352 S. Hayati and W. Song Chen [7], considered a different model for servo-hydraulic actuator dynamics, which is expressed in discrete-time domain as below TF.z/ D z ˛c:z .˛c 1/ (33.3) wherez is the z-transform variable and˛c is the ratio between the time that actuator needs to reach to command displacement and the controller time step. With the actuator dynamics given by Eq. (33.3), the corresponding compensator can be obtained by directly inverting it as [7]. GFF.z/ D 1 TF.z/ D ˛c:z .˛c 1/ z (33.4) Chen et al. [8], implemented this inverse compensation method in RTHS. Obtained results have shown successful tracking performance with command signal, which was an earthquake record, in presence of accurate value of ˛c. 33.2 Discrete-Time Compensator The aforementioned feedforward compensators have been successfully applied in RTHS to reducing servo-hydraulic time delay for input signals with a relatively low frequency bandwidth. To advance the use of RTHS further to other dynamic testing applications, compensators that can provide effective reduction in time delay for command inputs with relatively high frequency contents are needed. For instance, certain equipment or mechanical devices are tested with relatively high frequency excitations. In this paper, a newly developed discrete-time based compensator is presented to reduce actuator time delay when the input has a broader frequency range (0 30 Hz). This compensator is designed based on an Auto-Regressive with Exogenous model (ARX model) which can be written as below y.t/ Ca1y.t 1/ C Canay.t na/ Db1u.t/ C Cbnbu.t nb C1/ Ce.t/ (33.5) In Eq. (33.5), y is the output of the dynamic system; in this study, it is the measured displacement of the actuator, u is the input of the dynamic system; in this study, it is the command displacement signal sent into the controller, eis the disturbance, na is the number of system poles, and nb is the number of system zeros plus one. Transfer function of the servo-hydraulic actuator in z domain can be written using ARX model as TF z 1 D b1 C Cbnbz nbC1 1Ca1z 1 C Canaz na (33.6) If the transfer function of the actuator is minimum phase, the corresponding compensator (GFF) can be obtained directly by inverting Eq. (33.6) as GFF z 1 D 1 TF.z 1/ D 1Ca1z 1 C Canaz na b1 C Cbnbz nbC1 (33.7) However, most of the time, the transfer function of the actuator is nonminimum phase, which implies a direct stable inverse of the original servo-hydraulic actuator does not exist. To solve this problem, the unstable part of the compensator transfer function can be replaced by a Finite Impulse Filter (FIR). To demonstrate, Eq. (33.6) can be rewritten in the form of poles and zero as TF z 1 D BC z 1 B z 1 A.z 1/ (33.8) In this equation, BCandB are the terms in the numerator with stable and unstable zeros respectively andAcontains system poles. After inverting Eq. (33.8), the inverse of B will be replaced by a FIR.
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