equivalence between adaptive feedforward cancellation and disturbance rejection using the internal...

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2010; 24:211–218Published online 1 April 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/acs.1117

Equivalence between adaptive feedforward cancellation and disturbancerejection using the internal model principle

Xiuyan Guo1 and Marc Bodson2,∗,†

1QSecure, 333 Distel Cir, Los Altos, CA 94022, U.S.A.2Department of Electrical and Computer Engineering, University of Utah, 50 S Central Campus Dr Rm 3280,

Salt Lake City, UT 84112, U.S.A.

SUMMARY

The paper shows that a large class of adaptive algorithms for disturbance cancellation yields control systems that areequivalent to compensators implementing the internal model principle (IMP). The fact has been known to be true forperiodic disturbances with fixed frequency. However, the paper shows that the result can be extended to disturbances oftime-varying frequency (i.e. frequency-modulated signals), regardless of the rate of variation of the frequency. In particular,several adaptive controllers are shown to be equivalent to linear time-varying compensators implementing the IMP. Further,a pseudo-gradient algorithm produces the same responses as a polytopic linear parameter-varying compensator. Copyrightq 2009 John Wiley & Sons, Ltd.

Received 12 July 2008; Revised 23 November 2008; Accepted 19 January 2009

KEY WORDS: adaptive control; augmented error; adaptive feedforward cancellation; periodic disturbances; internal modelprinciple; linear parameter-varying systems

1. INTRODUCTION

The paper considers the problem of rejecting sinusoidaldisturbances with time-varying frequency, i.e. signalsof the form

d(t)=�∗c cos(�d(t))+�∗

s sin(�d(t)), �d(t)=�d(t) (1)

∗Correspondence to: Marc Bodson, Department of Electrical andComputer Engineering, University of Utah, 50 S Central CampusDr Rm 3280, Salt Lake City, UT 84112, U.S.A.

†E-mail: [email protected]

Contract/grant sponsor: National Science Foundation; contract/grant number: ECS0115070

The definition of the disturbance is comparable to thedefinition of a frequency-modulated signal in commu-nication systems. The time-varying frequency �d(t)is called the instantaneous frequency of the signald(t). �d(t) is assumed to be bounded and known, andthe parameters �∗

c and �∗s are assumed to be constant

and unknown. Although the parameters �∗c and �∗

smay vary in some applications at the same rate as thefrequency, there is a class of problems where theycan be considered constant or slowly-varying. Theseproblems include those associated with eccentricityand non-circularity of rotating devices. Eccentricitycompensation is discussed in [1]. In compact disc

Copyright q 2009 John Wiley & Sons, Ltd.

212 X. GUO AND M. BODSON

players, the frequency of rotation decreases as musicis being played, so that eccentricity causes a periodicdisturbance of varying frequency [2]. Another examplearises in web transport systems, such as the papermachine considered in [3]. Experiments on the testbedof [3] showed that the spectrum of the web tension hadlarge components at the frequencies of rotation of thewinding and unwinding rolls, due to the eccentricityand the non-circularity of the rolls. The frequenciesvaried as paper was transferred from one roll to anotherat high speed.

The paper considers a class of adaptive feedforwardcancellation algorithms that can be applied to rejectsuch frequency-modulated disturbances. The mainresult of the paper is that the algorithms are exactlyequivalent to a set of compensators implementingthe internal model principle. The equivalence is verygeneral and, in the case of several standard adaptivealgorithms, it yields linear time-varying compensatorsimplementing the internal model principle (IMP). Inter-estingly, the proof of equivalence for the time-varyingfrequency case is no more complicated than the earlierproof that assumed a fixed frequency and could notbe extended to the time-varying case. Further, thenew proof is more general, by accounting exactly forarbitrary initial conditions. The equivalence providesan opportunity to apply knowledge gained either fromadaptive control or from robust linear control to theother field. For example, the robustness of adaptivesystems can be assessed in previously unknown waysusing the equivalent linear system. On the other hand,the adaptive theory provides interesting control algo-rithms, such as those based on an augmented error(AE) and on least-squares algorithms, which would notintuitively be found by using a linear control approachbased on the internal model principle. The adaptiveimplementation also enables the direct use of angularmeasurements, without reconstruction of the instanta-neous frequency of the disturbance. Although omittedfrom the paper for brevity and clarity, extension of theresult to disturbances with multiple sinusoidal distur-bances is straightforward, and yields compensatorswith multiple parallel paths, each similar to the singlesinusoid case. The paper concludes with examplesdesigned to illustrate the application of the theoreticalresults.

2. ADAPTIVE FEEDFORWARDCANCELLATION (AFC)

We consider the disturbance rejection problem for asystem described by

y(t)= p(t)∗(u(t)−d(t)) (2)

where ∗ denotes the convolution operation and p(t)is the impulse response of the system, or plant. Theplant is assumed to be linear time-invariant with transferfunction P(s). The signals y(t), u(t), and d(t) arethe plant output, the control input, and the disturbancesignal, respectively. The plant will be assumed to bestable for the application of the AFC scheme (not forthe equivalence result). If the plant is unstable, a stabi-lizing controller can be designed and the techniquescan be applied to the closed-loop system. The goal ofthe control system is to generate a control input u(t)such that y(t)→0 as t→∞.

Given a disturbance of the form (1), the control inputcan be chosen to be

u(t)=�c(t)cos(�d(t))+�s(t)sin(�d(t)) (3)

so that disturbance cancellation could, in theory, beachieved exactly by letting �c(t)=�∗

c , �s(t)=�∗s . As

the nominal parameters are unknown, the controlstrategy is to use adaptation so that the parametersconverge to values such that the disturbance is rejected.

Let the nominal and adaptive parameter vectors be

�∗ =(�∗c �∗

s )T, �(t)=(�c(t) �s(t))

T (4)

The regressor vector is defined to be

w(t)=(cos(�d(t)) sin(�d(t)))T (5)

Therefore,

d(t)=wT(t)�∗, u(t)=wT(t)�(t) (6)

and

y(t)=p(t)∗(u(t)−d(t))=p(t)∗(wT(t)(�(t)−�∗)) (7)

Equation (7) falls into the framework of adaptivecontrol theory [4], and several algorithms are availableto update the adaptive parameters. Commonly usedadaptive algorithms include the pseudo-gradient,

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2010; 24:211–218DOI: 10.1002/acs

EQUIVALENCE BETWEEN AFC AND DISTURBANCE REJECTION 213

gradient and AE algorithms. We briefly review thesealgorithms.

2.1. AFC with pseudo-gradient algorithm

The pseudo-gradient algorithm is simply given by

�(t)=−gw(t)y(t) (8)

where g>0 is the adaptation gain. If the plant transferfunction was the identity, this would be a gradient algo-rithm for the problem, hence the name pseudo-gradient.Adaptive control theory indicates that the Lyapunovstability of the overall system and convergence of theerror y(t) to zero is ensured if the plant transfer func-tion P(s) is strictly positive real (SPR) [4]. However,the SPR condition is rarely, if ever, satisfied in practice.

2.2. AFC with gradient algorithm

The adaptive parameters in a gradient algorithm areupdated according to

�(t)=−gwF(t)y(t) (9)

wF(t)= p(t)∗w(t) (10)

where p(t) is an estimate of the plant impulse responsep(t) and wF(t) is called the filtered regressor vector.Note that the algorithm follows the direction of thegradient under the assumption of slowly varying param-eters. It is known that the gradient algorithm is gener-ally not stable for all adaptation gains g>0. However,under the condition that p(t)= p(t), the stability of thegradient algorithm can be proved using an averaginganalysis for small gain [5].2.3. AFC with an AE algorithm

Instead of using y(t) to update the adaptive parame-ters in the gradient algorithm as shown in (9), the AEalgorithm is obtained by using an AE

ea(t)= y(t)+wTF (t)�(t)−uF (t) (11)

where

uF (t)= p(t)∗u(t) (12)

and wF(t) is the filtered regressor vector defined in(10). The AE algorithm is stable for all g>0 and theerror converges to zero, assuming that the plant is

exactly known (without requiring an SPR conditionon the plant). Exponential convergence of the adaptiveparameters is achieved if wF(t) is persistently exciting(PE) [6].

3. EQUIVALENCE BETWEEN AFCALGORITHMS AND LTV COMPENSATORS

IMPLEMENTING THE IMP

The main result of this paper is the equivalence betweenthe AFC algorithms and time-varying compensatorsimplementing the internal model principle. The equiv-alence is shown graphically in Figure 1, where R(t) isa matrix specified in the following fact.Fact: let an AFC algorithm be given by

u(t)=wT(t)�(t)

w(t)=(cos(�d(t)) sin(�d(t)))T

with arbitrary update law

�(t)=F(y(·),w(·)), �(0)=�0

where F(y(·),w(·)) is some causal operator describingthe adaptation mechanism. Then, the operator from �(t)to u(t) is equivalent to a linear time-varying systemdescribed by the state–space realization

x(t)= Ad(t)x(t)+R(t)F(y(·),w(·)), x(0)= R(0)�0

u(t)=Cdx(t)

1s

cos( )d

sin( )d

1s

d(t)

u(t).

.

R(t)z(t) x(t)= 0

d(t)d(t)0

-.

x(t) + z(t)

u(t)=( 0 1)x(t)

u(t)

wd

Figure 1. General LTV IMP equivalence of an AFCalgorithm.



where

Ad(t)=[

0 �d(t)

−�d(t) 0

], Cd =[0 1]

R(t) =[sin(�d(t)) −cos(�d(t))

cos(�d(t)) sin(�d(t))

]

�d(t) = d�d(t)

dt

ProofThe proof uses the linear transformation x(t)=R(t)�(t), which is well defined for all t�0, andits inverse R−1(t), which is also well defined sinceR−1(t)= RT(t). Given

R(t)=�d(t)

[cos�d(t) sin(�d(t))

−sin(�d(t)) cos(�d(t))

]

it follows that

x(t)= d

dt(R(t)�(t))= R(t)R−1(t)x(t)+R(t)�(t)

u(t)=wT(t)�(t)=wT(t)R−1(t)x(t)

The fact is obtained by checking that

R(t)R−1(t)= Ad(t), wT(t)R−1(t)=Cd �

The model of the disturbance obtained in the abovefact is

x(t)= Ad(t)x(t), d(t)=Cdx(t) (13)

and it can be easily shown that the disturbance in(1) is the solution of the nonautonomous system (13)with initial conditions x(0)=[−�∗

s �∗c ]T. In other

words, the state–space system (13) is a model of thefrequency-modulated disturbance, as required by theinternal model principle [7].

The result of the above fact was known to be truefor some time in the fixed frequency case [8], andinteresting consequences were observed [6, 9, 10].Preliminary discussion of the time-varying case wasfirst presented in [11].

3.1. LTV IMP equivalence for pseudo-gradientalgorithm

We now turn to the specialization of the results tothe AFC algorithms that were presented earlier. Thepseudo-gradient algorithm described by (6) and (8) isequivalent to the LTV controller �(t) with state–spacerealization

x(t) = Ad(t)x(t)−[0 g]Ty(t)u(t) = Cdx(t)

(14)

The result can be checked from the fact, given that

F(y(·),w(·))=−gw(t)y(t), R(t)w(t)=[0 1]T

The equivalence can be used for a number ofpurposes. Stability properties of the adaptive algorithmcan be determined for plants that are not SPR. In theAppendix, it is shown that, because the parameter �d(t)appears linearly in the LTV equivalent controller (14),the closed-loop system belongs to the class of poly-topic linear parameter-varying (PLPV) systems. Suchsystems are nonautonomous LPV systems such that

x(t)= A(�(t))x(t)

where A(·) is an affine function of a time-varyingparameter �(t) and �(t) varies in a polytopic set � withvertices �1 �2, . . . ,�N , that is

� :=Co{�1,�2, . . . ,�N }

={

N∑i=1

�i (t)�i :�i (t)�0,N∑i=1

�i (t)=1

}

Stability of the closed-loop adaptive system can beensured if quadratic stability of the PLPV system canbe proved. In turn, this property can be guaranteed ifthere exists a single positive definite matrix P such that

ATi P+PAi<0, i=1,2

for two matrices given in the Appendix. Such equationcan be solved using a linear matrix inequality (LMI)solver [12].

If the controller is designed so that the closed-loopsystem is a uniformly exponentially stable lineartime-varying system, the trajectories of the systemwill always be bounded, even if the frequency of the



disturbance is not known exactly (since the disturbancesignal is bounded). In the time-invariant case [9], it hasbeen found that the parameters of the adaptive systemvaried at a frequency equal to the difference betweenthe algorithm and disturbance frequencies. A similarresult is expected to hold in the time-varying case,although the tracking error may be as conveniently orprecisely quantified as is possible in the time-invariantcase.

3.2. LTV IMP equivalence for gradient algorithm

The AFCwith the gradient algorithm in (9) is equivalentto an LTV controller �(t) with

x(t)= Ad(t)x(t)−gB(t)y(t), u(t)=Cdx(t) (15)

where

B(t)= R(t)wF(t) (16)

It should be noted that the LTV equivalent controller in(15) is not an LPV system any more, since B(t) is notlinearly dependent on �d(t). Therefore, the ability touse linear time-varying system theory results is consid-erably reduced.

3.3. LTV IMP equivalence for AE algorithm

The adaptive system with AE algorithm is equivalentto the LTV system �(t) with

x(t) = Ac(t)x(t)+Bc(t)(y(t)−uF (t)), u(t)=Cdx(t)

(17)

where uF (t) and B(t) were defined in (12), (16), and

Ac(t)= Ad(t)−gB(t)BT(t), Bc(t)=−gB(t) (18)

3.4. Application of the results

The results enable the transfer of knowledge betweenthe areas of adaptive systems and robust linear control.From the adaptive system implementation, one gains anumber of algorithms that would not intuitively emergefrom linear control theory. In particular, the AE struc-ture provides an algorithm that is guaranteed to bestable (Lyapunov stable, in general, and exponentiallystable under PE conditions) no matter what the range

P (s)y(t)

w (s)I

^-

u(t)

d(t)

y (t)

(t)

y(t)^u (t)

Figure 2. Robust stability of an AFC algorithm with multi-plicative plant uncertainty.

and rate of variation of the frequency parameter are[6]. Least-squares algorithm can also replace gradientalgorithms, yielding an additional set of time-varyingcompensators unlikely to be obtained by other means.The adaptive implementation of the algorithms is alsouseful in cases where the angle �d can be directlymeasured, such as through the use of an encoder on thedevice that causes the disturbance. The AFC algorithmsdo not require the frequency to be computed from theangular measurements.

On the other hand, the equivalence to linearcontrollers enables one to obtain robustness estimatesthat would normally not be available from adaptivecontrol theory. A model of plant uncertainty is themultiplicative uncertainty

P(s)= P(s)(1+wI (s)�(s)) (19)

where wI (s) is some known stable transfer func-tion and �(s) can be any stable transfer functionwhose magnitude is less than or equal to one in thefrequency domain (i.e. ‖�(j�)‖∞�1). The closed-loopLTV system including the uncertainty is shown inFigure 2, where �(t) is defined by (14), (15), or (17)for the pseudo-gradient, gradient, and AE algorithms,respectively. If the applicable vector w(t) or wF(t) ispersistently exciting, the nominal closed-loop system(i.e. with �(s)=0) is known to be exponentially stable.From the small gain theorem, robust stability of thesystem can be guaranteed if the root mean-square(RMS) gain from y�(t) to u�(t) is less than 1. Thisapproach will provide an estimate of the robustnessmargin, although it may be difficult to compute and



quite conservative. For the pseudo-gradient algorithm,the system from y�(t) to u�(t) is a PLPV system, sothat the robust performance can be evaluated using theLMI control toolbox [13].

4. SIMULATION RESULTS

In this section, we give an example of applicationof the results by considering the linear two-mass-spring-damper system of [12]. The system is shown inFigure 3. In Figure 3, d(t) is the external sinusoidaldisturbance force acting on the second mass, and theoutput of the system y(t) is the acceleration of thesecond mass, or x2(t). Here, we consider the casewhere the control signal (force) is u2(t) and acts on thesecond mass, i.e. the control signal and the disturbanceact at the same location.

The plant parameters are selected to be the sameas in [12] where m1=m2=1, k1=k2=100, c1=c2=1.Then, the corresponding plant is

P(s)= s2(s2+2s+200)

(s2+0.382s+38.2)(s2+2.618s+261.8)(20)

The objective is to design robust AFC algorithms withfast convergence rate to eliminate the effect of thedisturbance d(t). The frequency is assumed to varybetween �min and �max and d(t) has constant magni-tude equal to 1.

The only designed parameter in the AFC algorithmsis the adaptation gain g, which can be constant or madea function of the frequency. In general, the conver-gence rate of the algorithms is improved if the gaing is increased. Therefore, a simple design objective is

m1 m2

c1

k1

c2

k2

x1 x2

d(t)

u2(t)u1(t)

Figure 3. Two-mass-spring-damper system.

0 2 4 6 8-1.5

1

-0.5

0

0.5

1

1.5

Plan

t Out

put y

(t)

Time (s)

Figure 4. Plant output with pseudo-gradient algorithm.

to maximize the adaptation gain while maintaining acertain level of robustness to uncertainties in the plantmodel.

Here, we consider the pseudo-gradient algorithmto cancel a periodic disturbance with time-varyingfrequency decreasing linearly from �max=33rad/s to�min=21rad/s in the first 4 s of the simulation andincreasing the other way during the following 4 s. Notethat the plant transfer function is not SPR, so thatLyapunov analysis of the adaptive system would notsupport the use of the algorithm. However, the LTVequivalence with a limited frequency range enablesa design with guaranteed stability properties, furtheraccounting for possible unmodeled dynamics. A fixedadaptation gain will keep the adaptive system stable forany frequency in the stated range, any rate of changein instantaneous frequency, and any uncertainty withinthe bounds.

To illustrate the application of the theory, we arbi-trarily select a multiplicative plant uncertainty wI (s)=( 180s+0.1)/((1/(4×80))s+1). Note that a plant of the

form P(s)=(�/(s+�))P(s) fits the uncertainty modelfor ��75. Using the LMI control toolbox with thePLPV system, one finds that the largest adaptation gainthat will ensure robust stability in the presence of themultiplicative uncertainty is g=1.1. The RMS gainfrom y�(t) to u�(t) of Figure 2 is 0.9990 . The result ofa simulation with g=1.1 is shown in Figure 4, which



indicates that the disturbance is rejected well despitethe rapidly varying frequency and a non-SPR plant.The application of the LTV equivalence with the PLPVtheory yields a powerful result of robust stability of theadaptive algorithm in the presence of arbitrarily fastfrequency variations and uncertainty, which could notbe obtained from standard adaptive control theory.

Unfortunately, the LPV theory does not apply tothe other algorithms, which are not linearly parame-terized in the frequency. More unfortunate still is thefact that the gradient and AE algorithms exhibit, ingeneral, faster convergence and greater robustness prop-erties than the pseudo-gradient [10]. As an alternative,one may apply the LTI equivalence assuming that thefrequency varies slowly within some range. In this way,one may obtain, frequency by frequency, the optimaladaptation gain that will ensure robust stability. In prac-tical implementation, the values of g for intermediatefrequencies can be computed by linear interpolation.

5. CONCLUSIONS

Adaptive feedforward cancellation schemes wereconsidered for the rejection of a sinusoidal distur-bance with time-varying frequency. General adaptivesystems were shown to be equivalent to time-varyingcompensators incorporating the internal model prin-ciple, without any requirement of stability, zero initialconditions, or limited range or rate of variation of thefrequency parameter. For some standard adaptive algo-rithms, the time-varying compensators were found tobe linear. The framework enabled the evaluation of therobustness properties of the AFC algorithms in waysthat have not been possible without the equivalenceresults. Simulations illustrated the application of thetheoretical results.

APPENDIX

In this appendix, we show that if the time-varyingfrequency �d(t) varies in a range [�min,�max], theclosed-loop system with pseudo-gradient algorithm fitsthe class of PLPV systems. Letting

x(t)= x(t)−xss(t), u(t)=u(t)−d(t)

where xss(t) is defined as

xss(t)=[−�∗

s cos(�d(t))+�∗c sin(�d(t))

d(t)

]= R(t)�∗

Denote Bd =[0 g]T, so that

dx(t)

dt= Ad(t)x(t)−Bd y(t), u(t)=Cd x(t)

Then, given a state–space realization of the plant

x p(t) = Axp(t)+B(u(t)−d(t))

y(t) =Cxp(t)+D(u(t)−d(t))

the closed-loop system is described by⎛⎜⎜⎝dxp(t)

dtdx(t)

dt

⎞⎟⎟⎠=

[A BCd

−BdC Ad(t)−BdDCd

](xp(t)

x(t)

)

={

(1−�(t))

[A BCd

−BdC Ad,min−BdDCd

]

+�(t)

[A BCd

−BdC Ad,max−BdDCd

]}

×(xp(t)

x(t)

)

� {(1−�(t))A1+�(t)A2}(xp(t)

x(t)

)

where

�(t)= �(t)−�min

�max−�min

Ad,min=[

0 �min

−�min 0

]and

Ad,max=[

0 �max

−�max 0

]

Thus, the closed-loop system is a PLPV system withtwo vertices �min and �max.



ACKNOWLEDGEMENTS

This material is based upon work supported by the NationalScience Foundation under Grant No. ECS0115070.

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