observers are most likely constructed in discrete timefor any real life control design. However, in order for theseobserver-based design to work, sampling rate has to be rela-tively high since the notion of continuous time sliding modeis still applied.

For Sliding Mode, the continuous time definition and itsassociated design approaches for discrete time control im-plementation have been redefined to cope with the finitetime update limitations of discrete time controllers. Dis-crete time Sliding Mode (DSM) was introduced [34] fordiscrete time plants. The most striking contrast betweenSM and DSM is that DSM may occur in discrete time sys-tems with continuous right hand side, thus discontinuouscontrol and Sliding Mode, are finally separable. In discretetime, the notion of VSS is no longer a necessity in dealingwith motion on a Sliding Manifold.IV. SAMPLED ATASLIDING ODECONTROL ESIGN

We shall limit our discussions to plant dynamics whichcan be adequately modeled by finite dimensional ordinarydifferential equations, and assume that an apriori band-width of the closed loop system has been defined. Thefeedback controller is assumed to be implemented in dis-crete time form. The desired closed loop behavior includesinsensitivity to significant parameter uncertainties and re-jection of exogenous disturbances. Without such a demandon the closed loop performance, it is not worthwhile toevoke DSM in the design. Using conventional design ruleof thumb for sampled data control systems, it is reason-able to assume that for the discretization of the contin-uous time plant, we include only the dominant modes ofthe plant whose corresponding corner frequencies are wellwithin the sampling frequency. This is always achievablein practice by anti-aliasing filters which are inserted at theplant outpu ts. Actuator dynamics are assumed to be ofhigher frequencies than the sampling frequency. Otherwise,actuator dynamics will have to be handled as part of thedominant plant dynamics. Thus, all the undesirable para-sitic dynamics manifest only in the between sampling plantbehavior, which by default of using sampled data control,is essentially the open loop behavior of the plant. Clearly,this removes any remote possibilities of chattering due tothe interactions of sliding mode control with the parasiticdynamics.

We begin to summarize sampled da ta sliding mode con-

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trol designs with the well understood sample and hold pro-cess. This may seem to be elementary at first glance, it ishowever worthwhile since the matching conditions for thecontinuous time plant are only satisfied in an approxima-tion sense in the discretized models. We shall restrict ourdiscussions to linear time-invariant plants with uncertain-ties and exogenous disturbances,

? = A x + B u + E d , Z E I R ~ ,L E I R ~ ,E I R ' (42)where A ,B, are constant matrices, and d ( t ) is the exoge-nous disturbance. For the plant (42 ) , we assume that thesystem matrices are decomposed into nominal and uncer-tain components,

A = A + A A , B = B + A B (43)

the sampling instants. Let the sliding manifold be definedby

Sk = c x k 0 . k = 0,1 , . . s ( k T ) = S k (52)Two different definitions of discrete time sliding mode havebeen proposed for discrete time systems. While these def-initions share the common base of using the concept ofequivalent control, the one proposed in [34] uses a defini-tion of discrete time equivalent control U = u(kT)whichis the solution of

.$,++I = 0 , k = 0, 1,. . (53 )On the other hand, u iq is defined in [36]as the solution of

Ai = sk+l - k = 0 , k = 0,1, . . (54)where A , B denote the nominal components. Let theadmissible parametric uncertainties satisfy the followingmodel matching conditions [35]

Note that (53) implies (54), however, the converse is nottrue . Herein, the first definition given by Eqn.(53) shall beused.rank([ B AA AB f E ] )= rankB . (44) A . DSM Control Design for Nominal Plants

Given the nominal plant with no external disturbance,In DSM, byhe DSM design becomes intuitively clear.

definition,The discrete time model is obtained by applying a sampleand hold process to the continuous time plant with Sam-pling period T , which to O ( T 2 ) ,s given by:

sk+l Cxkfl = c(FXk +GUk)= 0 , (55)Z k + l = F X k 4-GU k + Ddk , 50 5 ( t o ) , (45)and provided tha t CG is invertible, t he DSM control whichis also the equivalent control, is given by the linear contin-uous feedback control,

s (kT) = x k ' = u k d ( k T ) == d k ' ( 4 6 )where F,G and D result from integrating the solution ofEqn.(42) over the time interval t E [kT, k + 1)T]with uk = -[CG]-lCFxk. (56)

u(t)= u(kT) d ( t ) = d ( k T ) , (47)

F = exp(AT) , G = rB , (48)F = F i-A A , r = lT xp(AT)dT, (49)G = Gi-AB, D = FE (50)

This discrete time model is an O ( T 2 )approximation ofthe exact model which is described by the same F and Gmatrices, but because the exogenous disturbance is a con-tinuous time function, the sample and hold process yieldsa D matrix which renders the matching condition in thecontinuous time plant to be a necessary, but not sufficientcondition for the exact discrete time model. However, byadopting the above O ( T 2 ) pproximated model, it followsfrom (50) that, if the continuous time matching condition(44) is satisfied, th e following matching conditions for thismodel hold :

The only other complication is that since l/Gll = O ( T - l ) ,the required magnitude of thi s control may be large. Ifthe bounds U. on uk are taken into account, the followingfeedback control has been shown [19] to force the systeminto DSM:

-[CG]-1CF5k, if 1uk/< au k = { -iisgn(sk), if l t L k l 2 ii (57)

A.l DSM Control of the integrator plantFor the nominal integrator plant with parasitic sensor

dynamics, we design an DSM controller based on Eqn.(57).Let the sensor time constant T~ = 0.02, and the controlmagnitude E = 1. The desired closed loop bandwidth isgiven to be about one Hz . A good choice of the samplingfrequency would be about 10 Hz (T=0.1) since the sen-sor dynamics are of 50 Hz, and therefore can be neglectedinitially in the design. The DSM control takes the form of

From an engineering design perspective, the O ( T 2 )mod-els are adequate since the between sample behavior of t hecontinuous time plant is also O ( T 2 ) lose to the values at

where x i = z , (kT) is the sampled value of the sensor out-put z,(t). Note that due to the control bounds, a linearfeedback control law is applied inside a boundary layer of

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thickness 2T about the sliding manifold x i = 0. With-out sensor dynamics, the behavior inside the boundarylayer is that of a deadbeat controller. The sensor dynam-ics impose a third order discrete time system inside thisboundary layer, and its eigenvalues are inside the unit cir-cle at {-0.002,O.l -f 110.436). For reference, the discretemodel of the open loop nominal plant and the sensor dy-namics has a pair of double real pole almost at the origin(4.54 l o p5 ) , which result from sampling at a frequencymuch lower than the sensor's 3dB corner frequency, anda pole at unity which is due to the integrator plant. Thethird order system response can be seen in Figure 10 wherethe sample values of the error between the constant unityreference command and the sensor output is plotted. Notethat only the behavior inside the boundary layer is shown,and it agrees well with the predicted third order behav-ior. The steady state error magnitude of 0.05 is due to theconstant disturbance d ( t ) = 0.5 as applied to this plantbefore, and the effective loop gain being T-l = 10. Fig-ure 11displays th e continuous time error of the plant sta teand the discrete time error of the sensor output where thetime lag due to the sensor dynamics can be seen during thetransient period.B. DSM Control with Delayed Disturbance Compensation

The earlier DSM control design for nominal plants canbe modified to compensate for unknown disturbances inthe system [37], [38]. From the discrete model in Eqn.(45) ,the one step delayed unknown disturbance

ad i - l = D d k - 1 = X k - X , - , - G U k - 1 , (59)can be computed, given the measurements x k , x k - 1 andu k - 1 , and the nominal system matrices F , G . If we let

The effectiveness of this controller is demonstrated by ex-amining the behavior of the s k when the control signal isnot saturated,

Sk+l = CD(dk - k - 1 ) . (61)If the disturbance has bounded first derivatives, Le . , 1215d < 03, d k - k - 1 is of O ( T ) , nd from the definition givenin Eqn.(50), IlDll = O ( T ) ,hence / s k i = O ( T2) , mplyingthat the motion of the system remains within an O(T2)neighborhood of the sliding manifold. This controller hasalso been shown [19] to force the system into DSM if thecontrol signal is initially saturated.

On the sliding manifold, the system dynamics are, toO(T2) ,nvariant with respect to the unknown disturbance.Since similar matching conditions exists for the O ( T2) is-crete models we have adopted, it follows from continuoustime sliding mode [28] that by using a change of sta te vari-ables, the discrete model can be transformed into

(62)k + l F 1 l x k + p 1 2 4 ,xi+l = F 2 1 ~ :+ F 2 2 4 + G 2 U k + D z d k , (63)

with t he sliding manifold given bysk = c l x i + c 2 X i = 0, (64)

and C2G is nonsingular. By eliminating xz, the reducedorder sliding mode dynamics are O(T2) pproximated by

(65)k+ l = ( 4 1 - F l 2 C $ 1 ) x : .B.l Discrete time disturbance compensation for the inte-

We continue with the DSM control design using the samesampling frequency and system parameter values. The con-troller which takes into the one step delayed disturbanceestimates is given by

grator plant

Note the PID controller structure of this controller whenthe system is inside the boundary layer. Figure 12showsthe sampled error between the reference command and thesensor output. The practically zero steady state error ismuch better than our O (T2) stimate due t o the PID con-troller structure . The one step delayed disturbance esti-mate is given in Figure 13, showing convergence to the ex-pected value. Figure 14 displays the continuous time errorbetween the plant state and the reference, and it s discretetime measurements.C. D SM Control with Parameter Uncertainties and Dzs-turbances

With the presence of system parameter uncertainties, theabove approach using one step delayed disturbance esti-mates can be still be applied. However the one step delayedsignal contains both delayed state and control values,

af k - 1 = A F x k - l + A G U k - l + D d k - 1 = X k - F X k - 1 - G U k - 1 ,(67)where A F = I'AA, AG = rAB. The DSM control is of

the same form as Eqn.(60), with d i - l replaced by f k - 1 .The behavior of sk is prescribed by

Sk+l = c ( f k - k - 1 ) = C D ( d k - k - 1 ) ++ C A F ( X k - k - 1 ) + C A G ( u k - U k - 1 ) . (68)

SincellAFll = O ( T ) ,we haveis bounded, x k - x k - 1 is of O(T) , and since

~ k + l C A G ( U ~ k - 1 ) + O ( T 2 ) . (69)Due to the coupling between Sk and U k , it has beenshown [38], [39] that the behavior outside the sliding man-ifold is governed by the following second order differenceequation,

~ k + l= - C A G ( C G ) - ' [ 2 s k - k - 1 1 + O ( T 2 ) , (70)which has poles inside the unit circle for sufficiently small11 AB 1. The permissible control matrix uncertainties are

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dictated by the above stability condition which determinesthe convergence on the sliding manifold. Note that pro-vided th at the parameter uncertainties on the system ma-trix, they do not impact the convergence, nor they affectthe motion on the manifold.C.1 Compensation for gain uncertainties in integrator

plantWe shall introduce gain uncertainties in the integrator

plant to examine their effects on the convergence of thesliding manifold. The actual plant is given by

j.= (1+ 7 ) U + d ( t ) (71)where y represents the gain uncertainty in the integra-tor . The DSM controller in Eqn.(66) can be used againbecause the right hand side of the one step delayed sig-nal is the same regardless of the parametric uncertainties.The root locus of th e second order system governing themotion outside the manifold is plotted in Figure 15 for-1 5 y 5 0.34. For y = 1, there is a pair of double polesat unity, and for y = 1/3, one of the poles becomes -1.The case for y = -0.5, corresponding to a pole of complexpairs -0.5 +~0.5,s simulated with the same reference anddisturbance as in the previous studies. Figure 16 showsthe sampled error between the reference command and thesensor output which converges to zero. Figure 17 displaysthe estimates of the exogenous disturbance and the residuecontrol signal due to the gain uncertainty. The continuoustime error of the plant s tate and the discrete time error ofthe sensor output are shown if Figure 18 for comparison.

V. CONCLUSIONSWe have examined systematically SMC designs which are

firmly anchored in sliding mode for the continuous time do-main. Most of these designs are focused on guaranteeingthe robustness of sliding mode in the presence of practicalengineering constraints and realities, such as finite switch-ing frequency, limited bandwidth actuators , and parasiticdynamics. Introducing DSM, and restructuring the SMCdesign in a sampled data system framework are both appro-priate, and positive steps in sliding mode control research.It directly addresses the pivotal microprocessor implemen-tation issues; it moves the research in a direction which ismore sensitive to the concerns of practicing control engi-neers who are faced with the dilemma of whether to ignorethis whole branch of advanced control methods for fear ofthe reported implementation difficulties, or to embrace itwith caution in order to achieve system performance oth-erwise unattainable. However, as compared with the idealcontinuous time sliding mode, we should also be realisticabout the limitations of DSM control designs in reject-ing disturbances, and i n its abi l i ty to w i t h s t a n d p a r a m e t e rvariations. The real tes t for the sliding mode research com-munity in t he near future will be th e willingness of controlengineers to experiment with these SMC design approachesin their professional practice.

REFERENCESV. I. Utkin, Variable structure systems with sliding modes,IEEE Rans. Automat. Contr., Vol.AC-22, No.2, pp. 212-222,1977.Friedland, B. Advanced Control Syste m Design, Prentice Hall,Englewood Cliffs, N .J . , 1996.Asada, H. and J-J. E. Slotine, Robot Analysis and Control,pp. 140-157, John Wiley and Sons, 1986.Bondarev, A. G ., S. A. Bondarev, N. E. Kostyleva, and V. I.Utkin, Sliding Modes in Systems with Asymptotic S tat e Ob-servers, Automation and Remote Control, pp. 679-684, 1985.Young K. D. and U . Ozgiiner, Frequency Shaping CompensatorDesign for Sliding Mode, Special Issue on Sliding Mode Control,International Journal of Control, pp. 1005-1019, 1993.Young, K. D. and S. Drakunov, Sliding mode control with chat-tering reduction, in Proceedings of the 1992 American ControlConference, Chicago, Illinois, pp. 1291-1292, June 1992.Su, W. C., S . V. Drakunov, U. Ozgtinerand K. D. Young, Slidingmode with chattering reduction in sampled data systems , PTO-ceedangs of the 32nd IEE E Conference on Decision and Control,San Antonio, Texas, pp. 2452-2457, December 1993.Young, K. D., and S. V. Drakunov,Discontinnous frequencyshaping compensation for uncertain dynamic systems, Proceed-ings 12th IFAC World Congress, Sydney, Australia, pp. 39-42,1993.Young, K. D. (editor), variable Structure Control fo r Roboticsand Aerosvace Amlications. Elsevier Science Publishers. 1993..[lo] Zinober, A . S. (editor), Vdriable Structure and Lyapunov Con-trol, Springer Verlag, London, 1993.

[ll] F.Garofalo and L.Glielmo (editors), Robust Control via VariableStructure and Lyapunov Techniques, Lecture Notes in Controland In formation Sciences Series, Vol. 217, pp. 87-106, Springer-Verlag, Berlin, Heidelberg, New York, 1996.1121 Utkin, V. I., Variable Stru cture Systems: Present and Future,Avtomatika i Telemechanika, No. 9, pp. 5-25, 1983 (in Russian),English Translation, pp. 1105-1119.[13] DeCarlo, R. A, S. H. Zak, and G . P. Matthews, Variable struc-tur e control of nonlinear multivariable systems: A tutorial, PTOC.[14] Utkin, V. I., Variable Stru cture Systems and Sliding Mode -State of the Art Assessment,, Young, K. D. (editor), VariableStructure Control for Robotics and Aerospace Applications, pp. 9-32, Elsevier Science Publishers, 1993.[15] Hung, J . Y., W. B. Gaq and J. C Hung, Variable structurecontrol: A survey, IEEE nans. Ind. Electron., Vol. 40, No. 1,[16] Flugge-Lutz, I., Discontinuous Automatic Control, PrincetonUniversity Press, 1953.[17] Tikhonov, A. N., Systems of differential equations with a smallparameter multiplying derivations, Mathematicheskii Sbornik,Vol. 73, No . 31, pp. 575-586, 1952 (in Russian).[18] Kokotovic, P.V., H. K. Khalil, and J. OReiley, Singular pertur-bation methods in control : analysis and design, Academic Press,

1986.[19] Utkin, V. I. , Sliding Mode Control in Discrete-Time an dDifference Systems, Variable Stmcture and Lyapunov Control,A.S.Zinober (ed ito r), Springer Verlag, London, pp.83-103, 1993.[20] Utkin, V. I., Slzding Modes and their applications tn VariableStructure Systems, Moscow:MIR, 1978 (translated from Russian).[21] Young, K-K. D., P. V. Kokotovic, and V. I. Utkin, A SingularPerturbation Analysis of High Gain Feedback Systems,, IEEE7h ns . Auto. Contr., Vol. AC-22, No. 6,pp. 931-938, 1977.[22] Slotine J-J. and S. S. Sastry , Tracking Contro l of NonlinearSystems using Sliding Surfaces with Application to Robot Ma-

nipulator, Int. J . Control, Vol. 38, No. 2, pp. 465-492, 1983.[23] Burton, J. A., A. S. I . Zinober, Continuous approximation ofvariable structure control, Int. J . System Sci., Vol. 17, No . 6,[24] Young, K-K. D. and P. V. Kokotovic, Analysis of FeedbackLoop Interaction with Parasitic Actuators and Sensors, Auto-matzca, Vol 18 , September 1982, pp . 577-582.[25] Kwatny, H. G. , K. D. oung, The Variable Struc ture Ser-vomechanism,, Systems and Control Letters, Vol. 1, N o. 3,pp. 184-191; 1981.[26] Young, K. D., an d V. I. Utkin, Sliding Mode in Systems withParallel Unmodeled High Frequency Oscillations, Proceedings of

of IEEE, Vol. 76, NO . 3, pp. 212-232, 1988.

pp . 2-22, 1993.

pp. 875-885, 1986.

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the Third IFAC Symposium on Nonlinear Control Systems De-sign, Tahoe City, California, Jun e 25-28, 1995.[27] Young, K-K. D. and H. G. Kwatny, Variable Struc ture Ser-vomechanism Design and i ts Application to Overspeed ProtectionControl, Automatzca, Vo1.18, No. 4, pp . 385-400, 1982.[28] Utkin, V. I. , Sliding Modes in Control Optimizatzon, Springer-Verlag 1992.[29] Slotine, J-J. E., J. K. Hedricks, an d E. A. Misawa, On SlidingObservers for Nonlinear Systems, ASME J . Dynamic Systems,Measurement and Control, Vol. 109, pp. 245-252 , 1987.[30] X u, J-X., H. Hashimoto and F. Harashima, On the design of aVSS Observer for Nonlinear Systems, h n s . of the Society of In-strument and Control Engineers (SICE), Vol. 25, No. 2, pp . 211-217, 1989.[31] Korondi, P., H. Hashimoto, K. D. Young, Discretetime SlidingMode Based Feedback Compensation for Motion Control, P r o -ceedings of Pow er Electronics and Motion Control (PEMCSG),Budape st, Hungary, ,Sept. 2-4, 1996, Vo1.2, pp. 21244-2 /248,1996.

[32] Young K . D., U. Ozgiiner, and J-X. Xu, Variable StructureControl of Flexible Manipulators, Young, K. D. (editor), Vari-able Structu re Control for Robotzcs and Aerospace Applications,pp. 247-277, Elsevier Science Publishers, 1993.[33] Gutman, S. and Leitmann, G., Stabilizing Feedback Controlfor Dynamic Systems with Bounded Uncertainties, Proceedingsof IEEE Conference on Decision and Contro1,pp. 94-99, 1976.[34] Drakunov S. V. and V. I. Utkin, Sliding mode in dynamic sys-

tems, Interna tional Jour nal of Control, Vo1.55, pp. 1029-1037,1990.[35] Drazenovic, B., Th e invariance conditions in variable struc turesystems, Automatica, Vo1.5, No. 3, pp. 287-295, 1969.[36] Furuta, K. , Sliding mode control of a discrete system, Systemsand Control Letters,,,Vol.l4, pp. 145-152, 1990.

[37] R. G. Morgan and U. Ozgiiner, A Decentralized Variable Struc-tur e Control Algorithm for Robotic M anipulators, IEEE Journalof Robotics and Automation, 1,1, pp. 57-65, 1985.[38] W-C. Su, S. V. Drakunov and U . Ozgiiner, Sliding Mode Control

in Discrete Time Linear Systems, Prepnnts of IFAC 12th WorldCongress, Sydney, Australia , 1993.[39] Su, W. C., S. V. Drakunov, U. Ozgiiner, Implementationof Variable Structure Control for Sampled-Data Systems, Ro-bust Control via Variable Structure and Lyapunov Techniques,F.Garofalo and L.Glielmo (editors), Lecture Notes in Control

and In formation Sciences Series, Vol. 217, pp. 87-106, Springer-Verlag, Berlin, Heidelberg, New York, 1996. 4)w8

R m lLocus Sensor dynamm. au_s=O.Ol, ez=.w3-->1150

-

t 11150 -lw -50 50Rea l-250 -2w

Figure 1: Root locus of boundary layer control, crossovergain g,=200.

Sensor dynamics, au-& 01, ayer ez=O 01,O 35, dis3urbbance;O 5

0 - -08I -I0006 i 1O w lw2 I

04024004

Figure 3: Block diagram of Observer based Sliding ModeControl.

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observerbasedSMC, ez-2a-3, tau_s.Ol.h-10. d=O50 010.008OW6-..

o mo.w2-.

g4.032o m - .4.006-.o@yj-001

- . . . . ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . ' . . . : . . . I . . . . . . . . . . . . .

- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . : . .. -

0 - . . .: . . . . ..:.. . . . . . . . . :. . . . ; . . . . . . . . ; . . .-b-- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . : .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . : . . . . . . . . . . . . . . : . . . .-

1 I I0 0.5 1 1.5 2 2.5 3 3 5 4

SMDisRlrbance estimator- tau-= 01 au-fdlO2. d=O 50 8 , I

1.081.06j w . . . .m-{ .02

(nE 1.go.9 .5mm

iiOW-..0.94.

Figure 4: Observer based SMC: error between reference Figure 7: SM Disturbance estimator Control: distur-command and observer state. bance estimntp.observerbasedSMC, ezSa-3, tau_* 01, h=10.d a . 51 1 , Limited bandwidthactuatwwith SM dlstukma, esm"ar. - 1 . . . A . . . . . . . . . . . . . . . . . . . . . . . . . . .-. . . . . . . . . . . . . . . . . ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . ' . . . . . , . . . . . . , . . . . . I . . . . I . . . . . . . . . . . . . . . . . .. . . . . . . 1 . . . . . . . . . ; . . . . . . . . . . . . . . . . . . . : . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . : . . . . . . . . . L . . . . . . . . . . . . . . . . : . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . ., . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .

o , g 2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . .

0 8 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L . . . : . . . . .. .

0,6-.... i ! : : ! *. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 8 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L . . . : . . . . .. .0,6-.... i ! : : ! *. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , ..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .o . g - .0 8

1

Limitsdbandwlmhactuator wlth SM disturbance esbmator0.6

. . . . .

0 2 0 0.5 1 1.5 2 25 3 3.5Qme,sBc

0 4 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E280,3-.*o ,2- . . . . . . . . . . . . . . . . . .:. . . . . . .;.. . . . . :. . . . .:. . . . . . :. . . . . . .3

0 1 . . . . . . .:. . . . - 1 . . . . . 1. . . . . . .:. . . . . .:. . . . . . . . . . . . . . . . .O O B 4bme,rec oi 015 li5 d 215 A 3'5

1

4

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Discrete ime SMC- tau_% 01, =O5

0 6 -

l05-

iE O 4 -z303-I].4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . .1 : I r ;

. . . . . . . .

0.5 1 1 5 ,,:* 2.5 3 3.5O l C I

U. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

OO 5 10 m a 3 0 3 5l 5 umesteps

tween reference command and sensor output.02 , ,, , Dlruete,!imeSMC;lau_s=Ol d=O5

O 1 5 t \.g o 1 t I

O 14 15I" 05 1 15 bmesec 25 3 3 5

one step delaved disturbance estimateDiscreteb n m SMC wilh disturbanceemnmbon tau-* 01 d=O 5

01

, , L , , I0 05 1 15 2 25 3 3 5 42 tlme sec

Figure 11: DSM Control for nominal plant: continuous Figure 14: DSM Control with disturbance compensation:continuous time and discrete time error resDonses.ime and discrete time error responses.

D " t e lime SMCwlm distulrbanceBstimabon tau-- 01 d=O5

008006- ! \040 02$ 04 024 06

0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 6 . . . . . . . . . . . . . . . . . . . . . . . . . . .

+ + + + + I0.4 . . . ~ . . . . . . . . . . . + . . . . . . . .+ . . . . . . . . . .I # ! + i + .+ Io + + + + + +*PE42lji4 4 . . . . . . . , - + + + . . . ; - . + + . ++ + . , . . . 1

. . . . . . . . . . . . . . . . . . . . . . . . . .6 .... . . . . . - .4 8 . . . . . . . . . , . . . . . . . . . . . . . . . . . . .

1 1 4.5 0 0.5 1Real partFigure 12: DSM Control with disturbance compensation: Figure 15: DSM control with control parameter varia-

tions: root Locus for evaluating sliding manifold conver-gence.

error between reference command and sensor output.

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Discretehme SMCwith paramefetuncemnbes anddlslurbacneesbmabon1

0.4L

0.2

0

0 5 1 0 1 5 2 0 2 5 3 0 3 54.2 hmes step

-

-

Figure 16: DSM Control with control parameter varia-tions: error between reference command and sensor output .

Discrere bm e SMCwlm parametet u n c a " e s and dishlrbacneemmaton

-n, U5 I O 15 x) 25 30 35hmes step

Figure 17: DS M Control with disturbance compensation:one step delayed parameter and disturbance estimate..

Disnete bme SMC with parmeter uncerfamnbes and disturbacneemmaton1 , , I

1

Figure 18: DS M control with control parameter varia-tions: continuous time and discrete time error responses.

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a control engineers quide to sliding mode control

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