Bin Wang, Bernard Brogliato, Vincent Acary, Ahcene Boubakir, Franck Plestan. HAL Id: hal

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1 Experimental comparisons between implicit and explicit implementations of discrete-time sliding mode controllers: Towards cattering suppression in output and input signals Bin Wang, Bernard Brogliato, Vincent Acary, Acene Boubakir, Franck Plestan To cite tis version: Bin Wang, Bernard Brogliato, Vincent Acary, Acene Boubakir, Franck Plestan. Experimental comparisons between implicit and explicit implementations of discrete-time sliding mode controllers: Towards cattering suppression in output and input signals. VSS t IEEE International Worksop on Variable Structure Systems, Jun 214, Nantes, France. IEEE, Variable Structure Systems (VSS), t International Worksop on, pp.1-6, <1.119/VSS >. <al > HAL Id: al ttps://al.arcives-ouvertes.fr/al Submitted on 1 Nov 217 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. Te documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés.

2 Experimental comparisons between implicit and explicit implementations of discrete-time sliding mode controllers: towards cattering suppression in output and input signals B. Wang, B. Brogliato, V. Acary, A. Boubakir, F. Plestan Abstract Tis paper presents a set of experimental results concerning te sliding mode control of an electro-pneumatic system. Two discrete-time control strategies are considered for te implementation of te discontinuous part of te sliding mode controller: explicit and implicit discretizations. Wile te explicit implementation is known to generate numerical cattering [6], [7], [12], [13], te implicit one is expected to significantly reduce cattering wile keeping te accuracy. Te experimental results reported in tis work remarkably confirm tat te implicit discrete-time sliding mode supersedes te explicit ones, wit several important features: cattering in te control input is almost eliminated (wile te explicit and saturated controllers beave like ig-frequency bang-bang inputs), te input magnitude depends only on te perturbation size and is largely independent of te controller gain and sampling time. I. INTRODUCTION Consider te scalar system ẋ(t) =u(t)+d(t), wit u(t) sgn(x(t)), were sgn( ) is te set-valued signum function: sgn() = [ 1, 1], sgn(x) =1if x>, sgn(x) = 1 if x <. Let te disturbance d(t) satisfy d(t) δ < 1 for some δ. Using Filippov s matematical framework of differential inclusions, one deduces tat for any x(), te state x(t) reaces te sliding surface x =in a finite time t, and ten x(t) =for all t t. In te differential inclusions language, u(t) is a selection ξ(t) of te interval [ 1, 1] for t t, and it satisfies ξ(t) =u(t) = d(t) after t. In a sense, te set-valued controller acts as a disturbance observer once te sliding mode is attained. It is clear tat if one multiplies te signum by a gain a>, i.e. u(t) =a sgn(x(t)), ten one still as u(t) = d(t) in te sliding pase after t. However tis time te value of te selection ξ(t) inside te set-valued part of sgn(x(t)) is divided by a, i.e. ξ(t) = d(t) a. Let us now consider te Euler discretization of tis system. It reads: x k+1 = x k + u k + d k, were f k = f(t k ) for a Bin Wang was wit BIPOP team, INRIA Grenoble, 655 avenue de l Europe, Saint-ismier, France. Bernard Brogliato is wit BIPOP team, INRIA Grenoble, 655 avenue de l Europe, Saint-ismier, France. bernard.brogliato@inria.fr Vincent Acary is wit BIPOP team, INRIA Grenoble, 655 avenue de l Europe, Saint-ismier, France. vincent.acary@inria.fr Acene Boubakir is wit Laboratoire de Commande des Processus, Ecole Nationale Polytecnique d Alger, Algéria. Tis work was perfomed during a stay at IRCCyN, Nantes, France. a boubakir@yaoo.fr Franck Plestan is wit IRCCYN, Ecole Centrale de Nantes, 1 rue de la Noë, BP 9211, Nantes Cedex 3, France. franck.plestan@irccyn.ecnantes.fr Tis work as been performed wit te support of te Frenc National Researc Agency (ANR) project CaSliM (ANR 211 BS3 7 1). function f( ), and t k = t +k are te sampling times, > is te sampling period. In suc a simple case, te Euler and ZOH discretizations are te same, except for te disturbance d k = t k+1 t k d(t)dt for te ZOH metod. Our focus is on ow to coose u k. Te explicit metod yields u k sgn(x k ), yielding te closed-loop x k+1 x k d k sgn(x k ). As alluded to above, limit cycles exist wic create oscillations around te sliding surface (ere te origin), known as te numerical cattering in te output [6], [7], [12], [13]. One of te consequences is tat te explicit controller keeps switcing between te two values 1 and -1, and never attains any point inside ( 1, 1). In particular te explicit controller cannot approximate te continuous-time selection ξ( ) =u( ) wen te system evolves close to te sliding surface. If a gain a> premultiplies u( ) ten te explicit controller switces between two discrete values a and a, te switcing frequency being inversely proportional to te sampling period: tis is te numerical cattering in te input. It is noteworty tat te mere notion of a sliding surface does not exist in tis case, since te discrete trajectories cannot attain te origin, and te controller cannot take values in te set-valued part equal to ( 1, 1). One ten as to resort to so-called quasi-sliding surfaces [3], [11]. Te implicit metod is implemented as follows. Since d(t) is unknown, one first constructs a nominal unperturbed system wit state x k, from wic te input is computed: x k+1 = x k + u k, u k sgn( x k+1 ). Tis is a so-called generalized equation wit unknown x k+1. Its solution yields after few manipulations ( u k = proj [ 1, 1]; x k tat is te projection on te interval [ 1, 1], and is a causal input (not depending on future values of te state). Notice tat in te unperturbed case, x k and x k are te same. As proved in [1], [2], te implicit controller guarantees convergence of x k to te origin in a finite number of steps, and disturbance attenuation by a factor during te sliding mode. Most importantly, te control input takes values in ( 1, 1) once x k as reaced te origin, as may be seen from te generalized equation from wic it is calculated, and one as during tat pase u k = d k : u k is a selection ξ k of te discrete-time differential inclusion x k+1 = x k + u k, u k sgn( x k+1 ), and te discrete-time input observes te disturbance wen te sliding mode is attained. Similarly to te continuous-time case, if te controller is multiplied by a ) 1

3 gain a>, tenteselectionξ k = d k a. Terefore te implicit controller as te same features as its continuous-time counterpart. We may summarize tem as follows: (i) Wen tere is no perturbation, te sliding surface is reaced after a finite number of steps. Wen a perturbation acts on te system, te state of te nominal system reaces te sliding surface after a finite number of steps, wile te perturbation effect is attenuated by a factor on te system s state. (ii) Despite te system s state x k never attains its sliding surface due to te disturbance, te notion of discretetime sliding mode does exist, and corresponds to te nominal system s state x k vanising, or equivalently to te set-valued controller evolving strictly inside te interval [ 1, 1]. Intismodetecontrollercompensates for te disturbance, and is a copy of it. Its magnitude is terefore independent, in te sliding mode, of te controller gain, and tere is no need to adapt te gain (denoted as a above, and as G in te sequel) on-line. (iii) Teoretically tere is no numerical cattering during te sliding mode, neiter in te sliding variable, nor in te input. (iv) Te discrete-time controller keeps te simplicity of its continuous-time counterpart, wit no added gain to tune. (v) Computing te input at eac step boils down to solving asimplegeneralizedequation,equivalentlyaprojection on [ 1, 1], orsolvingaquadraticprogram.tisisquite easy to implement in a code. Te implicit algoritm extends to iger dimension systems, and wit sliding surfaces of codimension 2 [2]. Te main objective of tis note is to confirm tese features experimentally. II. DYNAMICS OF THE PLANT AND CONTROLLERS Te electropneumatic system used for te controllers evaluation consists in two actuators wic are controlled by two servodistributors (see Figure 1). Fig. 1: [1] te electropneumatic system - On te left and side is te main actuator wose its position is controlled. On te rigt and side is te perturbation actuator wose load force is controlled. Under some assumptions detailed in [1], te dynamic model of te pneumatic actuator can be written as a nonlinear system wic is affine in te control input [u P u N ] T, u P (resp. u N ) being te control input of te servodistributor connected to te P (resp. N) camber.temodelisdivided in two parts: two first equations concern te pressure dynamics in eac camber wereas te motion of te actuator is described by te two last equations. Ten te model of te electropneumatic experimental set-up reads as ṗ P = krt V P (y) [ϕ P + ψ P u P S rt p P v] ṗ N = krt V N (y) [ϕ N + ψ N u N + S rt p Nv] (1) v = 1 M [S (p P p N ) b v v F ] ẏ = v, wit p P (reps. p N ) te pressure in te P (resp. N) camber, y and v being te position and velocity of te actuator. Te force F is a disturbance. Note tat te previous system appears to ave two control inputs given tat tere is one servo distributor connected to eac camber. In te sequel, only te main actuator position is controlled: given tat tere is a single control objective, one states u = u P = u N. Te coice of sliding mode controller as been made because of its intrinsic features of robustness. Let us define te socalled sliding variable as σ(x, t) =ë + λ 1 ė + λ e (2) wit e = y y d (t), y d (t) being te desired trajectory, supposed to be sufficiently differentiable. As sown in [9], [8], te first time derivative of σ can be written as σ = Ψ(x, t)+φ(x)u = Ψ n (x, t)+ Ψ(t)+[ (x)+ Φ(t)] u suc tat Ψ n, are te nominal functions and Ψ, Φ are te uncertain terms. From [9], [8], functions Ψ and Φ are bounded in te pysical working domain (wic gives tat te uncertain terms are also bounded). Furtermore, one supposes tat Φ is sufficiently small wit respect to to ensure tat 1+ Φ >. Fromapracticalpointofview, tis assumption is not too strong: it simply means tat te uncertainties are small compared to te nominal values. Let us consider te control law 1 : u = 1 [ Ψ n + v]. (4) By applying (4) in (3), one gets σ = Φ Ψ n + Ψ+ (3) [ 1+ Φ ] v. (5) 1 As sown in [4], suc a control law allows to reduce te magnitude of te sliding mode controller by using te nominal informations in te controller. 2

4 Once u as been designed to linearize by feedback te system (1), te discontinuous part of te controller is defined as v Gsgn(σ) (6) wit G tuned sufficiently large. Te controller v as been implemented under its discrete forms as follows (wit k, σ k = σ(k), being te sampling period) Explicit sliding mode control (wit sgn( ) function) v k = Gsgn(σ k ), Implicit sliding mode control (wit sgn( ) multifunction) v k Gsgn(σ k+1 ) (implemented wit a projection as indicated in te introduction). III. EXPERIMENTAL RESULTS Te two controllers ave been implemented wit several feedback gains and sampling times. Te lengt of te interval of study is 2 seconds. Te comparisons are made mainly wit respect to: te input v magnitude and cattering, and te tracking error. A. Comparison of te tracking errors e Data in Tables I II and III V caracterize te position tracking error e obtained by te two different implementation metods, from te aspects of average, range, standard deviation and variation wit five different sampling periods. Te symbol Avg denotes te average of te tracking error over te duration of te test, abs is te absolute value of tracking error, Rge is te range. Te variation of a realvalued function f( ) defined on an interval [a, b] R is te quantity Var [a,b] (f) = N 1 i= f(t i+1 ) f(t i ) (9) were te set of time instants {t,t 1,,t N } is a partition of [a, b].intefollowing,tevariationsoftepositionerrore for te two implementation metods wit te different gains G, avebeencalculatedbycoosingtepartitiontimest i in (9), as te sampling times. Remark 3.1: Te variation in (9) as a quantity to caracterize te analyzed signals, is not common in Control Engineering. It is tougt ere in te context of sliding mode control, tat suc a quantity is useful to measure te cattering level of a signal, since it does represent ow muc te signal varies. However due to te partition tat as been cosen (te sampling times) te results are not comparable from one sampling period to te next, but only between te tree controllers for a fixed. In oter words, in Tables III Vdataavetobecomparedinsideasinglecolumn,butnot from one column to anoter one. All te data concerning e are reported in Tables I V and Figure 2. It is confirmed in Tables III V, tat te variation of te implicit input starts to be significantly smaller tan tat of te oter two, for 5 ms, te improvement being uge for =. Tese first data tend to indicate tat, in te case of te implicit input, its variation is drastically smaller for larger sampling periods (for = 15 ms: 1836 for te explicit metod, 8.1 for te implicit one wit G =1 4 ), confirming tat cattering on e is reduced wen te implicit controller is used. Te fact tat te output signal is smoot for te implicit metod, wile it catters for te explicit controller for large sampling time, is obvious in Figure 2. Similar conclusions were obtained for G =1 4 and are not reported ere because of lack of space. Tables I II concerns G =1 5.Tetwometodssow similar results in terms of average, range and standard deviation of e, te implicit one providing sligtly better results. Te variation values are given in Tables III V wit G =1 5,andisquitevisibleinFigure2:tevariationofe wit te implicit input is muc smaller tan wit te explicit controllers, except for =1ms were te obtained values are of same order. Tis indicates tat te cattering on e is drastically reduced wit te implicit input 2. A first conclusion, tat will be strengtened in te next paragrap, is tat te implicit control metod allows to take larger gains witout decreasing te performance (it means tat it is possible to reject/counteract larger perturbations/uncertainties witout more cattering). Te performance of implicit control is better wen G is larger, wile it is less good wit te explicit and saturation controllers. 1ms 2ms Avg(abs(e)) Rge e (-5.569, 5.627) ( , ) Stand. Dev. e (a) Explicit control 5ms 1ms Avg(abs(e)) Rge e ( , 4.61) ( , ) Stand. Dev. e (b) TABLE I: Comparisons of position error e wen G = Te results for too small sampling periods ( 2 ms) are not conclusive, because of te limited bandwidt of te filters used to compute ė and ë to construct σ. 3

5 Avg(abs(e)) Range of e ( , ) Standard Deviation of e (a) Explicit control Avg(abs(e)) Range of e ( , ) Standard Deviation of e (b) TABLE II: Comparisons of position error e wen G =1 5. 1ms 2ms Explicit control e e e e+3 (a) G =1 5 TABLE III: Variation of position error e. B. Comparison of discontinuous inputs v in and : Te features of te control inputs is a key-point in tis work, given tat one of te objectives is to sow te influence of implicit control to te cattering effect. Let us now pass to te control inputs comparisons, wit data reported in Tables VI VIII, Tables IX XI and Figure 3. Data given in Tables VI VIII and IX XI caracterize te switcing functions for tese tree metods. It includes te range and variation. Remark 3.1 applies also for te variation of te control, so tat in Tables VI VIII, data ave to be compared inside a single column, but not from one column to anoter one. Wat we call te switcing functions are sgn(σ k ) in, and sgn(σ k+1 ) in. For te implicit controller, tis is wat we called te selection ξ k in Introduction. Tis is not to be confused wit te discontinuous control v in (6). Comparisons of te inputs in te two metods are given in Tables VI VIII from range and variation aspects. In addition, te two controllers are depicted in Figure 3, for various time steps and gains. Globally, te experimental results sow tat te implicit metod drastically reduces te input cattering and magnitude compared wit te explicit metod. Te explicit switcing input keeps oscillating between te maximum and minimum values like a bang-bang controller (see data in Tables VI VIII, and Figure 3(a), wic is quite representative of all te switcing functions obtained wit explicit discretizations). In te tables, all te values used to caracterize te cattering in implicit metod are invariably muc less tan te explicit metod. Figures 3(b) 3(d), wic concerns te implicit controller switcing function for various sampling times, sow tat te implicit input v in is largely independent sampling time. From Table VI VIII, te data in te rows corresponding to te implicit controller allow to obtain a confirmation of tis fact. Te magnitudes of te switcing function for te implicit controller, for 6 different gains G and two different sampling periods, are reported in 5ms 1ms s Explicit control e e (a) G =1 5 TABLE IV: Variation of position error e. Explicit control 2.57e (a) G =1 5 TABLE V: Variation of position error e. Tables IX XI. It confirms tat te magnitude of te input v in (6), wic is te switcing function times te gain G, does not depend neiter on G nor on in tis range of sampling times. Tis insensitivity property is believed to be a fundamental property of te implicit metod introduced in [1], [2], compared to explicit implementations wic drastically differ wen and/or G are varied. Te results depicted in Figure 3 clearly demonstrate tat wereas te explicit controller tends to approximate a signal tat switces infinitely fast between two extreme values like bang-bang inputs, tis is not at all te case for te implicit controller tat beaves in a totally different way. Tis is a nice confirmation of bot teoretical and numerical predictions [1], [2], tat te implicit controller does represent te discrete-time approximation of te selection of te differential inclusion according to Filippov s matematical framework. Input cattering is also visible in Table VI VIII. Variation of te implicit switcing function is muc smaller tan te oter two. C. Summary Tese extensive experimental tests prove tat items (ii) (iii) (iv) and (v) in te Introduction, are not only teoretical and numerical predictions obtained in [1], [2], but significantly influence te discrete-time implemented slidingmode controller. Te implicit metod allows to drastically reduce te input cattering and magnitude, wile enancing te tracking capabilities (output cattering is almost entirely eliminated). It also allows te designer to coose larger sampling periods, wic may be of strong interest in practice. Peraps counter-intuitively for Control Engineers, te performance and robustness increase wen te gain G increases, wic is tougt to considerably simplify te controller gain tuning process. We ave also conducted extensive experiments wit a saturated explicit input. Te results we obtained are quite similar to te case witout saturation (te saturation seems to play a very tiny role in cattering effects), and are terefore not reported in tis note. 4

6 1ms 2ms s Explicit control (-1., 1.) (-1., 1.) (-.844,.973) (-.66,.545) ) (a) Range of te switcing function. 1ms 2ms Explicit control (b) Variation of te switcing function. TABLE VI: Switcing function, gain G =1 5. 5ms 1ms Explicit control (-1., 1.) (-1., 1.) (-.36,.417) (-.289,.349) (a) Range of te switcing function. 5ms 1ms Explicit control (b) Variation of te switcing function. TABLE VII: Switcing function, gain G =1 5. IV. CONCLUSION Experiments ave been conducted on an electropneumatic system, wit two different implementations of te sliding mode controller: explicit and implicit discretizations. Te results demonstrate tat te teoretical and numerical predictions of [1], [2] are true: te implicit implementation, wic consists merely of a projection on te interval [ 1, 1] and is tus very easy to implement in a code, drastically supersedes te oter one. Te output and input cattering are reduced in a significant way, witout canging te controller basic structure (i.e., no additional filter,observer, or dynamic controller is added compared to te original, basic sliding mode controller) and keeping its simplicity (in particular te gain tuning is easy, wic is a strong feature of te ECB-SMC metod). Te main feature of te implicit discretization, is tat it keeps, in discrete-time, te multivalued feature of te teoretical continuous-time sliding-mode controller, as it is matematically imposed in Filippov s framework. Te proposed implicit discretization metod is generic in te sense tat it could apply to any kind of sliding mode, set valued control. Future researc sould terefore concern similar experiments on te same and oter set-up, wit twisting and ig-order sliding-mode controllers. Explicit control (-1., 1.) (-.173,.247) (a) Range of te switcing function. Explicit control (b) Variation of te switcing function. TABLE VIII: Switcing function, gain G =1 5. G =5ms (.3,.35) (.5,.5) =1ms (.25,.3) (.5,.6) TABLE IX: Magnitude of implicit switcing function sgn(x k+1 ) for varying gains G and sampling period. REFERENCES [1] V. ACARY, B. BROGLIATO, Implicit Euler numerical sceme and cattering-free implementation of sliding mode systems, Systems and Control Letters, vol.59, pp , 21. [2] V. ACARY, B. BROGLIATO, Y. ORLOV, Cattering-free digital slidingmode control wit state observer and disturbance rejection,ieeetrans. Automatic Control, vol.57, no 5, pp , May 212. [3] BARTOSZEWICZ, A,Discrete-time quasi-sliding-mode control strategies, IEEE Transactions on Industrial Electronics, vol. 45, no 4, pp , [4] R. CASTRO-LINARÈS, S.LAGHROUCHE, A.GLUMINEAU, AND F. PLESTAN, Higer order sliding mode observer-based control, IFAC Symposium on System, Structure and Control, Oaxaca, Mexico, 24. [5] L. FRIDMAN, J.MORENO AND R. IRIARTE (EDITORS), Sliding Modes after te First Decade of te 21st Century. State of te Art, Lecture Notes in Control and Information Sciences, Volume 412, 212, DOI: 1.17/ , Springer Verlag Berlin Heidelberg. [6] Z. GALIAS, X. YU, Complex discretization beaviours of a simple sliding-mode control system, IEEE Transactions on Circuits and Systems II: Express Briefs, vol.53, no 8, pp , August 26. [7] Z. GALIAS, X.YU, Analysis of zero-order older discretization of two-dimensional sliding-mode control systems, IEEE Transactions on Circuits and Systems II: Express Briefs, vol.55, no 12, pp , December 28. [8] A. GIRIN, F. PLESTAN, X. BRUN, AND A. GLUMINEAU, Robust control of an electropneumatic actuator : application to an aeronautical bencmark, IEEE Transactions on Control Systems Tecnology, vol.17, no.3, pp , 29. [9] S. LAGHROUCHE, M.SMAOUI, F.PLESTAN, AND X. BRUN, Higer order sliding mode control based on optimal approac of an electropneumatic actuator, International Journal of Control, vol.79, no.2, pp , 26. [1] Y. SHTESSEL, M.TALEB, AND F. PLESTAN, A novel adaptive-gain supertwisting sliding mode controller: metodology and application, Automatica, vol.48, no.5, pp , 212. [11] SIRA RAMIREZ, H.,Non linear discrete variable structure systems in quasi sliding mode, Int,J.Control,vol.54,no.5,pp , [12] B. WANG, X. YU, G. CHEN, ZOH discretization effect on single-input sliding mode control systems wit matced uncertainties, Automatica, vol.45, pp , 29. [13] X. YU, B. WANG, Z. GALIAS, G. CHEN, Discretization effect on equivalent control-based multi-input sliding-mode control systems, IEEE Transactions on Automatic Control, vol.53, no 6, pp , July 28. 5

7 G =5ms (.3,.35) (.6,.63) =1ms (.25,.3) (.5,.6) TABLE X: Magnitude of implicit switcing function sgn(x k+1 ) for varying gains G and sampling period G =5ms (.3,.3) (.6,.65) =1ms (.25,.25) (.5,.5) TABLE XI: Magnitude of implicit switcing function sgn(x k+1 ) for varying gains G and sampling period (a) Explicit. sign(s k ). G =1 4, =5ms (a) =1ms. Explicit metod (b) Implicit. sign(s k+1 ). G =1 5, =5ms (b) =1ms. Implicit metod (c) Implicit. sign(s k+1 ). G =1 5, =1ms (c) =. Explicit metod (d) =. Implicit metod (d) Implicit. sign(s k+1 ). G =1 5, =. Fig. 3: Switcing function: Comparison between explicit metod (sign(s k ))andimplicitmetod(sign(s k+1 )). Fig. 2: Real position y (mm) in blue and y d (mm) in red, under =1ms and = for G =1 5. 6