1482 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 3, JUNE 2015

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1 482 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 3, JUNE 205 μ-synthesis-based Adaptive Robust Control of Linear Motor Driven Stages With High- Frequency Dynamics: A Case Study Zheng Chen, Bin Yao, Senior Member, IEEE, and Qingfeng Wang, Member, IEEE Abstract Existing control approaches for the precision motion control of linear motor driven systems are mostly based on rigidbody dynamics of the system. Since all drive systems are subjected to the effect of structural flexible modes of their mechanical parts, the neglected high-frequency dynamics resulting from these structural modes have become the main limiting factor when pushing for better tracking performance and higher closed-loop control bandwidth. In this paper, physical modeling and dynamic analysis that take into account the flexibility of the ball bearings between the stage and the linear guideways are presented with experimental verification. With the gained knowledge of these high-frequency dynamics, a novel μ-synthesis-based adaptive robust control strategy is subsequently developed. The proposed control algorithm uses adaptive model compensation having accurate online parameter estimation to effectively deal with various nonlinearity effects and to transform the difficult trajectory tracking control problem into a robust stabilization problem. The well-developed μ- synthesis-based linear robust control technique is then employed in the fast feedback control loop design to explicitly deal with the robust control issue associated with the high-frequency dynamics to achieve higher closed-loop bandwidth for better disturbance rejection. Comparative experiments have been performed and the results show the better tracking performance of the proposed algorithm over existing ones. Index Terms Adaptive robust control (ARC), high-frequency dynamics, linear motor, model compensation, μ-synthesis. I. INTRODUCTION LINEAR motors have been widely used in machine tools [], microelectronics, and semiconductor manufacturing equipment [2], because of their potential of achieving high- Manuscript received October 4, 203; revised February 5, 204; accepted March 5, 204. Date of publication March 4, 204; date of current version May 8, 205. Recommended by Technical Editor M. de Queiroz. This work was supported in part by the National Natural Science Foundation of China under Grant , the National Basic Research and Development Program of China under 973 Program Grant 203CB035400, and the Science Fund for Creative Research Groups of National Natural Science Foundation of China under Grant Z. Chen is with the State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 30027, China, and also with the Department of Mechanical Engineering, Dalhousie University, Halifax, NS B3H4R2, Canada ( cwlinus@gmail.com). B. Yao is with the State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 30027, China, and also with the School of Mechanical Engineering, Purdue University, West Lafayette, IN USA ( byao@ieee.org). Q. Wang is with the State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 30027, China ( qfwang@ zju.edu.cn). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 0.09/TMECH speed and high-accuracy linear movement [3] by eliminating gear-related mechanical transmission problems. But to realize their high-speed/high-accuracy potential, various model uncertainties due to parameter variations (e.g., unknown load inertia) and disturbances, significant nonlinearities, and structural flexible mode effect have to be well handled by the controller design. Many methods have been developed to deal with model uncertainties in the precision motion control of linear motors, such as disturbance observer [4], repetitive model predictive control [5], iterative learning control [6], and various improved sliding mode control [7]. An adaptive robust control (ARC) approach [8] [2] has been developed for the high performance control of uncertain nonlinear systems in the presence of both parametric uncertainties and uncertain nonlinearities, and successfully applied to the precision motion control of linear motors [3] [5]. To further improve the tracking performance of linear motor driven systems, various compensations of specific nonlinearities have also been carried out, such as cogging force [6] [9], friction [9] [2], and nonlinear electromagnetic effect [22]. All of these researches are based on the rigid-body dynamics of the system. As such, the high-frequency dynamics such as the structural flexible modes neglected in these existing researches have become the main limiting factor in pushing for better control performance. In [23], frequency identification experiments on a specific linear motor driven stage are carried out to verify the presence of high-frequency dynamics. The knowledge of these high-frequency dynamics is then used to guide the gain tuning of the existing controllers to make a better tradeoff in maximizing the achievable control performance, without exciting the neglected flexible modes of mechanical structures. However, to achieve higher closed-loop bandwidth for an even better tracking performance, a new controller design architecture is necessary to explicitly take into account these neglected structural flexible modes. In [24], active compensation of highfrequency dynamics caused by the structural flexible modes using pole/zero cancelation technique is used as a preliminary attempt, which is quite sensitive to the accuracy of the identified high-frequency dynamics. During the past decades, μ-synthesis has become a mature robust H type optimal control design technique for output stabilization of linear time-invariant (LTI) systems with uncertainties and, thus, possesses the potential to be used in the design of controllers for linear motor driven stages with high-frequency dynamics. In [25] and [26], H optimal control has been successfully applied to the motion control of linear motors. However, they are still based on the rigid-body IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See standards/publications/rights/index.html for more information.

2 CHEN et al.: μ-synthesis-based ADAPTIVE ROBUST CONTROL OF LINEAR MOTOR DRIVEN STAGES WITH HIGH-FREQUENCY DYNAMICS 483 dynamics of the system and cannot explicitly handle the inherit nonlinearities of linear motor drives. In this paper, physical modeling and dynamic analysis of the high-frequency dynamics due to the flexibility of the ball bearings between the stage and the linear guideways are presented. The proposed mathematical model is then validated through identification experiments in the frequency domain. A novel μ-synthesis-based ARC algorithm is subsequently developed to simultaneously deal with the inherited nonlinearities, the highfrequency dynamics due to the structural flexible modes, and modeling uncertainties of linear motors. Specifically, through the use of adaptive model compensation with accurate online parameter estimations, the inherit nonlinearities of linear motor systems can be effectively compensated. Doing so also makes it possible to convert the originally much more difficult problem of trajectory tracking control with nonlinearities and uncertainties into a standard robust stabilization problem that can be solved with the traditional robust control techniques for LTI systems. The well-developed μ-synthesis tools are then employed to design less conservative robust stabilizing controllers by incorporating the major high-frequency dynamics as part of the nominal model, rather than treating them as unmodeled dynamics as in the previous researches. By doing so, a closed-loop system with higher bandwidth and better disturbance rejection is obtained. Comparative experiments are conducted to show the better tracking performance of the proposed control algorithms over existing ones. II. DYNAMICAL MODELS When neglecting the fast electrical dynamics and various structural flexible modes, the rigid-body dynamics of a linear motor can be described by [3] Mÿ = F m Bẏ A f S f (ẏ)+f dis () where y, ẏ, and ÿ represents the displacement, velocity, and acceleration respectively. F m is the driving force with F m = A u, where u is the control input in voltage and A is the lumped force and current amplifier gain. M, B, and A f are the inertia, viscous friction coefficient, and the Coulomb friction coefficient, respectively. S f (ẏ) is a known smooth function used to approximate the discontinuous sign function sgn(ẏ). F dis represents the lumped modeling errors and external disturbances. For simplicity, the dynamical model () is rewritten as Mÿ = u Bẏ Āf S f (ẏ)+ F dis (2) where M =M/A, B =B/A, Ā f =A f /A, F dis = F dis /A. So far, there have been very little research done on the modeling of structural flexible modes for linear motor driven stages. The only few available results in the literature are on the vibration analysis of linear guideway type recirculating linear ball bearings used in the linear motor stage [28], [29]. Furthermore, the mechanical resonance modes studied in [28] and [29] are in the kilohertz range, way above the frequency of the major flexible modes of linear motor stages observed in the experimental studies, which is in the range of hundred hertz only [23], [24]. Fig.. Schematic diagram of the linear motor driven stage. Similarly, the structural flexible modes of the stage alone should not be the cause of the observed major high-frequency dynamics due to the very high stiffness of the stage in the linear motor drive system. In the following, a novel physical modeling based on the lumped parameter model of the ball bearing flexibility is carried out to show that the observed major high-frequency dynamics actually come from the rotational vibration of the stage due to the flexibility of ball bearings between the stage and the two linear guideways. The schematic diagram of the stage is shown in Fig., in which y, y c, and y 2 represent the axial displacement of the stage at the left guideway, the mass center, and the right guideway, respectively. l = l + l 2 is the distance between the two linear guideways with l and l 2 being the distance of the stage mass center to the two guideways. Let b and b 2 be the distance of the stage mass center to the front edge and the end edge of the recirculating ball bearings at the two guideways, respectively, when no actuation force is applied. The linear encoder scale is placed on the left guideway and the electromagnetic driving force F m = A u of the linear motor is applied to the stage at the left guideway as well. B ẏ and B 2 ẏ 2 represent the viscous frictions between the stage and the ball bearings at the two guideways, respectively. Without the loss of generality, it is assumed that two guideways and the stage are rigid during the entire motion but the ball bearings at the two guideways could endure some lateral deformations to cause the stage rotate as well, with the rotational angle of the stage denoted by α. Let w (ζ) and w 2 (ζ) be the lateral force per unit length applied to the stage by the ball bearings at the two guideways, respectively, with ζ being the axial geometric distance of the ball bearings to the stage mass center when the stage has no rotation. In the following, it is further assumed that b = b 2 = b = 2 b so that the lateral motion of the stage can be neglected for simplicity. Since α, assuming elastic deformations of the ball bearing and the rail with an equivalent stiffness of k f, one obtains y c = y l sin α y l α y 2 = y l sin α y lα w (ζ) = k f (l ( cos α)+ζ sin α) k f ζα,ζ [ b, b] w 2 (ζ) = k f (l 2 ( cos α) ζ sin α) k f ζα. (3) The dynamical model of the stage can thus be obtained as follows after ignoring the Coulomb friction and disturbances for

3 484 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 3, JUNE 205 time being: Mÿ c = A u B ẏ B 2 ẏ 2 J α = A ul B ẏl + B 2 ẏ 2 l 2 + b b b b w (ζ)(l α + ζ) dζ w 2 (ζ)( l 2 α + ζ) dζ. (4) Substituting (3) into (4) and ignoring the high-order terms of the rotational angle α, the transfer function from u to y can be obtained as y(s) u(s) = A [(J + Ml 2 )s2 + B 2 l 2 s + K] JMs 4 +(JB + B 2 ll 2 M )s 3 +(B 2 ll 2 B + MK B )s 2 + BK l s (5) Fig. 2. System identification of Case I in frequency domain. where K = 4 3 k f ( b 3 ), B = B + B 2, K B = K +(B l B 2 l 2 )l, and K l = K +(B l B 2 l 2 )lb 2 /B. As K (B l B 2 l 2 )l and K (B l B 2 l 2 )lb 2 /B in general, K B K and K l K. With these approximations, the transfer function (5) becomes y(s) u(s) = Ms 2 + Bs (J + Ml)s B 2 l 2 s + K Js 2. (6) + B 2 ll 2 s + K It is thus clear that the overall transfer function consists of the traditional rigid-body dynamics described by G (s) = Ms 2 + and the major high-frequency dynamics having lightly damped Bs poles and zeros described by G 2 (s) = (J +Ml2 )s2 +B 2 l 2 s+k Js 2 +B 2 ll 2 s+k, which are caused by the resonant mode of the stage rotation due to the flexibility of ball bearings. Furthermore, the resonant frequency ω r and antiresonant frequency ω ar of G 2 (s) can be predicted based on the physical parameters of the linear motor stage as K K ω ar = J + Ml 2, ω r = J. (7) III. SYSTEM IDENTIFICATION To verify the correctness of the proposed dynamical model and to determine the frequency range of its validity, system identification in the frequency domain has been carried out on the same experimental system as in [23], which consists of a commercial gantry powered by two iron-core linear motors with a linear encoder resolution of 0.5 μm. The experiments have been conducted on the Y-axis of the gantry with the X-axis motor locked at the different positions of the stage in Fig.. The same identification procedure as in [23] is used, so only the identification results are given below. Case I (X-axis motor locked at center position): The frequency responses of the Y-axis are shown in Fig. 2. With the MATLAB system identification toolbox and using the least square curve fitting in the frequency domain, the following overall transfer function is obtained P (s) =G (s)g 2 (s)g 3 (s) (8) where G (s) = 0.56s s G 2 (s) = 245.6(s2 +32s ) (s 2 +40s )(s + 800) G 3 (s) = (s )(s s ) (s )(s )(s s ) (s s )(s s ) (s s )(s 2 +84s ). The structure of G (s) and G 2 (s) matches well with the linear rigid-body dynamics and the major high-frequency dynamics due to the flexibility of ball bearings in (6), with the identified parameters of M =0.56, B =2.3, ωar = 285.5, and ω r = G 3 (s) represents other high-frequency dynamics higher than 000 rad/s. With the manufacturer data of M =47.4, l =0.82, b =0.8, l =0.4, the moment of inertia and the equivalent stiffness can be calculated to be J =4.76 and K = from (7), which agree well with the value range of these physical parameters. Case II (X-axis motor locked at right-side position): The frequency responses of Y-axis in Case II are shown in Fig. 3, with the overall identified transfer function given below G (s) = 0.56s s G 2 (s) = (s2 +23s ) (s 2 +40s )(s + 800) G 3 (s) = (s + 800)(s s ) (s )(s s )(s 2 +29s ) (s s )(s s ) (s s )(s 2 +89s ). (0) (9)

4 CHEN et al.: μ-synthesis-based ADAPTIVE ROBUST CONTROL OF LINEAR MOTOR DRIVEN STAGES WITH HIGH-FREQUENCY DYNAMICS 485 Fig. 4. Block diagram of μ-synthesis-based ARC. Fig. 3. System identification of Case II in frequency domain. error, measurement noise, external disturbance, adaptive model compensation control input, feedback control input, and online parameter estimator, respectively. The identified G (s) in both cases is the same, agreeing well with the theoretical prediction of the proposed model (6) that the rigid-body dynamics remain the same with the X-axis motor locked at different positions. The same structure of the identified major high-frequency dynamics G 2 (s) is obtained in Case II but with different resonant and antiresonant frequency of ω ar = 260 and ω r = 420, respectively. With the known structural parameters of M =47.4 and l =0.4465, the moment of inertia and the equivalent stiffness in Case II are calculated to be J =5.87 and K = Again, these values agree well with the theoretical prediction of the proposed model (6); the equivalent stiffness in Case I and II are almost same, and the moment of inertia of Case II should be a little bit larger since the mass center moves away from the geometric center. As seen from Figs. 2 and 3, the proposed model P (s) in both cases fit their experimentally obtained frequency responses well up to 2000 rad/s. However, with the rigid-body dynamics only, the fitting would be valid only within 200 rad/s. Thus, for the rigid-body-dynamics-based existing controllers to function well in reality, it may be necessary to limit the targeted closed-loop bandwidth below 200 rad/s, due to the presence of the lightly damped flexible modes in G 2 (s) and G 3 (s). With the valid range of the rigid-body dynamics being known, the parameter identification of (2) in time domain is also carried out to obtain the nonlinear dynamic characteristics at low frequencies. The standard least square identification method is used [22] and the resulting identified parameters are M =0.6, B =0.23, and Āf =0.5. It is seen that the identified value of M correlates well with the frequency domain identification results. IV. μ-synthesis-based ARC DESIGN With an integrated consideration of the nonlinear rigid-body dynamical model (2) and the LTI dynamical model G 2 (s) (6) incorporating major high-frequency dynamics, a novel μ-synthesis-based ARC algorithm will be developed in this section. The block diagram of the overall closed-loop system is illustrated in Fig. 4, in which y, y d, e, n, d, u ff, u fb, and ˆθ represent the output displacement, desired trajectory, tracking A. Controlled Adaptive Model Compensation The adaptive model compensation is based on the nonlinear rigid-body dynamical model (2) and use accurate online parameter estimation to effectively deal with various nonlinearity effects. Specifically, (2) can be rewritten in the following linear regression form when considering F dis as a lumped constant: u = ϕ T θ () where θ =[θ,θ 2,θ 3,θ 4 ] T =[ M, B,Āf, F dis ] T is the unknown parameter set, and ϕ T =[ ÿ, ẏ, S f (ẏ), ] is the measurement regressor. The adaptive model compensation can then be designed as u ff = ϕ T d ˆθ (2) where ˆθ is the parameter estimate of θ. To reduces the effect of measurement noises, ϕ T d =[ ÿ d, ẏ d, S f (ẏ d ), ] is used, which depends on the reference trajectory y d only. The next step is to obtain accurate parameter estimates online. Due to the physical meanings of these unknown parameters, the following assumption can be made. Assumption. The parametric uncertainties are bounded with known bounds, i.e., θ Ω θ Δ = { θ : θmin θ θ max } (3) where θ min =[θ min,..., θ 4min ] T, θ max =[θ max,..., θ 4max ] T. With the above assumption, the projection type least square estimation algorithm [22] can be used to obtain accurate online parameter estimates with a controlled adaptation process. Specifically, ˆθ is updated by ˆθ = Projˆθ (Γτ), ˆθ(0) Ωθ (4) where Γ is a positive definite matrix, τ is an adaptation function to be determined later, and Projˆθ( ) is the standard projection mapping detailed in [8] and[9]. To reduce the effect of measurement noises and avoid the need of acceleration feedback, a stable filter H f (s) (e.g., H f (s) = (τ f s+) 2 ) is applied to the both side of (2) u f = ϕ T f θ (5)

5 486 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 3, JUNE 205 Fig. 5. Block diagram of μ-synthesis feedback control design. where f represents the filtered value of, and ϕ T f = [ ÿ f, ẏ f, S f (ẏ f ), f ]. Thus, by defining the prediction output and the prediction error as û f = ϕ T f ˆθ, ɛ =û f u f. (6) One obtains the following prediction error model ɛ = ϕ T f θ. (7) Γ and τ can be defined by the standard least square method { αγ +νϕ Γ = T f Γϕ Γϕ f ϕ T f f Γ, if λ max (Γ(t)) ρ M 0, otherwise (8) τ = +νϕ T f Γϕ ϕ f ɛ (9) f where α is the forgetting factor, ν 0 with ν =0leading to the unnormalized algorithm, and ρ M is the preset upper bound for Γ(t) to avoid the estimator windup. Lemma. [8] With the projection type least square estimation algorithm (4), the parameter estimates ˆθ are always within the known bounded set Ω θ, i.e., ˆθ(t) Ω θ t. In addition, if the following persistent excitation condition is satisfied, t+t t ϕ f ϕ T f dτ βi p t>t 0 for some T>0β >0 (20) then ˆθ converge to their true values θ. B. μ-synthesis Robust Feedback Controller Design With the above controlled adaptive model compensation, the trajectory tracking control problem can now be converted into the traditional stabilization control problem that can be solved readily with μ-synthesis-based robust control designs. The block diagram of μ-synthesis feedback loop design is shown in Fig. 5, in which e = y y d is the tracking error, u fb = u u ff is the feedback control input, n is the measurement noise, d represents the lumped external disturbances and the model compensation error of u ff, the physical plant is represented by a parameterized transfer function G(s) with multiplicative modeling error, K(s) is the feedback controller needs to be synthesized, and w d, w n, w u, and w e, are four weighting transfer functions for the closed-loop system inputs d, n, and the output u fb, e, respectively. By choosing G(s) =G (s)g 2 (s), the major highfrequency dynamics due to the bearing flexibility are explicitly Fig. 6. Bode diagram of the plant dynamics with uncertainties. taken into account in the design. Furthermore, the effect of uncertain physical parameters such as M, B, ω ar, ω r, and the damping coefficients can be explicitly quantified through the use of structured unknown parameters in G(s) with appropriate variation ranges. Other modeling uncertainties such as the neglected high-frequency dynamics G 3 (s) and the fitting errors in Figs. 2 and 3 are treated as unstructured uncertainties, which are represented in Fig. 5 by a known weighting transfer function w unc and the unity uncertain linear dynamics Δ. With a weighting transfer function of w unc =2(G 3 (s) ), the identified actual plant model of P (s) =G (s)g 2 (s)g 3 (s) is within the possible plant dynamics described in Fig. 5, as seen from the Bode diagram of the sample of possible plant dynamics in Fig. 6 as well, in which the red line is the nominal model G (s)g 2 (s) and the yellow line is the identified plant model P (s) in (8). The weighting functions for the controller design are chosen as 00 w d = 20 s +, w n = (s + 00) s w u = s s +, w e = 0000 (2) in which w d has a large weighting at low frequencies and decreases after 20 rad/s to represent the fact that the disturbances are mainly at low frequencies. As the measurement noise is usually of high frequency, w n is chosen to be very small at low frequencies and increases to the level of encoder resolution when higher than 500 rad/s. w u increases from a value of to 00 after 200 rad/s to reflect the changing control objective of focusing on the tracking error minimization at low frequencies to the balanced control input and the tracking error minimization at high frequencies. A small value of w e is used to reflect the different scales of the desired tracking error and the control input; the tracking error is in the order of less than m, while the control input is in the order of 0 V. By establishing the structure of Fig. 5 in MATLAB and specifying the weighting functions as in (2), the following feedback controller is obtained using the μ-synthesis in MATLAB robust

6 CHEN et al.: μ-synthesis-based ADAPTIVE ROBUST CONTROL OF LINEAR MOTOR DRIVEN STAGES WITH HIGH-FREQUENCY DYNAMICS 487 Fig. 7. Bode diagrams of the closed-loop transfer functions. Fig. 8. Bode diagrams of the input disturbance sensitivity functions. control toolbox K(s) = 9.97e7(s )(s + 453)(s )(s + 800) (s+4574)(s+304)(s )(s s ) (s s )(s s ) (s s )(s s ) (22) which achieves a robust stability margin of.03, indicating that the closed-loop stability is guaranteed for all possible modeling uncertainties. To make a fair comparison with the above μ-synthesis-based ARC controller, a PID feedback controller with the optimally tuned gains is used K(s) = k p + k d s + k i s, with k p = 5000, k d = 20, k i = (23) Using linear approximation of various nonlinearities with the adaptive model compensation assuming accurate online parameter estimations, Bode diagrams of the closed-loop transfer functions from y d to y in Fig. 4 are plotted in Fig. 7, where the blue lines represent the proposed μ-synthesis controller and the red lines represents the PID controller. Similarly, Bode diagrams of the input disturbance sensitivity functions from d to y in Fig. 4 are plotted in Fig. 8. Since the μ-synthesis-based ARC controller (22) incorporates the major high-frequency dynamics G 2 (s) as part of the nominal model, it is seen that a closed-loop bandwidth around 250 rad/s is achieved, which is larger than the less than 200 rad/s of the PID controller, and a better disturbance rejection is achieved with the proposed μ-synthesis controller at frequencies below 00 rad/s. V. COMPARATIVE EXPERIMENTAL RESULTS A sampling frequency of 5 khz is used in all the control experiments, which results in a velocity measurement resolution of m/s. The following three control algorithms are compared: C: The proposed μ-synthesis-based ARC algorithm given by (2) and (22). The lower and upper bounds of the parameter variations for θ are chosen as θ min =[0.5, 0.08, 0.05, ] T and θ max =[0.7, 0.45, 0.35, ] T, respectively. The least square type estimation algorithm of (8) and (9) is implemented with α =0.02, μ =0., ρ M = 000, an initial adaptation rate matrix of Γ(0) = diag{, 0, 00, 00}, an initial set of parameter estimates of ˆθ(0) = [0.5, 0.2, 0., 0] T, and τ f =0.004 for the filter function H f (s). C2: The integrated direct/indirect ARC controller in [27]. The feedback control parameters are chosen as k = 00, k s = 20, and γ d = For the online parameter estimation, the parameter bounds, initial values, and adaptive rates are set to be the same as in C. C3: The PID feedback controller (23) with the feedforward compensation of u ff = ϕ T d ˆθ(0). To verify the performance robustness of these controllers to external disturbances and parameter variations, the following sets of test experiments are performed: Set : Nominal performance of the controllers with no external disturbances and payload. Set 2: Performance robustness of the controllers to a sinusoid disturbance of d =0.5 sin(0t). Set 3: Performance robustness of the controllers to a payload of 5 kg mounted on the gantry. As in [3] and [6], the point-to-point motion with a travel movement of 0.4 m is used as the desired trajectory. The lowspeed experiment (the maximum velocity of 0.4 m/s) and the high-speed experiment (the maximum velocity of m/s) are carried out, respectively. Tables I and II show the tracking performance of the low-speed and high-speed experiments by quantitative measures, where e M, e F, e S, L 2 [e], and L 2 [u] represent the maximal transient tracking error, the final tracking accuracy during last 0 s, the steady-state tracking error, the L 2 norm of tracking error, and the average control effort. The magnified

7 488 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 3, JUNE 205 TABLE I TRACKING PERFORMANCE OF THE LOW-SPEED EXPERIMENTS e M (μm) e F (μm) e S (μm) L 2 [e](μm) L 2 [u](v) C (Set) C2 (Set) C3 (Set) C (Set2) C2 (Set2) C3 (Set2) C (Set3) C2 (Set3) C3 (Set3) TABLE II TRACKING PERFORMANCE OF THE HIGH-SPEED EXPERIMENTS e M (μm) e F (μm) e S (μm) L 2 [e](μm) L 2 [u](v) C (Set) C2 (Set) C3 (Set) C (Set2) C2 (Set2) C3 (Set2) C (Set3) C2 (Set3) C3 (Set3) Fig. 0. Parameter estimation of C at low-speed experiment (Set). Fig.. Magnified plot of tracking errors at high speed (Set). Fig. 9. Magnified plot of tracking errors at low speed (Set). plots of the tracking errors during one running period in Set are shown in Fig. 9 for low-speed experiments, and Fig. for high-speed experiments as well. It is seen from these results that, in general, due to the use of adaptive model compensation with converging online parameter estimates as shown in Fig. 0 for C, the tracking errors of both C and C2 become smaller after several periods of running, resulting in an improved performance over C3. Furthermore, due to the higher closed-loop bandwidth achieved by C, it exhibits a better transient tracking performance than C2. However, its tracking and disturbance rejection become worse at the resonant frequency ω r, leading to a steady-state tracking error of 2 μm oscillation at high-speed ex- periment. Further tuning of weighting functions in the design of μ-synthesis-based robust feedback controller may help remove this oscillation at high-speed experiments. In the disturbance rejection experiments of Set2, the steadystate tracking errors of C shown in Figs. 2 and 3 are almost half of those in C2 and C3, which verifies the better disturbance rejection performance of the proposed algorithm at low frequencies. With a 5 kg payload mounted on the gantry in Set3, the inertia estimate ˆθ of the proposed algorithm C quickly converges to its new actual value, as shown in Fig. 6. As a result, the same good tracking performance as in no load situation in Set is seen in Figs. 4 and 5, verifying the performance robustness of C to parameter variations. With the known inertial mass of M =47.4kgand the additional 5 kg payload, the accurate parameter estimates of θ in Set and Set3 can be verified by noting ˆθ (Set)/ˆθ (Set3) = 0.6/ /(47.4+5). VI. CONCLUSION In this paper, physical modeling and dynamic analysis of the major high-frequency dynamics seen in the linear motor driven

8 CHEN et al.: μ-synthesis-based ADAPTIVE ROBUST CONTROL OF LINEAR MOTOR DRIVEN STAGES WITH HIGH-FREQUENCY DYNAMICS 489 Fig. 2. Steady-state tracking errors at low speed (Set2). Fig. 5. Tracking errors at high-speed experiment (Set3). Fig. 3. Steady-state tracking errors at high speed (Set2). Fig. 6. Parameter estimation of C at high speed (Set3). Fig. 4. Tracking errors at low-speed experiment (Set3). stages are presented, which explicitly take into account the flexibility of the ball bearings between the stage and the linear guideways. System identification is also carried out to validate the proposed dynamical model. With the knowledge of both the nonlinear rigid-body dynamics and the structure of major high-frequency dynamics of the linear motor stages, a novel μ-synthesis-based ARC algorithm is proposed. Its use of controlled adaptive model compensation with accurate online parameter estimation effectively handles the inherit nonlinearities of the linear motor stages. As a result, the original more difficult tracking control problem is converted into a standard robust stabilization problem, which can be solved readily using the welldeveloped μ-synthesis-based linear robust control techniques. A robust feedback controller that achieves higher closed-loop bandwidth and better disturbance rejection at low frequencies is then obtained by incorporating the major high-frequency dynamics as a part of the nominal model in the μ-synthesis. Comparative experiments have been performed to verify the further improved control performance of the proposed algorithm.

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Lee, Dynamic analysis of a linear motion guide having rolling elements for precision positioning devices, J. Mech. Sci. Technol., vol. 22, no., pp , Zheng Chen received the B.Eng. and Ph.D. degrees in mechatronic control engineering from Zhejiang University, Zhejiang, China, in 2007 and 202, respectively. He is currently a Postdoctoral Researcher in the Department of Mechanical Engineering at Dalhousie University, Halifax, NS, Canada. His research interests include nonlinear adaptive robust control, precision motion control of mechatronic systems, bilateral teleoperation, and cooperative control of multiagent systems. Bin Yao (S 92 M 96 SM 09) received the B.Eng. degree in applied mechanics from the Beijing University of Aeronautics and Astronautics of China, Beijing, China, in 987, the M.Eng. degree in electrical engineering from the Nanyang Technological University of Singapore, Nanyang, Singapore, in 992, and the Ph.D. degree in mechanical engineering from the University of California at Berkeley, CA, USA, in 996. He has been with the School of Mechanical Engineering at Purdue University, Lafayette, IN, USA, since 996 and was promoted to the rank of Professor in He was honored as a Kuang-piu Professor in 2005 and a Changjiang Chair Professor at Zhejiang University by the Ministry of Education of China in 200 as well. Qingfeng Wang (M ) received the M.Eng. and Ph.D. degrees in mechanical engineering from Zhejiang University, Zhejiang, China, in 988 and 994, respectively. He then became a member of Faculty at Zhejiang University, where he was promoted to the rank of Professor in 999. He was the Director of the State Key Laboratory of Fluid Power Transmission and Control at Zhejiang University from 200 to 2005 and currently serves as the Head of the Institute of Mechatronic Control Engineering. His research interests include the electrohydraulic control components and systems, hybrid power system and energy saving technique for construction machinery, and system synthesis for mechatronic equipment.