Research Article  Open Access
Francisco PalaciosQuiñonero, Josep M. Rossell, Josep RubióMassegú, Hamid R. Karimi, "Structural Vibration Control for a Class of Connected Multistructure Mechanical Systems", Mathematical Problems in Engineering, vol. 2012, Article ID 942910, 23 pages, 2012. https://doi.org/10.1155/2012/942910
Structural Vibration Control for a Class of Connected Multistructure Mechanical Systems
Abstract
A mathematical model to compute the overall vibrational response of connected multistructure mechanical systems is presented. Using the proposed model, structural vibration control strategies for seismic protection of multibuilding systems can be efficiently designed. Particular attention is paid to the design of control configurations that combine passive interbuilding dampers with local feedback control systems implemented in the buildings. These hybrid activepassive control strategies possess the good properties of passive control systems and also have the highperformance characteristics of active control systems. Moreover, activepassive control configurations can be properly designed for multibuilding systems requiring different levels of seismic protection and are also remarkably robust against failures in the local feedback control systems. The application of the main ideas is illustrated by means of a threebuilding system, and numerical simulations are conducted to assess the performance of the proposed structural vibration control strategies.
1. Introduction
Over the last years, seismic protection of adjacent buildings has been attracting an increasing interest. For this kind of systems, the action of seismic excitations can produce interbuilding collisions (pounding), which can cause severe damage to the buildings structure and contents [1–5]. Consequently, structural vibration control (SVC) strategies for multibuilding systems must aim at mitigating not only the vibrational response of individual buildings, but also the negative interbuilding interactions.
The connected control method (CCM) is a SVC strategy for multibuilding systems that consists in linking adjacent buildings by coupling devices to provide appropriate reaction control forces. The application of the CCM using different types of passive [6–16], active [17–19], and semiactive [20–23] linking devices has been extensively investigated with positive results. Recently, more complex control configurations combining passive interbuilding dampers with local feedback control systems implemented in the buildings have been proposed [24, 25]. These activepassive SVC strategies combine the good properties of passive control systems and the highperformance characteristics of active control systems [26–28]. It should be highlighted, however, that most of the research effort undertaken to date has been directed at the twobuilding case, while more complex multibuilding problems still remain virtually unexplored. Obtaining a suitable formulation for the dynamical response of certain classes of connected multistructure mechanical systems is one of the major obstacles that has to be overcome in order to design SVC strategies for multibuilding systems. A preliminary work in this line presenting an activepassive SVC strategy for seismic protection of a threebuilding system can be found in [29].
The main contribution of the present paper is twofold: (i) a mathematical model to compute the overall vibrational response of connected multistructure mechanical systems is provided. (ii) Activepassive SVC strategies for seismic protection of multibuilding systems are designed using the proposed model and the CCM approach.
The paper is organized as follows: in Section 2, a general secondorder model for the unforced response of connected multistructure mechanical systems is provided. The forced response is also studied for some particular cases of special relevance in SVC. In Section 3, passive, active, and activepassive SVC strategies for seismic protection of multibuilding systems are discussed. The main ideas are presented by means of a threebuilding system. Finally, in Section 4, a set of numerical simulations is conducted to assess the effectiveness of the proposed control strategies.
2. Multistructure Connected System
In this section, we present a mathematical model to compute the dynamical response of the multistructure system schematically depicted in Figure 1. The overall system consists of parallel substructures . Each substructure is a massspringdamper system with degrees of freedom, and between adjacent substructures and , there is a linking system formed by a maximum number of springdamper elements. The aim of this section is to obtain a proper formulation of the secondorder equation that describes the overall motion of system in the form where is the global mass matrix; and are the total damping and stiffness matrices, respectively, including the internal stiffness and damping coefficients of the substructures as well as the stiffness and damping coefficients of the linking systems ; is the vector of external forces.
2.1. Unforced Response
Let us consider the th substructure displayed in Figure 2. The vector of relative displacements is where represents the relative displacement of the mass with respect to the fixed reference , which in this subsection is assumed to be an inertial frame.
A secondorder model for the substructure can be written in the form where denotes the vector of interstructure forces resulting from the interaction between adjacent substructures through the linking elements. The mass matrix is a diagonal matrix and the damping matrix has the following tridiagonal structure: The stiffness matrix has an analogous structure and can be obtained by replacing entries by in (2.5). We also define the damping and stiffness matrices of the linking system as follows:
The main difficulty in obtaining a simple formulation for the overall secondorder model (2.1) arises from the fact that adjacent substructures have, in general, different number of masses. This problem can be conveniently solved by extending the damping and stiffness matrices of the linking systems with a proper number of zero rows and columns. The benefits of this simple resource are twofold: (i) a plain and elegant matrix formulation of equation (2.1), and (ii) an extremely easy computational implementation. Next, we introduce the zeroextension of matrices and provide a simple Matlab function to compute it.
Definition 2.1. Given an matrix and two integers and , we define the zeroextension of as the matrix
obtained from by adding final zerorows and final zerocolumns.
The following Matlab function computes the matrix zeroextension:
Function M=zex(A,m1,n1)
[m,n]=size(A);
M=[A zeros(m,n1−n)
zeros(m1−m,n1)].
For the matrix
the zeroextension can be computed with the command zex(A,3,5), resulting
To obtain the expression for the vector of linking interstructure forces , we consider three different cases corresponding to the relative position of the substructure : (a) initial substructure , (b) interior substructure , , and (c) final substructure . For the initial substructure , from the force diagram in Figure 3, we have Equation (2.3) for takes now the form Note that, for simplicity, the explicit dependence on time has been omitted in (2.10), (2.11), and Figure 3, and notations like and have been used instead of and . The same will be done in the sequel when convenient.
Analogously, from the force diagram in Figure 4, it results and the secondorder model for can be written as
Finally, from Figure 5, we get and the corresponding secondorder model is
From (2.11), (2.13), and (2.15), we can now obtain an overall secondorder model for the unforced response of the multibuilding coupled system in the form where is the overall vector of displacements. To this end, we express the global damping and stiffness matrices in the form where matrices and correspond to the internal damping and stiffness of the substructures, respectively, and have the following block diagonal form: and matrices , have the form given in (2.5). The damping matrix corresponds to the linking systems and has the tridiagonal block structure shown in Figure 6, the stiffness matrix has the same structure as and can be obtained by replacing the entries by . Finally, the global mass matrix is the block diagonal matrix where , are the substructure mass matrices given in (2.4).
2.2. Forced Response
Now, we assume that some external excitations are acting upon the substructures . Specifically, we will turn out our attention to the particular case schematically depicted in Figure 7, where represents the acceleration of the reference frame , and the element is a force actuation device implemented between the adjacent masses and that produces a pair of opposite forces of magnitude as indicated in the figure. This case is particularly relevant for structural vibration control of seismically excited buildings, where the external acceleration corresponds to the seismic ground acceleration, and the actuation devices are interstory force actuators that implement suitable control forces to mitigate the vibrational response of the building.
A secondorder model for the vibrational response of the substructure can now be written in the form where the term is the vector of control forces acting on , and contains the inertial forces resulting from the fact that is now an accelerated reference frame. Denoting by the column vector with entries equal to 1, the vector of inertial forces can be written as For the vector of control actions, we consider the control location matrix of size and the vector of control actions to obtain Finally, considering (2.21), (2.22), (2.25), and the results presented in the previous subsection, we can derive a secondorder model for the overall vibrational response of the multistructure system in the following form: where is the overall displacement vector defined in (2.17); matrices , , are given in (2.18), (2.19), (2.20), and Figure 6; is the external acceleration, and is the external disturbance matrix; represents the overall vector of actuation forces and is the overall location control matrix defined as is the total number of degrees of freedom, and is the number of substructures. If no active control system has been implemented in the subsystem , can be taken as a zero vector and as a zero matrix of appropriate dimensions. The proposed model includes the action of external acceleration disturbance and active control systems implemented in the substructures and, moreover, is formally analogous to the usual formulation used in singlestructure SVC problems.
3. Structural Vibration Control Strategies for Multibuilding Systems
In this section, we are interested in designing SVC strategies for seismic protection of multibuilding systems. For clarity and simplicity, the main ideas are presented through the threestory building system schematically depicted in SubFigure 8(a), where the central fivestory building is assumed to require a special level of seismic protection. For this particular multibuilding system, four control configurations are considered: (a) activepassive, (b) passive, (c) uncoupledactive, and (d) uncontrolled. In the activepassive control configuration (see SubFigure 9(a)), an active local statefeedback control system with the actuation scheme presented in SubFigure 8(b) has been implemented in the central building. Moreover, two passive dampers have been placed as interbuilding linking elements: one at the thirdfloor level between buildings 1 and 2 and the other at the secondfloor level between building 2 and building 3. The passive control configuration (SubFigure 9(b)) only comprises the interbuilding passive dampers. In the uncoupledactive control configuration (SubFigure 9(c)), an active local feedback system has been implemented in the central building, but no passive interbuilding elements have been installed. Finally, no seismic protection is provided in the uncontrolled control configuration (SubFigure 9(d)), which will be used as a reference in the performance assessments.
(a) Connected multibuilding model
(b) Actuation scheme in
(a) Activepassive
(b) Passive
(c) Active
(d) Uncontrolled
The section has been structured in three parts. First, the results presented in Section 2 are applied to obtain a secondorder model for the threebuilding system. Next, suitable statespace models are derived. Finally, a statefeedback LQR controller is designed to drive the active local feedback control system implemented in building 2.
To compute the LQR local controller, the following particular values of the building parameters have been used: kg, Ns/m, N/m, N/m, N/m, for , , , , . The linking elements are considered as pure dampers with a damping constant = Ns/m and null stiffness; the value = indicates that no linking element exists at the th level between buildings and . The actuation elements , are assumed to be ideal force actuation devices, which are able to implement exactly the control actions producing the opposite pairs of control forces represented in Figure 8(b). These values will also be used in the numerical simulations conducted in Section 4.
3.1. SecondOrder Model
Let us consider the threestory building system displayed in Figure 8(a) as a lumpedmass planar system with displacements in the direction of the ground motion. In this case, the multibuilding system can be represented by the connected multistructure system shown in Figure 10. Using the results presented in the previous section, a secondorder model in the form to describe the buildings motion can be easily obtained. The overall vector of story displacements with respect to the ground is where represents the displacement of the th story in the th building. The mass matrix is with The total damping matrix can be written in the form where and the matrix corresponding to the linking elements has the following block tridiagonal structure: where denotes the zeroextension of , for example To obtain the total stiffness matrix matrices , can be computed replacing the damping coefficients , by the corresponding stiffness coefficients , in (3.7), (3.8), (3.10), and matrices , by , in (3.6), (3.9). For the activepassive control configuration depicted in Figure 9(a), the vector of control actions is and the control location matrix to produce the corresponding control forces can be written as follows: Finally, the disturbance input matrix is
3.2. FirstOrder StateSpace Model
Now, we take the state vector and derive the firstorder statespace model where the state, control, and disturbance input matrices are, respectively, Regarding the output, we consider two different cases: interstory drifts and interbuilding approaches. The interstory drifts represent the relative displacements between consecutive stories in the th building and are defined by where is the number of stories in building . The vector of interstory drifts can be obtained with the output matrix where The interbuilding approaches describe the approaching between the stories placed at the th level in the adjacent buildings , and are defined by where . The vector of interbuilding approaches can be computed with the output matrix Finally, let us suppose that the statefeedback controller has been computed to drive a local active control system in . We write the local vector of control actions as where matrices , are obtained by splitting the control matrix after the th column. The seismic response of the overall threebuilding system for different activepassive control configurations can be computed using the closedloop statespace model as follows: where the state matrix can be obtained using the matrices , , given in (3.18), and the overall control matrix with In particular, for the activepassive control configuration depicted in Figure 9(a), the overall control matrix has the form
3.3. Local StateFeedback Controller Design
To compute a local statefeedback LQR controller [30] for the actuation system in building , we consider the local secondorder model where is the vector of story displacements relative to the ground, is the vector of control actions, and matrices , , , have been given in the previous subsection. From (3.32), we obtain the firstorder statespace model with local state vector state matrix and control input matrix To obtain the local vector of interstory drifts we take the matrix given in (3.22) and define the local output matrix Next, we consider the weighting matrices and define the quadratic cost function to compute a local statefeedback LQR controller with the following control gain matrix:
4. Numerical Simulations
In this section, the vibrational response of the threebuilding system presented in Section 3 is computed for several control configurations. Specifically, the maximum absolute interstory drifts and maximum interbuilding approaches are computed for three control configurations: (a) activepassive, (b) passive, and (c) uncoupledactive, which are schematically depicted in Figures 9(a), 9(b), and 9(c). The vibrational response of the uncontrolled system (SubFigure 9(d)) is also computed, and it is used as a natural reference in the performance assessment. In all the cases, the fullscale NorthSouth El Centro 1940 seismic record obtained at the Imperial Valley Irrigation District substation in El Centro, CA, during the Imperial Valley earthquake of May 18, 1940, has been used as a ground acceleration input (see Figure 11).
The maximum absolute interstory drifts are displayed in Figure 12. Looking at the central graphic, the excellent behavior of the activepassive (black asterisks) and the uncoupledactive (blue circles) control configurations can be clearly appreciated. In fact, the data in Table 1 indicate that these active control configurations attain reductions of about 70% in the peak interstory drift values with respect to the uncontrolled response. For the lateral buildings, however, the situation is totally different. In this case, the activepassive control configuration produces a lower but still significant reduction of the interstory drifts, while no seismic protection is provided by the uncoupledactive configuration.

Regarding the interbuilding approaches, we can see in Figure 13 that interbuilding separations of about 7.5 cm would have resulted in interbuilding collisions for the uncontrolled configuration. In contrast, interbuilding separations of about 2.5 cm can be considered safe for the activepassive control configuration. An important reduction in the interbuilding approaches is also achieved by the uncoupledactive configuration, but the data in Table 2 indicate that the percentages of reduction obtained by this configuration are about 25 points inferior to those obtained by the activepassive control configuration.

To complete the comparison between the activepassive and the uncoupledactive configurations, the corresponding maximum absolute control efforts are presented in Table 3. The values in the table indicate that the activepassive configuration requires a slightly higher level of control effort. However, considering the superior performance exhibited by the activepassive configuration, the extra cost is certainly small.

The behavior of the passive control configuration is also remarkable. Despite its simplicity and null power consumption, percentages of reduction in the interstory drifts peak values of about 45% are achieved in buildings 2 and 3 and around 20% in building 1. Reductions of about 60% are also produced for the interbuilding approaches.
Finally, it should be highlighted the robustness of the activepassive control configuration against failures in the local active control system. Actually, in case of a full failure of the active control system, the passive level of seismic protection can still be guaranteed by the passiveactive control configuration. In contrast, the same kind of failure in the uncoupledactive configuration would produce a total loss of seismic protection.
5. Final Remarks and Conclusions
In this work, a mathematical model to compute the overall vibrational response of connected multistructure mechanical systems has been presented. Using the proposed model and following the connected control method approach, structural vibration control strategies for seismic protection of multibuilding systems can be efficiently designed. As a practical application of the new ideas, different control configurations for seismic protection of a particular threebuilding system have been designed. For these control configurations, numerical simulations of the threebuilding system vibrational response have been conducted using the fullscale NorthSouth 1940 seismic record as a seismic excitation. The simulation results come to confirm the excellent properties of control configurations that combine passive interbuilding dampers with local feedback control systems implemented in the buildings. These hybrid activepassive control strategies possess the good properties of passive control systems and also have the highperformance characteristics of active control systems. Moreover, activepassive control configurations can be properly designed for multibuilding systems that require different levels of seismic protection and are also remarkably robust against failures in the local feedback control systems. Finally, it is worth highlighting that the proposed activepassive control strategy is compatible with practically any control design methodology of the local feedback control systems and also with semiactive implementations of the actuation systems. Consequently, further research effort needs to be aimed at exploring more complex scenarios involving issues of practical interest such as wireless implementation of the communications systems [31], actuator saturation [32], actuation and sensor failures [33], structural information constraints [34, 35], uncertain stochastic networked systems [36–38], or limited frequency domain [39].
Acknowledgments
This work has been partially supported by the Spanish Ministry of Economy and Competitiveness through the Grant DPI201125567C02 and by the Norwegian Center of Offshore Wind Energy (NORCOWE) under Grant 193821/S60 from the Research Council of Norway (RCN).
References
 S. A. Anagnostopoulos, “Building pounding reexamined: how serious a problem is it?” In Eleventh World Conference of Earthquake Engineering, Elsevier Science, 1996. View at: Google Scholar
 K. T. Chau, X. X. Wei, X. Guo, and C. Y. Shen, “Experimental and theoretical simulations of seismic poundings between two adjacent structures,” Earthquake Engineering and Structural Dynamics, vol. 32, no. 4, pp. 537–554, 2003. View at: Publisher Site  Google Scholar
 P. Komodromos, P. C. Polycarpou, L. Papaloizou, and M. C. Phocas, “Response of seismically isolated buildings considering poundings,” Earthquake Engineering and Structural Dynamics, vol. 36, no. 12, pp. 1605–1622, 2007. View at: Publisher Site  Google Scholar
 D. LopezGarcia and T. T. Soong, “Assessment of the separation necessary to prevent seismic pounding between linear structural systems,” Probabilistic Engineering Mechanics, vol. 24, no. 2, pp. 210–223, 2009. View at: Publisher Site  Google Scholar
 P. C. Polycarpou and P. Komodromos, “Earthquakeinduced poundings of a seismically isolated building with adjacent structures,” Engineering Structures, vol. 32, no. 7, pp. 1937–1951, 2010. View at: Publisher Site  Google Scholar
 Y. L. Xu, Q. He, and J. M. Ko, “Dynamic response of damperconnected adjacent buildings under earthquake excitation,” Engineering Structures, vol. 21, no. 2, pp. 135–148, 1999. View at: Google Scholar
 W. S. Zhang and Y. L. Xu, “Dynamic characteristics and seismic response of adjacent buildings linked by discrete dampers,” Earthquake Engineering and Structural Dynamics, vol. 28, pp. 1163–1185, 1999. View at: Google Scholar
 Y. Q. Ni, J. M. Ko, and Z. G. Ying, “Random seismic response analysis of adjacent buildings coupled with nonlinear hysteretic dampers,” Journal of Sound and Vibration, vol. 246, no. 3, pp. 403–417, 2001. View at: Publisher Site  Google Scholar
 Z. Yang, Y. L. Xu, and X. L. Lu, “Experimental seismic study of adjacent buildings with fluid dampers,” Journal of Structural Engineering, vol. 129, no. 2, pp. 197–205, 2003. View at: Publisher Site  Google Scholar
 V. A. Matsagar and R. S. Jangid, “Viscoelastic damper connected to adjacent structures involving seismic isolation,” Journal of Civil Engineering and Management, vol. 11, no. 4, pp. 309–322, 2005. View at: Google Scholar
 A. V. Bhaskararao and R. S. Jangid, “Seismic response of adjacent buildings connected with friction dampers,” Bulletin of Earthquake Engineering, vol. 4, no. 1, pp. 43–64, 2006. View at: Publisher Site  Google Scholar
 J. Kim, J. Ryu, and L. Chung, “Seismic performance of structures connected by viscoelastic dampers,” Engineering Structures, vol. 28, no. 2, pp. 183–195, 2006. View at: Publisher Site  Google Scholar
 M. Basili and M. De Angelis, “Optimal passive control of adjacent structures interconnected with nonlinear hysteretic devices,” Journal of Sound and Vibration, vol. 301, no. 12, pp. 106–125, 2007. View at: Publisher Site  Google Scholar
 K. Makita, R. E. Christenson, K. Seto, and T. Watanabe, “Optimal design strategy of connected control method for two dynamically similar structures,” Journal of Engineering Mechanics, vol. 133, no. 12, pp. 1247–1257, 2007. View at: Publisher Site  Google Scholar
 C. C. Patel, “Dynamic response of viscous damper connected similar multidegree of freedom structures,” International Journal of Earth Sciences and Engineering, vol. 4, no. 6, pp. 1068–1071, 2011. View at: Google Scholar
 H. P. Zhu, D. D. Ge, and X. Huang, “Optimum connecting dampers to reduce the seismic responses of parallel structures,” Journal of Sound and Vibration, vol. 330, no. 9, pp. 1931–1949, 2011. View at: Publisher Site  Google Scholar
 R. E. Christenson, B. F. Spencer, N. Hori, and K. Seto, “Coupled building control using acceleration feedback,” ComputerAided Civil and Infrastructure Engineering, vol. 18, no. 1, pp. 4–18, 2003. View at: Google Scholar
 Z. G. Ying, Y. Q. Ni, and J. M. Ko, “Stochastic optimal couplingcontrol of adjacent building structures,” Computers and Structures, vol. 81, no. 3031, pp. 2775–2787, 2003. View at: Publisher Site  Google Scholar
 R. E. Christenson, B. F. Spencer, E. A. Johnson, and K. Seto, “Coupled building control considering the effects of building/connector configuration,” Journal of Structural Engineering, vol. 132, no. 6, pp. 853–863, 2006. View at: Publisher Site  Google Scholar
 H. Zhu, Y. Wen, and H. Iemura, “A study on interaction control for seismic response of parallel structures,” Computers and Structures, vol. 79, no. 2, pp. 231–242, 2001. View at: Publisher Site  Google Scholar
 R. E. Christenson, B. F. Spencer, and E. A. Johnson, “Semiactive connected control method for adjacent multidegreeoffreedom buildings,” Journal of Engineering Mechanics, vol. 133, no. 3, pp. 290–298, 2007. View at: Publisher Site  Google Scholar
 S. D. Bharti, S. M. Dumne, and M. K. Shrimali, “Seismic response analysis of adjacent buildings connected with MR dampers,” Engineering Structures, vol. 32, no. 8, pp. 2122–2133, 2010. View at: Publisher Site  Google Scholar
 M. S. Shahidzadeh, H. Tarzi, and M. Dorfeshan, “TakagiSugeno fuzzy control of adjacent structures using MR dampers,” Journal of Applied Sciences, vol. 11, no. 15, pp. 2816–2822, 2011. View at: Publisher Site  Google Scholar
 F. PalaciosQuiñonero, J. Rodellar, J. M. Rossell, and H. R. Karimi, “Activepassive control strategy for adjacent buildings,” in Proceedings of the American Control Conference, San Francisco, Calif, USA, 2011. View at: Google Scholar
 F. PalaciosQuiñonero, J. M. Rossell, J. Rodellar, and H. R. Karimi, “Activepassive decentralized H_{∞} control for adjacent buildings under seismic excitation,” in Proceedings of the 18th IFAC World Congress, Milano, Italy, 2011. View at: Google Scholar
 B. F. Spencer and S. Nagarajaiah, “State of the art of structural control,” Journal of Structural Engineering, vol. 129, no. 7, pp. 845–856, 2003. View at: Publisher Site  Google Scholar
 Y. Ikeda, “Active and semiactive control of buildings in Japan,” Journal of Japan Association for Earthquake Engineering, vol. 4, no. 3, pp. 278–282, 2004. View at: Google Scholar
 H. Li and L. Huo, “Advances in structural control in civil engineering in China,” Mathematical Problems in Engineering, Article ID 936081, pp. 1–23, 2010. View at: Google Scholar
 F. PalaciosQuiñonero, J. M. Rossell, J. Rodellar, and R. PonsLópez, “Passiveactive vibration control for connected multibuilding structures,” in Proceedings of the 8th International Conference on Structural Dynamics (Eurodyn), pp. 1931–1938, 2011. View at: Google Scholar
 K. Ogata, Modern Control Engineering, Prentice Hall, Upper Saddle River, NJ USA, 3rd edition, 1997.
 R. A. Swartz and J. P. Lynch, “Strategic network utilization in a wireless structural control system for seismically excited structures,” Journal of Structural Engineering, vol. 135, no. 5, pp. 597–608, 2009. View at: Publisher Site  Google Scholar
 H. Du and J. Lam, “Energytopeak performance controller design for building via static output feedback under consideration of actuator saturation,” Computers & Structures, vol. 84, no. 3132, pp. 2277–2290, 2006. View at: Publisher Site  Google Scholar
 W. Zhang, Y. Chen, and H. Gao, “Energytopeak control for seismicexcited buildings with actuator faults and parameter uncertainties,” Journal of Sound and Vibration, vol. 330, no. 4, pp. 581–602, 2011. View at: Publisher Site  Google Scholar
 F. PalaciosQuiñonero, J. M. Rossell, and H. R. Karimi, “Semidecentralized strategies in structural vibration control,” Modeling Identification and Control, vol. 32, no. 2, pp. 57–77, 2011. View at: Google Scholar
 J. RubióMassegú, F. PalaciosQuiñonero, and J. M. Rossell, “Decentralized static outputfeedback H_{∞} controller design for buildings under seismic excitation,” Earthquake Engineering and Structural Dynamics, vol. 41, no. 7, pp. 1199–1205, 2012. View at: Google Scholar
 H. Dong, Z. Wang, and H. Gao, “Distributed filtering for a class of timevarying systems over sensor networks with quantization errors and successive packet dropouts,” IEEE Transactions on Signal Processing, vol. 60, no. 6, pp. 3164–3173, 2012. View at: Google Scholar
 Z. Wang, B. Shen, H. Shu, and G. Wei, “Quantized ${H}_{\infty}$ control for nonlinear stochastic timedelay systems with missing measurements,” IEEE Transactions on Automatic Control, vol. 57, no. 6, pp. 1431–1444, 2012. View at: Publisher Site  Google Scholar
 Z. Wang, B. Shen, and X. Liu, “${H}_{\infty}$ filtering with randomly occurring sensor saturations and missing measurements,” Automatica, vol. 48, no. 3, pp. 556–562, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Y. Chen, W. Zhang, and H. Gao, “Finite frequency ${H}_{\infty}$ control for building under earthquake excitation,” Mechatronics, vol. 20, no. 1, pp. 128–142, 2010. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2012 Francisco PalaciosQuiñonero 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.