Mathematical Problems in Engineering

Volume 2015, Article ID 393572, 8 pages

http://dx.doi.org/10.1155/2015/393572

## Modeling and Identification of Discrete-Time Nonlinear Dynamic Cascade Systems with Input Hysteresis

Faculty of Electrical Engineering and Information Technology, Slovak Technical University, Ilkovicova 3, 812 19 Bratislava, Slovakia

Received 22 February 2015; Accepted 29 April 2015

Academic Editor: Hiroyuki Mino

Copyright © 2015 Jozef Vörös. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simple approach to modeling and identification of discrete-time nonlinear dynamic systems having an input hysteresis in cascade with a linear dynamic system is presented. A special form of Coleman-Hodgdon model for the hysteresis is considered, which is linear in parameters. For the cascade system parameter estimation, an iterative method with internal variable estimation is proposed. Simulation studies of cascade systems identification using special periodic inputs are included.

#### 1. Introduction

Hysteresis is a special type of multivalued nondifferentiable nonlinearity and is encountered in a variety of processes where memory effects are involved between the input and output variables. The hysteresis is a dynamic nonlinearity, because the current output depends also on the history of the input [1, 2]. It can be found in biology, optics, electronics, ferroelectricity, magnetism, mechanics, and so forth.

In many control applications, the presence of the hysteretic behavior in sensors and actuators causes a hard nonlinear relationship between inputs and outputs. This phenomenon occurs in all the smart material-based actuators such as piezoceramics, magnetostrictive, and shape memory alloys [3–5]. For example, the piezoelectric devices are widely applied to the positioning systems because of their high positioning accuracy, large driving forces, and rapid response capability [6]. However, the existence of hysteresis often limits their performance. The hysteresis in cascade with linear dynamic systems can lead to instability in closed-loop operations and complicates the task of controller design and analysis [7]. Therefore, modeling and identification of such nonlinear cascade systems is a significant problem that should be solved for the application of these devices.

It is of great importance to find the best models approximating the systems with hysteretic nonlinearities. To describe the behavior of hysteretic processes, several mathematical models have been suggested and a survey may be found in [8]. In addition, more approaches have been proposed for the identification and control of different types of hysteretic systems; see, for example, [9–13].

For a broad class of hysteretic systems, a first-order scalar time-domain differential equation can be used to describe the system behavior [14]. A relatively simple differential model of hysteresis, which is appropriate for the representation of rate independent hysteretic systems, is the so-called Coleman-Hodgdon model studied in [15–17]. This model is able to capture, in an analytical form, a range of shapes of hysteretic loops, which match the behavior of a wide class of hysteretic systems. Applications of this differential equation model based on more or less complex solutions of the differential equation were used, for example, in [6, 18–22].

In this paper, a new and simple approach to modeling and identification of discrete-time cascade systems with an input hysteresis followed by a linear dynamic system is presented. It means that the discussed system is a cascade of* nonlinear dynamic* and* linear dynamic* subsystems. A special form of the Coleman-Hodgdon model is considered for the hysteresis. This is based on using piecewise-linear “material functions,” introducing an appropriate internal variable, and rearranging the model equation. The resulting description of the whole cascade system is linear in parameters. For the cascade model parameter estimation, an iterative method with internal variable estimation is proposed. This enables performing the identification of cascade systems with input hysteresis on the basis of available input/output data. Simulation studies of cascade systems identification using special periodic inputs are included to demonstrate the feasibility of proposed approach. To the author’s knowledge, no work dealing with this problem was published up to now.

#### 2. Coleman-Hodgdon Hysteresis Model

The differential model of hysteresis (the so-called Duham model) is a representation of a rate-independent dynamic effect in the form of the first-order nonlinear differential equation in the time domainwhere is the output and is the input. Both and are real-valued functions of time with piecewise continuous derivatives and . In the following, the Coleman-Hodgdon model of hysteresis, based on function that is affine in and function that is constant in will be considered. Then the hysteresis model can be written aswith being constant. It is assumed that(i)the real-valued function is odd, is monotonely increasing, and is piecewise continuously differentiable with a finite limit for its first-order derivative at positive infinity;(ii)the real-valued function is even and piecewise continuous and at infinity of such a finite value that(iii)the real-valued functions and are such that

The solutions of differential equation (2) move on the curves given by and this can be transformed into the following discrete form by replacing the differentials with the corresponding differences and rearranging as follows:where is the discrete time. This description can be used for a broad class of hysteretic systems by an appropriate choice of the “material” functions and shaping the hysteresis. In [15], the following forms of piecewise-linear functions and were proposed (see Figure 1):where , are the slopes of and the constant determines the range of the central segment of as well as that of with the constant values . Evidently, the functions and agree with conditions (i)–(iii) and the major hysteresis loop and some minor loops generated by hysteresis model (6) based on these functions using triangular inputs with different amplitudes are shown in Figure 2. Note that the parts of the graph connecting the loops are not plotted to improve the visibility of loops.