Identification of linear time-invariant system using exponential curve fitting

Abstract

RCD networks, respectively. If the exciting voltages are made the same Vn = en (6) as well as the initial conditions, then the (1) and (5) are the same and have the same solution. Thus 4, im (7) and hence the currents j m in the derived RCD network are related to the currents i, in the Irc network by the relation j m = T 7 . aim This relation is valid for the forced as well as for the transient part of the solution. AS for the transient solution im has such terms as E S P ' (for poles sp of the first degree lying in the negative half of the s-plane, or on the f w axis, Le., with Re [sp] < 0), and as tmesPt, m = O,l,. . . , M -1 (for poles rn of the Mth degree lying in the negative half of the s-plane, Le., G t h Re [sp] < 0). The time derivative of all such terms, which will be terms of j,, are all ultimately nongrowing with time. it follows that the transient of the RCD derived network is ultimately nongrowing in spite of the second degree pole at the origin or at infiity. IV. REALIZABILITY CONDITIONS There are several ways of stating the necessary and sufficient conditions of a function P(s) to be the driving point immittance of a passive kc network. We shaIl take the following set of conditions: 1) P(s) is a real function of s, Le., P(s) = real for s = real; 2) P(s) is a rational function of s, Le., the ratio of two f h t e poly-3) PO) shall possess no poles in the right half of the s-plane; 4) Poles of P(s) on the jw axis, if any, should be simple, Le., first de-5 ) Re [P(jw)] 3 0 for -2 w 3 0. The necessilIy and sufficient conditions for a function Q(s) to be driving point admittance of an RCD network, or to be the driving point impedance of an FLR network, can be simply stated that (l/s)Q(s) should satisfy the above mentioned conditions Alternatively, the conditions can be stated directly on Q(s) and are as follows: nomials in s (for finte lumped networks); gree and should possess positive real residues; 1) Q(s) should be a real function of I, Le., Q(s) = real for s = d; 2) Q(s) should be a rational function of s, Le., the ratio of two finite 3) Q(s) shall pospess no poles in the right half of the s-plane. 4) Poles of Q(s) on the positive jw axis (excluding 0 and -) should be simple, i.e., fisst degree and have positive imaginary residues; 5 ) At the origin Q(s) may be regular or have a zero of the fmt or s e e ond degree; at infinity -Q(s) may be regular or have a pole of the first or second degree; polynomials; 6 ) Im [Q(jw)] 2 0 for > w b 0. REFERENCES [ 1 ] L. T. Bruton, "Frequency selectivity using positive impedance converter type networks," New applications of the generalised converter in element replacement.' to appear in Rqp. ZEEE. (71 -A new generalised inverter and its use in element nplace-(81 K. Panzer, "Aktive ,,Bandfilter minimaler Kondemtoren mit mer/t," to appear in LYE J. Electron. -it ~y s r . Impedanzkonvertern, Nachrichtertech. Z., vol. 27, pp. 379-382, Oct. 1974. [ 9 ] K. Panzer, "Active band-pass filters with a minimum number of capacitors using FDNR's and a gyrator," in h o c . ZEEE ...