Learning-based Adaptive Control for Nonlinear Systems

Transcription

1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES Lerning-bsed Adptive Control for Nonliner Systems Benosmn, M. TR14-64 June 14 Abstrct We present in this pper preliminry result on lerning-bsed dptive trjectory trcking control for nonliner systems. We propose, for the clss of nonliner systems with prmetric uncertinties which cn be rendered integrl Input-to-Stte stble w.r.t. the prmeter estimtion errors input, tht it is possible to merge together the integrl Input-to-Stte stbilizing feedbck controller nd model-free extremum seeking ES lgorithm to relize lerning-bsed dptive controller. We show the efficiency of this pproch on mechtronic exmple. Europen Control Conference ECC 14 This work my not be copied or reproduced in whole or in prt for ny commercil purpose. Permission to copy in whole or in prt without pyment of fee is grnted for nonprofit eductionl nd reserch purposes provided tht ll such whole or prtil copies include the following: notice tht such copying is by permission of Mitsubishi Electric Reserch Lbortories, Inc.; n cknowledgment of the uthors nd individul contributions to the work; nd ll pplicble portions of the copyright notice. Copying, reproduction, or republishing for ny other purpose shll require license with pyment of fee to Mitsubishi Electric Reserch Lbortories, Inc. All rights reserved. Copyright c Mitsubishi Electric Reserch Lbortories, Inc., 14 1 Brodwy, Cmbridge, Msschusetts 139

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3 14 Europen Control Conference ECC June 4-7, 14. Strsbourg, Frnce Lerning-bsed Adptive Control for Nonliner Systems Mouhcine Benosmn Abstrct We present in this pper preliminry result on lerning-bsed dptive trjectory trcking control for nonliner systems. We propose, for the clss of nonliner systems with prmetric uncertinties which cn be rendered integrl Input-to-Stte stble w.r.t. the prmeter estimtion errors input, tht it is possible to merge together the integrl Input-to-Stte stbilizing feedbck controller nd model-free extremum seeking ES lgorithm to relize lerning-bsed dptive controller. We show the efficiency of this pproch on mechtronic exmple. I. INTRODUCTION Extremum seeking ES is well known pproch by which one cn serch for the extremum of cost function ssocited with given process performnce under some conditions without the need of detiled modelling of the process, e.g. [1], [], [3]. Severl ES lgorithms with their stbility nlysis hve been proposed, e.g. [4], [5], [], [6], [3], [6], [1], [7], [8], nd mny pplictions of ES lgorithms hve been reported, e.g [9], [1], [11], [1], [13]. On the other hnd, clssicl dptive control dels with controlling prtilly unknown processes bsed on their uncertin model, i.e., controlling plnts with prmeters uncertinties. Clssicl dptive methods cn be clssified s direct, where the controller is updted to dpt to the process, or indirect, where the model is updted to better reflect the ctul process. Mny dptive methods hve been proposed over the yers for liner nd nonliner systems, we could not possibly cite here ll the design nd nlysis results tht hve been reported, insted we refer the reder to e.g. [14], [15] nd the references therein for more detils. Wht we wnt to underline here is tht these results in clssicl dptive control re minly bsed on the structure of the model of the system, e.g. liner vs. nonliner, with liner uncertinties prmetriztion vs. nonliner prmeteriztions, etc. Another dptive control prdigm is the one which uses lerning schemes to estimte the uncertin prt of the process. Indeed, in this prdigm the lerning-bsed controller, bsed either on mchine lerning theory, neurl networks, fuzzy systems, etc. is trying either to estimte the prmeters of n uncertin model, or the structure of deterministic or stochstic function representing prt or Mouhcine Benosmn m benosmn@ieee.org is with Mitsubishi Electric Reserch Lbortories, Cmbridge, MA 139, USA. totlity of the model. Severl results hve been proposed in this re s well, nd we refer the reder to e.g. [16] nd the references therein for more detils. We wnt to concentrte in this pper on the use of ES theory in the lerning-bsed dptive control prdigm. Indeed, severl results were recently developed in this direction, e.g. [17], [18], [19], [], [9], [1], [1], [13]. For instnce in [17], [18] the uthors used model-free ES, i.e., only bsed on desired cost function, to estimte prmeters of liner stte feedbck to compenste for unknown prmeters for liner systems. In [19], [] n extremum seeking-bsed controller for nonliner ffine systems with liner prmeters uncertinties ws proposed. The controller drives the sttes of the system to unknown optiml sttes tht optimize desired objective function. The ES controller used in [19], [] is not model-free in the sense tht it is bsed on the known prt of the model, i.e., it is designed bsed on the objective function nd the nonliner model structure. A similr pproch is used in [9], [1] when deling with more specific exmples. In [1], the uthors used for the cse of electromgnetic ctutors, model-free ES, i.e., only bsed on the cost function without the use of the systems model, to lern the best feedbck gins of pssive robust stte feedbck. Similrly, in [1], bckstepping controller ws merged with model-free ES to estimte the uncertin prmeters of nonliner model for electromgnetic ctutors. Although, no stbility nlysis ws presented for the full controller i.e., bckstepping plus ES estimtor, very promising numericl results where reported. In this work we propose to generlize the ide of [1], for the clss of nonliner system with prmetric uncertinties which cn be rendered iiss w.r.t. the prmeter estimtion errors. The ide is bsed on modulr design, where we first design feedbck controller which mkes the closed-loop trcking error dynmic iiss w.r.t. the estimtion errors nd then complement this iiss-controller with mode-free ES lgorithm tht cn minimize desired cost function, by tuning, i.e., estimting, the unknown prmeters of the model. The modulr design simplifies the nlysis of the totl controller, i.e., iiss-controller plus ES estimtion lgorithm. We first propose this formultion in the generl cse of nonliner systems nd then show cse-study on mechtronic exmple. This pper is orgnized s follows: Section II is used to EUCA 9

4 recll some nottions nd definitions. In Section III we present the min result of this pper, nmely the ES-bsed lerning dptive controller. Section IV is dedicted to n ppliction exmple, nd the pper ends with Conclusion in Section V. II. PRELIMINARIES Throughout the pper we will use. to denote the Eucliden norm; i.e., for x R n we hve x = x T x. We will use. for the short nottion of time derivtive. We denote by C k functions tht re k times differentible. A function is sid nlytic in given set, if it dmits convergent Tylor series pproximtion in some neighborhood of every point of the set. A continuous function α : [, [, is sid to belong to clss K if it is strictly incresing nd α =. A continuous function β : [, [, [, is sid to belong to clss KL if, for ech fixed s, the mpping βr, s belongs to clss K with respect to r nd, for ech fixed r, the mpping βr, s is decresing with respect to s nd βr, s s s. Let us now introduce the definition of Locl Integrl Inputto-Stte Stbility iiss. Definition 1 Locl Integrl Input-to-Stte Stbility [1]: Consider the system ẋ = ft, x, u 1 where x D R n such tht D, nd f : [, D D u R n is piecewise continuous in t nd loclly Lipschitz in x nd u, uniformly in t. The inputs re ssumed to be mesurble nd loclly bounded functions u : R D u R m. Given ny control u D u nd ny ξ D D, there is unique mximl solution of the initil vlue problem ẋ = ft, x, u, xt = ξ. Without loss of generlity, ssume t =. The unique solution is defined on some mximl open intervl, nd it is denoted by x, ξ, u. System 1 is loclly integrl input-to-stte stble LiISS if there exist functions α, γ K nd β KL such tht, for ll ξ D nd ll u D u, the solution xt, ξ, u is defined for ll t nd α xt, ξ, u β ξ, t + t γ us ds for ll t. Equivlently, system 1 is LiISS if nd only if there exist functions β KL nd γ 1, γ K such tht t xt, ξ, u β ξ, t + γ 1 γ us ds 3 for ll t, ll ξ D nd ll u D u. III. LEARNING-BASED ADAPTIVE CONTROLLER Consider the system 1, with n dditionl rgument representing prmetric uncertinties R p ẋ = ft, x,, u 4 We ssocite with 4, the output vector y = hx 5 where h : R n R h. The control objective here is for y to symptoticlly trck desired smooth vector time-dependent trjectory y ref : [, R h. Let us now define the output trcking error vector s e y t = yt y ref t. We then ssume the following Assumption 1: There exists robust control feedbck u iss t, x, ˆ : R R n R p R m, with ˆ being the dynmic estimte of the uncertin vector, such tht, the closed-loop error dynmics ė y = f ey t, e y, e 6 is iiss from the input vector e = ˆ to the stte vector e y. Remrk 1: Assumption 1 might seem too generl, however, severl control pproches cn be used to design controller u iss rendering n uncertin system iiss, for instnce bckstepping control pproch hs been shown to chieve such property for prmetric strict-feedbck systems, e.g. [15]. This is preliminry report, nd we do not pretend here to present detiled solution for ll the cses. A more detiled study of how to chieve Assumption 1 for specific clsses of systems nd how to use it in the context of ES lerning-bsed dptive control, will be presented in our future reports. Let us define now the following cost function Q ˆ = F e y ˆ 7 where F : R h R, F =, F e y > for e y. We need the following ssumptions on Q. Assumption : The cost function Q hs locl minimum t ˆ =. Assumption 3: The initil error e t is sufficiently smll, i.e., The originl prmeters estimtes vector ˆ is close enough to the ctul prmeters vector. Assumption 4: The cost function is nlytic nd its vrition with respect to the uncertin vribles is bounded in the neighborhood of, i.e., Q ξ, ξ >, V, where V denotes compct neighborhood of. Remrk : Assumption simply mens tht we cn consider tht Q hs t lest locl minimum t the true vlues of the uncertin prmeters. Remrk 3: Assumption 3 indictes tht our result will be of locl nture, mening tht our nlysis holds in smll neighborhood of the ctul vlues of the prmeters. We cn now present the following Lemm. 91

5 Lemm 1: Consider the system 4, 5, with the cost function 7, then under Assumptions 1,, 3, nd 4, the controller u iss, where ˆ is estimted with the multi-prmeter extremum seeking lgorithm ẋ i = i sinω i t + π Q ˆ ˆ i = x i + i sinω i t π, i {1,..., p} 8 with ω i ω j, ω i + ω j ω k, i, j, k {1,..., p}, nd ω i > ω, i {1,..., p}, with ω lrge enough, ensures tht the norm of the error vector e y dmits the following bound e yt β e y, t+α t γ β e, t+ e mxds, i=p where e mx = ξ1 ω + i=1 i, ξ 1, ξ >, e D e, ω = mx i {1,...,p} ω i, α K, β KL, β KL nd γ K. Proof: Consider the system 4, 5, then under Assumption 1, the controller u iss ensures tht the trcking error dynmic 6 is iiss between the input e nd the stte vector e y, which by Definition 1, implies tht there exist functions α K, β KL nd γ K, such tht, for ll e D e nd e D e, the norm of the error vector e dmits the following bound e y t β e y, t + α t γ e ds 9 for ll t. Now, we need to evlute the bound on the estimtion vector, to do so we use the results presented in []. First, bsed on Assumption 4, the cost function is loclly Lipschitz, i.e., η 1 >, s.t. Q 1 Q η 1 1, 1, V. Furthermore, since Q is nlytic it cn be pproximted loclly in V with qudrtic function, e.g. Tylor series up to second order. Bsed on this nd on Assumptions nd 3, we cn write the following bound [], pges : e t dt e t dt β e, t + ξ1 e t β e, t + ξ1 ω e t β e, t + ξ1 ω + + dt i=p i=1 i, with β KL, ξ 1 >, t, ω = mx i {1,...p} ω i, dt = [ 1 sinω 1 t + π,..., psinω p t + π ]T. which together with the bound 9 completes the proof. Remrk 4: The estimted prmeters upper bounds used in Lemm 1 re correlted to the choice of the first order multi-vrible extremum seeking MES 7 nd 8, however, these bounds cn be esily chnged by using other MES lgorithms, e.g. [3], [4], which is due to the modulr design of the controller, tht uses the iiss robust prt to ensure boundedness of the error dynmics nd the lerning prt to improve the trcking performnce. Remrk 5: We point out here tht ISS cn be substituted for iiss if we re deling with time-invrint systems nd ω, solving regultion problem insted of time-vrying trjectory trcking. A. Controller design IV. CASE STUDY We study here the exmple of electromgnetic ctutors modelled by the nonliner equtions [4] m d x dt = kx x η dx u = Ri + di b+x dt dt i b+x + f d i dx b+x dt, x x f 1 where, x represents the rmture position physiclly constrined between the initil position of the rmture, nd the mximl position of the rmture x f, dx dt represents the rmture velocity, m is the rmture mss, k the spring constnt, x the initil spring length, η the dmping coefficient, b+x i represents the electromgnetic force EMF generted by the coil,, b being constnt prmeters of the coil, f d constnt term modelling unknown disturbnce force, e.g. sttic friction, R the resistnce of the coil, L = b+x the coil inductnce ssumed to be dependent on the i dx position of the rmture, b+x dt represents the bck EMF. Finlly, i denotes the coil current, di dt its time derivtive nd u represents the control voltge pplied to the coil. We consider there the control problem of the electromgnetic system ssuming uncertinties on the spring constnt k, the dmping coefficient η, nd the dditive disturbnce f d. Let us define the stte vector z := [z 1 z z 3 ] T = [x ẋ i] T. The objective of the control is to mke the vribles z 1, z robustly trck sufficiently smooth i.e. C time-vrying position nd velocity trjectories z ref 1 t, z ref t = dzref 1 t dt tht stisfy the following constrints: z ref 1 t = z 1int, z ref 1 t f = z 1f, ż ref 1 t = ż ref 1 t f =, z ref 1 t = z ref 1 t f =, where t is the strting time of the trjectory, t f is the finl time, z 1int is the initil position nd z 1f is the finl position. To strt, we first write the system 1 in the following form ż 1 = z ż = k m x z 1 η m z ż 3 = R b+z 1 z 3 + z3 b+z 1 z + mb+z 1 z 3 11 u b+z 1 The uthors in [5] hve shown tht under the following 9

6 feedbck controller u= Rb+z 1 b+z 1 z 3 z z 3 b+z z 3 mb+z 1 z zref c z 3 ũ + mz ˆk z 3 m x z1 ˆη m z+ ˆf dm +c 3z 1 z ref 1 +c 1z z ref ż ref +κ 1z z ref ψ ż ref + mb+z 1 z 3 ˆk c1+κ 1 ψ m ˆη ˆη m żref z ˆk x z 1 z m + κ z 3 ũ mb+z 1 z 3 m x z1 ˆη m z+ f ˆ dm mb+z 1 z 3 + mb+z 1 z 3 κ 1z z ref z 3 mb+z 1 m m x z 1 ˆη m z + ˆf dm [ c1+κ 1 ψ ˆη m + κ1z z ref ] z m κ 3z 3 ũ mb+z 1 z ψ 3 +mb+z 1 z ˆk 3 m z zref +c 3z z ref with ũ= mb+z 1 +c 1z z ref ż ref ˆk m x z1 ˆη m z+ ˆf dm +c 3z 1 z ref 1 + mb+z 1 κ 1z z ref ψ, ψ, 1 13 where ˆk, ˆη, ˆfd re the system prmeter estimtes, nd ψ x z1 m, z m, 1 m T ; the system 1, 1 nd 13, is loclly iiss- LiISS. Next, we define the cost function tf tf Q ˆ = q 1z 1s z 1s ref ds+ q z s z ref s ds, 14 where q 1, q >, nd ˆ = ˆ k, ˆ η, ˆ fd T represents the vector of the lerned prmeters, defined such tht ˆkt = k nominl + ˆ k t ˆηt = η nominl + ˆ η t ˆf d t = f d nominl + ˆ fd t 15 where k nominl, η nominl, f d nominl re the nominl vlues of the prmeters, nd the ˆ i s re computed using discrete version of 8, given by x k k + 1 = x k k + k t f sinω k k t f + π Q ˆ k k + 1 = x k k k sinω k k t f π, x η k + 1 = x η k + η t f sinω η k t f + π Q ˆ η k + 1 = x η k η sinω η k t f π, x fd k + 1 = x fd k + fd t f sinω fd k t f + π Q ˆ fd k + 1 = x fd k fd sinω fd k t f π, 16 eventully, we conclude bsed on Lemm 1, tht the controller 1, 13, 15 nd 16, ensures tht the trcking error norm is bounded with decresing function of the estimtion error. B. Numericl results In this section, we illustrte our pproch for the nonliner electromgnetic ctutor modelled by 1, using the system prmeters given in Tble I [6]. The reference trjectory is designed to be 5 th order polynomil, x ref t = 5 i= i t t f i where the coefficients i re selected such tht the following conditions re stisfied: 93 Prmeter m R η x k b Vlue.7 [kg] 6 [Ω] 7.53 [ kg s ] 8 [mm] 158 [ N mm ] [ Nm [m] TABLE I SYSTEM PARAMETER VALUES A ] x ref =. mm, x ref.5 =.7 mm, ẋ ref =, ẋ ref.5 =, ẍ ref =, ẋ ref.5 =. We consider the uncertinties given by k = 4.5, η =.7 nd f d = 7.5. To mke the simultion cse more chllenging we lso introduced n initil error x =.1 mm on the rmture position. We implemented the controller 1 nd 13 with the coefficients c 1 = 1, c = 1, c 3 = 5, κ 1 = κ = κ 3 =.5, together with the lerning lgorithm 14, 15 nd 16 with the coefficients k =.5, ω k = 7.5, η =., ω η = 7.4, fd = 1, ω fd = 7.3, q 1 = q = 1. For more detils bout the tuning of the MES coefficients we refer the reder to [], [], [1], however, we underline here tht the frequencies ω i, ı = 1,, 3 hve been selected high enough to ensure efficient explortion on the serch spce nd ensure convergence nd tht the mplitudes i, i = 1,, 3 of the dither signls, hve been chosen such tht the serch is fst enough for this ppliction. Here due to the cyclic nture of the problem, i.e., cyclic motion of the rmture between nd x f, the uncertin prmeters estimte vector ˆk, ˆη, ˆfd T is updted for ech cycle, i.e., t the end of ech cycle t t = t f, the cost function Q is updted, nd the new estimte of the prmeters is computed for the next cycle. The purpose of using MES scheme long with iiss-bckstepping controller is to improve the performnce of the iiss-bckstepping controller by better estimting the system prmeters over mny cycles, hence decresing the error in the prmeters over time to provide better trjectory following for the ctutor. As cn be seen in Figures 1 nd 1b, the robustifiction of the bckstepping control vi extremum seeking gretly improves the trcking performnce. Figure b shows tht the cost function decreses below 1 within itertions. It cn be seen in Figure tht the cost strts t n initil vlue round 9, nd decreses rpidly fterwrds. Moreover, the estimted prmetric uncertinties ˆ k, ˆ η nd ˆ fd converge to regions round the ctul prmeter vlues, s shown on Figure 3. The number of itertions for the estimte to rech the ctul vlue of the prmeters my pper to be high. The reson behind tht is tht the llowed uncertinties in the prmeters re lrge, hence the extremum seeking

7 position [mm] x w/o ESC x ref x w ESC Cost function Q time [s] Obtined Armture Position vs. Reference Trjectory Number of itertions Cost function- zoom 1.5 v w/o ESC v ref v w ESC 5 velocity [mm/s] 1.5 Cost function Q time [s] b Obtined Armture Velocity vs. Reference Trjectory Number of itertions b Cost function Fig. 1. Obtined trjectories vs. Reference Trjectory- Cse with uncertin k, η, f d Fig.. Cost function- Cse with uncertin k, η, f d scheme requires lot of itertions to improve performnce. Furthermore, we purposely tested the chllenging cse of three simultneous uncertinties, which mkes the spce serch for the lerning lgorithm lrge note tht this cse of multiple uncertinties could not be solved with other clssicl model-bsed dptive controller [7], due to some intrinsic limittions of model-bsed dptive controller. However, In rel-life pplictions uncertinties ccumulte grdully over long period of time, while the lerning lgorithm keeps trcking these chnges continuously. Thus, the extremum seeking lgorithm will be ble to improve the controller performnce in much fewer itertions. Finlly, the control voltge is depicted on Figure 4, which shows n initil high vlue due to the reltively lrge simulted initil condition error on the rmture position. V. CONCLUSION In this pper we hve studied the problem of lerningbsed dptive control for nonliner systems with prmetric uncertinties. We rgued tht for the clss of nonliner systems which cn be rendered iiss w.r.t. the prmeter estimtion errors, by robust feedbck controller, it is possible to combine the iiss feedbck controller with model-free ES lgorithm to obtin lerning-bsed dptive controller. We showed detiled ppliction of this pproch on mechtronic exmple nd reported encourging numericl results. In this preliminry pper, we introduced the ide in generl setting, however, further investigtion re needed to nlyze specific nonliner systems clsses which cn be stbilized in the iiss sense w.r.t. to the estimtion error of the uncertin prmeters, nd show for these specific clsses constructive control design pproch, in the context of the lerning-bsed dptive control presented here. Further work will lso del with using different ES lgorithms with less restrictive conditions on the dither signls mplitude nd frequencies, e.g.[8], [4], together with investigting other model-free lerning lgorithms such us reinforcement lerning lgorithms, nd compring the obtined controllers to the vilble clssicl dptive controllers. REFERENCES [1] K. Ariyur nd M. Krstić, Rel-time optimiztion by extremum-seeking control. Wiley-Blckwell, 3. [] K. B. Ariyur nd M. Krstic, Multivrible extremum seeking feedbck: Anlysis nd design, in Proc. of the Mthemticl Theory of Networks nd Systems, South Bend, IN, August. [3] D. Nesic, Extremum seeking control: Convergence nlysis, Europen Journl of Control, vol. 15, no. 34, pp , 9. 94

8 ˆ η ˆ k Number of itertions Prmeter k estimte Number of itertions ˆ f d b Prmeter η estimte Number of itertions Fig. 3. U [v] c Prmeter f d estimte Prmeters estimtes- Cse with uncertin k, η, f d time [s] Fig. 4. Control voltge- Cse with uncertin k, η, f d [4] A. Scheinker, Simultneous stbiliztion of nd optimiztion of unkown time-vrying systems, in Americn Control Conference, June 13, pp [5] M. Krstic, Performnce improvement nd limittions in extremum seeking, Systems & Control Letters, vol. 39, pp ,. [6] Y. Tn, D. Nesic, nd I. Mreels, On non-locl stbility properties of extremum seeking control, Automtic, no. 4, pp , 6. [7] M. Rote, Anlysis of multivrible extremum seeking lgorithms, in Proceedings of the Americn Control Conference, vol. 1, no. 6. IEEE,, pp [8] M. Guy, S. Dhliwl, nd D. Dochin, A time-vrying extremumseeking control pproch, in Americn Control Conference, 13, pp [9] T. Zhng, M. Guy, nd D. Dochin, Adptive extremum seeking control of continuous stirred-tnk biorectors, AIChE J., no. 49, p , 3. [1] N. Hudon, M. Guy, M. Perrier, nd D. Dochin, Adptive extremumseeking control of convection-rection distributed rector with limited ctution, Computers & Chemicl Engineering, vol. 3, no. 1, pp , 8. [11] C. Zhng nd R. Ordez, Extremum-Seeking Control nd Applictions. Springer-Verlg, 1. [1] M. Benosmn nd G. Atinc, Multi-prmetric extremum seekingbsed lerning control for electromgnetic ctutors, in Americn Control Conference, 13, pp [13], Nonliner lerning-bsed dptive control for electromgnetic ctutors, in Europen Control Conference, 13, pp [14] I. D. Lndu, R. Lozno, M. M Sd, nd A. Krimi, Adptive Control Algorithms, Anlysis nd Applictions, ser. Communictions nd Control Engineering. Springer-Verlg, 11. [15] M. Krstic, I. Knellkopoulos, nd P. Kokotovic, Nonliner nd dptive control design. John Wiley & Sons New York, [16] C. Wng nd D. Hill, Deterministic Lerning Theory for Identifiction, Recognition, nd Control, ser. Automtion nd control engineering series. Tylor & Frncis Group, 6. [17] P. Hghi nd K. Ariyur, On the extremum seeking of model reference dptive control in higher-dimensionl systems, in Americn Control Conference, 11, pp [18] K. B. Ariyur, S. Gnguli, nd D. F. Enns, Extremum seeking for model reference dptive control, in Proc. of the AIAA Guidnce, Nvigtion, nd Control Conference, 9, doi: 1.514/ [19] M. Guy nd T. Zhng, Adptive extemum seeking control of nonliner dynmic systems with prmetric uncertinties, Automtic, no. 39, p , 3. [] V. Adetol nd M. Guy, Prmeter convergence in dptive extremum-seeking control, Automtic, no. 43, p. 1511, 7. [1] H. Ito nd Z. Jing, Necessry nd sufficient smll gin conditions for integrl input-to-stte stble systems: A Lypunov perspective, IEEE Trnsctions on Automtic Control, vol. 54, no. 1, pp , 9. [] M. A. Rote, Anlysis of multivrible extremum seeking lgorithms, in Americn Control Conference, June, pp [3] W. Nose, Y. Tn, D. Nesic, nd C. Mnzie, Non-locl stbility of multi-vrible extremum-seeking scheme, in IEEE, Austrlin Control Conference, November 11, pp [4] K. Peterson nd A. Stefnopoulou, Extremum seeking control for soft lnding of electromechnicl vlve ctutor, Automtic, vol. 4, pp , 4. [5] G. Atinc nd M. Benosmn, Nonliner lerning-bsed dptive control for electromgnetic ctutors with proof of stbility, in IEEE, Conference on Decision nd Control, 13, pp [6] N. Khveci nd I. Kolmnovsky, Control design for electromgnetic ctutors bsed on bckstepping nd lnding reference governor, in 5th IFAC Symposium on Mechtronic Systems, Cmbridge, September 1, pp [7] M. Benosmn nd G. Atinc, Nonliner dptive control of electromgnetic ctutors, in SIAM Conference on Control nd Applictions, 13, pp