SCIENCE CHINA Technological Sciences • RESEARCH PAPER •

November 2011 Vol.54 No.11: 3054–3063 doi: 10.1007/s11431-011-4561-3

Double-loop robust tracking control for micro machine tools FAN ShiXun1*, NAGAMUNE Ryozo2 & FAN DaPeng1 1

College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha 410073, China; 2 Department of Mechanical Engineering, the University of British Columbia, Vancouver, BC V6T1Z4, Canada Received February 24, 2011; accepted July 26, 2011; published online September 19, 2011

This paper addresses double-loop robust tracking controller design of the miniaturized linear motor drive precision stage with mass and damping ratio uncertainties. As an inner-loop, a disturbance observer (DOB) is employed to suppress exogenous low frequency disturbances such as friction and cutting force. To further eliminate the residual disturbance and to guarantee the robust tracking to the reference input, a -synthesis outer-loop controller is designed. For eliminating the steady state error, a technique is proposed to design the -synthesis outer-loop controller with an integrator. A guideline to select the bandwidth of the Q-filter in the DOB is provided. Simulations using a model of a prototype micro-milling machine indicate that the proposed outer-loop synthesis scheme is superior to the H suboptimal control in disturbance rejection performance and steady state tracking performance. Furthermore, it is shown experimentally that the proposed double-loop robust tracking controller improves the tracking performance of the stage by 29.6% over PID control with a DOB inner-loop. micro machine tools, robust tracking, -synthesis, disturbance observer Citation:

1

Fan S X, Nagamune R, Fan D P. Double-loop robust tracking control for micro machine tools. Sci China Tech Sci, 2011, 54: 30543063, doi: 10.1007/s11431-011-4561-3

Introduction

High-accuracy miniaturized components are increasingly in demand for various industries, such as aerospace, biomedical, electronics, environmental, communications, and automotive industries [1–5]. Micro-component fabrication requires reliable, repeatable and cost-effective methods. Micro-mechanical machining is an important fabrication method for creating miniature devices and components with features that range from tens of micro meters to a few millimeters in size. Comparing with miniature component manufacturing methods based on semi-conductor processing techniques, micro-mechanical machining has many advantages in that it is capable of fabricating of fabricating 3D free-form surfaces, and of processing a variety of metallic

*Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2011

alloys, composites, polymers and ceramic materials. Attention has been paid increasingly to micro-mechanical machining in recent years. Similar to conventional machining operations, micromechanical machining shapes the surface of materials using miniaturized cutting tools. Miniaturized linear motor drive precision stage (i.e. XY table), with a high performance controller and a high speed spindle, plays an important role in positioning the materials and/or cutting tools accurately. Precision stages for micro-mechanical machining usually work in a very low speed which is below 30 mm s1. Robust performances of tracking accuracy and motion smoothness in the whole workspace with the existence of model uncertainties and friction are vital to machining requirements as accuracy, surface roughness and dimensional repeatability. It is well known that the tracking behavior of a direct drive design is prone to uncertainties such as disturbances and model parameter variations. Therefore, much attention must tech.scichina.com

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be paid to the robustness in the controller design. Last decades have witnessed a sustained interest in the techniques for robust tracking control of the linear motor machine tool drives. Alter and Tsao [6, 7] employed the H controller to maximize the servo-dynamic stiffness, as well as to minimize the chatter, where they focused on nominal performance, not on robustness. Braembussche, Swevers and Brussel [8] performed a comparison study between the H controller and the discrete-time sliding mode controller for a prototype machine tool axis, and they proposed an alternative performance weighting function to improve the robust performance. They used unstructured dynamic uncertainty (multiplicative uncertainties and additive uncertainties) but not parametric uncertainty to represent the uncertainty of the system, and applied the conventional H loop shaping synthesis method to design optimal robust controllers. Yen [9] proposed a two-loop robust controller for compensation of the variant friction force in an over-constrained parallel kinematic machine. A disturbance observer (DOB) was adopted as the inner-loop compensator to reduce the effects of the variant friction force. An outer-loop 2-DOF H loop shaping controller was utilized to provide the overall system with sufficient robustness in terms of stability and tracking performance. He also used unstructured multiplicative uncertainty model to represent the system uncertainties. In robust control, the performance of the system needs to be improved as much as possible, while the controller has to be robust against model uncertainties and variations in dynamics. However, it is well known that robustness and performance are incompatible. Therefore, the only way to improve the robust performance is to reduce the size of uncertainties [10, 11]. For micro-mechanical machining, feed rate of each axis of the miniaturized precision stage is commonly slow and the frequency components of the reference command are usually distributed in low frequency region with a range below 10 Hz. High frequency dynamic uncertainties, which are mainly distributed above 150 Hz, have little influence on low frequency tracking performance. The most important factors that degrade the nominal tracking performance are the uncertain load mass and viscous damping ratio. In conventional researches [6–9], the authors used an unstructured multiplicative dynamic uncertainty to cover all the uncertainties involving uncertain mass, uncertain damping ratio and unmodeled high frequency dynamics, and guaranteed robustness by shaping the complementary sensitivity function. Due to the conservative nature of unstructured uncertainties, a significant amount of performance will be sacrificed in the resulting closed loop system. Instead of the unstructured uncertainties, real parametric uncertainties can be used to precisely represent the uncertain load mass and viscous damping ratio. The uncertain model with parametric uncertainties can be of much smaller size than that with unstructured uncertainties. Therefore, with parametric uncertainty representations, we can expect

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the improvement of robust tracking performance. For the system perturbed by structured parametric uncertainties, the robust controller design problem can be solved by using the analysis and synthesis techniques of the structured singular value  [11–13]. In this paper, to design a tracking controller for the miniaturized linear motor drive precision stages for micro-mechanical machining, a 2-DOF -synthesis framework is proposed. The parametric uncertain model of the linear motor drive is obtained. A double-loop tracking control strategy is introduced. As inner-loop control, a DOB loop is employed to suppress exogenous low frequency disturbances such as friction and cutting forces. On the other hand, for the outer-loop control, a  controller is designed to further eliminate the residual disturbance and guarantee the robust tracking to the reference input. For eliminating the steady state error, a technique is proposed to design the -synthesis outer-loop controller with an integrator. The key of the DOB loop design is to select a proper low-pass filter. Since the DOB loop is just a part of the overall closed loop system, the robust stability of the whole closed loop system should be considered during DOB loop design. A low-pass filter selection guideline which considers the robust stabilities of both the DOB inner-loop and the whole closed loop system is introduced in the paper. The comparative simulation between the introduced integral -synthesis outer-loop design scheme and the H mixed-sensitivity control based on unstructured uncertain model [8, 11–15] was conducted. When the same performance weighting functions were chosen, -synthesis scheme based on the parametric uncertain model to design outer-loop controller could give much better disturbance rejection performance than H mixed-sensitivity control based on the unstructured uncertain model. Additionally, the simulation also shows that the H mixed-sensitivity controller without the integrator can lead to a larger steady state error of the step response. The comparative experiments between the proposed double-loop robust tracking controller, the PID controller and the PID with DOB inner-loop controller were conducted on a prototype micro-milling machine. This paper is organized as follows. Section 2 introduces the parametric uncertainty modeling of the miniaturized linear motor drive precision stages. In Section 3, the double-loop robust tracking control scheme is presented. The design procedure of the -synthesis controller with an integrator is discussed, and the DOB low-pass filter selection criterion is given. In Section 4, simulations and experiments on a prototype micro-milling machine show the effectiveness of the proposed method.

2 Parametric uncertainty modeling The servo amplifiers of the linear motor machine tool drives usually work under current mode. There is an inherent

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closed current loop in the amplifier. Since the bandwidth of this current loop is usually above 1 kHz, it can be approximated as a constant gain. The block diagram of the linear dynamic model of the direct drive is illustrated in Figure 1. In the figure, the signal Fd denotes disturbance forces including nonlinear friction and cutting force, and Fm represents motor force. The corresponding state space model can be written as  0  0 1   x1    x1     b     K a K t  (u  d ),  x     2  0     x2   m x  m    AG

BG

(1)

x  y  1 0  1  .   x2  C

Here, the state vector consists of the stage position x1 and velocity x2, u denotes control input, which is the voltage input of the servo amplifier, d  Fd /( K a K t ) denotes the equivalent disturbance that has the same unit as u, i is the motor current provided by the servo amplifier. The scalars Ka and Kt are the amplifier gain and the motor force constant, respectively. They are treated as constant values, and their uncertainties are omitted. AG and BG are uncertain matrices with parametric uncertainties, due to bounded uncertain parameters m and b, while C is the constant output matrix. m is a parameter which depends on the types of manufacturing, spindles, clipping devices, and workpieces. The

Figure 1 The block diagram of direct drive.

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feed drives used in micro-milling machine centers usually have small size. For example, the mass can be around 2.5 kg, but the spindles, clipping devices and workpieces usually have relatively large masses, whose typical values are from 1 to 2 times of the feed drive. b is an uncertain parameter which is only known approximately by identification and may vary during operations depending on the work region, mass of the load and lubrication condition, but its varying range is usually small. Therefore, the parametric uncertainties for general purpose micro-machining machine tool feed drive systems can be represented, for example, as m  m0 (1  0.6 m ),

 m  1,

b  b0 (1  0.15 b ),

 b  1,

(2)

where m0 and b0 are the nominal values of the parameters, which can be calculated as m0 

mmax  mmin b b , b0  max min , 2 2

(3)

where mmax, mmin, bmax and bmin are identified by using unbiased linear square technique [16]. m and b are normalized uncertain parameters. The existence of Fd can deteriorate the motion smoothness, leading to undesirable large tracking errors. The existence of the nonlinear friction force, especially Coulomb frictions, can change the linear dynamics in the low frequency region significantly. As can be shown in Figure 2(b), if the nonlinear friction is treated as dynamic uncertainty, the performance of the controller will be too conservative to be used. In previous researches, the DOB control was introduced to deal with these kinds of disturbances. Since the high frequency uncertainties which are distributed above 150 Hz do not affect low frequency tracking performance too much, they can be omitted in the model. Figure 2 shows the comparison between the frequency response functions for the identified parametric uncertain model set (Figure 2(a)) and those for experimental data with various masses (Figure 2(b)). The robust tracking problem for this parametric uncertain

Figure 2 Low frequency dynamic uncertainty. (a) Parametric uncertain model; (b) experimental bode plots.

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plant with the existence of nonlinear friction can be described as 1. Design a 2-DOF robust tracking controller for the uncertainty plant (1) of micro-machining machine tool drive such that the tradeoffs between tracking performance, disturbance rejection performance and control effort can be optimized and guaranteed against the parametric uncertainties of load mass and damping ratio in eq. (2). 2. Design a DOB working as an inner-loop to enhance the rejection performance of the low frequency disturbance forces like Coulomb friction, and guaranteeing the robust stabilities of both the DOB loop and the closed loop system.

3 Double-loop robust tracking controller design 3.1 Double-loop robust tracking strategy for micro machine tool drives To solve the formulated problem in the previous section, we will combine the DOB and -synthesis technique. The DOB is used to compensate for low frequency nonlinear dynamic uncertainties to enhance low speed motion smoothness. As for high frequency dynamic uncertainties, since the bandwidth of the current loop is sufficiently high, they can be directly omitted. Hence, model uncertainties are only structured parametric uncertainties of m and b. -synthesis is utilized to deal with these structured uncertainties. The control structure is shown in Figure 3. The controller design process has two steps. 1) Design a robust outer-loop feedback controller with -synthesis for the parametric uncertain plant to guarantee the robust tracking performance. 2) Design a DOB inner-loop compensator to compensate for low frequency components of nonlinear friction and

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cutting forces, and guarantee the robust stabilities of both the DOB loop and the closed loop system. Each step is explained in detail in the next two subsections. 3.2 Robust outer-loop controller design with -synthesis Figure 4(a) shows the closed loop block diagram used for robust outer-loop tracking controller synthesis. The inner structure of the robust outer-loop tracking controller K, which has been shown in Figure 3, is also illustrated. The plant G is the parametric uncertain model represented by eq. (1). The control purpose is to make the stage position y precisely track the reference signal r in the presence of disturbance d by using a bounded control effort u. This is equivalent to designing a controller to make the tracking error e insensitive to both the reference input r and the disturbance d in an interested frequency region with limited control effort u. For this purpose, weighting functions WP and WU are added in the block diagram. WP is used to shape the two sensitivity functions from r to e and d to e together, while WU is used to limit the control effort. The selection of WP and WU can tradeoff the closed loop bandwidth and the control effort. The weighting function WP is tuned to reach the desired tracking and disturbance rejection performance, while WU is chosen based on the physical plant specifications. Traditional machine tool servo control applications have verified that an integrator should be included in the control loop to eliminate the steady state error. However, the direct application of the -synthesis technique will not result in a controller with an integrator. For this reason, an integrator is added to the input path of the controller. The structure of the robust tracking controller K is illustrated in Figure 4(a),

Figure 3 Double-loop robust tracking control strategy block diagram.

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where the one-input-one-output controller K is substituted for an integrator plus a two-inputs-one-output controller K . The existence of the integrator can guarantee steady state error to converge to zero, while the second input of the K is to make the control structure more general. To design a robust tracking controller K, we first design K by the -synthesis technique, and then combine it with the integrator. Prior to the design of K , the configuration shown in Figure 4(b) must be derived from Figure 4(a). To this end, the uncertain plant G is combined with the integrator and the weighting functions WU and WP to generate the model P in Figure 4(b). The mathematical expression of the configuration shown in Figure 4(b) is given by   x       xI     xP        A   xU        C1   eP    C  eU    2     e    e    

B1 D11 D21

 x      xI     x  B2  P   x  D12    U   ,   0    r     d       u  

(4)

where subscripts I, P and U stand for the dynamics of the integrator and the weighting functions WP and WU in the

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state space form, respectively, and  AG  B C A :  I G   BP CG   0

B  :  B1

0 0

0 0

AP 0

D D :  11  D21

0   0  , 0   AU 

0  B B2    I  BP   0

  DP CG   C1   0 C :    C 2    DI CG   CG  DP D12   0  0   DI  1

BG   0  , 0   BU 

BG 0 0 0 0

CP

0 CI 0

0 0 0

0  C U  , 0   0 

0 0  0 DU  . 0 0   0 0 

The role of the weighting function WP is to shape the closed loop sensitivity function to meet the following criteria: 1) The gain in low frequencies is small to lead to small tracking error and a good disturbance rejection capability; 2) The cross-over frequency is maximized to result in a fast response and good tracking performance; 3) The gain peak is relatively small to provide a large stability margin and less oscillatory time-domain response. The weighting function WU is chosen based on the physical plant specifications. This will also limit the cross-over frequency of WP. Increasing WU may lead to higher cross-over frequency, but with more control effort. A tradeoff between the tracking performance and the control input magnitude has to be taken by tuning these two weighting functions. For our experimental setup, WU is selected as 0.001, which means that the largest controller gain will be less than 60 dB. Corresponding to this constraint, WP is tuned by trial and error to arrive at the best achievable performance. WP used here is WP 

Figure 4 Block diagram. (a) The configuration to design  controller with an integrator; (b) general framework for multiobjective optimization problem.

0 AI

0.09s 2  87.15s  2581 . s 2  3.69s  2.581

(5)

Mixed sensitivity H optimization based on the unstructured model uncertainties usually gives a conservative controller design. However, for the parametric uncertain plant, -synthesis can be used to solve K with satisfactory tracking performance and disturbance rejection performance for machine tool drives. It can be implemented by using the

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DK iteration dksyn of the Robust Control Toolbox in MATLAB. Usually, dksyn gives us K with a high order, thus, model reduction needs to be done. MATLAB function reduce is used here to reduce the order of the controller K . For the example system, a seven order controller K is obtained after model reduction. The sensitivity functions of the design result are shown in Figure 5. By solving the -synthesis problem, K is obtained, which can be written in the state space form as x K  AK x K   BK 1 u  C K x K   DK 1

 e  BK 2    , e   e  DK 2    . e 

By comparing Figures 4(a) and 3, it can be seen that the tracking controller K, which will be used in a real system, is the combination of K and an integrator. It can be written in the state space form as x I  AI xI  BI e, e  C I xI  DI e.

 xK   AK BK 1C I   xK   BK 1 DI  BK 2    x    0  e, AI   xI   BI I          x K

u

(7)

By combining eqs. (6) and (7), the tracking controller K is written as

xK

AK

BK

(8)

x    C K DK 1C I   K    DK 1 DI  DK 2  e.    xI   C D K

K

It is an eighth-order controller. 3.3

(6)

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DOB inner-loop design

The DOB provides a way of rejecting disturbances which cannot be modeled in advance. The structure of the DOB is shown in Figure 3. It can be designed on the nominal model. The key procedure is to select the filter Q(s) [17, 18]. It can be written as Q( s) 

2 s  1 .  s  3 2 s 2  3 s  1 3 3

(9)

It was shown by researchers that a third-order low-pass filter often provides good performance with an appropriate choice of . Various guidelines were suggested for the selection of the Q-filter, and most of them focused on the robust stability of the DOB loop itself [17–20]. However, since the DOB loop is just a part of the overall closed loop system, the robust stability of the closed loop system should

Figure 5 Sensitivity functions of the integral  outer-control loop.

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also be considered during DOB loop design. Next, a guideline of the choice of  will be presented, by considering the robust stabilities of both DOB loop and the whole closed loop system. Suppose that the model uncertainty can be treated as a multiplicative perturbation as G ( s )  Gnom ( s )(1  W ( s )  ( s )),   H  , 



 1,

(10)

where W(s) is a stable transfer matrix. Equation (10) can be rewritten as W ( s )  G ( s ) / Gnom ( s )  1.

(11)

Considering the closed loop structure shown in Figure 3, and assuming G=Gnom, the output y of the closed loop system can be written as y : x 

Gnom ( s ) K ( s )  Q( s ) . 1  Gnom ( s ) K ( s )

(12)

The sensitivity function S(s) and the complementary sensitivity function T(s) of the closed loop system are respectively defined as S ( s) 

By observing Figure 2, the divergences caused by the unmodeled disturbance in the low frequency region between the linear models and the experimental models are distributed below 10 Hz. Therefore, the 3 dB cut-off frequency of the Q(s) filter around 30 Hz is chosen. The corresponding  is 0.005. The controller K is designed by using the proposed outer-loop design method presented in Subsection 3.3. Considering eq. (13), the singular values plot of T, Q and the inverse of WT are drawn in Figure 6. It can be seen that both of the two inequalities are fulfilled. Therefore, the closed loop system is robustly stable.

4

Simulations and experiments

4.1 Simulation comparison between the outer-loop  controller with an integrator and H suboptimal controller

Gnom ( s ) K ( s ) G ( s )(1  Q( s )) r  nom d 1  Gnom ( s ) K ( s ) 1  Gnom ( s ) K ( s ) 

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An H mixed-sensitivity control scheme is shown in Figure 7. The model of the system can be described by unstructured multiplicative uncertainties as eq. (10).

G ( s ) K ( s )  Q( s) 1  Q( s) , T ( s )  nom . (13) 1  Gnom ( s ) K ( s ) 1  Gnom ( s ) K ( s )

Define a stable weighting function as follows: WT ( s )  W ( s ) H high pass ( s ).

(14)

Hhigh pass(s) is used for noise rejection. According to small gain theorem [11] the DOB loop and the whole closed loop are robustly stable if QWT



WTT



 1,

(15)

 1.

(16)

The above criterion is used as a guideline in selecting . The Q-filter bandwidth selection procedure is 1) Select weighting function WT. 2) Choose  according to the frequency characteristics of the disturbance and the requirements of the disturbance rejection performance. 3) Check robust stability of the closed loop system according to eqs. (15) and (16). If the inequalities are not fulfilled, decrease  or relax the specifications of the weighting function and check robust stability again. For our experimental setup, we have obtained the functions W and WT as W

0.645( s  0.1) 3.225( s  0.1)( s  1257) , WT  . ( s  1.62) ( s  1.62)( s  6283)

Figure 6 T (solid), Q (dashed) and WT (dash-dot).

(17) Figure 7 Standard H mixed-sensitivity control scheme.

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Define S  1/(1  Gnom K ), R  K /(1  Gnom K ), T  1  S .

(18)

For robust performance, the weighed mixed-sensitivity design approach [11, 12] can be used, where the control synthesis procedure aims at simultaneously optimizing nominal performance and robust stability by minimizing the following norm of stacked weighed transfer functions: WP S

(19)

WU R . WTT



WP and WT determine the shapes of sensitivity and complementary sensitivity. WU is used to limit the control effort. WP and WU are chosen as the same as in Section 3. WT is designed as eq. (20) here which is the same as W at low frequencies but rolls off outside the control bandwidth. The roll-off of WT is to ensure sufficient stability margin for the unstructured uncertainties. WT 

3.225( s  0.1)( s  1257) . ( s  1.62)( s  6283)

(20)

Theoretically, robust performance is only guaranteed when eq. (19) is smaller than 1/ 2 . Here, LMI method is used to solve the H weighed mixed-sensitivity minimization problem [14, 15]. A sixth-order controller K was ob-

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tained.  equals 0.7, which is smaller than 1/ 2 , hence, the robust stability can be guaranteed. The sensitivity functions of the perturbed closed loop systems are shown in Figure 8. By comparing Figures 5 with 8, one can see that the sensitivity function from r to e of the integral  outer-loop ( S r e ) has a very low frequency gain below 150 dB, that is, 80 dB smaller than the H controller. The sensitivity function from d to e ( S d e ) of the integral  outer-loop is decreasing with frequency decreasing and is below the weighting function, but S d e of the H control can not be directly restricted by the weighting function. For comparing the steady state tracking performance and the disturbance rejecting performance, simulations on the nominal system with Coulomb and Stribeck friction model, as well as a random input noise were conducted. Figure 9 shows the steady state tracking errors of step responses of the two outer-loop control systems. Due to the effect of the input noise, the tracking error of  controller is oscillating around zero. The H controller shows better noise rejection performance than the  controller. However, it indicates that the  controller with an integrator has zero steady state error while the H controller has a steady state error of about 0.7 m. Sinusoidal signal tracking errors are shown in Figure 10. Due to the affection of the friction force, spikes in the errors occur when the direction of motion reverses. The error spikes of the integral  controller have the

Figure 8 Sensitivity functions of the H outer-control loop.

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Figure 9 Simulation results of step response steady state errors.

Figure 11 Micro-milling machine tool drive experiment setup.

Figure 10 Simulation results of sinusoidal signal tracking errors.

amplitude around 30 m and can quickly converge to extremely small values within 250 ms, while the amplitudes of the spikes of the H controller are 3 times larger than those of integral  controller and converge much slower, which will also lead to a much larger root mean square (RMS) tracking errors. 4.2 Experimental comparison between the double-loop robust tracking controller and PID with DOB inner-loop Figure 11 shows the experimental setup, i.e. a prototype micro-milling machine. The precision stage is Aerotech XY linear drive ALS10020. The position of the work table is measured by a linear scale with a resolution of 1 micro meter. The total mass of moving part is around 2.5 kg. The amplifier is Aerotech BA10-40 operating in the current control mode. The control unit is the dSPACE digital controller 1103 equipped with 16-bit AD and DA and 24-bit incremental encoder inputs. The sampling period is selected to be 0.1 ms. In order to compare the tracking performances of the controllers, a trajectory for the machine table position is generated. Considering the requirement of the micro-machining process and the test bed limits, the following values are chosen in the trajectory generation: stroke of 100 mm, velocity of 30 mm s2, acceleration of 200 mm s2. The trajectory profiles are shown in Figure 12. For performance comparison, the PID controller was designed to have similar closed loop bandwidth around 50 Hz

Figure 12 Desired line trajectory.

for the nominal plant (with 6 kg mass load). If a higher bandwidth is selected, the closed loop system for the plant with 3 kg mass load will easily become unstable. The experimental tracking errors of PID, -synthesis with an integrator, PID with DOB and the proposed double-loop robust tracking control for the case with 3 kg mass load are shown in Figure 13. The unit of the error axis is m. The spikes in the errors occurring around 4.25 s are caused by nonlinear friction, when the direction of motion reverses. The peak of the spike of the proposed double-loop robust tracking controller is smaller than those of all the other three controllers and converges more quickly, which leads to a much smaller RMS tracking error. It can reduce the error spike by 60% compared with PID and by 39% compared with PID with DOB. A numerical comparison based on the RMS error is given in Table 1. An improvement of 63.4% over PID and an improvement of 29.6% over PID with DOB have been achieved when the proposed double-loop robust tracking control is used.

5

Conclusions

In this paper, the parametric uncertain model of the linear motor drive for the micro-machining precision stage is obtained. A double-loop robust tracking control based on DOB and -synthesis techniques for this parametric uncertain

Fan S X, et al.

Sci China Tech Sci

November (2011) Vol.54 No.11

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access to the micro-milling machine setup. This work was conducted while the first author was a visiting student at UBC, with the financial support from the China Scholarship Council. 1 2

3 4 5 6

Figure 13 Tracking errors.

7 Table 1 Controller’s tracking error results Load mass (kg) 3 6 7.5 9

PID (m) 3.9 3.5 4.2 5.2

PID with DOB (m) 0.9 1.2 2.2 2.7

 controller with DOB (m) 0.8 1.1 1.4 1.9

model are designed. A guideline to select the bandwidth of the Q-filter in the DOB is presented, by considering the robust stabilities of both DOB loop and the whole closed loop systems. When the same performance weighting functions are chosen, the proposed 2-DOF -synthesis controller with an integrator can obtain steady state tracking performance and disturbance rejection performance superior to the H-infinity mixed-sensitivity controller. The designed controller was implemented on a prototype micro-milling machine with open CNC based on the dSPACE digital controller. It was experimentally demonstrated that the tracking performance was improved significantly by using the proposed double-loop robust tracking control strategy. An improvement of 63.4% over PID and an improvement of 29.6% over PID with DOB have been achieved. The proposed method is generally applicable to typical micromachining precision stage drives. This work was supported by the Canada Foundation for Innovation (CFI) and the National Natural Science Foundation of China (Grant No. 50875257). The authors would like to thank Professor Y. Altintas of the Manufacturing Automation Laboratory, the Department of Mechanical Engineering, the University of British Columbia (UBC), for providing

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