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Robust digital tracking controller design for high-speed positioning systems
Control Eng. Practice, Vol. 4, No. 4 pp. 527-536, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/96 $15.00 + 0.00
Pergamon PII:S0967-0661(96)00036-6
ROBUST DIGITAL TRACKING CONTROLLER DESIGN FOR HIGH-SPEED POSITIONING SYSTEMS S. Endo*, H. Kobayashi*, C.J. Kempf*, S. Kobayashi*, M. Tomizuka** and Y. Hori*** *NSK Ltd., 78 Toriba machi, Maebashi, Gunma-Prel%Japan 371 **Department of Mechanical Engineering, Universityof California at Berkeley. Berkeley, CA 94720, USA ***Department t~fElectrical Engineering, Universityvf Tokyo. 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan 113
(Received July 1995; in final form January 1996)
Abstract: In this paper, a robust digital tracking controller for high-speed positioning is proposed. For high-speed tracking problems, the combination of a feedforward controller and a robust feedback controller is desirable, because the feedforward controller anticipates and compemates for closed-loop dynamics and the feedback controller compermates mexr~amieal nonlinem'ities, parameter variations, and disturbances. A disturbance observer and PD compensation was used as a robust feedback controller and a zero-phase error-tracking controller was used as the f~edforwardcontroller. This method was applied to an X-Y table, and experimental results demonstrate that this method yields excellent tracking performance and robustness to plant parameter variations.
Keywords:
Digital control, disturbance rejection, feedforward compensation, positioning systems, robust performance, tracking characteristics.
1. INTRODUCTION
respect to the chip and lead ~ame acxawately and rapidly. The X, Y and Z axes must be coordinated to achieve an optimum tracking profile to avoid bonding failures. This e~ample requires high-speed and high-precision positioning and tracking.
High-speed multi-axis coordinated motion control is of vital importance in the automation of modem mmufacturing and assembly processes. In particular, high-speed position tracking appears in many applications.
Many X-Y tables are driven by a rotary DC servomotor coupled to a ballscrew. The mechanical design of such systems presents a series of trade-o~. A ballscrew and beating structure that has high sti~ess also generally has a large amount of t'iction. This can hinder tracking per~rmance, due to the
A wire bonder, which must attach free wires to VLSI and ASIC chips, is a good example. A typical wire bonder has three axes, X, Y and Z. The X and Y axis servos must position the bonding tool with 527
528
S. Endo et al.
"quadrant glitches" that occur when an axis changes direction. Furthermore, a long-lead ballscrew is desirable for rapid motion, but this makes the system more sensitive to parameter variations such as changes in payload. To overcome these problems, high-gain feedback controllers are typically used. In most cases, the resonant modes of the system place a limit on the bandwidth that can be achieved. For high-accuracy tmc&ing, the input-output transfer ffmction between the desired output and the actual output must be as close to unity as possible over a wide frequency range. By processing the desired output by a feedforword controller which acts as an inverse of the closed-loop system, the effective bandwidth of the overall system can be improved compared to using feedback alone. Such a technique can be applied in situations where the desired trajectory is known in advance and the model of the closed-loop system is accurate. If the closed-loop system has unstable zeros or lightly damped zeros, a stable inverse system cannot be obtained. For this problem, Tomizuka (1987) has proposed the zero-phase error-tracking controller, ZPETC, which eliminates phase error introduced by uncancellable zeros. When the design of the feedforw~'d controller is based on the inverse of a closed-loop system, tracking perforrnance may be sensitive to modeling errors and parameter variations of the plant. Furthermore, the feedforward controller is not capable of rejecting disturbances. Therefore, the feedback controller should be robust, in terms of stability and performance, when parameter variations and disturbances axe present. A few attempts have been made to apply ZPETC to actual mecla~ical systems. Suzuki and Tomizuka (1991) described the implementation of ZPETC with model-based friction compensation on a machine-tool serve system. Tsao and Tomizuka (1987) developed an ~aptive zero-phase error-tracking controller. Tung and Tomizuka (1993a, b) presented a feedforw~l controller design based on the precise identification of the closed-loop system over a frequency range relevant to tracking. In the area of robust controller design, Ohnishi (1987) introduced a disturbance observer which estimates and cancels disturbance effects. This technique elfectively increases the disturbance rejection while leaving the input-output characteristics essentially unchanged. Umeno and Hori (1990) have proposed a two-degree-ogff~eedom serve structure, extending the idea of the disturbance observer. The disturbance observer is effective for parameter uncertainties as well as disturbances. The disturbance observer can make the actual plant
behave like the nominal plant over a range of fequendes. Therefore, it provides ideal feedback characteristics for the design offeedforwat-d controllers based on the idea of system inversion. This paper describes a high-specd tracking controller which consists of a disturbance observer as an inner loop, a PD controller as an outer loop, and a ZPETC as a fecdforward controller. This follows the methods previously presented for a single-axis table application (Endo, et al., 1993) and work on friction compensation and high speed tracking in the twoaxis case (Endo, et al., 1994). The remainder of this paper is organized as follows: In Section 2, the system is described in detail and a model is developed. Section 3 covers the design process and application for the case of the disturbance observer design based upon discretization of a continuous-time system. The results are presented and discussed. Section 4 is an extension to the methods of Section 3, in which the disturbance observer is designed directly in discrete-time. Again, experimental results are presented and comparisons between the two methods are made. Conclusions are given in Section 5.
2. SYSTEM MODEL AND PRACTICAL PROBLEMS IN HIGH-SPEED TRACKING 2.1
Mechanical Characteristics
Figure 1 shows a configuration of a feed drive system for one axis. Notice that the system has two large inertias, i.e. a motor inertia and a table inertia, and that they are connected by a ballscrew. The primary sources of elasticity in the system are the ballscrew, flex coupling, and bearing supports. Figure 2 is a simplified model of the feed drive system. The goal is to control 02, but only 01 is measurable. Consequently, it is important that the feed drive dynamics are not excited by the controller output, and the elasticity in the structure places a limit on the useful bandwidth of the system. The total spring constant of the feed drive system Kro r is calculated as follows (NSK Ltd, 1993): 1
Krcrr
1
+
1
+
1
(I)
Kcp KBR Kp
Here, K~p is the coupling stiffness, KaR is the axial stiffness of the support beating and K F is the axial stiffness of the ballscrcw. A block diagram of the system is shown in Figure 3. B1 arid B2 correspond to viscous damping, acting on inertias J~ and J2. The effects of static friction
Robust Digital TrackingController Motor
Guide--
Coupling \
Ballscrew
Beari~// '
Tabl7
(3)
P(s) = P,(s)[1 + A(s)].
The nominal plant, P,(s), is detined by considering a system with no elasticity
/ /
Support ---7
529
P.(s)= lim P(s)= Kr---~.**
(Jx + Jz) sz + (B1 + B9)s"
(4) Motor and Coupling Inertia Equivalent Table Inertia KF Ballscrew Spring Constant KBR Support Bearing Spring Constant KCp Coupling Spring Constant J1
The measured transEr Emotion for the system described by Equation (2) is shown in Figure 4. The corresponding nominal plant is also shown. From Equations 2 and 3, A(s) can be lbund to be of the form
J2
~Xz(s) = b°sz + b~s + bE
Fig. 1. Feed drive system.
(5)
s 2 + als + a 2
7"1
where the parameters al, a2, bo, bl, and b2 are •nctions of Jl, J2, B1, B2, and K r . Knowledge of this perturbation, which arises ~-om the resonant mode near 300Hz, will be used to assure robust stability of the disturbance observer loop. Section 3 will show that the resonance places an upper limit on the ~equencies over which the disturbance observer is effective.
T2
K ror
Fig. 2. Simplified model of feed drive system. P Ib
r~
:%--.
=,oo
Ol
1
.go
Jl s 2 +B 1 s +
('~10100.~
%
rL
, '
I
! i i;i!i;i i i iiiiii[ 100
101
10a
02
[ friction 10"I
i0 °
101
102
Frequency (rad/sec)
Fig. 3. Block diagram of the feed drive system. are not included in the model, since these will be regarded as e0tternal disturbances. From the block diagram, the system transfer function is written,
Fig. 4. System identification results.
_Ol e(s)-
7.1= Jzs 2 + B2s+ Kr
2
~ 0 4
Frequency (rad/sec)
2
3"
(JtJ2s + KTJ1+KrJ ~)s +(BtB2s+Kr Bt + KrB2)s+(JIB2 +J2BI )$
(2) For design of the disturbance observer, it is useful to consider the plant to be a nominal plant with a multiplicitive perturbation,
Fig. 5. Quadrant glitches.
103
104
530 2.2
S. Endo e t
al.
Tracking Error G~dy
In multi-dimensional tracking, tracking errors can be defined in several ways. For circular paths, the radial tracking error is defined as
Gr~ Gey
AR -- ~fX-2+ y2 - R
(6) Fd
where x and y are the x- and y-axis table positions and R is the desired radius. In circular path tracking, large tracking errors appear when either the x-axis velocity or y-axis velocity is near zero. These carom are caused by fiction forces and are called "quadrant glitches". Figure 5 shows q - ~ r m t glitches. The radial tracking error will be emphasized in the present experiment to evaluate the effectiveness of the disturbance observer in the presence of friction.
r
e
e'
u
[d
q Y
ZPETC
L disturbance observer Fig. 6. Proposed controller.
2.3 Robustness Against Parameter Variations The equivalent table inertia is calculated as
J2 = ( ~ ) 2 ( M + AM)
(7)
wherep is the lead screw pitch, M is the mass of the table with no load, and A M is the mass of the load. Equation (7) shows that system will be sensitive to variations of the mass on the table when a long lead pitch feed screw is used. This underscores the that the design of the controller should be based on robust performance criteria.
.
3.1
DESIGN OF A TRACKING CONTROLLER BASED ON A CONTINOUS TIME DISTURBANCE OBSERVER
Controller Structure
Figure 6 shows the block diagram of the complete control system. The controller consists of three parts: An inner-loop controller, which is a disturbance observer;, an outer loop, which is a PD controller;, and a ZPETC as the feedforward controller. The gmction of the inner loop is to linesrize the system and to achieve Ge,y = In. The inner loop is designed in the continuous time domain and discretized by bilinear transformation. The outerloop PD controller stabilizes the system; as in many positioning applications, the plant transtx gmction has a pole at s=O. The outer loop is designed in the discrete time domain, based on the zero-orderhold equivalent of the nominal plant as shown in Figure 7.
Fig. 7. Equivalent controller. The feedforward controller design is based on G'ry which consists of the PD controller and nominal plant shown in Figure 7 to attain G,dy in a relevant ~lUency. A precise identification of P(s) is not necessary for the design of the feedforward controller because the robust inner-loop design forces the inputoutput character to be very close to Pn(s). From the practical design viewpoint, it is useful to take advantage of the prior knowledge of the desired trajectory and make use of the f~fforward controller. In this manner, stability problems arising t'om a high-gain feedback appro~w.h can be avoided and the system can still deliver very high bandwidth.
3.2 Disturbance Observer Design The disturbance observer as the inner-loop controller is designed in the continuous time domain. The dismrb~mce observer is inside the dotted line shown in Figure 6. Referring to Figure 6, for Q~ 1,
V(s)=E'(s)-{-v(s)+ lpn(s) P(s)(V(s)+O(s))} (8)
Robust Digital Tracking Controller
which implies
P~(S)E'(s)_D(s).
U(s)= P(s)
(9)
531
Robust stability of the disturbance observer loop is assured if the following condition (Doyle, et al., 1992) is satisfied:
m .axlr(s)A(s)I < m .axlr(s~lm(s)I < 1. Substituting this relation into the plant equation, i.e. Y(s) = P(s)(V(s) + D(s)), gives
Y(s)=Pn(s)E'(s).
s=J¢O
(13)
$=JO~
From this the following restriction on Q is obtained:
(10)
[Q(s)[ < Equation (10) shows that the disturbance observer compensates for the dismrbanom mad parameter variation of the plant, thus making the inner-loop system behave like the nominal plant. However, this scheme, with Q=I, does not work in practice because 1/P~ is normally unrealizable. Therefore, a low-pass filter Q in the disturbance observer is used so that 1/P~ is realizable. In the selection of Q, consideration must be given to obtaining internal stability of the controller and to reducing sensor noise. Thus, the disturbance observer design is essentially a matter of selecting Q.
1
Vs = jog.
(14)
Notice that in Equation 11 Q(s)= 1 is desirable for disturbance rejection. On the other hand, Equation (14) shows that keeping the Q(s) is desirable for robust stability. Therefore, Q(s) should be selected taking account of this trade-off. For this problem, Umeno and Hofi (1990) suggest the following form for Q(s) : N-2
1 + ~.~ ak(sz) k Q(s)
Recall that the plant dynamics were parameterized as P = Pn(1 + A(s)). Then, Y(s) can be expressed as
=
(15)
~=t N
1 + Z ak (s~')~: k=l
Y(s) = P. (s)(1 + A(s)) E'(s) -I (1 - Q(s))P~(s)(1 + A(s)) D(s). 1+ A(s)Q(s) 1 + A(s)Q(s)
(II) Notice that Y(s) = P(s)E'(s) for Q(s) = 1 and Y(s) = P(s)(E'(s) + D(s)) for Q(s) = 0. Therefore, by proper selection of Q(s), the actual plant caia be made to behave like the nominal plant over a specified frequency region.
For this application, N = 3 ,
aI =3,
a 2 = 3 , and
a 3 = 1 were selected. Figure 8 shows the sensitivity £mction and complementary sensitivity limction of this choice of Q(s). It can be seen that this choice of Q(s) results in the same rate of roll-ot~ -40 dB per decade, for both S and T. This implies that performance robustness and robust stability are balanced.
For internal stability and realizability (Umeno and Hofi, 1990) it is necessary that
Q(s) ~ RHo~,
Q(s)/P~ ~ R H . .
(12)
~.~ -20 - i..~
o.¢0|
.i.L- i
! i~i
[ iSiiT;ii
The inner loop formed by the disturbance observer, Ge.y, has an open-loop gain of Q / O - Q ) , where it is assumed that P=P~. Thus, the sensitivity finetion, S, for this inner loop is given by S = 1-Q and the complea-aentary sensitivity limction, T, is simply Q. The use of S and T in the design of the disturbance observer is particularly helplial. First, they can be used directly in the formulation of a robust design, as will be shown below. Second, they offer physical insight into the design behavior of the system. S is a measure of how much a disturbance will contribute to the system output, and T is a measure of how much sensor noise will contribute to the output.
10'
:: iii::-::.
ii
i il !!~
..... + 10 2
!
!iii
i i il q
...... i 10 3
10 +
10 5
F r e q u e n c y (rad/sec)
Fig. 8. Frequency response of T(s) and S(s).
i i i iiii:i m ........i+i+ i i+.iLiil 0
f
i i i iiiiii i i i i~ ~: L L : Lii?~i .....L+i_LLiiii (
%,
10 2
Frequency
10 3
(rad/secl
Fig. 9. Frequency response of 1 / A(s) and Q(s).
10 4
S. Endo et al.
532
~" is tuned on the basis of Equation (14). Figure9 shows a Bode plot of 1/A and Q. In this case, ~ = 0.0015, which is equivalent to a cutoff frequency is 667 rad/sec. The continuous-time inner-loop compensation was discretized by applying a bilinear transformation prewarped at the Q filter cutoff frequency. The experimental plant is represented by a zero-orderhold equivalent. However, the nominal plant model has been discretized by applying the bilinear transformation. A discrepancy between the two is observed in the form of an extra zero in the bilinear transformation. To reduce this etfeet, the sampling ~equency is sdected to be much lkster than that of the cutoff ~equency of Q. In this application, the sampling period was set to 0.2 re_see.
3.3
PD Controller
The outer-loop feedback controller is designed in the discrete time domain. A zero-order-hold equivalent of P. is obtained for this purpose and the design is based on the feedback structure in Figure 7. Notice that p,, in Figure 7 has a pole at 1 because of a pure integrator. To stabilize the feedback loop in Figure 7, the Eedback controller is selected to be of PD type. The PD controller moves the open-loop pole of I to the inside of the unit circle, making the system well-damped. For the design of the feedforward controller, it is important to take account of the coefficient quantization errors in the feedforward controller and the dosed-loop controller.
Gff(z_l)= y(k) = zdA(z-I)B-(z)
where B-(z) is obtained by replacing z -t in B-(g -1) by z. The overall ~equency response ~ m the desired output to the actual output is shown in Figure 10. It can be seen that almost peri~ tracking is attained up to 667 rad/see. For additional details regarding ZPETC design, see (Tomizuka, 1987).
3.5
Experimental Results
Figure 11 shows a schematic diagram of the experimental system. The positioning system used was an NSK CT-0505 X-Y table designed for wire bonding applications. The position transducer is a 4000 epr (counts per revolution) optical encoder. This encoder, coupled with 10 mm pitch ball-screw, provides a linear resolution of 2.5pan in table displacement. A can'mat driver was designed to control the motor torque directly. The dynamics of the driver can be approximated as a first-order low-pass filter. The time constant of the driver was of the order of O.1 msec. This is very ~st in comparison to the mechanical dynamics, so the driver dynamics were ignored in modeling the system.
a
Zero-Phase Error-Tracking Controller
::.iiii!il
!:!:;iii:
!i!i!!':~:~:......~...!..+.i.ii.li
~-,o ..... i.ii.~.~!i ...... !-r.i-.ii!i!~ •.4
3.4
i !!:.ilrii
i il ili!i iiii '101
10 =
Frequency (rad/sec)
Fig. 10. Overall transfer function.
mot°r/enc°de~ 7
G(z-1 ) = y(k) = z-aB(z -i ) _ z-aB - (z-1)B +(z-I) r(k) A(z -1) A(z -1) (16) where d is the number of delay steps, B+(z -l) represents the caneellable zeros and B-(z -1) represents the uncancellable zeros. The ZPETC is given by
i
ii~ii i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 °
The feedforward controller design is based on G~ instead of Gry to reduce the order of the feedforward controller and to simplify the design. Then a nearperfect tracking performance is attained within the bandwidth of Q(s), that is up to 667 tad/see. ZPETC as a feed~rward controller is designed as follows. Consider the asymptotically stable closedloop plant
(17)
rd(k) B+(z-')[B-(1)] 2
host computer
l
Fig. 11. Experimental system.
10 3
I1~
Robust Digital Tracking Controller The controller was implemented on a TMS320C30 digital signal processor. The sampling period was 0.2 msec.
533
Trackln¢l Error
5.0
The performance of this controller was compared with that of a P-PI controller which is commonlyused in positioning applications. The P-PI controller parametees were tuned so as not to excite the mechanical resonance. All the tests involved a desired trajectory consisting of a l0 mm diameter circle which was to be Iraeed in 0.5 seconds. Figure 12 shows the radial tracking error of the poPI controller, and Fig. 13 shows the radial tracking error of the proposed controller. It can be seen that the magnitude of the tracking error is significantly reduced by the proposed controller, particularly the qu~rant glitches. Although the proposed controller cannot completely eliminate the quadrant glitches, they have been reduced to a level that is suitable ~r most practical applications.
E c
E0 o t
-5.0 ..... ....
....
•
Fig. 14. Tracking error with the P-PI controller.
Trackln¢l Error $.0
Trackln~l Error • ..... +O.~lwl
%0
E e-
0
i5 o
i5 -5.0
."..i?..-..................:ii. " .
-$.0
Fig. 15. Tracking error with the proposed controller.
Fig. 12. Tracking error with the P-PI controller. Tracklm:l Error • 5.0
.""
.' """
• ' ....... -~o.~lnml. ° ..... O.al.. ; ~"
"'. '"'
°..
,
". ° " "
.
° .
•
A load mass on the table was added, which increased and etFeetively doubled J2. Figures 14 and 15 show the experimental results of the perturbed case. It can be seen that the tracking error under the proposed controller is almost the same as the nominal case. On the other hand, the traddng errors increased for the P-PI case.
.
4
4.1
:.i i. Fig. 13. Tracking error with the proposed controller.
DISCRETE-TIME-DOMAIN DESIGN OF DISTURBANCE OBSERVER Discrete time disturbance observer design
Although the perfonnance of the proposed controller was very good, there are practical limitations. In particular, implementation of such a controller generally requires a high-performance CPU. This is a direct result of the fist sampling rates needed to minimize errors in the discretization of the inner-loop compensation. Both computational speed and
S. Endo et al.
534
wordiength become important at high sampling rates. In this section, a discrete time domain design of the disturbance observer is proposed to address this problem. When the disturbance observer is designed directly in discrete time, the advantages are: I) The sampling ~queney can be reduced with less degradation in performance, 2) The elects of coe~cient quantization errors are reduced because the filter to be implemented has poles and zeros that are no longer crowded near the z=l, and 3) Robust stability is improved since there is no longer a discrepancy arising ~om representing the plant in discrete time using a bilinear transformation rather than a zero-order hold. In the discrete time domain design, the nominal plant is defined in the form of the zero-order-hold equivalent model which yields
p(z-1)=
bl z-I +b2 z-2 l+a~z -1 +a2 z-2 •
Let the discrete-time Q tilter be denoted by Qd(Z -1 ) with a numerator polynomial NQ(Z -t) and denominator polynomial DQ(Z-~).
For the product
Q~(z-1)P;l(z -1) to be causal, it is necessary to have deg(NQ)
Qd(Z -1 ) ---- bql Z-I -,I..bq2z-2 -I- bq3Z-3 1 + aqlZ-I + aq2Z-2 + aq3Z-3 "
+,-= E e2<,o,
o - )I'+
N,,
NQ(eJ~°ff )12
I
k=0
(19) which corresponds to an equation error formulation. f(.) is the frequency weighting function. Q~(z-i) denotes the target tilter obtained by discretizing Q(s) with a bilinear translbrm. The titting technique employs a straightforward least-squares minimization at the set off~equencies w0,cot,~o2... CON/2. The frequency weighting ~hnetion was chosen to be a simple first-order low-pass filter with a cutoff • xtuency set to half of the cutoff ~equency of the Q filter. This sdection provided a reasonably good match at high ~equeneies, while forcing the low • equency gain to be very dose to 1. For this problem, the equation-error optimization method yielded a stable approximation to Q~(z-1) that was a good match; Figure 16 shows the fiequency response of both Qt(z -1) and Qa(z-i). When a digital filter is designed, poles and zeros close to the unit circle are undesirable. This is particularly troublesome when the compensator dynamics are dose to z= 1 due to quantization error problems. Although these effects can be reduced by choosing the lbrm of the filter realization caregtlly (see, for example, (Phillips and Nagle, 1984)), it is best to avoid this problem entirely. One technique is to reduce the sample rate in order to move some of the dynamics toward the interior of the unit circle. Figure 17 shows the poles/zero map of the outer dosed-loop system described in Figure 7. Here, sampling ~a.~quenciesare changed and P and D gains are tuned to keep the cutoff f~luency of the closedloop system constant in each sampling Eequency. Notice that the dominant poles move inside the unit circle as the sampling frequency is reduced.
(18)
The ~mdamental issue in the design process is the selection of eoet~dents aql , aq2 , aq3 , bq9 bq2 ,
and bq3. To accomplish this, a ~lUency weighted curve fitting technique was used. Although such techniques are well known and commonly used, the particular method employed in this case will be briefly reviewed.
0
! ! i i i i i l l .........i i G i i i !
3 -'° .........i i l}ii
)
101
10 ~
i i iiiiiii
~ i : iiil}
:
:
......... ?
-2001
10 ~
Q,~ i i+i!i/
I0 z
t- I : .... :
~'-100
iiii+i
i i ii i i ~ i i
G)
The fit is accomplished by minimizing a penalty function of the foUowing form
..........i............i i i+
i
i
~ :!iii:
i
! !
+ ' + " i ' ! i i i ' ! + i .......... ~'" ' ~ T ? ~
i
i !!i!ii
:
i
':!~
i ii+iiTiI++ii' ~ i i!i::::::
~
~ +i.
! : ili:i
10 2 Frequency
10 4
i
} i i i il
:
;
10 3 (rad/$ec)
Fig. 16. Frequency response ff Qt and Q~
:::iii.
10 (
Robust Digital Tracking Controller
535
4.2 Experimental Results Figure 18 shows the experimental results of the proposed controller combined with a discrete-time domain design of the disturbance observer. In this case, the same design was used as in the previous section, except that the sampling time was increased to 1 msec.
Trackln¢l Error
......... ~ r 4 = = ! •
~
5.0
°l°°°
I,~~ °
o-
JN-.°~'"'.°,
i°
i
s, . .
" 'o,
.,'" °"
"x "LL,°
g : -~
Figure 19 shows the experimental result with the proposed controller combined with a continuous-time domain design of the disturbance observer. Again, the sampling rate was increased to 1 reset. In this case, oscillations on the order of 20/an are dearly visible.
o.''"
~
.
.
~
'
~
.
~
:
•
|t~o
1¢
-5.O •.°........ : ....... '...
These oscillations are caused by the reduction of stability margin as the sampling is slowed and discrepancies between the actual plant and discretized nominal plant are no longer negligible.
Fig. 19. Tracking error of the continuous-time design. Trscklnq Error
o, ...... J('o~o2lm=l • " " °'
°1 • " "" ° °
el...
e" " ° " " • • •
5.0 •" . ,
,~ .~_~.',"*'~;..."
". x ' - "
".
0.8! 0.6
E
0.4 E
-~ 0.2 <
g
3
0
E -- `0°2
`0.4 "
`0.6
/
"
-5.0
-0.8 -1 Real
Fig. 20. Tracking error of the discrete-time design.
Axis
Comparing Figures 18 and 19, it can be seen that the direct discrete-time design technique sutta's far less performance degr~_~ i o n when a slower sampling rate is used.
Fig. 17. Pole and zero map. Tracking
5.0
E
•• °" :="°" ~' r "
.' ,
~
Error
~
'*°', .. ", ,., •
As a test of robustness, the discrete-time design was tested with the inertia of -/2 doubled. Results for this case are shown in Figure 20. From this, it can be seen that the results are essentially the same as for Figure 18, an indication that the design is robust to parameter variations.
•
g
c@ E @ ao D
i5 5
'"--/:-...!.!!!!.!!iii-i;i.!!.-::¢. "'" ""....... '........"i""
'
"
Fig. 18. Tracking error with the discrete-time design.
CONCLUSIONS
In high-speed tracking applications, mechanical vibration, fiiction and parameter variations are serious problems. This paper has demonstrated that a controller which combines the disturbance observer as an inner loop, a PD controller as an outer loop, and ZPETC as a Eedforward controller is a highly
S. Endo et al.
536
effective method of addressing these problems. Disturbance observer designs in both the continuous and discrete-time domains were considered. Although the bandwidth of the dosedqoop system is limited be,:a__useof the mechanical resonance in the structure, the use of feedforward in combination with a robust inner-loop design yields performance fit superior to that of a standard P-PI controller. The proposed controller can reduce the tracking error to about 5 #m in tracking a circular path at high speed, roughly one fourth of the errors that occur with a well-tuned P-PI controller. The proposed controller also exhibited excellent mbusmess to variations in the table inertia. The proposed controller with a discrete-time disturbance observer shows almost the same performance as that of the continuous time domain design even if its sampling period is five times slower than that of continuous time domain design. Overall, this approach has been shown to be effective for improving the performance of a ballscrew-ddven X-Y table used in wire bonding. It is anticipated that the tracking performance of other systems, such as machining and assembly machinery, could be improved through the application of the proposed controllers.
REFERENCES Doyle, J.C., B.A. Francis and A.R. Tannenbaum (1992). Feedback Control Theory, pp. 47-52, Macmillan. Endo, S., M. Tomizuka and Y. Hori (1993). "Robust Digital Tracking Controller Design far High-Speed Positioning System", In: Proceedings of the 1993 American Control Conference, vol. 3, pp. 2494-2500. Endo, S., H. Kobayashi, Y. Yoshida, S. Kobayashi, M. Tomizuka, and Y. Hori (1994). "Robust Digital Tracking Controller Design for Two Dimensional High Speed Positioning System", In; Proceedings of the 3rd International Worl~hop on Advanced Motion Control, pp. O13-648.
NSK, Ltd (1993). Precision Machine Parts and Linear Motion Products Catalog. Ohnishi, IC (1987). "A New Servo Method in Mechatronics", Transactions of the Japanese Society of Electrical Engineering, vol I07-D, pp. 83-86. Phillips, C.L. and H.T. Nagle (1984). Digitial Control System Analysis and Design, PrenticeHall. Suzuki, A. and M, Tomizuka (1991). "Design and Implementation of Digital Servo Controller for High Speed Mac,kine Tools", In: Proceedings of the 1991 American Control Conference, pp. 1246-1251. Tomizuka, M. (1987). "Zero Phase Error Tracking Algorithm for Digital Control", ASME Journal of Dynamic Systems, Measurement, and Control, vol 109, no. 1, pp. 65-68. Tsao, T.C., and M. Tomizuka (1987). "Adaptive Zero Phase Error Tracking Algorithm For Digital Control, ASME Journal of Dynamic Systems, Measurement, and Control, vol 109, no. 4, pp. 349-354. Tung, E. and M. Tomizuka (1993a). "Low Velocity Friction Compensation for Machine Tool Feed Drive", In: Proceedings of the 1993 American Control Conference, vol 2, pp.19321936. Tung, E. and M. Tomizuka (1993b). "An Algorithm for High-Speed Control of Machine Tools", In: Proceedings of the 1993 American Control Conference, vol 2, pp. 1966-1970. Umeno, T. and Y. Hori (1990). "Two Degree of Freedom Controllers for Robust Servomechanism - Their Application to Robot Manipulators without Speed Sensors", Proceedings of the 1990 International Worichop on Advanced Motion Control, pp.179-188.
Pergamon PII:S0967-0661(96)00036-6
ROBUST DIGITAL TRACKING CONTROLLER DESIGN FOR HIGH-SPEED POSITIONING SYSTEMS S. Endo*, H. Kobayashi*, C.J. Kempf*, S. Kobayashi*, M. Tomizuka** and Y. Hori*** *NSK Ltd., 78 Toriba machi, Maebashi, Gunma-Prel%Japan 371 **Department of Mechanical Engineering, Universityof California at Berkeley. Berkeley, CA 94720, USA ***Department t~fElectrical Engineering, Universityvf Tokyo. 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan 113
(Received July 1995; in final form January 1996)
Abstract: In this paper, a robust digital tracking controller for high-speed positioning is proposed. For high-speed tracking problems, the combination of a feedforward controller and a robust feedback controller is desirable, because the feedforward controller anticipates and compemates for closed-loop dynamics and the feedback controller compermates mexr~amieal nonlinem'ities, parameter variations, and disturbances. A disturbance observer and PD compensation was used as a robust feedback controller and a zero-phase error-tracking controller was used as the f~edforwardcontroller. This method was applied to an X-Y table, and experimental results demonstrate that this method yields excellent tracking performance and robustness to plant parameter variations.
Keywords:
Digital control, disturbance rejection, feedforward compensation, positioning systems, robust performance, tracking characteristics.
1. INTRODUCTION
respect to the chip and lead ~ame acxawately and rapidly. The X, Y and Z axes must be coordinated to achieve an optimum tracking profile to avoid bonding failures. This e~ample requires high-speed and high-precision positioning and tracking.
High-speed multi-axis coordinated motion control is of vital importance in the automation of modem mmufacturing and assembly processes. In particular, high-speed position tracking appears in many applications.
Many X-Y tables are driven by a rotary DC servomotor coupled to a ballscrew. The mechanical design of such systems presents a series of trade-o~. A ballscrew and beating structure that has high sti~ess also generally has a large amount of t'iction. This can hinder tracking per~rmance, due to the
A wire bonder, which must attach free wires to VLSI and ASIC chips, is a good example. A typical wire bonder has three axes, X, Y and Z. The X and Y axis servos must position the bonding tool with 527
528
S. Endo et al.
"quadrant glitches" that occur when an axis changes direction. Furthermore, a long-lead ballscrew is desirable for rapid motion, but this makes the system more sensitive to parameter variations such as changes in payload. To overcome these problems, high-gain feedback controllers are typically used. In most cases, the resonant modes of the system place a limit on the bandwidth that can be achieved. For high-accuracy tmc&ing, the input-output transfer ffmction between the desired output and the actual output must be as close to unity as possible over a wide frequency range. By processing the desired output by a feedforword controller which acts as an inverse of the closed-loop system, the effective bandwidth of the overall system can be improved compared to using feedback alone. Such a technique can be applied in situations where the desired trajectory is known in advance and the model of the closed-loop system is accurate. If the closed-loop system has unstable zeros or lightly damped zeros, a stable inverse system cannot be obtained. For this problem, Tomizuka (1987) has proposed the zero-phase error-tracking controller, ZPETC, which eliminates phase error introduced by uncancellable zeros. When the design of the feedforw~'d controller is based on the inverse of a closed-loop system, tracking perforrnance may be sensitive to modeling errors and parameter variations of the plant. Furthermore, the feedforward controller is not capable of rejecting disturbances. Therefore, the feedback controller should be robust, in terms of stability and performance, when parameter variations and disturbances axe present. A few attempts have been made to apply ZPETC to actual mecla~ical systems. Suzuki and Tomizuka (1991) described the implementation of ZPETC with model-based friction compensation on a machine-tool serve system. Tsao and Tomizuka (1987) developed an ~aptive zero-phase error-tracking controller. Tung and Tomizuka (1993a, b) presented a feedforw~l controller design based on the precise identification of the closed-loop system over a frequency range relevant to tracking. In the area of robust controller design, Ohnishi (1987) introduced a disturbance observer which estimates and cancels disturbance effects. This technique elfectively increases the disturbance rejection while leaving the input-output characteristics essentially unchanged. Umeno and Hori (1990) have proposed a two-degree-ogff~eedom serve structure, extending the idea of the disturbance observer. The disturbance observer is effective for parameter uncertainties as well as disturbances. The disturbance observer can make the actual plant
behave like the nominal plant over a range of fequendes. Therefore, it provides ideal feedback characteristics for the design offeedforwat-d controllers based on the idea of system inversion. This paper describes a high-specd tracking controller which consists of a disturbance observer as an inner loop, a PD controller as an outer loop, and a ZPETC as a fecdforward controller. This follows the methods previously presented for a single-axis table application (Endo, et al., 1993) and work on friction compensation and high speed tracking in the twoaxis case (Endo, et al., 1994). The remainder of this paper is organized as follows: In Section 2, the system is described in detail and a model is developed. Section 3 covers the design process and application for the case of the disturbance observer design based upon discretization of a continuous-time system. The results are presented and discussed. Section 4 is an extension to the methods of Section 3, in which the disturbance observer is designed directly in discrete-time. Again, experimental results are presented and comparisons between the two methods are made. Conclusions are given in Section 5.
2. SYSTEM MODEL AND PRACTICAL PROBLEMS IN HIGH-SPEED TRACKING 2.1
Mechanical Characteristics
Figure 1 shows a configuration of a feed drive system for one axis. Notice that the system has two large inertias, i.e. a motor inertia and a table inertia, and that they are connected by a ballscrew. The primary sources of elasticity in the system are the ballscrew, flex coupling, and bearing supports. Figure 2 is a simplified model of the feed drive system. The goal is to control 02, but only 01 is measurable. Consequently, it is important that the feed drive dynamics are not excited by the controller output, and the elasticity in the structure places a limit on the useful bandwidth of the system. The total spring constant of the feed drive system Kro r is calculated as follows (NSK Ltd, 1993): 1
Krcrr
1
+
1
+
1
(I)
Kcp KBR Kp
Here, K~p is the coupling stiffness, KaR is the axial stiffness of the support beating and K F is the axial stiffness of the ballscrcw. A block diagram of the system is shown in Figure 3. B1 arid B2 correspond to viscous damping, acting on inertias J~ and J2. The effects of static friction
Robust Digital TrackingController Motor
Guide--
Coupling \
Ballscrew
Beari~// '
Tabl7
(3)
P(s) = P,(s)[1 + A(s)].
The nominal plant, P,(s), is detined by considering a system with no elasticity
/ /
Support ---7
529
P.(s)= lim P(s)= Kr---~.**
(Jx + Jz) sz + (B1 + B9)s"
(4) Motor and Coupling Inertia Equivalent Table Inertia KF Ballscrew Spring Constant KBR Support Bearing Spring Constant KCp Coupling Spring Constant J1
The measured transEr Emotion for the system described by Equation (2) is shown in Figure 4. The corresponding nominal plant is also shown. From Equations 2 and 3, A(s) can be lbund to be of the form
J2
~Xz(s) = b°sz + b~s + bE
Fig. 1. Feed drive system.
(5)
s 2 + als + a 2
7"1
where the parameters al, a2, bo, bl, and b2 are •nctions of Jl, J2, B1, B2, and K r . Knowledge of this perturbation, which arises ~-om the resonant mode near 300Hz, will be used to assure robust stability of the disturbance observer loop. Section 3 will show that the resonance places an upper limit on the ~equencies over which the disturbance observer is effective.
T2
K ror
Fig. 2. Simplified model of feed drive system. P Ib
r~
:%--.
=,oo
Ol
1
.go
Jl s 2 +B 1 s +
('~10100.~
%
rL
, '
I
! i i;i!i;i i i iiiiii[ 100
101
10a
02
[ friction 10"I
i0 °
101
102
Frequency (rad/sec)
Fig. 3. Block diagram of the feed drive system. are not included in the model, since these will be regarded as e0tternal disturbances. From the block diagram, the system transfer function is written,
Fig. 4. System identification results.
_Ol e(s)-
7.1= Jzs 2 + B2s+ Kr
2
~ 0 4
Frequency (rad/sec)
2
3"
(JtJ2s + KTJ1+KrJ ~)s +(BtB2s+Kr Bt + KrB2)s+(JIB2 +J2BI )$
(2) For design of the disturbance observer, it is useful to consider the plant to be a nominal plant with a multiplicitive perturbation,
Fig. 5. Quadrant glitches.
103
104
530 2.2
S. Endo e t
al.
Tracking Error G~dy
In multi-dimensional tracking, tracking errors can be defined in several ways. For circular paths, the radial tracking error is defined as
Gr~ Gey
AR -- ~fX-2+ y2 - R
(6) Fd
where x and y are the x- and y-axis table positions and R is the desired radius. In circular path tracking, large tracking errors appear when either the x-axis velocity or y-axis velocity is near zero. These carom are caused by fiction forces and are called "quadrant glitches". Figure 5 shows q - ~ r m t glitches. The radial tracking error will be emphasized in the present experiment to evaluate the effectiveness of the disturbance observer in the presence of friction.
r
e
e'
u
[d
q Y
ZPETC
L disturbance observer Fig. 6. Proposed controller.
2.3 Robustness Against Parameter Variations The equivalent table inertia is calculated as
J2 = ( ~ ) 2 ( M + AM)
(7)
wherep is the lead screw pitch, M is the mass of the table with no load, and A M is the mass of the load. Equation (7) shows that system will be sensitive to variations of the mass on the table when a long lead pitch feed screw is used. This underscores the that the design of the controller should be based on robust performance criteria.
.
3.1
DESIGN OF A TRACKING CONTROLLER BASED ON A CONTINOUS TIME DISTURBANCE OBSERVER
Controller Structure
Figure 6 shows the block diagram of the complete control system. The controller consists of three parts: An inner-loop controller, which is a disturbance observer;, an outer loop, which is a PD controller;, and a ZPETC as the feedforward controller. The gmction of the inner loop is to linesrize the system and to achieve Ge,y = In. The inner loop is designed in the continuous time domain and discretized by bilinear transformation. The outerloop PD controller stabilizes the system; as in many positioning applications, the plant transtx gmction has a pole at s=O. The outer loop is designed in the discrete time domain, based on the zero-orderhold equivalent of the nominal plant as shown in Figure 7.
Fig. 7. Equivalent controller. The feedforward controller design is based on G'ry which consists of the PD controller and nominal plant shown in Figure 7 to attain G,dy in a relevant ~lUency. A precise identification of P(s) is not necessary for the design of the feedforward controller because the robust inner-loop design forces the inputoutput character to be very close to Pn(s). From the practical design viewpoint, it is useful to take advantage of the prior knowledge of the desired trajectory and make use of the f~fforward controller. In this manner, stability problems arising t'om a high-gain feedback appro~w.h can be avoided and the system can still deliver very high bandwidth.
3.2 Disturbance Observer Design The disturbance observer as the inner-loop controller is designed in the continuous time domain. The dismrb~mce observer is inside the dotted line shown in Figure 6. Referring to Figure 6, for Q~ 1,
V(s)=E'(s)-{-v(s)+ lpn(s) P(s)(V(s)+O(s))} (8)
Robust Digital Tracking Controller
which implies
P~(S)E'(s)_D(s).
U(s)= P(s)
(9)
531
Robust stability of the disturbance observer loop is assured if the following condition (Doyle, et al., 1992) is satisfied:
m .axlr(s)A(s)I < m .axlr(s~lm(s)I < 1. Substituting this relation into the plant equation, i.e. Y(s) = P(s)(V(s) + D(s)), gives
Y(s)=Pn(s)E'(s).
s=J¢O
(13)
$=JO~
From this the following restriction on Q is obtained:
(10)
[Q(s)[ < Equation (10) shows that the disturbance observer compensates for the dismrbanom mad parameter variation of the plant, thus making the inner-loop system behave like the nominal plant. However, this scheme, with Q=I, does not work in practice because 1/P~ is normally unrealizable. Therefore, a low-pass filter Q in the disturbance observer is used so that 1/P~ is realizable. In the selection of Q, consideration must be given to obtaining internal stability of the controller and to reducing sensor noise. Thus, the disturbance observer design is essentially a matter of selecting Q.
1
Vs = jog.
(14)
Notice that in Equation 11 Q(s)= 1 is desirable for disturbance rejection. On the other hand, Equation (14) shows that keeping the Q(s) is desirable for robust stability. Therefore, Q(s) should be selected taking account of this trade-off. For this problem, Umeno and Hofi (1990) suggest the following form for Q(s) : N-2
1 + ~.~ ak(sz) k Q(s)
Recall that the plant dynamics were parameterized as P = Pn(1 + A(s)). Then, Y(s) can be expressed as
=
(15)
~=t N
1 + Z ak (s~')~: k=l
Y(s) = P. (s)(1 + A(s)) E'(s) -I (1 - Q(s))P~(s)(1 + A(s)) D(s). 1+ A(s)Q(s) 1 + A(s)Q(s)
(II) Notice that Y(s) = P(s)E'(s) for Q(s) = 1 and Y(s) = P(s)(E'(s) + D(s)) for Q(s) = 0. Therefore, by proper selection of Q(s), the actual plant caia be made to behave like the nominal plant over a specified frequency region.
For this application, N = 3 ,
aI =3,
a 2 = 3 , and
a 3 = 1 were selected. Figure 8 shows the sensitivity £mction and complementary sensitivity limction of this choice of Q(s). It can be seen that this choice of Q(s) results in the same rate of roll-ot~ -40 dB per decade, for both S and T. This implies that performance robustness and robust stability are balanced.
For internal stability and realizability (Umeno and Hofi, 1990) it is necessary that
Q(s) ~ RHo~,
Q(s)/P~ ~ R H . .
(12)
~.~ -20 - i..~
o.¢0|
.i.L- i
! i~i
[ iSiiT;ii
The inner loop formed by the disturbance observer, Ge.y, has an open-loop gain of Q / O - Q ) , where it is assumed that P=P~. Thus, the sensitivity finetion, S, for this inner loop is given by S = 1-Q and the complea-aentary sensitivity limction, T, is simply Q. The use of S and T in the design of the disturbance observer is particularly helplial. First, they can be used directly in the formulation of a robust design, as will be shown below. Second, they offer physical insight into the design behavior of the system. S is a measure of how much a disturbance will contribute to the system output, and T is a measure of how much sensor noise will contribute to the output.
10'
:: iii::-::.
ii
i il !!~
..... + 10 2
!
!iii
i i il q
...... i 10 3
10 +
10 5
F r e q u e n c y (rad/sec)
Fig. 8. Frequency response of T(s) and S(s).
i i i iiii:i m ........i+i+ i i+.iLiil 0
f
i i i iiiiii i i i i~ ~: L L : Lii?~i .....L+i_LLiiii (
%,
10 2
Frequency
10 3
(rad/secl
Fig. 9. Frequency response of 1 / A(s) and Q(s).
10 4
S. Endo et al.
532
~" is tuned on the basis of Equation (14). Figure9 shows a Bode plot of 1/A and Q. In this case, ~ = 0.0015, which is equivalent to a cutoff frequency is 667 rad/sec. The continuous-time inner-loop compensation was discretized by applying a bilinear transformation prewarped at the Q filter cutoff frequency. The experimental plant is represented by a zero-orderhold equivalent. However, the nominal plant model has been discretized by applying the bilinear transformation. A discrepancy between the two is observed in the form of an extra zero in the bilinear transformation. To reduce this etfeet, the sampling ~equency is sdected to be much lkster than that of the cutoff ~equency of Q. In this application, the sampling period was set to 0.2 re_see.
3.3
PD Controller
The outer-loop feedback controller is designed in the discrete time domain. A zero-order-hold equivalent of P. is obtained for this purpose and the design is based on the feedback structure in Figure 7. Notice that p,, in Figure 7 has a pole at 1 because of a pure integrator. To stabilize the feedback loop in Figure 7, the Eedback controller is selected to be of PD type. The PD controller moves the open-loop pole of I to the inside of the unit circle, making the system well-damped. For the design of the feedforward controller, it is important to take account of the coefficient quantization errors in the feedforward controller and the dosed-loop controller.
Gff(z_l)= y(k) = zdA(z-I)B-(z)
where B-(z) is obtained by replacing z -t in B-(g -1) by z. The overall ~equency response ~ m the desired output to the actual output is shown in Figure 10. It can be seen that almost peri~ tracking is attained up to 667 rad/see. For additional details regarding ZPETC design, see (Tomizuka, 1987).
3.5
Experimental Results
Figure 11 shows a schematic diagram of the experimental system. The positioning system used was an NSK CT-0505 X-Y table designed for wire bonding applications. The position transducer is a 4000 epr (counts per revolution) optical encoder. This encoder, coupled with 10 mm pitch ball-screw, provides a linear resolution of 2.5pan in table displacement. A can'mat driver was designed to control the motor torque directly. The dynamics of the driver can be approximated as a first-order low-pass filter. The time constant of the driver was of the order of O.1 msec. This is very ~st in comparison to the mechanical dynamics, so the driver dynamics were ignored in modeling the system.
a
Zero-Phase Error-Tracking Controller
::.iiii!il
!:!:;iii:
!i!i!!':~:~:......~...!..+.i.ii.li
~-,o ..... i.ii.~.~!i ...... !-r.i-.ii!i!~ •.4
3.4
i !!:.ilrii
i il ili!i iiii '101
10 =
Frequency (rad/sec)
Fig. 10. Overall transfer function.
mot°r/enc°de~ 7
G(z-1 ) = y(k) = z-aB(z -i ) _ z-aB - (z-1)B +(z-I) r(k) A(z -1) A(z -1) (16) where d is the number of delay steps, B+(z -l) represents the caneellable zeros and B-(z -1) represents the uncancellable zeros. The ZPETC is given by
i
ii~ii i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 °
The feedforward controller design is based on G~ instead of Gry to reduce the order of the feedforward controller and to simplify the design. Then a nearperfect tracking performance is attained within the bandwidth of Q(s), that is up to 667 tad/see. ZPETC as a feed~rward controller is designed as follows. Consider the asymptotically stable closedloop plant
(17)
rd(k) B+(z-')[B-(1)] 2
host computer
l
Fig. 11. Experimental system.
10 3
I1~
Robust Digital Tracking Controller The controller was implemented on a TMS320C30 digital signal processor. The sampling period was 0.2 msec.
533
Trackln¢l Error
5.0
The performance of this controller was compared with that of a P-PI controller which is commonlyused in positioning applications. The P-PI controller parametees were tuned so as not to excite the mechanical resonance. All the tests involved a desired trajectory consisting of a l0 mm diameter circle which was to be Iraeed in 0.5 seconds. Figure 12 shows the radial tracking error of the poPI controller, and Fig. 13 shows the radial tracking error of the proposed controller. It can be seen that the magnitude of the tracking error is significantly reduced by the proposed controller, particularly the qu~rant glitches. Although the proposed controller cannot completely eliminate the quadrant glitches, they have been reduced to a level that is suitable ~r most practical applications.
E c
E0 o t
-5.0 ..... ....
....
•
Fig. 14. Tracking error with the P-PI controller.
Trackln¢l Error $.0
Trackln~l Error • ..... +O.~lwl
%0
E e-
0
i5 o
i5 -5.0
."..i?..-..................:ii. " .
-$.0
Fig. 15. Tracking error with the proposed controller.
Fig. 12. Tracking error with the P-PI controller. Tracklm:l Error • 5.0
.""
.' """
• ' ....... -~o.~lnml. ° ..... O.al.. ; ~"
"'. '"'
°..
,
". ° " "
.
° .
•
A load mass on the table was added, which increased and etFeetively doubled J2. Figures 14 and 15 show the experimental results of the perturbed case. It can be seen that the tracking error under the proposed controller is almost the same as the nominal case. On the other hand, the traddng errors increased for the P-PI case.
.
4
4.1
:.i i. Fig. 13. Tracking error with the proposed controller.
DISCRETE-TIME-DOMAIN DESIGN OF DISTURBANCE OBSERVER Discrete time disturbance observer design
Although the perfonnance of the proposed controller was very good, there are practical limitations. In particular, implementation of such a controller generally requires a high-performance CPU. This is a direct result of the fist sampling rates needed to minimize errors in the discretization of the inner-loop compensation. Both computational speed and
S. Endo et al.
534
wordiength become important at high sampling rates. In this section, a discrete time domain design of the disturbance observer is proposed to address this problem. When the disturbance observer is designed directly in discrete time, the advantages are: I) The sampling ~queney can be reduced with less degradation in performance, 2) The elects of coe~cient quantization errors are reduced because the filter to be implemented has poles and zeros that are no longer crowded near the z=l, and 3) Robust stability is improved since there is no longer a discrepancy arising ~om representing the plant in discrete time using a bilinear transformation rather than a zero-order hold. In the discrete time domain design, the nominal plant is defined in the form of the zero-order-hold equivalent model which yields
p(z-1)=
bl z-I +b2 z-2 l+a~z -1 +a2 z-2 •
Let the discrete-time Q tilter be denoted by Qd(Z -1 ) with a numerator polynomial NQ(Z -t) and denominator polynomial DQ(Z-~).
For the product
Q~(z-1)P;l(z -1) to be causal, it is necessary to have deg(NQ)
Qd(Z -1 ) ---- bql Z-I -,I..bq2z-2 -I- bq3Z-3 1 + aqlZ-I + aq2Z-2 + aq3Z-3 "
+,-= E e2<,o,
o - )I'+
N,,
NQ(eJ~°ff )12
I
k=0
(19) which corresponds to an equation error formulation. f(.) is the frequency weighting function. Q~(z-i) denotes the target tilter obtained by discretizing Q(s) with a bilinear translbrm. The titting technique employs a straightforward least-squares minimization at the set off~equencies w0,cot,~o2... CON/2. The frequency weighting ~hnetion was chosen to be a simple first-order low-pass filter with a cutoff • xtuency set to half of the cutoff ~equency of the Q filter. This sdection provided a reasonably good match at high ~equeneies, while forcing the low • equency gain to be very dose to 1. For this problem, the equation-error optimization method yielded a stable approximation to Q~(z-1) that was a good match; Figure 16 shows the fiequency response of both Qt(z -1) and Qa(z-i). When a digital filter is designed, poles and zeros close to the unit circle are undesirable. This is particularly troublesome when the compensator dynamics are dose to z= 1 due to quantization error problems. Although these effects can be reduced by choosing the lbrm of the filter realization caregtlly (see, for example, (Phillips and Nagle, 1984)), it is best to avoid this problem entirely. One technique is to reduce the sample rate in order to move some of the dynamics toward the interior of the unit circle. Figure 17 shows the poles/zero map of the outer dosed-loop system described in Figure 7. Here, sampling ~a.~quenciesare changed and P and D gains are tuned to keep the cutoff f~luency of the closedloop system constant in each sampling Eequency. Notice that the dominant poles move inside the unit circle as the sampling frequency is reduced.
(18)
The ~mdamental issue in the design process is the selection of eoet~dents aql , aq2 , aq3 , bq9 bq2 ,
and bq3. To accomplish this, a ~lUency weighted curve fitting technique was used. Although such techniques are well known and commonly used, the particular method employed in this case will be briefly reviewed.
0
! ! i i i i i l l .........i i G i i i !
3 -'° .........i i l}ii
)
101
10 ~
i i iiiiiii
~ i : iiil}
:
:
......... ?
-2001
10 ~
Q,~ i i+i!i/
I0 z
t- I : .... :
~'-100
iiii+i
i i ii i i ~ i i
G)
The fit is accomplished by minimizing a penalty function of the foUowing form
..........i............i i i+
i
i
~ :!iii:
i
! !
+ ' + " i ' ! i i i ' ! + i .......... ~'" ' ~ T ? ~
i
i !!i!ii
:
i
':!~
i ii+iiTiI++ii' ~ i i!i::::::
~
~ +i.
! : ili:i
10 2 Frequency
10 4
i
} i i i il
:
;
10 3 (rad/$ec)
Fig. 16. Frequency response ff Qt and Q~
:::iii.
10 (
Robust Digital Tracking Controller
535
4.2 Experimental Results Figure 18 shows the experimental results of the proposed controller combined with a discrete-time domain design of the disturbance observer. In this case, the same design was used as in the previous section, except that the sampling time was increased to 1 msec.
Trackln¢l Error
......... ~ r 4 = = ! •
~
5.0
°l°°°
I,~~ °
o-
JN-.°~'"'.°,
i°
i
s, . .
" 'o,
.,'" °"
"x "LL,°
g : -~
Figure 19 shows the experimental result with the proposed controller combined with a continuous-time domain design of the disturbance observer. Again, the sampling rate was increased to 1 reset. In this case, oscillations on the order of 20/an are dearly visible.
o.''"
~
.
.
~
'
~
.
~
:
•
|t~o
1¢
-5.O •.°........ : ....... '...
These oscillations are caused by the reduction of stability margin as the sampling is slowed and discrepancies between the actual plant and discretized nominal plant are no longer negligible.
Fig. 19. Tracking error of the continuous-time design. Trscklnq Error
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/
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Fig. 20. Tracking error of the discrete-time design.
Axis
Comparing Figures 18 and 19, it can be seen that the direct discrete-time design technique sutta's far less performance degr~_~ i o n when a slower sampling rate is used.
Fig. 17. Pole and zero map. Tracking
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E
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.' ,
~
Error
~
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As a test of robustness, the discrete-time design was tested with the inertia of -/2 doubled. Results for this case are shown in Figure 20. From this, it can be seen that the results are essentially the same as for Figure 18, an indication that the design is robust to parameter variations.
•
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Fig. 18. Tracking error with the discrete-time design.
CONCLUSIONS
In high-speed tracking applications, mechanical vibration, fiiction and parameter variations are serious problems. This paper has demonstrated that a controller which combines the disturbance observer as an inner loop, a PD controller as an outer loop, and ZPETC as a Eedforward controller is a highly
S. Endo et al.
536
effective method of addressing these problems. Disturbance observer designs in both the continuous and discrete-time domains were considered. Although the bandwidth of the dosedqoop system is limited be,:a__useof the mechanical resonance in the structure, the use of feedforward in combination with a robust inner-loop design yields performance fit superior to that of a standard P-PI controller. The proposed controller can reduce the tracking error to about 5 #m in tracking a circular path at high speed, roughly one fourth of the errors that occur with a well-tuned P-PI controller. The proposed controller also exhibited excellent mbusmess to variations in the table inertia. The proposed controller with a discrete-time disturbance observer shows almost the same performance as that of the continuous time domain design even if its sampling period is five times slower than that of continuous time domain design. Overall, this approach has been shown to be effective for improving the performance of a ballscrew-ddven X-Y table used in wire bonding. It is anticipated that the tracking performance of other systems, such as machining and assembly machinery, could be improved through the application of the proposed controllers.
REFERENCES Doyle, J.C., B.A. Francis and A.R. Tannenbaum (1992). Feedback Control Theory, pp. 47-52, Macmillan. Endo, S., M. Tomizuka and Y. Hori (1993). "Robust Digital Tracking Controller Design far High-Speed Positioning System", In: Proceedings of the 1993 American Control Conference, vol. 3, pp. 2494-2500. Endo, S., H. Kobayashi, Y. Yoshida, S. Kobayashi, M. Tomizuka, and Y. Hori (1994). "Robust Digital Tracking Controller Design for Two Dimensional High Speed Positioning System", In; Proceedings of the 3rd International Worl~hop on Advanced Motion Control, pp. O13-648.
NSK, Ltd (1993). Precision Machine Parts and Linear Motion Products Catalog. Ohnishi, IC (1987). "A New Servo Method in Mechatronics", Transactions of the Japanese Society of Electrical Engineering, vol I07-D, pp. 83-86. Phillips, C.L. and H.T. Nagle (1984). Digitial Control System Analysis and Design, PrenticeHall. Suzuki, A. and M, Tomizuka (1991). "Design and Implementation of Digital Servo Controller for High Speed Mac,kine Tools", In: Proceedings of the 1991 American Control Conference, pp. 1246-1251. Tomizuka, M. (1987). "Zero Phase Error Tracking Algorithm for Digital Control", ASME Journal of Dynamic Systems, Measurement, and Control, vol 109, no. 1, pp. 65-68. Tsao, T.C., and M. Tomizuka (1987). "Adaptive Zero Phase Error Tracking Algorithm For Digital Control, ASME Journal of Dynamic Systems, Measurement, and Control, vol 109, no. 4, pp. 349-354. Tung, E. and M. Tomizuka (1993a). "Low Velocity Friction Compensation for Machine Tool Feed Drive", In: Proceedings of the 1993 American Control Conference, vol 2, pp.19321936. Tung, E. and M. Tomizuka (1993b). "An Algorithm for High-Speed Control of Machine Tools", In: Proceedings of the 1993 American Control Conference, vol 2, pp. 1966-1970. Umeno, T. and Y. Hori (1990). "Two Degree of Freedom Controllers for Robust Servomechanism - Their Application to Robot Manipulators without Speed Sensors", Proceedings of the 1990 International Worichop on Advanced Motion Control, pp.179-188.