Choice of the sampling interval for the identification of continuous-time systems from samples of input/output data Prof. N.K. Sinha, B.Sc.(Eng.), Ph.D., C.Eng., F.I.E.E., P.Eng., and S. Puthenpura, B.Tech. Indexing terms: Control systems, Control theory, Mathematical techniques Abstract: A criterion is introduced for determining the optimum sampling interval for the identification of system parameters. It increases the accuracy and stability in identification procedures. This is done by employ- ing a bilinear transformation from the z-plane to the w-plane. Some results of simulations are also included. These reveal the usefulness of the suggested algorithm even in the presence of measurement noise. 1 Introduction Digital computers are often used for the identification of process models (both discrete and continuous) from samples of input and output data. The continuous model of systems can be obtained basically by two methods. The first one is the so-called indirect method, which can be subdivided into two parts: (i) obtaining the discrete-time model of the system from samples of observations [1], and (ii) determination of a continuous-time model correspond- ing to the discrete-time model obtained [2, 3]. The other approach is the direct method, which is based on obtain- ing piecewise-constant solutions of linear differential equa- tions in between sampling intervals. The most notable approaches are the Walsh function method, block pulse function method and the trapezoidal pulse function method [3-5]. The main advantage of the indirect method is the availability of considerable literature and relatively less computation compared to the direct method. Hence, often the indirect method is preferred due to its simplicity and ease of use. A careful analysis of the direct and indirect methods can be found in Reference 3, by which the above mentioned points can be substantiated. It is also noteworthy that the main disadvantage of the indirect method is that the input is not always maintained constant between sampling intervals. This will be discussed in sub- sequent Sections. It is very important to note that all these procedures of identification are based on the fact that the sampling inter- val has been selected 'properly'. A rule of thumb, which is commonly used, is that the sampling interval T should be chosen in such a way that [8] X m T<0.5 (1) where X m is the magnitude of the largest eigenvalue of the continuous-time mode. In practice, however, one does not know the value of X m a priori. All that can be done is to make an intelligent guess based on the expected-time con- stants of the process under consideration. It may appear that the problem can be easily solved by making the sam- pling interval too small. This is not always true and may cause two other severe problems. First of all we may have too many samples; as a result, the computational burden is increased. This is, obviously, highly undesirable for online identification. Secondly, the identification problem becomes ill conditioned as will be shown later with the help of examples, both in the direct and indirect methods. Paper 4203D (C8), received 6th March 1985 The authors are with the Department of Electrical & Computer Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7 In this paper, we propose a method to overcome this difficulty. First, we employ the simple bilinear transform- ation from the z-plane to the w-plane. A criterion is then developed for obtaining the computationally optimum sampling interval, on the basis of the location of the eigen- values in the w-plane, for an assumed value of the sam- pling interval. Results of simulations are included to illustrate that better accuracy in parameter estimation can be obtained, for sampling intervals chosen, based on the proposed criterion, even in the presence of moderate amounts of measurement noise. 2 Statement of the problem Consider the discrete-time model of a system described by + 1 + 1 where m < n. The equivalent continuous-time model can be written as H ® = -5- (3) where r <n. Alternatively, the corresponding state-space forms are x(kT + T) = Fx(kT) + Gu{kT) y(kT) = Cx(kT) ' (4a) and x = Ax + Bu y = Cx {4b) respectively. The identification procedure is to estimate the par- ameters of eqns. 2 or 3 from the samples of input and output data. After this, eqns. 4a and 4b can be easily obtained with the help of canonical forms. In the indirect method of identification of continuous- time systems, first we estimate the parameters of eqn. 2 by any standard method of identification [1], and utilise the following relationships to obtain ean. 4b and hence eqn. 3: F = exp (AT) and G -r Jo exp (/4t)B dt (5) IEE PROCEEDINGS, Vol. 132, Pt. D, No. 6, NOVEMBER 1985 263