Shock and Vibration

Volume 2017, Article ID 1427270, 18 pages

https://doi.org/10.1155/2017/1427270

## An Observer-Based Controller with a LMI-Based Filter against Wind-Induced Motion for High-Rise Buildings

^{1}School of Civil and Environment Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China^{2}Department of Civil and Environmental Engineering, University of Surrey, Guildford GU2 7XH, UK

Correspondence should be addressed to Zuo-Hua Li; nc.ude.tih@auhouzil

Received 23 December 2016; Revised 21 March 2017; Accepted 11 April 2017; Published 25 May 2017

Academic Editor: Jiming Xie

Copyright © 2017 Chao-Jun Chen 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

Active mass damper (AMD) control system is proposed for high-rise buildings to resist a strong wind. However, negative influence of noise in sensors impedes the application of AMD systems in practice. To reduce the adverse influence of noise on AMD systems, a Kalman filter and a linear matrix inequality- (LMI-) based filter are designed. Firstly, a ten-year return period fluctuating wind load is simulated by mixed autoregressive-moving average (MARMA) method, and its reliability is tested by wind speed power spectrum and correlation analysis. Secondly, a designed state observer with different filters uses wind-induced acceleration responses of a high-rise building as the feedback signal that includes noise to calculate control force in this paper. Finally, these methods are applied to a numerical example of a high-rise building and an experiment of a single span four-storey steel frame. Both numerical and experimental results are presented to verify that both Kalman filter and LMI-based filter can effectively suppress noise, but only the latter can guarantee the stability of AMD parameters.

#### 1. Introduction

Active mass damper (AMD) is used to control the dynamic response of highly flexible buildings horizontally under environmental loadings such as strong wind [1–5]. Generally, a vector composition of displacement and velocity in the horizontal direction is used as a feedback signal for AMD control system [6, 7], but the whole displacements and velocities of each floor are too difficult to be measured directly. Therefore, a state observer design method is of great importance to the implementation of AMD control system in high-rise structures. The references showed the state observers can solve the problem for linear uncertain systems [8–10] and nonlinear systems [11–15]. Compared with displacement and velocity, [16] shows that the acceleration signal is easier to be measured and control system based on acceleration feedback is more robust. Unfortunately, the problem in the design process of an observer is that accelerometers may lead to a large estimation error that is regarded as noise. Therefore, filters for noise have to be considered.

At present, such filter process is often based on Kalman filter. In [17], a Kalman filter technique was used to estimate effective signal to noise ratio (SNR) in wireless sensor network (WSN) systems. Based on a maximum-likelihood criterion, Kalman filter for discrete-time systems was presented in [18]. In addition, an optimization-based adaptive Kalman filtering method was proposed in [19]. Moreover, a hybrid Kalman filter was established to denoise fiber optic gyroscope (FOG) sensors signal for discrete-time system in [20]. By unscented Kalman filter (UKF), extended Kalman filter (EKF), or particle filter (PF), the interacting multiple sensor filter (IMSF) had been presented in [21]. Similarly, based on filter and particle filter (PF), mixture Kalman filter (MKF) was built for conditionally linear dynamic systems in unknown non-Gaussian noises by [22]. A robust cubature Kalman filter (CKF) was designed for multisensors discrete-time systems with uncertain noise variances in [23]. Generally, Kalman filter, considering the disturbance as the observation input, can be used to estimate the system state by output data and is often applied in linear, discrete-time and finite dimensional systems [24–27]. Normal Kalman filter cannot consider input excitation during state estimation. The state derivative of a general AMD control system includes the velocity and acceleration responses, which are closely related to the external excitation. As a result, it leads to a large estimation error when neglecting the influence of external excitation. Furthermore, since the Kalman filter is strongly dependent on the statistical properties of noise and the selected Kalman filter gain is not a global optimal solution, the problem of control forces and strokes that are oversized output in an AMD system with Kalman filter should be considered. Therefore, a new real-time filter with optimal Kalman filter gain that considers external excitation can be designed for high-rise buildings based on linear matrix inequality (LMI) approach [28].

In this paper, a state observer design method based on structural acceleration is proposed for high-rise buildings under strong wind firstly. For comparative analysis, a Kalman filter and a LMI-based filter that consider input excitation are presented to reduce the adverse influence of noise on AMD control systems. Specifically, based on variable substitution method [29, 30], the design problem of the LMI-based filter can be transformed into a group of nonlinear matrix inequalities, which can be turned into a group of convex and easily solved linear matrix inequalities. Finally, a numerical example of a high-rise building and an experiment of a single span four-storey steel frame are presented to verify the efficiency of the proposed filters. The result shows that only the control system with a LMI-based filter can guarantee the stability of the AMD parameters and effectively filter out noise.

#### 2. An Observer-Based Controller with a Filter and Numerical Verification

##### 2.1. An Observer-Based Controller Design

The force equilibrium of a multi-degree-of-freedom (MDOF) system iswhere , , and are the mass, damping, and stiffness matrix of the system, respectively. is the control force. and are the location matrices of control force and strong wind, respectively. And , , and are the acceleration, velocity, and displacement of the system, respectively.

System state includes displacement and velocity. Then, (1) can be expressed into the state-space equation aswhere and are the control force and the input excitation, respectively. , , and are the state matrix, the excitation matrix, and the control matrix, and , , and are the state output matrix and the direct transmission matrix of excitation and control force, which can be expressed as

The control force of the system is

Substituting (4) into (2) leads towhere , , , and . A brief form of (5) is

The second equation of (6) can be written in the form of a partitioned matrix.where is a vector of displacement and velocity of the structure and its AMD and is a vector of acceleration, respectively. According to (7), the external excitation vector can be written as

Substituting (8) into (6) and (7) leads towhere , , , and .

Equation (9) can be written as

The state observer is

Substituting the second equation of (11) into the first equation leads towhere is the feedback gain of the observer. and can be used to estimate the estimated states of the structure and its AMD. is then used to calculate the control force.

##### 2.2. The Simulation of Wind-Induced Motions of a High-Rise Building

In this paper, a high-rise building called KingKey Financial Center (KK100) shown in Figure 1(a) has a height of 441.8 m, and its slenderness ratio is 10.2. Its structural periods and frequencies are listed in Table 1. Moreover, the lumped mass method is used for establishing the mass matrix of KK100 whose total mass is 5.79 × 10^{5} tons. Its stiffness matrix that has taken into account structural flexural and shear deformations is built based on unit-displacement method, and its structural damping ratio is 0.015. The first four natural mode shapes of KK100 along the minor-axis are given in Figure 2.