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E TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-30. NO. 6,UNE 1985 585AQ , A p E I n e x n e are positive diagonal, A & A&p, an d theof A are the eigenvalues of M . Consequently,

Q=7Q, P = A . (2.19)Iu . PROOF OF THE M A I N THEOREM

proof proceeds exactly as in [ 6 ] . Using the fact that a, is open, theof the Lagrange multiplier theorem can be used toO=AQ+QP+ , (3.1)

B = [ LTRLLTRCe-CTRLCTR2Cp

(n + ne ) X (n +ne)&,p a r e part it ioned into n X n, n X ne , and n,ne subblocks as

(3.2) yieldsO=AQl+QIAT+V1,

O=AQ~~+QIZA:+QI(B,C)+IZBT, (3.7)o=ApQ2+Q:A:+BeCQ~2+Q:2(BeC3T+B,V2B: , (34

O = A T P ~ + P ~ A + ( B , C ) T P : z + P 1 2 B , C + L T R L , (3.9)O = P I A ~ + A ~ P ~ ~ + ( E , C ) ~ P ~ - L ~ R C , ,3.10)

0=AFP2+P2Ae+C,TK2Ce. (3.11)omitted. Writing (3.8) as (see

O=(A,+B,CQIZQ~)QZ+QZ(A~+B~CQIZQ~)~+B~V~B~.Q2fis the M oore-Penrose or Drazin generalized inverse of Q 2 , t[l , Lemmas 2 .1 and 12.21 that Q2 is positive definite.(3.1 1) implies that P2 is positive definite. This justifies (3.4)

n X n nonnegativedefinite matrices (see [13], [14])Q = Q I-QnQT1QT2, Q = Q I ~ Q ; Q : ~ ,= P I ~ P T ~ P ~ ,

) implies (2.5) and (2.6) withG=QfQL, M=QzP2, r = P - y r

Q2P2= P? 2(P!:2Q2P:i3P:2, is positive semisimple. Sylves-(2.13). Note (2.19) and the identities

Q ~ = Q + Q , (3.12)

Q~=rQr T , p 2=GPG T . (3.14)Q12=QrT,P12= -PCT, (3.13)

(3.4) and (3.5) yield (2.8) and (2.9). Also, th eof (3.8) and (3.7) yield (2.7). Substituting (2.7)-(2.9)

superfluous. Finally, linear combinations of the remaining three( 3 . 6 ) , 3.7), and (3.10) yield (2.10)-(2.12).

IV. CONCLUDINGEMARKSThe question of m ultiple local m inima satisfying the optimal projectionequations for reduced-order state estimation and the problem of construct-ing numerical methods for solving these equations are beyond the scope ofthis note. It should be pointed out, however, that promising numericalresults for he model-reduction and fixed-order dynamiccom pensationproblems have been obtained by means of iterative algorithms that takefull advantage of the presence and structure of theoptimal projection [2],Finally, the results of this paper can be extended to include thefollowing related problems: 1) discrete-time s ysted discre te-tim e estima-tor; 2) infinite-dimensional system/finite-dimensional estimator [5]; an d3) parameter uncertainties [I], [151, [161.

[41, [71.

ACKNOWLEDGMENTThe authors w ish to thank Dr. F. M . Ham for directing their attention

to the reduced-order state-estimation problem as a fruitful appiication ofthe optimal projection approach.REFERENCES

D. C. Hyland, Optimality conditions for fmed-order dynamic compensation offlexible spacecraft with uncertain parameters, AIM 20th Aerosp. Sci. Meet.,Orlando, FL, Jan. 1982, paper 82-0312.methods and illustrative results, AIAA 2 1 s Aerosp. Sci. Meet., Reno. NV, Jan.-, The optimal projection approach to fmed-order compensation: NumericalD . C . Hyland and D. S . Bernstein, Explicit optimality conditions for fixed-order1983. paper 83-0303.dynamic compens ation, in Proc. 22nd IEEE Conf. Decision Conir . , SanD . C . Hyland, Comparison of various controller-reduction methods: SuboptimalAntonio, TX, Dec. 1983, pp. 161-165.versus optimal projection, in hoc. AIAA Dynam. Special is ts Conf. , PalmD. S . Bemstein and D. C. Hyland, The optimal projection equations for fixed-Springs, CA, May 1984, pp. 381-389.orde r dynamic compe nsation of distributed parameter systems, presented at theAIAA Dynam. Specialists Conf.. Palm Springs, CA . May 1984.D. C. Hyland and D . S. Bemstein, The optimal projection equations for fixed-order dynamic compensation, IEEE Trans. Autom at. Contr., vol. AC-29, pp.10341037, 1984.-. The optimal projection approach to model reduction and the relationshipbetween the methods of Wilson and Mo ore, in Proc. 23rd Conf. DecisionC . S . Sims, Reduced-order modelling and filtering, in Control and DynamicContr . , Las Vegas, N V , Dec. 1984.Systems, Vol. 18. C. T. Leondes, Ed . , 1982, pp. 5s-103 D. A. Wilson and R. N . Mishra, Design of low order estimators using reducedK. Kwakemaak and R. Sivan, Linear Optimal Control Systems. New York:models, Int . J . Contr . , vol. 23, pp. 447456, 1979.

W. M . Wonham, Linear Multivariable Control: A GeometricApproach.Wiley-Interscience, 1972.

S . L. Cambell and C.D. Meyer, Jr.. Generalized Inverses of LinearNew York: Springer-Verlag. 1974.Transformations. London: Pitman, 1979.A. Albert, Conditions for positive and nonnegative definiteness in terms ofE. Kreindler and A. Jameson, Conditions for nomegativeness of partitionedpseudo inverse, SIAM J . Appl. Math. , vol. 17, pp. 434440, 1969.

P. J . McLane, Optimal l i near filtering for linear systems with statedependentmatrices, IEEE Trans. Automat. Contr., v d . AC-17, pp. 147-148, 1972.noise, Inr. J. Contr . , vol. 10, pp. 42-51, 1969.D. S . Bernstein and D . C. Hyland, Optimal pmjectiodmaximum entropyModel Error and Concepts and Compensation. Boston, MA, June 1985.stochastic modeling and reduced-order design synthesis, IFAC Worksh op on

Legendre Series Approach to Identification and Ana lysisof Linear Systems

P . N. ARASKEVOPOULOSAbstract-In this paper the Legendre operational matrix of integrationis introduced,and it is subsequently used fo r the dentificationand

This paper is based on a prior submission of March 11, 1983.Manuscript h e i v e d October31. 1983; revised May 22, 1 984and December 18. 1984.

of Thrace, Xanthi. Greece. He is now with the Division of Computer Science,The author was with the Department of Electrical Engineering, Democritus UniversityDepartment of Electrical Engineering, National Technical University of Athens, Athens,Greece.

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586 IEEERANSACTIONSNUTOMATICONTROL, VOL. AC-30. NO. 6 . JUNE 1985analysis of linear systems. Th e algorithms proposed are easily implement- following will be used:able in a digital computer, and they areanalogous to those alreadyderived for other types of orthogonal functions. PPI(t)- o n - =(2n+l)pn(r), n B (4a)1)

I. INTRODUCTIONIn the recent years, a new technique has b een established for hesolution of various problems, such as system identification and analysis,using orthogonal functions. The main characteristic of this new techniqueis that it reduces these problems to that of solving a system of algebraic 1'I p n ( f ) p m V )dt=equations, thus greatly simplifying the problem so lution. The key idea ofthis new technique is based on the following integral property of the basisvector &):

The Legendre polynomials p,( t ) and p,,,(t) are orthogonal in theinterval [- , I], i .e . ,0 , if n f m

( 5 )I- if n=m.n + 1 'A functionflt) may be expanded into a Legendre series as follows:-times

where P is a square constant matrix and pT(t) = [po(t),pl(t), ..-,pr-&)], where the elements po(t),pl(t), - * *, ~ ~ - ~ ( t )re basis functionswhich areorthogonal on a certain interval. Clearly, the form f P ependson the particular choice of the basis vector p(t). Up to now, several suchbasis functions have been used, namely, the Walsh [1]-[7], the block-pulse [8] -[ll ], the Laguerre [12]-[14], the Chebyshev [15], the Herm ite[16], and the sine-cosine functions [17].

In this correspondence we introduce the Legendre polynomials forwhich the operational matrix of integration P is first derived. Subse-quently, using property (l ) , the problems of identification and analysis oflinear systems are studied. It is shown that, following the same procedureas in the case of other rthogonal functions, these problems may similarlybe solved.The Legendre polynomials may have advantages over other orthogonalfunctions. This is shown by way of an example at the end of Section V ,where Legendre polynomials converge to the exact solution of adifferential equation faster than other types of orthogonal functions, as,for example, Walsh, Hermite, and Laguerre.II.LEGENDRE o L m o M m s

The Legendre polynomials are defined for the time interval t E [- ,11 and they have the following analytical form [181 (Rodrigues' formula)

p,,(t)=--(t*-l)", n=O, 1, 2, ..e.1 d "2"n! dt"Using the above expression for a&), one may readily determine the firstfew Legendre polynomials.

rpo(r)=1Pdt)=

p ~ ( t ) - )p3(t)=1/2(5t3- t). (3 )

The Legendre polynomials have several interesting properties. Thep * = [ p : PII : : : : :00 00 - 012 \: 00 0 -a3 \ 13- [ i . . - -0 0 00

m. EGENDREPERATIONAL MATRIXOF INTEGRATIONIn this section, the operational matrix P of relation (1) for the case ofLegendre polynomials will be derived. To this end, consider therecurrence relation (4a), i.e., the relation

(2n+ l)pp.(t)=p~ll(t)-p~!,(r), n? 1. (8)Integration of (8) yields

From (4b) it follows that p f l A l ( - ) - pn- l (- ) = ( - I)n+l( - l ) n - l = ( - l ) f l -1 [ ( - l )2 - 11 = 0. Thus, (9) becomes*, 1

Relation (10) holds far n = 1, 2, 3, . .. . For n = 0, it isp0(u)du= r +1 =po(t)+do .

Using (10) and (1 l) , we write the following integral expression:

where P* is an r x ( r + 1) matrix having the form

\..... . -

0 0 0 : o0 0 0 : o0 0 0 : o0 0 0 , o

8 .

. I .. ' .

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TRANSACTIONSN AUTOMATICONTROL, VOL. AC-30, NO. 6 , JUNE 1985 587P an d p have dimensions r X r an d r X 1, respectively, and

p r ( t ) in the vector ( ~ , + ~ ( t ) ,.e., if we drop theP,elation (12) becomes the approximate relation1' PA 4 du=PPrW (15)j', '.. 1' P r b ) (do)*=PpkPr,(0. (16)

- I

P is specified in (13). Fo r k integrations, we have- I-

k timesthat the structure of the operational matrix P is appealing forsince all its elements are zero, except for its

w . ARAMET ER AND INITIAL CONDITION IDENTIFICATIONe class of systems considered are the single-input single-output linear

y("'+a,- ly("-')+..+aly(~)+aoy=boou(f);tE[-1, 11 (17)

1 . W e s t a r t b ythe differential equation (17) into an integral one throughas follows 1151: jl1 .. j'- I-n- ) times

n imes

- -times (n- ) times+. .+dl j' &+do. (18)- 1

s do, dl, . d, - appearing in (18) are due to the initialco, c , , .. c,- I and they satisfy the following relationship:

AC = d

C=

A =

... 0 00 00 0

......

Note that A is always nonsingular. Thus, once the vector d is determined,the initial condition coefficient vector c may readily be determined from(19).To proceed with the identification, we expand the input and outputsignals in truncated Legendre series as follows:

where, for simplicity in notation, we dro p from here on the index r .Introducing (20) into (18) and making use of relation (16), w e arrive at thefollowing equation [15]:y r ~ ( t ) + ~ , _ l y T P ~ ( t ) + ~ ~ ~ Z y r P P Z ~ ( t )+... aly'PP"~1p(t)+aoyrPP"(p(t)

=b# rPnp(t)+dn - erPn-lp(t)+...+dl eV p ( t )+doerp(t) (21)where eT = (1 , 0, 0, . a , 0). Now, equating the coefficients of p(t) ofboth sides in (21) and after some algebraic manipulations, we arrive at thefollowing relationship:

Qa =Y (22)where

BT= [ u n - 1 , an-2. ..., 0 , bo. dn-I, d n - 2 , ...,doland

e = [ - p r Y y j - p ) r y ...;- P " ) y (Pn)u (Pn-l)e i (pn- ') re j . . e] .

Relationship (22) is a linear system of r equations with 2n + 1 unknowns.Thus, we must have that r 2 2n + 1. An estimate for the parametervector B will then bea* =(QTQ)-'Qry. (23)

If QTQ s not invertible, then on e may increase, iteratively, the numberof the Legendre coefficients by one, until QTQbecomes invertible. It isnoted that there might be cases where the matrix QT Qmay never becomeinvertible. These cases app ear to be very difficult to detect by establishingsome type of criteria or conditions on Q , due to the complexity of thestructure of Q . Indeed, consider, for example, the first column of Q fo r r+ 1 elements in y . in which case the first column of Q is - L yrA .Partition PT+ and yr+ as follows:

The above expression shows that, if we increase r by one, then allelements of the first column of Q undergo a change, and similarly forother columns of Q. This means that if QTQr is not invertible, it is noteasy to tell whether or not Q T - P Q r + k , k = 1, 2, . . . re invertible.It is remarked that the Legendre polynom ials have been involved in thepast in the identification problem [19]. The method proposed in [19],however, is based on the numerical inversion of Laplace transform, andhence, it is entirely different from the present approach. The Legendrepolynomials appearing in [19] are involved only w ith regard to hemeasurement points ti of he input-output data, where t i = -log r,,where ri re the roots of the shifted Legendre polynomials.

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588 IEEE TRANSACTIONS ON AUTOMATIC C ONTROL, VOL. A C-30, NO. 6, JUNE 1985v. SOLUTION OF STATE-SPACE EQUATIONS

In this section we consider solving the state-space equationsi ( f )=Ax( t )+Bu( t ) ; ( - l ) = q , f E [ - I , 1 1 . (24)

To this end, the known input vector u( t ) s approximated as follows:

u( t )= E] ' =H q ( f ) . (25)Next, assume that the unknown state vector x( t ) may also be expanded inLegendre series as follows:

x ( t )=

= f0 ;fl I 2 . . f r - I l d t ) =F d O . (26)The idea in (26) is that instead of having x(t) as unknown, we now have Fas unknown.Now, returning to (24) and upon integration, we have

x ( t ) -x ( - ) = A s' x(.) d u+ B s' u(u) du . (27)- 1 - I

Intrcducing (25) an d (26) in (27) and using the property (15), we get

where

E =X1( - l ) 0 0 ... 0x*(-1) 0 o . - . o: I = [ X ( - l ) i O j O / ...:01.-f - ) columnsx"( - l ) 0 0 . *-

Equating the coefficients of a(t) n (28) we haveAFP-F=D, where D = -BHP-E. (29)

Clearly, (29) is a linear system of algebriac equations with F being theunknown. To facilitate the solution of (29), it is written in the followingsimpler form by using the Kronecker product [15]:Mf=d (30)

0.20.3

I0.40.5

: 0.60.70.80.91.0L

TABLE IVALUESOF SOLUTION OF (31) FOR t E O, 11e x a c t

solution1,0000000.9048370.8187300.7408180.6703200.6065300.5488110.4365850.4493280.4065690.367879

vi aL e g e n d r e0.9996320.9049310.8199750.7408450.6702240.6063940.5487390.4966400.4494810.4066440.367511

0.7408050.6701610.6063290.5486890.4966220.4495090.4067300.367667

0.7238C0 ' 0 745031 ' 0.6913580.664746 I 0.686833 I 0.6913580.613333 ~ 0.631510 .691358I .

0.426666.395833.418228- xact solutlonDo0Legendre and Chebyshev

series solutiol---_ ermite series solutionI,-!alsh series solutionL a y e r r e s e r i e s s o l u t i o n

0.5

,t0.5

Fig. 1. Graphical presentation of solution of (31) for r E (0, 1).

where

M = A 0 P'-I, f = ["Ir- 1 , d =wheref, and diare the ith columns of F an d D , respectively. Solving (30),one may then construct x ( t ) according to (26).It is remarked that the results of this paper may readily be extended tocover the more general case where the time interval of interest is (0, T ) ,following the same steps as in the case for Chebyshev polynomials [15].Example: Consider a system described by the differential equationx= -x , d o ) = 1. (3 1 )

The exact solution of (31) s x(t) =e-'. To apply the proposed technique,let x(t) =f rq(t ) . Then, relation (30)takes on the simple formMf=d, withM=-(P'+I) and d = - [ 1 0 0 ... 01'. (32)

To facilitate the comparison with other orthogonal functions which areorthogonal in intervals other than [- , 11, as, for example, the Walshfunctions, the analysis will be carried o ut in the interval t E [0, 11. To thisend,one may readily apply shifted Legendre polynomials which aredefined for t E [0, I] , and similarly with the Chebyshev polynomials.Following thi s, the approximate solution of (32) derived via orthogonalseries, for r =4, is summarized in Table I and Fig. 1. From these resultsit is clear that Chebyshev and Legendre ser ies give excellent results, m uchbetter than Hermite, Lag uerre, and Walsh series.

VI. CONCLUSIONSIn thi s paper, he problems of system identification and analysis oflinear systems are studied using Legendre polynomials. The algorithms

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589

as Walsh, block-pulse, Chebyshev, etc.paper covers only the case of linear time-invariant systems. Thebe extended to cover other types of systems

as time-varying, time-delay, distributed parameter systems, etc., byready used o study this type ofof orthogonal functions. Furthermore, theof optimal control may be studied along the same lines, as, fo r[q nd [15]. ACKNOWLEDGMENT

to G . Kekkeris for the numerical work on the example.REFERENCES

C.Chen and C. H. Hsiao, Timedomain synthesis via Walsh functions, Proc.IEE, vol. 122, pp. 565-570. 1975.G. P. Rao and L . Sivakumar, System identification via Walsh functions, Proc.IEE, vol. 122, pp. 1160-1161, 1975.S. G. Tzafestas, Walsh series approach to lumped and distributed systemidentification, J. Franklin Inst., vol. 305. pp. 199-220, 1978.P. N. Paraskevopoulos and A. C.B ou rn , Distributed parameter systemidentification via Walsh functions, Int. J . Syst. Sci., vol. 9, pp. 75-83, 1978.F. Chen and C . H. Hsiao, A state-space approach to Walsh series solution oflinear systems, Int. J . Sysl . Sci., vol. 6 . pp. 833-858, 1975.M . S . Corrington. Solution of differential and integral equations with Walshfunctions, IEEE Trans. Circuit Theory, vol. 20, pp. 470-476. 1973.C. F. Chen and C. H. Hsiao, Design of piecewise constant gains for optimalcontrol via Walsh functions, IEEE Trans. Autom at. C ontr., vol . 20. pp. 596-603, 1975.K . R.Pala nism y and D. K. B hattachatya, System identification via block-pulsefunctions, Inl . J. Syst . Sci., vol. 12. pp. 643-647, 1981.Sannuti, Analysis and synthesis of dynamic systems via block-pulse

S . Hsu and B. Cheng, Analysis and optimal control of time-varying linearfunctions, Proc. IEE, vol. 124, pp. 569-571. 1977.systems via block-pulse functions. Int. J . Contr . . vol. 33, pp. 1107-1122,1981.L. S . Shieh, C. K . Yeung. and B. G . McInnis. Solution of state-space equationsvia block-pulse functions, Int . J. Contr . , vol. 28, pp. 383-392, 1978.R. E. King and P. N. Paraskevopo ulos, Param etric identification of discrete-timeSISO systems, Int . J . Contr . , vol. 30, pp. 1023-1029. 1979.P. R. Clement, L aguerre functions in signal analysis and parameter identifica-tion, J . Franklin Inst., vol. 313, pp. 85-95, 1982.C. Hw ang and Y. P. Shih, Parameter identification via Laguerre polynomials,Int. J. Sys l . Sci., vol. 13, pp. 209-217, 1982.P. N. Paraskevopoulos, C hebyshev s eries approach to system identification,analysis and optimal control. J. Franklin Inst., vol. 316, pp. 135-157, 1983.P. N. Paraskevopoulos and G. Th . Kekkeris. Hermite series approach to systemidentification, analysis and optimal control, in Proc.Meas.Contr.Conf.,P. N. Paraske vopulos ,P. D. Sparis, and S. G. Mouroutsos, Fourier seriesAthens, Greece, 1983. vol. 2, pp. 146-149.

W . W . Bell. Special Functions fo r Scientirts and Engineers. Princeton, NJ:operational matrix of integration, Int. J. Syst. Sci., to be published.

R.Bellman, Identification of linear systems via numerical inversion of LaplaceVan Nostrand, 1968.transforms, IEEE Trans. Automat. Contr., vol. AC-IO, pp. 111-112, 1965.

ersistency of Excitation for Nonminimal Models ofSystems Having Purely Deterministic DisturbancesG R A H A M C . G O O D W I N , J O H N P. N O R T O N , ANDM. .V I S W A N A T H A Nact-All results to ate on the question of persistency ofon depend on the model being minimal. Thus the results do not

Australian Research Grant Committee.ved March 31, 1983: revised November 8. 1984 and December 18,

of Newcastle, Newcastle, NSW. Australia.C. Goodwin is with the Department of Electrical and Computer Engineering,

on leave at the Department ofP. Norton is with the Depamnent of Electronic and Electrical Engineering,of Newcastle, Newastle, NSW,with the Hunter District Water Board, Newcastle, NSW,

0018-9286/85/06004585

apply to systems having purely deterministic disturbances since theseautomatically give rise to uncontrollable modes on the unit circle. Thisnote presents new results on persistency of excitation applicable tononminimal models of systems having pu rely deterministic disturbances.I . INTRODUCTION

There has been a great deal of research on the question of parameterconvergence subject to a persistency of excitation condition on the inputsignal. For example, in the open-loop minimal-model case, it is knownthat it suffices for the input to contain enough sinusoidal components (see,for example, [l]).These results have also been extended (e.g., [2]) to model-referenceadaptive control algorithms by exploiting the fact that boundedness of thesystem variables has been established by other means. General results ofthis nature have recently been given in [3]. Similar results are known forthe stochastic case ; see, for example, [4], [5].It has also recently been shown that it is possible to achieve persistencyof excitation w ithout first requiring signal boundedness [6], [;1. In all ofthe above work, it has been assumed that the model is minimal. Thus theresults are not applicable to systems having purely deterministic distur-bances, since these give rise to nonminimal m odels having uncontrollableroots on the unit circle [8].The purpose of the current aper is to develop p ersistency of excitationconditions for nonminimal m odels of systems having purely deterministicdisturbances.

u. M O D E L DESCRIPTIONWe shall consider a single-input, single-output system described inDARMA form as

where

and q - s the unitdelay operator.We will nor assume that A ( q - ) nd B ( q - ) re relatively prime since,as explained in detail in [9 , ch. 21, purely deterministic disturbances giverise to common factors having roots on the unit circle. In the latter casewe can also write the model (2.1) as

where A(q- ) nd & - I ) are relatively prime w ith A 5 n , together witha disturbance model

where D(q-) s of degree n - A. We then have

As an illustration, if the disturbance {d( t ) ] s a periodic disturbance ofperiod p , then the common factor D(q- ) as the form

which has roots on the unit circle at e (m+ l ) r p ;m = 1, ...,P-

III. PARAMETERSTIMATIONLGORITHMSThe nonminimal model (2.1) can be fitted to input-output data usingprecisely the same algorithms as are used for cases when no disturbances

are present. Fo r example, the recursive least-sqaures algorithm for~$01 .000 985 IEEE