*Received 12 August 1996, 4 March 1997; received in finalform 29 May 1998. The original version of this paper waspresented as a plenary paper at the 13th IFAC World Congress,which was held in San Francisco, U.S.A., during 30 June—5 July1996. This paper was recommended for publication in revisedform by Editor M. Morari. Corresponding author ProfessorBrian D. O. Anderson. Tel. #61 26 279 8667; Fax #61 26 2798688; E-mail brian.anderson@anu.edu.au.-Research School of Information Sciences and Engineering

and Cooperative Research Centre for Robust and AdaptiveSystems, Australian National University, ACT 0200, Australia.

PII: S0005–1098(98)00103–4Automatica, Vol. 34, No. 12, pp. 1485—1506, 1998( 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0005-1098/98 $—see front matter

Survey Paper

From Youla—Kucera to Identification,Adaptive and Nonlinear Control*

BRIAN D. O. ANDERSON-

Key Words—Linear systems; nonlinear systems; identification; adaptive control; Youla—Kucera parameter.

Abstract—Some 20 years ago, formulae were presented for theset of all linear time-invariant controllers stabilizing a lineartime-invariant plant. This paper traces the development of manyideas from these formulae, covering linear H

2and H

=control,

identification, adaptive control and nonlinear systems. ( 1998Elsevier Science Ltd. All rights reserved.

1. WHAT IS THIS PAPER ABOUT?

This paper is about a useful way of parametrizingplants and controllers — in the first instance linearplants and controllers. The idea had its origindecades ago, is still giving rise to theoretical devel-opments, for example in problems of closed-loopidentification, adaptive control, and nonlinear sys-tems, and is being carried over into practice, onerecent reported case being in controller design forflotation circuits.

We begin with motivating material, concerningthe desirability of a parametrization of all stabiliz-ing controllers for a prescribed linear time-invari-ant plant. The Youla—Kucera parametrization ispresented, using fractional descriptions of plantsand controllers via stable rational transfer func-tions. Examples of its applications are noted. Wethen turn to more recent developments. For linearsystems, these include the incorporation of theparametrization into state-space formulas (with ap-plications to approximate loop tranfer recovery,and to direct adaptive control); we also note thesolution of H

2and H

=problems with constrained

pole positions.A dual parametrization (of all plants stabilized

by a fixed controller) is the basis for solving closed-

loop identification problems by their reduction tostandard open-loop problems, and the applicationof this concept to an adaptive control methodology(windsurfer adaptive control).

It is possible to consider Youla—Kucera para-meters simultaneously associated with plant andcontroller. We explore this concept, consideringhow a small change in a plant (represented byintroduction of a nonzero Youla—Kucera para-meter) should give rise to a corresponding control-ler change, when the design issue is stability,H

2optimality or H

=gain limiting.

Finally, we describe very recent work on non-linear systems, including closed-loop identific-ation, and Youla—Kucera parametrizations forgeneral nonlinear plant-controller interconnec-tions.

2. THE BASIC IDEA AND SOME OF ITS HISTORY

Let P(s) be the rational transfer function of astable open-loop plant with P(R)"0. How can allstabilizing controllers be characterized? Knowingone stabilizing controller C(s) and P(s), we candefine

Q(s)"C (s)

1#C(s)P (s), (2.1)

while if Q (s) and P (s) are known, we could recoverC(s) by

C(s)"Q (s)

1!P(s)Q(s). (2.2)

It is easily checked that if C(s) is stabilizing for P (s)and is proper, i.e. C(R) is finite, then Q(s) is stableand proper. Conversely, if Q(s) is any stable propertransfer function, and C(s) is defined by equation(2.1), then C (s) is necessarily a stabilizing controllerfor P (s).

Thus, stabilizing controllers are parametrized interms of the set of all stable proper transfer functions.

The result probably does not seem very exciting.But there is an important bonus: the closed-loop

1485

Fig. 1. Stabilized feedback loop showing plant P(s) and control-ler C(s).

transfer function G(s) is given in terms of P (s) andQ(s) by

G(s)"P (s)Q(s). (2.3)

The linearity in the free stable transfer function Q(s)is what gives the added appeal. Also, Q(s) does haveintrinsic significance: it is the transfer function fromr to u (see Fig. 1).

It is now well recognized that many closed-loopspecifications are convex in G(s) and hence convexin Q(s), but not of course convex in C(s); the con-vexity is of great assistance in design (Boyd andBarrett, 1991).

The idea of reparametrizing a controller designproblem to obtain linearity in the design transferfunction is probably an old one; for example, itappeared in (Newton et al. (1957). Formula (2.2)appears in Zames (1981) and is well known inchemical process control as ‘‘internal model con-trol’’ — because the controller is viewed as aninterconnection of Q(s) in a forward loop andP(s) in a positive feedback loop, see Morari andZafiriou (1989).

As expressed above, the idea is limited to scalarplants, which are open-loop-stable. How can thissubstantial limitation be lifted?

3. YOULA—KUCERA PARAMETRIZATION

Youla et al. (1976) and Kucera (1975) appear tohave independently explained how the ideas ofSection 2 extend to MIMO plants which are notnecessarily stable.

There are two key features in their idea. First,they assume one stabilizing controller is a prioriknown, so the task becomes one of characterizingall, already knowing one. [If the plant is a prioristable, C(s)"0 is immediately known to bestabilizing.] Second, they describe plants usingpolynomial fractional representations.

Thus, if P (s) is the rational transfer function ofa scalar plant, one thinks of it as n (s)/d (s), wheren and d are polynomials, both coprime to avoiddifficulties. If P (s) is multivariable, it can be re-garded as N(s)D~1(s) or DM ~1(s)NM (s), where N, D,etc., are matrices of polymonials; a form of copri-meness condition is imposed.

It became evident that a tidier formulation wasavailable if one moved away from polynomial frac-

tional representations to stable transfer functionfractional representations. Thus, a plant with trans-fer function 1/(s!1), instead of being regarded asa fraction with numerator and denominator 1 ands!1, respectively (both polynomials), can be re-garded as a fraction with numerator 1/(s#1) anddenominator (s!1)/(s#1) (both stable rationaltransfer functions).

Because stable proper rational transfer functionsform an algebraic entity known as a Euclideandomain [an observation certainly going back toForney (1970) and exploited by Desoer et al. (1980),etc., and Vidyasagar (1985) for the present pur-poses] notions such as coprimeness, and greatestcommon divisor determination via a Euclideanalgorithm and the like can be used.

Let S denote the set of stable proper rationaltransfer functions (in continuous or discrete time).Two matrices N, D with entries in S (henceforthwritten with some abuse of notation as N, D3S)and with the same number of columns are said tobe right coprime if there exist matrices XM ,½M 3Swith XM N#½M D"I. The main result is as follows,see e.g. Vidyasagar (1985). As with the earlier resultfor scalar, open-loop stable plants, it characterizesall stabilizing controllers using as a parameter anarbitrary stable proper transfer function (matrix).

¹heorem 1. Let a plant P"ND~1, with N and Dcoprime over S, be stabilized by a controller (ina negative feedback loop) C"X½~1, with X,½coprime over S. Then the set of all stabilizingcontrollers for P is given by M(X#DQ)(½!

NQ)~1 :Q3SN.

The idea of Section 2 is included here: supposeP is scalar and stable. Then N"P,D"1. [Thispair (N,D) is coprime.] Also C"0 is stabilizing,so X"0, ½"1. [The pair (X,½) is coprime.]Then the formula yields Q(1!PQ)~1, as before inequation (2.2).

It is certainly not obvious that the above para-metrization yields a closed-loop transfer functionmatrix which is linear in Q; the calculation is easywhen P and C are scalar, and the closed-loop trans-fer function is computed to be

G"

PC

1#PC"

NX

NX#D½

#

ND

NX#D½

Q ,

(3.1)

which is affine in Q—the next best thing to linear Q.In the MIMO case, the result is

G"[N 0] CI Q

0 ID CD !X

N ½ D~1

C0

ID (3.2)

and again the affine property holds. (The inversewill exist if the closed-loop is well-defined, which is

1486 Survey Paper

Fig. 2. More general plant-controller interconnection, with dis-turbances w and regulated variables z.

the case if both P and C are proper with one strictlyproper; the inverse is in S if the closed-loop isstable.) If XM , ½M , DM and NM are such that

C½M XM

!NM DM D CD !X

N ½ D"CI 0

0 ID , (3.3)

then equation (3.2) simplifies significantly to

G"N(XM #QDM ). (3.4)

Digression. Equation (3.3), which is known as the(double) Bezout identity (Vidyasagar, 1985), hasembedded within it many ideas, a few of which welist here:

f DM ~1NM and ½M ~1XM equal ND~1 and X½~1, andare left coprime fractional descriptions of theplant and initial stabilizing controller.

f The stability of the plant-controller loop is equi-valent to the invertibility (over S) of each of thematrices on the left of equation (3.3).

f (X#DQ) (½!NQ)~1"(½M !QNM )~1 (XM #QDM ).f For scalar plants and controllers, one can take

N"NM , D"DM , X"XM , ½"½M and then thedouble Bezout identity is equivalent to XN#

½N"1.

Equations (3.2) and (3.4) show the affine depend-ence of the closed-loop transfer function (matrix)linking r to y on the parameter Q(s)3S. It is nothard to verify that the transfer function (matrix)linking either r or d to any of e, u and y are similarlyaffine in Q(s). Of particular interest in many designsare the output sensitivity, viz., the transfer functionmatrix linking d to y:

S (s)"(I#PC)~1"(½!NQ)DM (3.5)

and the transfer function matrix from the externalinput r to the plant input u, viz.,

Gur

(s)"C(I#PC)~1"(X#DQ)DM . (3.6)

The affine dependence carries over into even moregeneral structures; for example, suppose that thereexists one controller C (s) producing internal stabi-lity for the set-up of Fig. 2. Here, w denotes thedisturbance signals, z the regulated variables, u theactuator inputs and y the sensor outputs (normallyincluding set-point data). Then the set of all con-

trollers achieving internal stability can be para-metrized very similarly to the set of Theorem 1in terms of a free proper Q(s)3S, and the closed-loop transfer function (matrix) from w to z has theform

Gzw"¹

1(s)#¹

2(s)Q(s)¹

3(s), (3.7)

with ¹1, ¹

2, and ¹

33S [and independent of Q(s)].

Formula (3.7) is behind one approach to solvingH

=problems where the task is to choose Q so that

DDGzw

DD=

lies below a nominated bound, or is mini-mized; see Zhou et al. (1995) and Green andLimebeer (1995). Equally, the formula can be usedas a basis for tackling H

2problems, where the task

is to choose Q to minimize DDGzw

DD2; see Doyle et al.

(1992).

4. FLOTATION CIRCUIT DESIGN EXAMPLE

The Youla parametrization has a reputation ofbeing a somewhat underutilized tool as far as prac-tical control system design is concerned. In thissection, as part counter to that idea, we summarizea practical example, in which the key is to exploitthe affine occurrence of an adjustable Q in theclosed-loop transfer function. The paper by Rossand Swartz (1995) describes a multivariable systemapplication of the Q-parametrization of a controllerto flotation circuit design, the model being like thatof Fig. 2. In a flotation cell, see Fig. 3, a froth isproduced through the injection of air, and the frothlayer contains a higher proportion of valuablematerial and a lower proportion of gangue than thefeed. It is used to generate the concentrate. Normal-ly, a series of cells are connected, generally withfeedback between them.

For an individual cell, the correspondences withFig. 2 are

f u : 2-vector comprising air flow rate and addi-tional water flow rate;

f w : change in total feed solids flow rate (G&#Z

&);

f z : concentrate valuable material;f y : same as z.

A discrete-time formulation is used, and the i!jelement of the matrix Q is

Qij"

L+k/0

qij(k)z~k. (4.1)

Thus, an optimization is carried out over a finite-impulse-response (FIR) Q(z~1). The quantity opti-mized is a time-weighted (weight"square of time)sum-of-squares-error of the outputs to a step ina feed disturbance

'"

NZ+r/1

ND+s/1

¼rs

L+k/1

k2[ssetrs

!srs(k)]2 . (4.2)

Survey Paper 1487

Fig. 3. Flotation cell.

Here NZ, ND refer to the dimensions of z and w (forall cells in a bank), ¼

rsis a weighting matrix, s

rs(k)

is the response at time k of zr, to a step at time zero

in ws, and s

setrs(k) is the corresponding reference

value.Normally, some cells will have their concentrates

used as input to other cells. One or more cells willprovide the final concentrate, and ¼

rswill reflect

the importance attaching to these last cells.Because of equation (3.7), the index (4.2) can be

expressed in terms of the free parameters in Q(z~1).Key to the optimization being so easy is the quad-ratic nature of equation (4.2) not just in Q (z~1), butin the individual parameters q

ij(k) making up

Q(z~1), as well as the finite-dimensional nature ofthe optimization.

5. A MORE ACADEMIC EXAMPLE

The advantage of the affine parametrizationis brought out in the following example (Doyleet al., 1992). The plant, which is to be stabilizedwith a series compensator in a negative feedbackloop, is

P"

1

(s!1)(s!2)"ND~1,

N"

1

(s#1)2,

D"

(s!1)(s!2)

(s#1)2.

The problem is to find a proper C(s) achievingclosed-loop stability; also the final value of y is 1when r is a unit step and d"0, and the final valueof y is zero when r"0 and d is a sinusoid of10 rad/s.

The Bezout equation XM N#½M D"1 is satisfiedby

XM "19s!11

s#1, ½M "

s#6

s#1

and one stabilizing controller is given by X½~1

with X"XM , ½"½M . The double Bezout identityholds by setting also NM "N and DM "D.

A Q(s)3S will ensure closed-loop stability forthe controller C"(X#DQ) (½!NQ)~1. The unitstep response requirement is that G(0)"1, or byequation (3.4),

Q(0)"6.

The requirement to suppress a sinusoidal outputdisturbance of 10 rad/s from equation (3.5) is

S ( j10)"0or

Q( j10)"!94#70j.

Since Q (s) is constrained by three real interpolationconditions, we can assume

Q(s)"a1#

a2

s#1#

a3

(s#1)2

and choose aito ensure the interpolation data are

fulfilled. This leads to a fourth-order controller

C(s)"X#DQ

½!NQ

"

!60s4!598s3#2515s2!1794s#1

s(s2#100)(s#9).

The denominator zeros of 0 and $j10 are consis-tent with the internal model principle and the data.

Professor S. Manabe has pointed out in a privatecommunication that the choice of denominator forN,D,XM ,½M and Q of (s#1) or (s#1)2 results ina very nonrobust system, while other choices maybe far more satisfactory.

6. RECENT DEVELOPMENTS OF THE

Q PARAMETRIZATION

In earlier sections, we have described materialinvolving the Youla—Kucera parametrization thatis reasonably well known. In this section, we look at

1488 Survey Paper

Fig. 4. (a) 2-Degree of freedom controller (b) Equivalent arrang-ment as in a.

some more recent developments, including twodegrees-of-freedom controllers, state-variable ver-sions of the earlier ideas, and controller design withconstrained closed-loop pole positions.

6.1. ¹wo-degrees-of-freedom ControllersA two-degree-of-freedom arrangements is de-

picted in Fig. 4a; the specialization C1"C

2"C

would recover the scheme of Fig. 1. Finding aQ(s)3S parametrization of all stabilizing C is nothard, using a trick (Moore et al., 1986).

Work with an augmented plant:

Pa(s)"C

0

P (s)D"N (s)D~1(s)"C0

N2D D~1

"CI 0

0 DM ~12D~1

C0

NM2D"DM ~1NM . (6.1)

Suppose a specific controller (in a negative feed-back loop) for P

a(s) is

[C01(s) C0

2(s)]"½M ~1[XM

1XM

2]"X½~1 (6.2)

where equation (3.3) holds. From the fact thatNM X#DM ½"I and NM and DM are structured, weconclude that

½"CI 0

½12

½2D ,

for some ½2.

The set-up of Fig. 4a is exactly what is obtainedwhen P

a(s) of Fig. 4b has a series compensator (in

a unity negative feedback loop), the first externalinput is r and the second external input is zero. Thefirst output of P

a(s) is always zero, and the second is

y of Fig. 4a; see also Fig. 4b. The closed-loop ofFig. 4b is stable exactly when C

2(s) stabilizes P(s).

The set of all stabilizing controllers [C1

C2] is

given, with Q1,Q

23S, by

[½M !(Q1

Q2)NM ]~1[(XM

1XM

2)#(Q

1Q

2)DM ]

"[½M !Q2NM

2]~1[XM

1#Q

1XM

2#Q

2DM

2].

(6.3)

Thus, the set of C2(s) is the same as the set of C(s)

arising in the one degree of freedom of problem,viz., (½M !Q

2NM

2)~1(XM

2#Q

2DM ) for arbitrary Q

23S;

the set of C1(s) is richer, and contains a further

parameter, viz, Q13S. The transfer function from

r to y can be computed as

G (s)"N2(XM

1#Q

1) (6.4)

and the transfer function from d to y is

S (s)"(½2!N

2Q

2)DM

2. (6.5)

There is a helpful separation here: Q1

affects track-ing but not disturbance behaviour, and Q

2affects

disturbance but not tracking behaviour.

6.2. Introducing Q(s) into a controller instate-space form

The idea of introducing Q(s) into a controllerobtained by combining a state estimator and a statefeedback law goes back at least to Tay and Moore(1988). The key is to pick nice associated fractionaldescriptions of the plant and controller.

Consider a plant P (s) with minimal realization

P (s) s"CA B

C 0D . (6.6)

[The notation is to be understood as saying P (s)"C(sI!A)~1B.] Let F be such that Re j

i(A#BF)

(0 and H such that Reji(A#HC)(0. It is well-

known that an observer-based stabilizing control-ler is defined by

xR#"(A#HC) x

#!Hy#BFx

#, (6.7)

u"Fx#. (6.8)

This controller is depicted in Fig. 5. Coprimefactorizations ND~1"DM ~1NM of the plant andX½~1"½M ~1XM of the controller obeying theBezout identity (3.3) are given by (Nett et al., 1984).

CD(s) !X(s)

N (s) ½(s) D s"

A#BF B !H

F I 0

C 0 I

, (6.9)

C½M (s) !XM (s)

!NM (s) ½M (s) D s"

A#HC !B H

F I 0

C 0 I

.

(6.10)

(More complicated formulas hold if there are directfeedthrough terms present.) It is not hard to verifyfor Fig. 5 that

½(s)0"[I!C(sI!A!BF)~1H]0"!y,

X(s)0"F(sI!A!BF)~1H0"u,

from which the right fractional representation½X~1 of the controller follows. The Youla—Kucera

Survey Paper 1489

Fig. 5. Observer-based stabilizing controller.

Fig. 6. Introduction of a Youla—Kucera parameter Q (s)3S into an observer-based stabilizing controller.

parameter Q(s)3S is introduced as shown inFig. 6, and it is not hard to check that

[½(s)!N(s)Q(s)]0"!y

[X(s)#D(s)Q(s)]0"u

so that

u"![X#DQ] [½!NQ]~1 y.

Similarly, it can be argued that

u"![½M !QNM ]~1[XM #QDM ]y.

A number of applications of this idea can be noted;for example:

f For a nonminimum phase plant, an LQG orH

2design cannot achieve loop transfer recovery.

However, a Q(s)3S can be selected to trade offH

2optimality and loop transfer recovery (Moore

and Tay, 1989).f Low-order adaptive adjustments can be made of

a high-order controller connected to a nominalplant. An adaptive Q (of low-order) can be usedto augment a fixed (possibly high order) control-ler connected to a nominal plant, where there areplant perturbations or uncertainties. In the dis-crete time case, Q can be taken as a finite impulseresponse transfer function; see Chakravarty andMoore (1986) and Tay and Moore (1991).

Above, for a controller of a particular structure,we identified in equation (6.2) left and right coprimefractional descriptions of the plant and controller,in state-space form and satisfying the Bezout iden-tity (3.3). However, it is possible for any stabilizingcontroller to construct such fractional descriptions(Chakravarty and Moore, 1986). Suppose the con-troller is

C0(s) s"CA# B#

C# D#D . (6.11)

Let F and F# be stabilizing state feedback gainssuch that Re j

i(A#BF)(0 and Re j

i(A##B#F#)

(0. Then

CD(s) !X(s)

N(s) ½(s) D

s"

A#BF 0 B 0

0 A##B#F# 0 !B#

F C#!D#F# I !D#

C F# 0 I

, (6.12)

C½M (s) XM (s)

!NM (s) DM (s)D

s"

A#BD#C BC# B !BD#

!B#C A# 0 !B#

!(F#D#C) C# I D#

C !F# 0 I

. (6.13)

1490 Survey Paper

Fig. 7. (a) Controller C0(s), displaying right coprime factorization (b) Plant P0 (s), displaying right coprime factorization.

Fig. 8. Stabilizing controller, using Youla parameter Q (s) andright coprime factorization of nominal controller and of plant.

There is a second pair of such definitions, usingH and H# such that Re j

i(A#HC)(0 and

Re ji(A##C#H#)(0.

If C0(s) is in fact F(sI!A!HC!BF)~1H thenthe choice F#"!C yields the pair (6.2).

Figures 7 and 8 illustrate the form of a generalstabilizing controller, based on the use of equa-tion (6.12). The order of a controller obtained whenintroducing a Q (s) has the potential to be large.It may be that a controller order reduction step(Anderson and Moore, 1990) must be employedafter determination of Q(s).

6.3. Pole positioning in ¸QG or H=

designConsider an LQG problem for the plant

xR "Fx#Gu#!w,

y"Hx#v,

where MF,G,HN is minimal, and [w@ v@]@ is azero mean, Gaussian white noise process with

covariance

CQ 0

0 RD d(t!s) where R is nonsingular.

Suppose one is interested in minimizing theexpected value of the quadratic index

»"P=

to

(xTQx#uTRu) dt,

where Q"QT50 and all the usual side conditionsapply. Suppose also one wishes all closed-looppoles to lie in Re[s](!a for some a'0. An olddevice is to replace the index by

»a"P=

to

e2at (xTQx#uTRu) dt ;

see Anderson and Moore (1970). Minimizing theexpected value of »a ensures that all closed-looppoles lie in Re[s](!a. One can then evaluateE(») for the design resulting from use of the»a index. It turns out that on occasions much lowervalues of E (») could be achieved from a controllerof prescribed order which achieves the closed-looppole constraint; thus minimizing E (»a) rather thanE(») may lead to very unsatisfactory results as faras E (») is concerned.

The Youla—Kucera parametrization provides analternative approach, which is closer to minimizingE(»), and which allows a wider choice of closed-loop pole positions (DeBruyne et al., 1995).

Suppose the closed-loop pole positions arerequired to lie in a domain D, contained withinRe[s](0, symmetric with respect to the realaxis, and (for technical reasons) containing atleast one point on the negative real axis. Forexample,

D"Ms :Re[s](!a, DIm[s]D4tan hDRe[s]D, a'0N.

Survey Paper 1491

Fig. 9. Set-up for open-loop identification.

Fig. 10. Set-up for closed-loop identification.

The plant can be represented as a coprime fractionND~1, where N, D3S

D, the set of proper rational

transfer function matrices with poles in D. Co-primeness is equivalent to the existence of X,½3S

Dwith XN#½D"I.

Now SD3S, the set of all stable proper transfer

functions. Therefore, the result of solving thenormal LQG problem with no closed-loop poleconstraint other than simple stability will bea stabilizing controller C(s) with fractional des-cription representable as (X#DQ)(½!NQ)~1 forsome Q3S.

Were Q3SD, the closed-loop poles will be in D.

In general, Q NSD. However, one can approximate

Q by some QD3SD, and secure thereby approxim-

ately the same value of E(»), but with all closed-loop poles in D. Indeed, if the degree of QD can bearbitrarily large, the approximation can be arbitra-rily accurate. In general, we do not want the degreeof QD and thus of C(s) to be arbitrarily large.

The same idea is valid for H=

design.

7. PARAMETRIZATION OF THE PLANT AND

CLOSED-LOOP IDENTIFICATION

In this section, we first consider the dual questionto that treated earlier of how to parametrize allplants (in addition to a nominal one) which arestabilized by a prescribed controller.

In a unity feedback loop with series comparator,with a scalar plant, it is clear that C(s) and P(s) canbe interchanged without affecting the closed-looptransfer function. This strongly suggests that thereshould be an analogous result to Theorem 1 deal-ing with the set of all plants stabilized by a fixedcontroller. Indeed, that is so:

¹heorem 2. Let a plant P"ND~1, with N and Dcoprime over S, be stabilized by a (negative feed-back) controller C"X½~1, with X and ½ coprimeover S. Then the set of all plants stabilized by theone controller C is given by M(N#½S)(D!XS)~1 :S3SN.

There is an extremely important application ofthis idea, to closed-loop identification in the pres-ence of noise.

7.1. Open and closed-loop identificationTo motivate the problem, first consider how

open-loop identification can be carried out, (seeFig. 9). In a common scenario, the input processu and noise process n are assumed independent andstationary. The plant is scalar, time-invariant andstable, and we can write

y"Pu#n. (7.1)

Measurements of u and y are available. Manyidentification schemes for estimating P effectivelyamount to cross-correlating with u and then solv-ing for P :

'yu

(s)"P(s)'uu

(s)#'nu

(s) (7.2)

"P(s)'uu

(s). (7.3)

Of course, equation (7.3) follows because u andn are made independent. In actual practice, thecalculation may be done in the time domain, andsample estimates (obtained over a finite interval)may be used instead of true expectations, etc., butthe basic principle is the same.

Now consider the stable closed loop of Fig. 10,with r and n independent processes. The task isto identify P(s) from measurements which do notinclude its noiseless output s. Equations (7.1) and(7.2) remain true. However, equation (7.3) is re-placed by

'yu

(s)"P (s)'uu

(s)!(1#C*P*)~1C*'nn

. (7.4)

The superscript asterisk denotes replacement ofs by !s, or complex conjugation on the ju axis.Evidently, the task of obtaining P (s) is considerablymore complicated. Also, even if '

nnis very small,

software packages may be predicated on an as-sumption that P(s) is stable, which need not be so.

One can of course seek to identify the closed-loop transfer function G"PC(1#PC)~1 from

'yr"G'

rr(7.5)

and then since P"G/C(1!G), one chooses anestimate PK of P in terms of an estimate GK of G as

PK "GK

C(1!GK ). (7.6)

These can be problems with this approach. Forexample, if C has an unstable pole, the loop com-prising PK and C will generally be unstable.

1492 Survey Paper

Fig. 12. Details of closed-loop identification.

Fig. 11. Alternative representation of noise-contaminated plant.

Also, parametrizations which are convenient forGK may be convenient for PK and vice versa, in view ofthe nonlinear connections; in some cases too, para-meter space regions corresponding to stable PK ,C pairs may contain disconnected sub-regions (VanDonkelaar and Van den Hof, 1996).

A clever resolution of these difficulties wasobtained independently in Tay et al. (1989), Hansen(1989), Hansen et al. (1989) and Schrama (1991).

Suppose that the controller C (s) is specified as½M ~1XM where XM ,½M 3S are coprime. Then by thecoprimeness there exist N,D3S with

XM N#½M D"I. (7.7)

This means that P0(s)"ND~1 is one plant (a

‘‘nominal’’ plant) stabilized by C(s). [Alternatively,one may start with a representation ND~1 of thenominal P

0(s) and then choose that particular rep-

resentation ½M ~1XM of the known stabilizing C(s) sothat equation (7.7) holds.] The set of all plantsstabilized by C(s) is given by

M[N(s)#½ (s)S (s)][D(s)!X (s)S(s)]~1 :S3SN.(7.8)

Here, X½~1 is a coprime right fractional repres-entation of C(s).

Suppose that DM ~1NM is a left coprime fractionaldescription of the nominal P

0(s) so that

C½M XM

!NM DM D CD !X

N ½ D"CI 0

0 ID . (7.9)

Then it is not hard to check that the set-up ofFig. 11, with inputs u and n and output y, implies

y"Pu#n,

where P is given by equation (7.8).Now the key identification idea is to identify S in

the first instance, rather than P. The perhaps sur-prising and pleasing aspect is that the identificationof S is a standard open-loop identification problem.To see this, observe from Fig. 12 that

(D!XS)x"u#X (DM !SXM )n

"r2#½M ~1XM r

1!½M ~1XM y

#X (DM !SXM )n,and

(N#½S)x"y!½(DM !SXM )n.

Multiplying the first equation by ½M and thesecond by XM and adding yields

x"XM r1#½M r

2. (7.10)

FurtherDx"u#Xz,

Nx"y!½z.

Multiplying the first equation by NM , and the secondby DM and subtracting yields

z"DM y!NM u, (7.11)

while also direct inspection of Fig. 12 shows that

z"Sx#(DM !SXM )n. (7.12)

Observe in equation (7.12) that (i) x and z areavailable from measurements on the closed-loop systems (via equations (7.10) and (7.11)); (ii)if n( )) is independent of r

1( ) ) and r

2( ) ), then

in equation (7.12), x ( ) ) and n ( )) are inde-pendent processes; and (iii) S is stable, via theYoula—Kucera theory. Hence, the identification ofS is a standard open-loop identification problem.So, for that matter, is the identification of a shap-ing filter, which when driven by white noise,will generate n.

Survey Paper 1493

Fig. 13. Second alternative realization of noise-contaminatedplant.

The blocks in Fig. 11, apart from S, are asso-ciated with right coprime fractional description ofa nominal plant and a controller stabilizing thenominal and the true plant. It is also possible towork with left coprime realizations. Fig. 13 depictsa representation of the plant using XM ,½M ,NM andDM instead of X,½, N and D. The signals labelledx and z and blocks labelled S in Figs. 11 and 13 arethe same, in that

P (s)"(N#½S)(D!XS)~1

"(DM !SXM )~1(NM #S½M )

and for both schemes, equations (7.10) and (7.12)are valid when the plant is connected in a feedbackloop with controller X½~1"½M ~1XM .

Since there is no convenient replacement ofequations (7.10) and (7.12) in which entities asso-ciated with right rather than left coprime factoriza-tion appear, it is perhaps tidier to work with thescheme of Fig. 13 than that of Fig. 11.

A number of other points should be made:

(i) If P (s) is known a priori to be strictly proper,and P

0(s) is strictly proper, S(s) must have this

property, and conversely.(ii) Suppose that for identification purposes a

finitely parametrized model set MSa ,a3AN isadopted and one obtains Sa* as some sort ofbest approximation to S. Let Pa* (s) be the cor-responding approximation to P(s). Then it canbe shown that

PC(I#PC)~1!Pa*C(I#Pa*C)~1

"½(S!Sa*)XM (7.13)

and if r2"0,

[PC(I#PC)~1!Pa*C(I#Pa*C)~1]r

"½(S!Sa*)x. (7.14)

In both equations, the left-hand side is a mod-elling error associated with a closed-looptransfer function. The right-hand side is theopen-loop modelling error associated with S,using a frequency weight ½.

A particular version of these formulas resultswhen Sa* is replaced by 0, corresponding to thenominal plant.

Then

PC(I#PC)~1!P0C (I#P

0C)~1"½SXM ,

(7.15)

[PC(I#PC)~1!P0C(I#P

0C)~1]r"½Sx.

(7.16)

The expressions relate S to a closed-loopquantity.

(iii) One can also note that

P!P0"DM ~1S (D!XS)~1, (7.17)

which shows that a frequency weighted versionof S yields the open-loop error between P andP0; one of the weights depends itself on S.

(iv) The identification of a P through P0

and thenS can give rise to a possibly large degree for P.If subsequent identifications refine knowledgeof P even further, the degree may increase ateach identification step. For this reason, it of-ten makes sense to do an order reduction of anidentified P. Since it is the closed-loop obtainedby using P with C which is relevant, one seeksa low-order PK so that a quantity such as thefollowing is minimized:

DDPC(I#PC)~1!PK C(I#PK C)~1DD=.

This is approximately (neglecting second-ordererror terms)

DD(I#PC)~1(P!PK )C(I#PC)~1DD=

and a frequency weighted model reductionproblem is to be tackled.

(v) It can be argued that a normalized coprimefractional description of P

0is best used (Van

den Hof et al., 1995). One of the reasons for thisis that it is possible to flag situations wherecontroller redesign after identification (Zanget al., 1995) may be problematic. We explainthis point in more detail. Let N, D be a nor-malized right coprime fractional description ofP0, so that N*N#D*D"I. Let PI be any

other plant, with any right coprime fractionaldescription NI , DI . Define the directed distancefrom P

0to P by

d (P0, P)"

*/&º3S KK C

N

DD!CNIDI Dº KK

=

.

Let

KK CP0C(I#P

0C)~1 P

0(I#CP

0)~1

(I#CP0)~1C (I#CP

0)~1 D KK

=

"c~1.

1494 Survey Paper

Then C stabilizes all P with d (P0,P)(c (Geor-

giou and Smith, 1990).Now suppose that we use the identification

procedure for P described above. In the pre-sence of noise, it may not yield the correct S,but rather some SK . Let the associated plant bedesignated by PK . If (N,D) is normalized, wehave

d(P0, PK )"

*/&º3S KK C

N

DD !CN#½SKD!XSK Dº KK

=

4KK CN

DD!CN#½SKD!XSK D KK

=

"KK A½

XBSK KK=

We know that C stabilizes P (by experiment),stabilizes P

0(by calculation) and stabilizes PK (by

the theorem on Youla—Kucera parametrization).However, a condition like

KK A½

XB SK KK='c~1

is a flag that d (P0, PK ) may also exceed c~1, so that

the difference between P0

and PK is in some sensesubstantial; redesign of a controller should there-fore proceed cautiously; see Bitmead et al. (1997).Even controllers close to C resulting from a re-design process could have stability problems.

7.2. Noisy identificationWe have explained that the basis of the identi-

fication problem is to use equation (7.12). It isimportant to consider at any one frequency whatthe signal-to-noise ratio is, and how it is related toquantities in the original actual and model closed-loop.

For simplicity, suppose r2"0 and that plant and

controller are scalar; thus DM "D, NM "N, etc. Thenin equation (7.12), the signal-to-noise ratio is

KS

D!SX K2 '

xx'

nn

"KSX

D!SX K2 '

r1r1'

nn

.

(Here ' is a generic symbol used to denote a spec-trum.) It is not hard to check the error between thatpart of the actual closed-loop output due to r

1and

the model output is

APC

1#PC!

P0C

1#P0CB r

1"SX½r

1,

while the noisy component of the actual closed-loop output is (1/(1#PC)) n"(D!XS)½n.

The associated signal to noise ratio is then

Ktracking error ‘‘signal’’

Noise K2"K

SX

D!SX K2 '

r1r1'

nn

,

which is the same quantity as the signal to noiseratio appearing in the identification step. This is animportant observation; it means that despite thecollection of filters and signal transformationswhich arise in setting up the open-loop identifica-tion, there are no choices that could improve orworsen the signal-to-noise ratio which constitutesa practical measure of the difficulty of identifying,and which is the SNR relevant in assessing theeffectiveness of a closed-loop identification fol-lowed by loop unravelling.

7.3. ¼indsurfer approach to adaptive controlThe ideas of Section 7.1 are crucial to the deve-

lopment of a methodology for adaptive controlwhich is much less based on identifying numeratorand denominator coefficients in plant or controllertransfer functions than in an iterative identificationand controller design scheme, tied to the frequencydomain (Lee et al., 1992, 1993, 1995).

The original motivation was to do adaptivecontrol robustly, i.e. without incurring ‘‘temporary’’instability in the learning phase. By way of exam-ple, conventional adaptive control can encountersubstantial difficulty if there are high-frequency dy-namics in the true plant not reflected in an a priorimodel of the plant, and possibly not even capable ofbeing reflected by adjusting coefficients in thea priori model, due to too low a degree having beenadopted for it.

Windsurfer adaptive control was first developedfor open-loop-stable plants (permitted though tohave a pole or poles at the origin). The key idea (aswhen a human learns windsurfing) is to initially usea controller defining a very small closed-loop band-width; design of this controller requires almost nomore prior knowledge than the sign of the DC gainof the plant.

A series of redesigns of the controller is theneffected, each pushing out the closed-loop band-width some more. At some stage, predicted andactual closed-loop performance becomes clearlydifferent; at this point, the plant is better identifiedusing the scheme of Section 7.1. This new identifica-tion will then cause the closed-loop transfer func-tion obtained using the new model of the plant andthat obtained using the actual model of the plant tobe far more similar. In effect, knowledge of theplant is developed over a wider bandwidth thanpreviously (with the quality of that knowledgelinked to closed-loop rather than open-loopbehaviour).

With the re-identified plant, the controller designis then adjusted, progressively expanding theclosed-loop bandwidth until again there is diver-gence between behaviour as predicted by the modeland the actual behaviour. A further (closed-loop)identification is executed, thereby further widening

Survey Paper 1495

the bandwidth over which the plant is adequatelydescribed (for the purpose of closed-loop control).Once again, closed-loop bandwidth is expanded viaa series of controller design, and so on.

The procedure thus gives rise to a sequence of(strictly) proper stable plant models, MG

0, G

1, 2N

and for each model Githere is a sequence of con-

trollers K0i, K1

i, 2Kf

igiving progressively larger

bandwidths for the closed-loop transfer functionG

iKj

i(I#G

iKj

i)~1.

A great number of important questions arise:

f Is there a scheme for progressively increasing theclosed-loop bandwidth? Yes, for stable plants atleast. The IMC method (Morari and Zafiriou,1989) is ideal, allowing direct control overclosed-loop bandwidth.

f When can the closed-loop bandwidth be in-creased with safety, i.e. without losing stability,while retaining use of the (possibly inaccurate)model G? There can be no sudden onset of insta-bility that is not preceded at a somewhat lesserdesign bandwidth by significant difference bet-ween predicted and actual closed-loop behaviour.

f What would we like to identify, in order that withthe new model, performance of the closed-loopsystem can be improved through controller re-design? It turns out that we would like to replacethe model G

iby a model G

i`1so that, with G the

true plant and ¹Mi`1

the designed closed-loop,

KKG!G

i`1G

i`1

¹Mi`1 KK

=

(1

and

KG!G

i`1G

i`1

¹Mi`1 K

is small for all frequencies above the lesser of thebandwidth of ¹M

i`1and the smallest frequency

corresponding to zeros if Gi`1

in Re[s]50.f What can we identify (as opposed to what would

we like to identify)? As we know, we can identifya Youla—Kucera parameter S, albeit in the pres-ence of noise. However, we can only identify itaccurately if a certain signal-to-noise ratio ishigh. ‘‘Signal’’ is the error between that part of thetrue plant output which is due to input excita-tions and the model output (which is due solelyto input excitations). ‘‘Noise’’ is that part of theplant output arising from disturbances. The signal-to-noise ratio is only likely to be high in certainfrequency ranges. Further, as pointed out in thelast subsection, it is this signal-to-noise ratio thatis also directly relevant in the open-loop identi-fication of the Youla—Kucera parameter S.

f When will there be a significantly high signal-to-noise ratio allowing identification of S in a fre-quency band? The instinctive answer is almost

correct: there must be a failure of the model tocorrectly predict closed-loop performance of thecontroller with the true plant. But there is a sur-prise qualification. Nonminimum phase zeros inthe passband of the plant and or model can meanthat noise disturbances cannot be well cancelledby the controller. Then ‘‘signal’’ and noise can behigh but the ratio may not be high, and reiden-tifiability is hard.

Roughly speaking it turns out that what we want toidentify and what we can identify coincide, untilnonminimum phase zeros appear in the closed-loop passband. Then the scheme will come to a stopin this situation; it may also stop when the open-loop bandwidth of the true plant is exceeded by theproposed design bandwidth, so that unreasonablylarge plant inputs are demanded.

Some important points should be made.

(i) The initial model may be very low order, evenif it is suspected that the plant has high-fre-quency resonances.

(ii) The scheme does not permit instabilities tooccur, nor for that matter adverse transientbehaviour; the latter especially is a concernfor conventional adaptive control schemes, es-pecially if there are unknown high frequencyresonances in the plant.

(iii) Generally, the Youla—Kucera parameter in theplant re-identification step need only be oforder 2—4, since a modest adjustment of themodel applicable over a limited bandwidth iswhat is required.

(iv) If the true plant is unstable, control of closed-loop bandwidth by adjustment of a singleparameter in the IMC design scheme is notstraightforward. Also, it may be hard to find aninitially stabilizing controller. Only prelimi-nary work has been done on the adaptive con-trol scheme for such plants.

Why there is iteration in the first place? Startingwith minimal information, the iterative approach,because it is gradual, allows an adaptive controllerto be found without risking ‘‘transient instability’’,i.e. the occurrence of massive signals, in practicalterms indistinguishable from those in an unstablesystem, during the learning phase of the adaption ofthe adaptive controller. Those ‘‘conventional’’adaptive controllers have great difficulty in protect-ing against transient instability, especially thosethat presuppose an order for the unknown plant.There is a risk that the assumed order is too low tocapture high-frequency resonances, and too high toallow inputs to be persistently exciting; either way,trasient instability can result.

New experiments are essential, to exceute pro-gressively wider bandwidths. At early iterations,

1496 Survey Paper

only low bandwidth inputs are applied to the plant.In later iterative steps, when higher-frequencybehaviour is being learnt, higher-frequency plantexcitation is required. This is achieved by ensuringthat the closed-loop external input is of sufficientbandwidth and also the controller bandwidth issuch as to let through signals of adequate band-width.

8. YOULA—KUCERA PARAMETERS IN PLANT

AND CONTROLLER

We have so far dealt with a set of controllersstabilizing one plant, and a set of plants stabilizedby one controller. We are now going to mix thesethemes.

8.1. ¹he stability issueWe begin with a question. Consider a nomi-

nal plant-controller pair, P0(s)"N(s)D~1 (s)"DM ~1(s) NM (s) and C0(s)"X (s)½~1(s)"½M ~1(s)XM (s).We assume that N,D, etc., are elements of S and

C½M XM

!NM DM D CD !X

N ½ D"CI 0

0 ID . (8.1)

For what pairs Q(s), S (s)3S will the plant P (s)"[N(s)#½(s)S (s)][D (s)!X(s)S (s)]~1 and control-ler C(s)"[X(s)#D(s)Q(s)][½(s)!N (s) Q(s)]~1

define a stable closed-loop? The answer is (Tayet al., 1989).

¹heorem 3. Let P0"ND~1"DM ~1NM and C0"

X½~1"½M ~1XM with equation (8.1) holding andN,D,2, 3S define a stable unity negative feed-back closed loop. Then P"[N#½S][D!XS]~1

and C"[X#DQ][½!NQ]~1 for S and Q3Sdefines a stable unity negative feedback closed loopif and only if S and Q together define a stable loop.

Proof. P and C will form a stable loop if and only if

CD!XS !X!DQ

N#½S ½!NQ D~1

3S

8GCD !X

N ½ D CI !Q

S I DH~1

3S

8GCI !Q

S I D~1

H3S ,

which holds if and only if S and Q together definea stable loop.

The fact that S, Q3S is not directly used. Notealso that it is not hard to vary the above argumentto show that generically the closed-loop poles ofthe P (s), C(s) system are the union of those of theP0(s), C0(s) systems and the S, Q system (Tay et al.,1989). h

The above result raises the following issue. Supposea controller design has been achieved for a nominalplant, and then more information concerning theplant becomes available (i.e. S(s) is learnt, at leastapproximately). How should the controller be ad-justed? Equivalently, how should Q(s) be chosen?

At the broadest level, the answer depends on thedesign criterion, and we look at two (LQG and H

=)

below. Of course, one could imagine repeating awhole design process with a new P (s) (involvingS(s)), obtaining C(s), from which one could thendevise Q(s). This however is not the point of thequestion; the key idea is to see whether Q(s) couldbe obtained from S(s) via some simple formula.Thus, we could imagine that if the nominal P0(s)and C0(s) were of high order, 40 say, and S(s)was second order (associated with the inclusion ofa further resonant mode in P(s) for example), thenQ(s) might be second order also. The discussionof closed-loop poles above shows that if theirpositions are the prime concerns, one could pro-ceed in exactly this way, i.e. computing Q(s) directlyfrom S(s).

8.2. ¸QG designThe problem of relating a plant perturbation

through to a controller perturbation for LQ designis treated in Anderson et al. (1994). We shall indi-cate here simply the nature of the answer. First, it isonly possible to get a simple relationship betweenS and Q when S is small. Then the form of the resultis (for a scalar plant)

QKA~1[A~*S*DD*']45!"-%

. (8.2)

In this formula, A is the minimum phase spectralfactor of an entity formed in solving the LQ pro-blem for P0 and C0,X*(s) denotes X(!s), ' is thespectrum of the exciting noises (and may be white),and [Z]

45!"-%denotes the stable part of a partial

fraction expansion of Z.This formula has not been reworked using state

variable calculations to try to relate the degrees ofS and Q.

Retaining the assumption of small S, it is possibleto obtain an expression for the change in quadraticcost. It is of the form

cost changeK1

2n P [B*Q#Q

*B] du. (8.3)

where B depends on P0 and C0.A second approach to LQ design is set out in Tay

et al. (1989). A design for P0 is done to yield C0.Then a separate design for S is done to yield Q; theinput and output of S appear in the index. No realguidance is given as to how to select weights for thissecond problem, although it is argued that forthe second problem, having constant rather than

Survey Paper 1497

freqency-dependent weights gives an index whichis logical in terms of the original objective of mini-mizing through choice of C (s) a quadratic indexinvolving P (s).

The degree of Q(s) is effectively that of S (s), whichis an advantage. An example in Tay et al. (1989)shows the efficacy of the approach.

It would certainly be of interest to quantify theloss of optimality of a design achieved by the two-step procedure (P0 gives C0 and S gives Q) incomparison with an optimal design (P gives C).

8.3. H=

designThe set-up here starts with a generalized plant

P0 (s)"CP11

(s) P12

(s)

P21

(s) P022

(s)D (8.4)

of the type shown in Fig. 2 and an associatedH

=controller C0(s). We suppose that P0

22"

ND~1"DM ~1NM and C0"X½~1"½M ~1XM , withthe usual Bezout equation

C½M XM

!NM DM D CD !X

N ½ D"CI 0

0 ID .

[Note the sign convention in Fig. 2 allows us toregard P0

22(s) and C0 as forming a unity negative

feedback loop.] All standard conditions for solva-bility of the H

=problem are assumed to be fulfilled

and also in the original reference (Yan and Moore,1993) P

12and P

21are assumed stable, though this

seems unnecessary if all unstable modes of P0(s)appear in P0

22(s).

Suppose now that P022

is replaced by P22"

(N#½S)(D!XS)~1 and let P (S) denote thecorresponding generalized plant. Then thereis a clever way of choosing Q(s), as follows.Define a second generalized plant involving Salone, by

PS"CPS11

(s) PS12

(s)

PS21

(s) PS22

(s)D"CC0 S

0 0D CS

ID[I!S] S D . (8.5)

Let FL(P, C) denote the system arising from inter-

connecting a generalized plant P with a controllerC, as illustrated in Fig. 2. Then (Yan and Moore,1993)

EFL(P (S),C(Q))!F

L(P0,C0)E

=4aEF

L(PS, Q)E

=,

(8.6)where

a"EP12

[D !X]E=KK C

XMDM D P

21 KK= . (8.7)

Evidently, the idea is to choose Q to keep theright-hand side of equation (8.6) small; then there issupposed to be good matching on the left-hand

side. The expression FL(PS,Q) evaluates as

FL(PS, Q)"C

SQ (I!SQ)~1 (I!SQ)~1S

Q (I!SQ)~1 !(I!QS)~1QSD .

It is undoubtedly clear that if ESE=

is small thenQ can be chosen with EQE

=small and EF

L(PS,Q)E

=also small. What is not clear is how much damagethe multiplier a could do. An examination of Yanand Moore (1993) shows that a is a crude boundingquantity; it may be possible to improve on thebound, and/or it may be possible to identify struc-tures where a is guaranteed to be small.

The above has one clear advantage; finding Qknowing S does not involve P0 (s) or C0 (s), and thedegrees of Q and S will be comparable if Q is foundvia H

=methods.

8.4. Indirect adaptive control and iterativecontroller design

The potential for indirect adaptive control usingthe ideas of this and the previous section should beclear. Methods tackling closed-loop identificationby reduction to an open-loop problem allow cor-rection of a nominal plant model P

0(s) by a stable

S(s); then a pole-positioning, LQG or H=

approachallows derivation of a Q(s) (Tay et al., 1989). Ofcourse, S (s) can be slowly varying, requiring on-linetechnology. Then Q(s) must be subject also toupdate.

In one sense, iterative controller design (Zanget al., 1995) is a slowed-up form of indirect adaptivecontrol. In a recent development of the ideas ofZang et al. (1995) and Bitmead et al. (1997), a cleverformula has been derived which neatly encapsu-lates the effects of plant and controller variation.In the scalar case, suppose

P0"D~1

0N

0, C

0"½~1

0X

0with

D0½

0#N

0X

0"1.

Suppose further that

P1"(D

0#Q

1X

0)~1 (N

0!Q

1½

0)"D~1

1N

1

is stabilized by C0

and P1

is also stabilized by

C1"(½

0#S

1N

1)~1(X

0!S

1D

1)

(Of course, D0, N

0, etc., are all stable transfer func-

tions). Then

T (P0,C

0)"C

11`P0C0

C0

1`P0C0

P0

1`P0C0P0C0

1`P0C0D"C

D0

N0D [½

0X

0]

and

T (P1, C

1)"A

X0

D0

!½0

N0B A

Q1

1 B (S 1!S1Q

1)

]AN

0!D

0½

0X

0B .

1498 Survey Paper

Note that

AX

0D

0!½

0N

0B A

N0

!D0

½0

X0B"A

1 0

0 1B .

8.5. Open issuesA key question raised at the start of this section

was: how should the adjustment Q(s) of a controllerbe determined, knowing an adjustment S (s) of theplant. The answers to this point have been incom-plete:

f Pole positioning. The incompleteness arises be-cause, firstly, pole positioning by itself is a limiteddesign objective and, secondly, the result is onlygeneric.

f ¸QG. The incompleteness also arises becauseeither one stays with the original performance indexand then only gets simplification, albeit limited,when adjustments are small; or one uses a differentperformance index for the S—Q optimization,which cannot give an overall optimal result.

f H=: The incompleteness arises because of an in-

equality in the key result, and an appearance ofa constant a with rather unclear intuitive content.In particular, even if an H

=controller C(Q) exists

for a plant P(S) (to achieve internal stability anda given performance level), there is no guaranteethat one can find it by using an index based ona plant PS (determined solely by S) and a feed-back controller Q.

Stabilizing P(S) with C(Q) knowing that P0 isstabilized by C0 is completely equivalent to stabiliz-ing S with Q; thus there is a decoupling of thestability problem for P (S), C(Q) into two, namelyP0,C0 and S, Q. Unfortunately, there is less tidydecoupling as soon as one introduces other goals ofcontrol apart from stability, and there may still bemuch scope for improving the decoupling. Twoideas which should be explored are the use of nor-malized coprime realizations, and consideration ofS which have significant values of ES ( ju)E onlywhere P0( ju) is approximately constant or evenvery small, thereby offering scope for a decouplingin the frequency domain that could feed throughinto the performance objectives.

8.6. Broader issues againThere are some linear system problems that

Youla—Kucera parametrizations cannot address.We give two examples. First, suppose the controlleris structured, e.g. decentralized, or hierarchical insome specific way; there is not a simple parametri-zation of all such stabilizing controllers. Second,suppose that a continuous-time plant is controlledby a sampled-data controller, and the plant is to beidentified in closed loop. Once again, the theory isin trouble.

These deficiences are the more serious in thatthey point to an apparent non-universality of theYoula—Kucera parametrization idea, and its non-applicability to classes of control systems of in-creasing importance (multi-agent/hybrid systems).The situation is however partly rescued by the factthat the idea does carry over to nonlinear systems,as we now explore.

9. NONLINEAR SYSTEMS

Many of the ideas applicable to linearsystems can be carried over to nonlinear systems,but a number cannot. Research aimed at esta-blishing just what is possible is in fact veryactive.

Before illustrating the role of a Youla—Kuceraparameter, it is however necessary to understandhow the concepts of right and left coprime realiza-tion arise.

We begin with right coprime factorizations; seeHammer (1985, 1987) and Verma (1988), andwe shall drive that idea as far as possible be-fore introducing the appropriate nonlinear general-ization of a left coprime realization. We shallsee that Youla— Kucera parametrization resultsusing right coprime factorization of nonlinearsystems clearly fall short of those for the linearcase.

9.1. Right coprime factorizationsWe begin with a system + , possessing a set of

input signals U (a subset of the time functions from[0,R) to Rk, k being the system input dimension)and a set of output signals y (a subset of the timefunctions from [0,R) to Rl, l being the outputdimension). We shall identify a subset of U, call itU4, with the set of stable input signals; often U4 is¸k2[0,R). Y4 is similarly defined. Stable systems

are those which map U4 to Y4 (causally) for anyinternal condition.

We say that + has a (stable) right factorization ifthere exists a set of time functions W with stablesubset W4 and stable operators D :WPU andN :WPY with D invertible such that +"ND~1.If the initial state is not zero, + should be paramet-rized by x (0)"x

0.

The factorization is termed right coprime if forall unbounded w3W, either Nw or Dw is alsounbounded, or equivalently if Nw3Y4 and Dw3U4

imply w3W4.A sufficient condition for this property is that

there exists a bounded operator ¸ :U]YPWsuch that

¸ CN

DD"I. (9.1)

Survey Paper 1499

Fig. 14. Representation of nonlinear system to allow generation of right factorization.

Note there is no requirement for ¸ to be separable,in the sense that

¸ (u, y)"¸1(u)#¸

2(y).

Such separability would imply

¸1N#¸

2D"I,

which is reminiscent of a necessary and sufficientcondition for coprimeness in the linear case. Ofcourse, if ¸ is known to be a linear operator, itautomatically has the separability property.

Example (Based on »an der Schaft, 1996). Considerthe system

xR "f (x)#g(x) u,(9.2)

y"h(x),

with f (0)"0, h (0)"0, and suppose that the con-trol law u"k (x)#w yields a stable closed-loopsystem from w to y. Figure 14 is a redrawing of thesystem, adding and subtracting a feedback k (x).The operators N and D are defined by

D~1 : uPw.xR "f (x)#g(x)u,(9.3)

w"u!k (x)or

D : wPu.xR "[ f (x)#g(x)k (x)]#g (x)w,(9.4)

u"k (x)#wand

N :wPy. xR "[ f (x)#g (x)k (x)]#g (x)w,(9.5)

y"h (x).

Is the factorization coprime? Under an observabi-lity condition, it is. Suppose w is unknown, but theassociated u( ) )"Dw( ) ) and y( ) )"Nw( ) ) areknown. Observability allows the recovery of x ( ) )via a stable operator from u ( ) ) and y ( ) ); then w( ) )can be recovered — see the second equation ofequation (9.3). Conversely, if a stable ¸ exists, w canbe recovered in a stable manner from u and y. Thenthe first equation of (9.4) allows x to be recovered ina stable fashion. Thus, observability is equivalent tothe existence of the stable operator ¸ in equation(9.1).

Example (Scherpen and »an der Schaft, 1994; »ander Schaft et al., 1995). Consider the same system

as that of the previous Example, and suppose thatthere is a state feedback law which minimizes theindex

P=

0

1

2[uTu#hT(x)h (x)] dt

for the system. Suppose further that with initialcondition x

0, the minimized value of the index is

»(x0)50, with » (0)"0 and

L»Lx

f!1

2

L»Lx

ggTLT»Lx

#

1

2hhT"0. (9.6)

(This is the standard Hamilton—Jacobi equationassociated with the problem).

Then we define [ND] by

xR "f (x)!g(x)gT(x)LT»Lx

#g (x)w,

y"h (x), (9.7)

u"!gT(x)LT»Lx

#w.

Notice that this is a special case of the previousExample; with the observability assumptionthat xR "f (x) and h(x),0 implies x,0,the stabilizing property of the feedback law canbe proved using the Hamilton—Jacobi equation.[The Lyapunov function for the unforced systemis »(x).] The Hamilton—Jacobi equation also isthe basis for establishing that

P=

0

(uTu#yTy) dt"P=

0

wTwdt.

This establishes coprimeness of the factorization,and in fact establishes that the factorization isa nonlinear equivalent of normalized realization,with unity gain from w to [uT yT]T.

Defining an explicit operator ¸ satisfiying equa-tion (9.1) requires a further assumption, regardingthe dual Hamilton—Jacobi equation (which is asso-ciated with the problem of transferring fromx(!R)"0 to a prescribed x

0, and minimizing

:0~=

12

[uTu#hT(x)h (x)] dt). Let ¼ (x)50, ¼ (0)"0 satisfy

L¼Lx

f#1

2

L¼Lx

ggTL¼T

Lx!

1

2hTh"0. (9.8)

1500 Survey Paper

Fig. 15. Nonlinear plant-controller interconnection.Fig. 16. With stable P, all stabilizing controllers are given by

stable Q.

Assume additionally that there exists a k%( ) ) such

thatL¼Lx

k%(x)"hT (x). (9.9)

We use it to define an operator ¸ by

pR "[ f (p)!k%(p)h(p)]#g (p)u#k

%(p)y,

f"gT(p)L»T

Lp(p)#u, (9.10)

p(0)"x (0).

The stability of ¸ is established using the Hamil-ton—Jacobi equation for ¼, with ¼ (p) a Lyapunovfunction for the unforced equation. The cascade ofequations (9.7) and (9.10) results in p(t)"x (t) for all¹ and then f(t)"w (t) for all t.

Note that while the signals u ( ) ) and y( ) ) enter¸ linearly, the nonlinearity of ¸ ensures that wecannot decompose the output ¸ (u, y) as ¸

1(u)#

¸2(y).The examples reveal that a wide class of finite-

dimensional nonlinear systems have a right cop-rime factorization. This motivates us to considerinterconnections of plants and controllers, bothwith right coprime factorization.

9.2. Plant-controller interconnectionsFigure 15 illustrates a plant-controller intercon-

nection, which we assume to be well-posed, i.e.w1, w

2exist and depend causally on r

1, r

2when the

latter are bounded time functions. Evidently,

Ar1

r2B"A

I !C

P I B Aw1

w2B

and the closed-loop is stable (in the obvious sense)just when

AI !C

P I B~1

exists and is bounded. (Existence of the inverse isa requirement for w

1, w

2to exist; obviously, we

could have chosen as outputs the outputs of C andP instead of w

1, w

2without significantly changing

the conclusion.)The following result connects the stability pro-

perty to right coprime factorization.

¹heorem 4 (»erma, 1988; Paice et al., 1992). Sup-pose that P"ND~1 and C"X½~1 are rightcoprime factorizations. The the closed-loop is wellposed if

CD !X

N ½ D~1

exists and the closed loop is stable if and only if thisinverse is a bounded operator. (If the inverse isknown to be stable for any plant and controllerright factorizations, which are not necessarilycoprime, coprimeness of the factorizations is auto-matically implied.)

The proof is very simple and most can be in-cluded. Observe that

AI !C

P I B~1

"AD 0

0 ½B AD !X

N ½ B~1

.

If the second operator on the right is bounded, thaton the left must be bounded. For the converse,suppose that

Az1

z2B"A

D !X

N ½ B~1

Ar1

r2B

with ribounded, one of z

iat least unbounded (say

z1) and w

1"Dz

1and w

2"½z

2are bounded. Since

Nz1#½z

2"r

2, Nz

1is bounded; with Nz

1and

Dz1, bounded, comprimeness requires z

1to be

bounded, a contradiction.

9.3. Characterizing the stabilizing controllers for anopen-loop-stable plant

We can now exhibit a Youla parameter for anopen-loop-stable plant, using ideas of Paice andVan der Schaft (1995a). The result is just like thatfor the linear case, given in Section 2.

Consider the scheme of Fig. 16; the controller iscomposed of an interconnection of an arbitrarystable operator Q and a model of the plant P. It isassumed that the initial state of the model and ofthe plant itself are the same. It is then evident thaty"PQr, so that C, given by

C"Q (I!PQ)~1 (9.11)

Survey Paper 1501

Fig. 17. Noisy identification problem with nominal nonlinear plant and linear stabilizing controller X½~1"½M ~1XM .

is stabilizing. (Note: the derivation of this formula issomewhat nontrivial in the nonlinear case.) Toexpress Q in terms of C and P, we have

Q"C(I#PC)~1. (9.12)

[This formula is not straightforward to derive bymanipulation from equation (9.11); it can be sub-stituted into equation (9.11) and the result verified.Alternatively, one can observe using Fig. 16 thatQ follows from C by connecting P in a negativefeedback loop, to undo the P in the positive feed-back loop generating C from Q.]

To establish that any stabilizing C necessarilycan be represented in the manner depicted, observethat if C is stabilizing, then QK "C(I#PC)~1 isstable (this is the operator from r to u). This formulacan be inverted to yield C"QK (I!PQK )~1, asrequired.

Actually, there is a substantial concealed defi-ciency in the above argument. Suppose there is anadditional external input r

1entering between C and

P. It is not possible to ensure that the state ofthe actual plant and that of the model within thecontroller are the same. Accordingly, the earlierargument cannot be carried through.

All is not lost however. If P has a Lipschitzcontinuity property, one can write (with u the con-troller output)

P (u#r1)"Pu#y

1, (9.13)

where y1obeys Ey

1E4KEr

1E for some K. Then the

scheme of Fig. 16 applies again, with r replaced byr!y

1.

9.4. Characterizing the set of stabilizing non-linearcontrollers for linear plants

When the plant is linear, and we have onestabilizing (in general nonlinear) controller, it isagain easy to parametrize all stabilizing controllers.

¹heorem 5 (»erma, 1988). Let P"ND~1 define alinear right coprime factorization of a linear plantand let C"X½~1 define a right coprime factoriza-tion of a possibly nonlinear stabilising controllerused in a negative feedback loop of the form ofFig. 15. Then the set of all stabilizing controllers is

given by C"(X!DQ)(½#NQ)~1 where Q isa stable operator, such that (½#NQ)~1 is welldefined.

9.5. Identification using right coprime realizationsIn this subsection, we will consider a nonlinear

version of the earlier result on identification.The key is to base the identification on a dual of

the idea of Section 9.4. We suppose that a linearcontroller C"X½~1"½M ~1XM stabilizes a nomi-nal nonlinear plant P0"ND~1; with noisy mea-surements on the true plant P connected in anegative feedback loop with C, we seek to findstable S such that the plant is (N!½S) (D#

XS)~1.Figure 17 illustrates the arrangement. As in the

linear case, we aim to produce an open-loop identi-fication problem, based on x, b and S.

We would like to be able to compute b and xfrom external measurements (of r

1, r

2, u and y);

using b and x, we would hope to infer S fromb"Sx. In the presence of noise, it turns out, aswe now establish, that x and b cannot be exactlycomputed from measurements, but only noisyapproximations of them.

To assist the calculation one shall introducea preliminary simplification. Because of the closed-loop stability of the nominal system, it follows that

R"½M D#XM N (9.14)

is stably invertible. Let us replace the fractionaldescription ND~1 of the nominal plant by NR(DR)~1.Then we obtain

I"½M D#XM N (9.15)

Computation of a noisy approximation of x. Theplant structure implies

Dx!Xb"u(9.16)

Nx#½b"y!v.

Multiply the first equation by ½M , the second byXM and add. The linearity of ½M and XM must be used,and there results [using equation (9.15)].

x"(½M D#XM N)x"½M u#XM y!XM v. (9.17)

1502 Survey Paper

Fig. 18. (a) Stable kernel representation, with u and y determin-ing z via a stable mapping (b) Plant represented by a stablekernel representation, so that u and z determine y; z"0 gives

normal plant description.

An alternative expression is

x"½M r2#XM r

1!XM v. (9.18)

Computation of a noisy approximation of b. BecauseX,½ are coprime and linear, there are linear stableoperators K and ¸ such that !KX#¸½"I.Now equations (9.16) and (9.18) yield

!Xb"u!D(½M r2#XM r

1!XM v),

½b"y!v!N (½M r2#XM r

1!XM v)

and so

b"K[u!D(½M r2#XM r

1!XM v)],

#¸[y!N(½M r2#XM r

1!XM v] (9.19)

!¸v

Obtaining S. A ‘‘normal’’ open-loop identificationproblem to find an operator S linking x and b viab"Sx would involve noiseless measurement ofx and noisy measurement of b (typically with thenoise additive). Our knowledge of r

1, r

2, u and

y shows that we have a noisy measurement of x,viz., ½M r

2#XM r

1as shown by equation (9.18), and

a noisy measurement of b, viz., K[u!D(½M r2#

XM r1)]#¸ (½!N(½M r

2#XM r

1)] as shown by equa-

tion (9.19). The same noise v is perturbing input andoutput measurements x and b, and it perturbs theoutput measurements in a nonlinear way.

In a high signal to noise situation, it is possible toobtain a more conventional identification problem.Let S

L, D

L, and N

Ldenote the linearizations of S, D

and N about the operating trajectory induced bythe input XM r

1#½M r

2. Then b"Sx and equation

(9.18) implies

b"S (XM r1#½M r

2)!S

LXM v,

while also equation (9.19) implies

b"MKu#¸y!KD(½M r2#XM r

1)!¸N(½M r

2#XM r

1)N

#(KDL#¸N

L)XM v!¸v

"known signal#(KDL#¸N

L) XM v!¸v.

The ‘‘known signal’’ is the contents of M2N, and isknown because r

1, r

2, u and y are presumed

measurable, and the operators K, ¸, D, etc., gene-rating the signal are known. Combining these twoequations yields

known signal"S(XM r1#½M r

2)

!(KDL#¸N

L#S

L) XM v#¸v

with a v independent of r1

and r2. Determination of

S is now conventional open-loop (albeit nonlinear)identification problem.

The above ideas are simple when P0 itself islinear, and are explained in Dasgupta and Ander-son (1996). Extensions can be found in Linard andAnderson (1996, 1997).

9.6. Kernel representationsIn general, left fractional representations do not

exist. As explained earlier, the fundamental reasonis that one cannot write ¸ (u, y)"¸

1(u)#¸

2(y) in

general, when ¸ is a nonlinear operator. There ishowever a form of representation that carries manysuch properties; this has been demonstrated ina series of papers; see (Paice and Van der Schaft,1994a, b; 1995b), as well as in a recent book by Vander Schaft (1996).

We call an operator R :u]yPz a stable kernelrepresentation of a system when R is a stable map-ping, and the equation R (u, y)"z for a prescribedx0, u ( ) ) and z( ) ) is solvable for y; further when

z"0, the system I/O mapping y( ) )"+x0

u( ) ) is tobe recovered. It is logical to use two different dia-grammatic representations for R(u, y)"z; seeFig. 18a and b, depending on whether u and y aredetermining z, or u and z are determining y.

Example. Let xR "f (x)#g (x)u y"h (x) be astable plant. Then

R : xR "f (x)#g (x)u,

z"y!h (x), (9.20)

is a stable kernel representation.Put another way, if a plant P is stable, and thus

has a right coprime factorization N (I)~1, it hasa kernel representation.

R(u, y)"y!Pu"z. (9.21)

Example. Suppose that k (x) has the property that

¸ : xR "f (x)!k(x)h (x)#g (x)u#k (x)y,

z"y!h (x), (9.22)

is a stable system, with u ( ) ), y ( ) ) arbitrary inputs.Now set z"0. Then this forces

xR "f (x)#g(x)u,

y"h (x).

Thus ¸ is a stable kernel representation.

Survey Paper 1503

Fig. 19. Construction of kernel representation of PSfrom kernel

representations of P and C (differently viewed) and from theYoula—Kucera parameter S.

[In the second example of Section 9.1, wegave a procedure which will generally, but notinfallibly, yield such a k (x), given an observabilityproperty. When k (x) is chosen according to thatprocedure, ¸ has unity ¸

2gain from [yT uT]T

to [zT zN T]T, where zN"u!gT (Lw/Lx).

Example. Let P be a linear plant, with left coprimefractional representation DM ~1NM . Then DM y!NM u"0 and the kernel representation is

[DM !NM ] Cy

uD"z.

A kernel representation is called coprime if it isstable and has a stable right inverse. A right inversefor ¸ in equation (9.10) is provided by

¸~1 : xR "f (x)!g(x)gT(x)+»T(x)#k(x)1

x(0)"p (0),

u"!gT(x)+»T(x),

y"h (x)#1. (9.23)

Notice that ¸~1 can be expressed as [XY], even

though ¸ cannot be expressed as [¸1

¸2]. Also, if

p(0)Ox (0), but ¸ forgets the initial state as timeevolves, then as tPR, ¸

p(0)¸~1

x(0)behaves more

and more like the identity operator.

Feedback is described in the following way. Con-sider a plant P with stable kernel representationR

P:U]YPZ

pand a controller C with stable

kernel representation RC:Y]UPZ

cwhich are

interconnected, for the moment with no externalinputs. (For convenience, sign inversions are as-sumed to be incorporated in the plant or controller,so the interconnection is a positive feedback inter-connection.) Then the loop is said to be null-well-posed if the equations

RP(u, y)"z

p,

RC(y, u)"z

c, (9.24)

have unique solutions for all zp, z

c; if the solutions

are stable for stable zp, z

cthe system is said to be

null-stable. Note that the operators RP, R

Ccan be

initial condition dependent.It is possible to show

f if a closed-loop system using stable kernel repre-sentation is null-stable, the kernel representationsare coprime and there exist right coprime factori-zations of plant and controller, and

f if a plant and controller have right coprime fac-torization and form a stable feedback system,coprime stable kernel representation can befound which together are null-stable. (They arenot unique.)

Although the definition of null-stability intro-duces certain extra signals, viz., z

pand z

c, these are

not the same as the external signals r1

and r2

of sayFig. 15, and the concept of null-stability is formallya different concept than stability.

½oula—Kucera parametrizations are introducedas follows:

f Assume P"ND~1 and C"X½~1 are describedby right coprime factorizations, and togetherform a stable closed loop. (For convenience,assume the connection is a positive feedbackcontroller)

f Form stable kernel representations of P and C,viz., R

P(u, y)"z

p, R

C(y, u)"z

c.

f Let S :ZcPZ

pand suppose S has a kernel

representation RS(z

c, z

p)"z

s.

f The modified plant PS

is defined by the kernelrepresentation

RPS

(u, y)"zs"R

S(R

C(y, u),R

P(u, y)).

Figure 19 shows the arrangement, probably muchmore clearly. Notice that the subblocks R

Pand

RC

require the interpretations of both Fig. 18b anda, respectively. The parallel with Fig. 13 (with noisesignal absent) should also be noted.

The main conclusions are

f The set of all plants stabilized by C is obtained byletting S range over all stable operators. [In thiscase, one may take R

S(z

c, z

p) as !Sz

c#z

pand

RPS

(u,y)"!SRC(y, u)#R

P(u, y).]

f The set of all controllers stabilizing P is obtainedby an obvious dual characterisation involvinga Q : z

pPz

c.

f PSis stabilized by C

Qif and only if S is stabilized

by Q.

Example. Suppose P and C are linear, with leftcoprime factorizations DM ~1NM and ½M ~1XM . ThenR

P(u,y)"DM y!NM u"z

pand R

C(y,u)"½M u!XM y"z

C.

Now RS(z

c, z

p) is !Sz

c#z

p"z

s, which implies

(DM !SXM )y!(NM #S½M )u"zsfor R

PS(u, y), and the

left factorization is (DM !SXM )~1(NM #S½M ) .

1504 Survey Paper

These results can be pushed to handle plantswith external inputs. By way of general comment, itturns out that

f Closed-loop stability is most easily handled withright coprime fractional representations.

f Youla—Kucera parametrizations are most easilyhandled with stable kernel representations.

f One requires results on the ability to switch bet-ween the two types of representation.

10. CRITIQUE

In this brief concluding section, we aim to recordsome of the issues that are not well addressed by theYoula—Kucera parameter idea and its variants:

f Decentralized controller design. There is no expli-cit parametrization of all stabilizing decentralizedcontrollers for a prescribed plant.

f ¸inear time-invariant plant identification in a sam-pled-data loop. It is clear that substantial modifi-cation of the earlier theory would be required tohandle identification of a continuous-time plantwhen a nominal model is known and a sampled-data controller is connected.

f Degree explosion: P (s) can have degree equal todegree P0#degree C0# degree S.

f Q design given a plant S, and nominal P0 and C0.This point has been well described.

f Keeping S simple when unknownness is simple.Suppose that a plant is completely known exceptfor the value assumed by two physical parametersappearing in the equation of the plant. It is gene-rally not straightforward to have a very simpleS capturing this unknownness. Again, supposeP comprises a memoryless saturating nonlin-earity (unknown) followed by a known lineardynamic part. It is not straightforward to haveS memoryless.

f More generally, utility of the nonlinear resultshas yet really to be established.

Acknowledgements—The author wishes to acknowledge thefunding of the activities of the Cooperative Research Centre forRobust and Adaptive Systems by the Australian Common-wealth Government under the Cooperative Research CentresProgram.

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