Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 5361702, 18 pages

https://doi.org/10.1155/2017/5361702

## Modeling and Control of Active-Passive Vibration Isolation for Floating Raft System

School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China

Correspondence should be addressed to Beibei Yang

Received 21 November 2016; Revised 16 January 2017; Accepted 23 January 2017; Published 21 February 2017

Academic Editor: Orest V. Iftime

Copyright © 2017 Beibei Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a new approach for constructing a mathematical model of the floating raft system directly from input-output measurements in the presence of noise. In contrast to the original OKID/ERA algorithm, which works through the observer Markov parameters, the new approach used observer output residuals to convert the initial stochastic identification to a virtually deterministic identification problem. The extension of deterministic algorithm to stochastic problems by proposed stochastic-to-deterministic conversion can be done with ease. A MIMO (multiple-input multiple-output) system state-space model and an associated Kalman filter gain can be identified. controller with high robustness to model error is designed to solve multifrequency varying vibration for floating raft system. Both simulated and experimental results confirm the validity and the benefits of the approach.

#### 1. Introduction

A floating raft system is a special double-layer isolation system aiming to reduce the level of noise and vibration and has been widely applied to many kinds of ships and submarines [1, 2]. It can isolate vibration of hosts and auxiliary machines and reduce the structural noise of ships and submarines effectively. It can also protect equipment and instruments in ships and submarines from being damaged and makes them operate properly when ships and submarines are subjected to external loads and sudden shocks [3]. However, in real applications, two and more machines are mounted on the floating raft system and their working frequencies are usually multifrequency and time varying [4]. The floating raft system includes vibrations of the flexible floating raft and of the rigid motors. Therefore, the floating raft system is a complex system generally subject to multifrequency varying vibration.

Theoretical modeling for floating raft systems can be summarized as multibody dynamics [5], impedance mode method [6], four-terminal parameter method [7], finite element method [8], statistical energy analysis [9], power flow method [10], and so on. The interactions between active actuators and active actuators on the floating raft system are ignored by these approaches. Meanwhile, the models are established on many assumptions that do not always hold.

In response to these obstacles, we take a system identification approach, constructing a mathematical model of the dynamic system directly from measured input-output data. The identified model can then be used to predict the response of the system to any excitation. In real applications, noise inevitably affects the measured data, making the identification problem stochastic. In the presence of noise, it would be desirable to identify from the experimental data not only the model of the system, but also the noise characteristics [11]. This presumes that the same sensors and actuators used in the identification tests will also be employed in the control system designed around the identified model. One of the most successful identification algorithms for linear state-space models is OKID/ERA (observer/Kalman filter identification) [12], which relies on an observer equation to compress the dynamics of the system and efficiently estimate its Markov parameters. The latter are then passed to the Eigensystem Realization Algorithm (ERA) [13] or some improved variants of it, for example, ERA with Data Correlation (ERA/DC) [14], to complete the identification process. The observer at the core of the method was proven to be the steady-state Kalman filter corresponding to the system to be identified and to the covariance of the process and measurement noise. A remarkable result of OKID/ERA is that the method provides simultaneously both the system matrices and the Kalman gain, extracting all the possible information present in the data.

OKID/ERA has been successfully applied for over twenty years, especially in the aerospace community (it was originally distributed by NASA for the identification of lightly damped structures), and it keeps receiving attention as researchers try to further improve it [15, 16] and apply it to linear time-varying problems [17] or even to nonlinear systems [18]. Additionally, OKID has been recently extended to the case where only output time histories are measured, leading to output-only observer/Kalman filter identification (O3KID) [19]. In this paper we show how ERA (or ERA/DC) is not the only method to complete the identification process. Thanks to a novel interpretation of the main OKID result, we prove that it is possible to use a Kalman filter to optimally transform a problem of identification from noisy data into a simpler, noise-free problem. As a result, we propose and demonstrate with examples several new OKID-based identification algorithms optimal in the presence of noise, which can be as many as the number of deterministic identification algorithms that one can find. We establish then the Kalman filter as the bridge from stochastic-to-deterministic system identification and OKID as a unified optimal approach to handle noisy data in system identification, paralleling the central role that the Kalman filter has in signal estimation.

In active noise control (ANC) for floating raft system [20], the filtered-x least mean square (FXLMS) algorithm [21] is the most popular adaptive feedforward algorithm to update the controller, as it enjoys good applicability for real-life applications in terms of both noise reduction performance and implementation cost. Some variants of the FXLMS algorithm [22–24] have been studied to reduce computational complexity or to improve convergence rate and the control of impulsive noise. As one of feedforward algorithms, FXLMS control algorithm has a common assumption that reference signal is stable periodic signal by default. However, in real applications, two and more machines are mounted on the floating raft system and their working frequencies are usually multifrequency and frequency varying.

In response to these obstacles, feedback control algorithms are mainly applied in structural vibration control and suppressing multifrequency varying vibration. LPV (Linear Parameter Varying) technology is used to update controller parameters according to the estimated frequency in real-time for suppressing the large range frequency varying vibration [25]. For the narrowband disturbance rejection problem of helicopter tail, LQR (Linear Quadratic Regulator) control algorithm and control algorithm can suppress frequency varying vibration in a small range, and experiment shows that control algorithm has the better performance than LQR control algorithm [26]. Adaptive LQ control algorithm with frequency estimator is applied to eliminate frequency varying disturbances of the diesel engine. Then, 3-input/3-output active vibration control experiments were conducted in the floating raft system [27]. The difference between identified model and system model can be described by using additive uncertainty and multiplicative uncertainty in control algorithm [28]. controller is based on high robustness, which can effectively solve the problem of overflow instability. Meanwhile, frequency response function of the closed-loop system can be reshaped by the design method based on the mixed sensitivity problem for control algorithm [29]. Thus, the multifrequency varying vibration problem can be effectively solved.

The paper is organized as follows. After rigorously formulating the stochastic system identification problem, the OKID core equation is derived, in a slightly different way with respect to [30] in order to better highlight the central role of the Kalman filter in system identification. Then the novel interpretation of the main OKID result is presented and it is shown how to convert the original stochastic problem into an equivalent deterministic form. The resulting new algorithms are outlined and their features are illustrated via a simple numerical example, which provides the ground to present the conceptual contribution of the work. An experimental example on a floating raft platform is given to show the method in action on a more realistic system. The MIMO system model can be identified and the order of identified model can be reduced by retaining pulse response weighting indices of every state element to the overall dynamics. controller can be designed by mixed sensitivity problem according to the reduced model identified by OKID. The identified model from input-output data and controller are demonstrated via both numerical and experimental examples on floating raft platform with varying frequency vibration source.

#### 2. Problem Description

Figure 1 shows that the analytical model of the floating raft system includes using springs and electromagnetic actuators. The overall system can be divided into several subsystems: vibration source ( and ), floating raft (), foundation (), and isolators (, and ). According to [31], the mathematical model of floating raft system with the dynamic equations of substructures can be expressed aswhere represents coordinate matrix to describe the motion of the floating raft system and is total mass matrix of floating raft system. is the modal stiffness matrix of the system. is the load matrix of system.where , , , , , , , , and are, respectively, translation displacements of mass center of , , and . , , , , , , , , and are, respectively, attitude coordinate system of , , and . In order to describe the elastic vibration of the floating raft, the former order mode is retained. , , and are, respectively, mass of subsystems , , and , , , , , , , , , and are, respectively, rotational inertia of objects , , and around the -, - and -axes. , , , , , , , , , , , , , , , , , and are, respectively, external disturbance forces of subsystems , , and .