We study the identification problem that arises in a linear hereditary system with distributed delay. This involves estimating an infinite-dimensional parameter and we use the method of sieves, proposed by Grenander, to solve this problem

Bagchi, Arunabha

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Identification of a hereditary system with distributed delay

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Systems & Control Letters 5 (1985) 339-345 North-Holland April 1985 Identification of a hereditary system with distributed delay Arunabha BAGCHI Deparmzenr a/ Applied Marhmrarics Twenfe Unicersicv oj Technolog?: P. 0. Bo.r 217, 75CXl A E Emchede, me Nerher/an& Received 7 September 1984 Revised 14 January 1985 We study the identification problem that arises in a linear hereditary system with distributed delay. This involves estimating an infinite-dimensional parameter and we use the method of sieves. proposed by Grenander. to solve this problem. Kq~.or&: Hereditary system. Distributed delay, Method of sieves. Consistent estimates. 1. Introduction Although extensive research has been done on the distributed system parameter identification problem (see [l] for a recent survey), relatively little attention has been paid, in the stochastic framework, to the problem of estimating unknown parameters that are infinite dimensional. This is not surprising, as Rozanov pointed out in his classic paper [2] that directly estimating infinite-dimensional parameters by the method of maximum likelihood either may not be possible or may turn out to be not consistent. To circumvent these difficulties, Grenander [3] suggested the method of sieves to estimate parameters in abstract spaces. We apply his idea to a canonical example where one wants to identify the distributed delay function of a linear hereditary system. Recently, Nguyen and Pham [4] applied the method of sieves to the problem of identifying a nonstationary diffusion model. 2. Problem formulation and estimation procedure Let X,, - co < t < co, be the stationary solution of a linear time-delayed system with distributed delay: dX,=l_Oba(B)X,+,d&‘dr+d~ (2.1) where, for simplicity, we assume X, to be a scalar process and { U$ - co < r < co} is a standard Brownian motion. It6 and Nisio [5] give a sufficient condition for the existence and uniqueness of the stationary solution X,, - 00 -z t < 00, of (2.1). Assume that X, is completely observed in an interval I, < I < T and, based on this observation, we want to estimate the function a(B) in [ -b,O]. We also assume that a( a) E L,[ -b,O], the space of square-integrable functions on [ -b,O], with (., 0) denoting inner product there. We could also start with the one-sided version of (2.1), i.e. for X,=w. -b<tgO, (2.2) where ((1) is a given element of L,[ - b,O]. If the system was stable, the following analysis would go through for the one-sided version of the problem, if we started our identification procedure after the system had reached the steady state. X, is mean-square continuous and has continuous sample paths. Let px be the measure induced by ( X,, ra Q I s T) on C[r,,T], the space of continuous functions on 0167-6911/85/53.30 0 1985, Elscvier Science Publishers B.V. (North-Holland) 339 Volume 5, Number 5 SYSTEMS 6: CONTROL LETTERS April 1985 [r,,T], with P,,, denoting the Wiener measure thereon. From Lipster and Shiryayev [6]. we know that ps is absolutely continuous with respect to p,,,, and the log-likelihood functional for the problem is given by Lr(a(‘))B dp -(X)=/‘(,” a(B)X,+8d6’)dX,-+~r-(/;$8)X,+~dB)’dr. II’ ‘0 -b ‘t1 (2.3) The underlying idea of the method of sieves is that, instead of tryin, 0 to estimate the function (u( .), one estimates a finite-dimensional projection of (u(a) of sufficiently high dimension to obtain a good enough estimate of the unknown parameter function. Basically, one constructs an increasing sequence V,, of subspaces in L,[ -b,O] of dimension err such that U,, >, ,, V is dense in Lz[ - b,O] and seeks a maximum of Lr(cr( 0)) only on the subspace v,. Suppose that (Pi, i = 1, 2,. . . , is a sequence of orthonormal vectors in L2[ -b,O] such that { +,, (pz,. . . , (p,,) forms a basis of I’,, for all II. If (u(e) E V,,, we may write a( 0) = 2 aj+j( -) (2.4) j-l and the log-likelihood functional can be expressed as =(T-t,){b (ma(T) _ 1 2a V)~/pa’T > where o(r) = ((u,, oz.. . . , a,,,)‘, 6”’ is a vector with j-th component (2.5) b!T” = 1 J -,‘(/” +J(8)X,+,dB)dX,. T-‘t, I” -b j= 1,2,...,+ and A”) is a matrix with ij-th component The ML (maximum likelihood) estimate of o(e) on l$ is then $T’( .) = z &;r)+j( .) (2.6) j-1 where s(T) - (iiy-1, . . . , cq,” ) is the solution of A (r)&(r) = b(T). If A(r) is nonsingular, &tT) = [A(T’]-‘b(r). Note that the equation A”‘&(r) = b”) can equivalently be expressed as 3. Consistency of the ML estimate In order to study the desired consistency property of our estimate, we introduce another orthonormal basis of V, relative to a different inner product. Let R(r) denote the covariance function of the stationary process X,. From [5, p. 521, we know that the process X, has a spectral density S(X) given by S(X) = 1 ’ I J ih - ’ eiXea(f?)dB -b I provided /!,a(Qde # 0. Clearly, S(X) > 0 for all A. (3.1) 340 Volume 5, Number 5 SYSTEMS & CONTROL LETTERS April 1985 In the space of measurable functions on [ -b,O], we introduce a new inner product = lrn (1” f( 8)eiA8dfI)’ S( X)dX by Fubini’s theorem --Q) -b Then (I, f)R = 0 implies l!,,f(8)eiAed8 = 0 for all A, since ~~,,f(f9)eiXedfI is a continuous function of x and S(X) > 0 for all A, and then f(8) = 0 a.e. Furthermore, R(r) being continuous, it is bounded on any compact set and (fvf)<cQ = (fvf)R<@J. (3.2) Let I.:[ -b,O] be the completion of L,[ -b,O] with respect to the metric induced by the inner product (0, s)~. Consider a sequence I$,, #2, . . . such that $,, I/J,,.. ., $J,,, form an orthonormal basis of V, for all II, relative to the new inner product. Let {IT’, &r), . . . , .$z) be the coordinates of B(r)(*) in the new basis ($1’ h..., I/.J,,,} of V,. From (2.7) it is clear that i(r)= ({I”, &r), . . . , &,T)) is the solution of F’T’$T’= g’T’ (3.3) where the matrix F(r) and vector g’” have components (3.4) 1 =-I’(/” ~j(8)X,+,d8)(~~ba(~‘)X,+,.dB)d~+~~T(~o T-r, I,, -b or, -6 +j(@)&+ed+K (3.5) Let atT)(*)= ~~r,.$j$j(=) be the orthogonal projection of ~(0) onto v, in the sense of the inner product (a, .)R. Then the first term on the right-hand side of (3.5) can be written as since l_Ob~~b~j(e)R(6-8’)[a(B’)-a’T’(B’)]dBdB’=0, j=l,2,...,n,. Therefore, (3.3) can be equivalently written as FtT’( [ (7-j _ (V-j) = /$T) where t(T) and /rtT) are vectors with components cj, i = 1, 2,. . . , /rr, and =- T(T) 1 J T-to to qj (r)dt +1 / T8j(‘)dW, T-to r, (3-6) (3.7) 341 Volume 5. Number 5 SYSTEMS t CONTROL LETTERS April 1985 where Z,(B, 8’)=X,+,X,+,.-R(B-fq, and Lemma 1. Asswnit~g that l?,lR(r)(‘dr < 03, EIg’- aij12 <& 0 for some constant C. Proof. ~l$?bS~~)~ = l JTj*~Xii([)Xij(S)dtds (T-ro)2 ‘II ‘0 EXjjtr)XijlsJ= ~~~~~i(e)~j(e~~~i~T~~j~~~~ .[” EX,+,X,+ds+,X,+,, -R(B-B’)R(T-7’)]dBde~d*d7’. By the formula for the four products of Gaussian random variables, EX,+,X,+ds+,Xs+,~ = EX,,,X,,,.EX,,,X,+,~ -I- EX,+,X,+,EX,+,.X,+,. + EX,,,X,+,EX,,,~X,+, =R(e-e’)R(T-T’)+R(t-s+e-T)R(t-s+e’-7’) +R(f-s+e-T’)z+-s+ef-7). Therefore, 0 EIh~‘-6,j12=& //// 0 -b ~i(e)~j(e')~;(T)1Cj(T') +R(t -s + 8 - T’)R(t - s + f?‘- T)dtds] d@de’dTdT’. = T-‘“R(t+e-T)R(t+e’-Tt) df 342 v~hme 5, Number 5 SYSTEMS & CONTROL LETTERS April 1985 A similar inequality holds for the other term and the results follows under the assumption that l:,IR(r)l*dr < co. •I Lemma 2. Under the same assumption as in Lemma 1, Proof. From (3.7), we get 1 TT // T- to tu to R(t-s+b-T)R(t-s+&-T')dfds< Irn iNr)i2dt -CO and we have, by the Schwartz inequality, On the other hand, =- $j(h+?(8-T)#j(T)d8dT since (#j, lG;.)R = 1, and the lemma follows. •I We are now in a position to prove our main result: Theorem. Under the assumption that n:/T + 0 as T + 00, II&!‘T’ - (‘T’ll + 0 in probability a~ T * CO. Proof. By jlA4jl we denote the operator norm of a matrix M; that is, IlMll = sup{ IIMxll, llxll< 1). Fix a number d, 0 c d-z 1. Let 343 Volume 5, Number S SYSTEMS 6: CONTROL LETTERS With & denoting the complement of A,, it follows from Lemma 1 P(&) f ECiCj( hj- C)’ t+ c d* d d*(T- t,) < E/2 for T> T,(E), for any E > 0 arbitrarily small. On the set A,, [(r)-[(r)= [F(T)]-‘/f(r). Since C~:,C~;,(l;:‘)- Sij)* < 1 on A,, it (see [4]) that April 1985 that F(r) is invertible and from (3.6) follows from results in matrix theory Therefore, for any 8 > 0, p({lli(T’-5’T’II>S}nAr)QP(IIh’T)II>G~) for T> T*(E), by Lemma 2. It follows then thar for T> max(T,( e), T2(e)), P(II~(~)-~(~)II>~)=P({II~(~)-~(~)~~>~}~~~)+~({I~I(~)-~(~)II~S}~~) dP({lli’T)-S(r)II>S}nAT)+P(~~)<~, proving the desired result. Corollary 1. If ny + 03 and rig/T --, 0 as T + 00, then ~imm~T[~~T~(e)-a(e)]R(e-e~)[B”‘(e’)-u(e’)]dede’=O + 0 in probability. Proof. We write Ilf 11; for (f, f)R and II f 11’ for (f, f ). The expression above can be written as lim Il~(r)(e)-~(e)ll~=~~~~l~~(r~(e)-~(T)(e)~~~+_lim~J(~(T)(e)-~(e)~~2, T+CO G lim Il~(T)-~(T)I12+,,li_mmIIa(r)(e)-,(e)l12. T-rCU T Since U ,, b ,y, is dense in L2[ - b,O], the result follows from the preceding theorem. 0 Corollary 2. Suppose that the operator 93 on L2[ - b,O] defined by 9f =g, g(e)=j_ObR(e-eqf(eydet, -bgeGo, is coercive, that is (f, f)2 Kllfll* f or some K > 0. Then, under the conditions of Corollary 1, +Iiim~Tp(T)(e)-u(e)J2de=o cl in probability. Proof. Obvious. 0 344 Volume 5, Number 5 SYSTEMS & CONTROL LETTERS April 1985 4. Conclusion Corollary 1 gives the natural consistency result for our problem. However, it is not totally satisfactory, since the function R involved in the inner product contains the unknown parameter a(-). What we ideally want is the result given in Corollary 2. It is, therefore, of interest to determine a sufficient condition ensuring coercivity of the operator 9. It is easy to extend the results to the vector case and also to the more general hierarchical model, where both point and distributed delays occur. References [l] M. Polis, The distributed system parameter identification problem: a survey of recent results, Proceedings of the 3rd IFAC Syn~posiwn on Control o/Distributed Parameter Sysfems (Preprints), Toulouse, France (1982). [2] Ju. A. Rozanov, In/i,lite Dimensional Gaassiaa Disfributions (American Mathematical Society, Providence, RI, 1971) (translation of the Russian original). [3] V. Grenander, Abstract Inference (John Wiley, New York, 1981). [4] H.T. Nguyen and T.D. Pham, Identification of nonstationary diffusion model by the method of sieves, SIAM J. Confrol Optim. 20 (1982) 603-611. [5] K. It6 and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Unio. 4 (1964) l-75. [6] R.S. Lipster and A.M. Shiryayev, Statistics o/Random Processes, I (Springer, New York, 1977). 345 

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