# Parameter Optimization for an Accurate Swept-Sine Identification Procedure of Nonlinear Systems
## 1 Introduction  
A number of different modeling techniques that are capable of representing the nonlinear behavior of physical devices have been proposed and, for each of these, there are sophisticated techniques for identifying the parameters that characterize the modeling technique. Modeling techniques of nonlinear devices can be grouped into three main categories that differ in the level of knowledge of the physical phenomenon that is represented in the model itself. The so-called white-box approaches require complete knowledge of the physics governing the nonlinear system [1]. Examples of these techniques are those that define the model of the real physical system by making use of differential equations [2] or wave digital filters [3, 4]. The complexity of a physical system often makes it necessary to carry out simplification techniques. Depending on the degree of simplification, the techniques are considered tobe part of the grey box or black box approaches. The grey box techniques imply partial knowledge of the physical phenomenon [5; 6], for example, in [7; 8]; the black box techniques, which in fact are among the most widely adopted, do not require prior knowledge of the physics of the system and the device is defined through its input-output relation [1; 9]. In the following, only nonlinear systems of the black box type are considered. In the case of black box techniques, the most widely adopted model is the Volterra series [10; 11]. For practical reasons, the series must be truncated to bring the number of model parameters to a finite number. Even in the case of a truncated Volterra series, the number of coefficients needed to define the model quickly becomes very large as the degree of the model increases. This becomes a serious limitation in that, in practice, it allows this technique to be used only with systems characterized by a limited degree of nonlinearity. Simplified models with respect to the Volterra series have been studied to deal with cases of strong nonlinearities. Among them, the Hammerstein and Wiener models are based on a split between a part of the model representing the dynamics of the response through linear filters, and a part of the model representing the nonlinear part of the response, which is considered static. In this way, these models provide accurate, though less general, representations of nonlinear systems even in the case of high-degree nonlinearities [12; 13]. There are numerous techniques in the technical literature for identifying such models [14; 15], and in particular, in the case of the Hammerstein model, a technique that has proven to be particularly effective is based on the use of appropriate swept-sine signals as input [16; 17; 18; 19]. This technique has yielded excellent results in numerous application areas, including acoustics and nondestructive testing and evaluation [20; 21; 22; 23]. The excellent results achieved with this identification technique are, however, always related to integral analyses performed in the frequency domain. The results of this identification method appear to be less brillant with regard to the time behavior of the model response, or equivalently of the impulse response of the filters present in the different branches associated with the different orders of the model. Anomalies in the time response are often found, among which the most frequent is related to oscillations at transitions. Artifacts of this type are known in the technical literature as Gibbs artifacts, and they manifest themselves in the form of spurious oscillations in the time domain response. It is well known that the cause of the occurrence of such spurious oscillations comes from bandwidth limitations of the performed measurements [24; 25]. In the present paper, the focus is on the optimization of the [PERSON]'s model identification procedure for the purpose of the optimization of its results as seen in the time domain.