IEEE iRANSACTIONS ON SYSTEiMS, MAN, AND CYBERNETICS, VoL.. SMC-4, NO. 6, NOVLMBER 1974

Stable Adaptive Schemes for System Identification

and Control Part II

KUMPATI S. NARENDRA AND PRABHAKAR KUDVA

Abstract-The extension of some of the results contained in Part I [1]to the case when only the plant outputs rather than all its state variablesare accessible for measurement is discussed. A unified approach to thesynthesis of an adaptive observer is presented whereby the plant state andparanmeters are simultaneously estimated. Uniform asymptotic stabilityof the scheme is proved using Lyapunov's direct method. The informationprovided by the adaptive observer is used to synthesize an adaptivecontroller for the plant. While all the principal results obtained here arefor the case of a single-input single-output plant, exteinsions to specialclasses of multivariable systems is indicated.

I. INTRODUCTIONJ 1¶E: ADL)APTIVE conitrol and identification of modelreference systems using Lyapunov's direct method is

discLussed in Part I [1] of this paper and deals exclusivelywith systerms whose entire state vectors are accessible formteasurement. A salient feature of this approach is thatstable adaptive laws are obtained in a simple fashiondirectly fronm the available signals, that is, the input, and theplant and model states, without the need to generateadditional auxiliary signals. While such procedures haveproved very effective, it is found that they are Inot directlyapplicable to inore realistic cases in which only the outputsof the plant are accessible. In such cases, it is necessary toestimiate the states of the plant prior to deriving the controlla-ws. However, since the parameters of the plant are un-known, thie state of the planit cantnot be estimated directlyusing a Lt enberger observer. This, in turn, leads to theconcept of an adaptive observer in which both the parametersand the state variables of the plant are estimated sin-iul-taneously.

Durinig the past year, several seemingly different versioinsof adaptive observers have appeared in the control litera-ture [2]- [5]. Carroll arnd lindorfl' [2] were the first toproxide suich an observer. Luders and Narendra suggestedani alternati\,e observer [4], and later modified their modelto have a siniple structure [5]. The paper by Kudva andNarendra 3] proposed yet another observer and appliedthe Kalman Yakubovich (K--Y) lemmia to prove its stability.

Manuscript received June 10, 1973; revised April 25, 1974, and June26, 1974. This work was supported by the Office of Naval Researchutnder Contract N0014-67-0097-0020 NR375-131.The authors are with the Becton Center, Yale University, New

flaven, Cor-n. 06520.

In this paper, it is shown that by applying the K--Y lemma,all of the previous results can be derived in a unified manner.A generalization of the type attempted here has ad-

vantages that go far beyond the problem of immnediateconcern. Since the nature of the stabilizatioin procedure isrendered transparent, it is felt that the method can be ex-tended to more general systems. In particular, while all theprincipal results presented in this paper are developed forsingle-input single-output systemns, it is possible to extenldthem (Section V) to special cases of multivariable systems.The problem of stable adaptive control of a system using

only input-output data is also considered here in this paper.For a single-input single-output system, it is shown that thesynthesis procedure for an adaptive observer may be directlyextended to that of an adaptive controller.

II. MATHEMATICAL PRELIMINARIES

The adaptive observer problem formulated in this paperis posed in the framework of the stability problem of aclass of time-varying systems. At the heart of the proof ofstability is the Kalman-Yakubovich lemma. We first presentspecial versions of the lemma [6].

i) Lemma I (Lefschetz): Given a stable matrix A and avector b such that the pair (A,b) is completely controllable,a real vector k, scalars / and ,i > 0, anid a positive definitematrix L = LT > 0, then there exist a real vectof g and apositive definite matrix P = PT > 0 satisfying

A7P + PA --gg7' -AL

Pb -- k -\/ I g

iff , is sufficiently small and the transfer function

H(s) -- 2 + k7'(jI - AY 1b

(Ia)(1 b)

(2)

is a strictlv positive real function.1ii) Lemma 2 (Mleyer): Given a stable matrix A aind a

vector b, a real vector k and a real positive scalar f, thenthere exist a real vector g and inatrices P = pT > 0,

I A rational function H(s) of the complex variable -m - tc-i iw ispositive real if i) H(a) is real and ii) Re H(J) > 0, for all ac > 0 andRe H(j) 2 0 for a = 0; the rational function H(j) is strictly positivereal if H(j - c) is positive real for some E > 0.

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NARENDRA AND KUDVA: SYSTiEM IDENTIFICATION AND CONTROL, PART II

1Al - MT,>, satisfying

ATP + PA = ggT _ M (3a)

Pb - k - \/l g (3b)

(gT,A) is completely observable (3c)iff the transfer function

H(,=)= I + kT(JI - A)lb (4)is a positive real function.

iii) Lemma 3 (Andersoni): Let H(j) be a square matrix ofrational transfer functions such that HT is finite, H(J) haspoles which lie in Re j < 0, or are simple for Re j = 0.If {C,A,B,H. } is a minimal realization of H, then H(G) =Hco + C(,iI - A) - 'B is a positive real matrix if a positivedefinite matrix P and matrices G and S exist such that

A7P + PA = -GGT (5a)

PB=CT-GS (5b)

STS = H + HOJ. (5c)

If HO = 0, (5b) reduces to

PB = CT. (6)

The three lemmas establish the positive realness of ascalar (or a matrix) function as the necessary and sufficientcondition for the existence of a positive definite matrix Pand a vector g (or a matrix G), which simultaneously satisfya matrix and a vector equation (or two matrix equations).The existence of P and g (or G), in turn, is used to prove thestability of a class of time-varying differential equations.The following theorem forms the core for the synthesis ofall the stable adaptive control and identification schemesdeveloped in this paper.

Theorem 1

Given a dynamical system represented by the completelycontrollable and completely observable triple {hT,K,d},where K is a stable rnatrix, a positive definite matrixF = FT > 0o, and a vector v(t) of arbitrary boundedfunctions of time. Then the system of differential equations

£ = Ke + doTv, e= hTe

S = -Fe,v (7b)is stable provided that H(J) A hT(d - K)-'d is strictlypositive real. If, furthermore, the components of the vectorv(t) are sinusoidal signals with distinct frequencies, thesystem (7) is uniformly asymptotically stable.

Proof: Consider the following quadratic function as acandidate for the Lyapunov function for the system (7):

V = 2 TPe + 4TFb]. (8)Differentiating (8) with respect to time, and substituting (7)into the result yields

V = ET(K P + PK)e -J Tp dbTV + TF-1%

If H(s) is strictly positive real, then Lemma 1 guaranteesthat there exist a vector g and matrices p = pT > 0,L = LT > 0 satisfying

KTP + PK = _ggT - pL

Pd = h.

(lOa)

(lOb)Hence V is negative semidefinite and the system (7) is stable.The proof of uniform asymptotic stability parallels that ofTheorem 2 in [1].

If, however, the pair (K,d) is not completely controllable,Theorem 1 can still be proved by using Meyer's lemma [7].

Corollary 1-1: Theorem 1 also holds when v = S(p),where x is an arbitrary bounded function of time, p is thedifferential operator d/dt, and S(j) is a vector of stabletransfer functions.

Corollary 1-2: Consider the completely controllable andcompletely observable triple {C,K,D}, where K is a stable(n x n) matrix, and C and D are (m x n) and (n x m)matrices, respectively. The system of n(m + 1)-differentialequations

£ = Ke + DDv(t), c0 = Cc

@ = - FeOvT(t)(Ila)

( llb)

is stable if F = FT > o, H(i) A C(jI - K)'1D is apositive real matrix, and H:, = 0.

If, furthermore, D is offull rank and the components ofthe vector v(t) are sinusoidal signals with distinct frequen-cies, the system (11) is uniformly asymptotically stable.The proof of this corollary follows along lines identical

to that of Theorem 1 using Anderson's lemma.In the following sections, e(t) represents the error between

the state vectors of the observer and the plant, and +(t) avector of parameter errors. While both e(t) and 0(t) areunknown and cannot be measured, the designer does haveaccess to ¢(t), which is determined solely by the adaptivelaws. The aim of the design, then, is to implement /(t) usingonly available signals so that limtOO e(t) = 0. In par-ticular, if the error equation were indeed (7a), with Ereplaced by e, the adaptive laws for 4 could be chosen tobe (7b) so as to achieve the desired convergence.

However, as is seen from the following section, if aminimal realization of the plant is considered, the errorequation invariably assumes the form

e = Ke + ok%, e, = hTe (12)

in which el(t), the output error, and 4(t) are the only acces-sible signals. In this case, the choice of an adaptive law forq with assured stability becomes a formidable problem.Fortunately, this difficulty can be circumvented with theaddition of a suitable vector signal w (which can be gen-erated using available signals) to (12). With this additionof w(t), (12) becomes

e = Ke + Ox + w, e, = hTe. (13)

= IET(KTP + PK)e + gT(Pd h)OTV. (9)The following proposition establishes the existence of suit-able signals w and v, which result in the same output e,(t)

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IEEE TRANSACT'IONS ON SYSTEMS, MAN, AND CYBERNETICS, NOVEMBER 1974

for both systems (7a) and (13). This implies, therefore, thatsince the asymptotic stability of (7a) and (7b) is guaranteed,so is the stability of (13) with the same adaptive law for 4.

Proposition 1

Given an arbitrary bounded function of time x(t). Thenthere exist vector signals v and w, with v = !N(p)< andw = w(Q,v), such that the following systems:

e = Ke + Ox + w,

== K£ + d4Tv,

e, = hTe

el = hTe

(14)

(15)

are equivalent in terms of the input (X)-output (e1) repre-sentation provided that the pair (hT,K) is completelyobservable.

Proof: The proof of this proposition consists of de-termining i) a suitable transfer vector C(S) that relates x tothe vector v, and ii) a vector w, which is a function only ofv and 4 (as mentioned earlier, Q is an available signal).Without loss of generality, the proof can be established

for the following observable canonical form for the pair(hT,K):

K= [-k ] and hT = (1 0 0 * 0). (16)

Hence with this choice of v and w, (14) and (15) have thesame input-output relation which completes the proof ofProposition 1.

Special case: A particular case of interest arises when thetriple {hT'K,d} is specified by

d T= h = (1 0 0 0)

(23)K = -AlK X A1

where the [(n - 1) x 1] vector

=(1 1 ... I)and

A = diag (-)2,-3 'n)withAj > 0 andRj 7& Aj, i =A j, for allQ e {112, -,n}.The corresponding vectors v and w are [4]

vi = [P1Jx. i=2,3,. nnl

Vl =

andw = [0,42V2,V)3v3, ...nvn]

(24)

(25)

(26)

(27)

Equations (14) and (15) imply that the following conditionshould be satisfied:

hT(pI- K)-'[Ox + W- dOTV] = 0 (17)

where {qil are the elements of the vector 4.

III. THE ADAPTIVE OBSERVER (SINGLE-INPUT SINGLE-OUTPUT PLANT)

Consider the linear time-invariant system described by

x = Ax + bu Y = hTX.n

pnE [4yix + wi - di OTv] = 0 (18)it=

where the subscript i denotes the ith element of the corre-

sponding vector.By associating {wi} with terms containing only v and the

first derivative of 4, and equating the coefficients of 4 tozero [3], the following choice of d, v, and w satisfies (18):

d T= ( d2 d3 ... dn) (19)

viLn-l+ d7pn-2 + +dlp

w = [0,qTA2v,TA3v, * *.,sT AnV]n-m + 1

i= 1,2, -n

(21)

In (28), x is an nth-order state vector, u is a scalar input, ya scalar output, and the triple {hT,A,b} is completelycontrollable and observable.

It is desired to build a suitable adaptive observer to

estimate the state vector and identify the triple {hT,A,b}, or

their equivalents, when only the input and output signals,u and y, are accessible for measurement.

Three models for the adaptive observer are presented in

the following sections. In each case, the important featuresare: i) choice of a suitable internal state representation forthe plant and the observer; ii) derivation of the error equa-tion, which has the form of either (12) or (15), and iii) choiceof the adaptive laws and the feedback signal v to guaranteethat the observer is stable.

m-2

-dm -dm+i -d 0 00 -dm . -d_ -dn O° 0* * s * * ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

0 0 -dm

d2 dm I

1 d2 * * dm_ 2 dm- 1

1 d2 d3

0

0

-dn00

}m-1

} n-m + 1

00 0

m = 23, ,n. (22)

or (28)

00

000

Am =010

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NARENDRA AND KUDVA: SYSTEM IDENTIFICATION AND CONTROL, PART II

TABLE I

Canonical Auxiliary signals Stabilizing feedbackCase 1 2 1 2 Positive real condition

Form v and v signals w and w

V nil TT -Inl+nn1Adn-2l+.+d l 1 w =[0,0'A2v . Anv1T hT (I-K)d-

j~~~~~~~2 njn-i n-2n-i ~~~~~~~~+d1 + ... +dn

2 p 2 ~~T T2nv-_n- u w [_ 2

A v*2 2T |n + kn-I + + k

+d2p + ..... ....... b+dn ..strictly positive reali e {1,2,3. n} with A as defined in eqn. (22)m

-T-A t- [1 1l 1 1 T T -l

2 = L~~~~~vJ + x1 w V2. v) h G,Il-K)- d2 A ~ T _ _ 2 2.. 2. 2T 1

|k (-i '°'''"°)T v2 = Pu w2 '[°;2 v2 ,, 2 v2 T ...0) [O 2 V2.-"n nj

i {(2,3,...,n} strictly positive real1 2

[ IVlV1 ; V1=UIX

In all the three cases, the development of the observerequations is concise and emphasis is placed mainly on thesethree features.

Structure of the Adaptive Observer

Cases I and 2 [3], [4]For cases 1 and 2, the following minimal realization for

the plant is adopted:

x = [-a A]x + bu y = h x =xl (29)

where: a,b are the unknown parameter vectors to beidentified, hT = (1 0 0 ... 0), and A is an arbitrary known[n x (n - 1)J matrix.Equation (29) can be rewritten as

x = Kx + (k - a)xl + bu

whereK= [-kIA]

is a stable matrix.The adaptive observer is designed to

structure:

k = K. + (k -a)xl + bu + wl + w2

The error e A [x - x] satisfies the d

e= Ke + 41x1 + 4)2u + wI + w2,

where

I) _ [a - a] and 4)2 _ [bUsing Proposition 1 and Theorem 1,

can be selected so as to obtain the followlaws:

a = -k0 = Flelvl, F-=

b =_2=-r2elv2, F2 =

Fig. 1. Observer scheme for Cases 1 and 2 for single-input single-output plant.

The particular canonical forms used in the two cases, thecorresponding auxiliary signals v',v2, and the feedbackstabilizing signals w',w2 are displayed in Table I. In both

= XI (30) cases, a sufficient condition for uniform asymptotic stabilityis that the input u have n distinct frequencies [2]-[4].The schematic diagram for the adaptive observer schemes

for these two cases is provided in Fig. 1. In Case 1, theefollowing plant is represented in the standard observable canonical

have the form [2]; the observer, however, is somewhat unwieldyfrom the point of view of implementation because of the

h= = *1. complexity of the feedback signals w1 and w2. In Case 2,(31) on the other hand, the internal representation of the plant

is chosen particularly to obtain the vectors w' and w2 in alifferential equation simpler form. The question naturally arises then, whether

e, = hTe (32) it is possible to eliminate completely the signals wl and w2by choosing a suitable representation for the plant triple{hT,A,b}. While the answer to this is negative for any

- b]. (33) minimal realization of the triple, one can dispense with1 2 1 2 these signals, as we shall see below, when the nth-order

v , v , w , and w plant is represented by an equivalent (2n- l)th-orderving stable adaptive system with (n - 1) unobservable states. In this case, theobserver is found to have the simplest structure.

]FT > 0 The fact that using a (2n - l)th-order system with (n- 1)unobservable states, we obtain the simplest observer struc-

F2T > 0. (34) ture may at first sight appear contradictory. However, it

555

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, NOVEMBER 1974

and where the caret denotes the estimates of the correspond-ing parameters and states.The state errors are defined as

eAA [X1 - xl] e' A [Xl - Xl] e2 [ 2 (38x J.

(38)

Fig. 2. Plant representation (nonminimal realization) for Case 3.From (36) and (37) and the initial conditions

0l (3.must be noted that in this scheme the observer also playsthe role of the auxiliary signal generator that is required inCases 1 and 2. It is this fact, coupled with the eliminationof the feedback signals w1 and w2 that contributes to thesimplification of the overall structure.

Case 3 [5]

Any completely controllable and observable nth-ordersingle-input single-output system can be equivalently repre-sented by the following (2n - 1)th-order model:

-[a, -aT T bl Xw]x= A 1x + [ u x(O) =0

(35)y - X 1

where the (2n - l)th-order state vector x - (x1,xlT,x2T)T,xl being a scalar and x' and x2, each (n- 1)-dimensionalvectors; the scalars a,,bl and the (n - 1)-dimensionalvectors a and b are the unknown parameters. ( and A aredefined in (24) and (25), and cT = (1 0 0 ... 0).

In general, the pair (IT A) can be replaced by any com-pletely observable pair (r',F) [5].The block diagram representation of this system is

provided in Fig. 2. In terms of the components of x, thesystem (35) can also be written as

xl --alxl - a x + blu + bTx, xl(O) - xJO

xl = Ax1 + {X1,

X2 = AX2 + (u,

X'(0) = 0

x2(0) = xO . (36)

If, further, the parameter error vectors are defined as

l = (01, 0'T) A [a, - al, (a - a)T]

0 = (&2 'PT) A [b1 - bl, (b -)T] (40)

the error equations become

-,= -)jej + 011x1 + FTx + qi2u + 2TX2 + bT2

e2 = Ae2. (41)

Finally, upon defining the error vector

E A (ei,ee ) (42)

-= Ke + d(4'Tv' + k2TV2), e = hTe (43)

where

K=[2' ~bT]K = -l TO A-

d = h = (1 0 0 O)T

VI = (xi, (T)T

v2 = (U ,X2T)T.This equation has a form similar to that of (7a) and thetransfer function hT(iI - K)-'d = l/(- + X1) is strictlypositive real.Hence the stable identification laws can be chosen as

follows:'1 =-Flelvl,2 = -F2e1V2,

F, = F1T > 0

F = 1T > 0. (45)A schematic diagram of the observer is shown in Fig. 3.

To estimate the parameters, an observer is developedwhose structure is similar to that of the plant. The observerequations are

2 = -xalx - aTx ± b1u + b x x- X1(1 - )X1 > 0

Xl = Ax' + /x1

X2 = Ax2 + (u

9 = 91 (37)with

A

21(°)= ° (0) =x2(o) =

C(ommentsa) As mentioned in the previous section, the various

stages in the design involve the choice of i) the observerstructure, ii) the auxiliary signal generator, and iii) theadaptive gains. The particular structure chosen also uniquelydetermines the feedback signals w1 and w2 required toassure stability.

b) The designer has considerable freedom in choosingthe poles of the observer and the auxiliary signal generatorsand the adaptive gains (F, and F2). In Cases 2 and 3, partof this freedom is lost since the poles of the auxiliary signalgenerators are the same as those of the observer.

c) The choice of parameters in b) together with themagnitude and frequency content of the inputs determine

556

s y=XI

(39)

(44)

557NARENDRA ANI) KUDVA: SYSTEM IDENTIFICATION AND CONTROL, PART It

p..-

Fig. 3. Observer structure for Case 3.

the speed of convergence of the overall scheme as well asits noise rejection properties. At the present time theseparameters are chosen by trial and error and there iscurrently considerable research activity in this area.

d) For purposes of simulation, the observer time con-stants are chosen to be approximately those of the system.When the eigenvalues of the matrix K in (31) are real, inCase 1, the positive real condition requires that the poles ofthe auxiliary signal generator and those of the observeralternate on the negative real axis. Hence once the matrixK is chosen, the speed of response of the observer is con-trolled essentially by the adaptive gains, and it is theseparameters that are varied in the trial and error methodmentioned in c).

e) In schemes 1 and 2, where canonical representationsof the plant are used, its state variables can be directlyestimated. In scheme 3, however, a suitable transformationof the (2n -- 1) observer state variables is required toestimate the state of the plant.

IV. ADAPTIVE CONTROLLER (SINGLE-INPUT SINGLE-OUTPUTPLANT)

The adaptive observers described in the previous sectionestimate both the plant state and the plant parameters andcan therefore be used to synthesize a suitable controllerfor the plant. If the desired transfer function of the overallsystem is specified, the observer state is fed back to adjustthe poles of the overall system asymptotically and a pre-filter is used to locate the zeros. In particular, the prefilterzeros are the desired zeros of the overall system, while thepoles of the prefilter cancel the zeros of the plant. Theapproach is consequently feasible only for plants with zerosin the left half of the complex plane.

Since the same inputs are fed back into both plant andobserver, the error equation for the observer is unchangedand thereby the stability of the observer is assured. It isnoted that only an imiplicit model of the overall desiredsystem is employed in this case, and that the prefilter andfeedback controller parameters are adjusted directly usingcurrent estimates of the plant parameters as provided bythe observer.

> - PREFILIER _

I_

yY

*O PLANT -

FEEDBACK - iCONTROLLER

Fig. 4. Adaptive control scheine for a single-input single-output plant.

The following is intended to illustrate the entire procedure(Fig. 4):

model (implicit) -b l*p + b2*p2 + a,*p + a2

plant -bi p-4 b2_p + alp + a2

prefilter bi*p +b1p + b2

feedback controller (a - a1)p + (a2 a2)bi*p + b2*

Simulation Results

Using scheme 1 computer simulations have been carriedout to identify the parameters of and to control a second-order plant.

i) Plant:

-a , [b1]

-a2 Lb2Jy = x1

a1, a2, b1, and b2 are the unknown plant parameters. Forpurposes of simulation, the following values were set:

a, = 5, a2 = 10, b1 = 1, b2 = 2.ii) Model:

- [-at 1] Xm + [u:] u Ym = Xml

with a1* = 7, a2* 10, bl* = 1,b2* = 1.iii) Observer:

X -k2 o] + [k2 a2(t,) XI +62(t) r

- e1 [ 17FrA 1l] - Lv[V2ToAVI9- elJTe

wherek1=-6 k2=8

e, = 9- y

A2= [ -d2 [O -3]

36 0 [5.5 00 4.3] 2 [0 0.45k

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, NOVEMBER 1974

5

4

3a1t2

0 L0 4 8 12 16 20 24 28 32 36 40

TIME (sec)

(G)

2.0-

1.6 A-J12 [J (t) b10.8

0 4 8 12 16 20 24 28 32 36 40

(c)Fig. 5. Convergence of obs

iv) Auxiliary signals:

V2' = -3v2' + X1 = V1

2 = -3v 2 + r=V12=

v) Reference input: u = periodic rectangular pulses ofamplitude 5 and fundamental frequency of 1/(27c) Hz.

vi) Adaptive laws:

(ai) = Flel (vi)

(br) = -f2e, (v2)

The results of the simulation are presented in Fig. 5.

V. MULTIVARIABLE SYSTIEMS

So far, we have discussed single-input single-outputplants. The adaptive procedure can be directly extended tothe case where the plant has many inputs and a singleoutput, so that the vector b is replaced by a matrix B.The case where the plant has several outputs is consider-

ably more difficult and direct extension of the resultsobtained so far is possible only in special cases. If the sys-tem is represented by the completely controllable and com-pletely observable triple {C,A,B}, the identification processinvolves determination of the triple to within an equivalenceclass. The main difficulty here lies in the choice of a suitablecanonical form and the prior information regarding thesystem that is needed to justify the choice. In the particularblack box situation where one has access to only the rinputs and Wl outputs of the system, and where an upperbound N on its order n is also specified, the identificationprocedure may involve determination of an mNth-ordersystem of which only n states are completely controllableand completely observable. Such a model is neither interest-ing nor practically useful. The problem of parametrizationof multivariable systems is currently an active area of

12 0

116

'11 .2

1G.8

10.4

10.0

a 2 (t)a210

4 8 12 16 20 24 28 32 36 40TIME (secs)

(b)

b2 22.0

1.8 X,,16 b2 (t)

12-

1.0 I_-LW .1 L10 4 8 12 16 20 24 28 32 36 40

(d)server parameters in example.

research and only when this problem is completely resolvedcan the synthesis of an adaptive observer be attempted forthe general multivariable case. In what follows, we brieflyindicate several special cases which are of practical interest.

Case 1 (A is cyclic)

Consider a completely controllable and completely ob-servable multivariable system of order n with an (r x 1)input vector u and (m x 1) output vector y. Let us furtherassume that the system is completely observable throughthe first output y1. In such a case (which has also beenconsidered in [5]), the system can be represented by thetriple {C,A,B }, where A is in the observable canonical formanid the first row of the matrix C (1 0 0 0). UJsingthe input vector u and the first output y1, the matrices Aand B can then be identified by using the procedure de-scribed in Section III. Since the estimates of the statevariables are generated by this process, the elements of theremaining (rm - 1)-rows of C can be determined simul-taneously by correlating these estimates with the correspond-inig (m - 1)-output errors.

Obviously, the preceding procedure can be extendedtrivially to the case where the system can be observedcompletely through any one output. In a fnore realisticcase, however, where the system is completely observable,but not through any single output, almost any arbitrarylinear combination of the outputs yo = cTy, yields a newoutput yo with the preceding property. This follows fromrtthe fact that the pair (cTC,A) is completely observable foralmost any vector c, if A is cyclic [9]. In such a case, yotogether with any (m -- 1) elements of the output vector yconstitute a new output vector y which may then be usedin the identification process as described in Section III.

Case 2

Corollary 1-2 which deals with the stability of an

n(m + l)th-order differential equation forms the basis for

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NARENDRA AND KUDVA: SYSTEM IDENTIFICATION AND CONTROL, PART II

known stable transfer functions, and K is an (m x r)matrix of unknown but constant gains. To identify theparameters of the matrix K, an adaptive observer is usedthat has the same structure as that of the plant with Kreplaced by a matrix K (t) of adjustable gains, so that Kiis an estimate of K. In state vector representation

x = Ax + BKq y = Cx plant

x = AR + BKf(t)q 9 = C5 observer (47b)

where i) q = G(p)u, u being the input vector; ii) the triple{C,A,B} is such that C(I - A)-'B = W(j) is a positivereal matrix and WOO3 = 0; iii) B is of full rank and A is astable matrix; and iv) i is an estimate of the plant statevector x. The error c A x( - x satisfies the differentialequation

£ = Ac + B'Dq; go A 9 - y = CB

where ID = [K (t) - K].Using Corollary 1-2, the stable identification laws are

obtained as

K = D = -rFoq (t); F = FT > 0. (49)(c)

(d)

Fig. 6. Identification of multivariable systems (Case 2). a) Errormodel. b) Open-loop representation. c) Closed-loop representationfor single-input single-output system. d) Closed-loop representation.

a different approach to the design of observers for multi-variable systems. We present this approach, which appearsto have considerable potential, in the following steps.

1) Consider a multivariable system (Fig. 6(a)) with a

positive real transfer matrix W(j) and let the input to thesystem be (D(t)q(t), where q(t) is a vector of independenttime functions and D(t) a matrix whose elements are un-

known but whose time derivatives can be adjusted. Corol-lary 1-2 assures that the output vector co(t) of the systemand the matrix 4D(t), both tend to zero asymptotically if thefollowing adaptive law is used:

lb = -FE0(t)qT(t), F = FT > 0. (46)

2) This result can be directly used in the identification ofa linear time-invariant plant (Fig. 6(b)) having a transfermatrix W(il)KG(J), where W(S) is a known (m x m)positive real transfer matrix, G(W) is an (r x m) matrix of

It is worth noting here that the vector input q(t) is commonto both the plant and the observer.

3) Feeding back identical signals at the input of both theplant and the observer does not affect the error equation ofthe overall system. For example, the scheme 3 treated inSection III for a single-input single-output system wouldcorrespond to a particular case of the problem described inStep 2, where q is a 2n-dimensional vector and the input isa two-dimensional vector whose first element is the actualinput to the system and the second element is the systemoutput. This is shown in Fig. 6(c).

4) We can generalize the results of the previous step tothe multivariable case as follows. If a multivariable systemhas a representation2 shown in Fig. 6(d), where W(j) is a

positive real transfer matrix and G(J) is a matrix of stabletransfer functions, the adaptive laws for adjusting theparameter matrix K of the observer are the same as thosegiven by (49).The transfer matrix of the plant given in Fig. 6(d) is

Z(j) = [I + W(j)K2G2()] 1 W(j)K1G1(j) (50)

where

G(S) = [~Y~O G2()]and K = [K1 -K2]. As mentioned in Step 3, every single-input single-output system can be represented in this formand the realization is nonminimal. It is therefore onlynatural to expect the representation to be nonminimal inthe multivariable case also.Whether or not every multivariable system has a non-

minimal representation of the form described in Step 4

2 The authors are indebted to Prof. B. D. 0. Anderson for bringingthis structure to their attention.

(a)

(b)

PLANT

(47a)

(48)

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNEriC.S, NOVEMBER 1974

remains an open problem. If the answer to this question isnegative, it appears worth investigating as to what class ofmultivariable systems would admit of such a representation.

VI. CONCLUSIONS

In Part I of this paper, schemes are developed for theidentification and control of multivariable systems in whichall the state variables are accessible. In this paper, the mainattempt has been to remove this stringent requirement. Fora single-input single-output system, a unified procedure isdeveloped for designing an adaptive observer that simul-taneously estimates both the state of the system and itsparameters. These estimates can then be used to determinesuitable feedforward and feedback controllers for thesystem.The problem of synthesizing adaptive observers for multi-

variable systems is considerably more difficult and is closelyrelated to the choice of a suitable canonical representationfor the plant. However, the methods developed for thesingle variable case can be extended to certain classes ofmultivariable systems, which are of practical interest; thegeneral problem remains unresolved.At this stage, questions relating to the choice of the

adaptive gains and the associated problem of the speed ofconvergence, the effect of observation noise and the de-termination of reasonable bounds on the parameter errors

when the plant parameters vary witlh tiiine also remainunanswered, although preliiinary efforts are already under-way. Resolution of these important questions is essentialbefore the procedure) outlined here emerge as viable tech-niques for use in practical situations. Ilowever, tlhere is littledoubt regarding the power and versatility of the entireapproach and the great potential it appears to possess forthe synthesis of stable adaptive multivariable systeniis.

REFLRENCES[1] K. S. Narendra and P. Kudva, "Stable adaptive schemes for

system identification and control- -Part I," this issue, pp. 542 -551.[2] R. L. Carroll and D. P. Lindorff, 'An adaptive observer for single-

input single-output linear systems," IEEE Trans. Automat. Contr.,vol. AC-18, pp. 428-435, Oct. 1973.

[3] P. Kudva and K. S. Nareindra, '*Synthesis of an adaptive obsei verusing Lyapunov's direct method," Irti. J. Contr., vol. 18, pp. 1201-1210, Dec. 1973.

[41 G. Ltiders and K. S. Narendra, "An adaptive observer arid ideiiri-fier for a linear system," IEEE Irans. Automat. Contr., vol. A- 18,pp. 496-499, Oct. 1973.

[5] , "A new canonical form for an adaptive observer," IEEETrans. Automat. Contr., vol. AC-19, pp. 117--119, Apr. 1974.

[6] K. S. Narendra and J. H. Taylor, Frequienicy Domain Criteria orAbsolute Stability. New York: Academic, 1973.

[7] K. R. Meyer, "On the existence of Lyapunov functions for theproblem of Lur'e," SIALMJ. Conrr., vol. 3, no. 3, pp. 373-383, 1965.

[8] R. V. Monopoli, "The Kalman--Yacubovich lemma in adaptivecontrol system design," IEEE Tranis. Automiat. Contr. (Corresp.),vol. AC-18, pp. 527-529, Oct. 1973.

[9] P. E. Caines, "The parameter estimation of state variable modelsof multivariable linear systems," The University of ManchesterInst. of Science and Technology, Control Systems Centre Rep. 146,Apr. 1971.

Correspondence

A Study on Aircraft Map Display Location and Orientation

D. L. IBATY, T. E. WEMPE, AND E. M. HUFF

Abstract-Six airline pilots participated in a fixed-base simulator

study to determine the effects of two horizontal situation display (HSD/map) panel locations relative to the vertical situation display (VSD), and

of three map orientations on manual piloting performance. Pilot com-

ments and opinions were formally obtained. Significant performancedifferences were found between wind conditions and among pilots, but

not between map locations and orientations. The results also illustrate the

potential tracking accuracy of such a display. Recommendations concern-

ing display location and map orientation are made.

-INTRODUCTION

Many advanced aircraft display designs include the use of

cathode ray tubes (CRT) to present attitude information on a

vertical situation display (VSD) anid navigation information on

a horizontal situation display (HSD). Due to the size of these

Manuscript received October 18, 1973; revised June 26, 1974.The authors are with the Man-Machine Integration Branch, Ames

Research Center (NASA), Moffett Field, Calit. 94035.

tubes and tnounting structures, there is often somie restrictionon their placement in the aircraft panel, which in somne cases imayrequire that they be positioned side-by-side (most pilots seem

a priori to prefer over-under placement).A simulator study was conducted to investigate the effect of

the relative position of these two displays on manual performanceand included, as the other major variable, three (3) sariations in

HSD map orientation to test for interactions. Although therehave been studies using projected map and CRT displays, e.g.,[1 ]- [3 ], no comparative studies on location or orientation havebeen done. The results of this experiment, which deal specificallywith different map display locations and orientations, should be

applicable not only to CRT map displays but also to two otihermajor types of map displays, namely, film projection and rear

projection CRT's.The simulator piloting task consisted of iriaking a series of

right and left procedure turns in level flight both in the preseilceand absence of cross winds. Pilot perforrnance >/as mieasured bycomputing both lateral and vertical rms errors. At the end of the

experiment each pilot completed a detailed questionniaire aboutthe experiment. The results of both the pilot performance data

and the questionnaire data are presented and discussed.

560