HECHNJfHIP•- -^ • » s -« £ y tj ^ j.^ j ^ j-"<.-eugncvyg ^^ ^BASIC THEOREMS ON SLIDING MODECONTROLLERSRomano M. DeSantisSeptembre 1995Tous droits réservés. On ne peut reproduire ni diffuser aucune partie du présent ouvrage, sous quelqueforme ou par quelque procédé que ce soit, sans avoir obtenu au préalable l'autorisation écrite des auteurs.Dépôt légal, Septembre 1995Bibliothèque nationale du QuébecBibliothèque nationale du CanadaBasic Theorems on Sliding Mode ControllersRomano M. DeSantisDépartement de Génie Electrique etGénie InformatiquePour se procurer une copie de ce document, s'adresser au:Service des EditionsEcole Polytechnique de MontréalCase postale 6079, Succursale Centre-VilleMontréal (Québec) H3C 3A7Téléphone: (514) 340.4473Télécopie: (514) 340-3734Compter 0,10$ par page et ajouter 3,00$ pour la couverture, les frais de poste et la manutention. Régleren dollars canadiens par chèque ou mandat-poste au nom de l'Ecole Polytechnique de Montréal.Nous n'honorerons que les commandes accompagnées d'un paiement, sauf s'il y a eu entente préalabledans le cas (rétablissements d'enseignement, de sociétés ou d'organismes canadiens.BASIC THEOREMS ON SLIDING MODE CONTROLLERSDeSantis, R.M.,Ecole Polytechnique de MontréalAbstract.Basic results are présentée! that play an essential rôle inthe analysis and design of sliding mode controllers. They includenecessary and sufficient conditions for a controller to be on asliding mode régime, characterization properties of the ensuingfeedback System dynamics, and sensitivity properties concerning theinfluence of parameter and external perturbations over thisdynamics. Although perhaps excessively short and necessarilyincomplète, this présentation offers the advantage to be carriedout under a unified setting and should provide a suitableintroduction to this mathematically difficult subject.1. IntroductionIn thèse last few years sliding mode controllers have beenrecognized as a useful addition to the already formidable arsenalof techniques that are relevant to the control of both linear andnonlinear plants(see, for example, Utkin 1992, DeCarlo Zak andMatthews 1988, and Buhler 1986). By inducing a feedback behaviorthat can potentially be made independent of plant parametervariations and external perturbations, this usefulness appears tobe particularly promising in applications where the controlledSystem is required to have a high tracking accuracy and a highspeed of response and where, at the same time, fast parametervariations and high level nonstationary perturbations may beprésent.It follows that one of the most important reasons for theadoption of sliding mode controllers is that they offer aparticularly robust performance with respect to certain classes ofperturbations and parameter variations (see, for example, El-Ghezawi et al. 1883, DeCarlo et al. 1988, Utkin 1991, Slotine1992). A second important reason is that they play a significantrôle in the performance improvement of existing controllers such asadaptive and model following controllers (Balestrino et al. 1984).A third reason is that they provide a natural vehicle for theanalysis and design of discrète controllers (in particularmultivariable relay controllers as illustrated in Buhler 1986 andDeSantis 1992) as well as for the improvement of gain tuningprocédures for more standard controllers (DeSantis 1988, 1989).Finally, yet an additional important reason is that ideas andtechniques relevant to sliding mode controllers are also relevantan provide analogous benefits in the realm of state estimationand parameter identification problems (Edwards and Spurgeon 1994).A sliding mode controller is a «variable structure controller»operating in a «sliding mode régime». Intuitively, a variablestructure controller is a controller of which the structuredépends on the belonging of the state of the plant to certainrégions in the state space; opération in a sliding mode régime isan opération where the state of the plant is constrained to evolve(slide) on the surface defined by the intersections of thèserégions (sliding surface). As suggested by thèse characterization,the basic idea governing the analysis and design of thèsecontrollers unfolds as follows. First, the desired behavior of theSystem to be controlled is translated in terms of the convergenceof the state to an appropriately selected sliding surface. Second,the central strategy is designed so as to guarantee that thisconvergence takes place. Third, care is taken to ensure that thebehavior of the feedback system remains satisfactory incorrespondence to the (realistic) situation where plant parametervariations external perturbations and non-modelled dynamics areprésent/ and where the envisioned control strategy is onlyimplemented within a certain approximation and the stateconstraints are only satisfied within a certain tolérance.In what follows we présent the éléments of the theory ofsliding mode controllers that are essential to carry out thèsesteps. Thèse éléments consist of a formai définition of slidingmode controllers and five basic theorems. Thèse theorems concernthe following aspects: necessary and sufficient conditions for acontroller to be of the sliding mode type, necessary and sufficientconditions for the System dynamics to be invariant to the présenceof perturbations, characterization of state sensitivity to theprésence of unmodelled behavior, stability of the dynanucs under asliding mode régime, and spécial sliding mode characterizationproperties in the case of linear plants. Two additional theoremsillustrate the application of sliding mode controllers to thedesign of discrète controllers. While for brevity of exposition wewill not discuss detailed applications attention is called to thefact that examples of convincing engineering applications of thèsetheorems abound, and we encourage the reader to look for them inthe cited literature (some of the applications developed at theEcole Polytechnique may be found in Serfass 1987, DeSantis 1992,Haromada and El Ferik 1993, Krau 1994).2. Formai Définition of a Sliding Mode ControllerGiven a (time invariant, linear in the control) dynamicSystem,x = f(x) + B(x)u(x,t) ,with x and u n- and m-dimensional state and control vectors, asliding mode controller is a controller with the following threeproperties:l) it has a variable structure: the control is described by u(x,t)=u(x,t)~, whereUi(x,t) = Ui+(x) for Si (x) > 0 i=l...m,Ui(x,t) = Ui-(x) for Si (x) < 0 i=l...m,where: Ui , Ui" and Si are pre-assigned continuous functions;2) its structure is continuously switching: no trajectory, x(t)te [0, °°) , entirely generated by one of thèse 2m potential controlstratégies is contained in the setS:= {x(t)|x(t) such that Si(x(t)) == 0 for i=l,..m};3) its action on the System may be viewed as the limit of aséquence of control actions which, from a physical point of view,are easier to implement than the action provided by the controllerdescribed by l) and 2) . More formally: there exists an open set 0.in R", HnS^O, such that: given any e>0, there exists A(s)>0, ô(8)>0such that: for any Xoeîîy such that |s(Xo) |<ô, the trajectory of•x = f(x) + B(x)u(x,t), x(to) = Xowith u(x,t) such thatUi = ui' if | si (x, t) | > Amin(Ui,u~i) < Ui < max(Ui,u~i) otherwise,has the property thatl S(X)|<8.Note that vector function s(x):=[si(x) ... Sm(x)]r is usuallyreferred to as the sliding mode function.Remark 1. Properties l) and 2) make a sliding mode controllersomewhat "idéal" and difficult to implement in practice. However,property 3), stipulâtes that this idéal behavior must be attainablewithin any desired degree of tolérance in terms of a morerealistic, physically implementable controller.3. On Necessary and Sufficient Conditions for ExistenceThe following theorem states that, for controller to be in asliding mode , there must exist an équivalent controller that doesnot necessarily have a variable structure and which must satisfy acertain property with respect to the original variable structurecontroller.Theorem 1:(A Necessary Condition for the Existence of a Sliding Mode Controller).Given a dynamic system,x = f(x) + B(x)u(x,t) ,a necessary condition for a variable structure controller definedbyUi(x,t) = Ui (x) for Si (x) > 0 i=l...p,Ui(x,t)~ = Ui~(x) for Si (x) < 0 i=l...p,to be a sliding mode controller is that an équivalent control Uequ^solution ofG(X) [f(x) + B(x)Uequ]=0,G(x):=gradx(s)exists and satisfies the inequalityin{u+i, u~i} < Uequi < max{u+i, u~i}. •The next theorem stipulâtes that to check whether a variablestructure controller is indeed a sliding mode controller, the taskof verifying the validity of property 3) required by its définitionmay be replaced by the considerably simpler task of verifying thata certain Lyapunov-type function exists.Theorem 2:(A Sufficient Condition for a Controller to be of the Sliding Mode Type).Given a dynamic system,x = f(x) + B(x)u(x,t) ,a sufficient condition for a variable structure controllerdescribed byUi(x,t) = Ui (x) for Si (x) > 0 i=l...m,Ui(x,t)' = Ui~(x) for Si (x) < 0 i=l...m,to be a sliding mode controller is that there exist a scalarfunction v(s,x,t), continuously differentiable with respect to eachof its arguments, such that in a certain open set Q contained inR", QnS^O, with S:= {x|x such that Si (x) = 0 for i=l,..m}, thefollowing properties hold:i. v(s,x,t) is positive definite with respect to s; v(0,x,t)=0;ii. for a sufficiently small R > 0 and for all x s £î and any t:inf v(s,x,t) = hp sup v(s,x,t) = Hp|s(x) l = R |s(x) | = Rwith hp < Hp only dépendent on R;iii. a total time derivative of v is négative everywhere thisfunction is defined andsup v(s,x, t) = -TÛR.l s(x) l = Rwith mp > 0 only dépendent on R.It should be noted that while theorems l and 2 imply that thesliding mode vector function eventually converges to zéro in spiteof the eventual présence of parameter or external perturbationsthis property does not necessarily imply that the dynamics of thesystem is unaffected by such a présence. However, as established bythe following theorem, this implication holds if thèseperturbations satisfy some appropriate "matching conditions".Theorem 3 (Dynamics Invariance when the Matching Conditions are Satisfied).A necessary and sufficient condition for the dynamics of aSystem•x = f(x) + B(x)u(x,t) + D(x)^,under sliding mode control to be invariant with respect to theperturbation, t,, is that the range of D(x) be contained in therange of B(x). •In the same circle of ideas as above, the fact that the valueof the sliding mode vector function is close to zéro, |s(x(t))|<8for arbitrarily selected e, does not necessarily imply that thetrajectory of the system state is close to the state trajectoryobtained when the sliding function is equal to zéro (s(x(t)=0).However, as established by the following theorem, this is the caseunder appropriate hypothèses.Theorem 4 (State Trajectory Sensitivity to Sliding Mode Controller Approximation).Suppose that:i) Given any 8i>0, there exists ôi(8i)>0 such that on an interval [0,T] any solution ofx = f(x) + B(x)u(x,t) , x(to) = Xowith u(x,t) such thatUi = Ui" if | si (x) | > ôimin(Ui,u~i) < Ui < max(Ui,u~i) otherwise,has the property that l s(x) |<8i;ii) partial derivatives of B (x) [G(x)B(x)] with respect to allarguments exist and are bounded in any bounded demain;iii) there exist positive quantifies M and N such that for anysolution, x(t), of the system in i) one has|f(x) + B(x)u(x,t) |< M + N|x|;iv) the functionf(x*) - B(x*) [G(x*)B(x*)]-lG(x*)f(x*),with x' solution of (the System under équivalent control)x = f(x) - B(x) [G(x)B(x)]-lG(x)f(x), x* (to) = Xo*is Lipschitz continuous;Then, given any S2>0, there exists a 62 (82) >0 such that|x(0)-x*(0) |<Ô2implies|x(t)-x*(t) |<82, t8[0,T]. •The following theorem 5 specializes the ensemble of the aboveresults to the case where the plant is linear and time invariant.In particular: the characterization of an équivalent model todescribe plant dynamics under sliding mode; a procédure to designthe sliding mode function so that the dynamics of the plant undersliding mode has a certain behavior; the explicit characterizationof a family of control laws that allow the sliding mode to beattained.Theorem 5 (Spécial Properties in the Case of a Linear Plant).Given a linear scalar-input system,•x = Ax + Bu (x, t) + D^,consider the sliding mode scalar function s(x)=Cx. Then:i) under sliding mode, a necessary and suffiaient condition for thestability of the system dynamics is that the System obtained byusing the équivalent controllerx = [I - B(CB)-lC]Axis asymptotically stable (its non zéro eigenvalues have négativereal part);ii) if the dynamics associated with the équivalent controller isasymptotically stable, then the trajectory characterized by|s(x)|i=Bi if XiS(x) >0ô> l [CB]-1CD^|to be a sliding mode controller is thatai ^ min /tmax / ^ 61twhere: ai is the i-th column of A;v) a necessary and sufficient condition for the System dynamicsunder sliding mode to be invariant with respect to theperturbation, Ç, is that the range of D be contained in the rangeOf B. . •Remark 2. A detailed illustration of the application of theorem 5 tothe design of a sliding mode controller for a position serve withan elastic link may be found in Serfass and DeSantis 1987.4. Implementation of a Continuous Controller Via a Discrète ControllerControllers are usually désignée! by assuming the controlaction to be arbitrarily selectable within a certain continuousrange of values. However, on occasion, practical considérations maymake it convenient or necessary to implement the controller byimposing that the control action be restricted to have one out of adiscrète set of values. We must then deal with the problem ofmodifying an already designed (continuous) controller into acontroller which, while subject to the discrète set of valuesconstraint, does nevertheless produce an effect équivalent to thatof the designed controller. The following theorems suggest thatadoption of an appropriate sliding mode controller may provide anatural solution to this problem.Theorem 6. (Discrète Implementation of a Continuous Nonlinear Controller)Under the assumption that Bo(x,t) is of a full rank, Bo(x,t)Bo+(x, t)is independent of x and t, and the inverse of Bo(x, t)'B (x, t)exists;then: the dynamics of the Systemx(t) = f(x,t) + B(x,t)u(t) + p(t), x(0) = xowhere x(t) s R" represents the plant state, u(t) s Rm is thecontrol, p(t) s R" a disturbance; f (x, t) and B(x,t), appropriatelydimensioned real valued vector and matrix functions, are given byf (x,t) := fo(x,t) + ôf(x,t)B(x,t) := Bo(x,t) + ôB(x,t)where fo(x,t) and Bo characterize the "nominal" behavior of theplant, and ôf(x,t) ôB(x,t) describe the influence of parametervariations;submitted to a discrète control satisfyingUi'(t) 8 {Un, ..., UiNi}, i= l, .. , mSGN{Ui*(t)-UDi(x,t)+Ui(t) }:= - SGN{[Bo'o(t)]i}U(t) := Bo+{ôf(x,t) + ÔB(x,t)u(t)+p(t)}CT(t) := Bo(x,t)Bo+(x,t)S(t)Bo (x,t):= pseudo-inverse of Bo(x,t),ft .S(t) := (x(t) - fo(x,t) - Bo(x,t)UD(x,t) }dtJohas the following properties:i) the state trajectory of the system is described by•x(t) = fo(x,t) + Bo(x,t)UD(x,t)+ [I- B[Bo+B]-lBo+] {- BoUD(x,t) + ôf(x,t) + p(t)}ii) if the columns of ôf(x,t), ôB(x,t) and p(t) are a linearcombination of the columns of Bo(x,t), then•x(t) = fo(x,t) + BoUD(x,t),iii) if the columns of ôf(x,t), ôB(x,t) and p(t) are orthogonal tothe columns of Bo(x,t), thenX(t) = f(X,t) + B(X,t)UD(X,t) + p(t),iv) if only the columns of §B(x,t) are a linear combination of thecolumns of Bo(x,t), then»x(t) = fo(x,t) + BoUD(x,t) + v(t),wherev(t) := [I- [BoBo+]]{ôf(x,t) +p(t)}. . •In the case of a linear plant, the above result may besimplified as follows.Theorem 7: (Discrète Implementation of a Continuous Linear Controller).Under the assumption that matrix Bo is of a full rank, andthat the inverse of Bo'B exists;then: the dynamics of the system described byx(t) = Ax(t) + Bu(t) + p(t), x(0) = xowhere x(t) e R" represents the plant state, u(t) e Rm is thecontrol, p(t) s R" a disturbance; A and B, appropriatelydimensioned real matrices, are given byA:= Ao + ÔAB:= Bo + ÔBwhere Ao and Bo characterize the "nominal" behavior of the plant,and ÔA, ÔB describe the influence of parameter variations;submitted to a discrète control satisfyingUi~(t) 8 {Un, ..., UiNi}, i= l, .. , mSGN{Ui*(t)-UDi(x,t)+Hi(t) }:= - SGN{[Bo'o(t)]i}U(t) := Bo+{ôAx(t) + ÔBu(t)+p(t) }O(t) := Bo(x,t)Bo+(x,t)S(t)Bo (x,t):= pseudo-inverse of Bo(x,t),ft .S(t) := (x(t) - AoX(t) - BoUo(x,t) }dtJohas the following properties;i) the state trajectory of the System is described by•x(t) = AoX(t) + BoUo(x,t)+ [I- B[Bo+B]-lBo+] {- BoUD(x,t) + ôAx(t) + p(t)}ii) if Rank( [5A|ôB|p(t) |Bo]=Rank(Bo) then•x(t) = Aox(t) + BoUD(x,t),iii) if[I- [BoBo+]]ô=ô for Ô8SPAN{ôA|ÔB|p(t) }thenx(t) = Ax(t) + BuD(x,t) + p(t),iv) if Rank( [ÔB|Bo] ) = Rank(Bo)thenx(t) = AoX(t) + BoUD(x,t) + v(t),wherev(t) := [I- [BoBo+]] {ôAx(t) +p(t)}. . •Remark 3. An application of theorem 6 to the design of a bang bangcontroller for an overhead Eucledian crane may be found in DeSantisand Krau 1994. DeSantis and DeSantis 1993 apply theorem 7 to designa bang bang controller for a flexible beam.ConclusionsUsing the concept of an équivalent control together with thenominal model of the plant, we can use theorems l, 2, 5, 6 and 7to design a controller so as to attain a prescribed sliding moderégime and a desired state trajectory behavior. However,unaccounted for non-idealities, in both the physical plant and inthe components of the controller, often prevent the sliding modeconditions to be satisfied exactly and the state trajectory toconditions to be satisfied exactly and the state trajectory tocorrespond to that predicted by the theory. We must then be carefulto verify that, in spite of the influence of thèse non-idealities,the value of the sliding mode function remain small and that thetrajectory produced by the real controller remain close to thatproduced by the idéal controller under nominal operatingconditions.In the case of spécial non-idealities (memorylessapproximations of an idéal switching function, perturbationssatisfying the matching or mis-matching conditions, and similar),theorem 3 helps us establish the extent within which this is indeedthe case. However, this theorem is not of a great help inpredicting what happens in the case of other more général non-idealities such as time delays, hysteresis and neglected actuatordynamics. It becomes then useful, in such cases, to explore theinfluence of thèse éléments over the actual behavior of the Systemby means of a direct expérimentation, or of extended simulationsor by mean of both approaches.RéférencesEl-Ghezawi, 0.M.E., Zinober, A.S.I., Sillings, S.A., Analysis andDesign of Variable Structure Systems Using a Géométrie Approach,Int Journal of Control, Vol. 38, N.3, pp 657-671, 1983.DeCarlo, R.A., Zak, S.H., Matthews, G.P., Variable StructureCentral of Nonlinear Multivariable Systems: A Tutorial, Proceedingsof the IEEE, Vol 76, N. 3, 1988.Utkin, V.I., Sliding Mode and Their ^plication to VariableStructure Systems, MIR Publishers, Moscow 1978.Utkin, V.I., Sliding Modes in Control and Optimization, Berlin,Springer-Verlag, 1992.Balestrino, A., De Maria, G., Zinober, A.S.I, Nonlinear AdaptiveModel Following Control, Automatica, Vol. 20, N.5, pp 559-568,1984.Slotine, J.J., Li,W., J^plied Nonlinear Control, Prentice Hall1991.Buhler, H., Réglage par Mode de Glissement, Presses PolytechniquesRomandes, 1986.Edwards, C., Spurgeon, S.K., On the Development of DiscontinuousObservers, Int. Journal of Control, Vol 59, N. 5, pp. 1211-1229,1994.DeSantis, R.M., Krau, S., Bang-Bang Control of a 3-D EuclideanCrane, Robotica, Vol.12, Sept '94, pp. 449-554.DeSantis, R.M., A Novel PID for Speed and Position Control, ASMEJ. of Dynamic Systems, Measurements and Control, Vol. 116, Sept'94, pp. 315-323.DeSantis, R.M., On the Implementation of a Continuous ControllerVia a Discrète Valued Controller, ASME J. Dynamic SystemsMeasurements and Control, Vol. 115, Sept'92, pp.400-406.DeSantis, R.M., An Adaptive PI/Sliding Mode Controller for aSpeed Drive, ASME J. Dynamic Systems Measurements and Control,Vol. 111, Sept '89, pp. 409-415.Serfass, C., DeSantis, R.M., Contrôleur à Structure Variable pourune Articulation Robotique Elastique, EPM/RT-87/19, RapportTechnique de l'Ecole Polytechnique de Montréal, (mai 1987).DeSantis, R.M., DeSantis, S., A Bang Bang Controller for VibrationRéduction in a Rotating Flexible Beam, EPM/RT/92/19, RapportTechnique de l'Ecole Polytechnique de Montréal, (Automne 1992).Hammada, Y., El-Ferik,S., DeSantis, R.M., Application del'Algorithme de Li-Slotine à un As servi s sèment de PositionAngulaire, EPM/RT/93/21, Rapport Technique de l'Ecole Polytechniquede Montréal, (Automne 1993).DeSantis, R.M., 7\n Adaptive PID/Sliding Mode Controller for aPosition Servo, Advances in Communications and Control Systems,Springer Verlag Lecture Notes, pp. 195-205, 1988.ECOLE ..P.OLYJEÇHNJOyE^ 01; ,M.O!^I^AL3 -CJ334 00nfi311 lyEcole Nyteàmque de MoiNalC.P. 6079, Succ. Centrc-viUe

Basic theorems on sliding mode controllers

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