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μ-Synthesis Based Adaptive Robust Control of Linear-Motor-Driven Stages with High-Frequency Flexible Modes*
6th IFAC Symposium on Mechatronic Systems The International Federation of Automatic Control April 10-12, 2013. Hangzhou, China
µ-Synthesis Based Adaptive Robust Control of Linear-Motor-Driven Stages with High-Frequency Flexible Modes ⋆ Zheng Chen ∗,∗∗∗ Bin Yao ∗∗,∗ Qingfeng Wang ∗ ∗
The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, 310027, China ∗∗ School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. ∗∗∗ Department of Mechanical Engineering, Dalhousie University, Halifax, NS, Canada B3J 2X4
Abstract: Linear motors have good potential to achieve high speed and high accuracy by eliminating gear related mechanical problems. By using the effective model compensation of nonlinear rigid-body dynamics, the close-loop bandwidth limitation due to the appearance of neglected high-frequency flexible modes has to be the main issue to maximize the achievable performance. In this paper, the modeling and identifications are carried out to verify the existing rigid-body and high-frequency dynamics. The µ-synthesis based adaptive robust control strategy is developed. The adaptive feedforward loop is used to keep good online parameter estimation and nonlinear model compensation, and also transfers the trajectory tracking problem into the traditional regulation problem which µ-synthesis control can more easily deal with. Since considering the fundamental flexible mode as a part of the nominal plant model, the µ-synthesis feedback loop can achieve higher close-loop bandwidth with the appearance of various model uncertainties. Comparative experiments have been done and the results show the excellent tracking performance of the propose algorithm. Keywords: Linear motor, Precision motion control, µ-synthesis, Adaptive control, High-frequency dynamics. 1. INTRODUCTION Linear motors have been widely applied in machine tools Yao et al. (1997), microelectronics manufacturing equipments and semiconductor manufacturing equipmentsYao et al. (2012), because of their potential of achieving highspeed and high-accuracy linear movement by eliminating gear related mechanical transmission problems. But to realize its high-speed/high-accuracy potential, various model uncertainties due to parameter variations (e.g., unknown load inertia) and disturbances, significant nonlinearities, and high-frequency flexible effect have to be considered into the controller design. Many control methods have been developed to deal with model uncertainties in the precision motion control of linear motors, such as disturbance observer Komada et al. (1991), neural-network-based control Otten et al. (1997), repetitive model predictive control Cao and Low (2009), iterative learning control Ding and Wu (2007); Wu et al. (2011) and various improved sliding mode control Wu et al. (2011); Huang and Sung (2010); Lin et al. (2012). An adaptive robust control (ARC) approach Yao (2009, 2010); Hu et al. (2011a); Gayaka and Yao (2011); Mohanty and Yao (2011a,b); Mohanty et al. (2012) is developed for the high-performance control of uncertain nonlinear ⋆ E-mail addresses: [email protected] (Zheng Chen), [email protected] (Bin Yao), [email protected] (Qingfeng Wang)
978-3-902823-31-1/13/$20.00 © 2013 IFAC
systems in the presence of both parametric uncertainties and uncertain nonlinearities, and successfully applied to the precision motion control of linear motors Xu and Yao (2001); Yao and Xu (2002); Hu et al. (2011b). 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 Yao et al. (2011); Lu et al. (2008); Hu et al. (2010); Chen et al. (2010), friction Jamaludin et al. (2009); Lu et al. (2009); Chen et al. (2010) and nonlinear electromagnetic effect Chen et al. (2013a). But all of these researches are focused on the rigid-body dynamics, and the maximum achievable performance of these controllers has been pursued by considering various model uncertainties and nonlinearities. Thus by now the largest performance limitation is the limited closeloop bandwidth due to the appearance of high-frequency flexible structural modes. In Ohta and Hayashi (2000); Yi et al. (2008) the resonant modes and vibration of linear guideway are discussed, and the physical modeling and identification of our specific linear motor driven stage have been carried out in Chen et al. (2013b). The knowledge of these high-frequency dynamics is used to guide the tuning of the controller gains to make a better trade-off in maximizing the performance while without exciting those high-frequency modes. The additional pole/zero cancelation is implemented as a early try in the control design
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of high-frequency modes Chen et al. (2012). µ-synthesis control is a kind of robust H∞ optimal control algorithms applied in the output feedback control of high order linear dynamics with model uncertainties, and as such has its definite advantage when considering the known flexible modes into the control design. In Alter and Tsao (1996); Liu et al. (2005), H∞ optimal control has been successfully developed for the motion control of linear motors, while they are more focused on the rigid-body dynamics.
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In this paper, the dynamical model, including nonlinear rigid-body dynamics and high-frequency flexible modes, is developed for linear motor driven stages. The identification of our experimental stage has been carried out, and verified the effectiveness of the proposed model. The µ-synthesis based adaptive robust control strategy is developed, where the adaptive model compensation of nonlinear rigid-body dynamics is designed as the feedforward loop, and thus transfers the tracking problem into the traditional regulation problem. The µ-synthesis control is subsequently used in the feedback loop design, which can achieve the optimal control performance with the appearance of various model uncertainties. Since the identified high-frequency model is considered as a part of the nominal model, the higher close-loop bandwidth can be achieved. Comparative experiments have been done and the results verify the excellent performance of the proposed algorithm. 2. DYNAMICAL MODELS In the traditional modeling of linear motors, researches are usually focused on the rigid-body dynamics. As such, when the fast electrical dynamics and structural flexible modes are ignored, the dynamical model of a linear motor can be described by Xu and Yao (2001) M y¨ = u − B y˙ − Af Sf (y) ˙ + Fdis
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where y, y˙ and y¨ represents the displacement, velocity and acceleration respectively. u is the control input, M is the inertia, B and Af are the viscous friction coefficient and the Coulomb friction coefficient. Sf (y) ˙ is a known smooth function used to approximate the discontinuous sign function sgn(y), ˙ such as Sf (y) ˙ = arctan(β y) ˙ with the value of β being large enough. Fdis represents the modeling errors and external disturbances. For mechanical and electronic systems, high-frequency dynamics are always existing and quite complicated due to various structural flexibilities of the specific plants. After the dynamical characteristics analysis of our linear motor driven stages, it is found that the stage rotation due to the flexibility of ball bearing is the fundamental flexible mode when compared to others such as electrical dynamics and mechanical resonance. The schematic diagram of the stage can be shown in Fig. 1. y represents the displacement of the stage at the left guideway, where the linear motor is driving and the position signal is measured. yc and y2 are the displacements of the stage at the mass center and the right guideway. α is the angle of the stage rotation. l = l1 + l2 , b = b1 + b2 are geometric sizes of the stage. w1 and w2 are the normal bearing pressure at the left and right guideway. Fm = u is the driving force, and B1 y˙ and B2 y˙ 2 are viscous frictions.
Fig. 1. Schematic diagram of the linear motor driven stage Since the high-frequency modes have to be described in the form of high order linear dynamics, the system model needs to be linearized and various nonlinearities and model uncertainties are ignored here. Thus, one obtains yc = y − l1 α y2 = y − lα (2) w1 = kf xα, x ∈ [−b2 , b1 ] w2 = kf xα, x ∈ [−b1 , b2 ] where kf is the equivalent stiffness due to the elastic deformations of the bearing ball and the rail. And the dynamical model of the stage can be obtained as following M y¨c = u − B1 y˙ − B2 y˙ 2 ∫b1 ∫b2 (3) Jα ¨ = ul1 − B1 yl ˙ 1 + B2 y˙ 2 l2 − w1 x dx − w2 x dx −b2
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Substituting (2) into (3) and assuming that B2 l2 ≈ B1 l1 , the transfer function from u to y can be achieved y(s) 1 (J + M l12 )s2 + B2 l2 s + K = (4) u(s) M s2 + Bs Js2 + B2 ll2 s + K where K = 23 kf (b31 + b32 ) and B = B1 + B2 . G1 (s) = 1 M s2 +Bs represents the linearized model of the rigid-body (J+M l2 )s2 +B l2 s+K
2 1 dynamics in (1), and G2 (s) = is the Js2 +B2 ll2 s+K fundamental flexible mode with the lightly damped poles and zeros. The resonant frequency ωr and anti-resonant frequency ωar of G2 (s) are √ √ K K ωar = , ω = (5) r J + M l12 J
The dynamical model (4) explicitly illustrates the fundamental flexible effect of the linear motor driven stage, and simultaneously holds the structural characteristics of the linear rigid-body dynamics. 3. SYSTEM IDENTIFICATION A commercial Anorad Gantry by Rockwell Automation has been set up Lu et al. (2008), and powered by Anorad LC-50-200 iron core linear motors. The linear encoders with a resolution of 0.5µm are used as position measurement signals. The experiments have been conducted on Y-axis, and a dSPACE CLP1103 controller board is used to implement real-time control algorithm.
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To verify the proposed high-frequency dynamical model and determine the valid range of the rigid-body dynamics, system identification in frequency domain has been carried out Chen et al. (2013b). The excitation signal is sum of 5000 sinusoidal signals ranging from 1Hz to 5KHz with a sampling frequency of 10KHz. The frequency responses of the system is achieved in Fig.2. By using the standard system identification toolbox in MATLAB and through iterations focusing on different frequency ranges, the following overall transfer function is obtained
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4.1 Adaptive Feedforward Model Compensation
where θ = [M, B, Af , Fdis ]T is the unknown parameter set, and φT = [−¨ y , −y, ˙ −Sf (y), ˙ 1] is the measurement regressor. The feedforward controller can be easily designed as uf f = −φTd θˆ (9)
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Fig. 2. System identification in frequency domain P (s) = G1 (s)G2 (s)G3 (s)
4. µ-SYNTHESIS BASED ARC CONTROL DESIGN
Before the feedback control design, feedforward model compensation is usually used to achieve better tracking performance. Compared to the high-frequency flexible modes, the nonlinear rigid-body dynamics with the accurate parameter estimates can be much more important to obtain good compensation effect. Thus, the feedforward controller design in this subsection is dependent on the nonlinear rigid-body model, and (1) can be rewritten in the linear regression form when considering Fdis as a lumped constant. u = −φT θ (8)
Blue: experimental data Red: curve fitting of P(s)
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least square identification method is used Chen et al. (2013a) and the resulting identified parameters are M = 0.61, B = 0.23, Af = 0.15. It is seen that the estimated value of M is correlates well with the frequency domain identification results. But B may not match well because of the lack of excitation signal in low frequency range.
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where 1 0.56s2 + 12.3s 2145.6(s2 + 32s + 8.15 × 104 ) G2 (s) = 2 (7) (s + 40s + 2.186 × 105 )(s + 800) 7.825 × 103 (s + 2000)(s2 + 180s + 1.474 × 106 ) G3 (s) = (s + 3000)(s + 4000)(s2 + 120s + 1.774 × 106 ) (s2 + 144.5s + 2.324 × 106 )(s2 + 103s + 4.9 × 106 ) × 2 (s + 112.3s + 2.436 × 106 )(s2 + 84s + 5.066 × 106 ) G1 (s) =
The structure of G1 (s) and G2 (s) matches well the linear rigid-body model and the fundamental flexible mode in (4), with the identified parameters M = 0.56, B = 12.3 ωar = 285.5 and ωr = 476.5. G3 (s) represents other uncertain high-frequency dynamics higher than 1000rad/s. As seen from Fig.2, the obtained P (s) fits the experimentally obtained frequency response well up to 2000rad/s. For the traditional feedback control, the achievable close-loop bandwidth is limited due to presence of the flexible modes G2 (s) and G3 (s). Within the frequency range of 200rad/s, only the rigid-body dynamics need to be considered and all flexible modes could be neglected, which is assumed in the previous researches. Thus, for those controllers to function well in reality, it is necessary to limit the targeted closed-loop bandwidth below 200rad/s. With the above discussion of the valid range of the rigid-body dynamics, the time domain identification of (1) is subsequently carried out to clarify the nonlinear dynamic characteristics at low frequencies. The standard
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. Then, the next key point is how to achieve the accurate parameter estimate 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., ∆
θ ∈ Ωθ = { θ : θmin ≤ θ ≤ θmax } (10) where θmin = [θ1min , · · ·, θ4min ]T , θmax = [θ1max , · · ·, θ4max ]T . A projection type least square estimation algorithm Chen et al. (2013a) is developed to achieve the accurate and bounded online parameter estimate. Specifically, θˆ is updated by: ˙ ˆ ∈ Ωθ θˆ = P rojθˆ (Γτ ) , θ(0) (11) where Γ is a positive definite diagonal matrix, and τ is an adaptation function to be determined later. The projection mapping P rojθˆ(•) is defined by ◦ •, if θˆ ∈Ωθ or nTθˆ • ≤ 0 nθˆnTθˆ P rojθˆ(•) = (12) (I − Γ )•, θˆ ∈ ∂Ωθ and nTθˆ • > 0 T nθˆ Γnθˆ ◦
in which • ∈ R4 , Ωθ and ∂Ωθ denote the interior and the boundary of Ωθ respectively, and nθˆ represents the outward unit normal vector at θˆ ∈ ∂Ωθ .
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To reduce the effect of measurement noises and avoid the need of acceleration feedback, a stable filter Hf (s) is used to the both side of (1). uf = −φTf θ
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Γ and τ can be defined by the standard least square method 1 αΓ − Γφf φTf Γ, if λmax (Γ) ≤ ρM 1 + νφTf Γφf Γ˙ = (16) 0, otherwise 1 τ= φf ϵ (17) 1 + νφTf Γφf where α is the forgetting factor, ν ≥ 0 with ν = 0 leading to the unnormalized algorithm, ρM is the pre-set upper bound for ∥Γ(t)∥ to avoid the estimator windup. Lemma 1. When using the projection type least square estimation algorithm (11), the parameter estimate θˆ are ˆ ¯ θ , i.e., θ(t) always within the known bounded set Ω ∈ ¯ Ωθ , ∀t. In addition, if the following persistent excitation (PE) condition is satisfied, t+T ∫
φf φTf dτ ≥ βIp , ∀t > t0 for some T > 0, andβ > 0 (18) t
then θˆ converges to its true value θ. 4.2 µ-synthesis Feedback Design µ-synthesis control is a kind of robust H∞ control algorithms to achieve the optimal control performance and simultaneously guarantee the robust stability at the appearance of model uncertainties. With these definite advantages in the output feedback control of high order linear dynamics, it is developed as the feedback controller of our system which consider the identified high-frequency model as a part of the nominal model. Due to the adaptive feedforward design in the above subsection, the position tracking of desired trajectory can be transferred into the traditional regulation problem which µ-synthesis control can more easily deal with. The block diagram of µ-synthesis design setup is shown in Fig.3. G(s) = G1 (s)G2 (s) represents the nominal model with the high-frequency mode identified in (7). e = y − yd is the tracking error, uf b = u − uf f is the feedback control input. ∆ represent the uncertain linear dynamics with the bounded magnitude of 1, and wunc is its weight to describe the uncertainty magnitude at difference frequencies. n is the measurement noise. d represents both external disturbances and the calculated error of the dynamic inversion uf f due to model uncertainties,
Fig. 3. The block diagram of µ-synthesis design setup unknown parameters and the flexible mode. K(s) is the feedback controller needs to be designed. wd , wn , wu , we are four weights for the system inputs d, n and output u, e respectively. The main issue of µ-synthesis control is to choose proper weights such that the control performance is optimized and the robust stability can simultaneously guaranteed for these model uncertainties. Thus, the weights in our system are chosen as follows: wunc = 2(G3 (s) − 1) 100 wd = 1 20 s + 1 0.5 × 10−6 (s + 100) wn = (19) s + 500 1 s+1 wu = 200 1 20000 s + 1 we = 10000 G3 (s) in (7) is the high-frequency dynamics higher than 1000rad/s where digital controllers can not implemented, so wunc has to consider it as the main model uncertainty and be extended twice due to various other uncertainties. wd has large weighting at low frequencies and decreases after 20rad/s to present the disturbance effect mainly locates at low frequencies. Oppositely the measurement noise usually locates at the high frequency range, as such wn is chosen to be very small at low frequencies and come to the level of encoder resolution when higher than 500rad/s. With the disturbance and noise rejection objective to be less than 0.0001, we is set to be 10000. The time constant of wu is chosen to be 1/200, which means the focus of control design is the tracking error at low frequencies but gradually transferred to the control input limitation when larger than 200rad/s. By using the MATLAB robust control toolbox, the feedback controller K(s) can be achieved 9.75 × 107 (s + 2 × 104 )(s + 4362) (s + 4422)(s + 3012)(s + 0.5574) (s + 800)(s + 3000)(s2 + 122.9s + 7753) (20) × 2 (s + 120.4s + 82270)(s2 + 3987s + 5.783 × 106 ) (s2 + 38.2s + 2.137 × 105 ) × 2 (s + 2199s + 4.683 × 106 ) K(s) =
The robust stability margin of the resulting controller is 1.13 > 1, which guarantee the stability for all values
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of its modeled uncertainties. Theoretically, setting larger turnover frequency of wd and wu will further improve the control performance due to higher achievable close-loop bandwidth. However it will also excite the high frequency uncertainties, and finally make the system unstable. So it is a trade-off to achieve the balance between control performance and robust stability.
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As in Xu and Yao (2001); Lu et al. (2008), the point-topoint motion with a travel movement of 0.4m is used as the desired trajectory, which is typical in manufacture industry. The low speed experiment (the maximum velocity of 0.4m/s) and the high speed experiment (the maximum velocity of 1m/s) are carried out respectively. The experimental results are shown in Fig.4-7. Table 1 and Table 2 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 seconds, the steady-state tracking error, the L2 norm of tracking error and the average control effort. It is seen that the transient tracking performance of C1 is obviously better than C2 in the both low speed and high speed experiments. Due to the lack of integral term in µ-synthesis control, the steady-state tracking error of C1 is with a 2µm oscillation. The online parameter estimations are shown in Fig.5 and Fig.7, where the inertial θ1 converges to its true value, and the friction coefficients θ2 and θ3 may vary a little at different speeds.
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with kp = 15000, kd = 120, ki = 300000. And the static feedforward compensation is designed as uf f = ˆ −φT θ(0).
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Fig. 4. Tracking errors at the low speed experiment
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C1: The proposed µ-synthesis based adaptive robust controller (20) and (9). The lower and upper bounds of the parameter variations for θ are chosen as θmin = [0.5, 0.08, 0.05, −1]T and θmax = [0.7, 0.45, 0.35, 1]T , respectively. The least square type estimation algorithm (16)and (17) is implemented with α = 0.02, µ = 0.1, ρM = 1000, and the initial adaptation rates are Γ(0) = diag{1, 10, 100, 100}. The initial parameter estimates ˆ are θ(0) = [0.5, 0.2, 0.1, 0]T . The filter function Hf (s) is set as τf = 0.004 and a relative degree equal to 2. C2: The PID feedback controller
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During the following control experiments, the dSPACE controller’s sampling frequency is selected to be 5kHz, which results in a velocity measurement resolution of 0.0025m/s. The following two control algorithms are compared:
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5. COMPARATIVE EXPERIMENTAL RESULTS
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Fig. 6. Tracking errors at the high speed experiment Table 1. Tracking performance of the low speed experiment by quantitative measures. C1 C2
eM (µm) 67.9 84.7
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Table 2. Tracking performance of the high speed experiment by quantitative measures.
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Fig. 7. Estimation of important parameters at the high speed experiment 6. CONCLUSION The integrated modeling of the linear motor driven stage including nonlinear rigid-body dynamics and high frequency flexible modes is carried out, and verified by the specific system identification in time domain and frequency domain. The µ-synthesis based adaptive robust control strategy is developed, where the adaptive feedforward model compensation of nonlinear rigid-body dynamics is designed to transfer the tracking problem into the traditional regulation problem. The µ-synthesis technology is subsequently used as the feedback loop design to achieve the optimal control performance with the appearance of model uncertainties. By considering the fundamental flexible mode as a part of the nominal model, the µ-synthesis controller can obtain higher close-loop bandwidth. Comparative experiments have been done, and the results shows the excellent performance of the proposed algorithm. REFERENCES Alter, D.M. and Tsao, T.C. (1996). Control of linear motors for machine tool feed drives: design and implementation of h∞ optimal feedback control. ASME J. of Dynamic systems, Measurement, and Control, 118, 649–656. Cao, R. and Low, K. (2009). A repetitive model predictive control approach for precision tracking of a linear motion system. IEEE Transactions on Industrial Electronics, 56(6), 1955–1962. Chen, S.L., Tan, K., Huang, S., and Teo, C. (2010). Modeling and compensation of ripples and friction in permanent-magnet linear motor using a hysteretic relay. IEEE/ASME Trans. Mechatronics, 15(4), 586–594. Chen, Z., Yao, B., and Wang, Q. (2012). Adaptive robust precision motion control of linear motors with high frequency flexible modes. Proceedings of The 12th IEEE International Workshop on Advanced Motion Control. Sarajevo. Chen, Z., Yao, B., and Wang, Q. (2013a). Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect. IEEE/ASME Transactions on Mechatronics, 18(3), 1122–1129.
Chen, Z., Yao, B., and Wang, Q. (2013b). Adaptive robust precision motion control of linear motors with integrated compensation of nonlinearities and bearing flexible modes. IEEE Transactions on Industrial Informatics, 9(2), 965–973. Ding, H. and Wu, J. (2007). Point-to-point motion control for a high-acceleration positioning table via cascaded learning schemes. IEEE Transactions on Industrial Electronics, 54(5), 2735–2744. Gayaka, S. and Yao, B. (2011). Accommodation of unknown actuator faults using output feedback based adaptive robust control. International Journal of Adaptive Control and Signal Processing, 25(11), 965–982. Hu, C., Yao, B., and Wang, Q. (2010). Coordinated adaptive robust contouring control of an industrial biaxial precision gantry with cogging force compensations. IEEE Transactions on Industrial Electronics, 57(5), 1746–1754. Hu, C., Yao, B., and Wang, Q. (2011a). Adaptive robust precision motion control of systems with unknown input dead-zones: a case study with comparative experiments. IEEE Transactions on Industrial Electronics, 57(6), 2454–2464. Hu, C., Yao, B., and Wang, Q. (2011b). Global task coordinate frame based contouring control of linearmotor-driven biaxial systems with accurate parameter estimations. IEEE Transactions on Industrial Electronics, 58(11), 5195–5205. Huang, Y. and Sung, C. (2010). Function-based controller for linear motor control systems. IEEE Transactions on Industrial Electronics, 57(3), 1096–1105. Jamaludin, Z., Brussel, H.V., and Swevers, J. (2009). Friction compensation of an xy feed table using frictionmodel-based feedforward and an inverse model-based disturbance observer. IEEE Transactions on Industrial Electronics, 56(10), 3848–3853. Komada, S., Ohnishi, K., and Hori, T. (1991). Hybrid position/force control of robot manipulators based on acceleration controller. Proc. IEEE Conf. on Robotics and Automation, 48–55. Lin, F., Chou, P., Chen, C., and Lin, Y.S. (2012). Dspbased cross-coupled synchronous control for dual linear motors via intelligent complementary sliding mode control. IEEE Transactions on Industrial Electronics, 59(2), 1061–1073. Liu, Z., Luo, F., and Rahman, M. (2005). Robust and precision motion control systems of linear-motor direct drive for high-speed x-y table positioning mechanism. IEEE Transactions on Industrial Electronics, 52(5), 1357–1363. Lu, L., Chen, Z., Yao, B., and Wang, Q. (2008). Desired compensation adaptive robust control of a linear motor driven precision industrial gantry with improved cogging force compensation. IEEE/ASME Transactions on Mechatronics, 13(6), 617–624. Lu, L., Yao, B., Wang, Q., and Chen, Z. (2009). Adaptive robust control of linear motors with dynamic friction compensation using modified lugre model. Automatica, 45(12), 2890–2896. Mohanty, A., Gayaka, S., and Yao, B. (2012). An adaptive robust observer for velocity estimation in an electrohydraulic system. International Journal of Adaptive Control and Signal Processing. (accepted and available
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on line with DOI 10.1002/acs.2291). Mohanty, A. and Yao, B. (2011a). Indirect adaptive robust control of hydraulic manipulators with accurate parameter estimates. IEEE Transaction on Control System Technology, 19(3), 567–575. Mohanty, A. and Yao, B. (2011b). Integrated direct/indirect adaptive robust control of hydraulic manipulators with valve deadband. IEEE/ASME Transactions on Mechatronics, 16(4), 707–715. Ohta, H. and Hayashi, E. (2000). Vibration of linear guideway type recirculating linear ball bearings. Journal of Sound and Vibration, 235(5), 847–861. Otten, G., Vries, T., Amerongen, J., Rankers, A., and Gaal, E. (1997). Linear motor motion control using a learning feedforward controller. IEEE/ASME Transactions on Mechatronics, 2(3), 179–187. Wu, J., Xiong, Z., Lee, K.M., and Ding, H. (2011). High-acceleration precision point-to-point motion control with look-ahead properties. IEEE Transactions on Industrial Electronics, 58(9), 4343–4352. Xu, L. and Yao, B. (2001). Adaptive robust precision motion control of linear motors with negligible electrical dynamics: theory and experiments. IEEE/ASME Transactions on Mechatronics, 6(4), 444–452. Yao, B. (2009). Desired compensation adaptive robust control. ASME Journal of Dynamic Systems, Measurement, and Control, 131(6), 1–7. Yao, B. (2010). Advanced motion control: from classical PID to nonlinear adaptive robust control. The 11th IEEE International Workshop on Advanced Motion Control, 815–829. Nagaoka, Japan. Yao, B., Al-Majed, M., and Tomizuka, M. (1997). High performance robust motion control of machine tools: An adaptive robust control approach and comparative experiments. IEEE/ASME Trans. on Mechatronics, 2(2), 63–76. Yao, B., Hu, C., Lu, L., and Wang, Q. (2011). Adaptive robust precision motion control of a high-speed industrial gantry with cogging force compensations. IEEE Transaction on Control System Technology, 19(5), 1149– 1159. Yao, B., Hu, C., and Wang, Q. (2012). An orthogonal global task coordinate frame for contouring control of biaxial systems. IEEE/ASME Trans. on Mechatronics, 17(4), 622–634. Yao, B. and Xu, L. (2002). Adaptive robust control of linear motors for precision manufacturing. International Journal of Mechatronics, 12(4), 595–616. Yi, Y., Kim, Y., and et al. (2008). Dynamic analysis of a linear motion guide having rolling elements for precision positioning devices. Journal of Mechanical Science and Technology, 22(1), 50–60.
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µ-Synthesis Based Adaptive Robust Control of Linear-Motor-Driven Stages with High-Frequency Flexible Modes ⋆ Zheng Chen ∗,∗∗∗ Bin Yao ∗∗,∗ Qingfeng Wang ∗ ∗
The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, 310027, China ∗∗ School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. ∗∗∗ Department of Mechanical Engineering, Dalhousie University, Halifax, NS, Canada B3J 2X4
Abstract: Linear motors have good potential to achieve high speed and high accuracy by eliminating gear related mechanical problems. By using the effective model compensation of nonlinear rigid-body dynamics, the close-loop bandwidth limitation due to the appearance of neglected high-frequency flexible modes has to be the main issue to maximize the achievable performance. In this paper, the modeling and identifications are carried out to verify the existing rigid-body and high-frequency dynamics. The µ-synthesis based adaptive robust control strategy is developed. The adaptive feedforward loop is used to keep good online parameter estimation and nonlinear model compensation, and also transfers the trajectory tracking problem into the traditional regulation problem which µ-synthesis control can more easily deal with. Since considering the fundamental flexible mode as a part of the nominal plant model, the µ-synthesis feedback loop can achieve higher close-loop bandwidth with the appearance of various model uncertainties. Comparative experiments have been done and the results show the excellent tracking performance of the propose algorithm. Keywords: Linear motor, Precision motion control, µ-synthesis, Adaptive control, High-frequency dynamics. 1. INTRODUCTION Linear motors have been widely applied in machine tools Yao et al. (1997), microelectronics manufacturing equipments and semiconductor manufacturing equipmentsYao et al. (2012), because of their potential of achieving highspeed and high-accuracy linear movement by eliminating gear related mechanical transmission problems. But to realize its high-speed/high-accuracy potential, various model uncertainties due to parameter variations (e.g., unknown load inertia) and disturbances, significant nonlinearities, and high-frequency flexible effect have to be considered into the controller design. Many control methods have been developed to deal with model uncertainties in the precision motion control of linear motors, such as disturbance observer Komada et al. (1991), neural-network-based control Otten et al. (1997), repetitive model predictive control Cao and Low (2009), iterative learning control Ding and Wu (2007); Wu et al. (2011) and various improved sliding mode control Wu et al. (2011); Huang and Sung (2010); Lin et al. (2012). An adaptive robust control (ARC) approach Yao (2009, 2010); Hu et al. (2011a); Gayaka and Yao (2011); Mohanty and Yao (2011a,b); Mohanty et al. (2012) is developed for the high-performance control of uncertain nonlinear ⋆ E-mail addresses: [email protected] (Zheng Chen), [email protected] (Bin Yao), [email protected] (Qingfeng Wang)
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systems in the presence of both parametric uncertainties and uncertain nonlinearities, and successfully applied to the precision motion control of linear motors Xu and Yao (2001); Yao and Xu (2002); Hu et al. (2011b). 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 Yao et al. (2011); Lu et al. (2008); Hu et al. (2010); Chen et al. (2010), friction Jamaludin et al. (2009); Lu et al. (2009); Chen et al. (2010) and nonlinear electromagnetic effect Chen et al. (2013a). But all of these researches are focused on the rigid-body dynamics, and the maximum achievable performance of these controllers has been pursued by considering various model uncertainties and nonlinearities. Thus by now the largest performance limitation is the limited closeloop bandwidth due to the appearance of high-frequency flexible structural modes. In Ohta and Hayashi (2000); Yi et al. (2008) the resonant modes and vibration of linear guideway are discussed, and the physical modeling and identification of our specific linear motor driven stage have been carried out in Chen et al. (2013b). The knowledge of these high-frequency dynamics is used to guide the tuning of the controller gains to make a better trade-off in maximizing the performance while without exciting those high-frequency modes. The additional pole/zero cancelation is implemented as a early try in the control design
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of high-frequency modes Chen et al. (2012). µ-synthesis control is a kind of robust H∞ optimal control algorithms applied in the output feedback control of high order linear dynamics with model uncertainties, and as such has its definite advantage when considering the known flexible modes into the control design. In Alter and Tsao (1996); Liu et al. (2005), H∞ optimal control has been successfully developed for the motion control of linear motors, while they are more focused on the rigid-body dynamics.
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In this paper, the dynamical model, including nonlinear rigid-body dynamics and high-frequency flexible modes, is developed for linear motor driven stages. The identification of our experimental stage has been carried out, and verified the effectiveness of the proposed model. The µ-synthesis based adaptive robust control strategy is developed, where the adaptive model compensation of nonlinear rigid-body dynamics is designed as the feedforward loop, and thus transfers the tracking problem into the traditional regulation problem. The µ-synthesis control is subsequently used in the feedback loop design, which can achieve the optimal control performance with the appearance of various model uncertainties. Since the identified high-frequency model is considered as a part of the nominal model, the higher close-loop bandwidth can be achieved. Comparative experiments have been done and the results verify the excellent performance of the proposed algorithm. 2. DYNAMICAL MODELS In the traditional modeling of linear motors, researches are usually focused on the rigid-body dynamics. As such, when the fast electrical dynamics and structural flexible modes are ignored, the dynamical model of a linear motor can be described by Xu and Yao (2001) M y¨ = u − B y˙ − Af Sf (y) ˙ + Fdis
(1)
where y, y˙ and y¨ represents the displacement, velocity and acceleration respectively. u is the control input, M is the inertia, B and Af are the viscous friction coefficient and the Coulomb friction coefficient. Sf (y) ˙ is a known smooth function used to approximate the discontinuous sign function sgn(y), ˙ such as Sf (y) ˙ = arctan(β y) ˙ with the value of β being large enough. Fdis represents the modeling errors and external disturbances. For mechanical and electronic systems, high-frequency dynamics are always existing and quite complicated due to various structural flexibilities of the specific plants. After the dynamical characteristics analysis of our linear motor driven stages, it is found that the stage rotation due to the flexibility of ball bearing is the fundamental flexible mode when compared to others such as electrical dynamics and mechanical resonance. The schematic diagram of the stage can be shown in Fig. 1. y represents the displacement of the stage at the left guideway, where the linear motor is driving and the position signal is measured. yc and y2 are the displacements of the stage at the mass center and the right guideway. α is the angle of the stage rotation. l = l1 + l2 , b = b1 + b2 are geometric sizes of the stage. w1 and w2 are the normal bearing pressure at the left and right guideway. Fm = u is the driving force, and B1 y˙ and B2 y˙ 2 are viscous frictions.
Fig. 1. Schematic diagram of the linear motor driven stage Since the high-frequency modes have to be described in the form of high order linear dynamics, the system model needs to be linearized and various nonlinearities and model uncertainties are ignored here. Thus, one obtains yc = y − l1 α y2 = y − lα (2) w1 = kf xα, x ∈ [−b2 , b1 ] w2 = kf xα, x ∈ [−b1 , b2 ] where kf is the equivalent stiffness due to the elastic deformations of the bearing ball and the rail. And the dynamical model of the stage can be obtained as following M y¨c = u − B1 y˙ − B2 y˙ 2 ∫b1 ∫b2 (3) Jα ¨ = ul1 − B1 yl ˙ 1 + B2 y˙ 2 l2 − w1 x dx − w2 x dx −b2
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Substituting (2) into (3) and assuming that B2 l2 ≈ B1 l1 , the transfer function from u to y can be achieved y(s) 1 (J + M l12 )s2 + B2 l2 s + K = (4) u(s) M s2 + Bs Js2 + B2 ll2 s + K where K = 23 kf (b31 + b32 ) and B = B1 + B2 . G1 (s) = 1 M s2 +Bs represents the linearized model of the rigid-body (J+M l2 )s2 +B l2 s+K
2 1 dynamics in (1), and G2 (s) = is the Js2 +B2 ll2 s+K fundamental flexible mode with the lightly damped poles and zeros. The resonant frequency ωr and anti-resonant frequency ωar of G2 (s) are √ √ K K ωar = , ω = (5) r J + M l12 J
The dynamical model (4) explicitly illustrates the fundamental flexible effect of the linear motor driven stage, and simultaneously holds the structural characteristics of the linear rigid-body dynamics. 3. SYSTEM IDENTIFICATION A commercial Anorad Gantry by Rockwell Automation has been set up Lu et al. (2008), and powered by Anorad LC-50-200 iron core linear motors. The linear encoders with a resolution of 0.5µm are used as position measurement signals. The experiments have been conducted on Y-axis, and a dSPACE CLP1103 controller board is used to implement real-time control algorithm.
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To verify the proposed high-frequency dynamical model and determine the valid range of the rigid-body dynamics, system identification in frequency domain has been carried out Chen et al. (2013b). The excitation signal is sum of 5000 sinusoidal signals ranging from 1Hz to 5KHz with a sampling frequency of 10KHz. The frequency responses of the system is achieved in Fig.2. By using the standard system identification toolbox in MATLAB and through iterations focusing on different frequency ranges, the following overall transfer function is obtained
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where θ = [M, B, Af , Fdis ]T is the unknown parameter set, and φT = [−¨ y , −y, ˙ −Sf (y), ˙ 1] is the measurement regressor. The feedforward controller can be easily designed as uf f = −φTd θˆ (9)
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Before the feedback control design, feedforward model compensation is usually used to achieve better tracking performance. Compared to the high-frequency flexible modes, the nonlinear rigid-body dynamics with the accurate parameter estimates can be much more important to obtain good compensation effect. Thus, the feedforward controller design in this subsection is dependent on the nonlinear rigid-body model, and (1) can be rewritten in the linear regression form when considering Fdis as a lumped constant. u = −φT θ (8)
Blue: experimental data Red: curve fitting of P(s)
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least square identification method is used Chen et al. (2013a) and the resulting identified parameters are M = 0.61, B = 0.23, Af = 0.15. It is seen that the estimated value of M is correlates well with the frequency domain identification results. But B may not match well because of the lack of excitation signal in low frequency range.
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where 1 0.56s2 + 12.3s 2145.6(s2 + 32s + 8.15 × 104 ) G2 (s) = 2 (7) (s + 40s + 2.186 × 105 )(s + 800) 7.825 × 103 (s + 2000)(s2 + 180s + 1.474 × 106 ) G3 (s) = (s + 3000)(s + 4000)(s2 + 120s + 1.774 × 106 ) (s2 + 144.5s + 2.324 × 106 )(s2 + 103s + 4.9 × 106 ) × 2 (s + 112.3s + 2.436 × 106 )(s2 + 84s + 5.066 × 106 ) G1 (s) =
The structure of G1 (s) and G2 (s) matches well the linear rigid-body model and the fundamental flexible mode in (4), with the identified parameters M = 0.56, B = 12.3 ωar = 285.5 and ωr = 476.5. G3 (s) represents other uncertain high-frequency dynamics higher than 1000rad/s. As seen from Fig.2, the obtained P (s) fits the experimentally obtained frequency response well up to 2000rad/s. For the traditional feedback control, the achievable close-loop bandwidth is limited due to presence of the flexible modes G2 (s) and G3 (s). Within the frequency range of 200rad/s, only the rigid-body dynamics need to be considered and all flexible modes could be neglected, which is assumed in the previous researches. Thus, for those controllers to function well in reality, it is necessary to limit the targeted closed-loop bandwidth below 200rad/s. With the above discussion of the valid range of the rigid-body dynamics, the time domain identification of (1) is subsequently carried out to clarify the nonlinear dynamic characteristics at low frequencies. The standard
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. Then, the next key point is how to achieve the accurate parameter estimate 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., ∆
θ ∈ Ωθ = { θ : θmin ≤ θ ≤ θmax } (10) where θmin = [θ1min , · · ·, θ4min ]T , θmax = [θ1max , · · ·, θ4max ]T . A projection type least square estimation algorithm Chen et al. (2013a) is developed to achieve the accurate and bounded online parameter estimate. Specifically, θˆ is updated by: ˙ ˆ ∈ Ωθ θˆ = P rojθˆ (Γτ ) , θ(0) (11) where Γ is a positive definite diagonal matrix, and τ is an adaptation function to be determined later. The projection mapping P rojθˆ(•) is defined by ◦ •, if θˆ ∈Ωθ or nTθˆ • ≤ 0 nθˆnTθˆ P rojθˆ(•) = (12) (I − Γ )•, θˆ ∈ ∂Ωθ and nTθˆ • > 0 T nθˆ Γnθˆ ◦
in which • ∈ R4 , Ωθ and ∂Ωθ denote the interior and the boundary of Ωθ respectively, and nθˆ represents the outward unit normal vector at θˆ ∈ ∂Ωθ .
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To reduce the effect of measurement noises and avoid the need of acceleration feedback, a stable filter Hf (s) is used to the both side of (1). uf = −φTf θ
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Γ and τ can be defined by the standard least square method 1 αΓ − Γφf φTf Γ, if λmax (Γ) ≤ ρM 1 + νφTf Γφf Γ˙ = (16) 0, otherwise 1 τ= φf ϵ (17) 1 + νφTf Γφf where α is the forgetting factor, ν ≥ 0 with ν = 0 leading to the unnormalized algorithm, ρM is the pre-set upper bound for ∥Γ(t)∥ to avoid the estimator windup. Lemma 1. When using the projection type least square estimation algorithm (11), the parameter estimate θˆ are ˆ ¯ θ , i.e., θ(t) always within the known bounded set Ω ∈ ¯ Ωθ , ∀t. In addition, if the following persistent excitation (PE) condition is satisfied, t+T ∫
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then θˆ converges to its true value θ. 4.2 µ-synthesis Feedback Design µ-synthesis control is a kind of robust H∞ control algorithms to achieve the optimal control performance and simultaneously guarantee the robust stability at the appearance of model uncertainties. With these definite advantages in the output feedback control of high order linear dynamics, it is developed as the feedback controller of our system which consider the identified high-frequency model as a part of the nominal model. Due to the adaptive feedforward design in the above subsection, the position tracking of desired trajectory can be transferred into the traditional regulation problem which µ-synthesis control can more easily deal with. The block diagram of µ-synthesis design setup is shown in Fig.3. G(s) = G1 (s)G2 (s) represents the nominal model with the high-frequency mode identified in (7). e = y − yd is the tracking error, uf b = u − uf f is the feedback control input. ∆ represent the uncertain linear dynamics with the bounded magnitude of 1, and wunc is its weight to describe the uncertainty magnitude at difference frequencies. n is the measurement noise. d represents both external disturbances and the calculated error of the dynamic inversion uf f due to model uncertainties,
Fig. 3. The block diagram of µ-synthesis design setup unknown parameters and the flexible mode. K(s) is the feedback controller needs to be designed. wd , wn , wu , we are four weights for the system inputs d, n and output u, e respectively. The main issue of µ-synthesis control is to choose proper weights such that the control performance is optimized and the robust stability can simultaneously guaranteed for these model uncertainties. Thus, the weights in our system are chosen as follows: wunc = 2(G3 (s) − 1) 100 wd = 1 20 s + 1 0.5 × 10−6 (s + 100) wn = (19) s + 500 1 s+1 wu = 200 1 20000 s + 1 we = 10000 G3 (s) in (7) is the high-frequency dynamics higher than 1000rad/s where digital controllers can not implemented, so wunc has to consider it as the main model uncertainty and be extended twice due to various other uncertainties. wd has large weighting at low frequencies and decreases after 20rad/s to present the disturbance effect mainly locates at low frequencies. Oppositely the measurement noise usually locates at the high frequency range, as such wn is chosen to be very small at low frequencies and come to the level of encoder resolution when higher than 500rad/s. With the disturbance and noise rejection objective to be less than 0.0001, we is set to be 10000. The time constant of wu is chosen to be 1/200, which means the focus of control design is the tracking error at low frequencies but gradually transferred to the control input limitation when larger than 200rad/s. By using the MATLAB robust control toolbox, the feedback controller K(s) can be achieved 9.75 × 107 (s + 2 × 104 )(s + 4362) (s + 4422)(s + 3012)(s + 0.5574) (s + 800)(s + 3000)(s2 + 122.9s + 7753) (20) × 2 (s + 120.4s + 82270)(s2 + 3987s + 5.783 × 106 ) (s2 + 38.2s + 2.137 × 105 ) × 2 (s + 2199s + 4.683 × 106 ) K(s) =
The robust stability margin of the resulting controller is 1.13 > 1, which guarantee the stability for all values
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of its modeled uncertainties. Theoretically, setting larger turnover frequency of wd and wu will further improve the control performance due to higher achievable close-loop bandwidth. However it will also excite the high frequency uncertainties, and finally make the system unstable. So it is a trade-off to achieve the balance between control performance and robust stability.
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As in Xu and Yao (2001); Lu et al. (2008), the point-topoint motion with a travel movement of 0.4m is used as the desired trajectory, which is typical in manufacture industry. The low speed experiment (the maximum velocity of 0.4m/s) and the high speed experiment (the maximum velocity of 1m/s) are carried out respectively. The experimental results are shown in Fig.4-7. Table 1 and Table 2 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 seconds, the steady-state tracking error, the L2 norm of tracking error and the average control effort. It is seen that the transient tracking performance of C1 is obviously better than C2 in the both low speed and high speed experiments. Due to the lack of integral term in µ-synthesis control, the steady-state tracking error of C1 is with a 2µm oscillation. The online parameter estimations are shown in Fig.5 and Fig.7, where the inertial θ1 converges to its true value, and the friction coefficients θ2 and θ3 may vary a little at different speeds.
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C1: The proposed µ-synthesis based adaptive robust controller (20) and (9). The lower and upper bounds of the parameter variations for θ are chosen as θmin = [0.5, 0.08, 0.05, −1]T and θmax = [0.7, 0.45, 0.35, 1]T , respectively. The least square type estimation algorithm (16)and (17) is implemented with α = 0.02, µ = 0.1, ρM = 1000, and the initial adaptation rates are Γ(0) = diag{1, 10, 100, 100}. The initial parameter estimates ˆ are θ(0) = [0.5, 0.2, 0.1, 0]T . The filter function Hf (s) is set as τf = 0.004 and a relative degree equal to 2. C2: The PID feedback controller
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Fig. 7. Estimation of important parameters at the high speed experiment 6. CONCLUSION The integrated modeling of the linear motor driven stage including nonlinear rigid-body dynamics and high frequency flexible modes is carried out, and verified by the specific system identification in time domain and frequency domain. The µ-synthesis based adaptive robust control strategy is developed, where the adaptive feedforward model compensation of nonlinear rigid-body dynamics is designed to transfer the tracking problem into the traditional regulation problem. The µ-synthesis technology is subsequently used as the feedback loop design to achieve the optimal control performance with the appearance of model uncertainties. By considering the fundamental flexible mode as a part of the nominal model, the µ-synthesis controller can obtain higher close-loop bandwidth. Comparative experiments have been done, and the results shows the excellent performance of the proposed algorithm. REFERENCES Alter, D.M. and Tsao, T.C. (1996). Control of linear motors for machine tool feed drives: design and implementation of h∞ optimal feedback control. ASME J. of Dynamic systems, Measurement, and Control, 118, 649–656. Cao, R. and Low, K. (2009). A repetitive model predictive control approach for precision tracking of a linear motion system. IEEE Transactions on Industrial Electronics, 56(6), 1955–1962. Chen, S.L., Tan, K., Huang, S., and Teo, C. (2010). Modeling and compensation of ripples and friction in permanent-magnet linear motor using a hysteretic relay. IEEE/ASME Trans. Mechatronics, 15(4), 586–594. Chen, Z., Yao, B., and Wang, Q. (2012). Adaptive robust precision motion control of linear motors with high frequency flexible modes. Proceedings of The 12th IEEE International Workshop on Advanced Motion Control. Sarajevo. Chen, Z., Yao, B., and Wang, Q. (2013a). Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect. IEEE/ASME Transactions on Mechatronics, 18(3), 1122–1129.
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