Copyriglu © IF:\( : Idclllili( ; llioll alld Parallll'llT Estilll<lliOII York , l ' K. RECURSIVE-ITERA TIVE IV ESTIMATION OF " MULTIPLE-INPUT TRANSFER FUNCTION MODELS K. Thomaseth*, P. Young** and C. Cobelli*** *L4.DSEB·(;.\'R , Plu/m 'll , !ta/\ ' "''''D I'/)(II'IIIII' III of l: :IIl·iml/lllI'l/la/ Scil' l/o'. L'lIil'nsily ur Lall cw ler, Lw/w, lln, L' /\ " ';";' F(/I'/)/I(/ tli I IIgl'g ll l'I'i(/, f.llilulo di E/dlmll'o/ic(J I' tli 1: ' /l'lImllim, PIlI/IIl ' (/,llaly Abstract. The refined instrumental variable (RIV) approach to the identification and estimation of multiple input transfer function models has some deficiencies when applied to short time-series obtained from impulse-response-type experiments of the used in oiomedical and environmental science. This paper proposes a modifica- tion of the RIV method which corrects these deficiencies by a novel application of "the Kalman filter. The practical utility of the method is demonstrated through it s application to the analysis of time-series data obtained from experiments on the glucose insulin system in humans and dogs. Keywords. functions; Instrumental variable; recursive estimation; Kalman filter; biomedical. multiple inputs; transfer INTRODUCTION A(z'-l) 1 -1 -n + alz + .... + a z n D(z-l) 1 + d,z -1 + •••• + d z-p = P B (z -1) b bliz -1 b = oi + + .... + .z , m' -mi Tfie theoretical and practical difficulties asso- ciated with the identification of full multi- variable (multi-input. multi-output. MIMO) model structures have often encouraged approaches to multivariable system identification based on simpler multi-input. single output (MISO) methods. For analytica1 convenience. it is normally assumed that the transfer functions in the MISO model have id£ntica£ denominator characteristics. thus reduc- ing the model to the following form. Clearly. equation (1) can be written in the follow- ing alternative form. y(k) where 2 E Zi(k) + t;(k) i=l ( i ) (l) -1 u,.(k) (ii) and t;(k) = e, A(z-l) , In these equations. y(k) is the observed output time-series (k = 1 to N) ui(k) are the various known input time-series (i = 1. 2 ..... 2) t;(k) is the coloured noise affecting the output measurement and. e(k) is a white noise source. with the u sua l 2 properties of zero mean and finite var,ance 0 -1 -1 -1 . A(z ). D(z ) and Bi(z ). ,=1.2 ..... 2 are poly- nomials in the backward shift operator of the following form 1345 -1 A(z )y(k) 2 -1 1 E B,(z )u.(k) + D(z- leek) i=l' , which will be recognised as the multi -input version of the well known auto-regressive. moving average. exogenous variables (ARMAX) model. first considered in the control literature by Astrom and Bohlin ( 1966). In practical terms. the multi-input model (1) has obvious disadvantages: it seems more natural to consider a model in which the inputs affect the output via pathways which can have independent transfer functions. as in the following model y(k) where now 2 E x(k) + t; (k) i=l ' B (z -1) x,(k) = -'-- u. (k) A. (z -1) , , -1 and t;(k) = e k C(z-l) (i) (i ;) (iii) -1 . Here. Ai(z ). ,=1.2 ..... 2. are the denominator polynomials for the various transfer functions. which are assumed to take the following f orm (2)