LPV observer-based strategy for rejection of periodic disturbances with time-varying frequency

Research Article LPV Observer-Based Strategy for Rejection of Periodic Disturbances with Time-Varying Frequency G. A. Ramos,1Ramon Costa-Castelló,2and John Cortés-Romero1 1Departamento de Ingenier´ ıa El´ ectrica y Electr´ onica, Universidad Nacional de Colombia, Bogot´ a, Colombia 2Institut d’Organitzaci´ o i Control de Sistemes Industrials, Universitat Polit` ecnica de Catalunya, 08028 Barcelona, Spain Correspondence should be addressed to Ramon Costa-Castell´ o; ramon.cost[email protected] Received 21 December 2014; Revised 29 April 2015; Accepted 4 May 2015 Academic Editor: Peter Dabnichki Copyright © 2015 G. A. Ramos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Rejection of periodic disturbances is an important issue in control theory and engineering applications. Conventional strategies like repetitive control and resonant control can deal adequately with this problem but they fail when the frequency of the disturbance varies with time. This paper proposes a Linear Parameter Varying (LPV) resonant observer-based control for periodic signal rejection which is able to deal with the changes in frequency of the disturbance signal. The observer includes, in an embedded way, an internal model of the disturbance that is based on its harmonic decomposition. In this way, the frequency of the disturbance signal constitutes a parameter that can be adjusted according to the variations of the signal. The resulting disturbance estimation is then used by a control law that cancels the periodic disturbance term while controlling a specified tracking task. The proposed scheme lets the control designer address the disturbance estimation and tracking problems separately. Experimental results, on a mechatronic test bed, show that the proposed LPV resonant observer-based control successfully rejects periodic disturbances under varying frequency conditions. 1. Introduction Rejection of periodic disturbances has been a subject of great interest in control theory and engineering applications. Periodic disturbances are present in many applications like robotics [1,2], power inverters [3], power active filters [4], andwindturbines[5], among others. The most common control strategies used for rejection of periodic disturbances are repetitive control (RC) [6,7] and resonant control [8]. RC constitutes a very efficient methodology in control applications that require tracking and/or rejection of periodic signals (see [9]). It is based on the internal model principle (IMP), thus requiring the inclusion of a periodic signal model in the control loop. However, one of the main drawbacks of RC appears when the frequency of the signals is uncertain or varies with time. In these cases, the traditional RC suffers from a significant performance loss [10]. In order to solve this problem, different strategies have been reported in the literature: a variable structure RC has been proposed in [11], where the frequency of the internal model is adapted to follow the exogenous signal changes; a High Order Repetitive Controller (HORC) is presented in [12]whichis robust against frequency variations and [13,14]proposea varying sampling controller for which the frequency discrete representation remains invariant. Similarly, based on the IMP, the resonant control [15] is dedicated to the tracking/rejection of selected harmonics present in a given signal. Alternatively, this problem can be treated using an observer-based control scheme. Under this approach, the observer is in charge of obtaining an estimate of the disturbancethatisthenusedbythecontrollawtorejectthereal disturbance. A review of disturbance observers design can be found in [16]. In this paper an LPV observer-based strategy is proposed aimed at rejecting periodic disturbances under variable frequencyconditions.Theproposedobserverisformulatedsuch that it includes an internal model of a periodic signal. This internal model is built from the decomposition of the periodic signal on its harmonic components. Thus, the observer is able to estimate the states of the plant and each of the selected frequency components. To overcome the frequency variation problem, the frequency of the signal, which is structurally Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 380609, 9 pages http://dx.doi.org/10.1155/2015/380609 2Mathematical Problems in Engineering embedded in the observer, is changed according to the exogenous variations. Furthermore, the observer gains are reaccommodated according to the different operating points caused by the varying frequency. The tuning of the system is a combined methodology that uses the pole placement technique for reference tracking and optimal Kalman-Bucy approach for the configuration of the resonant observer. Finally, stability analysis can be formulated in an LPV systems framework. In this way, since the closed-loop system is affine with respect to the varying frequency parameter a simple conditiontoestablishthestabilitycanbestated. The experimental validation of the proposal is carried out in a mechatronic platform. This is based on a DC motor exposed to a rotating periodic disturbance torque. Experimental results show that the proposed approach exhibits very high performance, reducing effectively the effect of the disturbances over the angular speed. It is also shown that the performance is preserved under variations of the disturbance frequency. Unlike classic control schemes for handling periodic signals, resonant and repetitive control, the proposed architecture offers the advantage of independently designing the disturbance rejection and tracking reference signals. Thispaperisorganizedasfollows.Section 2 describes the architecture of the proposed controller, Section 3 describes the platform and the experimental results, and finally conclusions and future work are proposed in Section 4. 2. Structure of Resonant Observer-Based Control This section describes the controller architecture of the proposed LPV observer-based strategy. The controller is composed by a disturbance observer, a state feedback control, and the reference internal model. The disturbance estimation is used to compensate the disturbance signal using the Active Disturbance Rejection Control (ADRC) philosophy. A complete stability analysis and some tuning criteria are provided. 2.1. Plant Model. Consider the following state-space linear plant model: x𝑝=A𝑝x𝑝+B𝑝𝑢+B𝑝𝜉𝑝, 𝑦=C𝑝x𝑝,(1) where x𝑝∈R𝑛is the state vector, 𝑢∈Ris the control action, 𝜉𝑝∈Ris the disturbance signal, and 𝑦∈Ris the system output. Similarly, A𝑝∈R𝑛×𝑛 is the state transition matrix, B𝑝∈R𝑛×1is the input vector, and C𝑝∈R1×𝑛 is the output vector.Thesystemdefinedby(A𝑝,B𝑝,C𝑝,0)is assumed to be a minimal representation being both controllable and observable. 2.2. Disturbance Model. In this work we are dealing with disturbances that can be written as 𝜉=𝜉1+𝜉2+⋅⋅⋅+𝜉𝑚(2) with 𝜉𝑘=𝑔𝑘sin (𝜔𝑘(𝑡)𝑡+𝜙𝑘),(3) where 𝜔𝑘(𝑡)is the frequency of each component and 𝑔𝑘and 𝜙𝑘are assumed unknown. In this work it is assumed that 𝑔𝑘and 𝜙𝑘are constant or piecewise constant. Under this hypothesis the frequency content of 𝜉is locally concentrated around 𝜔𝑘(𝑡). Although the values of 𝜔𝑘(𝑡)might be arbitrarily assigned, a particular case with great relevance is when 𝜔𝑘(𝑡) = 𝑘⋅ 2𝜋/𝑇𝑝(𝑡),where𝑘=1,...,𝑚. This case will be assumed from now in this work. Consequently 𝜔(𝑡)is defined as 𝜔(𝑡) = 2𝜋/𝑇𝑝(𝑡)and 𝜔𝑘(𝑡) = 𝑘⋅𝜔(𝑡).Incase𝑇𝑝(𝑡)is constant, 𝜉is a 𝑇𝑝-periodic signal and 𝜉𝑘are the different harmonic frequency components. Each sinusoidal term can be thought as generated by the following system, with appropriate initial conditions:  𝜉𝑘=−𝑘2𝜔2(𝑡)𝜉𝑘,(4) where 𝜔is the fundamental frequency which is assumed measurable (or known). In state-space this can be written as  z𝑘=𝜔(𝑡)A𝑘z𝑘, 𝜉𝑘=C𝑘z𝑘 (5) with z𝑘∈R2,A𝑘=[0𝑘 −𝑘 0],andC𝑘=[1,0]. Therefore, the disturbance signal, 𝜉,admitsthefollowing state-space representation:  z=𝜔(𝑡)A𝑧z, 𝜉=C𝑧z,(6) where z=[z𝑇 1z𝑇 2⋅⋅⋅ z𝑇 𝑚]𝑇∈R2𝑚and A𝑧=[ [ [ [ [ [ [ [ [ [ A100⋅⋅⋅ 0 0A 20⋅⋅⋅ 0 00A 3⋅⋅⋅ 0 .........d... 000⋅⋅⋅ A𝑚 ] ] ] ] ] ] ] ] ] ] , C𝑧=[C1C2C3⋅⋅⋅ C𝑚]. (7) 2.3. Augmented System. We can extend the plant model to include the disturbance signal using x=[x𝑇 𝑝z𝑇]𝑇∈R𝑛+2𝑚; thus we obtain the following augmented model: x=A(𝑡)x+B𝑢, 𝑦=Cx,(8) Mathematical Problems in Engineering 3 where A(𝑡)=[A𝑝B𝑝C𝑧 0𝜔(𝑡)⋅A𝑧], B=[B𝑝 0], C=[Cp0]. (9) This system, with appropriate initial conditions, is equivalent to (1) subject to (2) and (3). It is important to notice that (8) has no disturbance input and it is an observable system but it is not completely controllable system. The controllable subsystem corresponds to the plant while the noncontrollable subsystem corresponds to the disturbance model. Note that the disturbance is an exogenous signal and consequently it cannot be modified through the control action. 2.4. Resonant Observer. Inordertoobservethestateof(8) a Luenberger observer is proposed:  x=A(𝑡)x+B𝑢+L(𝑡)[𝑦−Cx] =(A(𝑡)−L(𝑡)C)x+B𝑢+L(𝑡)𝑦, (10) where x=[x𝑇 𝑝 z𝑇]𝑇is the augmented system state estimation and L(𝑡)=[L𝑇 𝑝(𝑡) L𝑇 𝑧(𝑡)]𝑇are the observer gains. The estimation error is defined as follows: e=x−x. Therefore, using (8) and (10) the estimation error evolution canbewrittenas  e=[A(𝑡)−L(𝑡)C]e.(11) 2.4.1. Stability Analysis. The stability of system (11) depends on the matrix: H(𝑡)=[A(𝑡)−L(𝑡)C] =[A𝑝−L𝑝(𝑡)C𝑝B𝑝C𝑧 −L𝑧(𝑡)C𝑝𝜔(𝑡)⋅A𝑧]. (12) In order to prove the stability of system (11),aLyapunov functioncanbeformulated: 𝑉𝑒=1 2e𝑇P𝑒(𝑡)e,P𝑒(𝑡)>0.(13) Consequently in order to guarantee closed-loop stability the following inequality must be fulfilled:  𝑉𝑒=e𝑇P𝑒(𝑡)H(𝑡)e+e𝑇H(𝑡)𝑇P𝑒(𝑡)e+e𝑇 P𝑒(𝑡)e <0,(14) so it is necessary that P𝑒(𝑡)H(𝑡)+H(𝑡)𝑇P𝑒(𝑡)+ P𝑒(𝑡)<0,∀𝑡>0.(15) Defining P𝑒(𝑡)=P0+𝜔(𝑡)P1,(16) where P0and P1are two symmetric matrices, the stability condition can be stated as P𝑒(𝑡)H(𝑡)+H(𝑡)𝑇P𝑒(𝑡)+ 𝜔(𝑡)P1<0,∀𝑡>0.(17) Using LPV theory [17,18], this condition can be checked in terms of an LMI which must be evaluated in four points defined by 𝜔∈{𝜔min,𝜔max}and  𝜔∈{ 𝜔min, 𝜔max}. 2.4.2. Tuning Procedure. In Section 2.4.1 observer stability conditions have been established. These conditions do not uniquely determine the observer gain, L(𝑡). An approach which provides a simple and convenient framework is optimal estimation. Although a theory for optimal estimation for timevarying systems exists [19], it implies difficult implementation and it is not easy to apply in practice. In Linear Time Invariant framework, it is well-known that the Kalman-Bucy filter constitutes the optimal Luenberger observer where the estimation error covariance is minimized [19]. Thus, the optimal observer gain is defined by L=PC𝑇V−1,(18) where Pis the unique positive-semidefinite solution of the algebraic Riccati equation: PA𝑇+AP −PC𝑇V−1CP +W=0,(19) where Vand Ware the spectral density matrices of the measurement and process noise, respectively. In this work an optimaltuning,basedontheKalmanfilter,isproposed.The observer gain is proposed to be linear varying as L(𝜔(𝑡))=L0+L1𝜔(𝑡),(20) where L0and L1are obtained by forcing L(𝜔max)=Lmax and L(𝜔min)=Lmin.Thus,Lmax and Lmin are obtained from (18) for 𝜔max and 𝜔min,respectively. Proposed approach guarantees stability at the extreme values of 𝜔(𝑡). Stability at intermediate points must be checked through conditions established in Section 2.4.1. 2.5. Closed-Loop System. In this section, the closed-loop stability of the system obtained using the observer estimation to close the loop is analyzed. Based on the state estimation, a state feedback control law is used; taking this into account the complete system takes the following form: x=A(𝑡)x+B𝑢,  x=(A(𝑡)−L(𝑡)C)x+B𝑢+L(𝑡)𝑦, 𝑢=−Kx, (21) where K=[K𝑝,C𝑧]correspondstothestatefeedbackgain. 4Mathematical Problems in Engineering Speed controller Full-bridge driver Power supply PWM Direction Period measurement Signal conditioning Real-time platform Fixed magnets Incremental encoder NSNS Rotating magnets Mechatronic system Mechatronic system DC motor r(t) 𝜔(t) + − 𝜔 Figure 1: General scheme and experimental setup of the case study. In order to simplify the analysis, these equations are writtenusingtheestimationerror: x=A(𝑡)x+B𝑢,  e=(A(𝑡)−L(𝑡)C)e, 𝑢=−K(x−e)(22) obtaining x=[A(𝑡)−BK]x+BKe,  e=(A(𝑡)−L(𝑡)C)e.(23) Asitcanbeseenthewell-knownseparationprinciplecanalso be applied to this LPV system. The dynamics of xdepends on [A(𝑡)−BK]=[A𝑝−B𝑝K𝑝0 0𝜔(𝑡)A𝑧], (24) which can be decomposed on the periodic signal generator dynamics and the closed-loop plant dynamics. Note that the closed-loop plant dynamics is described by an LTI system dynamics and only depends on A𝑝−B𝑝K𝑝,whichcanbe stabilized with a suitable selection of the constant gain K𝑝. 2.6. Closed-Loop System including Reference Internal Model. The controller which has been introduced in the previous section will be useful if we are interested in stabilizing the origin, but in most cases we are interested in tracking a reference. In order to guarantee this, the reference internal modelwillbeintroducedinthecontroller(seeFigure 3): xim =Aimxim +Bim (𝑟−𝑦),(25) 𝑢=Kimxim −Kx;(26) with this new control law the complete closed-loop system is defined by x=[A(𝑡)−BK]x+BKimxim +BKe, xim =−BimCx +Aimxim +Bim𝑟,  e=(A(𝑡)−L(𝑡)C)e.(27) The stability of this system can be analyzed in terms of the observer dynamics, A(𝑡)−L(𝑡)C, and the matrix, [A𝑝−B𝑝K𝑝B𝑝Kim −BimC𝑝Aim ]. (28) Note that this matrix is time invariant so it can be analyzed using regular methods, and Kim and K𝑝can be tuned using LTI methods. 3. Case Study The system used for the experimental validation of the proposed control strategy is a mechatronic system affected by nonlinear periodic disturbances. It consists of a Pulse Width Modulation (PWM) electronic amplifier, a DC motor, a 500 Pulses Per Revolution (PPR) incremental encoder, and a magnetic setup that generates a periodic torque under constant angular speed, 𝜔(𝑡).Thisdisturbancetorqueapplied to the plant is a nonlinear function of the angular position, 𝜃(𝑡). Hence, the control objective is to regulate the angular speed of the motor to a desired value despite the periodic torque disturbance. A detailed scheme of the mechatronic system, control loop, and the experimental setup can be observed in Figure 1. The reader is encouraged to read [20]for more detailed explanation of this system (roto-magnet plant). Mathematical Problems in Engineering 5 0 10 20 30 40 50 60 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Frequency (Hz) Magnitude 27.5 28 28.5 2.5 3 3.5 4 4.5 5 5.5 Time (s) Speed (rev/s) Figure 2: Open-loop system speed response and harmonic components at 4 rev/s. A dynamic model of the plant can be described as 𝑑𝜔(𝑡) 𝑑𝑡 =−𝑘𝑖𝑘𝑏 𝐽𝑅𝑎𝜔(𝑡)−𝐵 𝐽𝜔(𝑡)+2𝜋𝑘𝑖𝑘pwm 𝐽𝑅𝑎𝑢(𝑡) −2𝜋 𝐽𝜏𝑑(𝜃(𝑡)),(29) where 𝜔(𝑡)is the angular speed of the motor in rev/s, 𝜃(𝑡) is the angular position in rad, 𝑢(𝑡)is the control input in PWM percentage with range [−100,100],𝑘pwm =0.12 V/% is the conversion constant from PWM percentage to volts, 𝑘𝑖is the torque constant, 𝑘𝑏is the back-emf constant, 𝑅𝑎 is the armature resistance, 𝐽is the rotor inertia, 𝐵is the viscous-friction coefficient, and 𝜏𝑑(𝜃(𝑡))is the disturbance torque, which in this case is the nonlinear periodic torque produced by the magnetic setup (see the Appendix). Given the structural arrangement of the mechatronic system, the periodic perturbation varies according to the rotational speed of the rotating magnets. Such a time-varying disturbance is typical of rotary systems with eccentricity and imbalance problems among other typical problems. From a simple experimental open-loop step response of the mechatronic system (free of periodic torque), the transfer function of the plant was identified as 𝐺𝑝(𝑠)=Ω(𝑠) 𝑈(𝑠)=1.432 𝑠+1.613;(30) then (2𝜋𝑘𝑖𝑘pwm/𝐽𝑅𝑎)=1.432 and (𝑘𝑖𝑘𝑏/𝐽𝑅𝑎+𝐵/𝐽)=1.613. 4. Experimental Results The plant system is defined by (30); thus the state-space has the form of (1) with A𝑝=−1.613, B𝑝=1.432, and C𝑝=1. Figure 2 shows the open-loop response of the system at 4 rev/s. The speed time response and frequency spectrum are depicted. It can be seen that the disturbance torque significantly affects the speed response causing a large number of harmonic components in the speed signal. The reference signals used in the experiments are of step and ramp types; therefore, the reference internal model is designed to have two integrators; that is, Aim =[01 00 ],Bim = [0 1]. The controller follows the structure depicted in Figure 3. It can be noticed that the frequency of the disturbance is indirectly calculated using the reference signal. For this reason, in order to obtain a good estimation of the real frequency a suitable tracking performance is required. In other applications a different frequency estimation/calculation method may be required. As shown in Section 2.5,theobserverand the feedback controller can be designed independently due to the separation principle. The feedback controller is designed to place the closed-loop poles at −40 rad/s which is achieved by setting K𝑝=82.67 and Kim =[ −44692.73 3351.95]in control law (26). Proposed closed-loop poles are defined in order to provide fast tracking convergence while preserving the robustness margins. Looking at the disturbance spectrum (Figure 2)ithas been determined that at least 15 resonant elements will be necessary to reject it. The selected frequencies are the fundamental frequency (𝜔=2𝜋𝑓rad/s) and the subsequent 14 harmonic frequency components (𝜔𝑘=2𝜋⋅𝑓⋅𝑘rad/s with 𝑘=2,...,14), where 𝑓is the frequency in Hz which correspondswiththespeedvalueinrev/s. The procedure to select the speed dependent observer gainsisasfollows: (1) The minimum and maximum value of the speed is determined, in this case Vmin =2rev/s and Vmax = 8 rev/s, which corresponds with 𝜔min =4𝜋rad/s and 𝜔max =16𝜋rad/s, respectively. (2) A selection of the observer gain is performed for each of the previous operation points. The procedure to select the observer gains follows the standard Kalman-Bucy filter design as described in Section 2.4.2.WehaveselectedV=1andW= 𝛾[0C𝑧]𝑇[0C𝑧]where 𝛾represents the compromise between system noise and observer bandwidth. In this case, the parameter 𝛾has been selected such that 𝛾min =2.5⋅10−6and 𝛾max =5⋅10−7provide good disturbance rejection performance for Vmin and Vmax, respectively. It is important to note that a higher bandwidth has been designed for higher speeds since the disturbance harmonics components are then of higher frequency. Additionally, the quantification error produced by the encoder interface affects more the speeds measurement at higher velocities. (3) The stability of the LPV observer can be checked as described in Section 2.4.1.Afterfixing𝜔min = 4𝜋rad/s and 𝜔max =16𝜋rad/s, a symmetric acceleration range is imposed and beginning with a very small range it is increased until no solution is found. For this system, a frequency rate variation of ±10 rad/s2 (5/𝜋rev/s2)hasbeenobtained. To address the stability checking MATLAB Robust Control Toolbox [21]hasbeenused. Figure 4 shows the closed-loop response of the control system with the proposed LPV observer operating at 4 rev/s. 6Mathematical Problems in Engineering r − +e − − 2𝜋 −u + + 𝜉 𝜔 y  𝐱im =𝐀 im𝐱im +𝐁 ime 𝐱im 𝐊im 𝐂z 𝐊p  𝐳  𝐱p  𝐱p=𝐀 p𝐱p+𝐁 pu 𝐱p𝐂p   𝐱=(𝐀(t)−𝐋(t)𝐂)  𝐱+𝐁u Figure 3: LPV observer-based control system structure. 010 20 30 40 50 60 0 0.5 1 1.5 2 2.5 3 Frequency (Hz) Magnitude 12 14 16 3.98 4 4.02 Time (s) Speed (rev/s) ×10−3 Figure 4: Closed-loop system speed response and harmonic components at 4 rev/s. It can be noticed that the disturbance torque has been effectively rejected, providing a constant speed with very small harmonic components. Furthermore, the obtained error lies in the noise measurement magnitude. In the experiment, the results are obtained using a reference signal with speed variations described by 𝑟(𝑡)= { { { { { { { { { { { { { { { { { { { { { { { 4rev/s0s≤𝑡≤25 s −1 9𝑡+61 9rev/s25s≤𝑡≤34 s 3rev/s34s≤𝑡≤38 s 3 17𝑡−63 17 rev/s38s≤𝑡≤55 s 6rev/s𝑡≥55 s. (31) Two different configurations of the LPV observer have been used: a setup in which the frequency of the observer remains constant corresponding with 4 rev/s and the fully LPV observer configuration. Figure 5 presents the obtained experimental speed response. It can be seen that when the observer does not change the frequency the control system loses its performance as the speed deviates from the nominal setup. On the other side, the proposed LPV observer can 20 25 30 35 40 45 50 55 60 1 2 3 4 5 6 7 Time (s) Speed (rev/s) Observer with constant frequency 4Hz 20 25 30 35 40 45 50 55 60 Observer with variable frequency Tracking Reference 46.4 46.6 4.5 4.55 57 57.4 5.96 6 6.04 1 2 3 4 5 6 7 Speed (rev/s) Time (s) Figure 5: Closed-loop system speed response with variable speed. successfully reject the disturbance for all speed variations, thus tracking the reference signal with small error. As shown in Figure 5, even during a ramp type speed variation the system obtains very small error, provided the ramp slope is small enough. The control signal of the proposed LPV observer-based strategy is presented in Figure 6.Itcanbenoticedthatthe control signal profile changes its frequency following the speed changes, thus feeding the plant with the appropriate input to compensate the time-varying disturbances. Additionally, compared with the estimated disturbance shown in Figure 7,itisnoticeablethatthecontrolsignalshape corresponds with the disturbance one in order to provide the obtained rejection. 5. Concluding Remarks In this paper a new control architecture based on a resonant observer for estimation and rejection of periodic disturbances is proposed. The proposed observer estimates the harmonic decomposition of the disturbance and the Mathematical Problems in Engineering 7 Control signal 0 20 40 60 80 PWM (%) −20 20 25 30 35 40 45 50 55 60 Time (s) (a) 21 21.1 21.2 21.3 0 20 40 60 80 Control signal detail Time (s) PWM (%) 58.9 59 59.1 59.2 Control signal detail −20 0 20 40 60 80 PWM (%) −20 Time (s) (b) Figure 6: Obtained control signal using a variable speed profile. Top: the entire time interval. Bottom: detailed view for two different operating points. controller uses this estimation to cancel its effect on the system. The controller is designed to meet further tracking tasks. LPV resonant observer tuning is performed by applying the Kalman-Bucy filter. As a result, the performance of the estimation can be adjusted by a single parameter. Experimental results show that the proposed strategy effectively rejects the periodic disturbance imposing adequate tracking error dynamics also under varying frequency conditions. The authors are currently working to extend proposed controller to different types of electrical machines and developing new methodologies which guarantee stability in a constructive manner. Appendix Magnetic Torque Since the magnetic setup of the mechatronic system is composed by two rotating magnets and two fixed magnets, the disturbancetorqueobeysthefollowingnonlinearfunction [22]: 𝜏𝑑(𝑡,𝜃)=2 ∑ 𝑖=1 2 ∑ 𝑗=1𝑞𝑚𝑖𝑚𝑗𝜏𝑝(𝜃+𝜃𝑖,𝑗,𝑟𝑖,𝑥𝑗)(A.1) with 𝜏𝑝(𝜃,𝑟,𝑥)=𝑒 𝑚(sin (𝜌)𝐵𝑑+cos (𝜌)𝐵𝛼), 𝜌=arctan (𝑟sin (𝜃),𝑟cos (𝜃)−𝑥)−𝜃, 𝐵𝑑=𝜇0𝑚cos (arctan (𝑟sin (𝜃),𝑟cos (𝜃)−𝑥)) 2𝜋(−2𝑟cos (𝜃)𝑥+𝑥2+𝑟2)3/2, 𝐵𝛼=𝜇0𝑚sin (arctan (𝑟sin (𝜃),𝑟cos (𝜃)−𝑥)) 4𝜋(−2𝑟cos (𝜃)𝑥+𝑥2+𝑟2)3/2, (A.2) where 𝜃(𝑡)is the angular position of the motor in rad, 𝑞𝑚𝑖is themagneticintensityofthepoleforthe𝑖th moving magnet, 𝑚𝑗is the magnetic torque for the 𝑗th fixed magnet, 𝑟𝑖is the distance between the 𝑖th moving magnet and its rotation axis, 𝑥𝑗is the distance between the 𝑗th fixed magnet and its rotation axis, 𝜃𝑖,𝑗 istherelativepositionofthe𝑖th moving magnet with respect to the 𝑗th fixed magnet, 𝑒is the thickness of each moving magnet, and 𝜇0is the magnetic constant. For simulation purposes we use 𝜇0=4𝜋⋅10−7N⋅A−2,𝑒=0.004 m, 𝑟1=0.05 m, 𝑟2=0.05 m, 𝑥1=0.055 m, 𝑥2=0.055 m, 𝜃11 =0 rad, 𝜃12 =𝜋rad, 𝜃21 =𝜋rad, and 𝜃22 =0 rad. 8Mathematical Problems in Engineering 300 400 500 600 700 Disturbance estimation 20 25 30 35 40 45 50 55 60 Time (s) (a) 21 21.1 21.2 21.3 320 340 360 380 400 420 440 Disturbance estimation detail Time (s) 58.9 59 59.1 59.2 580 600 620 640 660 680 700 Disturbance estimation detail Time (s) (b) Figure 7: Disturbance estimation. Top: the estimation for the whole time interval. Bottom: detailed view of the estimation for two different operating points. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work has been partially funded by the following project: MICINN DPI2011-25649 (financed by the Spanish Ministry of Science and Innovation and EU-ERDF funds). 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