Overlapping Decentralized Guaranteed Cost Control Method for Discrete-Time Systems of Building Structures with Uncertain Parameters under Earthquake Action

This paper proposes a discrete-time overlapping decentralized guaranteed cost control algorithm via state feedback. Based on the inclusion principle, a structure system is decoupled into multiple subsystems, and the discretized subsystems’ controllers are designed by using the guaranteed cost control algorithm. Finally, the subsystems’ controllers are contracted into overlapping controllers of the original system, and the design of the control system is completed. This control method is introduced to the civil engineering field to solve the seismic control problem for discrete-time systems of building structures with uncertain parameters. The numerical analysis of the nine-story building structure model with uncertain parameters is carried out. The results show that the proposed control method can still effectively reduce the seismic response of discrete-time system when the structural parameters are uncertain, and the control effect is similar to the centralized control strategy; at the same time, the sampling rate and data transmission speed of the system are improved to ensure the reliability of the system. It illustrates the feasibility and effectiveness of the presented approach.


Introduction
e problem of real-time feedback control of control systems has been in focus for a long time [1][2][3]. A complete structural vibration control system generally consists of actuators, controllers, and sensors. e performance of the control system is determined by the real-time and accurate information transfer between these devices. e use of wireless transmission technology to replace traditional cable transmission can e ectively improve the real-time performance of system information transmission and reduce the cost of control system [4]. e emergence of advanced technologies, such as the wireless networked control systems (WNCS), digital computers, and micro controllers, promotes the popularization and application of control system in practical engineering [5,6]. e corresponding control force can be calculated in real time through the feedback information of each sampling, which is then converted into an analog signal in the form of a step function that is applied to the structure in the real-time feedback control system. In essence, the measurement information and input control force in the system are discrete-time, rather than the continuous-time function required by the continuous-time control algorithm [7]. e control algorithm designed by discrete-time method is more in line with the actual situation of the controlled system. erefore, it is particularly necessary to study the structural vibration control of discrete-time systems. Paul et al. [8] proposed a novel discretetime sliding mode control in order to attenuate the bidirectional vibrations of building structures, comparing the fuzzy sliding mode control with conventional controllers, which was found to be the most effective in mitigating bidirectional and torsional vibrations. Kemerli et al. [9] proposed the design and implementation of the discretetime sliding mode controller with a hybrid control strategy according to Gao's reaching law and variable rate reaching law and conducted simulated experiments with a 5-story building under seismic excitation; the results show that better results were achieved in terms of controller energy consumption and structural response compared to Gao's controller. Demir et al. [10] studied a method to stabilize the static output feedback (SOF) of a discrete-time linear timeinvariant (LTI) system by using a small number of sensors. Gómez et al. [11] discussed the active control of building structures subjected to tridirectional earthquake excitation, for which there is no available mathematical model, the discrete-time hysteretic nonlinear model was developed, and use the pole-placement design to control the building structure in bidirectional with vertical and torsional effects. ey performed a two-story building prototype to verify that the controller worked better than other horizontal and torsional actuators. e centralized control strategies are mostly used to suppress the vibration response of multidegree-of-freedom structures in previous studies. Although it can obtain good control effects, there are also disadvantages such as long feedback time and poor stability of the system. erefore, the decentralized control strategy is with the advantages of fast data transmission and less feedback delay and strong system reliability received extensive attention from researchers [12][13][14][15]. Liu et al. [16] proposed decentralized state feedback control based on substructure (S-DSFC) algorithm. A numerical study of a 6-story plane frame under seismic excitation indicated that both the substructure and the whole structural system could have better control and stability compared with the traditional decentralized control method. Fadhilah et al. [17] used the information from its own subsystem and other subsystems based on the interconnection to design decentralized controllers. e bilinear matrix inequality (BMI) problem is solved by the homotopy method, and the decentralized controller is found to have better attenuation of centralized disturbances compared to the state feedback controller by two numerical example simulations. Palacios-Quiñonero et al. [18] introduced a high-performance decentralized controller for vibration control of large building structures. A 35-story building is used as a research object to verify the flexibility and effectiveness of the decentralized control strategy. Experimental results show that the controller remains structured despite severe integration constraints. Aboudonia et al. [19] proposed a passive-based control scheme for discrete-time systems in which the control synthesis and operation are decentralized. e study chose appropriately cost functions for control, resulting in controllers that may lead to closed-loop behavior similar to LQR, then considering local conditions to ensure local passivity of all subsystems which implies asymptotic stability of the whole system. Zhu et al. [20] developed a control method of decentralized predictor based on large-scale systems with large input delay, and two methods for the delay compensation were proposed: the backstepping-based partial differential equation (PDE) approach and the reduction-based ordinary differential equation (ODE) approach. en, the Lyapunov-based dispersion analysis is proposed under the two prediction methods. e proposed methods are shown to be effective through two benchmark examples of coupled cart-pendulum systems when the input delay of the stabilization system is too large without predictor. Warsewa et al. [21] proposed a method for designing decentralized and distributed observers that can be used for the monitoring and control of large-scale adaptive structures. In scenarios where redundancy and decentralization are important, it provides a valuable approach for designing new types of observers for intelligent structures.
However, the above literature did not consider the existence of parameter uncertainty in the structural system when studying the system vibration control problem. is uncertainty can have a negative impact on the performance of the controlled system, so the influence of this uncertainty needs to be considered in the design of the controller. e guaranteed cost control algorithm provides an effective way to solve this class of control problems, and the method allows a defined upper bound on the performance index of the system while also ensuring its robust stability [22]. Zhang et al. [23] studied the guaranteed cost control problem for a kind of nonlinear discrete-time-delay system. A complete Lyapunov-Krasovskii function was constructed based on the Lyapunov matrix. e usefulness of the theoretical results is illustrated by presenting an example. Chen et al. [24] paid attention to the guaranteed cost control problem for a class of fractional-order (FO) uncertain linear systems with timedelay parameter uncertainties with norm-bounded. By using the linear matrix inequality (LMI) method and FO Razumikhin theorem, two design methods of guaranteed cost controllers with independent time delay and their guaranteed cost are given. Two numerical examples are studied to verify the effectiveness of theoretical formulas. Qi et al. [25] also studied the guaranteed cost control problem for a class of uncertain fractional-order (FO) linear systems with timedelay parameter uncertainties with norm-bounded by the linear matrix inequality (LMI) method and the FO Razumikhin theorem. Aiming at the admissible uncertainty, a time-delay and order-dependent design method with guaranteed closed-loop stability and cost are proposed. Chen et al. [26] analyzed the main influencing factors of the higher-order CSI effect. ese include the input frequency of the control voltage, the uncertainty of the structural parameters, and the control gain. In addition, a new time-delay compensation controller based on the guaranteed cost control (GCC) algorithm is proposed to consider multilevel higher-order CSI effects. Li et al. [27] proposed a state feedback controller based on guaranteed cost control (GCC) algorithm according to a Lyapunov stability theory and linear matrix inequality (LMI) method. A ten-story frame with AMD system is analyzed. e results show that the new controller can effectively suppress the dynamic response of high-rise buildings with parameter uncertainty. e paper studies the discrete-time system with uncertain parameters and proposes the overlapping decentralized guaranteed cost control algorithm based on state feedback. Taking a 9-story building structure under earthquake action as the research object, firstly, the guaranteed cost control equations in the discrete-time domain are derived considering the existence of uncertainties in the parameters of mass, damping, and stiffness of the structural system [27,28]. e system is extended and decoupled using the inclusion principle and then discretized, and subcontrollers are designed with guaranteed cost control algorithms. Finally, the contraction principle is applied to form the overlapping controller of the original system. e numerical analysis results show that the control method proposed in this paper can suppress the dynamic response of the structure well, and its control effect is comparable to that of the traditional centralized control strategy, which illustrates its effectiveness.

Mechanical Model of Building Structure with Uncertain Parameters
Considering the uncertainties of the mass, damping, stiffness, and other parameters of the building structure system, the motion equation of the structure under seismic excitation is obtained: where M, ΔM represent the mass matrix and variation matrix of the structure, respectively; C, ΔC represent the damping matrix and variation matrix of the structure, respectively; K, ΔK represent the stiffness matrix and variation matrix of the structure, respectively; x, _ x, € x represent the displacement, velocity, and acceleration vectors, respectively; D s , B u represent the position matrices of the external excitation and actuator, respectively; € x g represents the external excitation vector; and u represents the actuator control force vector. e structural form of the corresponding parameters in (1) is where α, β, c represent the rate of change of the mass, damping, and stiffness; and δ M , δ C , δ K are the unknown matrices of corresponding dimension. From the literature [29], we can obtain e state-space model is transformed by (1): where We define a new state vector: Mathematical Problems in Engineering e transformation matrix [22] T can be expressed as follows: e equation of state after conversion can be expressed as where

Guaranteed Cost Control Method for Discrete-Time Systems
Taking the sampling period ΔT � 0.02s, the continuous-time system (8) is discretized to obtain the state-space model of the discrete-time system, where A, B are constant matrices, and ΔA, ΔB are its corresponding unknown matrices; Z(k) ∈ R n represents the state vector of the system; and u(k) ∈ R m represents the control vector of the system. It is assumed that the uncertain matrices can be expressed as follows: where D, E 1 , E 2 represent the matrices of known constants with uncertain information. F ∈ R i×j is the unknown matrix satisfying the following condition: and it can be time-varying. A performance indicator is defined for system (10): where Q * , R * are the given symmetric positive definite weighting matrices.
Definition 1. For system (10) and the performance index (13), if there is a matrix G and a positive definite symmetric matrix P such that for all nonzero Z(k) and all uncertain matrices satisfying equation (12), we have e control law u(k) � GZ(k) is called a quadratic guaranteed cost control law with a performance matrix P for system (10).
Lemma 1 (see [30]). It is assumed that Y c , D c , E c are the known matrices and Y c is symmetric, then For all uncertain matrices F satisfying F T F ≤ I, if and only if there exists a constant ε > 0, such that Theorem 1 (see [31]). If u(k) � GZ(k) is a guaranteed cost control law with performance matrix P for system (15) and performance index (18), then the closed-loop system which is quadratically stable for all allowed uncertainties and the corresponding closed-loop performance index values satisfy the following form: Theorem 2 (see [31]). For system (15) and performance index (18), if the following optimization problem which have an optimal solution (ε opt , Y opt , X opt , W opt ), then u(k) � Y opt X − 1 opt Z(k) is the optimal quadratic guaranteed cost control law of system (15), and the corresponding upper bound of system performance is J * � Trace(X − 1 opt ).
Proof: According to Definition 1, there exists a quadratic guaranteed cost control law u(k) � GZ(k) for system (10). A matrix inequality holds if and only if there exists a matrix G and a symmetric matrix P > 0 such that for the allowed uncertain matrix F, the following matrix inequality holds: By the Schur complementary property of matrices, (20) can be equated to We define the matrix We convert (21) to According to Lemma 1, the matrix inequality (23) holds for all matrices F that satisfy F T F ≤ I. If and only if there exists a constant ε > 0 such that Multiplying the matrix on the left side of the above equation left and right by the matrix, respectively. We can obtain Let X � P − 1 , Y � GP − 1 and we get By the Schur complementary property of matrices, (28) can be equated to en, Applying the Schur complementary property of matrices, we get en, By applying the Schur complementary property of matrices again, it is found that (32) is equivalent to the constraint (i) in problem (19).

Mathematical Problems in Engineering
For the constraint (ii) in problem (19), it is equivalent to W > X − 1 > 0 by the Schur complementary property of matrices. So, the minimization of Trace(W) guarantees the minimization of Trace(X − 1 ), then it is the minimization of the upper bound of system performance. Since the objective function and constraints in problem (17) are convex functions of the variables, problem (19) is a convex optimization problem so that a global minimum can be achieved. e proof is obtained by eorem 2.

Overlapping Decentralized Control Method for Discrete-Time Systems
Consider the following state-space model of a linearly continuous-time-invariant system: where Z(t) ∈ R n , Z(t) ∈ R n represent the state vector of the system S and system S, respectively; u(t) ∈ R m , u(t) ∈ R m represent the input vector of the system S and system S, respectively; y(t) ∈ R l , y(t) ∈ R l represent the output vector of the system S and system S, respectively; A, B, C y , A, B, and C y are n × n, n × m, l × n, n × n, n × m, and l × n dimensional matrices, respectively. n ≤ n, m ≤ m, l ≤ l. Based on the inclusion principle, system S is extended to obtain system S, then where M A , N B , L C are the corresponding compensation matrices.
e system S is extended and decoupled into L substructures after eliminating the connection blocks [32]: where A ii , B ii represent the known constant matrices. ΔA ii , ΔB ii represent the unknown matrices of uncertain parameters in the model and have the following form: where D i , E i1 , E i2 represent the matrices of known constants with uncertain parameters. F i (t) represents an unknown matrix, and F T i (t)F i (t) ≤ I. Taking the sampling period ΔT � 0.02s, the continuoustime system (35) is discretized, then According to eorem 2, the controller design of the discrete-time model of the subsystem is carried out to obtain the state feedback gain matrices G i of the subsystems, which represented the state feedback gain matrices of a block diagonal: e overlapping controller can be obtained by transforming the gain matrices G based on the contraction principle [32]: (39)

Example Simulation and Analysis
Taking the 9-story Benchmark steel structure model (as shown in Figure 1) as the simulation example, the plan and elevation of the structure can be seen in literature [33,34]. e static coalescence method is used to reduce the order of the original finite element model, which reduces the degrees of freedom of vertical vibration and rotation and only retains nine translational freedoms of the original structure. e -u i+1 (t) corresponding parameters of the structure are listed in Table 1 [34]. e damping of the structure is Rayleigh damping, assuming the damping ratio of the first two orders of vibration is 0.02. Each layer of the structure is equipped with a drive using the interlayer drive method. e external excitation is a horizontal seismic load. e El Centro seismic wave (NS, 1940) is selected, peak acceleration is 3.0 m/s 2 , duration is 30 s, and sampling step is 0.02 s. e maximum variances that exist for the mass, damping, and stiffness of the structure are considered to be ±10%, ±15%, and ±15%, respectively. e equation of motion for uncertain parameters structural system under earthquake is shown in (1), and (2) are ΔM � α M, ΔC � β C, ΔK � c K. M, C, K represent the nominal mass, damping, and stiffness matrices of the structure, respectively. e corresponding uncertain parameters in (5) can be expressed as δ is an uncertain real scalar, |δ| ≤ 1.
For the discrete-time system of building structure with uncertain parameters, the discrete-time system state feedback overlapping decentralized guaranteed cost control method (Dis. Overlap.) and the discrete-time system state feedback centralized guaranteed cost control method (Dis.   Central.) are used for numerical analysis. e design of the controller is shown in Figure 2.
According to the design steps of overlapping decentralized control method described in Section 4, the 9-story structure is overlapped and decomposed into 2 substructures, and the overlapping part is chosen as the fifth layer, then the two decoupled subsystems are S        rate of 73.87%; the interstory displacement control effect of discrete-time overlapping decentralized guaranteed cost control method (Dis. Overlap.) is 72.34% ∼ 87.65% with an average control rate of 81.58%. From the above data, it can be seen that the proposed control method can achieve good control effects under the existence of uncertainty in structural parameters, and the average control rate of interstory displacement under each working condition is better than the centralized control strategy. e effectiveness and applicability of this method are illustrated. Figures 6-8 show the comparison of the acceleration time-history curves of the structural top floor with uncertain parameters under uncontrolled and discrete-time overlapping decentralized guaranteed cost control method (Dis. Overlap.). It can be seen from the figure that the proposed decentralized control method can effectively reduce the acceleration response of the structure when the structural parameters are uncertain. Figures 9 to 11 show the comparison of the maximum control force of the structure with uncertain parameters using the discrete-time overlapping decentralized guaranteed cost control method (Dis. Overlap.) and discrete-time centralized guaranteed cost control method (Dis. Central.). It can be seen from the figure that the output values required by the two control methods are within the range of the controller. Specifically, except for the overlapping layers, the control force required for each layer of substructure 1 is slightly greater than that of the centralized control method, and the control force required for each layer of substructure 2 is less than that of the centralized control method, in the case of comparable control effects. e reasons are as follows: (1) the overlapping decentralized guaranteed cost control method divides the original structure into two substructures which reduces the dimensionality of system.
(2) Each substructure controller in the overlapping decentralized control strategy is controlled independently and works in parallel. e overlapping layer acts as a connecting layer between two substructures, and the control force is influenced by both substructures. (3) Due to the existence of overlapping layers, substructure 1 and substructure 2 are similar to the relationship between the upper and lower floors. erefore, the control force of substructure 1 increases while the control force required for substructure 2 decreases, while the control effect is guaranteed to be constant.

Conclusions
is paper combines the overlapping decentralized control strategy and the guaranteed cost control algorithm and then proposes an overlapping decentralized guaranteed cost control method for discrete-time systems. e proposed method is first studied numerically with a nine-story parameters uncertainty building example. e following conclusions can be drawn from the present study: (1) e discrete-time overlapping decentralized guaranteed cost control algorithm based on state feedback proposed can achieve satisfactory control results under the uncertainty of the controlled structural parameters. It shows that the method is suitable for solving the vibration control problem of building structural systems with uncertain parameters. (2) e overlapping decentralized control strategy has a better average control rate of interlayer displacement compared with the centralized control strategy under the condition of comparable controller output, which illustrates the effectiveness and feasibility of the proposed method. (3) e overlapping decentralized control method realizes the dimensionality reduction of large systems.
In low-dimensional systems, the transmission speed of signals is rapid, the data processing is efficient, and the computational cost is low, which provides an effective idea for solving the vibration control problems of complex systems.

Data Availability
e data used to support the study are available from the corresponding authors upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.