Active Sound and Vibration Control: theory and applications Volume 31 || Adaptive harmonic control:...

Chapter 5

Adaptive harmonic control:tuning in the frequency domain

S. M. Veres and T. MeurersSchool of Engineering Sciences, University of SouthamptonSouthampton, UK, Email:[email protected]

This chapter discusses adaptive control based on control at individual harmon-ics for cancellation of periodic disturbances. It is shown that the original idea,which has been around for a long time, can be developed much further to pro-duce robust and highly adaptable controllers. First, a review is given of themost accessible literature, after which frequency-selective RLS and LMS meth-ods are presented. Simulations illustrate the effectiveness of the method.

5.1 Introduction

As in some applications the plant can be considered linear, a fundamental ideais to separate the control problem at each relevant harmonic of the disturbanceif that is dominated by a discrete set of frequencies. Then, by linearity thereis no interaction between the control solutions found at different harmonics.This idea can be described as follows for a single tonal active vibration orsound control problem. At frequency u denote the detection signal by X(OJ),

the control signal by u(u), the additive disturbance by d(uj) and the errorsignal by e(uj). Figure 5.1 describes the problem of producing u(u) so thate(cj) is as near to zero as possible.

Let CJI, CJ2, ..., cjn be a finite set of frequencies at which x or d haverelevant harmonic components. The objective is to design a W such that

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98 Active Sound and Vibration Control

DetectionSignal

X((O)Controller

W(JG)) U{(O)

Plant

am

d(G>)

y(co)

Figure 5.1 Block diagram for the harmonic control problem

e = d + GWx = 0 at these frequencies. At each frequency the harmoniccomponents of the signals, and also the response of the controller and theplant dynamics, can be characterised by single complex numbers, which willbe denoted by x(uk), u(u)k), d(uk), e(uk) and W(jwk), G(ju)k), respectively.Assuming that there is no feedback from the actuator to the detection, thecomponent of the error signal can be expressed as:

e(cjfc) = d(u)k) + W{juk)G(juk)x{uk), k = 1,2,..., n

Figure 5.1 shows the block diagram of the harmonic control system. Forcancellation the controller has to satisfy:

W(jojk) = — (5.1)

Control of periodic sound at individual harmonics has been around for sometime and Reference [219] gives a summary of the subject. William Conoverbuilt an electronic system to achieve this cancellation for the harmonics ofelectrical transformers [58] in 1956. To satisfy these conditions in Conover'swork the transfer function W is physically realised by bandpass filters, phaseshifters and amplifiers. Figure 5.2 shows a block diagram of Conover's sys-tem. The analogue circuits offered little flexibility in the selected frequencies,the amplifiers (amplitude control) and phase shifters were tuned manually.Later digital control solutions were sought when computer technology madethat a possibility. Various adaptive digital feedforward control methods havebeen proposed, among them the most widely used today is the filtered-x LMSmethod.


Adaptive harmonic control: tuning in the frequency domain 99

Referencepickup

—•»>

Band-pass

filter at CO i

Band-pass

filter atC02

Band-pass

filter at con

Phaseshifter

Phaseshifter

Phase

shifter

Amplitudecontrol

Amplitudecontrol

Amplitudecontrol

i f ]

Additioncircuit

Actuatoramplifier

Figure 5.2 Block diagram of Conover's harmonic controller

This chapter presents a study of adaptive frequency selective solutionsto periodic disturbance cancellation. The control schemes suggested are purefeedback control schemes, no detection signal is needed, as these are frequentlyunavailable in practical problems. Achievable performance is excellent com-pared to pure linear feedback solutions and the advantages can be summarisedas follows:

(i) For periodic disturbances with a nearly discrete spectrum the perfor-mance is far better than that which is achievable by linear feedbacksystems.

(ii) A high degree of adaptability for dynamical changes of the plant isachieved.

(iii) Stability is ensured despite a the high level of adaptation.

Compared to standard LMS-type feedforward control an obvious advantage isthat there is no need for a detection signal which is strongly correlated withthe disturbance. The only disadvantage of the scheme presented is that thedisturbance has to be dominated by a discrete spectrum (possibly slowly timevarying). This is a requirement that is satisfied by a large number of practicalvibration and sound control problems, hence the usefulness of the approach.



5.2 Problem formulation

T h e plant t o b e controlled is a linear t ime-varying two- input / two-ou tpu t sys-t em as presented in Figure 5.3: It is described by s table 2 x 2 transfer

Figure 5.3 Block diagram of the basic control scheme

functions:

so that at each time instant t:

y(t) = £ w*u(t - i ) , * > 0; y = [yl y*]T, u = [ul u2]T

and in matrix form we can also write:

where

Gk{q ' -

The following situation is practically important:

(i) u1 is an unmeasured excitation of the vibration,

(ii) u2 is a control input.

(5.2)

(5.3)

(5.4)

(5.5)



(iii) y1 is a measurable output which is to be regulated to zero.

(iv) y2 is a measurable output.

The regulation problem of yl to zero, described under conditions (i)—(iv), isdifficult or in general impossible to solve using mainstream adaptive controltheory because of the unmeasured ti1. In the following an RLS and an LMSapproach will be looked at to remedy this situation. To solve these problemssome simplifying assumptions will be made and the ideas of harmonic controlwill be used.

Control

FSF

FSF ^

FSF

KiJK

Figure 5.4 Frequency selective filtering (FSF) at a set of frequencies

(i) Regulation of y1 to zero is to be achieved at a given set of frequencies

uuu2,".,wn € [0,2TT]

(ii) The plant is slowly time varying so that ||Gt - Gt+i||i < c, t > 0 withi > 0 known a priori.

(iii) The outputs are measurable with a given accuracy p > 0.

A possible scheme to achieve this is as follows. Each output is led througha frequency selective filter (FSF) for each frequency CJI,CO2, . . . ,cjn as shown



in the block diagram in Figure 5.4. The signals coining from the filters areprocessed by the controller and a suitable input is synthesised. The mainresult of this section is dealing with the problem of how to compute a suitablecontrol input in the frequency domain.

First, some of the surrounding details will be clarified. The frequencyselective filters are of the form:

where r < 1, r ~ 1. The advantage of frequency selective filters is that thesettling time of the filters can typically be smaller than the length of an FFTbatch which could be used instead. Typically, the settling time of a satisfactoryFSF can be around 40 ~ 60 sampling periods and FFTs should be based on256 samples to have similar accuracy.

Notations:

), i = 1,2,... ,n

will be used for the steady-state complex amplitudes of the sine waves:

Hi(q)y\ Hi(q)y2, Hi(q)u\ Hi{q)u\ i = 1,2,..., n

respectively. Then (A) means that the control objective is to achieve:

The control signal will be computed as a linear combination of phase-shiftedsine-waves (an alternative method could be based on a robust HQO controllerbut there is more danger of instability in that case because of possibly falseuncertainty assessment of the frequency selective model used). Let the infinitetime horizon be split into periods of TV sampling instants, so that the kth.period is denoted by:

In many practical systems it might be realistic to assume that there is anaverage transfer function Gfc(^~1) valid over T{k) such that:

| |G*(e-^) - Gt(e-**)ll < v, t€ T(k)

with a priori known v > 0, where ||G|| denotes max|G;m|. Similarly, for thefiltered signals:

Hi(q)y\ Hi(q)y2, Hi(q)ul, Hi{q)u\ i = 1,2,... ,n,



the respective average complex amplitudes over time period T(k) will be de-noted by:

yi(eM), VIW"% «i(e*"), «2(e*"), « = 1,2,..., n

uKe?"*), i = 1,2,..., n, actually represent the components of the control inputso that u\ is defined as the sum of sine waves with complex amplitudes ul(e?Wi),i = 1,2,..., n during period T(k). This operation is clearly not linear filtering.

For further use let the components of Gk(e^Ui)[ul(ejUi) 1]T be denoted by:

5.3 A frequency selective RLS solution

Let Xfc, dfc and e* denote complex numbers representing the harmonic contentof the signals y2, Gllu\ and yi, during time period T(k), as described above atgiven frequency UJ and indicated in Figure 5.6 . Assume that G12 changes slowlyand good estimates g^ can be calculated for its complex gain at frequency w.(Let's postpone discussion of the estimability of G12 for a little while.) First asimple filtered-x algorithm can be introduced for each frequency CJ of interest.Introduce r* = x^g*. For any complex number c the notation c = c1 4- c*jwill be used. Then the equation:

dk + nk = ek (5.6)

can be rewritten in the form:

e\ = rlw\ - r2wl + d\ + nj, e\ = r 2 ^ + r1^^ + dk + n\ (5.7)

={ H kN< t < kN+N+l }

N sampling periods N sampling periods N sampling periods N sampling periods t-I 1 j : 1 1 j

k k+l k+2 fc+3

Figure 5.5 Periodic sectioning of the time scale for compensator design andfrequency response estimation



w*

\

g*

RLS

Figure 5.6 Block diagram of the RLS solution at a single frequency

for the output errors e* which are measured with error/noise n\ and n\. Tofind a suitable controller gain w^ represented by the coefficients wk and w%,the following exponentially weighted criterion can be optimised:

by the usual RLS algorithm with a forgetting factor A < 1. This can be carriedout in two stages for each k > 1:

Kk = = (I~ Kk(j>l)Pk

where the notations:(5.8)

9k = Pk = -rl

are used and Kk, wk, Pk are only intermediate variables in an updating step.Hence Wk, Pk are computed recursively with initial conditions dependent ona priori knowledge of the plant. The following simulation illustrates how thisalgorithm works.



Let:r, _ r / > * • % _ [ 1-2 - 0.2j 0.6-0.05u-u{e> ) - [ 0 8 + 0Aj 0.9-O.lj

where gk is time-varying gain, gk = c(cos0.02A:+jsin0.02fc) and c = 0.15, thedynamics are time varying.

Non-controlled Output (200 periods)

0.05 -

-0.05 -

-0.1-0.02

Real parts

Controlled Output (200 periods)

Figure 5.7 Complex amplitudes of the controlled and uncontrolled outputs for200 periods: (a) Non-controlled output, (b) Controlled output

The forgetting factor is set to A = 0.7 and the detection signal is a fixedconstant rk = 1, therefore no measured detection is used. As the forgettingfactor is low, reasonable performance is achieved as shown in Figures 5.7 and5.8. For a plant model the fixed and incorrect:

r - 0.9 + 0.4j 0.94 -O. l j

is used.



The simulation illustrates that the method can work as the complex ampli-tudes of the controlled output are small in Figures 5.7 and 5.8. Conditions ofstability with regard to G and xk are difficult to establish. The frequency selec-tive LMS solution presented in the next Section easily lends itself to stabilityconditions.

Non-controlled Output Non-controlled Output

50 100 150 200 50 100 150 200

-20Controlled Output Controlled Output

-60200 200

Figure 5.8 Amplitudes (dB) and phase (°) of uncontrolled outputs for 200periods: (a) Non-controlled output, (b) Controlled output


0.15


Change of G(1,2) during 200 periods

0.05 -

i -0.05 -

-0.1 -

-0.20.45

Figure 5.9 Change of the complex gain G12 during 200 periods

5.4 A frequency selective LMS solution

An adaptive law will be derived for the input complex amplitude at a singlefrequency and then its stability will be analysed in view of the uncertainty ofthe transfer function of the plant. It will be proved that the adaptation law isconvergent under large relative uncertainty of the plant transfer function.

Let the input complex amplitude during time period T(k) be denoted byu>k = u>r +uiJ- Similarly, the disturbance and output amplitudes will be denotedby dk = dj + dij and yk = y* + yfj, respectively. Then, at a single frequencythe model of the plant equation will be:

Vk dk

where g is a complex number representing the plant transfer at the givenfrequency.



The objective will be to bring yk to zero and therefore an instantaneouscriterion function for control will be:

The output can be rewritten as:

Vr + ViJ = 9rUkr + 9^ + dk

r + JigrVfl + ftfi* +

or in matrix form as:yk = Gu* + dr

where

The control law will be designed to move the control signal u*;+i in the negativegradient direction of the criterion c&. The gradient of the criterion functioncan be calculated as:

which finally gives:Vcfc = -2GT

yjfe

Note that for the computation of the gradient only an estimate is available asG is not known. With a fi > 0 step size this will define an adapted controlsignal as:

Ufc+it/u* - f*GTyk (5.9)

Let the estimated and the actual transfer matrices of the plant be denoted by:

* -«• 1 and G 0 ^ f § ' $ft Sr J [9i 9r

respectively. The actual output will be obtained from the equation:

y* = G0Ufc + d + ii* (5.10)

where n* is the measurement noise of the complex amplitude of the output.Substituting eqn. (5.10) into eqn. (5.9) gives that:

ufc+1 = (I - nGTG0)uk + ^GTd + nGTnk (5.11)



Lemma 5.4.1 (a) The adaptive law eqn (5.9) will be stable if the matrix:

has eigenvalues with modulus less than 1.(b) If the adaptive law eqn (5.9) is stable and n*. = 0 , then

lim I!* = Go ld

where u = G0"1d is the ideal control input to eliminate the vibration of the

output.

Proof: (a) is obvious from linear system theory. For (b) the steady state of thegiven linear stable system can be calculated as:

[ql - (I - /iGTG0)]-VGT(d + n,),,=1 = GjTM + Go'n*

which gives the desired result.The most interesting case, practically, is to find out how small the step sizejj, will have to be defined to ensure stability under a given G and its relativeerror. Let's denote the estimated complex gain associated with G and Go by:

8^9r+J9i ^dgo^9r+J9i

respectively. Then the following theorem holds.

Theorem 5.4.1 the control law eqn. (5.9) will be stable under any relativeerror less than 5 > 0 of the estimate g, (i.e. for |g0 — g| < <J|g[i, if and onlyif S < 1 and the condition

< 5 1 2 )

is satisfied.

Proof: first of all note that:

CTC _ [ 9r 9i] \ 9°r -9i] _ I" 9r9°r+9i9! ~9r9? + 9i9°r

^ ^ - " l - a 9r\ [SS 9°r J " 1-arf+ft.tf fttf + fctfIntroducing the notations a^grg® + fag? and bd= - grg? + fag? will give that:

\



with eigenvalues 1 — fj,(a ± bj). To prove the theorem one has therefore toprove that |1 — /x(a ± bj)\ < 1 for any relative error < 8 if and only if:

( 5 1 3 )

By definition it can also be seen that a + bj = ggj holds. Define the unitdisc C(l) = {z € C | |1 - z\ < 1}. In view of the relative error specificationthe true transfer can be written as g0 = g + tflglre*" with some r € [0,1] andu € [0, 2TT]. The uncertainty set of a + bj under maximum relative error 8 ofg is:

p u € [0, 2TT], r € [0,1]} (5.14)

Hence |1 - /z(a ± bj)\ < 1 will be satisfied under any relative error < 8 of g ifand only if D(8) C C(l). It can, however, be easily shown that D(8) C C(l)if and only if eqn. (5.12) holds. To see this notice that:

^ ) = {/x|g|2 + ^ | g | 2 r e ^ | U;E[0,2TT], r e [0,1]} (5.15)

and the shaded disk D(8) with radius 8\g\2 will be contained in C(l) if andonly if:

/x|g|2 + fy|g|2 < 2 and 8 < 1 (5.16)

which is equivalent to eqn. (5.12).

5.5 Simulation example

In this Section a combination of the above frequency-domain LMS and time-domain FSF will be used to illustrate the advantages and difficulties whichappear in this harmonic control approach.

The simulated plant is a twelfth-order linear dynamical system with polelocations and Bode plots shown in Figure 5.10. Its transfer function is approx-imately:

where the coefficients are rounded to four decimal points.The output of this system is affected by a periodic disturbance shown in

Figure 5.11 together with its estimated spectrum. The simulated disturbancehad harmonic components at the frequencies 1.5Hz, 5Hz, 7Hz, 4Hz, lOHz,11.5Hz and an additive white noise of amplitude 0.01.



Poles of the simulated plant in the complex unit circle

frequency (rad/s)

Figure 5.10 Poles and Bode plot of the plant simulated. Resonant and lightlydamped modes can be observed around 2 Hz, 3.5 Hz and 7 Hz

Disregarding the ripples, the estimate of the spectrum of the disturbancehas three high peaks at 1.5022Hz, 5.05Hz, 6.9951Hz and three low peaks at4.019Hz, 10.077 Hz and 11.554Hz. The control system will aim to cancel onlythree components of the disturbance, those around 1.5Hz, 5.5Hz and 7Hz.As the precise frequencies may not be known in a practical situation, thecontrol scheme will use an online estimation of the relevant frequencies of thedisturbance.The frequency selective adaptation is based on segmentation of the time axisinto equal periods T(k) of N = 600 samples. During each period T(k) theoutput is filtered through three FSFs concentrated at the estimated threemost relevant frequencies of the disturbance. The feedback input to the plantis a mix of three sine waves with constant phase and amplitude during eachT(k). The frequency selective RLS adaptive system described by the recursive



The output disturbance (300 samples of the time signal)

50

0

-100

-150

-200

-250

_ann

— •- ——,

Estimated spectrum of the disturbance by Hanning wind

• ^ • i r

: : : : : : : i

OW

-

frequency (Hz)102

Figure 5.11 A section of the time signal of the disturbance and the estimateof its spectrum by Hanning window, based on 500 samples

eqns (5.8) is applied at each frequency to determine suitable amplitudes andphases of the sine waves, based on the adaptation of w* by eqns (5.8). Asthe results of the method are very sensitive to errors in the frequencies ofthe harmonic components of the disturbance, a frequency estimation is alsocarried out at the end of each T(k). The frequency of the FSF outputs isestimated by nonlinear optimisation of the squared-sum error function on thebasis of the last 100 samples during each period T(k), which ensures that theFSF settled by that time during the previous 500 samples of T(k). To filterout the high-frequency noise of the estimates, a low-pass filter is also appliedto each LS frequency estimate fy. i = 1,2,3 to obtain smothered estimates by:

= 0.98/'+0.02/*, 2 = 1,2,3 (5.17)

is then used during time period T(k + 1) to set the frequency of the



Output during periods 2-3

200 400 600 800

Output during periods 28-30

1000 1200

0.5

-0-5 H

'IT IT IT IT IT II IT IT IT IT IT I1'1 I'll I111! I200 400 600 800 1000 1200 1400 1600 1800

Figure 5.12 Time signals of the output during periods T(2) — T(3) andT(28) - T(30)

respective controller sinewave. The length of the time-domain simulation is 29periods of T(k), k = 2,3, ...,30. Figure 5.12 shows the reduced vibration ofthe output during periods T(28), T(29), T(30) compared with the vibrationduring periods T(2) and T(3) displayed on the top plot. Figure 5.13 displaysthe complex gains of the control sinewaves during the periods from T(2) toT(30). The bottom plot in Figure 5.13 shows the amplitude of the three outputharmonic components, which were obtained by estimation of the amplitudeand phase of the sinewave outputs of the three FSF filters concentrated around1.5Hz, 5.5Hz and 7Hz. This figure shows that the harmonic components ofoutput y at frequencies 1.5Hz, 5.5Hz and 7Hz goes to near zero after 30 periods.The sinewave phase and amplitude estimation during T(k) is carried out byordinary least-squares fitting of a sinewave to the last 100 output samplesof each FSF applied to y during T(k). The top plot in Figure 5.13 showsthe convergence of the complex amplitudes of the three sinewave components



Complex gains of control inputs at three frequencies

Complex gains of outputs at three frequencies

Figure 5.13 The complex gains of the control inputs and the amplitudes ofthe estimated output harmonic components during time periodsT(2)- r (30)

of the control inputs, the computation of which was based on the FSF-RLSmethod of eqns (5.8) applied at each main harmonic component of disturbanced.

For the sake of completeness, and to illustrate the size of the control inputused, Figure 5.14 shows the control input during time periods T(2) — T(3) andT(28) - T(30). Note that in Figure 5.12 the output during periods T(28) -T(30) still contains the harmonic components of the disturbance at frequencies4Hz, lOHz and 11 Hz which were not intended to be cancelled by the controller.Hence, the controller achieved only about 20 dB reduction in the vibration ofthe output y.



Control input during periods 2-3

-20

-40

200 400 600 800 1000 1200 1400

Control input during periods 28-30

200 400 600 800 1000 1200 1400 1600 1800 2000

Figure 5.14 Time signals of the control input during periods T(2) - T(3) andT(28) - T(30)

5.6 Conclusions

The basic schemes of frequency selective RLS (FS-RLS) and LMS (FS-LMS)methods have been described for periodic disturbance compensation. For theFS-LMS method a stability robustness theorem was given. Advantages of themethods were pointed out as (i) no principle limits of performance as in linearfeedback control, (ii) high degree of adaptivity and (iii) high degree of stabilityrobustness. A disadvantage is that the scheme only works for disturbancesdominated by a discrete spectrum, although that can be slowly time varying.

This introductory research into the topic has some shortcomings whichfuture research will resolve:

• In the methods presented the discrete spectrum of the disturbances wasassumed to be known. Preliminary automatic spectrum analysis of the



disturbed output can be used to set the frequencies of interest. Sensi-tivity to estimation errors in the disturbance frequencies is high. Howto use the best estimates and sensitivity analysis should be the topicof future research. Detection of changes in an online system is of greatpractical interest.

• The efFect of white noise disturbances should be analysed more preciselyalthough the simulation shown indicates practically acceptable low sen-sitivity.

• The effect of nonlinear dynamics added to the basically nonlinear dy-namics is a practically relevant question.

• Best numerical implementation can be investigated, parallel computa-tional architectures can be made use of and large numbers of frequencypoints can be handled while performance is monitored and control actionis supervised.

• Extension to multi-input multi-output systems is a future task which islikely to result in a stable and highly adaptive control scheme to numer-ous applications where the disturbance vector is dominated by a discretespectrum.


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