adaptive tracking control by system inversion.pdf

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Proceedings of the American Control ConferenceSan Diego, California June 1999

Adaptive Tracking Control by System Inversion

Zongxuan Sun and Tsu-Chin Tsao

Department of Mechanical And Industrial EngineeringUniversity of Illinois at Urbana-ChampaignUrbana IL 61801, USA

Abstract

This paper presents a discrete time adaptive inversionscheme for linear systems and its usage in adaptive feedforwardand feedback controllers. Two parameter adaptation algorithms(PAA) were used in the proposed schem es. The firstPA A appliesthe extended bias-eliminating least-squares (EBELS) algorithm forplant estimation to ensure convergence toa tuned model undercolored noise and unmodeled dynamics. The secondPA A appliesEBELS or least mean squares (LMS), respectively for twoadaptive inversion schemes, to estimate a finite impulse response(FIR) plant inve rse filter.A feedforward and a repetitive feedbackcontrol algorithms are designed respectively by including theadaptive inverse filter.

1. Introduction

Most feedforward and some feedback tracking controldesigns, e.g. repetitive control, involve the determination of astable inverse of plant dynamics. The tracking controlperformance relieson the accuracy of the plant inversion, which inturns dependson the accuracy of the plant model estimation andthe plant inversion method. Adaptive inversion isa usefultechnique to obtain an accurate inverse model, which is lesssensitive to plant uncertainties and v ariations but also adjusts itselfto plant parameter changes.

canceller using adaptive inverse control in a feedback loop.However, there is a lack of stability analysis when systemsexperience colored noise and unmodeled dynamics. When theplant parameter estimation is biased, the adaptive inversecontroller may not be accurateany more.

When the bound of the noise is too large for the relativedead zone approach to be effectiveor the colored noise makes theLMS scheme invalid, the above mentioned adaptive inversionmethods cannot be applied. However, if the persistent excitationcondition is met, recent results on eliminating parameterestimation bias for input/output signals corrupted by coloredbounded noise( Feng and Zhang 1995, Zhang and Feng 1997) canbe utilized to ensure parameter convergence to a tuned model.This paper treats the adap tive feedforward and ad aptive repetitivecontrol problems using the extended bias-eliminating least-squares(EBELS) algorithm to solve the bias problem associated withbounded colored noise. Stability and robustness analysis arepresented and discussed along with simulation results.

2. System Model Description

Consider the system model as follows:

(2)

(3)

(4)

In indirect adaptive inversion schemes, a plant model is

line computation of plant inversion at every time step may be quite

B ( q - ' )

A w l )irst estimated and the plant inversion is either calculated or

estimated by a second parameter estimation scheme. Since theon-

computationally intensive, Tsao and Tom izuka (1988, 19 89, 1994)proposed the use of the zero phase error filter, which iscomputationally simple and efficient, as the approximation ofdelayed inverse plant in the indirect adaptive feedforward trackingcontroller, projectjon parameter adaptation schemes withfiltering, dead zones, and parameter space constraints were delay operator.employed in the PA A to achieve robust adaptive plant estimation

where the tuned is 0 =

A ( q - ) = 1 qq- ... anq-nB ( q - ' ) = b +qq-1 . bmq-A ( q - ' ) and B ( q - ' ) are coprime. q ( k ) and d ( k ) denotes the

unmodeled dynamics and colored noise respectively.q- is the

.

against bounded noise and unmodeled dynamics without assumingpersistency excitation condition.Sun and Tsao (1997) furtherapplied similar method in indirect adaptive feedback control,where an internal model wasalso included for asy mptotic trackingperformance. In particular, the adaptive zero ph ase error inversionwas used in adaptive repetitive control. The system stability andthe relations of the tracking error to the plant estimation error were

and the adaptive repetitive control schemes.

Remark 1: In system model I ) , the unmodeled dynamics~ ( k )can be mult ipl icat ive,additive or both, Note the bound for

dynamics will vary corresponding to the tuned modelstructure one chooses.In many tracking control applications, theupper bound for unmodeled dynamics can be determinedexperimentally Once the tunedmodel is fixed, so by choosing agood tuned model, the upper bound of unmodeled dynamics can

once the hardware of a system is fixed.provided for both the adaptive feedforward t racking be suppressed, Generally one can't suppress the boundfor noise

The idea of using a second PA A to estimate delayedplant inverse can mostly be found in the adaptive inverse controlproposed by Widrow and his associa tes. Widrow intro duced theadaptive inverse feedforward control (1985, 1986) using LeastMean Squares adaptation scheme. Widrow and Walach (1996)further discussed the Filtered-X algorithm and disturbance

Assumption stem I ) is astable,

Remark 2: We assum e the original open loop system is stable orcan be stabilized by a fixed cont ro~~er, his is a general in

system, i.e.A ( q - ' ) s strictlyis bounded by U and some constant.

0-7803-4990-6/99 10.00 1999 AACC 29


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many tracking control applications. Also note there is noassumptions made on the zero dynamics of the system model.

3. Param eter Adapta t ion Algor i thm

Our goal of plant estimation is to identify a tuned m odelinstead of the true model because of the existence of unmodeleddynamics. With both unmodeled dynamics and colored noise insystem model I ) , the tuned model will be biased if we useordinary least squares. Recent results on eliminating estimationbias for input output signals corrupted by colored bounded noiseFeng and Zhang 1995, Zhang and Feng 1997) can be utilized forplant estimation to ensure convergence to the tuned model. Thebasic idea behind the extended bias-eliminating least-squares(EBELS) algorithm is that by inserting some known parts into theestimated system, we can asymptotically estimate the bias causedby noise and then eliminate it. If we take the unmodeled dynamicsq ( k ) as a kind of noise which is correlated with both input andoutput in the identification process, we can then eliminate its biaseffect together with the bias caused by colored noise by usingEBELS algorithm.

Assumpt ion 2: Persistent excitation is ensured in the adaptivesystem.

By the properties of the extended bias eliminating least squaresalgorithm, we have following lemma:

h m m a 1: lim 8 ( N ) = 8 . w.p.1.Proof: See Zhang and Feng (1997).

N -

Remark 3: In the identification process, we take the unmodeleddynamics q ( k ) as a kind of noise because in most applications, itis impossible to distinguish unmodeled dynamics and noise fromexperimental data. Generally q ( k ) is correlated with u ( k ) , sonow both input and output are corrupted by noise. The ExtendedBias Eliminating Least Squares (EBELS) ensures an accurateestimate of the tuned model, i.e. G,,. Also notice that it can

be used in either open loop o r closedloop because this estimationalgorithm doesn't have any requirements on the noise model.

4. Adapt ive Feedforward Trackin g Cont ro l

Widrow and Walach (1996) proposed an adaptive

feedforward inverse control scheme (as shown in Fig.1 withF = I . Filtered-X algorithm was used to online adapt controllerparameters based on the estimated plant model. With nounmodeled dynamics and noise, the adaptive feedforward inversecontrol can achieve perfect tracking performance. For the case thatFigure 1 is at 'a' position, the estimated plant and therefore theadaptive inverse controller may be biased if there exist unmodeleddynamics or colored noise. Another approach is to put the switchat b position as shown in Fig.1 . In this case, the LMS adaptation

gain may need to be very small to ensure convergence. One wayto alleviate this problem is to introduce a low pass filter F in theFilterd-X algorithm. The Modified Filterd-X algorithm is

C ( k = C ( k )+ 2 a e ( k ) F i ( k ) 8)where

i ( k ) = [ ? ( k ) , ? ( k - l ) , . . . , ? ( k - M + l ) y, ? ( k ) = @ y r e f k ),

is the error signal as shown in Fig.1 a is the adaptation gain,F is a low pass filter, and the adaptive inverse controllerC is a

FIR filter with length .

Theorem : The proposed adaptive feedforward tracking controlscheme using EBELS and Modified Filterd-X algorithm canguarantee system stability and the adaptive controller willconverge to the inverse m odel of the original plant.

Proof: Suppose the inverse model of the original plantP is C* ,as shown in Fig. 1 (with switch at position b ), we have

where

E k )= (c - &k)T X k )- ( k ),

X ( k ) = [ x ( k )x ( k - l ) , . . . , x ( k - M + Z ) r , x ( k ) = pYref k )

& ( k ) - V T ( k ) X ( k ) - d ( k ) ,

& ( k )= - X T ( k ) V ( k ) - d ( k ) ,

V ( k+ = V ( k ) 2 U & ( k ) F i ( k )V ( k+ l ) = V ( k ) + 2 a ( - X T ( k) V ( k )- d ( k ) ) F i ( k )

V ( k+1) = I - a F i ( k ) X T k ) )V ( k )- a d ( k ) F i ( k )

where ~ ( k )&k) - c*(k)

Subtracting C* rom both sidesof Q 8), we get

Assume d ( k ) s not correlated w ith x , hen

E[V(k ] = E [ V ( k ) ] - 2 a E [ F i ( k ) X ( k ) ] E[ V ( k )]E[V(k+ 1 ] =E[V(k) ] -2aR E[V(k) ] ,

where R = E [ F Z ( k ) X ' ( k ) ]

we transform R into normal form-

E [ V ( k ) ]= TI z- u i p E [ V ( k ) ]E [ V. ( k + l ) ] = z - 2 a i ) E b ' ( k ) ] ,

E [ V ( k ) ]= F z- u; i rF- 'E[V(O)]

where V ' ( k )= T -'V(k)

E [ V ' ( k + l ) ] = ( Z - 2 a ; ir E [ V S ( O ) ]

1

we choose th e adaptation gain0 < a < where A,,, is the

maximum eigenvalue of R (also the maximum eigenvalue of ),then

lim I - 2 a L ) k + 0 , lim ~ ( k )-f c* .

Because C i s a FIR filter and it converges, stability of theadaptive system is guaranteed.

ax-

lim E [ v ( ~ ) ] 0 .k-+- k - 1 - k+-

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: lim y ( k )= y ( k - c ) + d k )k+-

Remark :

We ch oose the adaptive inverse controllerC as a FIR filter

so that it won't be biased as long as inputX is not correlated

with noise.If the structureof the tuned model G,, is carefully cho sen,P

may be very close to P (since i --f G,, by EB ELS), then kis nearly symmetric. We also kno w that unmodeled dynamicsare mostly concentrated at high frequencies, therefore by

appropriately filtering signalx , we can guarantee that k isalmost symmetricso that it is easy to determineh .Note in the proof, we have used the independenceassumption for theLMS algorithm. Also the convergence ofvariance can be guaranteed by carefully choosing theadaptation gain.

5. Adaptive Repetitive Control for Tracking Periodic Signals

To track or reject periodic signals, Tomizuka et. al.(1989) proposed a discrete time repetitive control scheme whichrequires the inverse plant model. In this section, we are going toapply the adaptive inverse control to the repetitive controlstructure to track or reject periodic signals.

5.1 Adaptive Repetitive Control Scheme I

The closed loop adaptive repetitive control system isshown in Fig.2 (with switch at position 'a' and F = I ) ,Uncertainties of the adaptive system are mainly from two sources.One is the unmodeled dynamics of the original plant; the other isdue to the plant inversion error which may rise because of limitedlength of the FIR filter and the adaptation time. Both uncertaintiescould deteriorate the performance and stability of the adaptiveclosed loop system.To enhance system robustness, the repetitivecontrol law is designed as follows:

u ( k ) = C ( k ) - (9)

where Q is a low pass filter, L is the period of the periodic

signals and C is som e positive constant smaller thanL .

The adaptation law fo r inverse controller is as follows:

t ( k +1) = ( k ) 2Cz&(k)X(k) (10)

where X ( k ) = [ x ( k ) , x ( k - l ) ; . . , x ( k - M + l ) ] T,x ( k ) = k@(k)is white noise, is the error signal andU is the adaptation

gain.

Theorem 2: The closed loo p adaptive system is BIBO stable. Byappropriately choosing Q , the closed loop transfer function will

converge closely to a pure delay.

Proof: Assume the original plant isp = 1 + A , ) , the closedloop characteristic polynomial be comes:

= 1- Q z - ~ Q z - ~ + ~ P 1 - Q z - ~ Q Z - ~ + ~ F ( ~ A, )Denote kF = z- (l + A , ) , where A, is due to the plant inversionerror.Then = l - Q z - L + Q z - ( O + A ~ X I + A , )

so we can design Q such that

IIQ(A1 + A 2 + A i d , ) I< 1

Thus I z k 1 , also C converges, then the closed loop systemstability canbe guaranteed (see Goodwin, et. al.,1986).

5.2 Adaptive Repetitive Control Scheme II

Adaptive repetitive control scheme I is conservative

because the adaptive inverse controllerC only approaches theinverse of estimated plant modelk instead of P So we need todesign Q to improve system robustness. Thisis the trade offbetween system robustness and performance. In this section, weare going to present an unbiased adaptive repetitive controlscheme which will result in better performance.

The closed loop ad aptive system is shown in Fig.2 (with switch atposition b). Th e repetitive control law is a s follows:

,.The goal of the adaptation law fo rC is to approach the inverse ofP . As shown in Fig. 2 (with switch at position 'b'), we closed theloop for inverse controller parameter adaptation. Now we use theactual output to adjust the controller parameters which willminimize tracking error variance. Since we have two closed loo ps

in the system, during the adaptation ford oth input X andoutpu t will be corrupted by noise and unmodeled dynamics.

Th e extended bias eliminatin g least squares is used again to get

the unbiased inverse controllerC * .Theorem 3: The closed loop adaptive system is BIBO stable. Theclosed loop transfer function will converge to a pure delay, i.e.lim y ( k )= y ( k - 5 +d ( k )

k+-

Proof : From Fig. 2 (with switch at position 'b'), we know thereference model is

j q k )= Fq-c i i (k )

j q k )= F f C * i i ( k )

j q k ) = FG,,C*ii(k) F q ( k )

where C* = 4-c is the exact inverse model of P , F is a lowpass filter to be designed such thatF q ( k ) -+ 0 .

P

The actual output 7 isj q k ) = FP&(k) F d ( k )

X k )= FG,,&(k) F q k ) + F d ( k )

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where G o is the tuned model. state errors (from 1 to 4) are 0.12, 0.07, 0.03 and 0.02respectively.

In the closed loop adaptive system, both i ( k ) and y ( k ) arecorrupted by noise. By applying the extended bias-eliminating

least-squares (EBELS) algorithm to the adaptation of e ,convergence to the tuned model is guaranteed.

: lim FG,k + FG,,C*, then we can conclude that

lim e + C* . The closed loop transfer function will converge toK -

k m

a pure delay q - L , so system stability is guaranteed and

lim y ( k )= y ref ( k - ) d ( k )k - w

Remark 5: Note the proposed ad aptive repetitive schemes can notonly guarantee perfect tracking performance for periodic signalsdue to the internal model principle but also provide good trackingperformance for non-periodic signals since the closed looptransfer function will converge to a pure d elay.

6. Simulation Results

The plant model for simulations is

P = 1oq-1-2.0q-2 which have unstable zeros at 2 and

infinity. Colored noise d ( k )= O . l w ( k )+ 0.05w(k- ) ( wherew ( k ) is white noise with variance1.0) was added to the output

during plant identification. A periodic siganl was used as referencesignal to test both adaptive feedforward and adaptive repetitivecontrol schemes. One period of the reference signal and therandom colored noise is show n in Fig.3

1 8q-+ 0.9q-2

For the plant p = 1.0q-1-2.0q-2 , we run the1- 3q-I 0.9q-2

EBELS and ordinary least squares10 times and get an averagedestimate P respectively.

For EBELS:A 1.0398q- - .0290q-2P =

1- .7951q- 0.8939q-2

0.9989q- - 1509q-21- .9555q- 0.2433q-2

For ordinary least squares: @=

We can see ordinary least squares estimateis seriously biasedwhile EBELS gives a very accurate estimate.

Since plant P has an unstable zero at2, we chose a29th order FIR filter as the inverse controller,i.e.

e = C o + C , q - + - - + C 2 9 q - 2 9

Denote C * as the opt imal inverse of P , then the inversion

l e k -c*k Ierrors are *loo , k =0,1, . . -29. Fig. 4 and 5

Ic*k

7. Conclusion

Adaptive tracking control schemes by system inversionusing extended bias-eliminating least-squares (EBLES) is

presented in this paper. Modified Filtered-X algorithm is proposedto compensate unmodeled dynamics in feedforward loop toachieve better tracking performance. The adaptive inversecontroller is then applied to the repetitive control structure to trackor reject periodic signals. Two adaptive repetitive control schemesare presented and compared. Simulation results showed theeffectiveness of the proposed adaptive tracking schemes.

Reference:

Widrow, B., Walach, E., 1996, Adaptive Inverse Control,Prentice Hall.

Zhang, Y., Feng, C., 1997, Unbiased Parameter estimation ofLinear Systems with Colored Noise, Automatica, Vol. 33, No.5,pp. 969-973.

Feng, C., Zhang, Y., 1995, Unbiased Identification of Systemswith Nonparametric Uncertainty, IEEE Trans. Automat. Contr.,Vol. 40, NO.5 pp. 933-936.

Zongxuan Sun , Tsu-Chin Tsao, 1997, Adaptive RepetitiveControl Design with Its applications to an ElectrohydraulicServo, 1997 ASME International Mechanical EngineeringCongress and Exposition, Fluid Power System and Technology,pp.39-53.

Tomizuka, M., Tsao, T.C., Chew, K.K., 1989, Analysis andSynthesis of Discrete-Time Repetitive Controllers, ASMEJournal of Dynamics, Measurement and C ontrol, vol.11 1, pp.353-358.

Goodwin, G. C., and Lozano-Leal, R., Mayne, D.Q.. andMiddleton, R. H. 986, Reapprochement between Continuousand Discrete Model Reference Adaptive Control, Automatica,Vol. 2, NO.2, pp. 199-207.

Tsao, T.-C., Tom izuka, M., 1994, Robust adaptive and repetitivedigital control and application to hydraulic servo for noncircularmachining ASME Journal of dynamic system, measurement andcontrol, Vol. 116, pp. 24-32.

show the tracking errors and plant inversion errors of adaptiveinverse feedforward control using filtered-X algorithm, adaptivefeedforward tracking control using m odified filtered-X algorithm,adaptive repetitive control scheme I and adaptive repetitivecontrol scheme I1 respectively. In Fig. 4, the maximum steady

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Reference Signal

: . . . . . . . T Istimation . .a

M ode l i ngignalILL -'TK+, ?

Fig. 1 Block Diagram of Adaptive Inverse Feedforward Control

Es t imat ion

M ode l i ngSignal

Fig. 2 Block Diagram of Adaptive Repetitive Control Schemes

0 50 100 150 200 25 300

Noise Signal b, Identification

-4'

0 5 100 150 200 250 300Count

v.

Fig. 3 Reference Sig nal and Noise in Identification process

Tracking E m rs o r A da p t ive Fe e d fow rd and Repetitive Control

:-1

200 400 600 800 1000l o

-1

-1 6 8 lo

: 2A

200- ' 6 8 1COO0 ZOO 400 6 800 1000

count

-1

Fig. 4 Tracking errors (from 1 to 4) for adaptive inversefeedforward control using filtered-X algorithm, adaptivefeedforward tracking control using modified filtered-X algorithm,adaptive repetitive control scheme I and adaptive repetitivecontrol sc hem e I1 respectively.

lnvers ion Errors for Adaptive Feed forw ard and Repeti t ive Control-2000F A . 1

& 10000 .

2000

& 10000 .

2000

2000

$ 1 0 0 0

0 5 10 1 5 20 2 5 30 Index for the Inverse Control ler Param eters

Fig. 5 Inversion errors (from 1 to 4) for adaptive inversefeedforward control using filtered-X algorithm, adaptivefeedforward tracking control using modified filtered-X algorithm,adaptive repetitive control scheme I and adaptive repetitivecontrol schemeI1 respectively.

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adaptive tracking control by system inversion.pdf

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