Nonlinear Backstepping Learning-based Adaptive Control of Electromagnetic Actuators with Proof of Stability

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1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES Nonlinear Backstepping Learning-based Adaptive Control of Electroagnetic Actuators with Proof of Stability Benosan, M.; Atinc, G.M. TR3-36 Deceber 3 Abstract In this paper we present a learning-based adaptive ethod to solve the proble of robust trajectory tracking for electroagnetic actuators. We propose a learning-based adaptive controller; we erge together a nonlinear backstepping controller that ensures bounded input/bounded states stability, with a odel-free ultiparaeter extreu seeking to estiate online the uncertain paraeters of the syste. We present a proof of stability of this learningbased nonlinear controller. We show the efficiency of this approach on a nuerical exaple. 3 IEEE Conference on Decision and Control CDC This work ay not be copied or reproduced in whole or in part for any coercial purpose. Perission to copy in whole or in part without payent of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by perission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgent of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payent of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c Mitsubishi Electric Research Laboratories, Inc., 3 Broadway, Cabridge, Massachusetts 39

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3 5nd IEEE Conference on Decision and Control Deceber -3, 3. Florence, Italy Nonlinear Backstepping Learning-based Adaptive Control of Electroagnetic Actuators with Proof of Stability Gökhan M. Atınç and Mouhacine Benosan Abstract In this paper we present a learning-based adaptive ethod to solve the proble of robust trajectory tracking for electroagnetic actuators. We propose a learning-based adaptive controller; we erge together a nonlinear backstepping controller that ensures bounded input/bounded states stability, with a odel-free ultiparaeter extreu seeking to estiate online the uncertain paraeters of the syste. We present a proof of stability of this learning-based nonlinear controller. We show the efficiency of this approach on a nuerical exaple. I. INTRODUCTION This work deals with the soft landing proble of electroagnetic actuators. Soft landing requires accurate control of the oving eleent of the actuator between two desired positions. It refers to attaining sall contact velocity, in turn ensuring a low-noise low-coponent-wear operation of the actuator. Furtherore, an actuator ust achieve soft landing over long periods of tie during which the coponents ay age. To overcoe these practical constraints, we developed a robust control algorith that ais for a virtually zero ipact velocity, hence achieving soft landing, and adapts to the syste aging via a learning-based algorith. Many papers have been dedicated to the soft landing proble for electroagnetic actuators, e.g. [], [], [3] and [4]. Linear controllers have been proposed for exaple in []. Linear controllers are usually designed to operate in a sall neighborhood of linearization points. To control the syste over a larger operation space, in this paper, we consider the nonlinear dynaics of the syste for control design. Various nonlinear controllers have been used in [], [3] and [4]. In [3], the authors proposed a nonlinear control based on Sontag s feedback to solve the proble of arature stabilization for an electroechanical valve actuator. However, this approach does not solve the proble of trajectory tracking and does not consider robustness of the controller with respect to syste uncertainties. In [5], authors designed a backstepping based controller for electroagnetic actuators. However, changes in syste paraeters are not considered in this paper. In [4], a nonlinear sliding ode approach was used to solve the proble of trajectory tracking for an electroagnetic valve actuator. Reported results show good tracking perforance; however, robustness with respect to paraetric uncertainties is not guaranteed. In [], authors used a single paraeter extreu seeking ES learning Gökhan M. Atınç gatinc@illinois.edu is with Mechanical Science and Engineering Departent and the Coordinated Science Laboratory, University of Illinois at Urbana-Chapaign, Mouhacine Benosan benosan@ieee.org is with Mitsubishi Electric Research Laboratories, Broadway Street, Cabridge, MA 39, USA. ethod to tune a scalar gain of the control online along with a nonlinear controller to solve the arature trajectory tracking proble for an electroechanical valve actuator. Due to the iterative nature of the learning process, the controller is intrinsically robust to odel uncertainties and paraeter drift over tie. However, an explicit proof of robustness with respect to odel uncertainties or syste s aging is not provided. In [6], authors proposed a nonlinear controller based on Lyapunov redesign techniques for electroagnetic actuators. The controller is copleented by a ultiparaeter extreu seeking MES control for tuning the feedback gains in order to provide robustness. In [7], authors designed a backstepping based controller which was robustified by an ES algorith to estiate uncertain paraeters. In [6] and [7], the effectiveness of the proposed schees was illustrated nuerically; however, no rigorous stability analysis was presented. In this work we use a nonlinear odel of the electroagnetic actuator to design a backstepping controller that ensures asyptotic trajectory tracking for the noinal syste, i.e., with no odel uncertainties. Subsequently, the controller is robustified by a MES algorith that is used to identify the odel s uncertain paraeters online, including paraeters that drift slowly over tie due to aging. Notice that contrary to [], [6], we use a MES approach to learn a vector of paraeters, and not the gains of the controller. In this sense, we are proposing a new learning-based adaptive control. Furtherore, we present a stability analysis of the overall control syste. This paper is organized as follows: We first recall soe useful definitions in Section II. Next, we present in Section III, a nonlinear odel of electroagnetic actuators. Then, in Section IV, we report the ain result of this work, naely, the learning-based adaptive nonlinear controller along with the stability analysis. Nuerical validation of the proposed controller is given in Section V, and finally, concluding rearks are stated in Section VI. II. PRELIMINARIES Throughout the paper we use. to denote the Euclidean nor. We denote an n n diagonal atrix by diag{,..., n } and k ties differentiable functions by C k. We now review soe definitions that we will use later. Definition Integral Input-to-State Stability [8]: Consider the syste ẋ = ft, x, u, where x D R n such that D, and f : [, D D u R n is piecewise continuous in t and locally Lipschitz /3/$3. 3 IEEE 77

4 in x and u, uniforly in t. The inputs are assued to be easurable and locally bounded functions u : R D u R. Given any control u D u and any ξ D D, there is a unique axial solution of the initial value proble ẋ = ft, x, u, xt = ξ. Without loss of generality, assue t =. The unique solution is defined on soe axial open interval, and it is denoted by x, ξ, u. Syste is locally integral input-to-state stable LiISS if there exist functions α, γ K and β KL such that, for all ξ D and all u D u, the solution xt, ξ, u is defined for all t and α xt, ξ, u β ξ, t + t γ us ds for all t. Equivalently, syste is LiISS if and only if there exist functions β KL and γ, γ K such that xt, ξ, u β ξ, t + γ t γ us ds for all t, all ξ D and all u D u. Definition iiss-lyapunov [8], [9]: A C function V : D R is called an iiss-lyapunov function for syste if there exist functions α, α, σ K, and a continuous positive definite function α 3, such that for all x D and 3 α x V x α x 4 V α 3 x + σ u 5 for all x D and all u D u. Definition 3 Weakly Zero-Detectability [9]: Let h : D R p with with h = be the output for the syste. For each initial state ξ D, and each input u D u, let yt, ξ, u = hxt, ξ, u be the corresponding output function defined on soe axial interval [, T ξ,u. The syste with output h is said to be weakly zerodetectable if, for each ξ such that T ξ, = and yt, ξ,, it ust be the case that xt, ξ, as t. Definition 4 Sooth Dissipativity [9]: The syste with output h is dissipative if there exists a C, proper and positive definite function V, together with a σ K and a continuous positive definite function α 4, such that V α 4 hxt, ξ, u + σ u 6 for all x D and all u D u. If this property holds with a V that is also sooth, syste with output h is said to be soothly dissipative. Finally, if 6 holds with h, i.e., if there exists a sooth proper and positive definite V, and a σ K, so that V σ u 7 holds for all x D and all u D u, the syste is said to be zero-output soothly dissipative. III. SYSTEM MODELLING We recall below a nonlinear odel of the electroagnetic actuator presented in []: = kx x η dx u = Ri + a di d x dt b+x dt dt ai b+x + f d ai dx b+x dt, x x f where, x represents the arature position physically constrained between the initial position of the arature, and the axial position of the arature x f, dx dt represents the arature velocity, is the arature ass, k the spring constant, x the initial spring length, η the daping coefficient assued to be constant, b+x ai represents the electroagnetic force EMF generated by the coil, a, b being constant paraeters of the coil, f d a constant ter odelling unknown disturbance force, e.g. static friction, R the resistance of the coil, L = a b+x the coil inductance assued ai dx to be arature-position dependent, and b+x dt represents the back EMF. Finally, i denotes the coil current, di dt its tie derivative and u represents the control voltage applied to the coil. In this odel we do not consider the saturation region of the flux linkage in the agnetic field generated by the coil, since we assue a current and arature otion ranges within the linear region of the flux. Based on this well known nonlinear odel of the electroagnetic actuator we will develop a backstepping nonlinear control and then we extend it to an adaptive version based on a MES algorith. 8 IV. LEARNING-BASED ADAPTIVE NONLINEAR CONTROL A. Backstepping Controller with Guaranteed Integral Inputto-State Stability Consider the dynaical syste 8. We define the state vector z := [z z z 3 ] T = [x ẋ i] T. The objective of the control is to ake the variables z, z track a sufficiently sooth at least C tie-varying position and velocity trajectories z ref t, z ref t = dzref that satisfy the following constraints: z ref t = z int, z ref t f = z f, ż ref t = ż ref t f =, z ref t = z ref t f =, where t is the starting tie of the trajectory, t f is the final tie, z int is the initial position and z f is the final position. Let us first write the syste 8 in the following for: ż = z t dt ż = k x z η z a b+z z 3 + f d ż 3 = R a z 3+ z 3 b+z z b+z + u a. 9 b+z In this section, we will first state a result discussed in [9] for autonoous systes, and then show that the sufficiency part of these results also hold for non-autonoous systes. Subsequently, we will ake use of these results to discuss the stability of the overall control syste. Theore Equivalent Characterizations of iiss [9]: Consider the autonoous syste ẋ = fx, u where x R n, f : R n R R n is locally Lipschitz and inputs are easurable and locally bounded functions u : 78

5 R R. The unique solution of the initial value proble ẋ = fx, u with x = ξ defined on soe axial open interval is denoted by x, ξ, u. The following properties are equivalent for the syste : [] The syste is iiss. [] The syste adits a sooth iiss-lyapunov function. [3] There exists an output that akes the syste soothly dissipative and weakly zero-detectable. [4] The syste is - GAS and zero-output soothly dissipative. Now we propose the following lea. Lea : Result Consider the non-autonoous syste. If there is soe output that akes the syste dissipative and weakly zero-detectable locally, then the syste is LiISS. Reark : Note that we will analyze the local stability properties of the electroagnetic actuator syste, hence we do not require conditions that give global iiss properties. To this purpose, we will odify the -GAS condition to -LUAS for the non-autonoous syste. Moreover, we only need sufficiency, hence soothness of iiss Lyapunov functions is not required. Thus, we odify properties 4 of Theore to the following ones for the non-autonoous syste : [a] The syste is LiISS. [a] The syste adits a continuously differentiable iiss-lyapunov function. [3a] There is soe output that akes the syste dissipative and weakly zero-detectable locally. [4a] The syste is -LUAS and zero-output dissipative. Reark : If we interpret Lea in ters of the odified conditions of Reark, then Lea states that for non-autonoous systes, if 3a holds, then a is true. To prove this lea, we will first show that 3a = 4a; then, we will show 4a = a, and finally we will prove that a = a. Now we proceed with the proof of Lea. Proof: 3a = 4a: Assue that there is soe output h that akes the syste weakly-zero detectable locally, and there exist a C positive definite function V, functions α, α K, σ K and a continuous positive definite function α 4 such that α x V t, x α x V α 4 hx + σ u hold for all x D and all u D u. With u =, we have V α 4 hξ, and since the syste is weaklyzero detectable, by LaSalle-Yoshizawa Theore [], we conclude that the syste is -LUAS. Also, we have V σ u fro, iplying, by Definition 4, that syste is zero-output dissipative. 4a = a: Assue 4a holds, and let V and σ be so that 7 holds. Since the syste is -LUAS, by a converse Lyapunov theore e.g., [], there exists a C function V for the syste such that V t α x V x α x 3 + V x ft, x, α x, x D 4 holds for soe continuous positive definite functions α, α, α K. If we take the derivative of V along the trajectories of the whole syste, we have V t + V x ft,x,u= V t + V x ft,x,+ V x [ft,x,u ft,x,]. 5 Since V is continuously differentiable and we consider x in a copact subset D, there exists a positive constant K V such that V K V, x D. 6 x Moreover, syste is locally Lipschitz in x and u, uniforly in t. This iplies that there exists a positive constant L u x such that ft, x, u ft, x, L u x u, 7 x D, u D u, t. Since x D, where D is copact, L uax := ax x D L u x exists. Thus, using the inequality 4, and the definitions for K V and L uax, we have V t + V x V ft, x, + x [ft, x, u ft, x, ] α x + K V L uax u. 8 After defining the K-function σ s = K V L uax s for s R, we rewrite 8 as V α x + σ u. 9 Thus, by Definition, V is an iiss Lyapunov function for the syste. a = a: Consider the iiss Lyapunov function V for syste satisfying 3 and 7. Then, by sufficiency discussion in [8] and [9], syste is LiISS. We now address the control proble of the dynaic syste with uncertain paraeters. Uncertain paraeters of the syste 8 are the spring constant k, the daping coefficient η, and the additive disturbance f d. To take into account the uncertainty, the backstepping controller is defined as u= a Rb+z b+z a z 3 z z 3 b+z + a z 3 b+z z zref c z 3 ũ + z ˆk z 3 x z ˆη z+ ˆf d +c 3z z ref +c z z ref ż ref +κ z z ref ψ ż ref + b+z z 3 ˆk c+κ ψ ˆη ˆη żref z ˆk x z z + κ z 3 ũ b+z z 3 x z ˆη z+ f ˆ d a b+z z 3 + b+z z 3 κ z z ref az 3 b+z x z ˆη z + ˆf d [ c+κ ψ ˆη + κz z ref ] z κ 3z 3 ũ b+z z ψ 3 +b+z z ˆk 3 z zref +c 3z z ref with ũ= b+z a +c z z ref ż ref ˆk x z ˆη z+ ˆf d +c 3z z ref + b+z a κ z z ref ψ, ψ, where ˆk, ˆη, ˆfd are the syste paraeter estiates, and ψ [ x z z T ]. We can now state the following lea. Lea : Result Consider the closed-loop dynaics given by 9, and, with constant unknown paraeters k, η, f d and consider the paraeter error vector 79

6 [ k ˆk η ˆη f d ˆf ] T d. Then, there exist positive gains c, c, c 3, κ, κ and κ 3 such that z t, z t are uniforly bounded and the syste 9 is locally integral input-to state stable LiISS with respect to,. Proof: Consider the full echanical subsyste that consists of only the first two equations with the virtual control input ũ := z3: ż = z ż = k x z η z + f d a b+z ũ. Defining the Lyapunov function V sub = c3 z z ref + z z ref, with c 3 >, we would like to design ũ so that Vsub = c z z ref along the trajectories of, but since the syste paraeters k, η and f d are unknown, we design the virtual input to be ũ given by. Substituting ũ into V sub, leads to V sub = c z z ref +z z ref k ˆkx z η ˆηz + f d ˆf d κ z z ref ψ 3 Using the definitions of ψ and, we can show the following: [ ] V sub c z z ref κ z z ref ψ + κ 4κ c z z ref + 4κ. 4 Note that we have ade use of the nonlinear daping ter κ z z ref ψ [] to attain a negative quadratic ter of [ ] ψ and i.e., κ z z ref ψ κ and a positive ter that is function of only 4κ, hence rendering V sub an iiss-lyapunov function for the echanical subsyste. Since we cannot directly control z3, we use backstepping to design the control input ut so that z3 converges to ũ, which in turn will render the echanical subsyste LiISS. Unfortunately, because of the uncertainty in syste paraeters, the best that can be done is to have z3 follow ũ with an error that is function of the uncertainty vector ; i.e., achieving inputto-state stability. To this purpose, we define the Lyapunov function for the full syste: V aug = V sub + z 3 ũ. Taking the derivative of V aug along the trajectories of the full syste, leads to the following inequality: V aug c z z ref + 4κ +z 3 ũ az zref b+z ũ +z 3 ũ z 3 Rb+z a z 3+ z z 3 b+z + b+z a u, 5 where ũ is the tie derivative of. By substituting the control input into 5, using the aforeentioned definitions of ψ and note that = [ ˆk ] T ˆη ˆfd and by aking use of the quadratic daping ters, e.g. κ z z ref ψ, we can show that Vaug satisfies the following inequality: V aug c z z ref + 4κ c z 3 ũ κ [ z 3 ũ b+z z 3 c+κ ψ ˆη z b+z + 4κ κ [ z 3 ũ κz z ref ] ψ κ ] ψ z 3 κ + 4κ κ 3 [ z 3 ũ b+z z 3 ψ κ 3 ] + 4κ 3. 6 Finally, fro 6, we deduce V aug c z z ref c z 3 ũ + 4κ + κ + κ 3. 7 We now express the uncertain syste in the following nonlinear tie-varying for: ė = ft, e,, 8 with e D e, D, where e := [z z ref z z ref z3 ũ] T and = [ ] T T T. By considering the output defined by h = [z z ref z3 ũ] T, we can show that the syste 8 with h is weakly zero-detectable i.e. by analyzing the zero-dynaics of 8 with h. Furtherore, inequality 7 satisfies 6, eaning that property 3a holds for 8. By the virtue of Lea, we conclude that syste 8 is LiISS with respect to the input, iplying that there exist functions α K, β KL and γ K, such that, for all e D e and D, and et β e, t + α t γ ds 9 for all t. Reark 3: In the rest of this paper, we will refer to the controller of Section IV-A as the ISS-backstepping controller. Reark 4: In this work, we assue that the unknown paraeters k, η, f d are constant. This is a realistic assuption since we are targeting the proble of aging in real applications, and usually aging happens very slowly over long period of tie. Hence, the slowly varying paraeters can be approxiated by constant uncertain paraeters. The analysis of the dynaical behavior of estiated paraeters is done via MES theory [3]. The obtained controller is described in the next section. B. Robustification of the ISS-backstepping Controller We now discuss how MES schee is utilized along with ISS-backstepping controller to render the control syste robust to uncertainties in syste paraeters. To this purpose we ake the following assuptions. Assuption : Consider the cost function Q=q z t f z t f ref +q z t f z ref t f, q,q > for the dynaical syste 9. Q has a local iniu at θ = [ ] T k η f d. Assuption : The initial error t is sufficiently sall, i.e., the original paraeter estiates vector θ = [ˆk ] T ˆη ˆfd is close enough to θ. Assuption 3: Q is analytic and its variation with respect to the uncertain variables is bounded in the neighborhood of θ, i.e., Q θ θ ξ, ξ >, θ Vθ, where Vθ denotes a copact neighborhood of θ. Reark 5: Assuption iplies that our result will be of local nature, eaning that our analysis holds in a sall neighborhood of the actual values of syste paraeters. Following [4], [3], we propose the MES algorith for the syste 9: ẋ p = a p sinω p t + π Q[x p + a p sinω p t + π ] θ p = x p + a p sinω p t + π, 3 8

7 where p =,, 3 corresponds to k, η and f d respectively. In addition, ω p = ω p, ω >, p =,, 3. Operation of the schee given in 3 is a ulti-paraeter analog of the schee proposed in []; here, the MES schee is ipleented throughout the travel of the oving part of the actuator, and at t = t f, Q is updated, and the whole cycle is reiterated with the new easureents. The purpose of using MES schee along with ISS-backstepping controller is to iprove the perforance of the ISS-backstepping controller by better estiating the syste paraeters over any cycles, hence decreasing the error in paraeters over tie to provide better trajectory following for the actuator. Now we can state the ain result on the proposed MESbased robust controller. Lea 3: Result 3 Consider the dynaical syste 9 with the ISS-backstepping controller given by and the cycle-to-cycle MES algorith given by 3 for estiation of syste paraeters k, η, and f d under Assuptions, and 3. Then, the following bound holds: t et β e,t+α γ β,t+ ax ds, 3 where e := [ ] T z z ref z z ref z 3 ũ, ax = ξ 3 i= a i + ax i {,,3}.5ξ a i, ξ, ξ >, e D e, α K, β KL, β KL and γ K. Proof: Based on Lea, we know that for the closedloop dynaics given by 9 and, there exist functions α K, β KL and γ K, such that, for all e D e and D, such that the following inequality holds for all t : et β e, t + α t ω + γ ds. 3 Now, in order to evaluate the bound on the estiation vector, we use the results presented in [4]. First, based on Assuption 3, the cost function is locally Lipschitz, i.e. η >, s.t. Qθ Qθ η θ θ, θ, θ Vθ. Moreover, since Q is analytic, it can be approxiated locally in Vθ by a second order Taylor series. Defining dt:= [ ] T a sinω t+β + π asinωt+β+ π a3sinω3t+β3+ π, by the virtues of Assuptions and, we can then write the following bound [4]: t dt t dt β,t+ ξ ω t β,t+ ξ ω + dt β,t+ ξ ω + 3 i= a i, with ξ >, for all t. Moreover, in [4], the MES algorith is shown to be a gradient-based algorith, such that the variation of θ is approxiated by θ R Q θ θ + dt, with R = li T T ds ds =.5 diag{a, a, a 3}. Using Assuption 3, we can write θ =.5 ax i {,,3} a i ξ + 3 i= a i. Finally, noting that +, we have β,t+ ξ ω + 3 i= a i +ax i {,,3}.5ξ a i, 33 which together with 3 copletes the proof. V. SIMULATIONS In this section, we illustrate our approach for the nonlinear electroagnetic actuator given by 8, with the syste paraeters given by: =.7 [kg], R = 6 [Ω], η = 7.53 [kg/sec], x = 8 [], k = 58 [N/], a = [N /A ], b = 4 5 []. The reference trajectory is designed to be a 5 th order polynoial, x ref t = 5 i= a i t t f i where the coefficients a i are selected such that the following conditions are satisfied: x ref =., x ref.5 =.7, ẋ ref =, ẋ ref.5 =, ẍ ref =, ẋ ref.5 =. Although several cases have been tested to validate the perforance of the proposed approach, due to space constraints, we present here only two cases. Case [k, η]: We consider the uncertainty in the echanical paraeters k and η. Uncertainties in k and η are given by k = and η =.. We set the paraeters of the extreu seeking algorith in the following way: a k =, ω k = 7.5, a η =., ω η = 7.4, q = q = 5. The results of this case are depicted in the figures,, 3 and 4. As can be seen in Figures and, without the robustification of the backstepping control via extreu seeking, the errors at t = t f are quite large; around.3 and.3 s. With the extreu seeking, the perforance of the control is significantly iproved. It can be seen in Figure 3 that after 5955 iterations, the cost decreases below a very sall value Q threshold =.. Moreover, the estiated paraetric uncertainties k and η converge to regions around the actual uncertainty values; these regions are approxiated by the extreu seeking algorith paraeters a k and a η as dictated by Lea 3. The nuber of iterations for the cost to decrease below the threshold level ay appear to be high; the reason is that the allowed uncertainties in the paraeters are relatively large, hence the extreu seeking schee requires a lot of iterations to iprove perforance. In real life applications, the uncertainty in paraeters accuulate gradually over a long period of tie, while the learning algorith keeps tracking these changes continuously. Thus, the extreu seeking algorith will be able to iprove the controller perforance relatively quickly, eaning that it will enhance the backstepping control in uch fewer iterations. In this paper, we intentionally report on challenging cases to show the adaptive ability the proposed ethod.case [b]: The siulations discussed here are designed to show that our schee works even for the situations where the syste is not linear with respect to the uncertain paraeters. To this purpose, we initially considered a case where the uncertainty is in the nonlinear paraeter b, and the uncertainty is given by b =.. Note that the backstepping control has not been explicitly designed to copensate for the uncertainty in b, but we wanted to test nuerically the challenging case where the estiated paraeters enters the odel in a nonlinear ter. We set the paraeters of the extreu 8

8 seeking algorith in the following way: a b =.5, ω b = 7.6, q = q = 5. The results of this case are depicted in the figures 5, 6, 7 and 8. It can be seen in Figures 5, 6 that the extreu seeking copensates for the uncertainty and iproves the perforance of the backstepping control. Without the estiation schee, since the paraeter b appears nonlinearly in the odel, the uncertainty in b deteriorates the position and velocity tracking of the syste significantly. With the addition of the extreu seeking schee to estiate b, the perforance is iproved iensely. It can be seen in Figure 7 that the cost starts at an initial value around, and after 638 iterations, it decreases below the threshold Q threshold =.. Moreover, the estiated uncertainty in b is.98 after 638 iterations, which is in accordance with the result of Lea 3. position [] x w/o ESC xref x with ESC tie [s] Fig.. Obtained Arature Position vs Reference Trajectory, a k =, a η=., ω k =7.5, ω η=7.4 velocity [/s] vref v w/o ESC v with ESC tie [s] Fig.. Obtained Arature Velocity vs Reference Trajectory, a k =, a η=., ω k =7.5, ω η=7.4 Cost function Q 5 Q=.999 after 5955 Iterations # of Iterations Fig. 3. Q=5xt f x ref t f +5vt f v ref t f, Q thres =. k η =.634 after 5955 Iterations η k = 9.45 after 5955 Iterations # of Iterations Fig. 4. k, η vs # of Iterations VI. CONCLUSION In this paper, we have proposed an adaptive controller based on a nonlinear backstepping and a odel-free MES algorith. We have proved the bounded input/bounded states stability of backstepping control and the stability of the cobined backstepping and MES controller. Future work will include taking explicitly into account nonlinear paraeters in the control law design and in the stability analysis, as well as coparing the perforance of this type of learning-based adaptive controllers to classical adaptive control ethods. REFERENCES [] W. Hoffann, K. Peterson, and A. Stefanopoulou, Iterative learning control for soft landing of electroechanical valve actuator in caless position [].8.6 xref.4 x with ESC x w/o ESC tie [s] Fig. 5. Obtained Arature Position vs Reference Trajectory, a b =.5, ω b =7.6 3 velocity [/s] v w/o ESC vref v with ESC tie [s] Fig. 6. Obtained Arature Velocity vs Reference Trajectory, a b =.5, ω b =7.6 Cost Function Q 5 5 Q=.998 after 638 Iterations # of Iterations Fig. 7. Q=5xt f x ref t f +5vt f v ref t f, Q thres =. b.. b =.968 after 638 Iterations # of Iterations Fig. 8. b vs # of Iterations engines, Transactions on Control Systes Technology, vol., no., pp , March 3. [] K. Peterson and A. Stefanopoulou, Extreu seeking control for soft landing of electroechanical valve actuator, Autoatica, vol. 4, pp , 4. [3], Rendering the electroechanical valve actuator globally asyptotically stable, in Conference on Decision and Control. Maui, HI: IEEE, Deceber 3, pp [4] P. Eyabi and G. Washington, Modeling and sensorless control of an electroagnetic valve actuator, Mechatronics, vol. 6, pp , 6. [5] N. Kahveci and I. Kolanovsky, Control design for electroagnetic actuators based on backstepping and landing reference governor, in 5th IFAC Syposiu on Mechatronic Systes, Cabridge, Septeber, pp [6] M. Benosan and G. Atinc, Multi-paraetric extreu seekingbased learning control for electroagnetic actuators, in Proceedings of the Aerican Control Conference. IEEE, 3, pp [7], Nonlinear learning-based adaptive control for electroagnetic actuators, in Proceedings of the European Control Conference, 3, pp [8] H. Ito and Z. Jiang, Necessary and sufficient sall gain conditions for integral input-to-state stable systes: A lyapunov perspective, IEEE Transactions on Autoatic Control, vol. 54, no., pp , 9. [9] D. Angeli, E. Sontag, and Y. Wang, A characterization of integral input-to-state stability, IEEE Transactions on Autoatic Control, vol. 45, no. 6, pp. 8 97,. [] W. Haddad and V. S. Chellaboina, Nonlinear dynaical systes and control: a Lyapunov-based approach. Princeton University Press, 8. [] H. Ito, A lyapunov approach to cascade interconnection of integral input-to-state stable systes, IEEE Transactions on Autoatic Control, vol. 55, no. 3, pp. 7 78,. [] M. Krstic, I. Kanellakopoulos, P. Kokotovic et al., Nonlinear and adaptive control design. John Wiley & Sons New York, 995. [3] K. B. Ariyur and M. Krstić, Real-tie optiization by extreuseeking control. Wiley-Blackwell, 3. [4] M. A. Rotea, Analysis of ultivariable extreu seeking algoriths, in Proceedings of the Aerican Control Conference, June, pp