Automatica 41 (2005) 563–574

www.elsevier.com/locate/automatica

Adaptive narrow band disturbance rejection applied to an activesuspension—an internal model principle approach�

Ioan Doré Landau∗, Aurelian Constantinescu1, Daniel ReyLaboratoire d’Automatique de Grenoble, ENSIEG, BP 46, 38402 Saint Martin d’Hères, France

Received 28 March 2003; received in revised form 11 March 2004; accepted 26 August 2004Available online 24 January 2005

Abstract

This paper presents a methodology for feedback adaptive control of active vibration systems in the presence of time varying unknownnarrow band disturbances. A direct adaptive control scheme based on the internal model principle and the use of the Youla–Kuceraparametrization is proposed. This approach is comparatively evaluated with respect to an indirect adaptive control scheme based on theestimation of the disturbance model. The comparative evaluation is done in real time on an active suspension system.� 2004 Elsevier Ltd. All rights reserved.

Keywords:Active vibration control; Active suspension; Adaptive control; Feedback control; Internal model principle

1. Introduction

One of the basic problems in active vibration control isthe attenuation (rejection) of narrow band disturbances ofunknown or varying frequency.2

Solutions for this important practical problem have beenproposed by the signal processing community and a num-ber of applications are reported (Elliott & Nelson, 1994;Elliott & Sutton, 1996; Beranek & Ver, 1992; Fuller,Elliott, & Nelson, 1995). However, these solutions (inspiredby Widrow’s technique for adaptive noise cancellation(Widrow & Stearns, 1985) ignore the possibilities offeredby feedback control systems and require an additional trans-ducer. The disadvantages of this approach are: (1) It requiresthe use of an additional transducer. (2) Difficult choice for

� The preliminary version of this paper has been presented at theIFAC World Congress, Barcelona, Spain, July 2002. This paper wasrecommended for publication in revised form by Associate Editor GangTao under the direction of Editor Robert R. Bitmead.

∗ Corresponding author. Fax:+33 4 76 82 6388.E-mail address:landau@lag.ensieg.inpg.fr(I.D. Landau).

1A. Constantinescu is now with the Ecole de Technologie Supérieure,Montréal, Canada, 2002.2 Disturbance with energy concentrated in a narrow band around anunknown or varying frequency.

0005-1098/$ - see front matter� 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2004.08.022

the location of this transducer. (3) It requires adaptation ofmany parameters.

To achieve the rejection of the disturbance (at leastasymptotically) without measuring it, a feedback solutioncan be considered. The common framework is the assump-tion that the narrow band disturbance is the result of a whitenoise or a Dirac impulse passed through the “model ofthe disturbance”.3 Several problems have been consideredwithin this framework leading to adaptive feedback con-trol solutions: (1) Unknown plant and disturbance model(Feng & Palaniswami, 1992). (2) Unknown plant modeland known disturbance (Sun & Tsao, 2000; Zhang, Mehta,Bitmead, & Johnson, 1998). (3) Known plant and unknowndisturbance model (Bodson & Douglas, 1997; Amara,Kabamba, & Ulsoy, 1999; Valentinotti, 2001; Marino,Santosuosso, & Tomei, 2003; Ding, 2003; Gouraud,Gugliemi, & Auger, 1997; Hillerstrom & Sternby, 1994).

The present paper belongs to the last category, since inthe context of active suspension the model of the plant canbe obtained by standard system identification and does not

3 Throughout the paper it is assumed that the order of the disturbancemodel is known but the parameters of the model are unknown (the ordercan be estimated from data if necessary).

564 I.D. Landau et al. / Automatica 41 (2005) 563–574

normally change during operation. The same situation isencountered in other application areas (Valentinotti, 2001;Amara et al., 1999). Among the various approaches con-sidered for solving this problem, one can mention: (1)Use of the internal model principle (Francis & Wonham,1976; Johnson, 1976; Bengtsson, 1977; Tsypkin, 1997;Valentinotti, 2001; Amara et al., 1999; Gouraud et al., 1997;Hillerstrom & Sternby, 1994) (2) Use of an observer forthe disturbance (Marino et al., 2003; Ding, 2003) (3) Useof the “phase-locked” loop structure (Bodson & Douglas,1997). Of course since the parameters of the disturbancemodel are unknown all these approaches lead to an adaptiveimplementation which can be ofdirect or indirect type.

From the user point of view and taking into account thetype of operation of existing adaptive disturbance compen-sation systems one has to consider two modes of operationof the adaptive schemes :

• Self-tuningoperation (the adaptation procedure starts ei-ther on demand or when the performance is unsatisfac-tory and the current controller is frozen during the esti-mation/computation of the new controller parameters).

• Adaptiveoperation (the adaptation is performed continu-ously and the controller is updated at each sampling).

The paper explores the use of the internal model prin-ciple for the rejection of time-varying unknown narrowband disturbances. The controller should incorporate themodel of the disturbance (Francis & Wonham, 1976;Johnson, 1976; Bengtsson, 1977; Tsypkin, 1997). Thereforethe rejection of unknown disturbances raises the problemof adapting the internal model of the controller and itsre-design in real-time.

One way for solving this problem is to try to estimate inreal time the model of the disturbance and re-compute thecontroller, which will incorporate the estimated model of thedisturbance (as a pre-specified element of the controller).While the disturbance is unknown and its model needs tobe estimated, one assumes that the model of the plant isknown (obtained for example by identification). The estima-tion of the disturbance model can be done by using standardparameter estimation algorithms (see for exampleLandau,M’Sirdi, & M’Saad, 1986; Ljung, 1999). This will lead toan indirect adaptive control scheme. This approach has beeninvestigated inBodson & Douglas (1997), Gouraud et al.(1997), Hillerstrom & Sternby (1994).

However, by considering the Youla–Kucera parametriza-tion of the controller (known also as theQ-parametrization)(seeFig. 1) it is possible to insert and adjust the internalmodel in the controller by adjusting the parameters of theQpolynomial. It comes out that in the presence of unknowndisturbances, it is possible to build a direct adaptive controlscheme where the parameters of theQ polynomial are di-rectly adapted in order to have the desired internal modelwithout recomputing the controller (polynomialsR0 andS0in Fig. 1 remain unchanged). The number of the controller

1 / S0

0

u(t) ++-

y(t)

Q

A

R

-

- +

Np / Dp

p (t)1

w(t)

δ (t)

Plant

Model Model

AdaptationAlgorithm^

Controller

q-dB / A

q-dB

Fig. 1. Direct adaptive control scheme for rejection of unknown distur-bances.

parameters to be directly adapted is roughly equal to thenumber of parameters of the denominator of the disturbancemodel. In other words, the size of the adaptation algorithmwill depend upon the complexity of the disturbance model.

This paper focuses on the development of the directfeedback adaptive control for the case of unknown andtime-varying frequency narrow band disturbances appliedto an active suspension. Specifically, variable frequencysinusoidal disturbances are considered. The direct adap-tive control scheme to be presented takes advantage of theYoula–Kucera parametrization for the computation of thecontroller. This algorithm takes its roots from an idea ofTsypkin (1991).4 A similar approach has been consideredin Valentinotti (2001)for an application to a chemical reac-tor but a theoretical analysis of the scheme is not provided.A related paper isAmara et al. (1999)where the applicationfield is the active noise control in an acoustic duct. However,the theoretical analysis is limited to a BIBO property of thescheme. In the present paper an analysis of the asymptoticproperties of the scheme in terms of disturbance rejectionand parameters convergence is provided. For evaluationpurposes (complexity and performance) an indirect adaptivecontrol scheme based on the Internal Model Principle hasbeen also developed.

The paper is organized as follows. In Section 2 the ac-tive suspension on which we shall test the algorithms ispresented. Section 3 is dedicated to a brief review of theplant, disturbance and controller representation as wellas of the Internal Model Principle. The direct adaptivecontrol scheme is presented in Section 4 and the corre-sponding stability analysis is given in Section 5. Section6 presents the results obtained in real time on the activesuspension. Some concluding remarks are given in Sec-tion 7. The indirect adaptive control scheme is presentedin Appendix A.

4 Note that the adaptive rejection of unknown disturbances using theYoula Kucera parametrization is not considered in the survey (Anderson,1998).

I.D. Landau et al. / Automatica 41 (2005) 563–574 565

1

23

4

primary force (disturbance)

elastomere cone

piston

residual force

mainchamber

hole

motor

support inertia chamber

controlleractuator

(piston position)

machine

Fig. 2. Active suspension system (scheme).

Fig. 3. Active suspension system (photo).

2. The active suspension

The structure of the system (the active suspension) usedin this paper is presented inFig. 2. Two photos of the sys-tem are presented inFig. 3 (Courtesy of Hutchinson Re-search Center and Laboratoire d’Automatique de Grenoble).It consists of the active suspension, a load, a shaker andthe components of the control scheme. The mechanical con-struction of the load is such that the vibrations produced bythe shaker, fixed to the ground, are transmitted to the upperside of the active suspension. The active suspension is basedon a hydraulic system allowing to reduce the over-pressureat the frequencies of the vibration modes of the suspension.Its components are: an elastomer cone (1) which marks themain chamber filled up with silicon oil, a secondary cham-ber (2) marked by a flexible membrane, a piston (3) attachedto a motor, an orifice (4) allowing the oil to pass betweenthe two chambers and a force sensor located between thesupport and the active suspension.

The controller will act upon the piston (through a poweramplifier) in order to reduce the residual force. The samplingfrequency(fs) is 800 Hz. The equivalent control scheme isshown inFig. 4. The system input,u(t) is the position of the

q B/A-dR/S

Secondary path

u(t) y(t)

+

+

-

Controller

q C/D-d 1

u (t)(disturbance)

p

Primary path

Residual force

p (t)1

Fig. 4. Block diagram of the active suspension system.

piston (seeFigs. 2and4), the outputy(t) is the residual forcemeasured by a force sensor. The transfer function(q−d1 C

D),

between the disturbance force,up, and the residual forcey(t)is calledprimary path. In our case (for testing purposes),the primary force is generated by a shaker controlled by asignal given by the computer. The transfer function(q−d B

A)

between the input of the systemu(t), and the residual forceis calledsecondary path. The input of the system being aposition and the output a force, the secondary path transferfunction has a double differentiator behavior.

The control objective is to reject the effect of unknownnarrow band disturbances on the output of the system (resid-ual force), i.e. to attenuate the vibrations transmitted fromthe machine to the support via the active suspension. Thephysical parameters of the active suspension system are notprovided by the manufacturer. The system has to be consid-ered as a “black box”.

3. Plant representation and controller structure

The structure of a linear time invariant discrete time modelof the plant (used for controller design) is

G(z−1) = z−dB(z−1)

A(z−1)= z−d−1B∗(z−1)

A(z−1),

where

d = the plant pure time delay in number of sampling periods,

A = 1 + a1z−1 + · · · + anAz

−nA,

B = b1z−1 + · · · + bnB z

−nB = q−1B∗,B∗ = b1 + · · · + bnB z

−nB+1.

A(z−1), B(z−1), B∗(z−1) are polynomials in the complexvariablez−1 andnA, nB andnB −1 represent their orders.5

The model of the plant is obtained by system identificationof the secondary path. Details on system identification ofthe model of the active suspension considered in this papercan be found inConstantinescu (2001), Landau, Karimi, &

5 The complex variablez−1 will be used for characterizing the systembehavior in the frequency domain and the delay operatorq−1 will beused for describing the system behavior in the time domain.

566 I.D. Landau et al. / Automatica 41 (2005) 563–574

Constantinescu (2001b), Landau, Constantinescu, Loubat,Rey, & Franco (2001a), Karimi (2002), Constantinescu &Landau (2003).

The controller to be designed is a RS-type polynomial(Landau, Lozano, & M’Saad, 1997; Landau, 2002) controller(seeFig. 4). The output of the planty(t) and the inputu(t)may be written as

y(t) = q−dB(q−1)

A(q−1)u(t) + p1(t), (1)

S(q−1) · u(t) = −R(q−1)y(t), (2)

whereq−1 is the delay (shift) operator(x(t)=q−1x(t+1)),p1(t) is the resulting additive disturbance on the output of thesystem andR(z−1), S(z−1) are polynomials inz−1 havingthe ordersnR, nS , with the following expressions:

R(z−1) = r0 + r1z−1 + · · · + rnRz

−nR

=R′(z−1) · HR(z−1), (3)

S(z−1) = 1 + s1z−1 + · · · + snS z

−nS

= S′(z−1) · HS(z−1), (4)

whereHR andHS are pre-specified parts of the controller(used for example to incorporate the internal model of adisturbance or to open the loop at certain frequencies).

We define the output sensitivity function (the transferfunction between the disturbancep1(t) and the output of thesystemy(t)) as

Syp(z−1) = A(z−1)S(z−1)

P (z−1), (5)

where

P(z−1) = A(z−1)S(z−1) + z−dB(z−1)R(z−1) (6)

defines the poles of the closed loop. In pole placementdesign,P(z−1) is the polynomial specifying the desiredclosed loop poles and the controller polynomialsR(z−1)

andS(z−1) are minimal degree solutions of (6) where thedegrees ofP, R andSare given by:nP �nA + nB + d − 1,nS = nB + d − 1, nR = nA − 1.

Using Eqs. (1) and (2), one can write the output of thesystem as

y(t) = A(q−1)S(q−1)

P (q−1)p1(t) = Syp(q

−1)p1(t). (7)

For more details on RS-type controllers and sensitivityfunctions seeLandau et al. (1997).

Suppose thatp1(t) is a deterministic disturbance, so itcan be written as

p1(t) = Np(q−1)

Dp(q−1)�(t), (8)

where�(t) is a Dirac impulse andNp(z−1) andDp(z

−1) arecoprime polynomials inz−1, of degreesnNp , nDp . We are

interested in the rejection of narrow band disturbances andin this case the roots ofDp(z

−1) are on the unit circle.6

The energy of the disturbance is essentially represented byDp. The contribution of the terms ofNp is weak comparedto the effect ofDp, so one can neglect the effect ofNp.

Internal Model Principle: The effect of the disturbancegiven in(8) upon the output

y(t) = A(q−1)S(q−1)

P (q−1)

Np(q−1)

Dp(q−1)�(t), (9)

whereDp(z−1) is a polynomial with the roots on the unit

circle andP(z−1) is an asymptotically stable polynomial,converges asymptotically towards zero if and only if the poly-nomialS(z−1) in the RS controller has the form

S(z−1) = Dp(z−1)S′(z−1). (10)

In other terms, the pre-specified part ofS(z−1) should bechosen asHS(z

−1)=Dp(z−1) and the controller is computed

using (6), whereP, Dp, A, B, HR andd are given.7

Using the Youla–Kucera parametrization (Q-para-metrization) of all stable controllers (Anderson, 1998; Tsyp-kin, 1997), the controller polynomialsR(z−1) andS(z−1)

get the form

R(z−1) = R0(z−1) + A(z−1)Q(z−1), (11)

S(z−1) = S0(z−1) − z−dB(z−1)Q(z−1). (12)

The (central) controller(R0, S0) can be computed by polesplacement (but any other design technique can be used).Given the plant model(A,B, d) and the desired closed-looppolesP one has to solve

P(z−1) = A(z−1)S0(z−1) + z−dB(z−1)R0(z

−1). (13)

Eqs. (11) and (12) charaterize the set of all stabilizable con-trollers assigning the closed loop poles as defined byP(z−1).For the purpose of this paperQ(z−1) is considered to be apolynomial of the form

Q(z−1) = q0 + q1z−1 + · · · + qnQz

−nQ. (14)

To computeQ(z−1) in order that the controller incorporatesthe internal model of the disturbance one has to solve thediophantine equation

S′(z−1)Dp(z−1) + z−dB(z−1)Q(z−1) = S0(z

−1), (15)

where Dp(z−1), d, B(z−1)and S0(z

−1) are known andS′(z−1) andQ(z−1) are unknown. Eq. (15) has a unique so-lution for S′(z−1) andQ(z−1) with nS0 �nDp +nB +d −1,nS′ = nB + d − 1, nQ = nDp − 1. One sees that the ordernQ of the polynomialQ depends upon the structure of thedisturbance model.

6 Since the external disturbance is narrow band, the filtering effect ofthe primary path around the central frequency can be approximated by again and phase lag which will be captured byNp .

7 Of course it is assumed thatDp andB do not have common factors.

I.D. Landau et al. / Automatica 41 (2005) 563–574 567

4. Direct adaptive control for narrow band disturbanceattenuation

The objective is to find an estimation algorithm whichwill directly estimate the parameters of the internal modelin the controller in the presence of an unknown disturbance(but of known structure) without modifying the closed looppoles. Clearly, theQ-parametrization is a potential optionsince modifications of theQ polynomial will not affect theclosed loop poles. In order to build an estimation algorithmit is necessary to define anerror equationwhich will reflectthe difference between the optimalQ polynomial and itscurrent value.

In Tsypkin (1997), by considering a time domain ap-proach to the “internal model” problem in the context of theQ-parametrization and using the sensitivity function, suchan error equation is provided and it can be used for devel-oping a direct adaptive control scheme. This idea has beenused inValentinotti (2001). A similar error equation has beendefined inAmara et al. (1999).

Using theQ-parametrization, the output of the system inthe presence of a disturbance can be expressed as

y(t) = A(q−1)[S0(q−1) − q−dB(q−1)Q(q−1)]

P(q−1)

× Np(q−1)

Dp(q−1)· �(t)

= S0(q−1) − q−dB(q−1)Q(q−1)

P (q−1)w(t), (16)

wherew(t) is given by (see alsoFig. 1)

w(t) = A(q−1)Np(q−1)

Dp(q−1)�(t)

=A(q−1)y(t) − q−dB(q−1)u(t). (17)

In the time domain, the internal model principle can be inter-preted as findingQ such that asymptoticallyy(t) becomeszero.

Assume that one has an estimation ofQ(q−1) at instantt,denotedQ(t, q−1). Defineε0(t +1) as the value ofy(t +1)obtained withQ(t, q−1). Using (16) one gets

ε0(t + 1) = S0(q−1)

P (q−1)w(t + 1)

− q−dB∗(q−1)

P (q−1)Q(t, q−1)w(t). (18)

One can define now the a posteriori error (usingQ(t + 1, q−1)) as

ε(t + 1) = S0(q−1)

P (q−1)w(t + 1)

− q−dB∗(q−1)

P (q−1)Q(t + 1, q−1)w(t). (19)

ReplacingS0(q−1) from the last equation by (15) one obtains

ε(t + 1) = [Q(q−1) − Q(t + 1, q−1)]× q−dB∗(q−1)

P (q−1)w(t) + v(t + 1), (20)

where

v(t) = S′(q−1)Dp(q−1)

P (q−1)w(t)

= S′(q−1)A(q−1)Np(q−1)

P (q−1)�(t)

is a signal which tends asymptotically towards zero.Define the estimated polynomialQ(t, q−1) asQ(t, q−1)=

q0(t) + q1(t)q−1 + · · · + qnQ(t)q

−nQ and the associated

estimated parameter vector:�(t) = [q0(t)q1(t) · · · qnQ(t)]T.Define the fixed parameter vector corresponding to theoptimal value of the polynomialQ as� = [q0q1 · · · qnQ ]T.

Denote

w2(t) = q−dB∗(q−1)

P (q−1)w(t) (21)

and define the following observation vector:

�T(t) = [w2(t) w2(t − 1) · · ·w2(t − nQ)]. (22)

Eq. (20) becomes

ε(t + 1) = [�T − �T(t + 1)] · �(t) + v(t + 1). (23)

One can remark thatε(t) corresponds to an adaptation error(Landau et al., 1997).

From Eq. (18) one obtains the a priori adaptation error

ε0(t + 1) = w1(t + 1) − �T(t)�(t),

with

w1(t + 1) = S0(q−1)

P (q−1)w(t + 1), (24)

w2(t) = q−dB∗(q−1)

P (q−1)w(t), (25)

w(t + 1) = A(q−1)y(t + 1) − q−dB∗(q−1)u(t), (26)

whereB(q−1)u(t + 1) = B∗(q−1)u(t).The a posteriori adaptation error is obtained from (19)

ε(t + 1) = w1(t + 1) − �T(t + 1)�(t).

For the estimation of the parameters ofQ(t, q−1)

the following parameter adaptation algorithm is used

568 I.D. Landau et al. / Automatica 41 (2005) 563–574

(Landau et al., 1997)

�(t + 1) = �(t) + F(t)�(t)ε(t + 1), (27)

ε(t + 1) = ε0(t + 1)

1 + �T(t)F (t)�(t), (28)

ε0(t + 1) = w1(t + 1) − �T(t)�(t), (29)

F(t + 1) = 1

�1(t)

F(t) − F(t)�(t)�T(t)F (t)

�1(t)

�2(t)+ �T(t)F (t)�(t)

. (30)

In order to implement this methodology for disturbancerejection (seeFig. 1), it is supposed that the plant modelz−dB(z−1)

A(z−1)is known (identified) and that it exists a controller

[R0(z−1), S0(z

−1)] which verifies the desired specificationsin the absence of the disturbance. One also supposes that thedegreenQ of the polynomialQ(z−1) is fixed,nQ=nDp −1,i.e. the structure of the disturbance is known.

The following procedure is applied at each sampling timefor adaptiveoperation:

(1) Get the measured outputy(t+1) and the applied controlu(t) to computew(t + 1) using (26).

(2) Computew1(t + 1) andw2(t) using (24) and (25) withP given by (13).

(3) Estimate theQ-polynomial using the parametric adapta-tion algorithm (27)–(30).

(4) Compute and apply the control (seeFig. 1):

S0(q−1)u(t + 1) = − R0(q

−1)y(t + 1)

− Q(t, q−1)w(t + 1). (31)

For theself tuningoperation of the adaptive scheme, theestimation of theQ-polynomial starts once the level of theoutput is over a defined threshold. A parameter adaptationalgorithm (27)–(30) with asymptotically decreasing adap-tion gain is used and estimation is stopped when the adap-tion gain is below a pre-specified level.8 During estimationof the new parameters, the controller is kept constant. Thecontroller is updated once the estimation phase is finished.

5. Stability analysis

We will make the following assumptions:

• H1: The available plant model(A, B) is identical to thetrue plant model (i.e.A = A, B = B) and the plant delayis known.

• H2: The model of the disturbance has poles on the unitcircle.

8 The magnitude of the adaptation gain gives an indication uponthe variance of the parameter estimation error—see for exampleLandauet al. (1997).

• H3: The order of the denominator of the disturbance modelnDP

is known.

Eq. (23) combined with Eqs. (27) through (30) lead to anequivalent feedback system (by defining�(t + 1)= �(t + 1)− �) which is formed by a strictly passive linear block (aunit gain) in feedback with a time varying passive block andwith v(t + 1) as an external input . However,v(t + 1) isa vanishing signal (see Section 4 after Eq. (20)) and doesnot influence the asymptotic behavior ofε(t + 1). Eq. (23)combined with Eqs. (27)–(30) (withv(t + 1) a vanishingsignal) have the form considered in Theorem 3.3.2 fromLandau et al. (1997, pp. 97–103)and therefore this theoremcan be used to analyze the system. Applying the above-mentioned theorem one concludes

limt→∞ ε(t + 1) = 0, (32)

limt→∞ [�(t + 1) − �]T�(t) = 0, (33)

[�(t) − �]TF(t)−1[�(t) − �]<M <∞, (34)

limt→∞

[ε0(t + 1)]21 + �T(t)F (t)�(t)

= 0. (35)

Under assumption H1 the signalw(t) can be written as

w(t) = A(q−1)p1(t) = A(q−1)Np(q−1)

Dp(q−1)�(t). (36)

SinceDP has the roots on the unit circle and the coefficientsof A and NP are bounded one concludes thatp1(t) and,respectively,w(t) are bounded.

The components of the regressor vector�(t) have theform

w2(t − i) = q−dB∗(q−1)

P (q−1)w(t − i). (37)

Since the polynomialP is asymptotically stable by design,one concludes thatw2(t − i) and�(t) are bounded. There-fore, from Eq. (35) one concludes that also9

limt→∞ ε0(t + 1) = 0. (38)

Sinceε(t + 1) goes to zero asymptotically, whent → ∞one can write from (33)

limt→∞ ε(t + 1)

= limt→∞

nDP

−1∑i=0

(qi − qi (t + 1))q−i

w2(t) = 0. (39)

Since asymptoticallyqi , i = 0,1, . . . , nDP− 1 will be

a constant, Eq. (39) can hold either ifqi − qi = 0, i =0,1, . . . , nDP

− 1 or if w2(t) is a solution of the differ-

ence equation[∑nDP

−1i=0 (qi − qi )q

−i]w2(t) = 0. In the

9 The convergence towards zero ofε0 can be proven also with the“bounded growth” lemma of Goodwin and Sin (Landau et al., 1997).

I.D. Landau et al. / Automatica 41 (2005) 563–574 569

presence of the external disturbance,w2(t) which is a fil-tered version of the disturbance will be characterized by adifference equation of ordernDP

and it cannot be a solu-tion of a difference equation of ordernDP

− 1. Therefore,lim t→∞ ε(t+1)=0 implies also the parametric convergencein the presence of the disturbance.10

6. Results obtained on the active suspension

The narrow band disturbance rejection procedure usingthe direct adaptive control scheme proposed in Section 4is illustrated in real time for the case of the control of anactive suspension (presented in Section 2). In our case thedisturbance will be a time-varying frequency sinusoid, so weshall considernDp = 2 andnQ =nDp − 1= 1. Furthermore,a comparison with the results obtained in real time with theindirect adaptive control scheme proposed in Appendix Awill be provided.

The identification procedure in open and closed-loop op-eration for the active suspension is discussed in detail inConstantinescu (2001), Landau et al. (2001b, a), Karimi(2002), andConstantinescu & Landau (2003). The frequencycharacteristic of the identified primary path model (open-loop identification), between the signalup sent to the shakerin order to generate the disturbance and the residual forcey(t), is presented inFig. 5. The first vibration mode of theprimary path model is near 32 Hz. The frequency charac-teristic of the identified secondary path model (closed-loopidentification), is presented also inFig. 5. This model hasthe following complexity:nB = 14, nA = 16, d = 0. Theidentification has been done using as excitation of the pistona pseudo random binary sequence (PRBS) with frequencydividerp=4 (for details on the PRBS signals seeLandau etal., 1997). There exist several low damped vibration modeson the secondary path, the first one being at 31.8 Hz with adamping factor 0.07. The identified model of the secondarypath has been used for the design and implementation of thecontroller.

The central controller (without the internal model of thedisturbance) has been designed using the pole placementmethod and the secondary path identified model. A pair ofdominant poles has been fixed at the frequency of the firstvibration mode (31.8 Hz), with a damping� = 0.8, and theother poles of the model have been considered as auxiliarydesired closed loop poles. In addition a pre-specified partHR =1+q−1 (R=HRR

′) which assures the opening of theloop at 0.5fs has been introduced and 10 auxiliary poles at0.7 have been added to the desired closed-loop poles. Theresulting nominal controller has the following complexity:

10Alternatively, in the presence of the disturbance,w2(t) is a persis-tently exciting signal of ordernDP

, (since its spectral density is differentfrom zero in at leastnDP

points between−�����) and using a theo-rem of Aström and Wittenmark (Landau et al., 1997, Theorem 3.4.1) onearrives to the same conclusion.

0 50 100 150 200 250 300 350 400 -40

-30

-20

-10

0

10

20

30

40Primary and Secondary Path Models

Mag

nit

ud

e [d

B]

Frequency [Hz]

Secondary PathPrimary Path

Fig. 5. Frequency characteristics of the primary and secondary path.

nR = 14, nS = 16 and it satisfies the imposed robustnessconstraints on the sensitivity functions.11

In order to evaluate the performances of direct and indi-rect methods in real time, time-varying frequency sinusoidaldisturbances between 25 and 47 Hz have been used (the firstvibration mode of the primary path is near 32 Hz).

For both direct and indirect adaptive control methods, twoprotocols have been defined: one for aself-tuningoperation,the other for anadaptiveoperation.

• Protocol 1: Self-tuning operationThe system operates in closed loop with a frozen con-troller. As soon as a change of the disturbance is detected(by measuring the variance of the residual output), the es-timation algorithm is started with the last frozen controllerin operation. When the algorithm converges (a criterionhas to be defined—see below)), a new controller is com-puted and applied to the system. The adaptation algorithmis stopped and one waits for a change of frequency.

• Protocol 2: Adaptive operationThe estimation algorithm works permanently (once theloop is closed) and the controller is recomputed at eachsampling. The adaption gain in this case does not tendasymptotically to zero.

• Start up: For comparison purpose the system is startedin open-loop for both protocols. After 5 s (4000 sam-ples) a sinusoidal disturbance of 32 Hz is applied on theshaker. The model of the disturbance is estimated andan initial controller is computed (same initial controllerfor both direct and indirect adaptive control). In the caseof the self-tuning operation the adaptation algorithm is

11Any design method allowing to satisfy these constraints can beused.

570 I.D. Landau et al. / Automatica 41 (2005) 563–574

0 0.5 1 1.5 2 2.5 3 3.5 4

-6

-4

-2

0

2

4

6

Direct adaptation method in self−tuning operation

Samples

Res

idu

al f

orc

e [V

]

32 Hz 25 Hz 32 Hz 47 Hz 32 Hz

x 104

Fig. 6. Time domain results with the direct adaptation method inself-tuning operation.

0 0.5 1 1.5 2 2.5 3 3.5 4

-6

-4

-2

0

2

4

6

Indirect adaptation method in self−tuning operation

Samples

Res

idu

al f

orc

e [V

]

32 Hz 25 Hz 32 Hz 47 Hz 32 Hz

x 104

Fig. 7. Time domain results with the indirect adaptation method inself-tuning operation.

stopped while in the case of the adaptive operation theadaptation algorithm continues to be active.

After the start upends, every 15 s (8000 samples) sinu-soidal disturbances of different frequency are applied (32,25, 32, 47, 32 Hz).

6.1. Protocol 1: Self-tuning operation. Real timeexperimental results

The measured residual force obtained in self-tuning op-eration with the direct adaptation method is presented inFig. 6 and with the indirect adaptation method inFig. 7.

0 50 100 150 200 250 300 350 400 -70

-60

-50

-40

-30

-20

-10

0

20

10

Frequency [Hz]

Spectral densities of the residual force. Direct method in self−tuning operation

dB

[V

rms]

Open loop (25 Hz) Open loop (32 Hz) Open loop (47 Hz) Closed loop (25 Hz)Closed loop (32 Hz)Closed loop (47 Hz)

Fig. 8. Spectral densities of the residual force in open and in closed loop,with the direct adaptation method in self-tuning operation.

We note in general a faster convergence speed of the directadaptive control scheme compared to the indirect one (ex-cept for 47 Hz).

For the self-tuning protocol, the spectral densities of theresidual force obtained in open loop, respectively, in closedloop using the direct adaptation scheme (after the algorithmconverges) are presented inFig. 8. The results are given forthe three frequencies used: 25, 32 and 47 Hz. We remark thatthe attenuations are larger than 49 dB for all the frequencies.Similar results are obtained with the indirect adaptation. Fordetails seeConstantinescu (2001).

We note the appearance of two harmonics of the first vi-bration mode of the primary path on the spectral densityin open loop when the frequency of the disturbance corre-sponds with the first resonance mode of the system (32 Hz).They appear in open loop because of the non-linearities ofthe system at large signals (there is an important amplifi-cation of the disturbance at the resonance frequency of thesystem in open loop). The harmonics do not appear in closedloop operation.

In self-tuningoperation, one uses an adaptation gainF(t)

with variable forgetting factor, with �0=0.97 and the initialforgetting factor�1(0)= 0.97 (the forgetting factor is givenby �1(t) = �0�1(t − 1) + 1 − �0, with 0< �0 <1). For thevariable forgetting factorthe adaptation gain tends asymp-totically towards zero. The convergence criterion has beenfixed as a threshold on the trace value of the adaptation gainmatrix. For details seeConstantinescu (2001).

The detection of a change of frequency is done usingthe variance of the measured residual force computed on asliding window of 50 samples.

6.2. Protocol 2: Adaptive operation. Real timeexperimental results

The measured residual force obtained in adaptive opera-tion is presented inFig. 9 for the direct adaptation methodand in Fig. 10 for the indirect adaptation method. An

I.D. Landau et al. / Automatica 41 (2005) 563–574 571

0 0.5 1 1.5 2 2.5 3 3.5 4

-6

-4

-2

0

2

4

6

Direct method in adaptive operation

Samples

Res

idu

al f

orc

e [V

]

32 Hz 25 Hz 32 Hz 47 Hz 32 Hz

x 104

Fig. 9. Time domain results with the direct adaptation method in theadaptive case (trace=3 × 10−9).

0 0.5 1 1.5 2 2.5 3 3.5 4

-6

-4

-2

0

2

4

6

Indirect method in adaptative operation

Samples

Res

idu

al f

orc

e [V

]

32 Hz 25 Hz 32 Hz 47 Hz 32 Hz

x 104

Fig. 10. Time domain results with the indirect adaptation method in theadaptive case.

adaptation gain withvariable forgetting factor combinedwith a constant trace(Landau et al., 1997) has been usedin order to be able to track automatically the changes ofdisturbance characteristics. The low level threshold of thetrace has been fixed at 3×10−9 for the direct algorithm andat 5× 10−7 for the indirect one (note that in the indirectadaptive scheme there are more parameters to estimate thanin the direct adaptive scheme). The attenuation obtainedwith the indirect adaptive scheme is less good than in theself tuning operation and less good than the one obtainedwith the direct adaptive scheme. This is certainly caused bythe phenomenon discussed in Appendix A.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 -4

-2

0

2

4

Time [s]

Res

idua

l for

ce [V

]

Chirp disturbance in open−loop

25Hz(const) 47Hz(const) 25 → 47Hz

25Hz(const) 47Hz(const) 25 → 47Hz

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 -4

-2

0

2

4

Time [s]

Res

idua

l for

ce [V

]

Direct adaptive control−chirp disturbance in closed loop

(a)

(b)

Fig. 11. Real-time results obtained with the direct adaptive method anda chirp disturbance: (a) Open loop; (b) Closed loop.

The spectral densities of the residual force (after the al-gorithm converges) are similar with those obtained inself-tuningoperation. SeeConstantinescu (2001).

According to the real time results presented above, onecan conclude that the direct adaptive control scheme givesbetter results than the indirect adaptive control scheme, fromthe point of view of the convergence speed and performance.In addition the direct adaptation scheme is much simplerthan the indirect one in terms of number of operations.

6.3. Direct adaptive control scheme under the effect ofsinusoidal disturbances with continuously time varyingfrequency

Consider now that the frequency of the sinusoidal distur-bance varies continuously and let us use a chirp disturbancesignal (linear swept-frequency signal) between 25 and 47 Hz.The tests have been done as follows: Start up in closed loopatt=0 with the central controller. Once the loop is closed, theadaptation algorithm works permanently and the controlleris updated (direct approach) at each sampling instant. After5 s a sinusoidal disturbance of 25 Hz (constant frequency) isapplied on the shaker. From 10 to 15 s a chirp between 25and 47 Hz is applied. After 15 s a 47 Hz (constant frequency)sinusoidal disturbance is applied and the tests are stoppedafter 18 s. The time-domain results obtained in open and inclosed-loop (direct adaptive control) are presented inFig.11. We can remark that the performances obtained are verygood.

6.4. Adaptation transients for direct adaptive control

Fig. 12 illustrates the adaptation transients on theinput and output when a step change of the frequency of the

572 I.D. Landau et al. / Automatica 41 (2005) 563–574

29.8 29.9 30 30.1 30.2 30.3 -3

-2

-1

0

1

2

3

Time(s)

Out

put (

resi

dual

forc

e)

29.8 29.9 30 30.1 30.2 30.3 -1.5

-1

-0.5

0

0.5

1

1.5

Inpu

t (co

ntro

l act

ion)

20 Hz

32 Hz

Fig. 12. Adaption transient in the direct adaptive control scheme for astep change of the disturbance frequency from 20 to 32 Hz.

disturbance occurs from 20 to 32 Hz, respectively. Onenotes that the convergence of the output requires less than0.25 s. This corresponds roughly to 6 periods for 32 Hz.Same duration of the adaptation transient are obtained forthe other frequencies step changes. These results have to becompared with the transients results given inBodson andDouglas (1997), Marino et al. (2003), Amara et al. (1999).

7. Conclusions

It was shown in this paper that the use of the internalmodel principle combined with the adaptation of the in-ternal model implemented in the controller allows a verygood rejection of the unknown narrow band disturbancesin active suspension systems without requiring the use ofan additional transducer. Two adaptive approaches (directand indirect adaptation) have been developed and testedcomparatively on an active suspension.

The results obtained in real time lead us to con-clude that the direct adaptive control scheme providesbetter performance than the indirect adaptive controlscheme. Furthermore, from the performances point ofview, the adaptive operation is more interesting than theself-tuning one for the direct adaptive control scheme.Moreover, the direct algorithm is much simpler than theindirect one.

Despite the interesting results obtained, some problemsshould constitute the object of further investigation. Amongthese problems we mention: (1) Assessment of the influenceof plant parameter variations. (2) Development of an anti-windup procedure for the direct adaptive control scheme(Zaccarian & Teel, 2000, gives solutions for the indirectadaptive control scheme).

Appendix A. Indirect adaptive control for narrow banddisturbance attenuation

The methodology proposed in this section concerns theindirect adaptive control for the attenuation of narrow banddisturbances and consists in two steps: (1) Identification ofthe disturbance model. (2) Computation of a digital con-troller using the identified disturbance model.

The disturbance is considered as a stationary signal havinga rational spectrum. As such it may be considered as the out-put of a filter with the transfer functionNp(z

−1)/Dp(z−1)

and a white noise as input

Dp(q−1)y(t) = Np(q

−1)e(t) or

y(t) = Np(q−1)

Dp(q−1)e(t), (A.1)

wheree(t) represents a Gaussian white noise and

Np(z−1) = 1 + np1z

−1 + · · · + npnNpz−nNp

= 1 + z−1N∗p(z

−1),

Dp(z−1) = 1 + dp1z

−1 + · · · + dpnDpz−nDp

= 1 + z−1D∗p(z

−1).

Therefore, the disturbance model can be represented by anARMA model. As we deal with narrow band disturbances,the filtering effect of the primary path in cascade with theoutput sensitivity function (when operating in closed loop)around the central frequency of the disturbance can be ap-proximated by a gain and a phase lag which will be capturedby theNp(z

−1)/Dp(z−1) model.

From Eq. (A.1) one obtains

y(t + 1) = −nDp∑i=1

dpiy(t − i + 1)

+nNp∑i=1

npie(t − i + 1) + e(t + 1). (A.2)

The problem is, in fact, an on-line adaptive estimation ofparameters in presence of noiseLandau et al. (1997, 1986).Eq. (A.2) is a particular case of identification of an AR-MAX model. One can use for example therecursive ex-tended least squaresmethod (Landau et al., 1997), whichis dedicated to the identification of this type of model. Theparameter adaptation algorithm given in (27)–(30) is used.The controller parameters are frozen while the disturbancemodel is identified. Once the disturbance model is identi-fied, the controller containing the disturbance dynamics iscomputed by solving the diophantine equations (6) and us-ing (4) withHS(z

−1)=Dp(z−1) (the identified model of the

disturbance). In order to apply this methodology we supposethat the plant model is known (can be obtained by identifi-cation). We also suppose that the degreesnNp andnDp ofNp(z

−1), respectively,Dp(z−1) are fixed.

I.D. Landau et al. / Automatica 41 (2005) 563–574 573

For theself tuningoperation a parameter adaptation algo-rithm with decreasing adaptation gain is used and estimationis stopped when the adaptation gain becomes smaller thana pre-specified level. Inadaptiveoperation, the parametersof the controller have to be re-computed at each samplinginstant based on the current estimation of the disturbancemodel (non vanishing adaptation gain). Unfortunately, whenthe estimated frequency approaches the true one, the outputsensitivity function will have a pair of complex zeros on theunit circle leading to a very strong attenuation of the mea-sured effect of the disturbance. This will make the estima-tion of the exact frequency almost impossible. As a conse-quence, inadaptiveoperation there will be a “bias” on theestimated frequency caused by the need to have a certainlevel of the measured output to carry on the estimation. Thisphenomenon has been clearly observed both on simulationand on the real system12 and has also been discussed byothers (Bodson & Douglas, 1997; Gouraud et al., 1997).

References

Amara, F. B., Kabamba, P. T., & Ulsoy, A. G. (1999). Adaptive sinusoidaldisturbance rejection in linear discrete-time systems-part 1 and 2.Journal of Dynamic Systems Measurement and Control, 121, 648–659.

Anderson, B. D. O. (1998). From Youla–Kucera to identification, adaptiveand nonlinear control.Automatica, 34, 1485–1506.

Bengtsson, G. (1977). Output regulation and internal models—a frequencydomain approach.Automatica, 13, 333–345.

Beranek, L. L., & Ver, I. L. (1992).Noise and vibration controlengineering: principles and applications. New York: Wiley.

Bodson, M., & Douglas, S. C. (1997). Adaptive algorithms for the rejectionof sinusosidal disturbances with unknown frequency.Automatica, 33,2213–2221.

Constantinescu, A. (2001).Commande robuste et adaptative d’unesuspension active. Thèse de doctorat, Institut National Polytechniquede Grenoble, décembre 2001.

Constantinescu, A., & Landau, I. D. (2003). Direct controller orderreduction by identification in closed loop applied to a benchmarkproblem.European Journal of Control, 9(1),

Ding, Z. (2003). Global stabilization and disturbance suppression of aclass of nonlinear systems with uncertain internal model.Automatica,39, 471–479.

Elliott, S. J., & Nelson, P. A. (1994). Active noise control. Noise/NewsInternational, pp. 75–98.

Elliott, S. J., & Sutton, T. J. (1996). Performance of feedforward andfeedback systems for active control.IEEE Transactions on Speech andAudio Processing, 4(3), 214–223.

Feng, G., & Palaniswami, M. (1992). A stable adaptive implementation ofthe internal model principle.IEEE Transactions on Automatic Control,37, 1220–1225.

Francis, B. A., & Wonham, W. M. (1976). The internal model principleof control theory.Automatica, 12, 457–465.

Fuller, C. R., Elliott, S. J., & Nelson, P. A. (1995).Active control ofvibration. New York: Academic Press.

12 One can use other signals for disturbance estimation in order toavoid this problem, but this will lead to an even more complicated system.As implicitly suggested by an anonymous reviewer, the signaly(t) can

be replaced byy(t) = y(t) − q−dBA

u(t).

Gouraud, T., Gugliemi, M., & Auger, F. (1997). Design of robust andfrequency adaptive controllers for harmonic disturbance rejection ina single-phase power network.Proceedings of the European controlconference, Bruxelles.

Hillerstrom, G., & Sternby, J. (1994). Rejection of periodic disturbanceswith unknown period—a frequency domain approach.Proceedings ofAmerican control conference, Baltimore(pp. 1626–1631).

Johnson, C. D. (1976). Theory of disturbance-accomodating controllers.In C.T. Leondes (Ed.),Control and dynamical systems(Vol. 12, pp.387–489).

Karimi, A. (2002). Design and optimization of restricted complexitycontrollers—benchmark. http://iawww.epfl.ch/News/EJC_Benchmark/.

Landau, I. D. (2002). Commande des systèmes-Conception,Identification et Mise en oeuvre. Hermes Science (WEB Address:http://landau-bookic.lag.ensieg.inpg.fr).

Landau, I. D., Constantinescu, A., Loubat, P., Rey, D., & Franco, A.(2001a). A methodology for the design of feedback active vibrationcontrol systems.Proceedings of the European control conference 2001,Porto, Portugal.

Landau, I. D., Karimi, A., & Constantinescu, A. (2001b). Direct controllerorder reduction by identification in closed loop.Automatica, 37, 1689–1702.

Landau, I. D., Lozano, R., & M’Saad, M. (1997).Adaptive control.London: Springer.

Landau, I. D., M’Sirdi, N., & M’Saad, M. (1986). Techniques demodélisation récursive pour l’analyse spectrale paramétrique adaptative.Revue de Traitement du Signal, 3, 183–204.

Ljung, L. (1999). System identification-theory for the user. (2nd ed.),Englewood Cliffs, NJ: Prentice-Hall.

Marino, R., Santosuosso, G. L., & Tomei, P. (2000). Robust adaptivecompensation of biased sinusoidal disturbances with unknownfrequency.Automatica, 39, 1755–1761.

Sun, Z., & Tsao, T. C. (2000). Adaptive control with asymptotic trackingperformance and its application to an electro-hydraulic servo system.Journal of Dynamic Systems Measurement and Control, 122, 188–195.

Tsypkin, Y. Z. (1991). Adaptive-invariant discrete control systems. In M.Thoma, & A. Wyner (Eds.), Foundations of adaptive control.LectureNotes in Control and Information Science(Vol. 160, pp. 239–268).Berlin: Springer.

Tsypkin, Y. Z. (1997). Stochastic discrete systems with internal models.Journal of Automation and Information Sciences, 29(4&5), 156–161.

Valentinotti, S. (2001).Adaptive rejection of unstable disturbances:application to a fed-batch fermentation. Thèse de doctorat, ÉcolePolytechnique Fédérale de Lausanne, April 2001.

Widrow, B., & Stearns, S. D. (1985).Adaptive signal processing.Englewood Cliffs, NJ: Prentice-Hall.

Zaccarian, L., & Teel, A. R. (2000). A benchmark example for anti-windupsynthesis in active vibration isolation tasks and anL2 anti-windupsolution.European Journal of Control, 6(5), 405–420.

Zhang, Y., Mehta, P. G., Bitmead, R., & Johnson, C. R. (1998). Directadaptive control for tonal disturbance rejection.Proceedings of theAmerican Control Conference, Philadelphia(pp. 1480–1482).

Ioan Doré Landau is Emeritus ResearchDirector at C.N.R.S. (National Centre forScientific Research) since September 2003and continues to collaborate with theLaboratoire d’Automatique de Grenoble(CNRS/INPG) of the Institut National Poly-technique de Grenoble.His research interests encompass theoryand applications in system identification,adaptive control, robust digital control andnonlinear systems. He has authored and co-authored over 200 papers on these subjects.He is the author of the books :Adaptive

574 I.D. Landau et al. / Automatica 41 (2005) 563–574

Control—The Model Reference Approach (Dekker, 1979) translatedalso in chinese,System Identification and Control Design(Hermès,1993, Prentice-Hall, 1990).Commande de systèmes: conception, identi-fication, mise en oeuvre(Hermes-Lavoisier, 2002) and co-author (withM. Tomizuka) of the bookAdaptive Control—Theory and Practice(inJapanese-Ohm 1981) as well as co-author (with R. Lozano and M.M’Saad) of the bookAdaptive Control(Springer Verlag, 1997).Dr. Landau received the Rufus Oldenburger Medal 2000 from theAmerican Society of Mechanical. He is “Doctor Honoris Causa” of theUniversité Catholique de Louvain-la- Neuve (2003).Dr. Landau was the first President of the European Community ControlAssociation (ECCA) from 1991 to 1993 (now EUCA) and he was Editorin Chief of the European Journal of Control from 1994 to end of 2002.

Aurelian Constantinescu was born in1974. He received an Aerospace Engineer-ing Degree from the Polytechnic Universityof Bucharest in 1992, a Master’s Degreein 1993 and a Ph.D. in 2001 from thePolytechnic National Institute of Grenoble.He is currently a postdoctoral researcher atEcole de Technologie Superieure, Montreal.His current research interests include globalnavigation satellite systems and robust,adaptive and active control.

Daniel Rey was born in Bergerac, France,in 1950. He received the master’s degreein electrical engineering in 1973 and inautomatic control in 1974 both from theNational Polytechnical Institute of Greno-ble, France (INPG). In 1978 he received thePh.D. degree from INPG for the applicationof a hierarchical control on a distillationpilot plant.Since 1980, he has been a CNRS ResearchEngineer in the Laboratoire d’Automatiquede Grenoble. He is currently in chargeof the technical staff of the Laboratoire

d’Automatique de Grenoble. His interests include implementation on real-time computers and experimentation of digital control on industrial pro-cesses.