Absolute Instantaneous Optimal Control Performance Index for Active Vibration Control of Structures under Seismic Excitation

In this paper, an instantaneous optimal control performance index for active control of structures under seismic excitation is analytically proposed. Absolute velocity and absolute displacement terms are implemented to the conventional state vector terms and eventually to the resulting performance index expression. (e seismic response reduction effectiveness of the proposed performance index is compared with the linear quadratic regulator control (LQR). For numerical verification of the performance index, an eight-story shear building with a fully active tendon controller system under unidirectional earthquake is considered as the first example. For a more complex model, a three-dimensional tier building under the effect of bidirectional earthquakes is selected as second numerical example. Unidirectional near fault and bidirectional near fault earthquakes are used in the simulations. (e control energy demand of each control method is also considered in the comparison. It is obtained from numerical simulations that the proposed performance index is as effective as LQR in attenuating structural vibrations. However, the resulting performance index does not require a priori knowledge of the seismic excitation like the LQR.(e nonlinear Riccati matrix equation solution of the LQR is not required in the proposed performance index as well.


Introduction
Active control has been an interesting topic of research for seismic mitigation of structures since smart control devices into structures were first implemented by Yao [1]. An active control device includes a real-time controller and a power supply, and it is integrated with control computers and sensors which are implemented to the structure. An active control algorithm should be used while operating an active controller device. e active control device could either be an active control actuator, active tendon controller, or active mass damper. ere are many interesting studies on new active control methods and new active control algorithms in the last years [2][3][4][5][6][7][8][9][10]. Shen et al. [2] investigated three control algorithms for active vibration control of the sting in wind tunnel.
ese algorithms were Classical Proportional and Derivative Algorithm, Artificial Neural Network Proportional Integral Derivative, and Linear Quadratic Regulator (LQR) Optimal Control Algorithm [2]. Active structural control was applied to an idealized two-span concrete highway bridge structure in [3]. ey idealized the continuous girder bridge as a two degrees of freedom system [3]. A decentralized scheme for active noise control from a game-theoretical perspective was studied by Quintana and Diego [4]; they formalized the Nash equilibrium in the interaction between the controllers in their interesting study. A force feedback-based control method for active tuned mass dampers was described by Wang and Yun [5]. Direct velocity feedback was implemented to their method [5]. A new performance index for active vibration control of threedimensional structures was presented numerically by Yanik et al. [6]. e problem of designing a state-feedback controller with both active structural control and passive base isolation was considered analytically/numerically by Miyamoto et al. [7]. A simple active and semiactive control integrated structural model for seismic vibration mitigation was numerically presented by Yanik and Aldemir [8]. For an active vibration isolation system, active hybrid control technology was applied in Wang et al. [9]. A mixed H 2 / H infinity -based active state feedback controller was described for mitigating vibrations in seismic excited container cranes in Azeloglu and Sagirli [10].
In addition to the control methods or control approaches defined above, one of the most popular active control algorithms is Linear Quadratic Regulator Control (LQR) [11,12]. However, there have been many criticisms on LQR controllers. It was stated by Yang et al. [13] that LQR controllers are known to be ineffective for systems suffering from variation of parameters and excitations which are broadband. In addition, it has been mentioned by Aldemir and Bakioglu [14] that only classical closed-loop control is applicable to structural control problems, and the Riccati equation is derived by not considering the seismic excitation term. It has been shown by Etedali and Tavakoli [15] that LQR did not show good performance in mitigating the seismic responses of an 11-story structure incorporating active tuned mass damper (ATMD) under strong seismic excitations. After comparing three active control methods with numerical examples that consist of buildings with different elevation including an ATMD on the top floor, it was obtained by Sareban [16] that, in LQR procedure, the applied force was limited. To overcome the deficiency of classical linear optimal control algorithms which are the applications of LQR [11,17], instantaneous optimal control algorithms for active structural control were first introduced Yang et al. [18]. It has been stated in [18] that instantaneous optimal control laws were defined for providing feasible control algorithms that can easily be implemented for applications to seismic-excited structures. Combination of instantaneous optimal control and iterative learning control has been analytically proposed by Tong [19], and a 20-story shear building was taken into account for numerical verification of the method. An instantaneous control algorithm for magnetorheological damping, which considered relative acceleration and velocity feedback, was analytically defined and numerically verified by Chandiramani and Purohit [20].
An absolute instantaneous optimal control performance index is analytically defined in this study. In this method, absolute velocity and absolute displacements of the structure are fed into the conventional state vector to obtain the resulting control force. A shear building and a 3D tier building are taken into account to perform the dynamic analysis under unidirectional and bidirectional earthquake excitation. e uncontrolled response reduction performance under seismic excitation, of the proposed performance index is compared with LQR. e formulation of the problem is given in Section 2.

Formulation of the Problem
An n story shear building with a fully active tendon controller system under the effect of unidirectional earthquake is schematically depicted in Figure 1.
is structure's equation of the motion can be presented as where X(t) � (X 1 , X 2 , . . . , X n ) T is the vector representing the relative displacement of each story relative to the ground, the dimension of this vector is n, M is the mass matrix with the dimensions of (n × n). e elements of mass matrix is m i � mass of i th story (i � 1, 2, . . ., n), C and K are damping and stiffness matrices in a respective way, both C and K are (n × n) dimensional matrices, E (n × 1) is the matrix related with the location of the excitation, and it can be written as E T � − (m 1 , . . ., m n ), active control location matrix is L, the dimension is (n × r), where r is the number of active controllers, U(t) is the force vector of active controllers and can be presented as U T (t) � (u 1 (t), . . . , u r (t)), the dimension of U(t) is r and f(t) is a scalar which can be defined as the unidirectional earthquake acceleration. Initial displacement and the initial velocity of this structure can be presented as If we define a conventional state vector as Z � (X, _ X) T , the state space form of equation (1) can be written as where t 0 (t � 0) is the time when the earthquake starts and t 1 is the duration longer than that of earthquake. e system matrices in equation (3) can be defined as where I is the identity matrix and the dimension is (n × n) and η � (1, . . ., 1) T is a vector which is constant with the dimension of n. e system expression given by equation (3) can be presented by considering Equation (2) as follows: To apply the active control force to the building, a control law should be chosen. In this paper, LQR is chosen for comparing the seismic effectiveness of the proposed instantaneous optimal control performance index. Section 2.1 includes information on LQR.

Linear Quadratic Regulator Control (LQR).
In LQR control, the classical integral type quadratic performance index is minimized and the optimal control force can be obtained as follows: where P is the solution of the following nonlinear matrix Riccati equation: 2 Shock and Vibration where Q C and R C are the weighting matrices, R C is positive de nite matrix, and Q C is a positive semide nite matrix. e numerical values of these weighting matrices should be selected by considering the adjustment between the power requirement of the actuators and response reduction. Large values of the elements of Q C matrix should be assigned if a signi cant structural response reduction is desired. In contrast, a designer should assign large values to R C weighting matrix if power consumption of the actuators is more important than the response reduction. We can also express equation (7) as where G C (n × 2n) is the control gain and can be expressed as e Riccati matrix P(t) obtained from equation (8) does not yield an optimal solution since the excitation term f(t) vanishes within the control interval [0, t 1 ] [21]. e LQR formulation is given very brie y in this section. More information on LQR can be obtained from [11,21]. In Section 3, the proposed performance index is de ned.

Absolute Instantaneous Optimal Control Performance Index
An instantaneous optimal control performance index (INS-ABS) for active control is analytically presented in this section. is performance index uses absolute velocity and absolute displacement information as feedback in the state vector. It should be noted here that this performance index is another application of the instantaneous optimal control algorithms that were de ned in the pioneering work of Yang et al. [18]. erefore, there are some similarities between this study and the formulation de ned in [18]. However, the big di erence in this performance index is the absolute state vector implementation into the resulting index and eventually to the resulting control force expression. Before dening the performance index, the absolute state vector is de ned as follows. e absolute accelerations for the given system can be written by using the parameters de ned in equation (1) as where v is the n dimensional location vector of earthquake excitation and can be written as v 1 1 1 1 . . . T . e absolute velocities can be presented as where where e time-dependent performance index considering absolute state vector can be stated as By using backward difference approximation and implementing Z abs (t) into the state space expression given with equation (3), one can obtain the following equation: e Hamiltonian of this instantaneous optimal control performance index H can be expressed as e necessary conditions for optimality are given below: As shown above, taking the derivatives of equation (16) by _ Z abs , U(t), and λ gives the equations as follows, respectively, with necessary operations on equations (18) and (19), the resulting proposed active control force can be written as e numerical examples are shown in Section 4.

Numerical Examples
In this section, firstly, the 2D shear building example is defined and the results obtained from dynamic simulations under earthquake excitation are given. For numerically verifying the proposed performance index in a more complex model, a tier building example is also presented. Tier buildings are simply three-dimensional shear buildings. Formulation of these buildings is presented in the appendix of this paper. More information on these buildings can be obtained in [6,22,23]. In a tier building, each story has three degrees of freedom. ese degrees of freedom are one rotational degree of freedom and two degrees of freedoms for the orthogonal axes.

Shear Building Example.
A shear building with active tendon controllers is considered for the numerical verification of the INS-ABS performance index as shown in Figure 2. e shear building is eight storied, and there are eight active tendon controllers in the building. e same example structure parameters given in Yang et al. [18] are used in this study. As this study is another simple application of instantaneous optimal control algorithms, the Yang et al. [18] model was chosen to be consistent with the first pioneering work on instantaneous controllers. e mass of each story of the structure is identical and chosen as m � 345.6 tons, while the stiffness k of each story is the same and k � 3.404 × 10 5 kN/m and the damping of each story is c � 2937 tons/s. e frequencies of the building are obtained as 0.92, 2.73, 4.45, 6.02, 7.38, 8.49, 9.32, and 9.82 Hz. e frequencies of this model are calculated by solving the eigenvalue problem on the shear building. e active control weighting matrices Q C and R C for LQR and Q and R for INS-ABS are composed and given below [24]. ere are 8 active tendon controllers in the building so that the size of Q matrix is 16 by 16 and R matrix is 8 by 8. Researchers are still working on to develop a proper selection of weighting matrices. However, there are no exact rules or concrete methods on how to assign the numerical values to the elements of the weighting matrices. erefore, within the framework of this paper, a case study is not carried out for the optimal selection of these weighting matrices. However, in this paper, Q C and R C for LQR and Q and R for INS-ABS are required to guarantee the stability of the controlled structure: e ground motion parameters are shown in Table 1. ese earthquake records are from the near fault regions of the corresponding earthquakes. With respect to the PGA information of these earthquakes, one moderate (El Centro) and two strong earthquakes are used in this study.
After running the simulations, time displacement and control force relationships as well as the maximum control energies of the structure are obtained. e top story time displacement and control force curve are shown in Figure 3 for El Centro earthquake. NC represents the uncontrolled conventional structure. Along with Figure 3, the displacement and absolute acceleration responses for all stories are defined with the following comparison measures: Comparison measures are calculated by normalizing the corresponding parameter with the uncontrolled first story displacement or absolute acceleration. Control energy (CE) is considered in the measures as well. Maximum absolute values of the corresponding parameters are taken into account in the calculation of the measures. e numerical values of the performance measures for El Centro earthquake are shown in Table 2.
As it can be indicated from Figure 3 and Table 2 that for El Centro earthquake, INS-ABS performance index is more effective in reducing uncontrolled absolute accelerations than LQR control algorithm with less control energy consumption. e reductions for structural displacements are at the same extent for both methods. e maximum displacement reduction percentages of different stories range from 10% to 47% for INS-ABS while they range from 14% to 48% for LQR. e reduction percentages for absolute accelerations are 39% to 56% for INS-ABS; however, these parameters range from 8% to 48% for LQR. e reductions of INS-ABS case are achieved with 12% less power consumption than LQR. e top story time displacement and control force curve is shown in Figure 4 for Northridge earthquake. e comparison measures are shown in Table 3 for the same earthquake.
It is shown in Figure 4 and Table 3 that under the effect of Northridge earthquake, INS-ABS performance index is more effective than LQR algorithm in reducing absolute accelerations with 5% less control energy. And, the uncontrolled displacement reduction percentages for both algorithms are at the same extent. Uncontrolled absolute acceleration response reduction percentages for different stories range from 49% to 35% for INS-ABS while the same parameters range from 12% to 35% for LQR. In addition, the uncontrolled story displacements range from 33% to 35% for LQR and 32% to 36% for INS-ABS. e same comparisons are carried out for Erzincan earthquake. e results are presented in Figure 5 and Table 4.
Like the two other earthquakes, INS-ABS is found to be more effective in reducing absolute accelerations than LQR with 19% less control energy consumption under the effect of Erzincan earthquake. e uncontrolled absolute acceleration reduction percentages range from 34% to 45% for INS-ABS performance index, while the same parameters range from − 1% to 34% for LQR. Moreover, the uncontrolled displacement reduction percentages are at the same extent for all earthquakes. 3D tier building example and simulation results are presented in Section 4.2.

Tier Building Example.
As shown in the previous section, El Centro earthquake resulted in the smallest structural responses. erefore, in this section, Erzincan and Northridge earthquakes are used in the simulations. e acceleration time histories of these earthquakes are shown in Figure 6. Faultnormal and fault-parallel components of these earthquakes are considered in the numerical simulations.
A six-story four-bay 2D building model is used for conversion to a 3D tier building model in this section. e six-story structure is assumed to be 50 m by 70 m in plan, and 27 m in elevation (6 × 4.5) as given in Figure 7     are 345 MPa steel. e floor system is comprised of 248 MPa steel wide-flange beams acting compositely with the floor slab [25]. e six-story structure has a total of 18 degrees of freedom with two orthogonal axes and a single rotational degree of freedom in each story. e frequencies of the building are calculated as 0.70, 1.04, 2.35, 2.54, 3.76, 4.53, 6.30, 6.73, 7.67, 8.53, 8.54, 9.35, 11.38, 12.67, 15.26, 21.21, 25.82, and 28.74 Hz. ese frequencies are also calculated by solving the eigenvalue problem relating to the stiffness and mass matrix of the tier building. e information on these matrices is given in the appendix of this paper.

Shock and Vibration
Tier building formulation is used in converting the 2D building into a simple 3D building. More information on 3D tier buildings and information on 2D to 3D conversion procedure in detail can be found in [6,22,23]. ese types of structural plans were studied for in-plane (2D) analysis of the benchmark structures in Ohtori et al. [25]. e weighting matrices for LQR and INS-ABS are selected as   Shock and Vibration 7 N-S direction is the direction where fault normal component of the earthquake acts. e maximum displacement in N-S direction and maximum rotation of each story for Erzincan bidirectional ground motion is shown in Figure 8. Tendon controllers are assumed to be operating in the N-S direction; therefore, both methods are not effective in reducing the displacements in E-W direction. It should be noted here that the displacements in E-W direction are almost three times smaller than the ones in N-S direction. It   can be indicated from Figure 8 that the INS-ABS method is more effective than LQR in reducing both uncontrolled displacements and rotations. e uncontrolled N-S displacement reduction percentages range from − 5% (for the second story LQR resulted in bigger story displacement than NC) to 5.5% for LQR and from 10% to 11% for INS-ABS. ese parameters for uncontrolled rotation range from − 0.63% to 2% for LQR and 11% to 15% for INS-ABS. e maximum absolute accelerations in N-S direction and E-W direction for each story are shown in Figure 9. As it is seen from Figure 9, LQR resulted in almost 37% to 2.54% bigger uncontrolled absolute accelerations in N-S direction. In E-W direction, LQR control resulted in 7% to 18% bigger accelerations than NC. Only one-story acceleration is found to be smaller than NC in E-W direction. However, INS-ABS is obtained to be effective in absolute acceleration reduction. In N-S direction, uncontrolled absolute acceleration reduction percentages range from 7% to 18% for INS-ABS.
ese results are achieved with control energies that are at the same extent in both methods. e same comparisons are carried out for the structure under Northridge bidirectional earthquake acceleration. Maximum displacements in N-S direction and rotations are shown in Figure 10 for this earthquake. It can be indicated from Figure 10 that maximum response reductions are achieved with INS-ABS. e uncontrolled displacement reduction percentages range from 4.8% to 8.5% for LQR. However, the same parameters range from 29% to 32% for the INS-ABS method. e rotations are at the same extent for LQR and NC. e uncontrolled rotation reduction percentages are around 14% for INS-ABS. e maximum absolute accelerations in N-S direction and E-W direction for each story are shown in Figure 11.
It is shown in Figure 10 that LQR resulted in almost 6% to 17% bigger uncontrolled absolute accelerations in N-S direction for some stories, and the absolute accelerations in E-W direction are at the same extent with NC. However, INS-ABS case has smaller absolute accelerations than NC in N-S direction. For N-S direction, the uncontrolled absolute acceleration reduction percentages range from 14% to 34% in INS-ABS. e control energy consumption in INS-ABS case is found to be extremely smaller than LQR for Northridge earthquake. LQR resulted in almost 20 times bigger control energy than INS-ABS. e conclusions obtained from this study are given in Conclusions.

Conclusions
An absolute instantaneous optimal control performance index for seismically excited structures is analytically defined and numerically tested in this paper. e major distinction of this performance index from LQR is, as the nonlinear matrix Riccati equation of LQR is obtained by ignoring the earthquake excitation term, LQR is approximately optimal and does not satisfy the optimality condition. LQR cannot be applied to earthquake-excited structures, because the whole earthquake ground acceleration history is not known apriori. In comparison with the LQR, INS-ABS does not require a priori knowledge of the seismic excitation. erefore, it can be applied to seismically excited structures. With respect to the numerical results, the proposed performance index reduces both the absolute accelerations and displacements of the uncontrolled structures. e numerical results also showed that for 3D tier buildings, INS-ABS performance index outperformed the LQR control algorithm in reducing both structural displacements and absolute accelerations in the fault normal direction of the earthquake. e structural rotations in INS-ABS case are also found to be smaller than LQR. In addition, it is obtained from shear building analysis that INS-ABS is more effective in reducing absolute accelerations than LQR with smaller control energy consumption. Control energy consumption is related with the power requirement of the active tendon controllers.

Appendix
In Figure 12, an n story tier building with identical story height (h) and floor lengths (L x and L y ) in each story is simply given. CM and CS in Figure 12 represent the center of mass and center of stiffness. respectively. e control forces acting in NS direction are shown very simply with U i . e displacements in NS and EW directions and rotation are given with X i , X i+1 and θ i for the ith story. Earthquake accelerations in NS and EW directions are denoted by f x (t) and f y (t), respectively. e mass and stiffness matrices for this building are shown below.   In the mass matrix, m 1 , m 2 , m i , m n− 1 , and m n are the mass of the first, second, ith, (n − 1)th, and nth (top) story in a respective way. e eccentricities for first, second, ith, (n − 1)th, and nth (top) story of the structure for NS and EW directions are given, respectively, as x c1 , y c1 , x c2 , y c2 , x ci , y ci , x c(n− 1) , y c(n− 1) , x cn , and y cn . e rotational masses for n th, (n − 1)th, and ith stories can be expressed as follows [6]: m nθ � m nθ x 2 cn + y 2 cn + ρ 2 n ; m (n− 1)θ � m (n− 1) x 2 c(n− 1) + y 2 c(n− 1) + ρ 2 (n− 1) , m iθ � m iθ x 2 ci + y 2 ci + ρ 2 i , where ρ is the radius of gyration of the corresponding story.
In the stiffness matrix, it is assumed that the moment of inertia of all stories is identical. Elasticity modulus E is also assumed to be identical for all stories. e stiffness matrix components can be written as

Data Availability
e Matlab-Simulink simulations and data used to support the findings of this study are available from the author upon request.

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