Discrete-Time Multioverlapping Controller Design for Structural Vibration Control of Tall Buildings under Seismic Excitation

Abstract

In this paper, a computationally effective strategy to obtain multioverlapping controllers via the Inclusion Principle is applied to design discrete-time state-feedback multioverlapping LQR controllers for seismic protection of tall buildings. To compute the corresponding control actions, the proposed semidecentralized controllers only require state information from neighboring stories. This particular configuration of information exchange allows introducing a dramatic reduction in the transmission range required for a wireless implementation of the communication system. To investigate the behavior of the proposed semidecentralized multioverlapping controllers, a proper simulation model has been designed. This model includes semiactive actuation devices with limited force capacity, control sampling times consistent with the communication latency, time-delayed state information, and communication failures. The performance of the proposed multioverlapping controllers has been assessed through numerical simulations of the seismic response of a 20-story building with positive results.

1. Introduction

Over the last decades, problems of ever increasing complexity have been considered in the field of Structural Vibration Control (SVC). Current SVC systems for seismic protection of tall buildings can involve a large number of sensors and actuation devices and a wide and sophisticated communication network [1–3]. Semidecentralized control strategies, which can operate using only state information from neighboring stories, are especially relevant for wireless implementations of the communication system. Semidecentralized state-feedback LQR controllers were proposed by Wang and Lynch in [4], and the study was extended to state-feedback 𝐻_∞ controllers by Wang et al. in [5]. The numerical and experimental results obtained in these works clearly indicate that the proposed semidecentralized control strategies are specially suitable for SVC of tall buildings with wireless communications. It should be highlighted, however, that important computational difficulties can arise when applying these control design strategies to large buildings. The LQR controller design presented in [4] uses a variant of the heuristic iterative procedure proposed by Lunze in [6], and the 𝐻_∞ controller design in [5] is based on a Linear Matrix Inequality formulation (LMI). For large-dimensional problems, a great computational effort is required by the iterative procedure used in the LQR design. Analogously, solving large-dimensional convex optimization problems with LMI constraints is also a costly computational task.

In this context, the design of semidecentralized controllers using multioverlapping decompositions based on the Inclusion Principle (IP) is a very interesting option [7–12]. Broadly speaking, the IP allows decomposing the original large-dimensional problem into a set of low-dimensional decoupled problems. This decomposition takes advantage of the particular structure of the original system and can help to significantly reduce the computational effort. Examples of successful applications of the IP to SVC can be found in [13–15]. Recently, an effective computational strategy to design semidecentralized multioverlapping controllers based on a sequential application of the IP was presented by Palacios-Quiñonero et al. in [16]. In that work, semidecentralized multioverlapping LQR controllers are designed for seismic protection of a four-story building with positive results. However, it has to be noted that all these applications of the IP to SVC have been conducted using small buildings, continuous-time models, and assuming highly idealized conditions, such as active force actuators with unrestricted force capacity and communication systems with no failures nor delays.

The main contribution of the present paper is to present a large-scale application of the IP to the design of semidecentralized controllers for SVC, paying special attention to some aspects of practical relevance. More specifically, the computational strategy proposed in [16] is applied to design discrete-time state-feedback multioverlapping LQR controllers to mitigate the seismic response of a 20-story building. Moreover, to gain a meaningful insight into the behavior of the proposed multioverlapping controllers, the models used in the numerical simulations include some factors of practical relevance such as control sampling rates, realistic implementation of the control actions, time-delayed state information, and communication latency and failures. One of the main difficulties encountered when applying the IP to discrete-time controller design is that the natural structure of the continuous-time model is lost in the discretization process. To overcome this difficulty, the discretization process has been carried out on the expanded decoupled subsystems. The results obtained in the numerical simulations confirm the excellent characteristics of the proposed semidecentralized multioverlapping controllers for SVC of large buildings.

The organization of the paper is as follows: In Section 2, a detailed derivation of the continuous-time state-space model for an 𝑛-story building is presented. Section 3 begins with a summary discussion on the design of discrete-time state-feedback centralized LQR controllers. Next, the main ideas involved in the design of discrete-time state-feedback multioverlapping LQR controllers are briefly presented. In Section 4, three mathematical models used to conduct the numerical simulations of the building seismic response are presented: (i) Basic building model, which consists in a discrete-time approximation of the continuous-time state-space model with a small sampling time and no control action. (ii) Centralized control model, which implements the discrete-time centralized LQR controller with perfect state knowledge, small sampling time, and ideal semiactive force actuators with limited force capacity. (iii) Multioverlapping control model. This case implements a discrete-time multioverlapping controller considering semiactive actuation devices with limited force capacity, a control sampling time consistent with the communication latency, time-delayed state information, and communication failures. Finally, in Section 5, the control design methodology presented in Section 3 and the simulation models introduced in Section 4 are applied to a particular 20-story building to assess the performance of the proposed multioverlapping controllers.

2. Continuous-Time Building Model

Let us consider the 𝑛-story building schematically displayed in Figure 1, which is modeled as a lumped-mass planar system with displacements in the direction of the ground motion. The building motion can be described by the second-order differential equation: 𝑀𝑞̈(𝑡)+𝐶𝑞̇(𝑡)+𝐾𝑞(𝑡)=𝑇_𝑢𝑢(𝑡)+𝑇_𝜔𝜔(𝑡), (2.1) where 𝑞(𝑡)=𝑞₁(𝑡),…,𝑞_𝑛(𝑡)^𝑇 (2.2) is the vector of story displacements with respect to the ground, and 𝑞_𝑖(𝑡) represents the displacement of the 𝑖th story. 𝑀, 𝐶, 𝐾 are the mass, damping, and stiffness matrices, respectively. The vector of control actions is 𝑢(𝑡)=𝑢₁(𝑡),…,𝑢_𝑛(𝑡)^𝑇, (2.3) where 𝑢_𝑖(𝑡) represents the control force exerted by the actuation device 𝑎_𝑖 (see Figure 2), and 𝑇_𝑢 is the control location matrix. The seismic ground acceleration is 𝜔(𝑡), and 𝑇_𝜔 denotes the disturbance input matrix. The mass and stiffness matrices have the following structures: 𝑚𝑀=¹⋯⋯𝑚_𝑛𝑘,𝐾=¹+𝑘₂−𝑘₂−𝑘₂𝑘₂+𝑘₃−𝑘₃⋯⋯⋯⋯⋯⋯−𝑘_𝑛−1𝑘_𝑛−1+𝑘_𝑛−𝑘_𝑛−𝑘_𝑛𝑘_𝑛, (2.4)(2.5) where 𝑚_𝑖 and 𝑘_𝑖 represent, respectively, the mass and stiffness of the 𝑖th story. When the values of the story damping coefficients 𝑐_𝑖, 1≤𝑖≤𝑛, are known, a damping matrix 𝐶 with the same structure as 𝐾 can be obtained by replacing 𝑘_𝑖 by 𝑐_𝑖 in (2.5). Alternatively, a tridiagonal damping matrix in the form 𝐶=𝛼₀𝑀+𝛼₁𝐾 (2.6) can be computed following the Rayleigh damping approach by setting the damping ratio values for two selected natural frequencies [17]. For the actuation system schematically depicted in Figure 2, the control location matrix has dimensions 𝑛×𝑛 and the following structure: 𝑇_𝑢=𝑇_{𝑢𝑖,𝑖}𝑇=1,for1≤𝑖≤𝑛,_{𝑢𝑖,𝑖+1}𝑇=−1,for1≤𝑖<𝑛,_{𝑢𝑖,𝑗}=0,otherwise, (2.7) where [𝑇_𝑢]_𝑖,𝑗 denotes the element in the 𝑖th row and 𝑗th column of 𝑇_𝑢. Finally, the disturbance input matrix is 𝑇_𝑤=−𝑀[1]_𝑛×1, (2.8) where [1]_𝑛×1 is a column vector of dimension 𝑛 with all its entries equal to 1.

Figure 1 

Building lumped-mass model.

Figure 2 

Actuation scheme.

Now, we take the state vector 𝑥_𝐼(𝑡)=𝑞(𝑡)𝑞̇(𝑡) (2.9) and derive a first-order state-space model 𝑆_𝐼∶𝑥̇_𝐼(𝑡)=𝐴_𝐼𝑥_𝐼(𝑡)+𝐵_𝐼𝑢(𝑡)+𝐸_𝐼𝜔(𝑡). (2.10) The state, control, and disturbance input matrices are, respectively, 𝐴_𝐼=[0]_𝑛×𝑛𝐼_𝑛−𝑀⁻¹𝐾−𝑀⁻¹𝐶,𝐵_𝐼=[0]_𝑛×𝑛𝑀⁻¹𝑇_𝑢,𝐸_𝐼=[0]_𝑛×1−[1]_𝑛×1, (2.11) where [0]_𝑟×𝑠 represents a zero matrix of dimensions 𝑟×𝑠, and 𝐼_𝑛 is the identity matrix of dimension 𝑛. Next, we consider a new state vector 𝑥(𝑡)=𝑥₁(𝑡),…,𝑥_2𝑛(𝑡)^𝑇, (2.12) which groups together the interstory drifts and interstory velocities in increasing order 𝑥(𝑡)=𝑥₁(𝑡)=𝑞₁(𝑡),𝑥₂(𝑡)=𝑞̇₁(𝑡),𝑥_2𝑖−1(𝑡)=𝑞_𝑖(𝑡)−𝑞_𝑖−1(𝑡)𝑥,for1<𝑖≤𝑛,_2𝑖(𝑡)=𝑞̇_𝑖(𝑡)−𝑞̇_𝑖−1(𝑡),for1<𝑖≤𝑛. (2.13) Using the change of basis 𝑥(𝑡)=𝑃𝑥_𝐼(𝑡), (2.14) defined by the 2𝑛×2𝑛 matrix 𝑝𝑃=^1,1=1,𝑝_2,𝑛+1𝑝=1,_{2𝑖−1,𝑖−1}=−1,𝑝_{2𝑖−1,𝑖}𝑝=1,for1<𝑖≤𝑛,_{2𝑖,𝑛+𝑖−1}=−1,𝑝_{2𝑖,𝑛+𝑖}𝑝=1,for1<𝑖≤𝑛,_𝑖,𝑗=0,otherwise, (2.15) we obtain the new state space model: 𝑆∶𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐸𝜔(𝑡), (2.16) where 𝐴=𝑃𝐴_𝐼𝑃⁻¹,𝐵=𝑃𝐵_𝐼,𝐸=𝑃𝐸_𝐼. (2.17) In this work, we restrict our attention to the interstory drifts as output variables. The output vector can then be obtained as 𝑦(𝑡)=𝑦₁(𝑡),…,𝑦_𝑛(𝑡)^𝑇=𝐶_𝑦𝑥(𝑡), (2.18) where 𝐶_𝑦 is a matrix of dimensions 𝑛×2𝑛 with the following structure: 𝐶_𝑦=𝐶_{𝑦𝑖,2𝑖−1}𝐶=1,for1≤𝑖≤𝑛,_{𝑦𝑖,𝑗}=0,otherwise. (2.19)

3. Control Design

To design a discrete-time centralized state-feedback LQR controller for the 𝑛-story building model presented in the previous section, we begin by considering the continuous-time system: 𝑆_𝑐∶𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡), (3.1) obtained from (2.16) by removing the disturbance term 𝐸𝜔(𝑡). The discrete-time system corresponding to the zero-hold approximation of (3.1) with sampling time 𝜏 is 𝑆_𝑐𝜏∶𝑥[𝑘+1]=𝐴_𝜏𝑥[𝑘]+𝐵_𝜏𝑢[𝑘], (3.2) where 𝐴_𝜏=𝑒^𝐴𝜏,𝐵_𝜏=^𝜏₀𝑒^𝐴𝑡𝐵𝑑𝑡. (3.3) Next, we consider the discrete-time state-feedback controller: 𝑢[𝑘]=−𝐺_𝜏𝑥[𝑘] (3.4) and the quadratic index: 𝐽(𝑥,𝑢)=^𝑘=∞_𝑘=0𝑥[𝑘]^𝑇𝑄𝑥[𝑘]+𝑢[𝑘]^𝑇𝑅𝑢[𝑘], (3.5) where 𝑄 is a symmetric positive semidefinite matrix, and 𝑅 is a symmetric positive definite matrix. The control gain matrix 𝐺_𝜏 that minimizes (3.5) under constraints (3.2) and (3.4) can be computed as 𝐺_𝜏=𝑅+𝐵^𝑇_𝜏𝑃𝐵_𝜏⁻¹𝐵^𝑇_𝜏𝑃𝐴_𝜏, (3.6) where 𝑃 is the solution of the discrete-time Riccati equation: 𝐴^𝑇_𝜏𝑃𝐴_𝜏−𝑃+𝑄−𝐴^𝑇_𝜏𝑃𝐵_𝜏𝑅+𝐵^𝑇_𝜏𝑃𝐵_𝜏⁻¹𝐵^𝑇_𝜏𝑃𝐴_𝜏=0. (3.7)

To design a multioverlapping controller that is able to compute the control actions 𝑢_𝑖[𝑘] using only state information corresponding to neighboring stories, we consider the 𝑛-story building decomposed into a sequence of 𝑛−1 two-story overlapped subsystems 𝑆^(𝑖)=𝑠_𝑖,𝑠_𝑖+1,1≤𝑖≤𝑛−1, (3.8) where 𝑠_𝑖 represents the 𝑖th story. This overlapping decomposition is schematically depicted in Figure 3. Following the sequential multioverlapping decomposition strategy proposed in [16], the initial continuous-time system (3.1) can be conveniently expanded to form a new continuous-time system: 𝑆∶𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡), (3.9) where the state matrix 𝐴 and the control input matrix 𝐵 are block diagonal. The expanded system 𝑆 can then be decomposed into a sequence of decoupled continuous-time subsystems: 𝑆^(𝑖)∶𝑥̇^(𝑖)(𝑡)=𝐴^(𝑖)𝑥^(𝑖)(𝑡)+𝐵^(𝑖)𝑢^(𝑖)(𝑡),1≤𝑖≤𝑛−1. (3.10) For the continuous-time subsystems 𝑆^(𝑖), we compute discrete-time zero-hold approximations with sampling time 𝜏: 𝑆^(𝑖)𝜏∶𝑥^(𝑖)[𝑘+1]=𝐴𝜏^(𝑖)𝑥^(𝑖)[𝑘]+𝐵𝜏^(𝑖)𝑢^(𝑖)[𝑘],1≤𝑖≤𝑛−1, (3.11) where 𝐴𝜏^(𝑖)=𝑒𝐴^(𝑖)𝜏,𝐵𝜏^(𝑖)=₀𝜏𝑒𝐴^(𝑖)_𝑡𝐵^(𝑖)𝑑𝑡 (3.12) and consider the local quadratic indexes: 𝐽̃^(𝑖)𝑥^(𝑖),𝑢^(𝑖)=^𝑘=∞_𝑘=0𝑥^(𝑖)[𝑘]^𝑇𝑄^(𝑖)𝑥^(𝑖)[𝑘]+𝑢^(𝑖)[𝑘]^𝑇𝑅^(𝑖)𝑢^(𝑖)[𝑘],1≤𝑖≤𝑛−1, (3.13) to compute local discrete-time LQR controllers 𝑢^(𝑖)[𝑘]=−𝐺𝜏^(𝑖)𝑥^(𝑖)[𝑘],1≤𝑖≤𝑛−1, (3.14) which minimize the indexes (3.13) under constraints (3.11) and (3.14). Finally, the sequence of expanded local control matrices 𝐺𝜏^(𝑖) is contracted back to a control gain matrix 𝐺𝜏 in order to define a discrete-time multioverlapping controller: 𝑢[𝑘]=−𝐺𝜏𝑥[𝑘] (3.15) for the original discrete-time system (3.2).

Figure 3 

Decomposition in two-story overlapping subsystems.

Remark 3.1. The expansion-contraction procedure associated to the design of multioverlapping controllers for large buildings is only outlined in this section. For clarity and simplicity, a detailed account of this procedure has not been included in the paper. However, a complete presentation of this background material together with some practical applications to SVC of small buildings can be found in [15, 16].

Remark 3.2. The expanded block-diagonal system (3.9) can only be computed when the matrices of the initial state-space system have a suitable zero-nonzero block structure. For the building model (2.1), the initial state-space system (3.1) has a proper structure. However, this structure is lost in the discretization process and the expansion-decoupling process can no longer be applied to the discrete-time state-space system (3.2). To overcome this difficulty, the expansion-decoupling process is first completed for the continuous-time system and, after that, the discretization process is carried out on the continuous-time expanded decoupled subsystems (3.10) to obtain the discrete-time expanded decoupled subsystems (3.11).

Remark 3.3. It should be noted that the control gain matrix 𝐺_𝜏=[(𝑔_𝜏)_𝑖,𝑗] given in (3.6) is a full matrix of size 𝑛×2𝑛, and the full state is required to compute the control action for the actuation device 𝑎_𝑖: 𝑢_𝑖[𝑘]=^𝑗=2𝑛_𝑗=1𝑔_{𝜏𝑖,𝑗}𝑥_𝑗[𝑘],1≤𝑖≤𝑛. (3.16) In contrast, the multioverlapping control matrix 𝐺𝜏=[(𝑔𝜏)_𝑖,𝑗] has a block-tridiagonal structure and only requires a reduced number of 4–6 states to compute the control action for each actuation device. More specifically, we have 𝑢[𝑘]=𝑢₁[𝑘]=∑^𝑗=4_𝑗=1𝑔𝜏_1,𝑗𝑥_𝑗[𝑘],𝑢_𝑖[𝑘]=∑^𝑗=2𝑖+2_{𝑗=2𝑖−3}𝑔𝜏_𝑖,𝑗𝑥_𝑗[𝑘]𝑢,for1<𝑖<𝑛,_𝑛[𝑘]=∑^𝑗=2𝑛_{𝑗=2𝑛−3}𝑔𝜏_𝑛,𝑗𝑥_𝑗[𝑘]. (3.17)

Remark 3.4. For clarity and simplicity, the controllers presented in this section have been computed following an LQR approach. However, it has to be highlighted that other control strategies are also possible. For example, an application of the IP to the design of semidecentralized static output-feedback controllers for SVC can be found in [14].

4. Simulation Models

One of the main objectives of the present work is to gain a meaningful insight into the behavior of semidecentralized multioverlapping controllers through numerical simulations. To this end, the simulation models have to include some relevant factors such as sampling rates, realistic implementation of the control actions, time-delayed state information, and communication latency and failures. Trying to achieve a proper balance between simplicity and accuracy, we have considered a simulation framework formed by three different models: (i) Basic building model, which consists in a discrete-time approximation of the continuous-time state-space model (3.1) with a small basic sampling time 𝜏 and no control action. (ii) Centralized control model, which implements the discrete-time LQR controller given in (3.4) with perfect state knowledge and the basic sampling time 𝜏. The actuation devices, however, are assumed to be ideal semiactive force actuators with limited force capacity. (iii) Multioverlapping control model. This case implements the discrete-time multioverlapping controller given in (3.15) considering semiactive actuation devices with limited force capacity, a control sampling time 𝜏>𝜏 consistent with the communication latency, time-delayed state information, and communication failures. In all the cases, the interstory drifts are taken as output variables.

The basic building model is: ℳ_𝜏∶𝑥[𝑘+1]=𝐴_𝜏𝑥[𝑘]+𝐸_𝜏𝜔_𝜏[𝑘],𝑦[𝑘]=𝐶_𝑦𝑥[𝑘], (4.1) where 𝐴_𝜏 is the discrete-time state matrix in (3.3); 𝐸_𝜏 is the discrete-time disturbance input matrix, which can be computed as 𝐸_𝜏=^𝜏₀𝑒^𝐴𝑡𝐸𝑑𝑡, (4.2) with 𝐴 and 𝐸 representing, respectively, the state and disturbance input continuous-time matrices; and 𝜔_𝜏[𝑘]=𝜔(𝑘𝜏) is the sampled disturbance. The vector of interstory drifts is computed with the output matrix 𝐶_𝑦 given in (2.19). A good approximation of the uncontrolled seismic response of the building can be obtained using the basic building model ℳ_𝜏 with a small sampling time 𝜏.

The centralized control model is: ℳ_𝜏∶𝑢[𝑘]=−𝐺_𝜏𝑥[𝑘],𝑥[𝑘+1]=𝐴_𝜏𝑥[𝑘]+𝐵_𝜏𝜎(𝑢[𝑘])+𝐸_𝜏𝜔_𝜏[𝑘],𝑦[𝑘]=𝐶_𝑦𝑥[𝑘], (4.3) where the sampling time 𝜏, the matrices 𝐴_𝜏, 𝐸_𝜏, and 𝐶_𝑦, and the sampled disturbance 𝜔_𝜏[𝑘] are the same as those used in (4.1); and 𝐵_𝜏 is the discrete-time input-control matrix defined in (3.3). The vector of control actions 𝑢[𝑘]=𝑢₁[𝑘],…,𝑢_𝑛[𝑘]^𝑇, (4.4) is computed using the discrete-time centralized control gain matrix 𝐺_𝜏 given in (3.6), and the vector of control forces is 𝜎(𝑢[𝑘])=𝜎𝑢₁[𝑘],…,𝜎𝑢_𝑛[𝑘]^𝑇. (4.5) In this section, the actuation devices 𝑎_𝑖 are modeled as ideal semiactive force actuators with maximum actuation force [𝑓_max]_𝑖. For a given control action 𝑢_𝑖[𝑘], the actual control force exerted by the actuation device 𝑎_𝑖 is 𝜎𝑢_𝑖[𝑘]=𝑢_𝑖[𝑘]if𝑢_𝑖[𝑘]⋅𝑣_𝑖[𝑘]<0,and𝑢_𝑖[𝑘]≤𝑓_max𝑖sgn𝑢_𝑖[𝑘]⋅𝑓_max𝑖if𝑢_𝑖[𝑘]⋅𝑣_𝑖[𝑘]<0,and𝑢_𝑖[𝑘]>𝑓_max𝑖0if𝑢_𝑖[𝑘]⋅𝑣_𝑖[𝑘]≥0, (4.6) where 𝑣_𝑖[𝑘] is the corresponding interstory velocity, and sgn(𝑥)=𝑥/|𝑥| is the signum function. In the centralized control model, we assume an ideal communication system, which can provide a perfect knowledge of the full state vector 𝑥[𝑘]. This model is used as a reference in the performance assessment of the multioverlapping controller.

In the multioverlapping control model, we consider the actuation-communication system schematically depicted in Figure 4, consisting in an actuation device 𝑎_𝑖, a sensor unit 𝑠_𝑖̂, a local control unit 𝑐_𝑖̂, and a wireless communication unit 𝑤_𝑖. The actuation device 𝑎_𝑖 produces the semiactive implementation of the control actions defined in (4.6). The sensor unit 𝑠_𝑖̂ is an ideal sensor that provides an exact measurement of the local state 𝑥_𝑖[𝑘]=𝑦_𝑖[𝑘]𝑣_𝑖[𝑘], (4.7) where 𝑦_𝑖[𝑘] and 𝑣_𝑖[𝑘] denote the local interstory drift and velocity, respectively. To model the operation of the local control unit, we introduce the controller sampling time 𝜏=𝑟𝜏̂,𝑟̂>1, (4.8) where 𝜏 is the basic sampling time used in (4.1) and (4.3), and 𝑟̂ can be understood as the maximum number of sampling steps that can be spent by 𝑐_𝑖̂ to collect state information from neighboring stories through the communication unit 𝑤_𝑖. We also consider the update-control times 𝑘_𝑗=𝑗𝑟,̂𝑗≥1, (4.9) and define the interval of state-information gathering Δ_𝑔𝑘_𝑗=𝑘_𝑗−1+1,…,𝑘_𝑗, (4.10) and the interval of control-action holding Δ_ℎ𝑘_𝑗=𝑘_𝑗,…,𝑘_𝑗+1−1, (4.11) which are schematically represented in Figure 5. The operation of the local control unit 𝑐_𝑖̂ has been modeled in accordance with the following set of basic principles: (P.1)The local control action 𝑢_𝑖[𝑘] is updated at the sampling times 𝑘=𝑘_𝑗, 𝑗≥1. (P.2)The local controller unit 𝑐_𝑖̂ has direct access to the sensing unit 𝑠_𝑖̂; consequently the local state 𝑥_𝑖[𝑘_𝑗] is assumed to be always available. (P.3) The state information of neighboring stories obtained through the wireless communication unit 𝑤_𝑖 has the form 𝑥_𝑖′[𝑘−𝛿_𝑖′[𝑘_𝑗]], where 𝑖^′=𝑖±1 and the time delay satisfies 0≤𝛿_𝑖′[𝑘_𝑗]<𝑟̂.(P.4)For a given time interval 𝐼 with length 𝜏, the events 𝐸_𝑖,𝑖′(𝐼,𝜏)=[𝑐_î obtains the neighboring state 𝑥_i′ in the time interval 𝐼 of length 𝜏] are independent random events with common probability Prob𝐸_𝑖,𝑖′(𝐼,𝜏)=𝑝𝜏. (4.12)(P.5)If 𝐼, 𝐼^′ are non-overlapping time intervals with respective lengths 𝜏 and 𝜏^′, then 𝐸_𝑖,𝑖′(𝐼,𝜏) and 𝐸_𝑖,𝑖′(𝐼^′,𝜏^′) are independent random events. (P.6)Through the time interval Δ_𝑔[𝑘_𝑗], the local control unit 𝑐_𝑖̂ tries to collect the state information required to compute the control action. If this state information is successfully acquired, the flag variable 𝜙_𝑖[𝑘_𝑗] is set to 1 and the new control action 𝑢_𝑖[𝑘_𝑗] is computed; otherwise, the flag variable 𝜙_𝑖[𝑘_𝑗] and the control action 𝑢_𝑖[𝑘_𝑗] are both set to 0. (P.7)The control action computed at the sampling time 𝑘=𝑘_𝑗 is held through the time interval Δ_ℎ[𝑘_𝑗]. According to the previous principles, the vector of control actions 𝑢[𝑘]=𝑢₁[𝑘],…,𝑢_𝑛[𝑘]^𝑇 (4.13) can be computed by setting the initial value 𝑢[0]=[0]_𝑛×1, (4.14) and, for 𝑘>0, using the expression 𝑢[𝑘]=−𝐹_𝜙[𝑘]𝐷𝐺𝜏𝑥[𝑘]𝐺𝜏𝑥[[𝑘(𝑟)̂𝑢[]+𝐿𝑘−𝛿]]ifmod𝑘,=0,𝑘−1,otherwise, (4.15) where mod(𝑘,𝑟)̂ represents the integer remainder after division, 𝐹_𝜙[𝑘] is the diagonal matrix 𝐹_𝜙[𝑘]=𝜙₁[𝑘]⋱𝜙_𝑛[𝑘], (4.16)𝐷𝐺𝜏 is the block-diagonal matrix 𝐷𝐺𝜏=𝐺𝜏_1,1⋱𝐺𝜏_𝑛,𝑛 (4.17) formed by the diagonal blocks of 𝐺𝜏𝐺𝜏_𝑖,𝑖=𝑔𝜏_{𝑖,2𝑖−1},𝑔𝜏_𝑖,2𝑖,1≤𝑖≤𝑛, (4.18) and the matrix 𝐿𝐺𝜏=𝐺𝜏𝐺𝜏−𝐷 (4.19) contains the out-of-diagonal blocks of the block-tridiagonal multioverlapping control matrix 𝐺𝜏. The notation 𝑥[𝑘−𝛿[𝑘]] represents the delayed state 𝑥[[𝑘=𝑥𝑘−𝛿]]¹𝑘−𝛿₁[𝑘⋮𝑥_𝑛]𝑘−𝛿_𝑛[𝑘], (4.20) where 𝛿_𝑖[𝑘] is the delay in the local state 𝑥_𝑖.

Figure 4 

Actuation-communication system for the multioverlapping control model.

Figure 5 

Time intervals for state information gathering and control action holding.

The multioverlapping control model can now be obtained by completing (4.14) and (4.15) with the state and output equations 𝑥[𝑘+1]=𝐴_𝜏𝑥[𝑘]+𝐵_𝜏𝜎(𝑢[𝑘])+𝐸_𝜏𝜔_𝜏[𝑘],𝑦[𝑘]=𝐶_𝑦𝑥[𝑘]. (4.21) It should be noted that expressions (4.16) and (4.20) are only evaluated at the update-control times 𝑘_𝑗=𝑗𝑟̂. Moreover, according to (P. 1), (P. 4) and (P. 6), the information-state flag variables 𝜙_𝑖[𝑘_𝑗] are independent random variables with Bernoulli distributions ℬ(𝑝). In particular 𝜙₁[𝑘_𝑗] and 𝜙_𝑛[𝑘_𝑗] have a Bernoulli distribution ℬ(𝑝𝜏), and 𝜙_𝑖[𝑘_𝑗] has distribution ℬ(𝑝𝜏²) for 1<𝑖<𝑛. Finally, it should also be noted that the probability of gathering the neighboring state information 𝑥_𝑖′ by the controller unit 𝑐_𝑖̂ in a time interval 𝐼^′ of length 2𝜏 is Prob𝐸_𝑖,𝑖′𝐼^′,2𝜏=2𝑝𝜏−𝑝𝜏². (4.22) This formula can be easily obtained by writing the time interval 𝐼^′ as union of two non-overlapping intervals 𝐼^′=𝐼^′₁∪𝐼^′₂ of length 𝜏. According to (P. 5), 𝐸_𝑖,𝑖′(𝐼^′₁,𝜏) and 𝐸_𝑖,𝑖′(𝐼^′₂,𝜏) are independent random events, and the probability of failing to acquire the state 𝑥_𝑖′ in the whole interval 𝐼^′ is (1−𝑝𝜏)². Analogously, it can be shown that the corresponding probability for a time interval 𝐼^′′ of length (1/2)𝜏 is Prob𝐸_𝑖,𝑖′𝐼^′′,12𝜏=1−1−𝑝𝜏. (4.23)

5. Numerical Simulations

In this section, the behavior of discrete-time multioverlapping LQR controllers is investigated through numerical simulations of the seismic response of a 20-story building. The parameter values for this particular building are collected in Table 1 and are similar to those used in [5]. The damping matrix has been computed as a Rayleigh damping matrix by setting a 5% of damping ratio for the 1st and 18th natural frequencies. The actuation system 𝑎_𝑖 implemented between the (𝑖−1)th and 𝑖th stories (see Figure 2) is assumed to be formed by a number of identical actuation devices that work coordinately as a single device. The force saturation level of a single actuation device has been taken as 1.2×10⁶ N. The total number of actuation devices and the maximum actuation force for the actuation systems 𝑎_𝑖, 1≤𝑖≤𝑛, is also presented in Table 1.

Table 1 

Particular parameter values for the 20-story building.

For this 20-story building, three different discrete-time LQR controllers are designed: a centralized controller 𝐺_𝜏, which has been obtained using the basic sampling time 𝜏=10⁻³ s; and two multioverlapping controllers 𝐺𝜏 and 𝐺𝜏^′, computed with sampling times 𝜏=40𝜏, and 𝜏^′=20𝜏, respectively. The particular values of the weighting matrices in (3.5) used to design the centralized controller are 𝑄=𝐼₄₀ and 𝑅=10^−17.5×𝐼₂₀. For the multi-overlapping controllers, the weighting matrices in (3.13) used to compute the local expanded LQR controllers 𝐺𝜏^(𝑖) and 𝐺^(𝑖)𝜏^′, 1≤𝑖<20, have been taken as 𝑄^(𝑖)=𝐼₄ and 𝑅^(𝑖)=10^−17.5×𝐼₂, for 1≤𝑖<20.

In the numerical simulations, the maximum absolute interstory drifts have been computed for different control configurations. The basic building model given in (4.1) with sampling time 𝜏=10⁻³ s has been used to compute the uncontrolled seismic response. The controlled response corresponding to the centralized controller 𝐺_𝜏 has been obtained with the centralized control model presented in (4.3). Finally, the multioverlapping control model defined in (4.14), (4.15), and (4.21) has been used to compute the controlled response for the multioverlapping controllers 𝐺𝜏 and 𝐺𝜏^′. In all the cases, the full scale 1995 Kobe North-South seismic record has been taken as ground acceleration (see Figure 6). This seismic record, obtained at the Kobe Japanese Meteorological Agency station during the Hyogoken-Nanbu earthquake of January 17, 1995, is a near-field record that presents large acceleration peaks which are extremely destructive to tall structures [18, 19]. In the multioverlapping control model, the reference value of 𝑝𝜏=0.95 has been set for the probability of obtaining state information from neighboring stories in a time interval of length 𝜏=40ms. According to (4.23), for the multioverlapping controller 𝐺𝜏^′ with control sampling time 𝜏^′=20ms, the probability of successfully gathering state information from neighboring stories can be taken as 𝑝𝜏^′=0.78.

Figure 6 

Full scale Kobe 1995 North-South seismic record.

In Figure 7(a), the red line with asterisks (Overlap. fail in the legend) presents the maximum absolute interstory drifts corresponding to the multioverlapping controller 𝐺𝜏 with controller sampling time 𝜏=40ms, probability of successful communication 𝑝𝜏=0.95, and state delay 𝛿=39ms. The blue line with circles (Overlap. in the legend), displays the values obtained with no communication failures, that is, with 𝑝𝜏=1. The interstory drifts peak values corresponding to the multioverlapping controller 𝐺𝜏^′ with controller sampling time 𝜏^′=20ms, and state delay 𝛿=19ms are presented in Figure 7(b). Here, the red line with asterisks corresponds to the probability 𝑝𝜏^′=0.78, and the blue line with circles presents again the results for 𝑝𝜏^′=1. In both cases, the graphics corresponding to the uncontrolled response (black line with triangles), and the controlled response for the centralized controller 𝐺_𝜏, with controller sampling time 𝜏=1ms, with no communication failures nor delays (black line with squares) have been included as reference. In Figure 8, the red line with asterisks (Over. delay in the legend) displays the maximum absolute interstory drifts corresponding to the multioverlapping controller 𝐺𝜏 with 𝜏=40ms, 𝑝𝜏=0.95, and state delay 𝛿=39ms, while the green line with circles (Over. no delay in the legend) presents the response obtained with null state delay.

(a) Controller sampling time 40 ms

(b) Controller sampling time 20 ms

Figure 7 

Maximum absolute interstory drifts for the 1995 Kobe North-South seismic record. Simulations with maximum state delay.

Figure 8 

Maximum absolute interstory drifts for maximum and minimum state delays (controller sampling time 40 ms).

The graphics in Figure 7(a) show the excellent performance of the proposed multi-overlapping controller for controller sampling times 𝜏 compatible with moderate values of communication latency and also compatible with moderate rates of communication failures. Moreover, the graphics in Figure 7(b) clearly illustrate the trade-off between the controller sampling time 𝜏 and the probability of successful communication 𝑝𝜏. Certainly, taking a smaller sampling time 𝜏 allows a more accurate implementation of the control actions; however, this also implies a reduction of the probability 𝑝𝜏 which, in the end, may result in an overall loss of performance.

Finally, the graphics in Figure 8 show the moderate influence of the state delay 𝛿 in the multioverlapping controller performance for reasonable values of the controller sampling time 𝜏.

Remark 5.1. It is worth to be mentioned that the behavior of the ideal discrete-time centralized controller 𝐺_𝜏 is very similar to the behavior exhibited by an ideal continuous-time centralized LQR controller. As mentioned in Remark 3.3, full state information is required by centralized controllers and this fact makes them unsuitable for SVC of large buildings with wireless communication systems. A detailed discussion of this point can be found in [5].

Remark 5.2. The proposed semidecentralized controllers can operate using only state information from neighboring stories. This fact makes it possible for them to successfully collect the required state information in a relatively small time interval. As a side effect, state delays are also small and have no significant impact on the controller performance.

Remark 5.3. Force saturation is an important issue in SVC. For large seismic excitations, the required control actions frequently exceed the force capacity of the actuation devices. Consequently, force actuation constraints should be considered when studying the controllers behavior. All the numerical simulations of the controlled responses presented in this paper have been conducted using the force saturation values displayed in Table 1.

6. Conclusions and Future Directions

In this paper, a computationally effective strategy has been used to design discrete-time state-feedback multioverlapping LQR controllers for seismic protection of tall buildings. This strategy, based on a sequential application of the Inclusion Principle, produces a block-tridiagonal control gain matrix that allows computing the corresponding control actions using only state information from neighboring stories. Due to this particular information exchange configuration of the multioverlapping controllers, the transmission range and the control sampling frequency in wireless implementations of the communication system can be dramatically improved. To investigate the behavior of the proposed semidecentralized multioverlapping controllers, a proper simulation model has been designed, which allows including semiactive actuation devices with limited force capacity, control sampling times consistent with the communication latency, time-delayed state information, and communication failures. To assess the performance of the proposed multioverlapping controllers, numerical simulations of the seismic response for a 20-story building model have been conducted with positive results.

For clarity and simplicity, the controllers presented in this paper have been designed following an LQR approach. In future works, further research effort should be addressed at exploring the effectiveness of the proposed control design strategy in more complex scenarios, which can involve issues of practical interest such as structural information constraints [20], actuator saturation [21], actuation and sensor failures [22], and limited frequency domain [23]. Other natural extensions of the present work should include a deeper treatment of some important practical aspects related to the communication system such as missing measurements [24–28], stochastic uncertainties [29], and stochastic nonlinearities [30–33].

Acknowledgments

This work has been partially supported by the Spanish Ministry of Economy and Competitiveness through the Grant DPI2011-25567-C02 and by the Norwegian Center of Offshore Wind Energy (NORCOWE) under Grant 193821/S60 from the Research Council of Norway (RCN).