224 S. Hayati and W. Song Fig. 27.1 Experimental setup in the large scale structural laboratory at the University of Alabama Fig. 27.2 Implementation of FIR compensator 27.2 FIR Compensator Hayati and Song [10] used an Auto-Regressive with Exogenous (ARX) model to represent the plant dynamic accurately. In this study, the plant was comprised of an actuator attached to a mass and a linear spring (see Fig. 27.1). Then, the inverse of the plant dynamic model was modified by using a Finite Impulse Response (FIR) filter to develop a new model-based compensator. Due to the usage of FIR filter, this compensator was named FIR compensator. Figure 27.2 depicts how the FIR compensator connects in series with the plant (Dactuator and the specimen attached to it). In this figure, r, uandyare reference signal, compensated signal and measured signal, respectively. Compared to the exciting compensators for RTHS, the FIR compensator can reduce the time delay more effectively when the actuator is subjected to inputs with a wider frequency bandwidth [0 30 Hz] input [10]. Implementing the FIR compensator in an RTHS test makes it a powerful technique to study the dynamic behavior of those systems that work under inputs with high-frequency bandwidth such as suspension system of a vehicle. Although the FIR compensator can reduce the time delay effectively, its performance can be deteriorated if the dynamic of the plant changes. Therefore, a feedback controller was designed to improve the robustness of the FIR compensator. 27.3 Feedforward-Feedback FIR Compensator In this extended abstract, the FIR compensator [10] is used as a feedforward controller for the actuator. To improve the robustness of the FIR compensator, a feedback controller is designed and combined with it. Then the performance of the combined system, feedforward and feedback controllers, was investigated through the numerical study. 27.3.1 Feedback Controller The feedback controller places the closed-loop poles of the plant at desirable locations such that the output, y, tracks the reference signal, r, accurately. In this study, the feedback controller was designed based on the modern control theory. In contrast to the classical control theory which utilizes the transfer function models in the frequency domain, the modern control theory uses the time domain state-space models [12]. For this research, the Linear Quadratic Regulator (LQR) control is used to determine the optimal feedback gain (K) by minimizing a cost function shown in Eq. (27.1) [13, 14] J D 1 2Z 1 0 xTQx CuTRu dt (27.1) Where, T is the transpose operator and Qand Rare weighting matrixes for states and control signal vectors, respectively. Then, the feedback gain is multiplied by the states of the deviation plant to obtain the feedback control signal as shown in Eq. (27.2). Here, the deviation plant is defined as a plant which its states are the error between the states of the real plant and an ideal plant (more information can be found in [13]).
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