µ-Synthesis-Based Adaptive Robust Control of Linear ... - IEEE Xplore

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

μ-Synthesis-Based Adaptive Robust Control of Linear Motor Driven Stages With HighFrequency 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 μsynthesis-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 INEAR motors have been widely used in machine tools [1], microelectronics, and semiconductor manufacturing equipment [2], because of their potential of achieving high-

L

Manuscript received October 14, 2013; revised February 5, 2014; accepted March 15, 2014. Date of publication March 14, 2014; date of current version May 18, 2015. Recommended by Technical Editor M. de Queiroz. This work was supported in part by the National Natural Science Foundation of China under Grant 51475412, the National Basic Research and Development Program of China under 973 Program Grant 2013CB035400, and the Science Fund for Creative Research Groups of National Natural Science Foundation of China under Grant 51221004. Z. Chen is with the State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China, and also with the Department of Mechanical Engineering, Dalhousie University, Halifax, NS B3H4R2, Canada (e-mail: [email protected]). B. Yao is with the State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China, and also with the School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]). Q. Wang is with the State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China (e-mail: qfwang@ zju.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2014.2369454

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]–[12] 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 [13]–[15]. 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 [16]–[19], friction [19]–[21], 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

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CHEN et al.: μ-SYNTHESIS-BASED ADAPTIVE ROBUST CONTROL OF LINEAR MOTOR DRIVEN STAGES WITH HIGH-FREQUENCY DYNAMICS

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 [13] ˙ + Fdis M y¨ = Fm − B y˙ − Af Sf (y)

(1)

where y, y, ˙ and y¨ represents the displacement, velocity, and acceleration respectively. Fm is the driving force with Fm = A1 u, where u is the control input in voltage and A1 is the lumped force and current amplifier gain. M , B, and Af are the inertia, viscous friction coefficient, and the Coulomb friction coeffi˙ is a known smooth function used to cient, respectively. Sf (y) approximate the discontinuous sign function sgn(y). ˙ Fdis represents the lumped modeling errors and external disturbances. For simplicity, the dynamical model (1) is rewritten as ¯ y¨ = u − B ¯ y˙ − A¯f Sf (y) ˙ + F¯dis M

(2)

¯ = M/A1 , B ¯ = B/A1 , A¯f = Af /A1 , F¯dis = Fdis /A1 . where M 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. 1.

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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. 1, in which y, yc , and y2 represent the axial displacement of the stage at the left guideway, the mass center, and the right guideway, respectively. l = l1 + l2 is the distance between the two linear guideways with l1 and l2 being the distance of the stage mass center to the two guideways. Let b1 and b2 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 Fm = A1 u of the linear motor is applied to the stage at the left guideway as well. B1 y˙ 1 and B2 y˙ 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 w1 (ζ) and w2 (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 b1 = b2 = ¯b = 12 b so that the lateral motion of the stage can be neglected for simplicity. Since α  1, assuming elastic deformations of the ball bearing and the rail with an equivalent stiffness of kf , one obtains yc = y − l1 sin α ≈ y − l1 α y2 = y − l sin α ≈ y − lα w1 (ζ) = kf (l1 (1 − cos α) + ζ sin α) ≈ kf ζα, ζ ∈ [−¯b, ¯b] w2 (ζ) = kf (l2 (1 − cos α) − ζ sin α) ≈ −kf ζα.

(3)

The dynamical model of the stage can thus be obtained as follows after ignoring the Coulomb friction and disturbances for

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time being: M y¨c = A1 u − B1 y˙ − B2 y˙ 2 Jα ¨ = A1 ul1 − B1 yl ˙ 1 + B2 y˙ 2 l2 − 



¯b

−¯b

w1 (ζ)(l1 α + ζ) dζ

¯b

+ −¯b

w2 (ζ)(−l2 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) A 1 [(J + M l12 )s2 + B 2 l 2 s + K ] = 4 u(s) J M s + (J B + B 2 ll2 M )s3 + (B 2 ll2 B + M K B )s2 + BK l s

(5) where K = 43 kf (¯b3 ), B = B1 + B2 , KB = K + (B1 l1 − B2 l2 )l1 , and Kl = K + (B1 l1 − B2 l2 )lB2 /B. As K  (B1 l1 − B2 l2 )l1 and K  (B1 l1 − B2 l2 )lB2 /B in general, KB ≈ K and Kl ≈ K. With these approximations, the transfer function (5) becomes 1 (J + M l12 )s2 + B2 l2 s + K y(s) = ¯ 2 . ¯ u(s) Js2 + B2 ll2 s + K M s + Bs

(6)

It is thus clear that the overall transfer function consists of the traditional rigid-body dynamics described by G1 (s) = M¯ s 21+ B¯ s and the major high-frequency dynamics having lightly damped (J +M l 12 )s 2 +B 2 l 2 s+K , poles and zeros described by G2 (s) = J s 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 ωr and antiresonant frequency ωar of G2 (s) can be predicted based on the physical parameters of the linear motor stage as   K K ωar = . (7) , ωr = 2 J + M l1 J 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. 1. 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) = G1 (s)G2 (s)G3 (s)

(8)

Fig. 2.

System identification of Case I in frequency domain.

where G1 (s) =

1 0.56s2 + 12.3s

G2 (s) =

2145.6(s2 + 32s + 8.15 × 104 ) (s2 + 40s + 2.186 × 105 )(s + 800)

G3 (s) =

7.825 × 103 (s + 2000)(s2 + 180s + 1.474 × 106 ) (s + 3000)(s + 4000)(s2 + 120s + 1.774 × 106 )

×

(s2 + 144.5s + 2.324 × 106 )(s2 + 103s + 4.9 × 106 ) . (s2 + 112.3s + 2.436 × 106 )(s2 + 84s + 5.066 × 106 ) (9)

The structure of G1 (s) and G2 (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 ¯ = 0.56, B ¯ = 12.3, ωar = 285.5, identified parameters of M and ωr = 476.5. G3 (s) represents other high-frequency dynamics higher than 1000 rad/s. With the manufacturer data of M = 47.4, l = 0.82, b = 0.18, l1 = 0.41, the moment of inertia and the equivalent stiffness can be calculated to be J = 4.76 and K = 10.13 × 105 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 G1 (s) = G2 (s) =

0.56s2

1 + 12.3s

2087.2(s2 + 23s + 6.76 × 104 ) (s2 + 40s + 1.764 × 105 )(s + 800)

G3 (s) = 2.575 × 107 (s + 1800)(s2 + 624s + 2 × 106 ) (s + 3000)(s2 + 5522s + 9.056 × 106 )(s2 +291s+2.583 × 106 )

×

(s2 + 252s + 4.5 × 106 )(s2 + 100s + 8.066 × 106 ) . (s2 + 146s + 5.055 × 106 )(s2 + 89s + 9.485 × 106 )

(10)

CHEN et al.: μ-SYNTHESIS-BASED ADAPTIVE ROBUST CONTROL OF LINEAR MOTOR DRIVEN STAGES WITH HIGH-FREQUENCY DYNAMICS

Fig. 4.

Fig. 3.

System identification of Case II in frequency domain.

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Block diagram of μ-synthesis-based ARC.

error, measurement noise, external disturbance, adaptive model compensation control input, feedback control input, and online parameter estimator, respectively. A. Controlled Adaptive Model Compensation

The identified G1 (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 G2 (s) is obtained in Case II but with different resonant and antiresonant frequency of ωar = 260 and ωr = 420, respectively. With the known structural parameters of M = 47.4 and l1 = 0.4465, the moment of inertia and the equivalent stiffness in Case II are calculated to be J = 5.87 and K = 10.35 × 105 . 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 G2 (s) and G3 (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 ¯ = 0.61, used [22] and the resulting identified parameters are M ¯ = 0.23, and A¯f = 0.15. It is seen that the identified value B ¯ correlates well with the frequency domain identification of M results. IV. μ-SYNTHESIS-BASED ARC DESIGN With an integrated consideration of the nonlinear rigid-body dynamical model (2) and the LTI dynamical model G2 (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, yd , e, n, d, uf f , uf b , and θˆ represent the output displacement, desired trajectory, tracking

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 = −ϕT θ

(11)

¯ , B, ¯ A¯f , F¯dis ]T is the unwhere θ = [θ1 , θ2 , θ3 , θ4 ]T = [M known parameter set, and ϕT = [−¨ y , −y, ˙ −Sf (y), ˙ 1] is the measurement regressor. The adaptive model compensation can then be designed as uf f = −ϕTd θˆ

(12)

where θˆ is the parameter estimate of θ. To reduces the effect of measurement noises, ϕTd = [−¨ yd , −y˙ d , −Sf (y˙ d ), 1] is used, which depends on the reference trajectory yd 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 1. The parametric uncertainties are bounded with known bounds, i.e., Δ

θ ∈ Ωθ = { θ : θm in ≤ θ ≤ θm ax }

(13)

where θm in = [θ1m in ,. . ., θ4m in ]T , θm ax = [θ1m ax ,. . ., θ4m ax ]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, θˆ is updated by ˙ θˆ = P rojθˆ (Γτ ) ,

ˆ ∈ Ωθ θ(0)

(14)

where Γ is a positive definite matrix, τ is an adaptation function to be determined later, and P rojθˆ (•) 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 Hf (s) (e.g., Hf (s) = 1 (τ f s+1) 2 ) is applied to the both side of (2) uf = −ϕTf θ

(15)

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Fig. 5.

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

Block diagram of μ-synthesis feedback control design.

where •f represents the filtered value of •, and ϕTf = [−¨ yf , −y˙ f , −Sf (y˙ f ), 1f ]. Thus, by defining the prediction output and the prediction error as ˆ u ˆf = −ϕTf θ,

=u ˆ f − uf .

(16)

Fig. 6.

Bode diagram of the plant dynamics with uncertainties.

One obtains the following prediction error model ˜  = −ϕTf θ.

(17)

Γ and τ can be defined by the standard least square method  αΓ − 1+ν ϕ1T Γϕ f Γϕf ϕTf Γ, if λm ax (Γ(t)) ≤ ρM f Γ˙ = 0, otherwise (18) τ =

1 ϕf  1 + νϕTf Γϕf

(19)

where α is the forgetting factor, ν ≥ 0 with ν = 0 leading to the unnormalized algorithm, and ρM is the preset upper bound for Γ(t) to avoid the estimator windup. Lemma 1. [8] With the projection type least square estimation algorithm (14), the parameter estimates θˆ are always within ˆ ∈Ω ¯ θ ∀t. In addition, if the ¯ θ , i.e., θ(t) the known bounded set Ω following persistent excitation condition is satisfied,  t+T ϕf ϕTf dτ ≥ βIp ∀t > t0 for some T > 0β > 0 (20)

taken into account in the design. Furthermore, the effect of uncertain physical parameters such as M , B, ωar , ωr , 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 G3 (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 wu n c and the unity uncertain linear dynamics Δ. With a weighting transfer function of wu n c = 2(G3 (s) − 1), the identified actual plant model of P (s) = G1 (s)G2 (s)G3 (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 G1 (s)G2 (s) and the yellow line is the identified plant model P (s) in (8). The weighting functions for the controller design are chosen as wd =

100 , 1 20 s + 1

wu =

1 200 s + 1 , 1 20000 s + 1

t

then θˆ 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 − yd is the tracking error, uf b = u − uf f is the feedback control input, n is the measurement noise, d represents the lumped external disturbances and the model compensation error of uf f , 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 wd , wn , wu , and we , are four weighting transfer functions for the closed-loop system inputs d, n, and the output uf b , e, respectively. By choosing G(s) = G1 (s)G2 (s), the major highfrequency dynamics due to the bearing flexibility are explicitly

wn =

0.5 × 10−6 (s + 100) s + 500

we = 10000

(21)

in which wd 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, wn is chosen to be very small at low frequencies and increases to the level of encoder resolution when higher than 500 rad/s. wu increases from a value of 1 to 100 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 we 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 0.0001 m, while the control input is in the order of 10 V. By establishing the structure of Fig. 5 in MATLAB and specifying the weighting functions as in (21), the following feedback controller is obtained using the μ-synthesis in MATLAB robust

CHEN et al.: μ-SYNTHESIS-BASED ADAPTIVE ROBUST CONTROL OF LINEAR MOTOR DRIVEN STAGES WITH HIGH-FREQUENCY DYNAMICS

Fig. 7.

Fig. 8.

Bode diagrams of the closed-loop transfer functions.

control toolbox K(s) = 9.97e7(s + 2 × 104 )(s + 4513)(s + 3000)(s + 800) (s+4574)(s+3014)(s+0.6527)(s2 +120.4s+8.219 × 104 )

×

(s2 + 122.8s + 7738)(s2 + 34.4s + 2.091 × 105 ) (s2 + 4123s + 5.715 × 106 )(s2 + 1933s + 4.785 × 106 )

(22)

which achieves a robust stability margin of 1.03, 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 1 K(s) = kp + kd s + ki , with kp = 15000, s kd = 120, ki = 300 000.

(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 yd 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 G2 (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 100 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

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Bode diagrams of the input disturbance sensitivity functions.

resolution of 0.0025 m/s. The following three control algorithms are compared: C1: The proposed μ-synthesis-based ARC algorithm given by (12) and (22). The lower and upper bounds of the parameter variations for θ are chosen as θm in = [0.5, 0.08, 0.05, −1]T and θm ax = [0.7, 0.45, 0.35, 1]T , respectively. The least square type estimation algorithm of (18) and (19) is implemented with α = 0.02, μ = 0.1, ρM = 1000, an initial adaptation rate matrix of Γ(0) = diag{1, 10, 100, 100}, an initial set of parameter esˆ = [0.5, 0.2, 0.1, 0]T , and τf = 0.004 timates of θ(0) for the filter function Hf (s). C2: The integrated direct/indirect ARC controller in [27]. The feedback control parameters are chosen as k1 = 100, ks = 120, and γd = 3000. For the online parameter estimation, the parameter bounds, initial values, and adaptive rates are set to be the same as in C1. C3: The PID feedback controller (23) with the feedforward ˆ compensation of uf f = −ϕTd θ(0). To verify the performance robustness of these controllers to external disturbances and parameter variations, the following sets of test experiments are performed: Set 1: 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(10t). Set 3: Performance robustness of the controllers to a payload of 5 kg mounted on the gantry. As in [13] and [16], 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 1 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 eM , eF , eS , L2 [e], and L2 [u] represent the maximal transient tracking error, the final tracking accuracy during last 10 s, the steady-state tracking error, the L2 norm of tracking error, and the average control effort. The magnified

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TABLE I TRACKING PERFORMANCE OF THE LOW-SPEED EXPERIMENTS

C1 (Set1) C2 (Set1) C3 (Set1) C1 (Set2) C2 (Set2) C3 (Set2) C1 (Set3) C2 (Set3) C3 (Set3)

e M (μm )

e F (μm )

e S (μm )

L 2 [e](μm )

L 2 [u ](V )

56.5 72.3 81.6 54.0 73.1 90.2 90.3 103.3 118.5

35.2 57.1 81.6 37.4 64.8 90.1 41.8 54.8 118.2

1.0 0.5 0.5 6.0 15.5 15.5 1.0 0.5 1.0

5.7 7.0 9.2 6.6 12.7 14.2 6.9 8.0 14.8

0.78 0.76 0.77 0.79 0.76 0.77 0.86 0.85 0.86

TABLE II TRACKING PERFORMANCE OF THE HIGH-SPEED EXPERIMENTS

C1 (Set1) C2 (Set1) C3 (Set1) C1 (Set2) C2 (Set2) C3 (Set2) C1 (Set3) C2 (Set3) C3 (Set3)

Fig. 9.

e M (μm )

e F (μm )

e S (μm )

L 2 [e](μm )

L 2 [u ](V )

64.8 65.3 80.1 52.4 77.6 97.6 92.9 104.1 118.0

52.2 57.8 77.5 52.2 63.9 89.8 45.3 57.6 115.9

2.0 1.0 1.0 5.5 16.0 16.0 1.5 1.0 1.0

8.1 8.0 10.3 8.6 13.4 15.0 8.7 8.1 16.7

1.33 1.29 1.30 1.33 1.29 1.30 1.48 1.46 1.47

Magnified plot of tracking errors at low speed (Set1).

plots of the tracking errors during one running period in Set1 are shown in Fig. 9 for low-speed experiments, and Fig. 11 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. 10 for C1, the tracking errors of both C1 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 C1, it exhibits a better transient tracking performance than C2. However, its tracking and disturbance rejection become worse at the resonant frequency ωr , leading to a steady-state tracking error of 2 μm oscillation at high-speed ex-

Fig. 10.

Parameter estimation of C 1 at low-speed experiment (Set1).

Fig. 11.

Magnified plot of tracking errors at high speed (Set1).

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 C1 shown in Figs. 12 and 13 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 θˆ1 of the proposed algorithm C1 quickly converges to its new actual value, as shown in Fig. 16. As a result, the same good tracking performance as in no load situation in Set1 is seen in Figs. 14 and 15, verifying the performance robustness of C1 to parameter variations. With the known inertial mass of M = 47.4 kg and the additional 5 kg payload, the accurate parameter estimates of θ1 in Set1 and Set3 can be verified by noting θˆ1 (Set1)/θˆ1 (Set3) = 0.6/0.68 ≈ 47.4/(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

CHEN et al.: μ-SYNTHESIS-BASED ADAPTIVE ROBUST CONTROL OF LINEAR MOTOR DRIVEN STAGES WITH HIGH-FREQUENCY DYNAMICS

Fig. 12.

Fig. 13.

Fig. 14.

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Steady-state tracking errors at low speed (Set2). Fig. 15.

Tracking errors at high-speed experiment (Set3).

Fig. 16.

Parameter estimation of C 1 at high speed (Set3).

Steady-state tracking errors at high speed (Set2).

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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[22] Z. Chen, B. Yao, and Q. Wang, “Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect,” IEEE/ASME Trans. Mechatron., vol. 18, no. 3, pp. 1122–1129, Jun. 2013. [23] Z. Chen, B. Yao, and Q. Wang, “Adaptive robust precision motion control of linear motors with integrated compensation of nonlinearities and bearing flexible modes,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 965–973, May 2013. [24] Z. Chen, B. Yao, and Q. Wang, “Adaptive robust precision motion control of linear motors with high frequency flexible modes,” presented at the IEEE 12th Int. Workshop Adv. Motion Control, Sarajevo, Bosnia and Herzegovina, 2012. [25] D. M. Alter and T. C. Tsao, “Control of linear motors for machine tool feed drives: Design and implementation of H ∞ optimal feedback control,” ASME J. Dyn. Syst., Meas. Control, vol. 118, pp. 649–656, 1996. [26] Z. Z. Liu, F. L. Luo, and M. A. Rahman, “Robust and precision motion control systems of linear-motor direct drive for high-speed X-Y table positioning mechanism,” IEEE Trans. Ind. Electron., vol. 52, no. 5, pp. 1357–1363, Sep. 2005. [27] B. Yao and R. Dontha, “Integrated direct/indirect adaptive robust precision control of linear motor drive systems with accurate parameter estimates,” in Proc. 2nd IFAC Conf. Mechatron. Syst., Berkeley, CA, USA, 2002, pp. 633–638. [28] H. Ohta and E. Hayashi, “Vibration of linear guideway type recirculating linear ball bearings,” J. Sound Vib., vol. 235, no. 5, pp. 847–861, 2000. [29] Y.-S. Yi and Y. Y. Kim, J. S. Choi, J. Yoo, D. J. Lee, S. W. Lee, and S. J. Lee, “Dynamic analysis of a linear motion guide having rolling elements for precision positioning devices,” J. Mech. Sci. Technol., vol. 22, no. 1, pp. 50–60, 2008. Zheng Chen received the B.Eng. and Ph.D. degrees in mechatronic control engineering from Zhejiang University, Zhejiang, China, in 2007 and 2012, 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 1987, the M.Eng. degree in electrical engineering from the Nanyang Technological University of Singapore, Nanyang, Singapore, in 1992, and the Ph.D. degree in mechanical engineering from the University of California at Berkeley, CA, USA, in 1996. He has been with the School of Mechanical Engineering at Purdue University, Lafayette, IN, USA, since 1996 and was promoted to the rank of Professor in 2007. 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 2010 as well. Qingfeng Wang (M’11) received the M.Eng. and Ph.D. degrees in mechanical engineering from Zhejiang University, Zhejiang, China, in 1988 and 1994, respectively. He then became a member of Faculty at Zhejiang University, where he was promoted to the rank of Professor in 1999. He was the Director of the State Key Laboratory of Fluid Power Transmission and Control at Zhejiang University from 2001 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.