## 1 Introduction  
A traditional discrete-time system is called a single-rate system, in which the input refreshing period equals the output sampling period [1,2]; In some complex nonlinear systems, the sampling rates of the output and the input are different due to the limitation of the measurement technology and method. The system, which has two different input-output operating frequencies, is called a dual-rate system [3,4]. Take the dual-rate warp yarn dyeing process with indigo dye for dyeing blue denim products as an example [5]; the measurement of leuco-indigo concentration in the process needs at least 15 min by the automatic redox titration method, while one can adjust/control the dosage of leuco-indigo more quickly (at any time) by feeding indigo solution with different concentrations into the indigo dyeing bath. In the area of dual-rate/multirate sampled system identification, [PERSON] proposed three gradient parameter estimation methods for dual-rate sampled systems [4]; [PERSON] _et al._ explored a hierarchical least squares method for dual-rate sampled systems [6]; [PERSON] _et al._ studied a hierarchical identification method for general dual-rate sampled systems [7]. By using T-S (Takagi and Sugeno) fuzzy models, [PERSON] _et al._ proposed a filtering method for multirate nonlinear sampled-data systems [8]. The piecewise-linear characteristic is often encountered in control systems, either alone or in cascade with linear dynamic systems to describe processes operating with different gains in different input intervals [9, 10, 11], such as in the nonlinear servomechanism [10], in the heating and cooling processes [11], _etc._ It is also well known that piecewise-linear functions can be used as a general tool to approximate nonlinear functions [10]. In previous applications, Hammerstein systems with a two-segment piecewise-linear nonlinearity have been studied in [12], and an extension to a discontinuous two-segment piecewise-linearity with preloads and dead-zones nonlinearity was discussed [12]. [PERSON] extended the key variable separation identification method for a Hammerstein system with a two-segment piecewise-linear nonlinearity [13] to a Hammerstein system with a multi-segment piecewise-linear characteristic [14] and with a time-varying backlash [15]. This paper deal with dual-rate Hammerstein systems with a two-segment piecewise-linear nonlinearity in cascade with a linear dynamic system. In the literature of single-rate nonlinear Hammerstein system identification, [PERSON] and [PERSON] studied a Newton iterative identification method for an input nonlinear finite impulse response system with moving average noise using the key variable separation technique [16]. [PERSON] explored a maximum likelihood estimation algorithm for Hammerstein CARARMA systems based on the Newton iteration [17]. [PERSON] and [PERSON] developed an auxiliary model-based recursive least squares algorithm for Hammerstein linear-in-parameters output error moving average systems [18]. [PERSON] _et al._ presented iterative algorithms to identify nonlinear MIMO (Multi-Input Multi-Output) Hammerstein systems with moving average noises [19]. Due to the structure of a nonlinear block plus a linear block, the identification model of a Hammerstein system naturally contains the products of the parameters of the nonlinear block and the linear block. Moreover, the traditional single-rate discrete-time model is not suitable for the dual-rate sampled-data of the Hammerstein system. These bring difficulties to directly using a standard least squares method. The intent of this paper is to study identification methods of Hammerstein nonlinear systems with dual-rate sampling period in input-output signals.