Adaptive Tuning to Frequency Response Specifications

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ADAPTIVE TUNING TO FREQUENCY RESPONSE SPECIFICATIONS Yu Tang and R. Ortega Di"isiol/ de Posgrado, Focultad de Ingenieria, Universidod .\'({ciOl/ol de Mexico. P.D. Box 70-256, 04510, D.F., Mexico

Keywords-Adaptive control, fl'equency response design.

robust

and Khargonekar, 1987) that to develop a useful theory for plants with unstructured uncertainty a key quest ion i s " Ho" to perform succesfu ll parameter adjustment for systems in frequency balls?". An alternative that has been explored is to develop new identification algorithms which not only generate a nominal plant model within the "structured" uncertai nty set, but also generate frequency domain bounds that can be used to represent the unstructured uncertainty. See e. g. (LaMaire et aI., 1987; Kosut, 1987). Another alternative (Krause and Khargonekar, 1988) is to derive non-conservative time-varying bounds on the energy of the unmodeled response from the weighted frequency domai n bounds of, e. g. ,stable proper factor perturbations of the plant. Some analytical issues of frequency response identification of an unknown plant are considered in (Ljung and Glover, 1981; Wahlberg and Ljung. 1986). For fur·ther references and comments see (Ortega and Tang. 1988) .

control.

Ahstract-The paper describes a procedure to design adaptive controllers when the specifications are given in terms of, a number of points of, a desired closed loop frequency response. The key idea is the use of a bank of "band-compensators". each one of which is intended to act on its correspond i ng frequency band. The compensator parameters are updated based on a f ini te impulse response on-l ine interpolation of the plant frequency response. The latter is carried out using a bank of comb filters and independent processing of the complex valued gain of each band. This frequency-domain approach obviates the need of parametric representations for both. plant and reference model. and inherits lhe robustness properties of frequency response designs. I . I NTRODUCTION The majori ly of the regulators used in industry are tuned using frequency response methods. There are at least two reasons for this: i) Frequency domain magnitude bounds on dynamical operators is a characterization of modeling errors widely accepted by control system designers. i i) In most industrial single-loop controller applications the specifications of interest. say. overshoot, settling time. and actuator authority. can be directly considered using frequency response reasoning. In spite of this. it is common experience that many regulators are in practice poorly tuned. basically because efficient robust melhods for regulator tuning are not available. Adaptive techniques for automatic tuning. have had a modest success in applications. usually requiring difficult and ad-hoc commissioning. One reason is that most of the research effort in adapl i ve control has concentrated on the problem of adaptive stabiUzation of "black-box" systems instead of the, practically more important. automatic tuning problem. Another reason is that in adaptive contro l the emphasis has been on time domain reasoning. The resul t is that the task of translating the engineering frequency domain information and constraints into the time-domain design parameters of adaptive control is almost impossible.

We study the following problem in the paper: Given a set of points of the desired closed loop frequency response. find an adaptive controller that asymptotically attains the frequency response matching. Our approach to solve this problem is conceptually simple and relies on the identification of points of the plant frequency response. The key idea is the use of banks of "band-compensators" and "band-estimators", each one of wh ich is intended to act on its corresponding frequency band. The compensator parameters are updated based on a fini te impulse response (FIR) on-line interpolation of the plant frequency response. The latter is carried out using a bank of comb filters and independent processing of the complex valued gain of each band. This paper has been strongly influenced by th~ pioneering work on identification of (Bitmead and Anderson, 1981) and the more recent (Parker and Bitmead. 1987). We borrow from these papers the central ideas of approaching the plant frequency response estimation problem as an interpolation problem, and the use of fixed frequency sampling filters (FSF) with adaptive gains. To the best of our knowledge, this is the first attempt to use these ideas for the control problem.

This paper addresses the problem of designing adaptive controllers when the plant prior information and the specifications are gi ven in terms of frequency responses. It is our belief that for regulat ion appl icat ions where slow adaptation is feasible, (i . e., simple, stable, slowly-varying plants and not very demanding specifications). it is sensible to formulate t he ident ificat ion and adapt i ve control problem in a frequency-domain setting. Our work provides a contribution, if modest, in this direction.

We shall give two separate treatments of the stability question for the indirect adaptive controller we propose . The first one is rigourous and consequently requires, to fulfill the technicalities of the stability proof. considerable modifications to the identification algorithm . The second one is Simpler, relies on persistency of excitation (PE) and total stability arguments. e. g. (Anderson et al . . 1986 ), but it has to appeal to heuristic reasoning . We however believe this reasoning to be sensible for the following distinctive features of the approach: i) The use of the bank of comb fi 1 ters and independent processing of each frequency band has

A brief review of 1 iterature related with this approach fo 11 ows. I t has been recogn i zed (Krause

427

the effect of breaking down the estimation problem to a collection of two parameter regression estimations with each regressor consisting of a narrow-band signal. The result is that, on one hand, each band-compensator of the bank only requires excitation in its corresponding frequency spectrum. On the other hand, the PE condition can be easi ly monitored to stop adaptation if PE is lost at a given band. i i) Since we are dealing with identification of points of the plant Nyquist locus, incorporation of prior knowledge to enhance the estimation is most natural and straightforward. No appeal need to be made to structured characterizations of the plant and only standard templates of its Nyquist locus are needed. iii) Ill-conditioning in the controller calculation, major stumbling block for "standard" indirect schemes, appears only at single well identified points and can easi ly be avoided via projections.

G

plant

are n

t

input

represents

the

and

(2.2) can be expressed as

(q)

(2.4)

\!wen

Controllers that satisfy (2.4) will be refered to as interpolating controllers. In the paragraph below we present a convenient implementation for this controllers. II.2 Frequency Sampling Filter Interpolating Controllers It is clear that, for each wen, (2.4) defines two algebraic equations that, given knowledge of the reference model and plant Nyquist locus, i.e. G (W·):=A(k)+jB(k)

(2 . 5a)

GO(w"):=A~(k)+jB~(k)

(2.5b)

p

N

N

can be sol ved to get N points of the compensator frequency response G (W·):=C(k)+jD(k) c

(2.6a)

N

via the relation

r A(k)

l-B(

k)

B(k)

1rAo(k)l

(2,6b)

A(k)JlB+(k)l

Notice that (2.6b) is ill-conditioned only at the points where IG (W·) 1=0. Therefore, without loss N

p

of general i ty, we wi 11 assume the zeros at the interpolation points.

plant has no

To get a controller implementation from C(k), D(k) we propose a "band-wise" approach, which roughly can be explained as follows. The error signal is first passed through a bank of comb filters to "spl it" the signal into its spectral components . The outcoming signals are fed into a bank of compensators connected in parallel, each one intended to compensate the open-loop frequency response at its specific frequency. That is, the control signal is synthesized as the following

output,

p

(2.3) )

They

effect

disturbances as seen at the output, and G

)

Jw

m

(2.1)

Yt

Ut'

respectively,

Jw

I-G (e

FormulatiQ~ Frequency Response I Ilterpolation. 'i.~ Consider a single-input single-output 1 inear time-invariant (LTI) stable plant

where

(e

m

Problem

Yt=G/q)ut+n t

(2.2)

If we denote with GO(e Jw ) the desired return-ratio frequency response, that is

11. NON-ADAPTIVE DESIGN 11.1

(e Jw ), \!wen

m

c

p

In addition to a presentation of specific results, this paper makes an attempt to be self-contained and not require familiarity with signal processing techniques. This task is simplified by limiting its scope to the basic ideas, rather than the technical ities in the analysis. A comprehensive treatment of the subject may be found in (Tang, 1988). For further information on signal processing see (Rabiner and Cold, 1975; Papoulis, 1977). The structure of the paper is as follows: in Section 11 we describe the non-adaptive design procedure; Sect ion I I I contains the proposed adapt i ve controller and its stabi 1 ity analysis. A simulat i on example is given in Section IV and some conclusions are presented in Section V. Due to space 1 imi t, all proofs are omi ted. are available under request.

=G

l+G (eJw)G (e Jw )

of is

the plant transfer function. Our control objective is given in terms of frequency response interpolation, and is stated as follows: Ci ven a set of Ne;! points in the complex plane and a +

N-l u=lu'(k)

corresponding set of frequencies wen, ru;;[ -rr , rr) . Design a feedback compensator such that the closed-l oop frequency response interpolates the points at the given frequencies. Furthermore, the desigl\ procedure should not rely on a parametric model for the plant with the only available prior information being the value of the plant frequency response at n.

t

. L=0

(2.7a)

t

(2.7b)

u~ (k)=F/q)e (k)

t

e/k)=A/q)e

(2.7c)

t

(2.7d)

Our approach is close in spirit to the quantitative feeback design of (Horowitz and Sidi, 1972) and is related with the idea of identifying the cri tical point, of the plant Nyquist locus, used in (Astrom and Hagglund, 1984) for controller tuning. For ease of presentation, we will assu!:'e the frequency set to be equally spaced, say W '

where

A.(q)

is

a

band pass

frequency W· and F N

•

(q)

filter

with

cent er

is the band-compensator to

(see notation). Also, the desired values of the closed loop frequency response wi 11 be expressed in terms of a reference model G (e Jw ).

be defined below. A convenient implementation of the bandpass fi lters is via frequency sampling filters (FSF), which are defined as (see (Bitmead and Anderson, 1981) for details), 1 l-q -N (2.8) A.(q):=-N l-W·q-l

We consider the standard unit feedback controller configuration. Let G (q) denote a controller such

It is easy to see that the frequency response of A sat isfies

that frequency response matching is achieved, i .e.

A

N

N

m

k

c

k

428

(i)={ N

1

0

t=k t.,k

(2.9)

One benefit of FSF is that el(k) is approximately the

component

of

e

at

l

frequency

Wk. N

We are in a position to present important stability result.

These

components do overlap sI ightly in frequency but have zero contri bution from other fi lters at the cent er frequency. The use of FSF to attain the spectral decomposition is fundamentally different from batch fast Fourier transform. since in the former case the spectral components are isochronous with the input.

u = t

N-l

L Ut(k)

c

(not necessarily FSF or of finite order). such that the closed-loop system is stable. Then. (2 .1 0)-(2.13) insures closed-loop stability if in(H ID) N>--=--in [~ ] c

(2. 15b)

(2.15c)

Ut (k)=c(k)e t (k)+d(k)e t _ 1 (k)

•••

(2. JOb)

The corollary of the theorem is that an FSF interpolat ing controller is stabi 1 izing if there exists a stabilizing controller (possibly infinite dimensional) that insures frequenc y matching (at the interpolation points) and the number of interpolation points is sufficiently large. How large is determined by the amplitude of the resonant peak of the transfer function ul~Yl (i . e .

(2 . JOc)

where -N

H (q)=_I_ l-q o N ----1 l-q

(2.11a) -N

H (q) =_1_ _ _ _ . :. I--.: -q:!..-_.----;; k N 1 2 1-2cos(wk)q +q

and the controller.

(2.12a)

c(k)=2C(k)

(2.

k

L t

l

C

L [c(k)+d(k)q

-1

JH (q)e

k=O

12b)

(2.

(2.13)

For a practical implementation of the controller. the following should be used: N -N

(2.11a')

When

N -N

H (q)=_I_ k

N

I-a q

1-2cos(w )aq k

-1

2 -2

(2.11b·)

+a q

typical application the working spectrum is at low frequencies. the interpolation frequencies should be distributed according to a logarithmic rule. 11.3 Stability Analysis In order to study the stabi 1 i ty of the plant in closed-loop with the FSF interpolating controller (2 .1 3) we need the following interpolating error bound of (Parker and Bitmead. 1987a).

G (e p

Jw

)

is

unknown.

the

controller

Ill . INDIRECT ADAPTIVE CONTROLLER 111.1 Identification Hodel. The main ingredient for the adaptive implementation of the controller is an on-line estimator of the plant frequency response points (2. 5a). To this end. we take. as for the controller realization. a "band-wise" FIR interpolation approach. That is. first. we propose to interpolate the plant frequency response points with an FSF interpolation model. Then. a bank of FSF is used to split the frequency components of the plant input and output and the complex valued gain of each band is processed independently .

Proposition 2.1 . Given G(z). a function defined in the complex z-plane and analytic on {z: Izl>r. r
Let us summarize

cocfficients c(k). d(k) can not be calculated by (2.4). As usual in adaptive control. there are two possible alternatives : direct estimation of the controller parameters or identification of the plant frequency response and subsequent calculation of the (certainty-equivalent) controller. The first approach has been explored in (Tang. 1989). in this paper we wi 11 propose an indirect formulation which is described below.

with a marginally small than one to guarantee the stability of the filter H/q). Also. since in a

IIG(eJw)-GI(eJw) Iloo"MR

k=O.l ..... L.

1. Choose the (odd) number of interpolating points (N). the de~ired closed-loop frequency response points (Gm(e Jw ) and the stability margin (a) of the FSF filters (2.11'). 2. Determine the frequency response points of the pl~nt lA(k).B(k)) and desired return-ratio (A (k). B (k)) us i ng (2. 3 ). ( 2 . 5 ) . 3.From (2.6) calculate the frequency response points of the interpolating controller. 4. Obtain the coefficients of the FSF controller (2.13) via (2.12).

k

1-aq l_aq-l

stabi 1 izing

the calculat ions needed for the determinat ion of the control law. The same procedure is used in the adaptive formulation replacing. throughout. the true parameters by time varying estimates.

12c)

Notice that it is an FIR N-th order filter.

H(q)= _ I_ o N

the

p

is known at w=wk'

The overall compensator will be refered to as FSF interpolating compensator and is thus given by (see Fig. 1) u =G (q)e:=

of

11.4 Design Procedure. jw The control law is readily implemented if G (e )

c(O)=C(o) •

d(k)=-2 [ cos(w )C(k)+sin(w )D(k)}

"smoothness"

m)

(2 .11b )

and the coefficients are given by

k

(2. 15a)

(2. JOa)

L'= . -2

k=O

following

Proposi tion 2.2 . Consider the plant (2.1). A'isume there exists an interpolating controller G (q).

From (2.8) we see that the "band compensators" are the complex gains (2 . 6) . As suggested in (Bitmead and Anderson. 1981). a possible implementation of the controller using real coefficients is obtained by grouping the terms k and N-k (N odd). which leads to L

the

(2 .14a )

wi th

Similarly to the controller implementation we define an FSF interpolation model of the plant as the FIR fi Iter

(2. 14b)

••• 429

1

C (q):= p

L

~ [a(k)+b(k)q k;Q

-I

}H (q)

difference here is that, roughly speaking, the gain of the "mismatch" transfer function i('(q)

(3. 1)

k

will decrease with increasing interpolation points. However, similarly to the robustness problem, to carry out the stabi 1 i ty analysis we must include in the estimator a normalization signal. To this end, we need the lemma below.

where a(k), b(k) are related with the plant frequency response points A(k), E(k) of (2.5a) via

o] -1

laCk)] b(k)

(3.2)

To avoid ill conditioning in the controller calculation, see (2.6), we will include a parameter projection. To this end, we assume known an Hoo ball for the plant Nyquist locus, from which we can determine the bounds b (k)sb(k)sb (k)

a (k)sa(k)sa (k); H

m

(3.9) Under these conditions

sT)

max

k=l, 2, ... , L,

(3.4)

hmmax(k):=2{[l+lcos

(3.5)

y~(k):=H~(q)Yt

W )

k

In the sequel, (. )'(k) denotes the signal filtered by lI~(q).

~

O(k):=[a(k),b(k)}T

(3.7a)

~ (k):=[u (k), u (k)]T t tt-I

(3.7b)

m

lI (q ):=(a(k)+b(k)q k

-1

-I

L

-1

)(l-cosw q

-I

k

e=o

-1

+h

m

max

(3. JO)

(k)

W

k

1+ ls inw IHe (e k

JW

P

) II + 00

+ ~ [l+ lcos wel+lsin well

M RN P P

e=o

Jw

p

))

I- cos

We]

~ oo [ I-cos

w k

(3.11)

••• Not i ce that the bound (3.10) is computabl e if we assume known bounds on the noise, radius of stability and Hoo norm of the plant . In

the

sequel

we

will

use

(~):=(. )IPt(k)

}

Proposition 3. 1 . Consider the normalized perturbed linear regression

)lI (q) e

(3 . 12)

(3.7d)

Notice

that

the

perturbation term '\ (k)

regression has components due approxiration error in C/q)-C/q), mainlobe error effect

Hi q ),

lines at w=w

k

e;
to

the H~(q)-l

When ut(k)

the

in

for

normalized signals . Our main stability result i s the proposition below. For further details on the identifier see (Lozano and Collado, 1989; Lozano and Ortega, 1987)

(3.7c)

}nt(k)

)[H~(q)-I}+2[I-cos(wk)q

[C (q)-C (q))+ \ 2(a(e)+b(e)q p p L

G (e

(3.6)

where

,~

max

e;
stra ightforward,

Yt(k)=~~(k)9(k)+~t(k)

=n

L

+(I-cos

1

(k) :

wi th the bound

and the k-th frequency band (normalized) component of the plant output as

~t(k):=Nk(q)ut(k)-2[I-cos(wk)q

(3.8)

Int Isn max Define a normalization signal as

At this point we find convenient to define the normalized bandpass filter (i.e., its frequency response is unit at the center frequency) as

After some I enghty, but calculations we can show that

noise

(3.3)

H

m

linear

perturbed Lemma 3. 1 . Consider the Assume the regression (3.6), (3.7). satisfies the bound

a nd the parameter update law

the

1\

noise

n , t interpolation and sidelobe

1\

°t(k)=9t_l(k)+~t(k)Ft(k)~t(k)Ct(k)

(3. 13a)

F-I(k)=F-II(k)+~ (k)~ (k)~T(k)

(3. J3b)

t

has only spectral

t-

t

t

t

ct (k)=yt(k)-e T (k)~ (k) t-I t

then the last three terms disappear.

(3.J3c) 1\

Assume that for given Fa(k»O, da(k»O and 9 (k) a

Our task now is to, based on the perturbed linear regression (3.6), define an identifier that would provide some desired stability properties. This question is addressed from two different perspectives. The first one is to provide a rigourous stability proof for a least squares algorithm with relative dead zone inspired from (Lozano and Co 11 ado , 1989), see also (Lozano and Ortega, 1987). The second one exploi ts the distinctive features of the problem and appeals to some heuristic (but hopefully sensible) arguments to infer stability of the algorithm. Discussion on the latter will be defered to Section 4.

1\

A.1

1\

8(k)e{8eIR 2 : [8-8 (k)]T F-I(k)[8-8 (k)] sd (k)} o

0

0

0

Under these conditions, 1. for all

times tl
1\

2 8(k)e{8cIR : [9-8 (k)lTF-I(k)[8-8 (k)lsd (k)) t t t t where d (k)=d (k)+~ (k)8 (k)2 tt-I t t T)

111.2 Identification Algorithm: Stability Analysis. From the first right hand term of (3.7c) we remark that the perturbation term is possibly unbounded. A similar situation appears in "standard" schemes in the face of unmodelled dynamics. An important

8 (k) :

t

max

(3. 14a)

(k)

Pt (k)

(3 . 14b)

2 . the parameter error 9t(k):=8t(k)-8(k) satisfies the bound

430

is a narrow band signal, therefore it can be approximately represented as a cosine. If its averaged energy is larger than m, and n is

3. Furthermore, if we choose the forgetting factor

"",~{ : '.,'"

','"

-

"

m (k) [1+~ (k) TF t

t

..,"',

t-l

t

t

t

(3.15a)

wi th m (k): =~ (k) TF2 (k)~ (k) t t t-l t

(3.15b)

t.t(k):=Ot(k)[Ot(k)+£l(l+a). a>O,£>O

(3 . 15c)

then, 3a) 3b )

A (k)
t

as

(k)~

in that frequency band (nt(k)),

then the corresponding parameter error wi 11 tend to a neighborhood of zero and consequent ly frequency response matching wi 11 be attained for that band. The richness of the plant input is clearly related with the excitation level of the reference signal. The distinctive feature here is that we impose a "band- wise" PE requirement that insures good "band-compensation" . Therefore, if PE is avai lable in the worki ng spectrum, good performance will be attained choosing a sufficiently large N (see Proposition 2.2). As is well known, (see e. g . (Ortega and Tang, 1988)), the main stumbl ing block in indirect adaptive control is how to avoid the regions in plant parameter space where the control design is ill-posed. In our case, this corresponds to, (see (2.6)), identification of the "zero plant". Let us note, because the point is not always appreciated, that this is not a technicality that can easily be avoided via simple projections or signal injection. Interestingly enough, various simulat ions we carried out wi thout provisions to avoid this problem exhibited instability. In the present deSign procedure it is straightforward to select a modification to the estimator, e.g. a hard projection, to avoid ill-conditioning on the control calculations. Another alternative is to freeze the controller parameters, as suggested in (Bai and Sastry, 1987), to get over the "difficult" patches. lie have prefered in this paper the soft projection (3.9c), which exhibited good behaviour, even in the case of "loose" prior estimate bounds.

otherwise

(k)~ (k) J

t

k

relatively small

if m (k)"t. (k)

t~.

A

dt(k), Ft(k) and St(k) converge.

••• The main feature of the proposed algorithm is that adaptation is turned-off when the "band PE" is below a time varying threshold. The level of "band PE" is monitored via mdk) and the threshold determined by an upperbound on the normalized disturbance t.dk) . The proposition insures boundedness and convergence of the signals of interest. Speciffically, it is shown that the bound on the parameter error is determined by the level of "band PE" and the size of the disturbance . Furthermore. we prove that adaptation

is asymptotically stopped. The description of the adaptive controller is completed wi th (3 . 2) and points _ 2-4 of the previous section. The estimates St(k) replace a(k), b(k), and the control law is given as

The proposed adaptive scheme with the estimator (4.1) was used to control the plant (2.1) with

(3.16)

The adaptive

~ontrol

2

scheme is depicted in Fig. 1.

q_ __ G (q)= __I_ _::--'O-'....:.7..:.62. p q-O.B q2-0.2Bq+O.04

IV. NUMERI CAL EXAMPLE

and nt=O. The model is chosen as

This section contains a simulated example that illustrates the main ideas of the frequency response matching approach. To carry out the simulations we used a simpler normalized gradient estimator with "band PE" monitoring and a soft projection. The estimator is described by 9

G(q)=~ m

The following parameters are used in the adaptive controller:

-f (k) t

(4. la)

t

t

T: L l

T=l-T

4>T(k)4> (k)"m T

T

where O<'o(k)
T

k

,

k

k

otherwise

a(O) real

width of the observation window and m the minimum

bounds

level of excitation, which is in its turn related with the signal to noise ratio . ft(k) is a (soft)

init ial

k

«k)=

{

t

The bounds

e

H

°

t

if

m

eml ,e Hl

a(

1)

- I. 05

b(

1)

1.4

(4,6) (-1.5,0) (0,2) 6

-0 . 4

I

a(2)

b(2)

-0.66

-0.46

(-1,0)

(-1,0)

-0. I

-0. I

1.

Our objective with this simulation is twofold: First, we want to sho w that the "band-estimators" and "band- co mpensators" effectively act only on their corresponding frequency band. Second , to show that, in spi te of the small number of interpolation poi nts, good behaviour is attained with the adaptive interpolating controller . To The reference sequence is of the form

H

etl (k)"Slm ,i=l,2

4.99

Table

projection vector whose components are of the form (Kreisselmeier, 1985) l el(k)_Si if (k)?:si e l (k) _Sl

k

(4.1b)

m/O are design parameters,

that specify the PE monitoring. Tk determines the

t

T =1

Simi larly to the signal processing problem, in appl ications N should range in 50-lOO. For N=5 there are three interpolation frequencies w =O, l 2rr/5, 4rr/5. The corresponding values of the plant Nyquist locus, estimated bounds and initial conditions in the estimator are summarized in the table below

with 't(k) the adaptation gain: if

(4.3)

q-O.2

N=5, a=O.99,

4> (k)£ (k) (k)=S (k)+, (k) t t tol t t 1+4>T(k)4> (k) t

(4.2)

(4 . 1c)

otherwise

are directly obtained from (3 . 3).

It is worth remar king t hat the band PE test (4.lb) can be impleme nted recursively. In practice , u t (k) 43 1

r

t

=

2 2sin(wl t) 2sin(w2t)

for te[O,50) for te[50, 100) for te[100,150)

2 -2

for te[150,200) for te[200,250)

P. J.. Parker and. R. R.. Bit_ad. -Adaptive frequency response Identification," Proceedings 26th eoc. Los Ana;eles, CA. Dec. 1981 L. R.• Rablner and B. Gold. Theory and ApplIcatIon of DIgHal SIgnal PrCCesslrll. Prentice-Hall. 1915 Y•• Tans. -On robust adaptive control -. Ph . D. theSis. National University or Hexlco. Nov. 1988. Y. Tang. "A frequency dOllaln approach to robust adaptive control,· Prcc . o1eerJcan Control Conf. Plttsbw-gh. 1989 B. 'lla.hlberg and L. LJW\&. - DeSign varIables for bias distribu tion I n transfer function estlutlon.· IEEE Trans. Automt. Contr. Vol. AC-J l . NO.2 . pp134-144. 1986

The idea is to operate the system in the three frequency bands at different times, starting with the zero frequency. In this way, the estimated parameters and corresponding band control signals will react at different times. In Fig. 2 we show the time evolution of the components ut(O), U (1) t and Ut (2) of the control signal. The estimated (.) Control and predlctlon

parameters are shown in Fig. 3 and the tracking error ' (4.4) is depicted in Fig. 4. The experimental results verify the foretelled parameter convergence. The jumps in the parameter estimates, specially a(O), in the last steps are due to the small number of interpolation points. Notice that, in spite of this fact, the tracking performance is fairly good.

r----·-···~·Oy-·------·--'I:-~;I-'·--~TO~

-----·-·-·f

.~§] "'''0-~~''''': '] ' ,", ~------~~

et(L)

:

1

j

Ut(L)

!

0~~

V. CONCLUSIONS lie close with some comments on the approach for adaptive controller design proposed in the paper. lie believe the method we have described to be a simple, systematic way to tune controllers to fr e quency response specifications when the desired closed-loop performance is not very demanding. It constitutes an attempt, if modest, to rigorously formulate a non-parametric, frequency-domain approach to adaptive control which exploites standard techniques of signal processing theory. This latter feature can hardly be overestimated, since it allows us to dispose of a vast array of hardware and software tools for the implementation of the control law.

I

-'--------_. __._ ...

_---_. __._-_.__ ._---_0... _-_._----_.- .------.. i -.~--~

Cc-(q,t) (b)

In spite of the boundedness and convergence reults presented in proposi tion 3.1, the issue of stability of the overall scheme remains essent ially open. Its establ ishment, with the level of general i ty of our formulat ion, seems a very difficul t task. The problem being that very lillle can be rigourously said about an LTI system (solely) from its frequency response. This aspect of the problem is independent of the adaptation. lie believe the formulation presented in the paper is worth pursuing for, at least, the following reasons: the successful use of frequency domain-based design methods in a big number of practical applications, the considerable expertise available on estimation techniques and the urgent need to dispose of an "automatic" procedure to carry out the designs

rsr

Interpolating Composator

Fig. 1. Adaptive Controller Implelllented by FSF

REFERENCES Ander-son et .. 1.. ~Stabl1Hy of adaptive sy ste/ZlS: PassivJ'Y and AvengJng An.slysls·. HIT Press, 1988. K. J. ¥.stro. and r. Hagglund. ~"utOllll.ttc tuning of sl.ple regulators with speciflca.tlons on phase and aaplltude -.rglns-, Automatic., Vol. 20, No. 5, pp. 645-651, 1984. £. \I. Bal and S. 5 .• Saslry. ~ Clobal slabll1ty proofs for contlnuous-t\.e Indirect adaptIve control sche..es, ~ 1£££ Trans. Automat. Contr., Vcl. AC-32, pp. 537-543, 1987 B. o.O.

R. R. 811Mead and 8.

D. O. Anderson, -Adaptive Frequency Se.apl1ng Filters,· IEEE Trans. Acoustics. Speech. and SIgnal Processlng. Vol. ASSP-29, pp. 884-893. Jun. 1981 I. H. Horowltz and H. S1d!, ·Synthesls of feedback systeas with l&rge plant ignorance for prescribed tl_-do_ln tolerances-. Int. J . Control, Vol. 18, 00. 2. pp. 287-309. 1972. R. L. J:osut. -Adaptive uncertainty .od.lins:: on-line robust control design," Proe. NX Hlnneapol1s ...... 1987 J. H. Xrause and P. P. Khargonekar. -Identiflcatlon under frequenc), do ... ln bounded dynaalcal perturbationa." 1987. (PrIvate correspoDUnce). C.. Irelnel.aler. -An approach to stable indirect adaptive control". Auta..tJca. Vol. 20, pp. 793-795, 1985. R. O. laMat". L. Yalavanl. H. Athans and G. Stein. "A frequency-do-.ln estl_tor tor use tD adaptive control syste• • " Proc. NX, Hlnneapoll ...... 1987 L. LJuna and ~. Clover, "Frequency do_In ve""~ ti_ do_In _thoda In systea ldenttrtcatlon." Auto-..tJca. Vol. 17. Ho . l. pp71-86. 1981 R. lozano and J. Co 11 ado. "Adaptive control tor ayste. with bounded dlsturt.nces". IEEE Tr~. Aut. Cont .• Vol. 34. No. 2. reb. 1989. R. lozano and IL Orhga. "Refor.ulation of the pe.ra.aeter lclentttlcatton problea for ayst ..19 with disturbanCes". Auta.atJc:., Vol. 23. ~. 1987. R. Ortep ud Y. Tang. "Robustnea: ot adaptlv. controllers: a survey". Autoutlca (to IIppear). Septe.ber 1989. A. Papoul1s. SJ6nIII Ana:lJ'sJs. HcCraw-Hlll 1977 P. J. Parker and R. R. Bit_ad. -Approxl_tlon of stable and unstable syste . . via frequency response lnterpole-tlon." Proced1nrs 10th IFIIC Con&r. Auta.at. Contr .. Vol. 10. Munich. pp. 358-363, Jul. 1987

Ot (')

"-t (11 b

t (.2)

. ~(O)

F;g. 3

t

t """~ ~ Iio

1= i 9· If

432

y~- G...Yt. , .. -t