Active Sound and Vibration Control: theory and applications Volume 49 || Adaptive methods in active...

Chapter 3

Adaptive methods in activecontrol

S. J. ElliottInstitute of Sound and Vibration Research, University of Southampton,Southampton, UKEmail: [email protected]

In this chapter adaptive feedforward and fixed feedback controllers are reviewedand their robust stability conditions contrasted for active control applications.

3.1 Introduction

Active control systems, and particularly active sound control systems, havea number of distinguishing features compared with other control problems.The disturbance to be rejected often has broadband and narrowband com-ponents, with the latter often being nonstationary. The plant is generallyhigh order, well damped and nonminimum phase, and its response can changerapidly, due for example to the movement of people within an enclosure. Thenarrowband disturbance components are often attenuated with a feedforwardcontroller which is made adaptive to track the nonstationarities. The filtered-reference LMS algorithm, which is generally used to adapt such controllers,is particularly robust to changes in plant response. If the plant is subject tounstructured multiplicative uncertainty, the algorithm is stable provided theupper bound of the uncertainty is less than unity. The algorithm can be madeeven more robust by introducing a leakage term into the update equation.Feedback controllers can be used to control the broadband components of the

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58 Active Sound and Vibration Control

disturbance, but their performance bandwidth is limited by propagation de-lay in the plant. Such feedback controllers must be robustly stable to plantchanges, and the internal model control (IMC) architecture allows the condi-tion for robust stability to be related back to that for the adaptive feedforwardsystem. The feedback controller can be made adaptive to nonstationary dis-turbances by adapting the control filter in the IMC arrangement. The robuststability condition can be maintained during adaptation by incorporating aconstraint into the adaptation algorithm, which can be implemented with theleast loss of performance by using an adaptation algorithm operating in thediscrete frequency domain.

Adaptive methods axe used in active control systems to automatically ad-just the response of the controller to compensate for changes in the responseof the plant or in the form of the excitation. Active systems for the control ofsound and vibration are generally designed to reject the disturbances gener-ated by the original, primary, course, rather than to track a command signal,and thus the design of the controller which gives the best performance willdepend on the statistical properties of the disturbance. If the disturbance isnonstationary, as it is in many applications of active control, the controllermust be adapted to track the changes in the properties of the disturbanceif good control is to be maintained. In many applications of active control,particularly active sound control, some of the changes which occur in the re-sponse of the plant are too rapid to identify effectively without introducingan unacceptably high level of identification noise. Externally generated iden-tification noise is generally necessary to measure the plant response in activecontrol since the disturbance is not sufficiently rich. For rapid changes inplant response the level of identification noise required to accurately track thechange can contribute more to the perceived acoustic output than the originaldisturbance, which rather defeats the object of the controller. If the plantresponse can only be identified under nominal conditions, but varies aboutthis nominal response in a way which cannot be measured, then it is neces-sary for the control loop and the adaptation algorithm which compensates thecontrol loop, for changes in the disturbance, to be robust to these changes. Inthis chapter, adaptation methods for active control systems will be describedwhich are robust to changes in the plant response. Because the controller canbe made adaptive, feedforward control is no longer open loop and is widelyused in active control when an independent reference signal is available, whichis well correlated with the disturbance. If the controller is implemented asa digital FIR filter, or an array of filters for the multichannel case, it canbe adapted to minimise a quadratic cost function using the filtered-referenceLMS algorithm. The convergence properties and robustness of this algorithm


Adaptive methods in active control 59

-h

x(n)Incidentplanesound wave

Reference Electronicsignal controller

Yu(n) e(n)

Secondary Erroractuator signal

Figure 3.1 Example of the application of a single-channel feedforward controlsystem to the active control of plane waves of sound in a duct.

are briefly described in both the single channel and multichannel cases. Inapplications where no independent reference signals are available active con-trol systems can be implemented using feedback control. In this case thereare two dangers from changes in the plant response: first that the feedbackloop itself may become unstable, and second that the adaptation algorithmmay become unstable. Both of these possibilities will be considered and it willbe demonstrated how the adaptation algorithm can be modified to minimisea quadratic cost function with the constraint that the feedback loop remainsrobustly stable.

3.2 Feedforward control

In this section we consider the performance and stability of adaptive controlalgorithms for feedforward control systems. The case of a single-channel sys-tem is considered first, with a single reference signal, secondary actuator anderror sensor, but with a general broadband excitation. Such a control systemis typically used in the active control of sound in ducts, for example. Second,the behaviour is considered of a system with multiple actuators and error sen-sors but only when it is excited by relatively narrowband excitation about aknown frequency. Such systems are used for the active control of sound inpropeller aircraft and helicopters.

3.2.1 Single-channel feedforward control

The application of a single-channel feedforward control system to the activecontrol of plant sound waves in a duct is illustrated in Figure 3.1. The wave-



form of the incident disturbance, which is assumed to be stochastic and broad-band, is measured by the detection sensor, a microphone in this case, and fedforward to the electronic controller the output of which drives the secondaryactuator, which is a loudspeaker in this case. This arrangement was first sug-gested by Lueg in 1936 [180]. In order to monitor the performance and providea feedback signal for the adaptation of the controller an error sensor is locatedfurther downstream to measure the residual soundfield. Typically, the elec-tronic controller is implemented digitally, since it is easier to adapt a digitalfilter, and as well as the usual data converters, effective antialiasing filters andreconstruction filters are included in the controller to prevent aliasing whichwould otherwise be audible. Also, the possible acoustic feedback path fromthe secondary actuator back to the detection sensor can be compensated forinside the controller, by using a digital filter to model this response and usingan arrangement similar to the echo canceller used in telecommunications sys-tems, as described for example in Reference [235]. The block diagram of thesingle channel feedforward control system can now be drawn as in Figure 3.2,from which the sampled error signal can be written as:

e(n) = d(n) + gTu(n) (3.1)

where d(n) is the disturbance, g is the vector of impulse response coefficientsof the plant and u(n) is the vector of past values of the input signal to theplant. If the control filter is linear and time invariant, the ordering of thecontrol filter and plant in Figure 3.2,(a) can be notionally transposed and theblock diagram redrawn as in Figure 3.2,(6), in which case the sampled errorsignal can be written in the alternative form:

(n) (3.2)

where w is the vector of coefficients of the FIR control filter and r(n) is thevector of past values of the reference signal filter by the plant response. Eqn.(3.2) allows a cost function equal to the mean square error signal:

J = E[e2(n)] (3.3)

where E denotes the expectation operator, to be written as a quadratic func-tion of the coefficients of the control filter. This may be differentiated withrespect to the filter coefficients and the instantaneous value used to updatethe control filters at each sample in a stochastic gradient algorithm called thefiltered-reference LMS algorithm

w(n + 1) = w(n) - ar(n)e(n) (3.4)



(a) x(n) —

Referencesignal

(b)x(n) —

Referencesignal

W(z)u(n)

G(z)

Feedforwardcontroller

Plant

G(z) W(z)

d(n)

Plant Filtered Feedforwardreference controller

signal

Disturbancesignal

Errorsignal

Disturbancesignal

Errorsignal

Figure 3.2 (a) Block diagram of a single-channel feedforward control sys-tem, (b) An equivalent block diagram for a linear time-invariantcontroller.

as described for example in Reference [337], where a is a convergence coef-ficient. In a practical implementation the filtered reference signal must begenerated by passing the reference signal through an estimate of the plantresponse, as shown in Figure 3.3, to generate the estimated filtered referencesignal f(n), and the practical form of the filtered-reference LMS algorithmbecomes:

w(n + 1) = w(n) - af (n)e(n) (3.5)

The convergence behaviour of this algorithm can be analysed by using eqn.(3.2) for e(n) in eqn. (3.5), assuming w(n) and r(n) are independent andtaking expectations to give:

E(w(n + 1) - wo,) = [/ - aE(f(n)fT(n))]E[w(n) - wTO] (3.6)

where w^ is the steady-state value of the filter coefficients if the algorithm isstable, which is given by:

Woo — E(r(n)d(n)) (3.7)

The stability of the algorithm depends on the eigenvalues of the matrix whichin general are complex. The condition for stability is that the real part of each



Adaptivecontroller Plant d(n)

x(n)

-»[ W{z)u(n)

G(z)

G(z)

Plantmodel

Figure 3.3 Block diagram of the practical implementation of the filtered-reference LMS algorithm, in which the plant model is used togenerate the filtered-reference signal.

of these eigenvalues must be positive [269], i.e.:

Re[Eig[£(r(n)r(n)T)]] > 0 (3.8)

In general, it is not straightforward to relate the eigenvalues of this matrix tothe difference between the responses of the physical plant and the plant model.It has, however, been shown [325] that a sufficient condition for convergenceis that the ratio of transfer functions of the plant model and plant, | M , isstrictly positive real (SPR) which implies that stability of the algorithm isassured providing:

Re[G(e'wT)G(e>wT)] > 0, for all uT (3.9)

If the plant model is only in error by a phase shift (j>(toT), so that G(eju}T) =G(ejU3T)(j>(ujT), then eqn. (3.9) reduces to:

cos(<j>{wT)) > 0, for all uT (3.10)

Stability is thus assured provided the phase error is less than 90° at eachfrequency, which is a generalisation of a previous result for a single frequencyreference signal [70]. Alternatively if the plant model can be assumed to beequal to the plant response under nominal conditions, but the plant itself is



assumed to be subject to multiplicative uncertainty, the magnitude of whichis bounded, then:

G(eju}T) = G0(ejujT) (3.11)

andG(^T) = G0(e>'wT)(l + A(e^T)) (3.12)

where| A ( ^ T ) | < B ( ^ T ) (3.13)

In this case the stability condition in eqn. (3.9) becomes:

l + Ke[A{ejuyr)] > 0 (3.14)

If the multiplicative uncertainty is unstructured, so that its phase is unknown,then the worst-case condition for stability will be where the phase of A(e^uT) is180° and its magnitude is equal to its upper bound. In this case the conditionfor the stability of the filtered-reference LMS algorithm reduces to:

B(e*"T) < 1 (3.15)

which can be directly compared with the condition for stability of a conven-tional feedback system, as described below. The filtered-reference LMS algo-rithm can be made more robust to differences between the plant and plantmodel by introducing some leakage into the adaptation eqn, which then be-comes:

w(n + 1) = (1 - a/3)w(n) - af{n)e{n) (3.16)

which minimises a modified quadratic cost function given by:

J = E[e2{n)) + pE[wT(n)w(n)} (3.17)

The convergence condition given by eqn. (3.9) is modified by the leakage termto be of the form:

Re[Gf(ejuT)G(ejwT)}^^ > 0 for all u)T (3.18)

If the plant model is again only in error by a phase shift (J>{UJT), then eqn. 3.18can be used to show that a sufficient condition for stability is:

cos(<t>(ujT)) > |G(~fr)|2 for all <jT (3.19)

So that if |G(eJwT)|2 is particularly small at some frequencies, in particularmuch less than /?, then the right-hand side of eqn. (3.19) will be less than 1



and the algorithm is stable for any phase error in the plant model at this fre-quency. Alternatively if we assume an unstructured multiplicative uncertaintyas in eqns (3.11), (3.12) and (3.13), then the sufficient condition for stability,assuming again the worst-case uncertainty so that A(e^uT) = —S(eJwT), isthat:

< l + lG(ir)|2 for <*

If we again consider the situation in which \G0(ejU}T)\2 is much less than 0

at some frequencies, then the upper bound on the multiplicative uncertaintycould be much greater than unity at these frequencies and yet the algorithmwould remain stable. The value of /? thus allows a trade-off to be made be-tween performance, low /?, and robust stability, high /?. A more precise controlover this trade-off could be achieved if the filtered-reference LMS algorithmwas implemented in the frequency domain and ft was allowed to vary as afunction of frequency. A more complete analysis of the filtered-reference LMSalgorithm, which includes the dynamics of the plant, can be performed forsinusoidal reference signals, which shows that the behaviour of the adaptivefeedforward algorithm can be exactly represented by an equivalent feedbackcontrol system in which the response of the equivalent feedback controller hasa peak at the frequency of the reference signal [269]. The equivalent feedbackcontrol system thus has a notch at this frequency in its response from the dis-turbance input to the error output. The maximum bandwidth of this notch,which gives the maximum bandwidth over which the disturbance can be ef-fectively controlled, is equal to the reciprocal of the delay in the plant [269],which is a condition also encountered below for conventional feedback controlsystems. Active feedforward control systems are widely used to control nar-rowband disturbances caused by rotating or reciprocating sources, because areference signal at the fundamental frequency can typically be obtained from atachometer signal. Active sound control systems with multiple secondary loud-speakers and multiple error microphones have been used to give good controlof the low-frequency tonal noise in the passenger cabin of propeller aircraft forsome time [83] and are now widely used commercially [45]. Such systems needto be adaptive to track nonstationarities in the primary field during differentflight conditions and typically use a fixed model G, of the matrix of plantresponses at the operating frequency G, in their adaptation equation. Thestability condition for such systems, with effort weighting parameter /?, nowtakes the form [9]:

Re[Eig[GHG + pI]] > 0 (3.21)



(a) (b)d(n) Disturbance

oc a

-mo

e(n) i

Plant

G(Z)

•H(Z)

y.—

Error

Primarysound field

Electricalcontroller

Controller

Figure 3.4 (a) Schematic of a feedback controller for the active suppressionof an acoustic disturbance, (b) Its equivalent block diagram.

which only has to be satisfied at the excitation frequency. Very small values of/3 ensure that the control system is stable for the changes in plant response typ-ically encountered in practice, due to people moving around in the passengercabin for example, without significantly degrading the performance.

3.3 Feedback control

3.3.1 Fixed feedback controllers

When no external reference signal is available that is well correlated with thedisturbance, then the output of the error sensor must be used to drive thesecondary actuator via a negative feedback controller, as illustrated in Figure3.4. The ratio of the output of the error sensor after control to that beforecontrol is equal to the sensitivity function of the feedback controller, which isgiven by:

" 1 + G(ju)H(juj)

We shall see in the following section how such an arrangement can be viewed asbeing equivalent to a feedforward system with an internal reference signal, butfor now we will briefly review the robust stability conditions for the feedbackloop. With the plant under nominal conditions the stability of the closedloop is determined by the well known Nyquist criterion applied to the polarplot of the open-loop frequency response. A simple geometric construction



in the Nyquist plane can also be used to derive a condition for the robuststability of a fixed feedback controller when the plant is subject to unstructuredmultiplicative uncertainty of the form given in eqns (3.12) and (3.13), whichcan be written as [95]:

\T(ju)B(ju))\<l for all u> (3.23)

where T(JUJ) is the complementary sensitivity function, which for the feedbackcontroller shown in Figure 3.4 is equal to:

)H(jw)) (3>24)

3.4 Internal model control

An interesting way of looking at the action of the feedback controller is toassume that it is implemented using internal model control [212], which isillustrated in Figure 3.5 for a sampled time implementation, to be consistentwith the discussion of feedforward controllers above. If we assume that theplant is under nominal conditions and that the plant model is equal to the plantresponse under these conditions, so that G(z) = G(z) = GQ(Z), it can be seenfrom Figure 3.5 that the block diagram reduces to an equivalent feedforwardsystem for which the sensitivity function is equal to:

S(z) = §^ = l + W(z)G0(z) (3.25)

Under these conditions the output of the plant model exactly cancels theoutput of the physical plant, and so the signal driving the controller is equalto the disturbance, d(n) = d(n), and this acts as an internal reference signalfor this arrangement. The complementary sensitivity function T(z) is equalto lS(z) which from eqn. (3.25) is given by:

T(z) = -W(z)G0(z) (3.26)

The robust stability condition, eqn. (3.23), can thus be cast in a particularlysimple form for an internal model control system as:

\W(^)G0(e^r)B(e^r)\ < 1 for all uT (3.27)

Eqn. (3.27) illustrates the fact that if the uncertainty B(ejujT) is particularlylarge at some frequencies then the gain of the control filter, |W(e*wr)|, must



W(z)•0»)

* G(z) *&e(n)

Plant model

Gfe) -K>

Figure 3.5 Block diagram of a negative feedback controller implemented usinginternal model control

be reduced correspondingly in order for the robust stability condition in eqn.(3.27) to be satisfied. If the control filter W(z) was an FIR device the optimumH2 controller, which minimises the mean square error, can be derived usingconventional Wiener theory to calculate the coefficients of W(z) under nominalconditions using eqn. (3.25). We have already seen how introducing a termsimilar to control effort into the quadratic cost function being minimised, eqn.(3.17), can reduce the magnitude of the control filter's frequency response atthe expense of a slightly increased mean square error. One way of minimisingthe mean square error while ensuring that the robust stability condition givenby eqn. (3.27) is satisfied would thus be to derive the Wiener filter whichminimised eqn. (3.17) for a sequence of increasingly large values of /J untileqn. (3.27) is satisfied at all frequencies. Another approach, which would givebetter performance than the weighted H2 approach above, is to take a discretegrid of frequency points, i.e. z = e?2irk/N, k = 0 , 1 , . . . , N — 1, and solve theconvex optimisation problem of minimising an H2 performance criterion equalto the mean square error, which is given by the sum of the values of the errorsignal's power spectral density, see(k), in each frequency bin:

min X; see(k) = + W(k)G0(k)\*SM(k)

while maintaining the robust stability constraint that:

\W(k)G0(k)B(k)\ < 1 for all uT

(3.28)

(3.29)



as suggested by Boyd et al. [35].

This method has been implemented using sequential quadratic program-ming by Rafaely and Elliott [19] to derive the frequency response of a feedbackcontroller which minimises the #2/^00 control problem defined by eqns (3.28)and (3.29) for an active acoustic headrest. A low-order controller was then fit-ted to this frequency response and implemented in practice. An advantage ofthis approach is that all the parameters required for the controller design, suchas the nominal plant response, Go(k), the disturbance power spectrum, S^k),and the multiplicative plant uncertainty, B(k), can be directly measured fromsimple experiments on the physical system under control. Also, if the controlloop is made robust to plant uncertainty, it will also tend to be robust toerrors introduced into the controller response when reducing its order. Thebandwidth over which significant reductions can be achieved with a feedbackcontrol system is inherently limited to be less than the reciprocal of the delaysin open loop. In an active sound control system the performance bandwidth isthus fundamentally limited by the propagation delay between the secondaryactuator and error sensor, but any additional delay in the controller will fur-ther reduce this bandwidth. Feedback systems designed to control relativelybroadband disturbances are thus typically implemented using analogue com-ponents to minimise this delay. In some active noise control applications, suchas inside helicopters, there are many narrowband components in the acousticdisturbance spectrum as well as the broadband component. The frequenciesof many of these narrowband components can also change with time and it isinconvenient to obtain external reference signals for each of them. Whereasthe frequency response of the optimal controller for the broadband noise isa relatively smooth function of frequency, which can readily be implementedwith analogue components, that for the narrowband components will onlyhave a high gain at the frequencies of the main tonal peaks, and in order tomake this adaptive, to track the nonstationarity, it is most convenient to im-plement this part of the controller digitally. The extra delay introduced in theantialiasing filters, data converters and digital processing will not affect theperformance for these disturbance components since their bandwidth is small.This reasoning leads to the design of a combined analogue/digital controller forsuch applications, with a fixed analogue controller applied directly round theplant to attenuate the broadband component of the disturbance, and an outeradaptive digital controller to attenuate the narrowband components. The de-sign of such an adaptive digital controller, which maintains robust stability, isdiscussed in Section 3.4.1.



Control filter

Figure 3,6 A simple form of adaptive feedback controller in which the controlfilter in an IMC arrangement is adjusted to maintain performancein the face of nonstationary disturbances.

3.4-1 Adaptive feedback control

Adaptive feedback controllers are generally designed to compensate for changesin the plant response, and generally require the injection of a probe signal, oridentification noise, into the system [1], In active control applications the plantresponse often changes relatively quickly and the levels of identification noiseneed to be relatively high if these changes are to be successfully tracked. Suchlevels of identification noise can increase the output of the plant to the extentthat the attenuation in the disturbance can be negated. It may still be nec-essary to make the feedback controller adaptive, however, to maintain goodperformance for nonstationary disturbances, but if the plant response cannotbe accurately identified at every instant the adaptation algorithm must be de-signed to ensure that the feedback controller is always robustly stable to theseplant changes. The internal model control arrangement provides a convenientframework within which such an adaptive controller can be designed. Considerthe adaptive feedback controller shown in Figure 3.6, in which an FIR controlfilter in an IMC arrangement is adapted to minimise the mean square error.

There are now two levels of feedback round the system, that due to thefeedback controller and that due to the adaptation algorithm, and in general itis very difficult to analyse the interaction between these two loops. If the plantmodel is perfect, the arrangement reduces to an adaptive feedforward system,



for which the stability conditions have been derived above. The sensitivityfunction of the controller is then given by eqn. (3.25), which shows that themean square error is a quadratic function of the coefficients of the FIR filterW(z), and thus the adaptive filtering algorithms described above can be useddirectly. It is, however, unrealistic to expect that the plant model will alwaysbe perfect since otherwise it could be perfectly compensated for and a feedbackcontroller would not be necessary. If the plant model is not assumed to beperfect, the sensitivity function of the internal model control system shown inFigure 3.5 is:

l + G(z)W(z)S{z)

The stability of the feedback loop is assured if \{G{e?"T) -G{e^T))W{e?uT)\ isless than unity, which is equivalent to the condition for robust stability in eqn.(3.27) above. If we assume that \G{ejuT) - G(e?wT)W(e*ujT)\ is considerablyless than unity, however, then the system reduces to the feedforward system,for which the mean square error is a quadratic function of the filter coefficients.Between these two conditions it has been found that the mean square errorappears to be a convex function of the filter coefficients even if it is not aquadratic function [18], and so in principle the gradient descent algorithmssuch as the filtered reference LMS can still be used to adapt the control filterW(z). In practice, however, the system may drift very close to the stabilityboundary of the feedback loop and stochastic variations in the filter coefficientsmay push the system into instability. A sensible strategy for adapting thecontrol filter to minimise the mean square error, while avoiding the danger ofinstability in the feedback loop, would be to use the leaky filtered referenceLMS algorithm, eqn. (3.16), and use the leakage parameter /? to preventthe frequency response of the control filter from becoming large enough toapproach the robust stability condition. A far more precise control of thisconstraint can be obtained by implementing the adaptive algorithm in thefrequency domain, in which case individual values of /3 can be used in eachfrequency bin [81]. By monitoring the parameter \W(k)Go(k)B(k)\ in eachfrequency bin during convergence, it can be detected when the robust stabilitycondition, eqn. (3.28), is approached and the relevant value of/3 then increasedto prevent the controller from converging too close to this constraint [81].

3.5 Conclusions

There are various characteristic features of the active control problem, partic-ularly the problem of active sound control, which can be listed as:



(i) Disturbance rejection rather than set-point following 2) disturbance mayhave both narrowband and broadband components and is generally non-stationary

(ii) Plant response is high order, well damped and non-minimum phase,including delays.

(iii) Plant response may change over short time periods.

(iv) Objective is minimising a mean square (H2) output.

Because the main object is to reduce the mean-square output of the system,e.g. the mean-square output of a pressure microphone which determines thesound pressure level, high levels of identification noise cannot be used to trackthe rapid changes in the plant response. The control methods that are usedmust thus be robust to these changes in the plant.

If an external reference signal is available adaptive feedforward controllersare widely used to control the narrowband disturbances. The adaptation al-gorithm is often of the filtered-reference LMS type in which the update isgenerated by multiplying the instantaneous error signal by the reference sig-nal filtered by an internal model of the plant. For slow adaptation rates thestability of this algorithm is assured provided the ratio of the transfer functionof the internal plant model to that of the plant is strictly positive real. Thisimplies that the phase error between the plant model and the physical plantmust always be less than 90° or can alternatively be interpreted as requiringthat the unstructured multiplicative uncertainty has an upper bound of lessthan unity. If a leakage term is introduced into the adaptation algorithm bothof these conditions are relaxed and the system becomes even more robust todifferences between the response of the internal plant model and that of thephysical plant. Adaptive feedforward systems are used to control the tonalcomponents of the sound in the passenger cabins of propeller aircraft [45, 83].

The broadband components of the disturbance can sometimes be controlledwith a feedback controller, although the bandwidth over which control can beachieved is fundamentally limited by the propagation time between the sec-ondary actuator and error sensor. The internal model control (IMC) architec-ture of the feedback controller allows a direct comparison of the performanceof the feedback and feedforward controllers. It also allows the robust sta-bility condition to be simply expressed as an upper bound on the frequencyresponse of the control filter within the IMC controller. A straightforwarddesign strategy can then be developed in the discrete frequency domain forthe calculation of the frequency response of the controller which minimises



the mean-square error while maintaining the robust stability constraint. This#2/^00 design method only uses parameters which can be directly measuredfrom the plant and disturbance. Such a method has been used in the designof a feedback controller for an active headrest [19]. The IMC architecture alsoinspires a strategy for an adaptive feedback controller, which maintains goodperformance in the face of nonstationary disturbances but is robust to plantvariations. If such an adaptive controller is implemented digitally, the delayin the processor will further increase the delay in the loop and thus reduce thebandwidth over which control can be achieved. It may thus be advisable to usea combination of an inner analogue control loop for the stationary broadbanddisturbances and an adaptive digital outer control loop or nonstationary nar-rowband disturbances. These features have been tested on an active headset,in which the secondary loudspeaker and error sensor were located inside theearshell [17].


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