Probing signals for model reference identification - IEEE Xplore

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[24] A. W. Starr and Y . C. Ho, “Nonzero-sumdifferentialgames,” J. MethodsinOptimization forEconomicStabilizationPolicy. Optimiz. Theory Appl., vol. 3, no. 3, 1969. Amsterdam: North-Holland, 1975. Y.-C. Ho, “Differential games, dynamic optimization and generalized control theory,” J. Optimiz. Theory Appl., Sept. 1970. Y.-C. Ho and K.-C. Chu, “Team decision theory and information structures in optimalcontrol problems-Part I,” IFEE Trans. Automat. Contr., vol. AC-17, pp. 15-22, I-eb. 1972. R. S. Holbrook, “Apractical methodforcontrolling a large non-linear stochastic system,” unpublished paper, May 1973. Robert S. Pindyck (S’68-M71) received the B.S. D.Kendrick and J. Majors,“Stochastic control with uncertain and M.S.degrees in electricalengineering and macroeconomic parameters,” unpublished paper, 1973. the Ph.D. in economics from the Massachusetts F. Kydland,“Noncooperativeanddominant playersolutions in Institute of Technology. Cambridge, in 1966, discrete dynamic games,” Int. Econ. Rm., vol. 16, June 1975. D. A. Livesey, “Optimizing short-term economic policy,” Econ. J . , 1967, and 1971. respectively. Sept. 1971. He is currentlyAssociateProfessor of ManJ. Marschak and R. Radner, EconomicTheory of Teams. New agement at M.I.T., and has been doing research Haven, C T : Yale Univ. Press, 1972. in applications of control theory to macre and R. S. Pindyck, Optimal Planning for Economic Stabilizarion. microeconomic policy. and in the economics of Amsterdam: North-Holland, 1973. energy and natural resources. He is the author -, “An application of the linear-quadratic tracking problem to of Optimal Planning for Economic Stabilization economic stabilization policy,” IEEE Trans. Automat. Contr.. vol. (Amsterdam: North-Holland, 1973), and co-author of The Economics of AC-17. DD. 287-300. June 1972. [21] -, “Optimal poli&es foreconomic stabilization,” Economerrica,theNatural Gas Shortage: 196&1980 (Amsterdam:North-Holland), Price Controb and the h’atural Gar Shortage (American InEnterprise May 1973. [221 M. and J. B. C m . “SamPled-data Nashcontrols in non- stitute), and EcommerricModelsandEconomics Forecasts (New York: zero sum differential games,” Int. J . Contr., vol. 17. no. 6, 1973. McGraw-H1ll). He has also written a number of articles on [23] -, “On the solution of the open-loop Nash Riccati equations in to economics. and On the economics Of energy markets linear quadratic differential games,” Int. J. Conrr., vol. 18, no. 1, Of and energy policy. 1973.

-,

~~

ProbingSignalsforModelReference Identification

Abstruct-This paper examines certain properties of probing signals employed in a model reference identification procedure which guarantee identification of theparameters of a linearmultivariablesystem.The property of persistent excitation is examined with respect to a new class of (persistently spanning) signals. In particular, this class of signals includes many (periodic or almost periodic)signals which havepreviouslybeen considered as effective probing signals. The results yield concrete guidelines useful in the practical design of probing signals.

I. INTRODUCTION PPLICATION of Lyapunov’s stability criteria to the A design of adaptive systems hasbeen the subject of many investigations in recent years [ 1]-[4]. In particular, the identification problem has been studied for the linear time-invariant system i(t)=Ax(t)+Bu(t) Manuscript receivedAugust24,1976;revised March 1,1977. Paper recommended by R. Monopoli, Chairman of the Adaptive.Learning Systems, Pattern RecognitionCommittee. This workwas supported in part by the National Research Council of Canada under Grant A-7399. J.S-C. Yuan is with SPAR Aerospace Products Ltd., Toronto, Ont.. Canada. W. M. Wonham iswith theDepartment of ElectricalEngineering, University of Toronto, Toronto, Ont., Canada.

am,

where x( t ) E W, u( t ) E assuming the unknown elements in the matrices A and B are fixed. In this paper we confine our attention to an identification scheme first studied by KudvaandNarendra [5] using periodic signals u ( . ) . Here, using a more rigorous approach, we shall prove convergence for a much broader class of input signals. More recently, both Anderson [6] and Morgan [7] have independentlyobtained necessary and sufficient conditions for exponential convergence. Their results involve a “nondegeneracy” condition on the combined input/output signals [x’( u’( .)I. Whereas these results are interesting in their own right, they are difficult to check a priori, since only the input u ( .) is subject to design. Although the results we present here are mainly sufficient conditions for convergence, they may be directly utilized in the design of probing signals. The paper is organized as follows. Section I1 describes the identification scheme on which our subsequent results are based. In Section I11 we present the main results which exploit the property of “persistent excitation” in the present scheme. These results bring out some interesting properties of the probing signals which play an important a),

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YUAN AND WONHAM: MODEL REFERENCE IDENTIFICATION

role in the convergence of the identification algorithm. In since E is stable, (7) determines r uniquely. Thus, along Sections IV and V we study the application of some of the solution to (4), the time derivative of V is given by these properties in the design of probing signals. f =- - e21' Q e + t r { @ ' ( l ? e x ' + F , ) + \ k ' ( T e u ' + F 2 ) } . (8) 11. AN IDENTIFICATIONSCHEME BASEDON Let LYAPUNOV'S STABILITY CRITERIA Theidentificationschemedescribedhere is due to Narendra et al. [ 11: [2]. Consider a plant whichis described by (I), where x ( t ) E %, u ( t ) E am, and the pair ( A , B ) has constant parameters. A model is constructed of the form ~ ( t ) = ~ y ( t ) + ( A ( t ) - ~ ) x ( t ) + B ( r ) u ( (2) t)

A(

where E is an arbitrary stable matrix. and E( contain adjustable parameters. We shall refer to (2) as the identifier for (1). Define the following error functions: e).

. -

F 1-- @ = A = - r e x ' and

.

(3a)

@(.)=A(*) -A

(3b)

*(.)=B(.)-B.

(3c)

-

F 2 = 9 = B = -re#'.

(9b)

1 V = - -e'Qe Q 0. 2

( 10)

Then

a )

e ( . ) = y ( . ) - x(.)

(9a)

The function V(e,cP,+) is then a Lyapunov function for system (4) in the error space. To show asymptotic identifiability it is necessary to prove global asymptotic stability of the null solution to (4) and (9), and for this we shall exploit the properties of V and u. 111. STABILITYANALYSIS

We assume the following. At each time t , consider ( e ( t ) ,@ ( t ) ,\ k ( r ) ) as a point in the 1) A is stable (Le.? if h is an eigenvalue of A , then "error-space" g n n2+ nm . In these coordinates, the identifi(1 1) Re h < 0). cation process is described by 2) { u ( t ) ,t 2 0} is bounded and (temporarily) piecewise P=Ee+@x+9u (4a) continuous. (12) (e,@.,*,t)&=F, (4b) Then, clearly, { x ( t ) . r > 0} is well defined and bounded. .k = F~ ( e ,@,9,t ) (4c) Our first stability result is obtained by direct application of known theorems of Lyapunov type (see, e.g., [8, Theowhere the identification laws F , and F2 are to be chosen rems X.3.1 and X.3.21). such that Proposition I : For the system described by (4) and (9): all trajectories { ( e ( t @( ) , t).\k(t)),t > 0} are bounded, and e(t)+O the null solution ( e (t ) ,@(t ) , \k( t ) )=0 is uniformly stable. @(t)+O as r+m. ( 5 ) Furthermore, e(t)+O as t+cc. The above result follows directly from the choice of r; 9(t)+O no special property has yet been assumed of the signals Clearly. for (4) to possess the null solution ( e ( t ) , @ ( t ) , u ( . ) and x ( . ) . However. to show asymptotic identificashow that @(r)+O and P(t)+O as \ k ( t ) ) = O , we must have F,(O.O.O.~)EO and F2(0,0:O1t)=tion.onemustalso 0. t + , x . At this point. we introducea special class of Definition 1: The identifier (2) is said to identzh the functions as follows. plant asynzptorically if the null solution (e(t). @ ( t ) ,\k(r))= Let be a set of points in [O. m ) for which there exists 0 of the system of errordifferentialequations (4)is r ' # t " implies I f - r''1 a 6 >0, such that for all t'. t" EC:~, globally asymptotically stable. 2 6. Denote by ? . [O. x ) the class of real-valued functions We shall next assign a specific structure to the adaptive on [0, cc) such that for every g E .?[O. 00). therecorrelaws F , and F,. Consider the function spond some 6 > 0 and some $8 with the following properties: V (e, @,9)= - e ' r e + tr (a'@ + \k'*k)} (a) g ( t ) and g ( t ) are continuous and boundedon ;{ (6) +

1

z8

where r is a symmetric positive definite n X n matrix. Let Q be a symmetric positive definite n X n matrix. Let r be chosen to satisfy the equation

E'T+TE+ Q=O;

[O. 0O)/i;',.

(b) For all r ' E Z8. g ( t ) and g ( r ) have finite limits as t J t ' and trt'.'

T ' his class of functions was also used by Anderson [6](who cited [IO]

(7) as the source), in obtaining results on exponential stability.

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-22, NO. 4, AUGUST

Thus, a function in S[O,co) has bounded continuous derivative almost everywhere on [0, co), and has an upper bound for the density of jump points or comer points on the t-axis. For instance, a (possibly aperiodic) square wave belongsto 9 [ 0 ,co) provided the time interval between successive jump points is bounded away from zero. Now write (4a) as P (I)= Ee(r)+ w ( f )

1977

fier (2), it is sufficient that #(.)E 9‘[0, 00) andthat [ x ‘ ( -),u’( .)] be persistently spanning.

For the proofwe need the following lemma, whose proof can be found in the Appendix. Lemma 2: Consider the functions g : [0, co)+CkN and$ [O, x ) + W X * ’ . Suppose f is piecewise continuousand f(t)+O as t + X . If g is persistently spanning, then f(t)+O

(13)

as t+co

provided

where

+

w ( t ) A @(t)x(t) \k(t)u(t).

Proposition 2: Consider the system given by (1 3) where E is stable, and w ( - ) ~ 9[0, ’ m).’ If e(r)+O as t+co, then w(t)+O

as r+co.

(15)

For the proof, we need the following lemma, whose proof can be found in the Appendix. Lemma I: If f ( - ) ~ 9 [ 0co) , and f(t)+O as t+m, then f ( t ) + O as r+m. Proof of Proposition 2: With w( E 9[O. co),then e( * )is bounded and continuous on r > 0. Hence, P( E 9[0, x). But fromLemma 1, e ( t ) + O as t+co. Thus, w(t)+O as t+co. 0 Note that for w( E 9 [0,m). it is sufficient that u( .) E 9 [ 0 ,a).Also, that e(t)+O as t+co follows directly from Proposition 1. Thus, wehave shownthat with input signals chosen from 9 [ 0 ,co),it is necessarily true that 0

f(t)g(r)+O as t+m.

( 14)

)

e)

a )

Proof of Theorem 1: Let g ( . ) Li [ x / ( - ) u’(-)]’. , Suppose [ e ( .), @(-),‘k(. )] is a solution function to (4) and (9). From Proposition 1, it is clear that every row f ( - ) of [@(-),‘k(.)] has the properties that f E 9’[0, co) and f(t)+O(r+co). From Proposition 2, we have f ( t )g(t)+O(t+co). Hence, the result followsLemma from 2. 0 It is now possible to show that certain almost periodic signals areadequateasprobing signals. Theapproach used here will differ from Anderson’s [4]. Let us first recall the definition of “almost periodicity” [12]; the continuity requirement that is sometimes imposed will be dropped. Definition 3: A bounded function g : [0, co)+gNis said to be almost periodic (ap) if for any E > 0, there is an L > 0 such that in any subinterval of length L in [0, m ) , there is a point T such that I I g ( t + T ) - g ( t )Ol l O such that persistently spanning. )IG,~-G-~II O such thatatany t > 0 there is a T, E [ r, r + e] for which

IV. PROBISGSIGNALDESIGN: A CONINUOUS SIGNAL Suppose the input u ( - ) to the system (1) is given by a finite sum of sinusoidal signals of different frequencies:

x K

Now fix any t > 0, and define

u(t)=

uksinykt

(20)

k=l

am,

where K and { uk E yk E ; k E K } are the design parameters. Assume the y k are all distinct and positive. Note that u( - ) is not necessarily periodic, but is nevertheless almost periodic. As in Section 111. we assume that A in (1) is stable. Then the steady-state response of the system (1) to the forcing input (20) is given by K

x(t)=

E

(tksinYkt+77kCOSYkt)

(21)

k=l

where then Clearly, u( .) and x ( are both ap. Hence, by Theorem 2 and Lemma 3, for asymptotic identification it is sufficient to show that for all a I E 9' and a2E %I x rn, 0

This implies

)

a,x(t)+a2u(t)=0,

t>O,

(23)

Thus,property (17) in the definition of ps functions is implies satisfied. 0 a 1= O and a 2 = 0 . Now it is known ([ 13, Theorem 73.11) that if the matrix A of system (1) is stable, and u ( . ) is ap and is sufficiently Now (23) implies well-behaved (e.g., belongs to ??[O, co)),then there exists a K unique ap response x( .). Hence. we can summarize the [(a15k+(Y2Uk)siny,t+a,77kcosy,f]=o: f>o. result in the following theorem. k=l Theorem 2: Assume condition (1 1) holds and that u E It follows (on multiplying by { ~ ~ ~ and p n taking ~ } the limit 9[0, x ) . Then for the asymptotic identification of system (1)by the identifier (2): it is sufficient that u be almost lim 1/ T i r * . ) that periodic, and that there exist a set of points {ti, i E n m} T + x such that the ( n m ) X ( n m ) matrix c ~ ~ ~ ~ + + ~ u k = k=1.2:**.K. ~ ~ ~ ~ = o .

+

+

+

From (22), this is equivalent to al(iykI-A)-'Buk+a2uk=0,

is nonsingular. FinaIly. for practical purposes, condition (18) may be replaced by condition (19) of the following lemma, whose straightforward proof is omitted. Lemma 3: Let g : [0, Then there exists a set of points { t,:i E N } such that det[ g ( t l ) .. . g(t,\,)]#O if and only if the following holds: 0 o ) + C k 2 ' ' .

In the following two sections, we shall utilize the results just obtained in the design of probing signals.

(24)

for k E K . Thus, the parameters { K ; u,,y,, k E K } must be chosen such that (24) implies a , = 0 and a2= 0. For this design. we have the following result. Proposition 4: Consider a stable n X n matrix A with cyclic index4 Y. and invariant factors of degrees n , , i E v. Suppose B is an n X m matrix. and the pair ( A ,B ) is controllable (hence, 1 < Y < m). There exist m linearly independent vectors { g,. i E m } in Ciim with the following property: If vectors { uk,k E K } ( K 2 n + m - Y) are defined according to 4The cyclic index of a matrixisthenumber of companionmatrix blocks initsrationalcanonicalform, i.e., thenumberofitsinvariant factors (see [ 14, p. 161).

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IEEE TRANSACTIONS OK AUTOMATIC CONTROL, VOL. AC-22, NO. 4, AUGUST

1977

Then CY(-~Y~Z-A)-IBU~+PU~=O

(kEK).

Subtracting, we get Im [ a ( i y k z - A ) - l B u k ] = O

(kEK);

equivalently, yk[

or since the where Bk ( k E K ) are arbitrary nonzero scalars, then forall aE and p E C k lx m , and arbitrary distinct positive real numbers yk ( k E K j , the equations

yk

=o

(kEK);

are nonzero.

a(y:1+A2)-'BLlk=O

'3,

a(iykZ-A)-'Buk+puk=o,

a(y:I+A2)-IBUk]

(kEK).

Write a=(a,,a,:.. .ap). where a i E ? $ l x q .By (25) and the rational canonical form of A , we have

kEK,

a I ( y ~ Z + A ~ ) - l b l = O . k=l;-. *n, imply a=O and p=O. This result is a generalization of the following lemma, a l ( y ~ Z + A ~ ) - ~ b 1 2 + a 2 ( y ~ Z + A ~ ) - ~ b 2 = 0 , whose proof can be found in the Appendix. k=n,+1:..,n,+n2 Lemma 4: Suppose A E W x n and b E ( 3 n . Let ( A ,6 ) be controllable, and A be stable. Let { y,, i E n} be a set of distinct positive real numbers. Then for all @ E u- 1 ai(y:Z+Az)-'~iu+a,(y~Z+A,Z)-ib,=O. @(y:Z+A2)-'b=0, kEn i= I k=n,+..* +nu-,+l;.-,n. implies @ = 0. Proof of Propsirion 4: By controllability, there exists Applying Lemma 4 to each of these equalities in turn, we a pair of bases for ( A ,B ) such that A has the rational get canonical f o r d al=o,a2=o,~~~.au=o. A =diag(A,; * .,A,),

x

A i ~ % q x T(

Finally. since there is at least one subset of m linearly independent vectors in the set { u k . k E K } . it follows that

i E v j , and

p=o.

0

Since the basis in which ( A . B ) takes the special form indicated is in general not unique, the vectors { g,, i E m } are not unique either. Indeed, almost any set of m linearly independent vectors gi chosen "at random" in 5irn-that is. excluding vectors on a finite number of fixed ( m- 1)dimensional hyperplanes-will have the property that the corresponding pair ( A . B ) is similar to the form assumed where b, E $4; thepairs ( A i . b i j . iE v, are controllable. in the proof (see [14, Lemma 1.21). Furthermore, if g,, i E Y are the first Y unit vectors in %"', In our present application the matrix A usually contains then ni and Y unknown elements.Consequently.thenumbers are not available to the designer. However, if we select the u k (k E nm) according to e,g,, Bkg,,

Let { g,,,; . . , g m } be the remaining unit vectors in qrn and define vectors { u k , k E K } according to (25). Now suppose a(iykz-A)-lBuk+pUk=O 'See [ 14, Theorem 1.21.

(k€K)

k = 1; . . ,n k=n+l;.-,2n p,

'k=Ii

B,g,

(26)

k=(m-l)n+l;--,nm

the result in Proposition 4 holds for all possible ni and Y. Hence, structural uncertainty about A can be taken care

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YUAN AND WONHAM: MODEL REFERENCE IDENTIFICATION

of simply byincludingmorefrequenciesintheinput signal. Thus, we have obtained a set of concrete design guidelines for selecting the parameters { yk, uk,K } of the input signal u ( . ) defined in (20).We summarize the results in the following theorem, which follows directly from Proposition 4 and Theorem 2. Theorem 3: Suppose the pair ( A ,B ) of the system (1) is controllable, and A is stable. Let the input u ( - ) be given by (20) where K > nm; the yk ( k E K ) are distinct positive real numbers; and within the set { uk, k E K } there are m linearly independent vectors, the spanof each repeating at least n times. Thentheidentifier (2)will (almostcertain1y)'j asymptotically identify the system (1). Remark: An obvious candidate for a probing signal is u(t)=col(u'(t), u 2 ( t ) ; - . , u " ( t ) ) where eachone of the components ui( contains n distinct frequencies (i.e., a total of nm distinct frequencies). In fact the choiceof uk in (26) is conservative. Since, in the generic case, the matrix A is cyclic (v = l), it is only (generically) necessary to incorporate K = n m - 1 distinct frequencies in u( .), i.e., only one of the m linearlyindependentvectors must repeat at least n times. e )

+

Then the steady-state response x j ( - ) is given by

where bj ii Be+ Now cn,=O(l/n) as Inl+co and (inoilA ) - ' = O(l/n) as Inl+co. Thismeans cn,(inwjI A)-'bjexp(inw,t)=0(1/n2) as Inl+co. Thus,the series (28) converges absolutely and uniformly for t 2 0. Therefore, the steady-state response to (27) can be written as

where

and the series (29) alsoconvergesabsolutely and uniformly for t > 0. Since u( is obviously ap, to show that [x'( - ) , u ' ( - ) ]is ps,itis sufficient(byLemma 3) to show thatfor all a I€ % I x n and a , € % I X m , e )

V. PROBING SIGNAL DESIGN: A PIECEWISE CONTINUOUS SIGNAL

Suppose u ( . ) E y [ O ,m) and hasperiodiccomponents; then each has a Fourier series representation so that

a,x(t)+a2u(t)=0,

(30)

t>O

implies a , = 0 and a2=O. From (27) and (29), condition (30) becomes rn

t > O (31) where a k E bk E %", and the yk are distinct numbers. Now take 6 > o and ea C 10, a)to Co~esPond tou(*)E where it is clear that the series converges piecewise uni5'[O,m). Let O < A < 6, and formly in [0,a),and uniformly has bounded partial sums. In order to eliminate the trigonometric terms in (31), we I(A)A ( t :t>o, need the following lemma, whose prooffound be in can 1 Appendix. the .. Lemma 5: Suppose sn( converges to s( E 9[O, co) For all such A, it is known that the Fourier series (27) piecewise in m), and { S n ( t ) } is convergesuniformlyin [0, c o ) / Z ( A ) . In this sense, thebounded.Then series (27) converges piecewise un!forormly in l0,co). Since E "'[O, &), it is clear that each one of its components is of boundedvariationonitsbaseperiod; lim -1J T s n ( t ) d t = T l iTs ( t ) d t n-m T 0 hencethepartialsums Ef=,(akcosykt+bksinykt)are bounded uniformly in K and t > 0. and the convergence is uniform for T E (0,co). Now the steady-state response of system ( I ) to the input Now multiply the terms in (31) by (say) sinymt and take the(27) canasbe written limit

am,

u

e )

e )

;(a)

J =1

xj(-) is the steady-state response of system (1) to the input Lemma 5 enables us to write u. E ., where 5 is thejth component in u ( - ) and is thejth . J N unit vector in '3,". Suppose u,( - ) has Fourier series J

lim lim

T-xa N+co

'?he ''almost certainty" is to be interpreted by the earlier remark on the choice of spanning vectors in (25).

' J T

T

2

0 k=O

[ e . .

lsiny,tdt=~.

By the uniformity of convergence with respect to T, we

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IEEE TRANSA(SI1ONS ON AUTOMATIC CONTROL, VOL. AC-22, NO.

u‘( -)I has a “persistently spanning” property (Definition 2) may be employed as probing signals. This class includes many conventionally used probing signals, such as periodic or almost periodic signals. Furthermore, using the properties of this class of signals, we can now actually design some of the required probing signals. For instance, we have shown that a suitably designed periodic square wave is indeed an acceptable probing signal. Althoughtheresultsdiscussedhere have beenbased uponaspecificidentificationscheme,it is naturalto expectthat some suchpersistentexcitationproperty is required in any convergent identification procedure. Amongthesignificantproblems left openforfuture research we mention that of determining good estimates of the rate of convergence, especially in cases where convergence is known on general grounds to be exponentially fast (cf. [6], [7]).Also of great interest would be a systematic study of the effects of random noise, especially as some experimentalresults (cf. [9]) indicatethatthe identification scheme is reasonably insensitive in this regard.

[x‘(

and since the yk are all distinct, there follows “*&+“2Uk=O1177k+(Y2bk=o,

all k .

From (29), (32)yields

Immediately we recognize that Proposition 4 furnishes the conditions sufficient for (33) to imply a I = 0 and a2= 0. The complete design for u( - ) can now be summarized by the following theorem. Theorem 4: Let the pair ( A ,B ) of system (1) be controllable and A be stable. Suppose u( .) E 9’[0, co)and has periodiccomponents. If in eitherone of the sequc&ces { a k } and { b k } of the series representation (27) of u ( - ) , there are m linearly independent vectors, the span of each repeating at least n times in the sequence, then the identifier (2) will (almost certainly)’ asymptotically identify the system (1). Remark: By virtue of the remark following Theorem 3, ifwe considerthegeneric case where thematrix A is cyclic, then the condition in Theorem 4 may be relaxed so that the span of only one of the m linearly independent vectorsneedstorepeatnot less than n times in the sequence. A common example of the type of probing signal discussed here is one whose components are periodic square wave signals with distinct base frequencies. As a simple illustration, consider the case m = 2 ; let the base frequenu ( . ) be 1 and 2 (rad/s), cies of thecomponentsin respectively. Then the set { yk} in the representation (27) is theset of all positive integers.Either of thesequences { a k } and { b k } will be of the form

where each “*” denotes a real number entry whose value depends on the corresponding Fourier coefficients of the component functions. Since the span of the vector (1 0)’ repeats an infinite number of times, the (relaxed) condition of Theorem 4 is then obviously satisfied.

4, AUGUST 1977

e),

APPENDIX

Proofof Lemma I: From the definition of 9 [ 0 ,a), there exist a 6 > O and a K > O such that l j ( t ) l < K for all t E[O, co)/C8. Suppose f(t)+O as t+co. Then there exist an E > O and a sequence { t k } , tkTco as kTco, at which (say) f ( t k ) 2 E > 0, for definiteness. The proof for the case f(t,) Q - E max{2, K ~ / E }At . each tk, one of the following two cases must arise: 1) tk B ea. Then f( exists and is bounded in either e)

[tk-(6/L),tkl 2) t k E

Or [ t k ? t k + ( 6 / L ) 1 .

E8. Then f(.) exists and is bounded in (tk,tk+

(a/ L)],. or

[ t k - ( 6 / ~ t)k ),, where bothlimlt,J(t) and limtl,, f ( t ) are finite. In either case, f ( t )> E - ( K 6 / L )> 0 for t in the corresponding subinterval in which f( .) exists and is bounded. But f(t)-+O as t+co implies the existence of a T > O such that If ( t ) l < ( e - ( K S / L ) ) 6 / 2 L e E‘, for t > T . Thus, for any t k > T + ( 6 / L ) , a) if j ( - )exists and is bounded in [ t k - (6,’ L).tk] (case 1) or [ t, - (6/ L ) ,t,) (case 2), then

= - E’ + 2E’ = e‘;

b) if f(.) exists and is bounded in [ t k ,tk + ( 6 / L ) ](case

1) or ( t k , t k + ( a / L ) ](case 2), then VI.

CONCLUSIONS

For a model reference identification procedure,we have studied the excitation properties that constitute an effec0 Contradiction results in both cases. tive probing signal. Inparticular, we findthatinput Proof of Lemma 2: By the properties of ps functions, signals u ( . ) for which the combined state and input vector there exists a triple ( M ,L , K ) such that ’See footnote 6.

It g ( q l Q M ,

t > 0,

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WAN AND WONHAM: MODEL REFERENCE IDENTIFICATION

and for all t > 0, there exists a sequence { T, i = 1,. . . ,N } witht0 ( j E n) are all distinct, then det Q ZO,and C = 0. This means the vectors { ( A* y,zI)-’b, j E n } are linearly independent, and the proof is complete. Proof of Lemma 5: Pick 6 > 0 and a set of points corresponding to s(*). It suffices to show uniform convergence for T €[a, 00). Clearly, there exists a K > 0 such that I s ( t ) l < K ( t > 0) and Isn(f)l0) for all n. Fix any O < - E < ~ Kand , choose

+

6 A = --E. 4k Hence, we have O < A < S. Now, by piecewise uniform convergence, there exists an N such that for all n > N and 1 E [O,w ) / l(A), Isn(t)--(t)I N ,

Joseph S-C. Yuan (S’67-M’76) was bornin Shanghai,China.He received the BASc. and M.A.Sc. degrees in electrical engineering in1970 and 1972, respectively, fromtheUniversity of British Columbia, Vancouver, Canada, and the Ph.D.degree in electrical enweering in 1976 from the University of Toronto, Toronto, Ont., Canada. He is currently with the systems control group at SPARAerospaceProducts Ltd., Toronto, Out., Canada.

Nowthere can be at most T .~/ S points from interval [S, TI. T h u s ,

in the

W. M. Wonham ( M m M ’ 7 G F 7 7 ) was born

Hence,

REFERENCES

[I]

K. S. N a r a & a a d p. Kudva,“Stableadaptive s h e m a for system identification and control-Parts I and 11,” IEEE Trans. Syst., Man, Cybern., vol. SMC-4, pp. 542-560, Nov. 1974.

in Montreal, P.Q., Canada. He received the B. Eng. degree from McGiU University, Montreal, P.Q., Canada and the Ph.D. degree in control University the engineering from of Cambridge, Cambridge,England in1956 and 1961,respectively. He has held teaching and research positions Institute with PurdueUniversity,theResearch forAdvancedStudies (MAS), BrownUniversity, and NASA. Currently, he is Professor of Electrical Engineering and Chairman of the Systems Control Group at the University of Toronto, Toronto, Ont., Canada. His research interests are centered on synthesis problems in multivariable control.