【正文】 l if a conventional model structure, ., ARX (autoregressive with exogenou s input) is adopted. In the steadystate, an ARX model approximates the nonlinear mapping in Fig. 3 by a straight line passing through the origin, by virtue of its linearity. A better approach would be to shift the origin to a point around which the input–output mapping is approximately symmetric. The new origin bees 60% duty cycle and 0 bar pressure output. Such a shift of the origin amounts to subtracting the input duty cycle by 60%. The relatively high signaltonoise ratio leads to the choice of the ARX model structure as a model set, due primarily to its simplicity:where uoffset is the aforementioned offset value for input data and na=nb are the orders of the model, which are determined later. Then, the step response of the hydraulic actuator is obtained in order to help to design the identification input. The natural frequency (in an approximate sense) turns out to be around 5 Hz. A safety factor of ‘2’results in the excitation frequency band of 10 Hz. 50 sinusoids with various binations of amplitudes and phases are generated within the 10Hz band. The length of data is chosen to be 1024 considering the frequency content of the model to be identified, and the sample rate of 100 Hz is used, which corresponds to the sampling rate of TCU in mercial vehicles. MATLAB System Identification Toolbox (Ljung, 1995) is used to process the data and to obtain an ARX model.The orders of the ARX model are initially set to be equal to those of the nonlinear model developed in Hahn (1999): na=nb= resulting tenthorder model has mean square prediction residual of , which appears to be sufficiently small for observer design application considering that the pressure of the hydraulic actuator usually fluctuates about bar (refer to Fig. 5). However, the relatively high order of the ARX model does not render itself amenable to efficient design of a nonlinear observer. In order to reduce the model order without promising the fidelity to a great extent, the identified ARX model is transformed into a balanced realization, where most controllable and observable states may be determined for model reduction (Skogestard amp。 Tanaka, 1996。 Vehicle power transmission control system。 Hydraulic actuator。 Hibino, Osawa, Yamada, Kono, amp。 the equivalent rotational inertia of the vehicle varies as the number of passengers changes, and only a rough bound on the friction coefficient is available.3. Identification of a hydraulic actuator. Motivation for empirical modelingFig. 2 shows the hydraulic actuator considered in this paper. It basically consists of three elements: a PWM type solenoid valve, a pressure modulator valve and a pressure control valve. The first and second regulating valves, not shown in , regulates the main pressure of the entire hydraulic circuit. The pressure modulator valve further decreases the output pressure of the first regulating valve (channel 7) to a lower pressure level.The output pressure of the pressure modulator valve at channel 9 is always regulated around bar in the steadystate by means of the feedback chamber 10 (Hahn, 1999). The voltage signal from the transmission control unit (TCU) drives the PWMtype solenoid valve so that the pressure at channel 11 assumes values between 0 bar and 4 bar. The pressure at channel 11 acts on the spool of the pressure control valve, which in turn generates engage/disengage pressures for the friction element in the mechanical subsystem. Since the engage/disengage pressures are applied to the same friction element from the opposite sides, the difference between engage and disengage pressures may be regarded as the output of the hydraulic actuator. A nonlinear mathematical model of the hydraulic actuator has been obtained in Hahn (1999) using the Newton’s second law of motion. Although the nonlinear model in Hahn (1999) matches the experimental results to a certain extent, it has some drawbacks when applied to the observer design problem considered in this paper:1. The model order is too high (E10).2. The governing differential equations are too stiff to numerically solve in realtime.3. There exist numerous unknown parameters that need to be estimated or tuned in order to obtain reasonable match between experimental results and model predictions.A possible alternative is to capture the dynamics essential to the design of a nonlinear observer and to obtain a lowerorder controloriented empirical model. In this paper, two empirical models are proposed based on the system identification (Ljung, 1999).1. Nonlinear.2. Mostly smooth, although its behavior is qualitatively different at a couple of points, . nearly discontinuous.3. Not onetoone when the duty cycle is around 60%. 4. Not even symmetric with respect to origin。 Zak (1987) and Zak (1990), where Q is positive definite as defined in (13) and e is a design parameter that specifies the desired accuracy of the output observation error (Misawa amp。 for example, the introduction of a nonlinear ARX (NARX) model structure to acmodate the nonlinear steadystate and dynamic behavior.6. Concluding remarksIn this paper, an observerbased algorithm is presented to monitor a hydraulic actuator which enables pressure estimation in a vehicle power transmission actuator together with a physical model for the mechanical subsystem provide a foundation upon which a robust observer may be designed to guarantee desired performance against parametric variations and torque estimation errors. The performance of the designed observer is verified via hardwareintheloop simulation results, which shows satisfactory pressure estimation accuracy. The proposed algorithm has potential to be widely applicable to advanced vehicle power transmission control and fault diagnosis.ReferencesChen, C. T. (1984). Linear system theory and design. London: Saunders College Publishing.Franklin, G. F., Powell, J. D., amp