【正文】 ion. The simpli?ed static model is validated again by experiments. Key words: Fluid mechanics。 Modeling。 Identi?cation。 Solenoid valve。 PWM。 Servo pneumatic。 Simulation1. Introduction Electropneumatic control valves are used as interfaces between electronic controls and ?uid ?ow. There are two types of electropneumatic valves to control the ?uid ?ow of a pneumatic actuator. These are the continuously acting servo/proportional valves and the on–off switching valves. Due to having proportional input–output behavior, pneumatic servo systems are usually realized by the continuously acting servo/proportional valves. But these valves have plex structures, are very costly and tend to be bulky pared to inexpensive, pact and lightweight switching ,servopneumatic systems utilizing fast switching valves have been developed . In order to use the discrete on–off switching valves instead of the continuously acting servo valves and to obtain similar proportional characteristics, pulse width modulation (PWM) techniques are used. Applying the PWM signal as the control input to a switching valve makes the valve on and off successively. As a result, ?uid is passed through a valve that is pletely open or pletely closed and is delivered to the actuator as discrete packets of pulse widths result in bigger packets of ?uid. If the time rate of delivery of these packets (PWM frequency) is considerably faster than the dynamics of the actuator and load, then the system ?lters the discreteness of the packets and responds to the average of the input signal, similar to the continuous case. PWMdriven fast switching valves are employed in several areas of industrial applications. A typical application of this kind for pneumatic systems is the realization of control functions in intelligent electropneumatic braking systems of motor vehicles . Many other pneumatic servo control applications in robotics and other areas have been pursued by researchers in order to reduce the cost, plexity and weight of the system by utilizing switching valves instead of servo valves. These techniques have been also successfully applied to hydraulic servo systems . The fully variable valve actuation in bustion engines can be mentioned as an application of PWMdriven hydraulic switching valves which improves the performance and ef?ciency of the engine. Dynamic characteristics of fast switching solenoid valves in?uence the successful operation of the entire ?uid power system. Therefore, it is of critical importance to elaborate a reliable dynamical model for these valves to be applied in design of the ?uid power hardware and its control. The model must accurately handle the three constituting subsystems of the valve, namely the electromagnetic, mechanical and ?uid subsystems. There are many works in which the modeling and control of pneumatic actuators are ,very few authors have treated the modeling of solenoid valves. Ye et al. investigated some static characteristics of a PWM solenoid valve and presented an equation for determining the maximum operating modulation et al. proposed a simple model for a 2/2 hydraulicsolenoid valve in which the model parameters are tuned to yield good agreement between the measured and the simulated variables. Yuksel et a new pneumatic solenoid valve and investigated its dynamic model to describe the switching characteristics of the investigations into the behavior of pneumatic fast switching valves can be found in [23,24]. In the above works,solenoid valves are studied from the viewpoint of valve design. However, the main objective of the present modeling is to emphasize on the applicability of the model in the controller design process for PWMdriven servopneumatic systems. In this Note, modeling and identi?cation of a PWMdriven pneumatic fast switching valve are presented. In the following, in Section 2, the internal structure of the valve is investigated and the electromagnetic, mechanical and ?uid subsystems are discussed including their interactions. In Section 3, governing equations of the valve are derived in the form of nonlinear state equations. In Section 4, identi?cation of the unknown parameters and validation of the model are performed by paring the simulated and measured currents of the solenoid. The nonlinear dynamical model obtained so far is a powerful simulation and putational tool。 however it should be simpli?ed in order to be effectively used in control design applications. This task is performed in Section 5 in which the moving average of the spool position is calculated and a simpli?ed static model is presented between the duty cycle of the PWM input and the averaged spool position. Finally, Section 6 contains some concluding results.2. Structure of the valve The solenoid valve under study is a fast switching spring return 3/2 valve. In Fig. 1, simpli?ed schematic views are shown to indicate the closed and open positions of the valve. In the absence of solenoid excitation, the valve is kept at its closed endposition by the return spring (Fig. 1(a)). When the solenoid is energized by DC voltage, the resultant magnetic force displaces the moving part against the return spring, valve opens (Fig. 1(b)) and a ?ow cross section develops through an ori?ce. A block diagram representation of the valve is given in Fig. 2 which indicates the signal ?ow of the model from the input voltage to the output ?ow, including the intermediate interactions. As shown in this ?gure, a solenoid valve can be deposed into three interacting subsystems。 the electromagnetic, mechanical and ?uid subsystems. The two ?rst subsystems, namely the solenoid with its electromagnetic equations, and the mechanical part consisting of a mass–spring–damper system, behave dynamically and have transient responses. In the next section, the nonlinear dynamic equations governing these subsystems will be derived in which the pressure forces are also included. The third subsystem,