Frequency Selective Weakly-Dual Adaptive Control

Copyright © IFAC Adaptive Systems in Control and Signal Processing, Budapest, Hungary, 1995

FREQUENCY SELECTIVE WEAKLY-DUAL ADAPTIVE CONTROL Sandor M Veres &hool 0/Electronic and Electrical Engineering The University o/Birmingham, Edgbaston. B15 2IT, UK e-mail: [email protected]. tel: 44-21-4144346,jar: 44-21-4144291

Istvan Valyi Institute o/Mathematics Budapest University 0/ Economics, H- 1093. Hungary Abstract. On-line interaction of worst-case identification and control is discussed in this paper on a case study of active noise control. Instead of using some external excitation of the system to improve identification, here the effect of an internal signal on identification is studied. Adaptation of the compensator is performed at a finite set of dominant frequencies of the noise. On-line identification is based on the calculation of polygon bounds of the feasibility sets of complex gains and uncertainties are taken into account. Updating of the compensator is carried out periodically, based on the uncertainty sets of the complex gains. Stability of the combined identification and control scheme is proved. The results presented are in many respect following the literature in the area of frequency selective filtering and adaptation. Keywords. Adaptive control, active noise control, adaptive filters, self-regulation.

In this paper the aim will be to show that some

1. INTRODUCTION

internal signals, or in other cases a periodic output reference, can sufficiently excite the closed-loop system to perform worst-case identification which in turn is accurate enough to maintain guaranteed control performance of slowly time-varying plants. Hence synergic interaction of identification and control is maintained. The main characteristic of the present study can be summed up as follows.

One of the most challenging problems in adaptive control is to guarantee that the control action taken supports the identification algorithm and vice versa. Dual controllers (Bertsekas and Rhodes, 1973, Wittenrnark, 1975, Astrom and Wittenrnark, 1989), synthesise this problem into one by using only one criterion, that of the controller. Solving the control problem by dynamic programming can be too hard both analytically and numerically, hence on-line algorithms can make little use of this approach. The difficulties associated with dual control have led to inventing schemes which explicitly separate identification and control but are globally convergent (Goodwin, a review, 1991; Goodwin et al, 1980; Kreisselmeier, 1985; Lozano-Leal and Goodwin, 1985; Ortega et al., 1989). Weak-duality, where a criterion of both identification and control is used explicitly, has been dealt with in some cases (Moore et al, 1989; Polak et al., 1987).

A non-parametric frequency domain approach is taken in the sense that the response of the plant is identified at an priori given set of frequencies. Advantages of combining classical design with methods of adaptive control has been discussed for instance by Bitmead and Anderson (1981), Wittenrnark and Per-Qlof Kallen (1991), Tang and Ortega (1993). Frequency selective filters and Lagrange compensators have been used. The new results in this paper, with respect to the available literature, are the proofs of stability and of synergic interaction between on-line worst-case identification and control. The results presented are also concerned with worst-case guarantees. This means that, under certain assumed disturbance bounds, all the feasible plant models are taken into account in the form of

a

This research has been supported by the British Council.

83

description of the dynamics of such a configuration can be given by the equations

set-models, in similar manner as in parameter bounding identification (Norton and Veres, 1990). The exact feasibility sets of some complex gains will be computed using polytope bounding. Worst-case performance will be evaluated by accounting for the worst possible performance over set models. Similar approach has been used in Polak et al., (1987) . Synergy acting between identification and control will be proved under certain conditions for a system which is an analogue of a simple active noise cancellation (ANC) system. The conditions of synergic interaction include the existence of stabilising compensators for each possible plant transfer function, lower bounds on the amplitude of noise levels and some inequality conditions formulated in terms of the filtering and measurement errors. A related previous work along similar lines of investigation has been in Veres and Vcilyi (1995), where the synergic interaction of on-line identification and control has been investigated in for the case of self-tuning compensators with periodic reference signals. Section 2 outlines the ANC system and the basic components of its dynamic equations, including the compensator used. Section 3 outlines the frequency selective adaptive scheme. Section 4 is concerned with the concise mathematical statement of results and conditions of stability and performance.

where F. H will be assumed to be transfer functions known from frequency response measurements. Their Bode plots will be considered to be experimentally known and G will be assumed to be variable. C and K will be digital filters to be designed by the adaptive scheme, and will be assumed as proper rational transfer functions of the backward shift operator q -I, so that for instance notation C=C{q-l) can be written. Similarly, F. G,

H are assumed to be functions of q -I in the infinite power series form, for instance co

F

=F{q-I) =Lfkq-k

For sake of simpler

k=O

notation, dependence on q-l will be suppressed at many places in the paper. To reduce the degree of freedom involved in the design, assume that K=O . Substituting the expression of e from the third equation into the second one gives

2. DESCRIPTION OF THE SYSTEM

y

To illustrate the principles, consider an active noise cancellation system as depicted in Fig. 1. Schemes for other applications, which are similar to the one presented, can be easily devised.

=Gn+(F -GH)u

(2)

Assume now that e consists of only a single frequency OE[O,1t]. Apart from stability conditions a controller C(q-J) with transfer

Noise source Cancellation speaker

M~roph~I~

f

Y

at Cl) would result in y=O from (2). Since -(1- CH) is the determinant of the left hand side matrix in (1), therefore the stability requirement on C is that

B Microphone II

Cancellation area

I I-CH

Fig. 1. An ANC configuration The aim is to use the cancellation speaker to achieve y=O using signals from the two microphones. To analyse this control problem assume that the transfer function from the cancellation speaker to the cancellation area is F, from the noise source to the cancellation area it is G and from the cancellation speaker to Microphone I it is H. Microphone I is not necessarily in the close proximity of the noise source but it is assumed to be somewhere between the noise source and the cancellation area. A simplified

(4)

must be a stable transfer function. If (3) holds, the transfer from n to y at frequency Cl) will be y= Gn+(F-GH)u

=(G+CF-CGH)n

and by (3) G+CF-CGHI

84

q=,JO

=0

(5)

Assume now that G is changing to some new G' and cancellation does not hold any more, i. e. G'+CF-CG'HI

q=el'J

=g'~O

the left hand side matrix in (I) is -(1-KF-eH) and hence the stability of l/( I-KF-eH) is required. Thus a suitable choice of K might increase the degree

(6)

of freedom of choosing the compensator

The idea of adaptation is very simple: by measuring u and y the complex gain g' could be estimated with some accuracy and hence C could be altered to make (5) hold again. The evaluation of uncertainties in estimating g' would have to be taken into account to ensure stability and good performance. It is now reasonable to ask whether g' can be estimated with sufficient accuracy in order to find a suitable new compensator C' which will then in turn support identification with the same accuracy at least Before going into any of the details the situation can be outlined as follows. The size of a bounding region

ea

for G'

(e

iQ

)

Consider the case where e has a discrete and finite spectrum with unknown amplitudes and phases, but with known frequencies. Let these frequencies be

co

g

, co L

and their discrete equivalents be

to calculate a compensator

C(q-·)

such that (3)

°°

holds for each of 0= 1 , 2 " " 'OL' with the estimate of G substituted. On-line calculation of the estimates and the compensator at each sampling period might be too demanding in real time. It may also be difficult to ensure the stability and performance of continuously operating scheme. Consider therefore a periodic scheme of compensation which estimates the complex gains of G over a time period T(k)=[kN. kN+N-I] and then applies the results of estimation during the next time period T(k+I)=[(k+I)N, (k+I)N+N-I] for every k=O,I,2,3, .... Assume that the gain of the applied

of g' . It might now

well be the case that n has large enough amplitude to ensure estimation of G with an accuracy that will allow the minimax design of a compensator which in turn will support re-estimation of G to provide a compensator ...etc. Hence, in a certain sense, the estimation-eontrol cycle can be closed even if not continuously, but at least periodically. During some time period the complex gains can be estimated, while the compensator obtained from these can be applied during the next time period. Asynergic interaction of identification and control is hoped for to come into effect. Fig. 2 shows the block diagram of the signals involved.

compensator has been C;1e , while estimate of the gain of G + Cle F -

g

has been the

Cle GH

(9)

over period T(k) . Then the compensator with gain (10)

will be applied over time period T(k+ I), which is obtained by first calculating an estimate of G from

~-n:}----~+y

e

=co I' co 2 ,...

~ = Tco;,i=1,2, ... ,L with sampling period T. Then the task of the controller is to estimate the complex gain of G at each of these frequencies and

(i) on the amplitude of signal n. (ii) on how the mapping

e

-1 ).

3. OlITLINE OF THE ADAPTIVE SCHEME

will mainly depend:

distorts the bounding region

C( q

g

+

as Ale

G_=.g_--=-C_F_ - I-CIeH and then substituting that into

u

F-GH

Fig. 2. Block diagram of theANC system

(11)

(12)

This latter formula has been derived from (5). The new compensator CIc+I will obviously affect estimation over the period T(k+ I) which will then in turn affect the new compensator. This latter will affect estimation over T(k+2) and so an. There will

If K =0 is selected, then the controller with a fixed C is a feed-forward compensator. Thus adaptation of C, based on measurements of n and y. is the only feedback involved. Using a K~ can be useful if it helps stabilising the system. The determinant of 85

be a sequential interplay between identification and control, the stability of which has to be proved. In the next section the aim will be to give this idea a more precise formulation. First bounding of the complex gains from n to y will be analysed for its errors. Later the adaptive algorithm will be defined and its stability proved.

{hi -J<

bI (k) = b" - 0c! I(lr.I ' c) e PY(k) (b" c:) e P"(k)}(16) I I 'I , I I I

transfer from n to y . Let the centre of denoted by e? and that of the following lemmas hold:

the

n;

= N;(q-I)n

and

0

1'

O 2 ,,,, ,0 L .

y; =N;(q-I )y, dia J pY (k)] S

"1. '

. n{ P;n( k )] S d/Q

Assume that the

maximum deviations of n; and y; from perfect sine waves of frequency

n;

can be bounded by

cos

to;n; )' /2

2E;

cos

(

n; /2

)'

(17) .

1=

1,2... ,n

Lemma 2. The diameter of the gain feasibility set satisfies

and 0; , respectively. Hence the inequalities

E;

by e; (k) . Then

n; = 2nl K with D3 integer. Then the diameters of the feasibility sets are bounded as

i=I,2, ... ,n . The next aim will be to estimate the complex gain of the transfer between n and y at

frequencies

p;Y (k)

Pt (k )

Lemma 1. Let

transfer from n to y at frequencies 0 1 ' O 2 ,,,, ,OL the controller will first apply notch filters to both n

andy to obtain

I

will represent the fesible set of complex gains of the

4. STABILITY AND PERFORMANCE In order to estimate the complex gains of

]

Iy;(t)-a; cos(tn j +,;)1 so;, (13)

In; (t )_a;n cos(tn; +,?)I S E;,

(18)

t eT(k) i = 1,2, ... ,n

hold where a;, ar, ,;, ,? are amplitudes and phases to be estimated. Since the inequalities in (13) are nonlinear in the equivalent forms

ai

cI>;, write them into

and

Iy;(t)-a; cos,; costn; +a; sin,; sintn;1 So;, i = 1,2,... ,n

In; (t)-a;n cos,?costn; +ar sin,? sintn;j SE;,

Assume now that G; cC, i = 1,2, ... ,n are aprIOri sets of possible complex gains of the transfer at

t e T(k)

which

bYI

are

=aY I

cos.l. Y , 't',

br = a;n cosejl?,

linear

in

2' =aY

the

new

parameters

sin.l. Y 't',

and

n = ar sin,?

Each of the set of

I

I

Cj

frequency n;. Let G be an transfer functions.

(v"

h(t) - bjn costn; +Cr sintn;j

(15)

SE;,

t e T(k)

Over each period T(k) the following recursive operations are performed at each frequency OJ,i = 1,2, ... ,n.

1. The feasibility sets

(b;n ,c;)

, respectively.

p;Y (k)

and

p;n (k)

are

calculated, based on measurements over T(k), and

will define a two-dimensional polytope, i.e. polygons

F/{k) and P;YI.) in the planes of

set of possible

The adaptive algorithm.

inequalities

Iy;(t)-b; costn; +c; sintn;1 ~o;, t eT(k)

apriori

h; (k) , i=I,2,... ,n. are defined as in (16).

(bi,ci)and

Clearly the set of complex

numbers

86

(23)

i =1,2, ... ,L

for (9).

(ii) There is a

. '-1 2,.....-, T are 3. The new compensator gams c;l'+1 ,1-, computed as

'V; > 0, i

l' r

°< a. < 1 and

positive quantities

=1,2,... , L such that

A

c l' +1 = c; .I; - gl' i=l 2 L ' / ; _ gl' ; , , ,...,

(20)

h

4. Finally a compensator Ck+1

(q-I)

is computed so

i=1,2,.. L.

that

tj( 1- C + H) l' 1

and

oy H;"g

is stable which is applied over time 'Y;(X)=I

period T(k+1) when steps 1-3 are repeated for k+1 in place of k.

where

Comments. Construction of a compensator in step 4 of the algorithm can be done using Lagrange compensators of the form

"

Ck+1 = L(b;k+1 +d;k+lq-I)H;(q)

"

cos(O; 12)E; -Edl-h;xrl

E; is the amplitude of

I'

i=1,2, ... ,L

the harmonic

component of e at frequency 0; and g is an upper bound of the amplitude-gains from n to y . (iii) The transfer G l' is constant over periods T(k), k=1,2,3,.... Changes from G l' to Gk+1 occur

(21)

slowly, so that the compensator Cl' (q

;=0

where

-I)

of

G l'

can be assumed to be stabilising for Gk+1 . Then the control scheme, based on the adaptive algorithm defined above, will be stable.

Comments. Condition (iii) is not a serious constraint in practice since intermediate changes can be

and b;k+1 +d;l'+le-jo, = Ck+I(ejo,

),i = O,l, ... ,L

incorporated into E; and 0; . Furthermore, the stabilising effect of a previous compensator to the next plant can be tested a priori. based on experiments and apriori assessing the possible error bounds of gain bounding. Because of lack of space, the details of this analysis are omitted here. but they are available in Veres (1994). Condition (ii) is related to maintaining and regenerating

as

suggested by Tang and Ortega (1993). To be able to retain stability of Lagrange

tj(l-Cl'+IH) in our scheme

compensators

"

Ck+1 = LQ;(q-I)H;(q)

of

the

form

simultaneous identification and control. E; approximated by

can be used under the

does again need assessment.

;=0

constraint that where

N; (q-I

can be

le. To test condition (i)

apriori experimental and analytical

Q;(q-I)=Ck+I(eJ"o' ),i=O,l, ... ,L,

Outline of the proof recursions

Q; (q -I) are rational functions with higher

degree of freedom.

Theorem 4.1 Assume that the following two conditions are satisfied. (i) For any transfer GeG there is a compensator C( q -I ) which stabilises

tj(1- Ck+ H) and satisfies I

87

First

the stability of the

D(;n.:;(;&al>.

can be proved where ejk denotes estimation error bounded by the diameter of the feasibility sets bj (k ). By substitution it follows that

_Gjk -e: 1(I-cjk

hI

c;

h;)

= /; -h;Gjk -e: hj 1(I-cjk h;)

(24)

and hence

hI

c;

I

-

_qk k

/; - hP;

1f,lle:l/ll-c:h;1 If, -hp;I(If, - hP; 1-111; lie: II 11- c:h; p

~ -:----r-...;l,.-...!.....!---;-~:---"""

(25) For stability it is sufficient to prove that if

Ic:-r;I:S;;'Vj

then

Ic;hl_r;I:s;;a'Vj

anu

1.0.

lUlOOes.

~ 1~

''').

;:,umClcnuy

informative functions and the minimax feedback control of uncertain dynamic systems. IEEE Trans. Automat. Control. AC-18. 112-123. Bitmead, RR and Anderson, B.D.O. (1981). Adaptive frequency sampling filters. IEEE Trans. Acousrics Speech and Signal Proc. ASSP-29. 684-693. Goodwin, G.C. (1991). Can we identify adaptive control? Proc. ECC'91. Grenoble. 1714-1725. Goodwin, G.C.• Ramadge. P.l and Caines. P.E. Discrete-time multivariable adaptive control.. IEEE Trans. ofAutomat. Control. AC-24. 1980. 449-456. Kreisselmeier. G. (1985). An approach to stable indirect adaptive control. Automatica. 21. 425431. Lozano-Leal and G.C. Goodwin, (1985). A globally convergent adaptive pole-placement algorithm without persistence of excitation requirement. IEEE Trans. Autom. Control. AC-30. 795-798. Moore. lB.• T. Ryall and L. Xia (1989). Central tendency adaptive pole assignment. IEEE Trans. Aut. Control. AC-34. 363-367. Ortega. R • R Kellyan R Lozano-Leal (1989). On global stability of adaptive systems using an estimator with parameter freezing. IEEE Aut. Control. AC-34. 343-346. Polak. E.• S.E. Salcudean, D.Q. Mayne (1987). Adaptive control of ARMA plants using worstcase design by semi-infinite optimization. IEEE Trans. Autom. Contr. • AC-32. 388-3%. Ryall. T. and lB. Moore (1989). Central tendency minimum variance adaptive control. IEEE Trans. Aut. Control. AC-34. 367-371. Tang. Y. and Ortega, R (1993). Adaptive tuning to frequency response specifications. Automatica. 29. 1557-1563. Tang, Y. (1989). Adaptive frequency response identification using the Lagrange filter. Automatica. 25. 451-455. Veres. S. M.. (1994). Parameter bounding identification in adaptive control. Int. J. of Adaptive Control and Signal Processing. 9, 3346. Veres. S. M. (1994). Weak duality in worst-case adaptive control. Proc. of ACC'94. June 29-July

with some

and r;k = _Gjk / /; -hjGjk . However. by condition (ii) an by (19) the right hand side of (25)

o
is overbound by a'V j and this proves the stability of the computations of the compensators applied over periods T(k) • k=1.2.3 •.... Finally. the compensated control system remains stable within each period T(k) • k=1.2.3 •... by assumptions (i) and (iii) of the theorem, which completes the proof. 0

5. CONCLUSION Synergic interaction of identification and control has been investigated for an analogue of an ANC system. An internal signal provided sufficient excitation for bounding the complex gains at different frequencies. It has been proved that the loop of identification and control can be periodicly closed and the adaptive scheme remains stable for time varying plant dynamics. Synergy of identification of control has been proved in the sense that, not only the identification will be satisfactory in itself. but the resulting control action will support its own identification. Future research will be to investigate other system configurations for synergy. for instance self-tuning compensators in the case of periodic reference signals.

1. Baltimore. Veres. S. M. and I. VaIyi. (1995). The synergic principle in adaptive control. Research Memorandum No. 41. School of E.&E. Eng.• Univ. of Birmingham, Edgbaston, B15 2TT. UK. Wittenmark. B.. (1975). Stochastic adaptive control methods: a survey. Int. J. Control. 21. pp. 705730. Wittenmark. B. (1975). An active suboptimal dual controller for systems with stochastic parameters.

REFERENCES Astrom, K.l (1993). Matching criteria for control and identification Proc. ECC'93. Groningen. 248-251. Astrom, K.l and B. Wittenmark (1989). Adaptive Control. Addison Wesley. Reading. MA.

Automatic Control Theory and Applications. 3. 13-19.

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