IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. IE-33, NO. 3, AUGUST 1986 Identification of Continuous-Time Systems Using Instrumental Variables with Application to an Industrial Robot SARAT C. PUTHENPURA AND NARESH K. SINHA Abstract-A new method of constructing instrumental variables for identification is introduced. Its usefulness in the identification of continuous-time systems is investigated. The technique is then applied for modeling the arm of an industrial robot used for welding purposes. Results showed that the proposed method of using instrumental variables is computationally simple and at the same time gives better performance in the presence of measurement noise as compared to existing methods. I. INTRODUCTION IDENTIFICATION of continuous-time models of systems is an important step in implementing control laws based on continuous-time state space concepts [1]. Digital computers are widely used for identification of system parameters from samples of input-output data. Basically, there are two ap- proaches for obtaining a continuous-time model of a process. One is the so-called "indirect method" which is performed in two parts: (1) the estimation of the parameter of a discrete- time model of the system, and (2) the transformation of the discrete-time model to an equivalent continuous-time model [2], [3]. The other is the "direct method" which provides continuous-time models of systems directly. This is done by interpolating between samples using block pulse functions [4], trapezoidal pulse functions [5], etc. This paper discusses the application of trapezoidal pulse functions (T.P.F.) along with the instrumental variable method for the identification of continuous-time models of systems from the samples of input and output data, where the measurements are contaminated with noise. A new computationally simple method of con- structing instrumental variables is introduced and its perform- ance is investigated. Simulated as well as real-life examples show that the proposed method gives better results in comparison with other existing methods when the data used for identification is noisy. II. TRAPEZOIDAL PULSE METHOD OF INTERPOLATION Consider a continuous-time function y(t), for which only the sampled values y(kT) (k = 0, 1, 2, * * *, Tis the sampling interval) are available. In the block-pulse function (B.P.F.) method, y(t) is approximated from y(kT) as a sequence of rectangular pulses of height yl(k) = 1/2 [y(kT) + y(kT + T)] and width Tover the interval kT c t c (k + 1)T. But in Manuscript received September 28, 1984; revised September 17, 1985. The authors are with the Department of Electrical and Computer Engineer- ing, McMaster University, Hamilton, Ont., Canada. IEEE Log Number 8607765. trapezoidal pulse function (T.P.F.), y(t) is approximated as varying linearly in between the sampling instants, i.e., 1 y(t) = - I [(k + 1) T- t] y(kT) + [t - kT] y(kT+ T)} T for (1) This is intuitively a better method compared to B.P.F. III. THE INSTRUMENTAL VARIABLE METHOD OF IDENTIFICATION The basic idea of the instrumental variable method is simple and straightforward. We have samples of input and output data u(kT) and y(kT), respectively, for k = 0, 1, 2, * . and T being the sampling interval. Normally, y(kT) is contaminated with noise, as a result we can write y(kT) =z(kT) + n(kT) k=O, 1, 2, ... (2) where z(kT) is the noise-free output of the system and n(kT) is the noise sequence, which is generally colored. The basic problem in using the instrumental variable (IV) technique is to construct the so-called "instrumental variables" {h(kT)} such that (1) {y(kT)}, {u(kT)}, and {h(kT)} are strongly correlated, and (2) {n(kT)} and {h(kT)} are not correlated. One commonly used method is to pass {u(kT)} through a deterministic system called "the auxiliary model" and use its output as {h(kT)}. Another computationally simple proce- dure is the use of {h(kT)} = {y(k - I)T}. It is to be noted that the latter method is not as effective as the first. Also, the success of the first method depends upon an intelligent guess of the parameters of the auxiliary model; the closer it is to the actual system the better the results [6], [7]. Hence, one can update the auxiliary model with the latest estimate of system parameters and iterate several times until the procedure converges to a set of parameter values. Naturally, the computational effort and time required by this method is quite substantial. This makes it somewhat unattractive for on-line applications. A simple, but very powerful method of constructing instrumental variables will now be proposed. Let {h(kT)} = {y(kT)} + {y(kT)} 0278-0046/86/0800-0224$01 .00 1986 IEEE (3) 224 kT< t < (k + 1) T.