Trajectory-adaptive digital tracking controllers for non-minimum phase systems without factorisation of zeros

M. M . Mustafa

Abstract: The paper discusses the design of a zero phase error feedforward digital tracking algorithm. The proposed algorithm can be used for minimum or non-minimum phase systems without requiring the zeros polynomial to be factored into minimum and non-minimum phase factors. To improve tracking when the zeros are close to the unit circle, a scheme called trajectory- adaptive zero phase error control is proposed in which the controller parameters are automatically tuned online to the current reference trajectory without requiring explicit information on the function to be tracked. The performance of the trajectory-adaptive controller is further improved by using a look-ahead learning scheme that enables the controller to make advance adjustment in anticipation of the upcoming trajectory.

1 Introduction

In many motion control applications, such as machining processes, perfect reference trajectory tracking is desired. Mathematically, this can be achieved by placing the inverse model of the system between the reference signal and the system. In practice this cannot be done because the presence of non-minimum phase zeros (zeros outside the unit circle) make the inverse model unstable.

A survey of various digital tracking algorithms is given in [ 11. A two-degrees-of-freedom controller consisting of feedback and feedfonvard controllers is usually employed in such a controller. The common strategy in all these controllers is to cancel all the poles of the closed-loop system and the zeros located inside the unit circle. To compensate for the dynamics associated with the uncan- celled zeros, following the original idea given in [ 2 ] , zero phase error tracking (ZPET) is achieved by incorporating the reciprocal polynomial of the uncancelled zeros poly- nomial in the feedforward controller. This strategy, however, does not eliminate the gain error. A scaling factor in the feedforward controller was suggested in [ 2 ] to ensure overall unity DC gain. This scaling method may not result in a bandwidth sufficiently wide to cover the frequency spectrum of the desired trajectory. To improve tracking for a wider frequency spectrum, several strategies have been proposed. Additional feedforward filters were suggested in [3] and [4, 51 proposed algorithms for accurate tracking of reference trajectories belonging to a certain class. [6] described a design to compensate the gain error based on frequency-weighted L2-norm optimisation

0 IEE, 2002 IEE Proceedirigs online no 200201 15 DOI I O 1049hp-cta 20020 1 15

Paper first received 10th April and in revised form 6th December 2001 The author is with the Department of Electrical, Electronic and Systems Engineering, Faculty of Engineering, University Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

IEE Proc -Control Theory Appl Vol 149 No 2 March 2002

and [7] proposed a method of predicting the tracking error, and using this prediction to modify the reference trajectory, but this method is accurate only if there is a single uncancellable zero.

While all the methods described require factorisation of the zeros polynomial into minimum and non-minimum phase factors, the method proposed here does not require this factorisation. To handle the case where the gain cannot be made unity for the desired frequency spectrum, a trajectory-adaptive ZPET controller is proposed. This trajectory-adaptive ZPET controller automatically retunes itself if the frequency component of the trajectory changes. This online retuning overcomes the limitation of earlier methods [S-101 in which explicit information on the function to be tracked is required a priori. The trajec- tory-adaptive ZPET controller described here is different from parameter-adaptive controllers [6, 9, 111 in which the latter refers to online estimation of system parameters, and these parameters are used to adjust the ZPET controller online. To improve tracking accuracy when there are changes in the dominant frequency content of the reference trajectory, a look-ahead learning scheme is proposed to enable the controller to make advance adjustment to track the anticipated reference trajectory.

2 Zero phase tracking error without factorisation of zeros

The configuration of two-degrees-of-freedom controller usually employed in the ZPET controller is shown in Fig. 1. Due to the feedforward nature of the ZPET controller, the overall performance of the controlled system will be affected if there is plant-model mismatch. By applying appropriate feedback around the plant, one can, to a certain extent, minimise this effect by ensuring that the frequency response of the closed-loop transfer function will not vary significantly within the bandwidth of the trajectory signal if there are changes in plant dynamics.

157

feedback system controller controller

Fig. 1 Two-degrees-of-freedoom contsoller

Let the system be represented by the following discrete time model

where

A(z- ' ) = 1 + up- ' + u2z-2 + . . ' + u n p

B(z-l) = bo + biz-' + b2zP2 + . . . + bnhzPnh

d = time delay

For our purpose here and referring to Fig. 1, it is assumed that the closed-loop transfer function (without feedfonvard control) is given by

z - ~ B, (z- ' ) A, (2-I )

G,(z-') =

Setting factor B,(z-') into minimum and non-minimum phase factors:

B,(z-') = B,+(z-')B,(z-')

where B:(z-') denotes the minimum phase factor and B,(z-') is the non-minimum phase factor.

The conventional ZPET feedforward controller reported in the literature can be divided into three blocks as shown in Fig. 2. The structure of the ZPET feedforward controller without zeros factorisation is given in Fig. 3 . As with all other ZPET controllers, the design focuses on the selection of an appropriate gain compensation filter to ensure that the overall gain is unity within the frequency spectrum of the reference trajectory. To ensure that the gain compensa- tion filter Fg does not introduce any phase error, using a

similar approach to that in [ti], the gain compensation filter is a symmetrical finite impulse response filter

n,

. ' k O F,(z, Z - ' ) = + z - ~ ) (3)

A suitable cost function to represent the error between the desired and actual frequency response is

The design objective reduces to finding a set of uk such that the cost function (4) is minimised. The optimal uk can be found by expanding (4) as ii polynomial in z and z-I, and equating coefficients of like: powers.

Example 1: Consider the following closed-loop discrete- time system

z-'(1 - o.lz-')(l + 2z-1) G,(z-') = ~ ( 5 ) (1 - 0.9~-')(1 - 0 . 2 ~ ~ ' )

The zeros polynomial anti its reciprocal are B,(z-')= ~ ~ ' ( 1 - O.lz-')(l +2z-') and B,(z)=z(l - O.lz)(l +2z). Consider the case where the order of F,(z-', z) is n,=4. The optimal ak obtained by minimising the cost function (4) is shown in Table 1.

The frequency response -For the overall system is shown in Fig. 4. For comparison, the frequency response when only DC gain compensation. is used is shown in Fig. 5. The frequency response shown in Fig. 4 can be improved if the order F,(z-', z ) is increased. Fig. 6 shows the frequency

Table 1: Optimal ak for fourth-order gain compensation filter

k O 1 2 3 4 ~~

~ ( k 0.3027 -0.1260 0.06347 -0.02852 0.01020

gain phase compensation compensation

filter filter Fig. 2 Conventional f e e d j o n s d zero phase error controller

stables inverse

gain phase compensation compensation

filter filter Feedforward zesu phase erros contsolles without factorisation of zesos Fig. 3

158 IEE Pro,.-Cuntrol Theory Appl., Vol. 149, No. 2, March 2002

0.8

-5 0.6- 0)

0.4

0.2

0

1.2r

-

-

-

I I I I I I I

% 0.6 ol

1.2-

1 .o

0.8

.% 0.6

0.4

0.2

0)

0 .

0.2 0'41

7

- -

-

- I I I i I I I

01 I I I I I I I

0.1 0.5 0.9 1.3 1.7 2.1 2.5 2.9 frequency

Fig. 5 tion

Frequency response of ZPET controller with DC gain compensa-

response when n, = 10. In this case the gain is very flat at unity, implying good tracking can be achieved for a wide range of frequencies. The optimal values of c!k in this case is given in Table 2.

The disadvantage of this scheme compared to the scheme with factorisation is that the value of n, will be larger than when the zeros are factored, as F, has to approximate the inverse of both the minimum and non- minimum phase zeros. This may not be a serious drawback as the increase in memory required to store these extra parameters will be insignificant compared to the amount required to store the program code. Furthermore, as will be shown later, satisfactory performance can still be achieved with relatively low-order Fg using trajectory-adaptive ZPET.

Table 2: Optimal (Yk for tenth-order gain compensation filter

3 Convergence of high-order zero phase error tracking controller

Intuitively we may think that by minimising the cost function (41, CEO ak(zk + zPk) will converge to 1 /B,(Z)B~(Z-'). This is definitely undesirable because l/B,(z)Bc(z-') is unstable as either B,(z) or B,(z-') will always contain unstable zeros. The result shown in Table 2, however, shows that 0 ak(zk + zPk) converges as a convergent power series. It will be shown that the optimal

mk(zk + zPk) converges to a convergent sequence and the controller parameters c(k can also be computed using Laurent series expansion. It is not the intention of this section to highlight this alternative method of computing Cxk as this method requires the zeros to be factored into minimum and non-minimum phase factors.

The following derivation of the optimal xk(zk + z?) follows closely the work reported in

[ 123. The cost fbnction (4) can be rewritten as

and

where K = supz = I B,(z-')B,(z) 1 . Therefore the left-hand side of (7) can be made as small as possible by minimising the cost on the right-hand side (RHS). Following the same approach given in [12], the term l/Bc(z)Bc(z-l) is then expanded into a Laurent series with both negative (positive power of z ) and positive (negative power of z ) sequences. These negative and positive sequences can be obtained by expanding l/B,(z)B,(zp') into a sum of stable and unstable rational transfer functions. The stable transfer function is expanded in powers of z-' to generate the causal positive sequence, and the unstable transfer function is expanded in powers of z for the non-causal negative sequence.

Using the orthogonality of the functions . . . zP3 , zP2, z-' , zo, z ' , z2, z 3 . . . , the RHS of (7) is minimised when

k=O

(8) 1

Laurent series expansion of Bc(z)Bc(z-')

Example 2: Consider again the discrete time model (5) :

B,(z)B,(z-') = (1 - O.lz)(l + 2z)(1 - o.lz-l)(l + 22-1)

Expanding 1 /B,(z)B,(z-') into stable and unstable terms

1 - -0 .3046~~ + 2.5652 B,(z)B,(z-') - (z - 1O)(z + 2)

0 . 3 0 4 6 ~ ~ - 0.006413 ( Z + O . ~ ) ( Z - 0.1) .

+

k O 1 2 3 4 5 6 7 8 9 10

~ ( k 0.3046 -0.1283 0.06654 -0.03303 0.01653 -0.008258 0.0041 19 -0.002040 0.0009901 -0.0004486 0.0001602

IEE Proc.-Control Theory Appl., Vol. 149, No. 2, March 2002 159

,/”adjust I I

Expand the stable term of negative powers of z

0.3046~2 -:0.006413z -

(Z + 0.5)(~ - 0.1)

k=O 5 (0.04OO8 (h) k+0.2645 (- i) ‘ ) z P k

Thus

= I0.3046, -0.1282,0.0662, -0.0330,0.0162,. . .)

Note that these values a k are very similar to these given in Table 1 or Table 2.

4 Trajectory-adaptive zero phase error tracking controller

In systems where the zeros are located close to the unit circle, it is difficult, using a low-order controller, to make the overall transfer function from reference trajectory to output approximately equal to 1 over the desired frequency range. Good tracking is only possible in such cases by explicitly incorporating the function to be tracked in the controller [8-lo]. The difficulty here is the problem of extracting this information online. A set of tuned filters to separate each frequency component of the periodic signals has been suggested, but its implementation is cumbersome [5]. Such schemes that explicitly incorporate information on the function to be tracked may not be effective if the trajectory cannot be decomposed into simple functions, or the trajectory may not even fall within the class of trajectories the controller is desi ned for.

by equating coefficients of like powers, we can solve for ak

using a parameter estimation algorithm such as recursive least squares. Let { ( k } be a sequence of excitation signals which is artificially generated. The optimum ak can be estimated by running the estimation algorithm using the scheme shown in Fig. 7.

Instead of solving B,(z),B(z- ? C;ro + z - ~ ) = 1

I 1

t+ Fig. 7 Computing controller parameters using recursive identification

Since the excitation signal ( k is artificially generated, this freedom can be exploited by choosing ( k to have a frequency spectrum similar or close to the frequency spectrum of the reference trajectory. Effectively, this is similar to using a frequency weighting function to give emphasis to the frequency spectrum of interest. The best way to capture the dominant frequency component of the trajectory signal is to use the trajectory signal itself as the excitation signal. The only precaution, required is that the trajectory signal may not be persistently exciting, and may cause covariance blow-up, especially if a forget- ting factor is employed in the estimation algorithm. Various methods, such as variable forgetting factor [13] and dead- zone [14], can be used to overcome such problems. In our case, the problem is much simpler to address as the excitation signal is artificially generated. To ensure that the excitation signal is always persistently exciting, a low- level noise signal which has its frequency spectrum close to the frequency spectrum of the reference trajectory can be superimposed on the reference trajectory. Note that the addition of noise to the reference trajectory will not add disturbance to the system as the ‘noisy’ reference trajectory will not pass through the system, as shown in Fig. 8.

Example 3: Consider the following discrete-time system which has an unstable zero close to the unit circle, and is hence more difficult to cancel using a low-order ZEPT controller

GC(z-’) = Z-yl ~ - O.lz-I)(l + 1.lz-1) (1 - 0.9~-’)(1 - 0.22-1)

n

Fig. 8

160

Trajectory-adaptive zero phase error controller

IEE Proc.-Control Ekeoty Appl., Vol. 149, No. 2, March 2002

ing the advantage of being able to preview the future reference trajectory, look-ahead learning is proposed.

In the conventional identification algorithm, estimates are based on past and current data. With look-ahead learning, past, current and future data are used. Here, look-ahead learning does not simply mean advancing the regressor vector in the estimator by a certain number of time steps, as this will not result in faster convergence. To -facilitate convergence, at each sampling interval we cycle the estimation process several times using past, present and hture data to obtain the current estimates. At each cycle the regressor is advanced by one time step. The length of the future horizon, N, will determine how far the estimator looks ahead. The length of future horizon should not be chosen too large as this can result in current estimated controller parameters that place too much emphasis on tracking the future trajectory at the expense of tracking the current trajectory. Experience indicates that a suitable value for N is half the number of controller parameters to be estimated. This provides a rough compromise between the need to make advanced preparation for the future while not forgetting the need to track the current trajectory.

Example 4: Consider now a more difficult system with a zero on the unit circle and one non-minimum phase zero:

-1.5 I L

0 25 50 75 100 125 150 time step

Sinusoidal re@rence trajectory with tinie-var?iingfreqiiency Fig. 9

0.2 0'3 [

-0.3 I I 1 0 25 50 75 100 125 150

time step Fig. 10 Tracking error using white noise as excitation signal

0.2 0'3 1

-0 3 25 50 75 100 125 150

time step

Fig. 11 Tracking error using reference trajectory as excitation signal

A sinusoidal reference trajectory with time-varying frequency, shown in Fig. 9, is chosen here to demonstrate the ability of the controller to track the trajectory with changing frequency component. Fig. 10 shows the tracking error when white noise is used as the excitation signal. The root mean square (RMS) tracking error is 2.40. Tracking error when the trajectory is used as the excitation signal is shown in Fig. 11. The RMS tracking error in this case is 1.02. The benefit of using the trajectory as the excitation signal to estimate the controller parameters is evident when comparing these two results.

5 Look-ahead learning

In the recursive least squares identification algorithm, a forgetting factor smaller than 1 is used to give more emphasis to the new data. Although a smaller forgetting factor will help the estimates to converge to the new values faster when there is a change in the system, at best, the time to convergence cannot be less than the number of estimated parameters. This can result in poor tracking accuracy during this period, especially if a high-order controller is used. To address this problem, again exploit-

IEE Pmc -Control Theoty Appl, Vol 149, No 2, March 2002

z-1(1 - z-1)(1 + 22-1) G,(z-l) =

( I - 0.9~-')(1 - 0.2z-I)

Because of the zero on the unit circle, no ZEPT controller can cancel the effect of this zero at all frequencies.

A 20th-order ZPET controller is used in this example. A high-order controller is used here due the presence of zeros close to and on the unit circle. Because of the large number of parameters to be estimated, convergence to final values will take some time when the frequency component of the trajectory signal changes. This is demonstrated in Fig. 12 when no look-ahead learning is employed. The result when look-ahead learning with N = 10 is employed is shown in Fig. 13. The advantage of using look-ahead learning can be

RMS error=1.64

0.3 r

0 25 50 75 100 125 150 time step

Fig. 12 ZEPT without look-ahead learning

RMS error=0.33 0.2 1

-o.2 -0.3 0 ~ 25 50 75 100 125 150

time step Look-ahead learning for N = I O Fig. 13

161

0.2 0’3 [ RMS error=0.40

-0.3 I I

0 25 50 75 100 125 150

ZEPTwith look-ahead learning, N= 20 time step

Fig. 14

clearly seen from these results. Fig. 14 corresponds to look-ahead learning with N=20. This figure shows the deterioration in tracking accuracy when very large N is used.

This look-ahead learning scheme can easily be extended to make it adaptive to changes in system parameters by running it in parallel with a system parameter estimator. Even without look-ahead learning, the ZPET proposed here clearly has advantages over the adaptive ZPET proposed in [ 1 13, because in the scheme proposed here there is no requirement for non-minimum phase zeros to be known a priori. In the scheme proposed in [l 13, the non- minimum phase zeros are assumed to be known, and only the minimum phase zeros are estimated.

In certain cases where the online computation capability of the controller is very limited, for a known fixed para- meter plant, the control input to the closed-loop system can be pre-computed off line and downloaded to the controller during actual runs.

6 Conclusions

A zero phase error tracking controller without requiring factorisation of zeros is presented. To handle the case where accurate tracking is not possible at all desired frequencies, trajectory-adaptive zero phase error tracking control has been proposed to tune controller parameters online to suit the frequency component of the trajectory signal. Further improvement in tracking accuracy was achieved by employing look-ahead learning strategies, in which the controller can make advance adjustment to its

parameters to cope with the upcoming trajectory. This helps to reduce tracking errors in the dominant frequency component of the trajectory during the transition.

7 Acknowledgment

The author would like to thank all the referees for their valuable and constructive comments.

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References

TOMIZUKA, M.: ‘On the design of digital tracking controllers’, AMSE J1 Dyn. Sysf., Meas. Control, 1993, 115, (l), pp. 412418 TOMIZUKA, M.: ‘Zero phase error tracking algorithm for digital control’, A S M E 1 Dyn. Syst., Meas. Control, 1987, 109, (I) , pp, 65-68 HAACK, B., and TOMIZUKA, M.: ‘The effect of adding zeros to feedforward controllers’, ASMEJ Dyn. Syst., Meas. Control, 1991,113.

JAYASURIYA, S., and TOMIZLIKA, M.: ‘ Feedforward controllers for perfect tracking that does not invert plant zeros outside the unit disc’. Proceedings of IFAC 12th Triennial World Congress, Sydney, Australia,

WALGAMA, K.S., and STERNEIY, J.: ‘A feedforward controller design for periodic signals in non-minimum phase processes’, In/. 1 Control,

YEH, S.S., and HSU, P.L.: ‘An o’ptimal and adaptive design of feedfor- ward motion controller’, IEEWASME Trans. Mechatron., 1999, 4, (4), pp. 428439 TORFS, D., SWEVERS, J., and DE SCHUTTER, J.: ‘Quasi-perfect tracking control of non-minimal phase systems’. Proceedings of the 30th Conference on Decision and Control, Brighton, England, 199 I , pp, 241-244 TORFS, D., SWEVERS, J., and DE SCHUTTER, J.: ‘Optimal feedfor- ward prefilter with frequency domain ppecification for non-minimum phase systems’, ASME Manu$ Sei. Eng., 1994, 68, (2), pp. 799-808 TSAO, T.C., and TOMIZUKA, M.: ‘Robust adaptive and repetitive digital tracking control and application to a hydraulic servo for noncir- cularmachining’, AMSEJ Dyn. Syst., Meas. Control, 1994,116, (I) , pp. 24-32 TUNG, E..D., and TOMIZUKA, M.: ‘Feedforward tracking controller design for high-speed motion control’. Proceedings of IFAC 12th Triennial World Congress, Sydney, Australia, 1993, pp. 299-302 TSAO, T.C., and TOMIZUKA, Id.: ‘Adaptive zero phase error tracking algorithm for digital control’, AIZISEJ Dyn. Kvst., Meas. Control, 1987,

pp. 6-10

1993, pp. 547-550

1995, 61, (3), pp. 695-718

109, (2), pp. 349-354 GROSS, E., TOMIZUKA, M., and MESSNER, W.: ‘Cancellation of discrete time unstable zeros by feedforward control’, A S M E 1 Dvn. Svst.,

~~

Meas. Control, 1994, 116, (i), pp. 33-38 SANOFF, S.P., and WELLSTEAD, P.E.: ‘Comments on: implementation of selftuning regulators with variable forgetting factors’, Automatica, 1983, 19, (3), p. 346 LOZANO-LEAL, R.: ‘Convergence analysis of recursive identification algorithm with forgetting factor’, Automatica, 1983, 19, (I) , pp. 95-97

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