PERTURBATION SIGNALS FOR SYSTEM IDENTIFICATION zyxwv GFED Generating piecewise-constant excitations with an arbitrary power spectrum zyxw WVU J. Schoukens P. Guillaume R. Pintelon zyxwvutsr FEDCBA Indexing t e r m : Arbitrary power spectrum, Continuous time linear dynamic systems, Periodic excitations zyxwvut Abstract: A comparison is made between zero- order-hold-noise (ZOH-noise) and ZOH-periodic excitations. Their ability to excite a system with maximum power and a specified power spectrum, while the peak value of the excitation is restricted, is studied. 1 lntroduction The identification of continuous time lineardynamic systems starting from a discrete set of measurements is important in a wide variety of problems. Because the dis- crete measurements do not contain all the information about the continuous time signals, additional assump- tions are required. Among the many existing possibilities the zero-order-hold (ZOH, the excitation signal remains constant between two successive samples, and the system has no poles above half the sampling frequency) and the band limited (BL, the measured signals have no energy above half the sampling frequency) assumptions are most popular. The first assumption leadsto discrete time models, while the second one results in a continuous time model [l, 21. Continuous time models are preferred if the physical interpetation of the model is important. zyxwvu EDCBA For control applications, the ZOH assumption is best suited to solving the problem because it leads to a discrete time model that allows the prediction of the system response at the sampling instances. A study of the design of broad- band excitation signals under the BL assumption has been made [3]. Although most of the conclusions from this study still remain valid under the ZOH assumption, some new aspects and possibilities appear due to the change from BL to ZOH. Under the BL assumption, binary sequences lose a lot of their advantages because their crest factor increases significantly through the band limiting filter operation, while under the ZOH assump- tion this advantage is completely preserved. In this paper, a new comparison between noise excitation (binary noise and clipped noise) and periodic excitations (discrete inter- val binary sequences [4, 51; ZOH multisines) is made, examining the possibility of creating signals with a speci- fied power spectrum and a low crest factor. It shouldbe notedthat a continuous time system excited with ZOH excitation can be exactly described by a discrete time model leading directly to the ‘classical’ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 0 IEE, 1995 Paper zyxwvutsrqponm 1845D (C8, C9), received 25th January 1995 The authors are with Vrile Universiteit Brussel, Department ELEC, Pleinlaan 2, 1050 Brussels, Belgium IEE Proc -Control Theory Appl, Vol 142, No 3, May 1995 discrete time identification methods [l]. Although a ZOII-excitation has an infinite bandwidth it turns out that all the properties of the discrete time identification techniques and their resulting estimates are described as a function of the spectrum of the underlying discrete time sequences from which the ZOH reconstruction is made. For this reason we will always consider the spectrum of the discrete time sequence in this paper. Because the spectrum of real discrete time sequences is periodic with period 277 (after normalisation of the sampling frequency to 1) it is sufficient to show the spectrum in the frequency band [0, zyxwv BA n]. 2 Design of a noise excitation with an arbitrary power spectrum Although in the authors’ opinion noise excitation suffers from many significant drawbacks [SI, it is still very popular. Its continuous power spectrum (as opposed to the discrete power spectrum of periodic excitations) is the major advantage of noise excitation. It guarantees the excitation of even very narrow resonance peaks while the risk of damaging the systemdue to high power concentrations at these frequencies is negligible. Two important drawbacks of noise excitation are the dificulty of controlling its power spectrum (for example, gener- ating a bandpass signal) while maintaining a low crest factor, and its stochastic behaviour (the properties of a singlerecord can be significantly different from the average behaviour). A number of methods have been pro- posed in the literature to solve the first problem. One possibility is to use random binary sequences switching between two levels + V and - V , with Poisson distrib- uted zero crossings (the telegraph signal). The bandwidth of this signal can be controlled by changing the param- eter of the Poisson distribution [SI. Because the switch- ing intervals of the telegraph signal can occur at arbitrary moments it is not very well suited for discrete time system identification. Tulleken [7] proposed a gener- alised discrete interval random binary sequence to over- come this problem. By selecting asymmetric switchingho-switching probabilities it is possible to This work is supported by the Belgian National Fund for Scientific Research, the Flemish govern- ment (GOA-IMMI), and the Belgian government as a part of the Belgian programme on Inter- university Poles of attraction (IUAPSO) initiated by the Belgian State,Prime Minister’s Office, Science Policy Programming. 241