IEEE TRANSACTIONS ON INSTRUMENTATIONAND MEASUREMENT, VOL. 53, NO. 4, AUGUST 2004 1245 Linearization of Nonlinear Dynamic Systems Johan Schoukens, József G. Németh, Gerd Vandersteen, Rik Pintelon, and Philippe Crama Abstract—In this paper we propose a method to linearize a non- linear dynamic system: the nonlinear distortions are reduced, and the linear dynamics are corrected to a flat amplitude and linear phase in a user defined frequency band. Index Terms—Linearization, nonlinear system. I. INTRODUCTION A N IDEAL instrument should not disturb the measurements in the frequency band of interest. To reach that goal it should have: — a transfer function with a flat amplitude and linear phase; — a linear behavior. In practice, it is very hard to meet these requirements. A number of methods were presented to compensate for the non ideal dynamics. The transfer function is modeled and next a dig- ital filter with an inverse characteristic within a known delay is designed to correct the undesired behavior [1]–[3]. A sim- ilar idea can be applied to correct for the nonlinear distortions. Static nonlinear corrections are popular to increase the linearity of analog-to-digital (ADC) and digital-to-analog (DAC) conver- tors [4]–[7]. It is clear, that in general we have to compensate for the linear dynamics and the nonlinear distortions at the same time since often the nonlinear distortions are not static. A first possibility would be to build dedicated nonlinear dynamic models for each problem. Although this is the most powerful approach, it is very time consuming because the whole modeling process should be restarted for each new design. The alternative is to use a black box model to describe the de- vice-under-test. This offers the advantage that general purpose packages can be used to do the identification. Unfortunately, such packages exist at this moment only for linear system identification [8], [9]. Recently, a flexible nonlinear black box structure was proposed. Although this structure cannot describe the whole variety of nonlinear systems, it allows to approximate quite well many different types of systems, while the identification procedure is not much more complicated than the linear identification approach. This methodology will be Manuscript received June 15, 2003; revised March 22, 2004. This work was supported in part by the FWO-Vlaanderen, the Flemish community (Concerted Action ILiNos, bilateral agreement BWS2002), and in part by the Belgian gov- ernment (IUAP-V/22). J. Schoukens, R. Pintelon, and P. Crama are with the Department ELEC, Vrije Universiteit Brussel, 1050 Brussels, Belgium (e-mail: johan.schoukens@vub.ac.be). J. G. Németh is with the Department of Management Information Systems, Budapest University of Technology & Economics, H-1521 Budapest, Hungary. G. Vandersteen is with the IMEC, 3001 Leuven, Belgium. Digital Object Identifier 10.1109/TIM.2004.831123 Fig. 1. Basic setup. used in this paper. Within the frequency band of interest, the nonlinear structure will be identified using the output of the system as the input of the model, while the (delayed) input will be used as the output for the model. Outside the frequency band of interest, the gain of the compensating model should be controlled. Gains that are too high would increase significantly the out of band noise. The paper is organized as follows: First, the setup is de- scribed. Next, the flexible nonlinear structure is introduced. This model will be used to equalize the device-under-test, keeping the out-of-band behavior under control. Finally, the method will be tested on simulations and experiments. II. PROBLEM SETUP The basic setup is given in Fig. 1. A dynamic nonlinear de- vice should be compensated starting from the sampled output (the sampling frequency is normal- ized to 1). In order to do so, a dynamic discrete time nonlinear system is designed such that within the fre- quency band of interest, the output is a delayed copy of the input (1) In practice, the choice of delay will be a part of the design problem. III. A FLEXIBLE BLACK BOX NONLINEAR STRUCTURE A. The Model The model structure that will be used for the compensating structure is given in Fig. 2. As it can be seen, it consists of a number of parallel branches, each taking care for a typical nonlinear behavior. It can also be noticed that only one linear model block is allowed. The major reason for this choice is the need for fast and simple identification methods. — The first branch (called linear branch) captures all linear dynamics. is a rational form in (continuous time model), or in (discrete time model), and a real number. — The second branch (called Hammerstein branch) is a Hammerstein system. is a static nonlinear system, and can be described using a simple poly- nomial model . It is also possible 0018-9456/04$20.00 © 2004 IEEE