Control Engineering Practice 19 (2011) 1168–1181

Contents lists available at ScienceDirect

Control Engineering Practice

0967-06

doi:10.1

$The

paper a

Confere� Corr

E-m

journal homepage: www.elsevier.com/locate/conengprac

Adaptive regulation—Rejection of unknown multiple narrow banddisturbances (a review on algorithms and applications)$

Ioan Dore Landau a,�, Marouane Alma a, Aurelian Constantinescu b, John J. Martinez a, Mathieu Noe c

a GIPSA-LAB, Department of Automatic Control, ENSIEG BP 46, 38402 Saint-Martin d’H�eres, Franceb E.E. Dept, Ecole de Technologie Superieure, Montreal, Canadac Paulstra - Vibrachoc, Active Noise and Vibration Control Department, 91028 Evry, France

a r t i c l e i n f o

Article history:

Received 18 February 2010

Accepted 2 June 2011Available online 14 July 2011

Keywords:

Direct adaptive regulation

Indirect adaptive regulation

Internal model principle

Youla–Kucera parametrization

Multiple narrow band disturbances

Active vibration control

61/$ - see front matter & 2011 Elsevier Ltd. A

016/j.conengprac.2011.06.005

preliminary version of this paper has been pr

t the MED-09, International Mediterranea

nce, Thessaloniki, Greece, June 2009.

esponding author.

ail address: ioan-dore.landau@gipsa-lab.greno

a b s t r a c t

The paper addresses the problem of attenuation (rejection) of unknown and time varying multiple

narrow band disturbances without measuring them. In this context the disturbance model is unknown

and time varying while the model of the plant is known (obtained by system identification) and almost

invariant. This requires to use an adaptive feedback approach. The term ‘‘adaptive regulation’’ has been

coined to characterize this control paradigm. Application domains include: mechanical and mecha-

tronics systems, active vibration and noise suppression systems, some types of bio-chemical reactors.

The paper reviews the various techniques proposed for solving this problem. It will focus on the

presentation of the direct and indirect adaptive regulation strategies using the internal model principle

and the Youla–Kucera parametrization which have been extensively used in applications.

The paper also reviews a number of applications including: active suspension systems, active vibration

control systems, active noise control, bio-chemical reactors, distributed flexible mechanical structures and

Blu-ray disc drives. Real time results obtained on various applications will illustrate the methodology.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

One of the basic problems in control is the attenuation(rejection) of unknown disturbances without measuring them.The common framework is the assumption that the disturbance isthe result of a white noise or a Dirac impulse passed through the‘‘model of the disturbance’’. The knowledge of this model allowsto design an appropriate controller. When considering the modelof a disturbance, one has to address two issues: (1) its structure(complexity, order of the parametric model), (2) the values of theparameters of the model. In general one can assess from data thestructure for such ‘‘model of disturbance’’ (using spectral analysisor order estimation techniques) and assume that the structuredoes not change. However the parameters of the model areunknown and may be time varying. This will require to use anadaptive feedback approach.

The classical adaptive control paradigm deals essentially withthe construction of a control law when the parameters of theplant dynamic model are unknown and time varying (Astrom &Witenmark, 1995; Fekri, Athans, & Pasqual, 2006; Ioannou & Sun,1996; Landau, Lozano, & M’Saad, 1997). However in the present

ll rights reserved.

esented as an invited plenary

n Control and Automation

ble-inp.fr (I.D. Landau).

context the plant dynamic model is known (can be obtained bysystem identification) and almost invariant and the objective isthe rejection of disturbances characterized by unknown and timevarying disturbance models. It seems reasonable to call thisparadigm ‘‘adaptive regulation’’. In what follows ‘‘adaptivecontrol’’ and ‘‘adaptive regulation’’ will be characterized withthe help of Fig. 1.

In classical ‘‘adaptive control’’ the objective is tracking/distur-bance attenuation in the presence of unknown and time varyingplant model parameters. Therefore adaptive control focuses onadaptation with respect to plant model parameter variations.The model of the disturbance is assumed to be known and invariant.Only a level of attenuation in a frequency band is imposed (with theexception of DC disturbances where the controller may include anintegrator). The important remarks to be made are:

�

No effort is made to estimate in real time the model of thedisturbance. � The disturbances have in general an undesirable effect upon

the adaptation loop (parameters drift, bursting, etc.) and aclass of ‘‘robust adaptation algorithms’’ has been developed inorder to reduce the negative impact of the disturbances uponthe ‘‘adaptation loop’’.

In ‘‘adaptive regulation’’ the objective is to asymptotically suppressthe effect of unknown and time varying disturbances. Thereforeadaptive regulation focuses on adaptation of the controller

Fig. 1. Plant model and disturbance model.

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–1181 1169

parameters with respect to variations of parameters of the dis-turbance model. The plant model is assumed to be known. It is alsoassumed that the possible small variations of the plant model canbe handled by a robust control design. The important remarks to bemade are:

�

dist

of p

equ

ban

No effort is made to estimate in real time the model of the plant.

� A correlated measurement of the disturbance (an image of the

disturbance) is not available.

To be more specific, in adaptive regulation the disturbancesconsidered can be defined as ‘‘finite band disturbances’’. Thisincludes single or multiple narrow band disturbances or sinusoidaldisturbances. Furthermore for robustness reasons the disturbancesshould be located in the frequency domain within the regionswhere the plant has enough gain (see explanation in Section 3).

Solutions for this problem, provided that a correlated mea-surement with the disturbance (an image of the disturbance) canbe obtained by using an additional transducer, have been pro-posed by the signal processing community and a number ofapplications are reported (Beranek & Ver, 1992; Elliott & Nelson,1994; Elliott & Sutton, 1996; Fuller, Elliott, & Nelson, 1995).However, these solutions (inspired by Widrow’s technique foradaptive noise cancelation, Widrow & Stearns, 1985) ignore thepossibilities offered by feedback control systems and require anadditional transducer. The principle of this feedforward compensa-

tion for adaptive rejection of unknown disturbances is explainednext. This measurement correlated with the disturbance isapplied to the control input of the plant through an adaptivefilter (in general a finite impulse response—FIR) whose para-meters are adapted such that the effect of the disturbance uponthe output is minimized. The disadvantages of this approach are:

�

It requires the use of an additional transducer. � Difficult choice for the location of this transducer (it is probably

the main disadvantage).

� It requires the adaptation of many parameters.

To achieve the rejection of the disturbance (at least asymptoti-cally) without measuring it, a feedback solution has to be con-sidered. As mentioned earlier, the common framework is theassumption that the disturbance is the result of a white noise or aDirac impulse passed through the ‘‘model of the disturbance’’.Throughout the paper it is assumed that the order of thedisturbance model is known (it can be estimated from data1)but the parameters of the model are unknown.

1 A simple example is the case of multiple narrow band or sinusoidal

urbances. Their number can be immediately defined by looking to the number

eaks of the power spectral density of the system output. The order will be

al to twice the number of peaks, since a second order is associated to a narrow

d or sinusoidal disturbance.

From the user point of view, one has to consider two modes ofoperation of the adaptive schemes:

�

Adaptive operation. The adaptation is performed continuouslywith a nonvanishing adaptation gain and the controller isupdated at each sampling. � Self-tuning operation. The adaptation procedure starts either

on demand or when the performance is unsatisfactory.A vanishing adaptation gain is used. The current controller iseither updated at each sampling instant once adaptation startsor is frozen during the estimation/computation of the newcontroller parameters.

The objectives of the present paper are:

�

To clearly state what are the problems to be solved. � To review the techniques available emphasizing those which

are used in practice.

� To illustrate the application of the methodology. � To review a number of applications.

The problem of known plant and unknown disturbance model hasbeen addressed in a number of papers (Amara, Kabamba, & Ulsoy,1999a, 1999b; Bodson & Douglas, 1997; Ding, 2003; Gouraud,Gugliemi, & Auger, 1997; Hillerstrom & Sternby, 1994; Landau,Constantinescu, & Rey, 2005; Marino, Santosuosso, & Tomei, 2003;Valentinotti, 2001) among others. The following approaches con-sidered for solving this problem may be mentioned:

1.

Use of the internal model principle (Amara et al., 1999a, 1999b;Bengtsson, 1977; Francis & Wonham, 1976; Gouraud et al., 1997;Hillerstrom & Sternby, 1994; Johnson, 1976; Landau et al., 2005;Tsypkin, 1997; Valentinotti, 2001; Valentinotti et al., 2003).

2.

Use of an observer for the disturbance (Ding, 2003; Marinoet al., 2003; Serrani, 2006).

3.

Use of the ‘‘phase-locked’’ loop structure considered in com-munication systems (Bodson, 2005; Bodson & Douglas, 1997).

Of course, since the parameters of the disturbance model areunknown, all these approaches lead to an adaptive implementa-tion which can be of direct or indirect type.

The various approaches mentioned will be briefly reviewed next.Use of the internal model principle: Using the internal model

principle, the controller should incorporate the model of thedisturbance (Bengtsson, 1977; Francis & Wonham, 1976; Johnson,1976; Tsypkin, 1997). Therefore the rejection of unknown distur-bances raises the problem of adapting the internal model of thecontroller and its re-design in real-time.

One way for solving this problem is to try to estimate in realtime the model of the disturbance and re-compute the controller,which will incorporate the estimated model of the disturbance (asa pre-specified element of the controller). While the disturbanceis unknown and its model needs to be estimated, one assumes

Fig. 2. Indirect adaptive control scheme for rejection of unknown disturbances.

q-dB/A1/S0u(t) +

+-y(t)

Q

A

R0

q-dB

-

- +

Np/Dp

p(t)

w(t)

d(t)

Plant

Model Model

Adaptation

Algorithm^

Controller

Fig. 3. Direct adaptive control scheme for rejection of unknown disturbances.

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–11811170

that the model of the plant is known (obtained for example byidentification). The estimation of the disturbance model can bedone by using standard parameter estimation algorithms (see forexample Landau, M’Sirdi, & M’Saad, 1986; Ljung, 1999), once agood observer for the disturbance has been built. This will lead toan indirect adaptive control scheme. The principle of such ascheme is illustrated in Fig. 2. The time consuming part of thisapproach is the redesign of the controller at each sampling time.The reason is that in many applications the plant model can be ofvery high dimension and despite that this model is constant, onehas to re-compute the controller because a new internal modelshould be considered. This approach has been investigated inBodson and Douglas (1997), Gouraud et al. (1997), Hillerstromand Sternby (1994), and Landau et al. (2005) and new experi-mental results are presented in this paper. It is important toindicate that when full rejection of disturbances is not possible forrobustness reasons, one should consider only a certain level ofattenuation. In such situations indirect adaptive regulation seemsto be the appropriate solution.

However, by considering the Youla–Kucera parametrization ofthe controller (known also as the Q-parametrization), it ispossible to insert and adjust the internal model in the controllerby adjusting the parameters of the Q polynomial (see Fig. 3).It comes out that, in the presence of unknown disturbances, it ispossible to build a direct adaptive control scheme where theparameters of the Q polynomial are directly adapted in order toget the desired internal model without recomputing the control-ler (polynomials R0 and S0 in Fig. 3 remain unchanged).The number of the controller parameters to be directly adaptedis roughly equal to the number of parameters of the denomi-nator of the disturbance model. In other words, the size of the

adaptation algorithm will depend upon the complexity of thedisturbance model.

Use of adaptive observers: The basic idea in Marino et al. (2003)and Marino and Tomei (2007, 2008) is to built an adaptiveobserver of the disturbance and incorporate this observer in thecontroller. Once again the parameters of the plant model areassumed to be known. The plant is assumed to be stable. Thestrategy seems to be dedicated to disturbances acting on theinput of the plant. Application of this approach to output typedisturbance will require additional hypothesis on the plant model(stable inverse). While this scheme does not directly makereference to the ‘‘internal model principle’’, one can howeverremark that through the adaptive observer the model of thedisturbance is incorporated in the controller. An extension of theabove approach with explicit reference to the internal modelprinciple for the case of nonlinear plants is discussed in Ding(2003). Within this approach one has to consider also Serrani(2006) where the objective is to reject the periodic disturbancesacting on the measurement of the plant output.

Use of the phase-locked loop structure: A direct approach for therejection of sinusoidal disturbances with unknown frequencies,based on the integration of a phase-locked loop for adaptivefeedback control with known plant model is presented in Bodsonand Douglas (1997). The disturbance frequency estimation andthe disturbance cancelation are performed simultaneously using asingle error signal. The elimination of the high-frequency compo-nents within the system is done by using a low-pass compensator,no additional filtering being necessary. The knowledge of thefrequency response of the plant in the frequency range of interestis required. Because of the lock-in range of the phase-lockedloops, there exist an upper limit on the initial errors of thedisturbance frequencies. An initialization scheme for providingrough initial estimates of the disturbance frequency is needed.Experimental results are mentioned in Bodson (2005).

The present paper is focused on the use of the internal modelprinciple which has generated significant applications in theadaptive context.

The paper is organized as follows. Section 2 is dedicated to abrief review of the plant, disturbance and controller representa-tion as well as of the Internal Model Principle. Some importantrobustness issues are addressed in Section 3. The direct and theindirect adaptive control schemes for disturbance rejection arepresented in Sections 4 and 5, respectively.. The application of themethodology to an active suspension system, is presented inSection 6. The application to an active vibration control systemusing an inertial actuator is presented in Section 7. Real timeresults are presented in both sections and in addition a compar-ison between direct and indirect adaptive regulation is done inSection 7. A number of other applications are reviewed in Section8. Some concluding remarks are given in Section 9.

2. Plant representation and controller structure

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

Gðz�1Þ ¼z�dBðz�1Þ

Aðz�1Þ¼

z�d�1Bnðz�1Þ

Aðz�1Þ, ð1Þ

with

d¼ the plant pure time delay in the number of sampling periods

A¼ 1þa1z�1þ � � � þanAz�nA ,

B¼ b1z�1þ � � � þbnBz�nB ¼ z�1Bn,

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–1181 1171

Bn ¼ b1þ � � � þbnBz�nBþ1,

where Aðz�1Þ, Bðz�1Þ, Bnðz�1Þ are polynomials in the complexvariable z�1 and nA, nB and nB�1 represent their orders.2 Themodel of the plant may be obtained by system identification.Details on system identification of the models considered in thispaper can be found in Landau and Zito (2005), Constantinescu(2001), Landau, Karimi, and Constantinescu (2001), Landau,Constantinescu, Loubat, Rey, and Franco (2001), Karimi (2002),and Constantinescu and Landau (2003).

Since this paper is focused on regulation, the controller to bedesigned is a RS-type polynomial controller (Landau & Zito, 2005;Landau et al., 1997)—see also Fig. 6.

The output of the plant y(t) and the input u(t) may be written as3

yðtÞ ¼q�dBðq�1Þ

Aðq�1Þ� uðtÞþpðtÞ, ð2Þ

Sðq�1Þ � uðtÞ ¼ �Rðq�1Þ � yðtÞ, ð3Þ

where q�1 is the delay (shift) operator (xðtÞ ¼ q�1xðtþ1Þ) and p(t) isthe resulting additive disturbance on the output of the system.Rðz�1Þ and Sðz�1Þ are polynomials in z�1 having the orders nR and nS,respectively, with the following expressions:

Rðz�1Þ ¼ r0þr1z�1þ � � � þrnRz�nR ¼ R0ðz�1Þ � HRðz

�1Þ, ð4Þ

Sðz�1Þ ¼ 1þs1z�1þ � � � þsnSz�nS ¼ S0ðz�1Þ � HSðz

�1Þ, ð5Þ

where HR and HS are pre-specified parts of the controller (used forexample to incorporate the internal model of a disturbance or toopen the loop at certain frequencies).

The following sensitivity functions are defined:

�

beh

des

sam

Output sensitivity function (the transfer function between thedisturbance pðtÞ and the output of the system y(t)):

Sypðz�1Þ ¼

Aðz�1ÞSðz�1Þ

Pðz�1Þ, ð6Þ

�

Input sensitivity function (the transfer function between thedisturbance pðtÞ and the input of the system u(t)):

Supðz�1Þ ¼ �

Aðz�1ÞRðz�1Þ

Pðz�1Þ, ð7Þ

where

Pðz�1Þ ¼ Aðz�1ÞSðz�1Þþz�dBðz�1ÞRðz�1Þ

¼ Aðz�1ÞS0ðz�1Þ � HSðz�1Þþz�dBðz�1ÞR0ðz�1Þ � HRðz

�1Þ ð8Þ

defines the poles of the closed loop (roots of Pðz�1Þ). In poleplacement design, Pðz�1Þ is the polynomial specifying the desiredclosed-loop poles and the controller polynomials Rðz�1Þ and Sðz�1Þ

are minimal degree solutions of (8) where the degrees of P, R andS are given by: nP rnAþnBþnHR

þnHSþd�1, nS ¼ nBþd�1 and

nR ¼ nA�1. For more details on RS-type controllers and sensitivityfunctions see Landau and Zito (2005).

Internal model principle and Youla–Kucera parametrization.Using Eqs. (2) and (3), one can write the output of the system as

yðtÞ ¼Aðq�1ÞSðq�1Þ

Pðq�1Þ� pðtÞ ¼ Sypðq

�1Þ � pðtÞ: ð9Þ

2 The complex variable z�1 will be used for characterizing the system’s

avior in the frequency domain and the delay operator q�1 will be used for

cribing the system’s behavior in the time domain.3 t¼1,2,y denote the normalized discrete time (real time divided by the

pling period).

Suppose that p(t) is a deterministic disturbance, so it can bewritten as

pðtÞ ¼Npðq�1Þ

Dpðq�1Þ� dðtÞ, ð10Þ

where dðtÞ is a Dirac impulse and Npðz�1Þ, Dpðz�1Þ are coprimepolynomials in z�1, of degrees nNp

and nDp, respectively. In the

case of stationary disturbances the roots of Dpðz�1Þ are on the unitcircle. The energy of the disturbance is essentially represented byDp. The contribution of the terms of Np is weak compared to theeffect of Dp, so one can neglect the effect of Np.

Internal model principle: The effect of the disturbance given in(10) upon the output:

yðtÞ ¼Aðq�1ÞSðq�1Þ

Pðq�1Þ�Npðq�1Þ

Dpðq�1Þ� dðtÞ, ð11Þ

where Dpðz�1Þ is a polynomial with roots on the unit circle and Pðz�1Þ

is an asymptotically stable polynomial, converges asymptoticallytowards zero if and only if the polynomial Sðz�1Þ in the RS controllerhas the form

Sðz�1Þ ¼Dpðz�1ÞS0ðz�1Þ: ð12Þ

In other terms, the pre-specified part of Sðz�1Þ should be chosen asHSðz

�1Þ ¼Dpðz�1Þ and the controller is computed using (8), where P,Dp, A, B, HR and d are given.4

Using the Youla–Kucera parametrization (Q-parametrization)of all stable controllers (Anderson, 1998; Tsypkin, 1997), thecontroller polynomials Rðz�1Þ and Sðz�1Þ get the form

Rðz�1Þ ¼ R0ðz�1ÞþAðz�1ÞQ ðz�1Þ, ð13Þ

Sðz�1Þ ¼ S0ðz�1Þ�z�dBðz�1ÞQ ðz�1Þ: ð14Þ

The central controller ðR0,S0Þ can be computed by poles placement(but any other design technique can be used). Given the plantmodel ðA,B,dÞ and the desired closed-loop poles specified by theroots of P one has to solve

Pðz�1Þ ¼ Aðz�1ÞS0ðz�1Þþz�dBðz�1ÞR0ðz

�1Þ: ð15Þ

Eqs. (13) and (14) characterize the set of all stabilizable con-trollers assigning the closed-loop poles as defined by Pðz�1Þ (it canbe verified by simple calculations that the poles of the closed loopremain unchanged). For the purpose of this paper Q ðz�1Þ isconsidered to be a polynomial of the form

Q ðz�1Þ ¼ q0þq1z�1þ � � � þqnQz�nQ : ð16Þ

To compute Q ðz�1Þ in order that the polynomial Sðz�1Þ given by(14) incorporates the internal model of the disturbance, one hasto solve the diophantine equation:

S0ðz�1ÞDpðz�1Þþz�dBðz�1ÞQ ðz�1Þ ¼ S0ðz

�1Þ, ð17Þ

where Dpðz�1Þ, d, Bðz�1Þ and S0ðz�1Þ are known and S0ðz�1Þ and

Q ðz�1Þ are unknown. Eq. (17) has a unique solution for S0ðz�1Þ andQ ðz�1Þ with: nS0

rnDpþnBþd�1, nS0 ¼ nBþd�1, nQ ¼ nDp

�1. Onesees that the order nQ of the polynomial Q depends upon thestructure of the disturbance model.

Summary of the hypotheses (for adaptive regulation):

�

The plant model is assumed to be known (can be obtained bysystem identification). � The zeros of the plant model (zeros of polynomial B) can be

stable or unstable.

� The order of the disturbance model denominator Dp is known

(can be estimated from data) but its parameters are unknown.

� Dp and B do not have common factors.

4 Of course it is assumed that Dp and B do not have common factors.

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–11811172

3. Robustness considerations

It is well known that the introduction of the internal model forthe perfect rejection of the disturbance (asymptotically) will haveas effect to raise the maximum value of the modulus of the outputsensitivity function Syp. This may lead to unacceptable values for themodulus margin (jSypðe�joÞj�1

max) and the delay margin if thecontroller design is not appropriately done (see Landau & Zito,2005). As a consequence, a robust control design should beconsidered assuming that the model of the disturbance and itsdomain of variation in the frequency domain are known.The objective is that for all situations, acceptable modulus marginand delay margin are obtained. If the number of narrow banddisturbances is too high or a finite band disturbance cover a toowide frequency region, it may not be possible to cancel their effectassuring in the mean time robustness of the closed loop (the wellknown ‘‘water bed’’ effect resulting from the Bode integral of Syp inthe frequency domain). In such cases one has to consider only anattenuation of the disturbance to an acceptable level.

Furthermore at the frequencies where perfect rejection of thedisturbance is achieved one has Sypðe�joÞ ¼ 0 and

jSupðe�joÞj ¼

Aðe�joÞ

Bðe�joÞ

��������: ð18Þ

Eq. (18) corresponds to the inverse of the gain of the system to becontrolled. The implication of Eq. (18) is that cancelation (or ingeneral an important attenuation) of disturbances on the outputshould be done only in frequency regions where the system gain islarge enough. If the gain of the controlled system is too low, jSupj

will be large at these frequencies. Therefore, the robustness vsadditive plant model uncertainties will be reduced and the stress onthe actuator will become important. Eq. (18) also implies thatserious problems will occur if Bðz�1Þ has complex zeros close to theunit circle (stable or unstable zeros) at frequencies where animportant attenuation of disturbances is required. It is mandatoryto avoid attenuation of disturbances at these frequencies.

Since on one hand it is not desired to react to very high-frequency disturbances and on the other hand it is desired to havea good robustness, it is often wise to open the loop at 0:5fs (fs is thesampling frequency) by introducing a fixed part in the controllerHRðq

�1Þ ¼ 1þq�1 (for details see Landau & Zito, 2005 and Section 2).

4. Direct adaptive regulation

The objective is to find an estimation algorithm which will directlyestimate the parameters of the internal model in the controller in thepresence of an unknown disturbance (but of known structure)without modifying the closed-loop poles. Clearly, the Q-parametriza-tion is a potential option since on one hand modifications of the Q

polynomial will not affect the closed-loop poles and on the otherhand the internal model can be implemented with appropriate valuesof the Q parameters. In order to build such an estimation algorithm itis necessary to define an error equation which will reflect thedifference between the optimal Q polynomial and its current value.

In Tsypkin (1997), such an error equation is provided and itcan be used for developing a direct adaptive control scheme. Thisidea has been used in Valentinotti (2001), Valentinotti et al.(2003), Amara et al. (1999a, 1999b), and Landau et al. (2005).

Using the Q-parametrization, the output of the system in thepresence of a disturbance can be expressed as

yðtÞ ¼Aðq�1Þ½S0ðq

�1Þ�q�dBðq�1ÞQ ðq�1Þ�

Pðq�1Þ�Npðq�1Þ

Dpðq�1Þ� dðtÞ

¼S0ðq

�1Þ�q�dBðq�1ÞQ ðq�1Þ

Pðq�1Þ�wðtÞ, ð19Þ

where w(t) is given by (see also Fig. 3)

wðtÞ ¼Aðq�1ÞNpðq�1Þ

Dpðq�1Þ� dðtÞ

¼ Aðq�1Þ � yðtÞ�q�d � Bðq�1Þ � uðtÞ ¼ Aðq�1ÞpðtÞ: ð20Þ

From Eq. (20) it results that w(t) is a filtered estimation of thedisturbance (see also Eq. (2)). In the time domain, the internalmodel principle can be interpreted as finding Q such thatasymptotically y(t) becomes zero. Assume that one has anestimation of Q ðq�1Þ at instant t, denoted Q ðt,q�1Þ. Definee0ðtþ1Þ as the measured value of yðtþ1Þ obtained with Q ðt,q�1Þ

(called a priori error). Using (19) one gets

e0ðtþ1Þ ¼ eðtþ1jQ ðt,q�1ÞÞ ¼S0ðq

�1Þ

Pðq�1Þ�wðtþ1Þ

�q�dBnðq�1Þ

Pðq�1ÞQ ðt,q�1Þ �wðtÞ: ð21Þ

One can define now the a posteriori error (which is computedusing Q ðtþ1,q�1Þ) as

eðtþ1Þ ¼ eðtþ1jQ ðtþ1,q�1ÞÞ ¼S0ðq

�1Þ

Pðq�1Þ�wðtþ1Þ

�q�dBnðq�1Þ

Pðq�1ÞQ ðtþ1,q�1Þ �wðtÞ: ð22Þ

Replacing S0ðq�1Þ from the last equation by (17) one obtains

eðtþ1Þ ¼ ½Q ðq�1Þ�Q ðtþ1,q�1Þ� �q�dBnðq�1Þ

Pðq�1Þ�wðtÞþvðtþ1Þ, ð23Þ

where

vðtÞ ¼S0ðq�1ÞDpðq�1Þ

Pðq�1Þ�wðtÞ ¼

S0ðq�1ÞAðq�1ÞNpðq�1Þ

Pðq�1Þ� dðtÞ

is a signal which tends asymptotically towards zero.Define the estimated polynomial Q ðt,q�1Þ as: Q ðt,q�1Þ ¼

q0ðtÞþ q1ðtÞq�1þ � � � þ qnQ

ðtÞq�nQ and the associated estimatedparameter vector: yðtÞ ¼ ½q0ðtÞ q1ðtÞ . . . qnQ

ðtÞ�T . Define the fixedparameter vector corresponding to the optimal value of thepolynomial Q as: y¼ ½q0 q1 . . . qnQ

�T . Denote:

w2ðtÞ ¼q�dBnðq�1Þ

Pðq�1Þ�wðtÞ ð24Þ

and define the following observation vector:

fTðtÞ ¼ ½w2ðtÞ w2ðt�1Þ . . .w2ðt�nQ Þ�: ð25Þ

Eq. (23) becomes

eðtþ1Þ ¼ eðtþ1jyðtþ1ÞÞ ¼ ½yT�y

Tðtþ1Þ� �fðtÞþvðtþ1Þ: ð26Þ

Eq. (26) expresses the regulation error as a function of thedifference between the optimal value and the current estimatedvalue of the Q-parameters. It has the standard form for anadaptation error allowing a parameter adaption algorithm to beimplemented (Landau et al., 1997). The objective of the parameteradaption algorithm will be to drive the estimated parameterssuch that this error goes to zero for any initial conditionsconcerning the estimated parameters and the value of the output.In addition it is expected that a quadratic criterion in terms of thiserror will be minimized.

With the above notations the a priori adaptation error given byEq. (21) becomes

e0ðtþ1Þ ¼ eðtþ1jyðtÞÞ ¼w1ðtþ1Þ�yTðtÞfðtÞ, ð27Þ

with

w1ðtþ1Þ ¼S0ðq

�1Þ

Pðq�1Þ�wðtþ1Þ, ð28Þ

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–1181 1173

w2ðtÞ ¼q�dBnðq�1Þ

Pðq�1Þ�wðtÞ, ð29Þ

wðtþ1Þ ¼ Aðq�1Þ � yðtþ1Þ�q�dBnðq�1Þ � uðtÞ, ð30Þ

where Bðq�1Þuðtþ1Þ ¼ Bnðq�1ÞuðtÞ.Taking into account the expression of w given by (20), it

results that

w1ðtþ1Þ ¼Aðq�1ÞS0ðq

�1Þ

Pðq�1Þ� pðtþ1Þ: ð31Þ

This means that w1 corresponds to the effect of the disturbanceupon the output when using the central controller, i.e. when theQ-parameters are null (Aðq�1ÞS0ðq

�1Þ=Pðq�1Þ is the output sensi-tivity function associated to the central controller). Therefore

e0ðtþ1Þ is a measure of the regulation error and the term yTðtÞfðtÞ

corresponds to the disturbance compensation introduced by theQ-parameters at instant t.

The a posteriori adaptation error given in (22) gets the form

eðtþ1Þ ¼w1ðtþ1Þ�yTðtþ1ÞfðtÞ:

For the estimation of the parameters of Q ðt,q�1Þ the followingparameter adaptation algorithm is used (Landau et al., 1997):

yðtþ1Þ ¼ yðtÞþFðtÞfðtÞeðtþ1Þ, ð32Þ

eðtþ1Þ ¼e0ðtþ1Þ

1þfTðtÞFðtÞfðtÞ

, ð33Þ

e0ðtþ1Þ ¼w1ðtþ1Þ�yTðtÞfðtÞ, ð34Þ

Fðtþ1Þ ¼1

l1ðtÞFðtÞ�

FðtÞfðtÞfTðtÞFðtÞ

l1ðtÞ

l2ðtÞþfTðtÞFðtÞfðtÞ

2664

3775, ð35Þ

1Zl1ðtÞ40; 0rl2ðtÞo2, ð36Þ

where l1ðtÞ,l2ðtÞ allow to obtain various profiles for the evolutionof the adaptation gain F(t) (for details see the end of this sectionand Landau et al., 1997; Landau & Zito, 2005 as well as Sections6 and 7). This algorithm will minimize a quadratic criterion interms of the regulation error having the form

Jð1,t1Þ ¼Xt1

t ¼ 1

f ½l1ðt1,tÞ,l2ðt1,tÞ�½w1ðtÞ�yTðt1Þfðt�1Þ�2, ð37Þ

where the weighting f ½l1ðt1,tÞ,l2ðt1,tÞ�40 depends upon thechoices of l1ðtÞ,l2ðtÞ (i.e. the profile of the adaptation gain F(t)).5

For a stability analysis of this scheme see Landau et al. (2005).In order to implement this methodology for disturbance rejection

(see Fig. 3), it is supposed that the plant model z�dBðz�1Þ=Aðz�1Þ isknown (identified) and that it exists a controller ½R0ðz

�1Þ,S0ðz�1Þ�

which verifies the desired specifications in the absence of thedisturbance. One also supposes that the degree nQ of the polynomialQ ðz�1Þ is fixed, nQ ¼ nDp

�1, i.e. the structure of the disturbanceis known.

The following procedure is applied at each sampling time foradaptive operation. At tþ1:

1.

dec

squ

Get the measured output yðtþ1Þ and the applied control u(t) tocompute wðtþ1Þ using (30).

2.

Compute w1ðtþ1Þ and w2ðtÞ using (28) and (29) with P givenby (15) as well as e0ðtþ1Þ using (27).

5 For example, if one takes l1ðtÞ ¼ 1 and l2ðtÞ ¼ 1 for all t one gets the

reasing adaptation gain algorithm. In this case f¼1 in the criterion (37) (least

ares type criterion).

3.

of t

Estimate the Q-polynomial using the parametric adaptationalgorithm (32)–(35).

4.

Compute and apply the control (see Fig. 3):

S0ðq�1Þ � uðtþ1Þ ¼�R0ðq

�1Þ � yðtþ1Þ�Q ðtþ1,q�1Þ �wðtþ1Þ: ð38Þ

In the adaptive operation one uses in general an adaptation gainupdating with variable forgetting factor l1ðtÞ (the forgetting factortends towards 1), combined with a constant trace adaptation gain.The equation for updating the variable forgetting factor is

l1ðtÞ ¼ l0:l1ðt�1Þþð1�l0Þ ð39Þ

with l1ð0Þ, l0 ¼ 0:9520:99.Once the trace of the adaptation gain is below a given value,

one switches to the constant trace gain updating. The trace of theadaptation gain F(t) is maintained constant by modifying appro-priately l1ðtÞ for a fixed ratio aðtÞ ¼ l1ðtÞ=l2ðtÞ. The correspondingformula is

tr Fðtþ1Þ ¼1

l1ðtÞtr FðtÞ�

FðtÞfðtÞfðtÞT FðtÞ

aðtÞþfðtÞT FðtÞfðtÞ

" #¼ tr FðtÞ, ð40Þ

where aðtÞ ¼ l1ðtÞ=l2ðtÞ is kept constant.The advantage of the constant trace gain updating is that the

adaptation move in an optimal direction (least squares) but thesize of the step does not goes to zero. For details see Landau andZito (2005) and Landau et al. (1997).

For the self-tuning operation of the adaptive scheme, theestimation of the Q-polynomial starts once the level of the outputis over a defined threshold. A parameter adaptation algorithm(32)–(35) with decreasing adaptation gain (l1ðtÞ ¼ 1 and l2ðtÞ ¼ 1)or with variable forgetting factor and decreasing adaptation gain isused. The estimation can be stopped when the adaptation gain isbelow a pre-specified level.6 The controller is either updated ateach sampling or is updated only when the estimation of the newQ-parameters is finished.

5. Indirect adaptive regulation

The methodology proposed in this section concerns the indir-ect adaptive regulation for the attenuation of disturbances andconsists of two steps: (1) Identification of the disturbance model.(2) Computation of a digital controller using the identifieddisturbance model.

For the identification of the disturbance model one has to setfirst a reliable observer of the disturbance.

Provided that the model of the secondary path is known(which is the case in the context of adaptive regulation) anestimation yðtÞ of the disturbance pðtÞ can be obtained by(Bodson & Douglas, 1997; Gouraud et al., 1997):

yðtÞ ¼ yðtÞ�q�dB

AuðtÞ: ð41Þ

However as it can be seen from (20)

wðtÞ ¼ Aðq�1Þ � yðtÞ�q�d � Bðq�1Þ � uðtÞ ¼ Aðq�1Þ � pðtÞ: ð42Þ

The signal w(t) (see also Fig. 3) can be viewed as an estimationof the disturbance filtered through the polynomial Aðq�1Þ. Bothsolutions for getting an estimation of the disturbances can beused. For numerical reasons it is recommended to use w(t) (sincethe polynomial Aðq�1Þ may have roots close to the unit circle).

6 The magnitude of the adaptation gain gives an indication upon the variance

he parameter estimation error—see for example Landau et al. (1997).

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–11811174

The disturbance is considered as a stationary signal having arational spectrum. As such it may be considered as the output of afilter with the transfer function Npðz�1Þ=Dpðz�1Þ and a white noiseas input:

Dpðq�1Þ � pðtÞ ¼Npðq

�1Þ � eðtÞ or pðtÞ ¼Npðq�1Þ

Dpðq�1Þ� eðtÞ, ð43Þ

where e(t) represents a Gaussian white noise, and

Npðz�1Þ ¼ 1þnp1

z�1þ � � � þnpnNpz�nNp ¼ 1þz�1Nn

pðz�1Þ,

Dpðz�1Þ ¼ 1þdp1

z�1þ � � � þdpnDpz�nDp ¼ 1þz�1Dn

pðz�1Þ:

Therefore the disturbance model can be represented by an ARMAmodel. For narrow band disturbances, the filtering effect of theprimary path in cascade with the output sensitivity function(when operating in closed loop) around the central frequency ofthe disturbance can be approximated by a gain and a phase lagwhich will be captured by the Npðz�1Þ=Dpðz�1Þ model.7

From Eq. (43) one obtains

pðtþ1Þ ¼�XnDp

i ¼ 1

dpipðt�iþ1Þþ

XnNp

i ¼ 1

npieðt�iþ1Þþeðtþ1Þ: ð44Þ

The problem is, in fact, an on-line adaptive estimation of para-meters in the presence of noise (Landau et al., 1986, 1997).Eq. (44) is a particular case of identification of an ARMAX model.One can use for example the recursive extended least squares

method (Landau et al., 1997), which is dedicated to the identifica-tion of this type of model. The parameter adaptation algorithmgiven in (32)–(35) is used. In order to apply this methodology, it issupposed that the plant model is known (can be obtained bysystem identification). It is also assumed that the degrees nNp

andnDp

of Npðz�1Þ respectively Dpðz�1Þ are fixed (they depend upon thestructure of the disturbance model). The controller containing thedisturbance dynamics is computed by solving the diophantineequation (8) and using (5)

Aðz�1ÞHSðz�1ÞS0ðz�1Þþz�dBðz�1ÞRðz�1Þ ¼ Pðz�1Þ ð45Þ

with HSðz�1Þ ¼ Dpðz�1Þ (the current estimated model of the dis-

turbance). However other controller design strategies can be used.In adaptive operation, the parameters of the controller have to

be re-computed at each sampling instant based on the currentestimation of the disturbance model (the disturbance estimationalgorithm will use a nonvanishing adaptation gain). For the self-

tuning operation the algorithm for disturbance model estimationuses a parameter adaptation algorithm with decreasing adapta-tion gain. One can either update the controller parameters at eachsample, or one can keep constant the parameters of the controllerduring disturbance estimation and up-date the parameters of thecontroller once the estimation has converged (a measure of theconvergence is the trace of the adaptation gain).

The drawback of this approach is the complexity of thecomputations which have to be done at each sampling sinceone has to solve a Bezout equation of a dimension which dependson the order of the model of the secondary path and the order ofthe disturbance model (often the order of the model of thesecondary path is very high). A reduction of the complexity isobtained if one uses a Youla–Kucera parametrization of thecontroller. In this case one has to solve Eq. (17) instead ofEq. (45). The usefulness of indirect adaptive regulation comes inview when instead of rejection of large number of narrow band

7 For a sinusoid one has nDp¼ 2 and nNp

¼ 2.

disturbances or wide band disturbances, one considers for robust-ness reasons only an attenuation of the disturbance.8

6. Application 1—adaptive rejection of narrow banddisturbances on an active suspension

6.1. The active suspension

The structure of the system (the active suspension) is presentedin Fig. 4. Two photos of the system are presented in Fig. 5 (Courtesyof Hutchinson Research Center, Vibrachoc and GIPSA-LAB, Grenoble).It consists of the active suspension, a load, a shaker and thecomponents of the control scheme. The mechanical construction ofthe load is such that the vibrations produced by the shaker, fixed tothe ground, are transmitted to the upper side of the activesuspension. The active suspension is based on a hydraulic systemallowing to reduce the over-pressure at the frequencies of thevibration modes of the suspension. The controller will act uponthe piston (through a power amplifier) in order to reduce theresidual force. The sampling frequency is 800 Hz. The equivalentcontrol scheme is shown in Fig. 6. The system input, u(t), is theposition of the piston (see Figs. 4 and 6), the output y(t) is theresidual force measured by a force sensor. The transfer function(q�d1 C=D) between the disturbance force, up, and the residualforce y(t) is called primary path. In our case (for testing purposes),the primary force is generated by a shaker controlled by a signalgiven by the computer. The plant transfer function (q�dB=A)between the input of the system, u(t), and the residual force iscalled secondary path. The input of the system being a positionand the output a force, the secondary path transfer function has adouble differentiator behavior. The sampling frequency is 800 Hz.

The control objective is to reject the effect of unknown narrowband disturbances on the output of the system (the residualforce). The system has to be considered as a ‘‘black box’’. Detailedexperimental results can be found in Landau et al. (2005), hereonly some of the most illustrative results will be presented.

6.2. Identification of the active suspension

The identification procedure in open- and closed-loop opera-tions for the active suspension is discussed in detail in Landau,Karimi, et al. (2001), Landau, Constantinescu, et al. (2001), andKarimi (2002). The frequency characteristic of the identifiedprimary path model (open-loop identification), between the signalup sent to the shaker in order to generate the disturbance and theresidual force y(t), is presented in Fig. 7. The first vibration modeof the primary path model is near 32 Hz. The frequency character-istic of the identified secondary path model (closed-loop identifi-cation) is presented also in Fig. 7. This model has the followingcomplexity: nB ¼ 14, nA ¼ 16, d¼0. The identification has beendone using as excitation of the piston a PRBS (pseudo-randombinary sequence) with frequency divider p¼4 (for details on thePRBS signals see Landau & Zito, 2005). There exist several very lowdamped vibration modes on the secondary path, the first onebeing at 31.8 Hz with a damping factor 0.07. The identified modelof the secondary path has been used for the design and imple-mentation of the controller. The central controller (without theinternal model of the disturbance) has been designed using thepole placement method and the secondary path identified model.

8 A simple solution, in the case of narrow band disturbances, is to replace in

Eq. (45) HSðz�1Þ ¼ Dpðz�1Þ with roots on the unit circle by HSðz

�1Þ ¼Dpðz�1Þ with

roots inside the unit circle.

controller

residual force

primary force (disturbance)

1

23

4

machine

support

elastomere cone

inertia chamber

piston

main

chamber

hole

motor

actuator

(piston position)

Fig. 4. Active suspension system (scheme).

Fig. 5. Active suspension system (photo).

q-dB/AR/S

Secondary path

u(t) y(t)

+

+

-

Controller

q-d1C/D

up (t)(disturbance)

Primary path

Residual force

p (t)

Fig. 6. Block diagram of active vibration suppression systems.

0 50 100 150 200 250 300 350 400−40

−30

−20

−10

0

10

20

30

40

Primary and Secondary Path Models

Mag

nitu

de [d

B]

Frequency [Hz]

Secondary PathPrimary Path

Fig. 7. Frequency characteristics of the primary and secondary paths.

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–1181 1175

The resulting central controller has the following complexity:nR ¼ 14, nS ¼ 16 and it satisfies the imposed robustness constraintson the sensitivity functions (for the design methodology seeLandau & Zito, 2005).9

6.3. Rejection of sinusoidal disturbances—real time results

Direct adaptive regulation under the effect of sinusoidal distur-

bances of various frequencies: In order to evaluate the performancesin real time, time-varying frequency sinusoidal disturbancesbetween 25 and 47 Hz have been used (the first vibration modeof the primary path is near 32 Hz). Only some results in theadaptive operation regime will be presented here (for more detailssee Landau et al., 2005). For the adaptive regime, the spectraldensities of the residual force obtained in open and in closed loop,respectively, using the direct adaptation scheme (after the algo-rithm converges) are presented in Fig. 8. The results are given forthree frequencies: 25, 32 and 47 Hz. The attenuations obtained arelarger than 49 dB for all the frequencies. Similar results are obtained

9 Any design method allowing to satisfy these constraints can be used.

in the self-tuning regime as well as with the indirect adaptationalgorithm.

Direct adaptive regulation under the effect of step changes in the

frequency of sinusoidal disturbances: The measured residual force

0 50 100 150 200 250 300 350 400−70

−60

−50

−40

−30

−20

−10

0

10

20

Frequency [Hz]

Spectral densities of the residual force.Direct method in adaptive 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 adaptive regulation scheme in adaptive operation.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

−6

−4

−2

0

2

4

6

Direct method in adaptive operation

Samples

Res

idua

l for

ce [V

]

32 Hz 25 Hz 32 Hz 47 Hz 32 Hz

Fig. 9. Time domain results with the direct adaptive regulation scheme in the

adaptive case.

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–11811176

obtained in adaptive operation is presented in Fig. 9 for the directadaptive regulation scheme. As it can be observed the adaptationtransient is very fast (less than 0.25 s—for a zoom see Landauet al., 2005). This explain the capabilities of the scheme to handlenarrow band disturbances with time varying frequency (up to acertain speed of variation). The system is started in open loop andthe loop is closed simultaneously with the application of the firstsinusoidal disturbance. Every 15 s (12 000 samples) sinusoidaldisturbances of different frequency are applied (32, 25, 32, 47,32 Hz). An adaptation gain with variable forgetting factor combined

with a constant trace (see Section 4 and Landau et al., 1997;Landau & Zito, 2005) has been used in order to be able to trackautomatically the changes of disturbance characteristics. The lowlevel threshold of the trace has been fixed at 3� 10�9 for thedirect adaptation algorithm.

Direct adaptive regulation under the effect of sinusoidal distur-

bances with continuously time varying frequency: Consider now

that the frequency of the sinusoidal disturbance varies continu-ously and let us use a chirp disturbance signal (linear swept-frequency signal) between 25 and 47 Hz. The tests have beendone as follows: Start up in closed loop at t¼0 with the centralcontroller. Once the loop is closed, the adaptation algorithmworks permanently and the controller is updated (directapproach) at each sampling instant. After 5 s a sinusoidaldisturbance of 25 Hz (constant frequency) is applied on theshaker. From 10 to 15 s a chirp between 25 and 47 Hz is applied.After 15 s a 47 Hz (constant frequency) sinusoidal disturbance isapplied and the tests are stopped after 18 s. The time-domainresults obtained in open and in closed-loop (direct adaptiveregulation) are presented in Fig. 10. The performances obtainedare very good. Excellent rejection of a sinusoidal disturbance witha speed of variation of the frequency around 4.4 Hz/s is assured.

7. Application 2—adaptive rejection of multiple narrow banddisturbances on an active vibration control system using aninertial actuator

7.1. The inertial actuator

In this application a different technological approach is used forsuppressing the effect of vibrational disturbances. Instead of usingan active suspension, one uses an inertial actuator which will createvibrational forces to counteract the effect of vibrational distur-bances (inertial actuators use a similar principle as loudspeakers).The structure of the system is described in Fig. 11. It consists of astandard passive damper and an inertial actuator fixed to thechassis where the vibrations should be attenuated. The testingsetting is exactly the same as for the active suspension (see Fig. 5).

The controller will act on the inertial actuator (through apower amplifier) in order to reduce the residual force.The equivalent control scheme is shown in Fig. 6. The systeminput is the position of the mobile part of the actuator. Like for theactive suspension, the secondary path has a double differentiatorbehavior. The system has to be considered as a ‘‘black box’’ andthe control objectives are similar to those for the active suspen-sion. The sampling frequency is 800 Hz.

7.2. Results obtained with the inertial actuator

The performance of the system for rejecting multiple unknowntime varying narrow band disturbances will be illustrated using thedirect adaptive control scheme presented in Section 4. Since twosimultaneous time varying frequency sinusoids will be consideredas disturbances, one should take nDp

¼ 4 and nQ ¼ nDp�1¼ 3.

Same procedure for system identification in open and closedloop, as for the active suspension, has been used. The frequencycharacteristics of the primary path (identification in open loop)and of the secondary path (identification in closed loop) areshown in Fig. 12. The secondary path has the following complex-ity: nB ¼ 12, nA ¼ 10, d¼0. The identification has been done usingas excitation a PRBS (with frequency divider p¼2 and N¼9).There exist several low damped vibration modes in the secondarypath, the first vibration mode is at 51.58 Hz with a damping of0.023 and the second at 100.27 Hz with a damping of 0.057. Onlythe ‘‘adaptive’’ operation regime has been considered for thesubsequent experimental results.

Direct adaptive regulation under the effect of two sinusoidal

disturbances: Fig. 13 shows the spectral densities of the residualforce obtained in open loop and in closed loop using the directadaptation scheme (after the adaptation algorithm has con-verged). The results are given for the simultaneous applicationof two sinusoidal disturbances (65 and 95 Hz). One can remark

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]

Resid

ual fo

rce [V

]

Chirp disturbance in open−loop

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]

Resid

ual fo

rce [V

]

Direct adaptive control−chirp disturbance in closed loop

25Hz(const) 47Hz(const)25→ 47Hz

25Hz(const) 25→ 47Hz 47Hz(const)

Fig. 10. Real-time results obtained with the direct adaptive regulation scheme and a chirp disturbance: (a) open loop; (b) closed loop.

Fig. 11. Active vibration control using an inertial actuator (scheme).

0 50 100 150 200 250 300 350 400−80

−60

−40

−20

0

20

Primary and secondary path models

Frequency (Hz)

Am

plitu

de (d

B)

Secondary pathPrimary path

Fig. 12. Frequency characteristics of the primary and secondary paths (inertial

actuator).

0 50 100 150 200 250 300 350 400−80

−70

−60

−50

−40

−30

−20

−10

0Power Spectral Density Estimate

Frequency (Hz)

PSD

Est

imat

e (d

B)

Direct Adaptive Control (65+95Hz)Open Loop (65+95Hz)

Fig. 13. Spectral densities of the residual force in open and in closed loop, with the

direct adaptive regulation scheme in adaptive operation (inertial actuator).

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–1181 1177

a strong attenuation of the disturbances (larger than 45 dB). Notethat there exists a significant measurement noise at 50 Hz (powernetwork) which is not amplified in closed loop.

Direct adaptive regulation under the effect of step changes in the

frequency of two sinusoidal disturbances: Time domain resultsobtained with direct adaptation scheme in ‘‘adaptive’’ operationregime are shown in Fig. 14. The disturbances are applied at 1 s(the loop has already been closed) and step changes of theirfrequencies occur every 3 s. Fig. 15 shows the correspondingevolution of the parameters of the polynomial Q. The convergenceof the output requires less than 0.4 s in the worst case.

Comparison between direct and indirect adaptive regulation

(adaptive regime): In Fig. 16 the results for direct adaptiveregulation for the case of a single sinusoidal disturbance in the

0 2 4 6 8 10 12 14 16−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (sec)

Res

idua

l for

ce (V

)

Adaptive Disturbances Rejection (sum of 2 sinusoids)

95Hz+65Hz 100Hz+70Hz 95Hz+65Hz 90Hz+60Hz 95Hz+65Hz

Fig. 14. Time domain results with direct adaptive regulation for simultaneous

step changes of two sinusoidal disturbances (inertial actuator).

0 2 4 6 8 10 12 14 16−5

−4

−3

−2

−1

0

1

2

3

4

5Convergence of Adaptation Parameters

Time (sec)

Ada

ptat

ion

para

met

ers

Fig. 15. Evolution of the parameters of the polynomial Q during adaptation

(inertial actuator).

0 2 4 6 8 10 12 14 16

−0.4

−0.2

0

0.2

0.4

0.6

Time (sec)

Res

idua

l for

ce (V

)

Direct adaptive disturbance rejection

95Hz90Hz95Hz100Hz95Hz

Fig. 16. Time domain results with direct adaptive regulation for a single

sinusoidal disturbance with step changes in frequency (adaptive regime).

0 2 4 6 8 10 12 14 16

−0.4

−0.2

0

0.2

0.4

0.6

Time (sec)

Res

idua

l for

ce (V

)

Adaptive disturbance rejection

95Hz90Hz95Hz100Hz95Hz

Fig. 17. Time domain results with indirect adaptive regulation for a single

sinusoidal disturbance with step changes in frequency (adaptive regime).

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–11811178

presence of step changes in frequency are shown. In Fig. 17 theresults obtained in the same context with an indirect adaptiveregulation scheme are shown. Contrary to the results given inLandau et al. (2005) for the adaptive operation, here the perfor-mances for indirect and direct scheme are quite similar.The explanation is that here an observer for the disturbancewas introduced while in Landau et al. (2005), one uses directly theoutput measurement as an estimation of the disturbance.

8. Review of other applications

Four applications using the internal model approach foradaptive regulation will be reviewed next.

8.1. An adaptive active noise control system for acoustic ducts

In Amara et al. (1999b) an adaptive controller design approachbased on the Youla–Kucera parametrization and the internal modelprinciple is implemented in order to solve a noise cancelationproblem in an acoustic duct. The design approach is presented inAmara et al. (1999a) and it is similar to the method presented inSection 4. The plant considered is presented in Fig. 18. It consists of aduct, a pair of speakers, a pair of microphones and their amplifiers,and an anti-aliasing filter. The first speaker is used to generate thenoise signal, the second one is used to generate the control signal tocancel the noise at a particular location in the duct (disturbance andcontrol speakers). The two microphones are used as measurementand performance evaluation. The model of the plant has beenobtained by system identification. The identification experimentwas performed by first exciting the disturbance speaker with whitenoise and collecting data from the output of the anti-aliasing filter.The same identification procedure was repeated for the controlspeaker. A spectrum analyzer provided the frequency response ofthe plant based on the collected data. A direct adaptation of theYoula–Kucera parameters is done. The experiment consideredconsists of exciting the disturbance speaker with a single tonesinusoidal signal and running the adaptive controller in order tocancel the noise at the performance microphone. The disturbancefrequency takes values from 100 to 300 Hz (with a frequency step of10 Hz). The sampling period used is 0.0011 s. The experimentalresults show a mixed performance for the adaptive systems withgood performance only in certain disturbance frequency ranges. Atsome ranges of the disturbance frequency, the adaptive systems

SpectrumAnalyzer

AntiAliasing

Filter

AdaptiveController

MeasurementMicrophone

PerformanceMicrophone

ControlSpeaker

DisturbanceSpeaker

Fig. 18. Block diagram of the noise cancelation system.

Gw

FC GpPsp P

w

Bioreactor

-

-

Fig. 19. Control strategy of the fed-batch fermenter.

Fig. 20. A flexible distributed mechanical structure with active disturbance

compensation (CARV)—photo.

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–1181 1179

achieved 20–50 dB reduction in the performance signal PSD (powerspectral density) at the disturbance frequency and also good overallreduction in the PSD of the signal. At some other disturbancefrequency ranges, the performance was not as good. It has beenconcluded that the unmodeled plant nonlinearities are most likely tobe responsible for the observed deterioration of the adaptive systemperformance at some frequencies in experiments as compared tosimulations.

8.2. Adaptive regulation of a fed-batch reactor

An algorithm very similar to the one presented in this paper fordirect adaptive regulation is presented in Valentinotti (2001) andValentinotti et al. (2003). The application considered to illustratethe performances of the proposed algorithm is the Saccharomyces

cerevisiae fed-batch fermentation. It is an example of a systemsubject to an unstable disturbance (but over a finite time).

The objective of the application was to maximize the biomassproductivity in the reactor. Hence, the substrate concentration inthe reactor needs to be kept at the critical value in order tomaintain a constant ethanol concentration. However, the substrateconcentration is unknown a priori and may change from experi-ment to experiment. So the problem of maximizing the productionof biomass is converted into that of regulating the concentration ofethanol. The plant is considered as an integrator that constantlyproduces ethanol. The substrate consumption by the cells ismodeled as an exponential disturbance which the controller hasto reject. The methodology has been successfully implemented inan experimental setup in the laboratory. So the control objective isto keep the concentration of this fermentation product constant.

The control strategy is presented in Fig. 19. The controller C

generates the substrate feed rate F that is capable of maintainingexponential cell growth and the desired production of ethanol.The controller will be able to reject the exponential substrateconsumption disturbance w, if the rate at which the substrate issupplied is sufficient. Since the cells grow exponentially, it isexpected that the substrate feed rate also be exponential.An adaptive disturbance rejection methodology is used to control

the fed-batch fermentation near its optimal operating point.The biomass and ethanol productions rates have been decoupled,as it can be seen in Fig. 19, where Gw relates the rate of substrateoxidation reaction to the rate at which the substrate is fed (thediscrete transfer function of the disturbance) and Gp relates thesubstrate feed rate to the ethanol production rate. S and P arethe concentrations of the substrate and ethanol, respectively, andPsp is the ethanol concentration setpoint.

The derived linear models have been used successfully toimplement the direct adaptive disturbance rejection methodologyusing the Q-parametrization (similar to the method described inSection 4). The ethanol concentration in the bioreactor is the onlymeasurement required. The only parameter that is estimated on-line is the exponential cell growth rate.

8.3. Adaptive rejection of narrow band disturbances in a distributed

flexible mechanical structure

A view of the mechanical structure (named CARV) is given inFig. 20 (courtesy of GIPSA-LAB, Grenoble). The structure of thesystem is presented in Fig. 21. The system consists of three mobilemetallic plates (M1, M2, M3) connected by springs. The first and thethird plates are also connected by springs to the rigid part of thesystem formed by two other metallic plates connected togetherrigidly. The upper and lower mobile plates (M1 and M3) areequipped with inertial actuators. The one on the top serves asdisturbance generator, the one at the bottom serves for disturbancecompensation. The system is equipped with a measure of theresidual acceleration (on plate M3) where the effect of thedisturbances has to be attenuated. The mechanical structureis representative for a variety of situations encountered in practice.

A direct adaptive regulation scheme of the type given in Section4 is used for the rejection of vibrations (narrow band disturbances).The identified transfer functions of the primary path and secondarypath are shown in Fig. 22. It can be observed that there are anumber of low damped vibration modes which make the systemsensitive to narrow band disturbances. Fig. 23 shows the resultsobtained for the adaptive rejection of a sinusoidal disturbance withvariable frequency (step changes). More details can be found inAlma, Landau, Martinez, and Buche (2009).

Residual force

Inertial actuator

Control signal

Controller

Power

Amplifier

M1

M2

M3

SupportPrimary force

(Disturbance)

Support

Fig. 21. A flexible distributed mechanical structure with active disturbance

compensation (CARV)—scheme.

0 50 100 150 200 250 300 350 400−80

−60

−40

−20

0

20

40

Frequency [Hz]

Mag

nitu

de [d

B]

Primary and secondary path models

Secondary pathPrimary path

Fig. 22. Frequency characteristics of the primary and secondary paths of the CARV

structure.

0 5 10 15 20 25−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time [sec]

Res

idua

l for

ce [V

]

Adaptive disturbance rejection

83Hz 86Hz 83Hz 80Hz 83Hz

Fig. 23. Rejection of sinusoidal disturbances with step changes in frequency on

the CARV structure.

Fig. 24. The DaTcube, a Blu-ray disc tester. Courtesy of DaTarius.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Time (sec)

Tra

ckin

g E

rror

(μm

)

Fig. 25. Tracking error at 4� rotational disc speed (adaptation starts at 0.2 s).

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–11811180

8.4. Adaptive suppression of main periodic disturbances in Blu-ray

disk drives servomechanisms

A relevant control problem in optical disk drives is the trackingcontrol of the radial lens position. Due to rotational nature of themechanism, deformations are sources of periodic disturbances. InBlu-ray discs, the main source of such disturbances is the disk

eccentricity. A direct adaptive regulation scheme has been designed,similar to the one described in Section 4. An extensive experimentalstudy has been carried out in order to validate this approach using aBlu-ray disc tester developed by DaTARIUS Technology (Austria).A photo of this equipment can be seen in Fig. 24. Fig. 25 showsexperimental results indicating a significative reduction of thetracking error at high playback speed, when the adaptive regulationscheme is connected. More details can be found in Martinez (2009).

9. Conclusions

The paper has reviewed direct and indirect adaptive regulationschemes using the internal model principle and the Youla–Kuceraparametrization of the controllers. The use of this methodology has

I.D. Landau et al. / Control Engineering Practice 19 (2011) 1168–1181 1181

been illustrated in detail on two applications: an active suspension,and an active vibration control system using an inertial actuator.Other applications of this approach have also been reviewed.The experimental results available on a variety of applicationsallow to conclude that this methodology is ready to be applied.

Extensions to the multivariable case of the approachesdiscussed in this paper have been considered. See for exampleFicocelli and Amara (2009).

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