Time-Domain Approximation by Iterative Methods

An iterative procedure is presented which permits the determination of a rational transfer function in the Laplace transform variable s which is optimal with respect to given input and output time-functions. The optimal system of a particular order is defined as the one whose output when subjected to the known input function is nearest in the time integral square sense to the desired output function. The method is thus applicable to a number of problems involving the minimization of an integral square error. To illustrate the technique, a set of optimal lumped-parameter delay lines is synthesized and their characteristics investigated; the behavior and convergence of the iteration in these problems is also studied. A comparison of other iterative methods applicable to the same problems leads to the conclusion that the proposed procedure has real advantages in computational simplicity and speed of convergence. T IME-DOMAIN approximation can be regarded as the problem of finding a network of given complexity which, when subjected to a particular input, will produce (at least approximately) a desired output as a function of time. If it is assumed that any rational function of given order can be synthesized as the transfer function of a network, the problem is reduced to finding the coeflicients of a ratio of polynomials in the transform variable, s, which best fits the required input-output relation. It is in this sense identical to the problem of identifying an unknown linear system from a sample of its input and output functions. Among the computational procedures proposed for linear system identification, Kalman’s [l] method for finding the coefficients of a rational z-transform function by linear regression is among the easiest to use. However, as a method for time-domain synthesis it has two disadvantages: it results in a sampled-data model which must then be approximated by a rational function of s; and it minimizes an artificial error criterion which does not correspond directly to any measurable error. It has been shown [2] that the second objection can be eliminated Manuscript received August 30, 1965i revised June 17, 1966. This work was partially supported by National Science Foundation Grant GK-270, and Army Research Office Contract DA-31-124 ARO-D292; it made use of computer facilities supported in part by Grant NSF GP-579. An abridged version of this paper was presented at the 1965 Third Allerton Conference on Circuit and System Theory, University of Illinois, Urbana. L. E. McBride, Jr., is with the Metals and Controls Division, Texas Instruments Incorporated, Attleboro, Mass. He was formerly with the Department of Electrical Engineering, Princeton University. K. Steiglitz is with the Department of Electrical Engineering, Princeton University, Princeton, N. J. H. W. Schaefgen is with the Aeronutronic Division, Philco Corporation, Newport Beach, Calif. He was formerly with the Department of Electrical Engineering, Princeton University. by using an iterative procedure, with the Kalman estimate as first iteration; if the iteration converges, as it does in many practical cases, the resulting transfer function minimizes the mean square error between desired and actual model output. It is the purpose of this paper to show that similar iterative procedures can be applied directly to continuous system identification and to give examples of their effectiveness in time-domain approximation. Let the system input be x(t) and the desired output function y(l), both defined for t > 0 [z(t) may be an ideal impulse, in which case y(t) is the desired network impulse response]. If v(t) is the actual response of the network to z(t), a natural error criterion is J = s,T [u(t) y(t)]’ dt = ~‘e2(t) dt. (1) The transfer function of the network to be synthesized has the form G(s) = a,+a,-,s+ ... +a& b, + b,-,s + 1 ‘. + sn a,s? + an-,sen+l + *. . + UO = bs-” + b,-ls-n+l + . . . + 1 =N(S>. D(s) w (In most problems a, is omitted because it is known that the network impulse response should not contain an impulse at t = 0; it is also possible to assume a numerator order different from that of the denominator, but this alternative will be ignored to simplify the notation.) Then if X(s) is the Laplace transform of s(t), etc., V(s) = gf$ X(s), E(s) = V(s) Y(s) (3)