Empirical multi-degree-of-freedom-generalized minimum variance control for buildings during earthquakes

Abstract

Structural control of a multi-degree-of-freedom building under earthquake excitation is investigated in this paper. The ARMAX model calculation is developed for a linear representation of multi-degree-of-freedom structure. A control approach based on the generalized minimum variance algorithm is developed and presented. This approach is an empirical method to control the story unit regardless of the coupling with other stories. Kanai-Tajimi and Clough–Penzien models are used to generate the seismic excitations. Those models are calculated using the specific soil parameters. In order to test the control strategy performances under real strong earthquakes, the structure has been subjected to EL Cento earthquake. RST controller form shows the stability conditions and the optimality of the control strategy. Simulation tests using a 3DOF structure are performed and show the effectiveness of the control method using of the empirical method.

Keywords

Generalized minimum variance control autoregressive moving average exogenous model multi-degree-of-freedom structural control multivariable systems

Introduction

The impact of control theory in the different domains of engineering and applied sciences has become increasingly important in the last few decades.^1,2 Researchers are very interested in control against the external disturbances (vehicles, buildings, sensors disturbances, etc.).^2–8 One of the important missions of structural control is to ensure the safety of structures and cities during large earthquakes.^9–13 In fact, Weng et al.¹⁴ have proposed the finite-time vibration control of earthquake-excited linear structures with input time-delay by considering the saturation. The objective of designing controllers is to guarantee the finite-time stability of closed-loop systems while attenuating earthquake-induced vibration of the structures. Gudarzi¹⁵ has presented a robust μ-synthesis output-feedback controller design for seismic alleviation of multi-structural buildings with parametric uncertainties and to tackle the instabilities and performance declines due to these uncertainties. Tınkır et al.¹⁶ have investigated a SolidWorks and SimMechanics-based dynamic modeling technique and displacement control of flexible structure system against the disturbance through theoretical simulation and experimental approach. PI and LQR controllers have been used as a control strategy in the active mode. Both simulation and experimental results have shown the reduction of vibration due to the disturbance effects.

A multi-degree-of-freedom (MDOF) structure can be considered as a large-scale system with interconnected subsystems, which are story units. The problem of interconnection between the story units and the structural properties of the building is considered as a whole and must be addressed. The interconnection effects can also be introduced in the control law formulation, or can be considered as external disturbances to be compensated by control actions. Another problem encountered in MDOF structural control is the number and location of actuators. Several structural models have been studied and developed for control perspective: state space, ARMAX (autoregressive moving average exogenous), etc. Alqado et al.¹⁷ developed five methods for structural identification approach, specifically ARX, ARMAX, BJ, OE, and state-space models, and have implemented them for the identification process. Furthermore, the paper shows that the ARMAX and the output error models had indicated an excellent performance to predict the mathematical models of vibration's propagation in the building. The ARMAX model can give an interesting representation of the system for a digital control perspective. In this case, the system is characterized as stochastic process described by a linear model subjected to external disturbances. Ying et al.¹⁷ proposed an algorithm for the decentralized structural control of tall shear-type buildings under unknown earthquake excitation. A decentralized control algorithm based on the instantaneous optimal control scheme was developed with limited measurements of structural absolute acceleration responses. The inter-connection effect between adjacent substructures was treated as ‘additional unknown disturbances’ at substructural interfaces to each substructure. However, the control algorithm is based on the minimization of some cost functions such as linear quadratic regulator (LQR) which requires a solution of the Riccati equation. This one is subjected to a boundary condition at the terminal time that leads to a sub-optimal solution. So, in such perturbed systems, the controller performances and robustness may be weak.¹⁸

The purpose of this paper is to develop a control strategy applied to an earthquake excited 3DOF structure. This algorithm attempts to minimize a generalized cost function including variance of both output and control effort. For this purpose, a specific model of the system to be controlled and its environment will be developed.

The GMV algorithm has been widely studied in literature. It was introduced by Clarke^19,20 as an extended version of the minimum variance (MV) algorithm initially developed by Åström.^21–23 Our approach presented in this paper is a generalization of the GMV algorithm for the multivariable case. A multivariable ARMAX model of the structure is used.

Our approach is a method that uses the mono-variable GMV controller for each story. The coupling terms appearing in the structural model are not introduced in control law formulation but rather they are considered as external perturbations to be compensated by the control actions. The ARMAX polynomial parameters are not tuned; they are calculated according to the structural parameters. Also, the soil characteristics described by the dynamical model (Kanai-Tajimi, Clough-Penzien) can be introduced within the structural model that leads to an optimal prediction and also defines the stability conditions that take the soil parameters into account. In other words, the poles of the polynomial C(q⁻¹) are the closed-loop poles which are related to the introduction of the soil parameters.

The paper is organized as follows: the next section deals with the development of the dynamic model of the structure, while the following section deals with the presentation of the MIMO ARMAX model. The GMV algorithm is introduced in the subsequent section. The seismic excitation models are then presented.. Then, in the next section, simulation results showed the effectiveness of the developed algorithms. Finally, some conclusions are given.

Dynamical model of an MDOF structure

The purpose of this section is to develop the dynamical model of an MDOF structure under seismic excitation. Figure 1(a) is a schematic representation of the structure. It is a multi-story building with active tendon controllers in each story unit. In order to establish the dynamical model, the following assumptions are considered

Figure 1.

Schematic representation of a multi-degree-of-freedom structure under seismic excitation (a) motion of the structure (b) forces equilibrium of the i^th story unit.

(1) Each story is supposed to be a lumped mass in the girder.

(2) The two vertical axes between two adjacent floors are weightless and inextensible in the vertical direction.

Figure 1(b) is a representation of the dynamic force equilibration at the ith story unit, which can be written as²⁴

f_{I_{i}} (t) + f_{A_{i}} (t) + f_{E_{i}} (t) - f_{A_{i + 1}} (t) - f_{E_{i + 1}} (t) = u_{i} (t) - u_{i + 1} (t)

(1)

where

$f_{I_{i}} (t)$ is the inertial force of the mass m_i

f_{I_{i}} (t) = m_{i} {\ddot{x}}_{i}^{t} (t)

(2)

$f_{A_{i}} (t)$ is the damping force of the i^th story

f_{A_{i}} (t) = c_{i} ({\dot{x}}_{i} (t) - {\dot{x}}_{i - 1} (t))

(3)

$f_{E_{i}} (t)$ is the elastic force of the i^th story

f_{E_{i}} (t) = k_{i} (x_{i} (t) - x_{i - 1} (t))

(4)

where relative displacement x_i(t).u_i(t) is the control force from the controller installed between the (i–1)^th story and the i^th story.

Absolute displacement $x_{i}^{t} (t)$ is defined as

x_{i}^{t} (t) = x_{i} (t) + x_{g} (t)

(5)

where x_g(t) is the ground motion.

The ith story unit is characterized by its characteristic parameters. In this system, it is assumed that the structural mass, mi, and the elastic stiffness, ki, have been concentrated in floors and columns, respectively. Internal viscous damping, ci, is also a parameter that describes the structural behavior.

Substituting equations (2) to (5) into equation (1), we obtain

m_{i} ({\ddot{x}}_{i} + {\ddot{x}}_{g}) + c_{i} ({\dot{x}}_{i} - {\dot{x}}_{i - 1}) + k_{i} (x_{i} - x_{i - 1}) - c_{i + 1} ({\dot{x}}_{i + 1} - {\dot{x}}_{i}) - k_{i + 1} (x_{i + 1} - x_{i}) = u_{i} - u_{i + 1}

(6)

with

i = \bar{1, n}

where

c_{n + 1} = 0

k_{n + 1} = 0

x_{0} = 0

x_{n + 1} = 0

u_{n + 1} = 0

, and

{\ddot{x}}_{g} (t)

represent the ground acceleration.

Equation (6) can be written in the matrix form as²⁴

M \ddot{X} (t) + C \dot{X} (t) + K X (t) = L U (t) - M I_{v} {\ddot{x}}_{g} (t)

(7)

where

$X^{T} (t) = [\begin{matrix} x_{1} (t) & x_{2} (t) & \dots & x_{n} (t) \end{matrix}]$ is the structural displacement vector

$U^{T} (t) = [\begin{matrix} u_{1} (t) & u_{2} (t) & \dots & u_{n} (t) \end{matrix}]$ control vector

M is the (n × n) mass matrix of the structure.

C is the (n × n) damping matrix of the structure.

K is the (n × n) stiffness matrix of the structure.

L is the (n × 1) matrix indicating the location of actuators.

I_v =[1 1……1] ^T unity vector of dimension (n × 1).

ARMAX model of the structure

We are interested in this section to derive the ARMAX model of the MDOF structure.

Structural model

Consider the state space equation of the MDOF structure which can be obtained from equation (7) by choosing $Z^{T} (t) = ⌊ Z_{1}^{T} (t) {Z_{2}}^{T} (t) ⌋ = [X^{T} (t) {\dot{X}}^{T} (t)]$ as a state vector.²⁴

{\begin{array}{l} \dot{Z} (t) = A Z (t) + B U (t) + E {\ddot{x}}_{g} (t) \\ Y (t) = D Z (t) \end{array}

(8)

where

A = [\begin{matrix} \underline{\underline{0}} & I \\ - M^{- 1} K & - M^{- 1} C \end{matrix}]

of dimension (2n × 2n),

B = [\begin{array}{l} \underline{\underline{0}} \\ M^{- 1} L \end{array}]

of dimension (2n × n)

$E = [\begin{array}{l} \underline{0} \\ I_{v} \end{array}]$ of dimension (2n × 1), $D = [\begin{matrix} I & \underline{\underline{0}} \\ \underline{\underline{0}} & \underline{\underline{0}} \end{matrix}]$ of dimension (2n × 2n)

I is the identity matrix of dimension (n × n)

$\underline{\underline{0}}$ is the null matrix of dimension (n × n)

$\underline{0}$ is the null vector of dimension (n × 1).

The digital control of the system needs the knowledge of the discrete representation model. Thus, the discrete model, described by equation (8), can be written in the following form^24–26

[\begin{matrix} A_{11} (q^{- 1}) & \dots & A_{1 n} (q^{- 1}) \\ ⋮ & ⋱ & ⋮ \\ A_{n 1} (q^{- 1}) & \dots & A_{n n} (q^{- 1}) \end{matrix}] X (t) = [\begin{matrix} B_{11} (q^{- 1}) & \dots & B_{1 n} (q^{- 1}) \\ ⋮ & ⋱ & ⋮ \\ B_{n 1} (q^{- 1}) & \dots & B_{n n} (q^{- 1}) \end{matrix}] U (t) + [\begin{matrix} C_{11} (q^{- 1}) & \dots & C_{1 n} (q^{- 1}) \\ ⋮ & ⋱ & ⋮ \\ C_{n 1} (q^{- 1}) & \dots & C_{n n} (q^{- 1}) \end{matrix}] \underline{e} (t)

(9)

where

$X (t) = {[\begin{matrix} x_{1} (t) & \dots & x_{n} (t) \end{matrix}]}^{T}$ is the n × 1 output vector

$U (t) = {[\begin{matrix} u_{1} (t) & \dots & u_{n} (t) \end{matrix}]}^{T}$ is the n × 1 input vector

$\underline{e} (t) = {[\begin{matrix} e_{1} (t) & \dots & e_{n} (t) \end{matrix}]}^{T}$ is the n × 1 zero-mean white noise vector with covariance matrix

E [e (t) e^{T} (t)] = r * I

A_{i i} (q^{- 1}) = 1 + \sum_{l = 1}^{n_{i i}} a_{i i_{l}} q^{- l}, A_{i j} (q^{- 1}) = \sum_{l = 1}^{n_{i j}} a_{i j_{l}} q^{- l}, B_{i j} (q^{- 1}) = q^{- d_{i j}} \sum_{l = 1}^{m_{i j}} b_{i j_{l}} q^{- l}

C_{i i} (q^{- 1}) = 1 + \sum_{l = 1}^{n c_{i i}} c_{i i_{l}} q^{- l}, C_{i j} (q^{- 1}) = \sum_{l = 1}^{n c_{i j}} c_{i j_{l}} q^{- l}

for

i = \bar{1, n} j = \bar{1, n} d_{i j} \geq 0

q⁻¹ shift operator is defined as $q^{- 1} x_{i} (t + 1) = x_{i} (t)$

Empirical model

Our calculation for the empirical model is based on a re-parameterization of the structural model. This is done to obtain a decoupled model where the interactions between stories are considered to be external perturbations. Considering the dynamical equation of motion of the i^th story

m_{i} {\ddot{x}}_{i} + c_{i} ({\dot{x}}_{i} - {\dot{x}}_{i - 1}) + k_{i} (x_{i} - x_{i - 1}) - c_{i + 1} ({\dot{x}}_{i + 1} - {\dot{x}}_{i}) - k_{i + 1} (x_{i + 1} - x_{i}) = u_{i} - u_{i + 1} - m_{i} {\ddot{x}}_{g}

(10)

Equation (10) can be written as

m_{i} {\ddot{x}}_{i} + (c_{i} + c_{i + 1}) {\dot{x}}_{i} + (k_{i} + k_{i + 1}) x_{i} = u_{i} - u_{i + 1} - m_{i} {\ddot{x}}_{g} + c_{i} {\dot{x}}_{i - 1} + c_{i + 1} {\dot{x}}_{i + 1} + k_{i} x_{i - 1} + k_{i + 1} x_{i + 1}

(11)

Introducing new notations, equation (11) has the form

m_{i}^{*} {\ddot{x}}_{i} + c_{i}^{*} {\dot{x}}_{i} + k_{i}^{*} x_{i} = u_{i}^{*} - m_{i}^{*} {\ddot{x}}_{g} + φ_{i}^{*}

(12)

where

m_{i}^{*} = m_{i}

c_{i}^{*} = c_{i} + c_{i + 1}

k_{i}^{*} = k_{i} + k_{i + 1}

$u_{i}^{*} = u_{i} - u_{i + 1}$ is the effective control force applied to the i^th story.

$φ_{i}^{*} = c_{i} {\dot{x}}_{i - 1} + c_{i + 1} {\dot{x}}_{i + 1} + k_{i} x_{i - 1} + k_{i + 1} x_{i + 1}$ contains all the coupling terms from the other story units.

Equation (12) can be interpreted as a new decoupled subsystem with new structural parameters, new control variable and a term of perturbations. The ARMAX model of each subsystem is determined from equation (12). By neglecting the term $φ_{i}^{*}$ and introducing the following notations: $ω_{0 i}^{2} = \frac{k_{i}^{*}}{m_{i}^{*}}$ , $ξ_{i}^{2} = \frac{c_{i}^{*}}{2 m_{i}^{*} ω_{0 i}}$ then applying Laplace transform to equation (12), we obtain

X_{i} (s) = \frac{\frac{1}{m_{i}^{*}}}{s^{2} + 2 ξ_{i} ω_{0 i} s + ω_{0 i}^{2}} U_{i}^{*} (s) - \frac{1}{s^{2} + 2 ξ_{i} ω_{0 i} s + ω_{0 i}^{2}} {\ddot{X}}_{g} (s)

(13)

where

X_{i} (s)

{\ddot{X}}_{g} (s)

and

U_{i}^{*} (s)

are the Laplace transform of

x_{i} (t)

{\ddot{x}}_{g} (t)

and

u_{i}^{*} (t)

respectively; s is the Laplace operator.

Figure 2 shows a bloc diagram of the i^th story model.

Figure 2.

Bloc diagram of the i^th story model.

Depending on the model of seismic excitation, we develop different ARMAX models that can be obtained, and the following cases arise.

Case 1

The seismic excitation model is unknown or is not taken into consideration. Equation (13) has the form

X_{i} (s) = \frac{{H_{1 N}}_{i} (s)}{{H_{1 D}}_{i} (s)} U_{i}^{*} (s) + \frac{{H_{2 N}}_{i} (s)}{{H_{2 D}}_{i} (s)} {\ddot{X}}_{g} (s)

(14)

where

{\begin{cases} {H_{1 D}}_{i} (s) = {H_{2 D}}_{i} (s) = s^{2} + 2 ξ_{i} ω_{0 i} s + ω_{0 i}^{2} \\ {H_{1 N}}_{i} (s) = \frac{1}{m_{i}^{*}} \\ {H_{2 N}}_{i} (s) = - 1 \end{cases}

The ARMAX model of the structure is obtained by the discretization of equation (14)

x_{i} (t) = \frac{B_{i} (q^{- 1})}{A_{i} (q^{- 1})} u_{i}^{*} (t) + \frac{C_{i} (q^{- 1})}{A_{i} (q^{- 1})} {\ddot{x}}_{g} (t)

(15)

where

A_{i} (q^{- 1}) = 1 + a q {i 1}^{- 1} + a_{i 2} q^{- 2}

B_{i} (q^{- 1}) = b_{i 1} q^{- 1} + b_{i 2} q^{- 2}

C_{i} (q^{- 1}) = c_{i 1} q^{- 1} + c_{i 2} q^{- 2}

By analytical discretization of equation (14) using the Z-transform, the polynomial parameters are given by⁵

{\begin{cases} a_{i 1} = - 2 α_{i} β_{i} \\ a_{i 2} = α_{i}^{2} \end{cases}, {\begin{cases} b_{i 1} = \frac{1}{m_{i}^{*} ω_{0 i}} [1 - α_{i} (β_{i} + \frac{ξ_{i} ω_{0 i}}{ω_{i}} γ_{i})] \\ b_{i 2} = \frac{1}{m_{i}^{*} ω_{0 i}} [α_{i}^{2} + α_{i} (\frac{ξ_{i} ω_{0 i}}{ω_{i}} γ_{i} - β_{i})] \end{cases}, {\begin{cases} c_{i 1} = - \frac{1}{ω_{0 i}} [1 - α_{i} (β_{i} + \frac{ξ_{i} ω_{0 i}}{ω_{i}} γ_{i})] \\ c_{i 2} = - \frac{1}{ω_{0 i}} [α_{i}^{2} + α_{i} (\frac{ξ_{i} ω_{0 i}}{ω_{i}} γ_{i} - β_{i})] \end{cases}

with

ω_{i} = ω_{0 i} \sqrt{1 - ξ_{i}^{2}}

α_{i} = \exp (- ξ_{i} ω_{0 i} T_{e})

β_{i} = \cos (ω_{i} T_{e})

γ_{i} = \sin (ω_{i} T_{e})

and T_e is the sampling period.

Case 2

The ground acceleration is described by the Kanai-Tajimi model, i.e.

{\ddot{X}}_{g} (s) = G_{1} (s) E (s)

(16)

where

G_{1} (s) = \frac{G_{1 N} (s)}{G_{1 D} (s)} = \frac{2 ξ_{g} ω_{g} s + ω_{g}^{2}}{s^{2} + 2 ξ_{g} ω_{g} s + ω_{g}^{2}}

E(s) is the Laplace transform of white noise.

Substituting equation (16) into equation (13), we obtain

X_{i} (s) = \frac{\frac{1}{m_{i}^{*}}}{s^{2} + 2 ξ_{i} ω_{0 i} s + ω_{0 i}^{2}} U_{i}^{*} (s) - (\frac{1}{s^{2} + 2 ξ_{i} ω_{0 i} s + ω_{0 i}^{2}}) (\frac{2 ξ_{g} ω_{g} s + ω_{g}^{2}}{s^{2} + 2 ξ_{g} ω_{g} s + ω_{g}^{2}}) E (s)

(17)

which can also be written as

X_{i} (s) = \frac{{H_{1 N}}_{i} (s)}{{H_{1 D}}_{i} (s)} U_{i}^{*} (s) + \frac{{H_{2 N}}_{i} (s) G_{1 N} (s)}{{H_{2 D}}_{i} (s) G_{1 D} (s)} E (s)

(18)

By reducing the second member of equation (18) to a common denominator, we obtain

X_{i} (s) = \frac{{F_{1 N}}_{i} (s)}{F_{D}_{i} (s)} U_{i}^{*} (s) + \frac{F_{2 N}_{i} (s)}{F_{D}_{i} (s)} E (s)

(19)

where

{\begin{cases} F_{D}_{i} (s) = {H_{1 D}}_{i} (s) G_{1 D}_{i} (s) = (s^{2} + 2 ξ_{i} ω_{0 i} s + ω_{0 i}^{2}) (s^{2} + 2 ξ_{g} ω_{g} s + ω_{g}^{2}) \\ {F_{1 N}}_{i} (s) = {H_{1 N}}_{i} (s) G_{1 D}_{i} (s) = \frac{1}{m_{i}^{*}} (s^{2} + 2 ξ_{g} ω_{g} s + ω_{g}^{2}) \\ F_{2 N}_{i} (s) = {H_{2 N}}_{i} (s) G_{1 N}_{i} (s) = - (2 ξ_{g} ω_{g} s + ω_{g}^{2}) \end{cases}

Block diagrams are shown in Figure 3 to illustrate the previous calculations.

Figure 3.

Block diagram of i^th story model under Kanai-Tajimi seismic excitation.

Figure 4.

Schematic representation of the LQR approach for a 3DOF structure. LQR: linear quadratic regulator.

Figure 5.

Schematic representation of the empirical LQR approach for a 3DOF structure. LQR: linear quadratic regulator.

Figure 6.

Schematic representation of the GMV approach for a 3DOF structure. GMV: generalized minimum variance.

Discretization of equation (19) gives the discrete ARMAX model of the structure under the Kanai-Tajimi ground acceleration¹⁸

LQR algorithm

Control algorithms for linear systems have been extensively studied. Optimal control algorithms are based on the minimization of a quadratic performance index whose objective is to maintain the desired system state while minimizing the control effort.¹

LQR for MDOF structure

The LQR is applied for the system represented in equation (8). In this case, the control gain matrix is determined as¹

G = - \frac{1}{2} R^{- 1} B^{T} P

(20)

where R is the weighting matrix for the control force vector.

P is the solution of Riccati equation given as

A^{T} P + P A - \frac{1}{2} P B R^{- 1} B^{T} P + 2 Q = \underline{\underline{0}}

(21)

where Q is the weighting matrix for the state vector,

LQR using the empirical model

In this case, we develop an LQR controller which is calculated for each story of the structure. The controller is designed based on the structural model represented by equation (12). Each controller is calculated using an SDOF structural model of the ith storey by ignoring the coupling terms, represented by the term, due to the interconnection with other subsystems.

m_{i}^{*} {\ddot{x}}_{i} + c_{i}^{*} {\dot{x}}_{i} + k_{i}^{*} x_{i} = u_{i}^{*} - m_{i}^{*} {\ddot{x}}_{g}

(22)

Using the state space concept, equation (22) can be written as

{\dot{Z}}_{i} (t) = A_{i} Z_{i} (t) + B_{i} U_{i} (t) + E_{i} {\ddot{x}}_{g} (t)

(23)

where

{\dot{Z}}_{i} = [\begin{array}{l} x_{i} (t) \\ {\dot{x}}_{i} (t) \end{array}], A_{i} = [\begin{matrix} 0 & 1 \\ - \frac{k_{i}}{m_{i}^{*}} & - \frac{c_{i}}{m_{i}^{*}} \end{matrix}], B_{i} = [\begin{array}{l} 0 \\ \frac{1}{m_{i}^{*}} \end{array}], E_{i} = [\begin{array}{l} 0 \\ - 1 \end{array}]

In this case, the control gain matrix is

G_{i} = - \frac{1}{2} R^{- 1} {B_{i}}^{T} P_{i}

(24)

where R is the weighting matrix for the control force vector.

P_i is the solution of Riccati equation given as

A_{i}^{T} P_{i} + P_{i} A_{i} - \frac{1}{2} P_{i} B_{i} {R_{i}}^{- 1} {B_{i}}^{T} P_{i} + 2 Q = 0

(25)

where Q is the weighting matrix for the state vector.

Generalized minimum variance control algorithm

The generalized minimum variance (GMV) algorithm was introduced by Clarke^20,21 to control the non-minimum phase systems. It is an extension of the MV algorithm²⁶ which, by choosing a certain performance criterion and a certain model of the controlled system and its perturbations, attempts to minimize the variance of the output.

GMV control algorithm for MDOF structure

A generalization of the GMV algorithm for multi-input-multi-output (MIMO) systems has been proposed for structures under earthquakes. The controlled system in this case is assumed to be described by a linear vector difference equation including a moving average of white noise.

The criterion to be minimized is

J = E {[{‖ P \underline{y} (t + d + 1) - R_{w} \underline{w} (t + d + 1) ‖}^{2} + {‖ {Q^{'}}_{m} \underline{u} (t) ‖}^{2}]}^{}

(26)

where

E is the expected value w(t + d + 1) is the n × 1 vector defining the reference signal.

P, R_w, ${Q^{'}}_{m}$ are the weighing polynomial matrices of dimension n × n each.

In the MIMO case, we proceed by discretization of the state space model described by equation (8), we obtain full matrix model.

The discrete multivariable ARMAX model is then obtained. It is given by

\bar{A} (q^{- 1}) X (t) = \bar{B} (q^{- 1}) U (t) + \bar{C} (q^{- 1}) I_{v} {\ddot{x}}_{g} (t)

(27)

where

\begin{array}{l} \bar{A} (q^{- 1}) = [\begin{matrix} {\bar{A}}_{11} (q^{- 1}) & {\bar{A}}_{12} (q^{- 1}) & 0 \\ {\bar{A}}_{21} (q^{- 1}) & {\bar{A}}_{22} (q^{- 1}) & {\bar{A}}_{23} (q^{- 1}) \\ ⋱ \\ 0 & {\bar{A}}_{n (n - 1)} (q^{- 1}) & {\bar{A}}_{n n} (q^{- 1}) \end{matrix}] \\ \bar{B} (q^{- 1}) = [\begin{matrix} {\bar{B}}_{11} (q^{- 1}) & {\bar{B}}_{12} (q^{- 1}) & 0 \\ {\bar{B}}_{22} (q^{- 1}) & {\bar{B}}_{23} (q^{- 1}) \\ ⋱ \\ 0 & {\bar{B}}_{n (n - 1)} (q^{- 1}) & {\bar{B}}_{n n} (q^{- 1}) \end{matrix}] \\ \bar{C} (q^{- 1}) = [\begin{matrix} {\bar{C}}_{11} (q^{- 1}) & 0 \\ ⋱ \\ 0 & {\bar{C}}_{n n} (q^{- 1}) \end{matrix}] \end{array}

With this model, the MIMO GMV controller is calculated.

As in the mono-variable case, we first have to derive the optimal predictor $ϕ^{*} (t + d + 1)$ of $ϕ (t + d + 1) = P \underline{y} (t + d + 1)$ and since it is a future information, it is given by

ϕ^{*} (t + d + 1) {\tilde{C}}^{- 1} (q^{- 1}) [\tilde{F} (q^{- 1}) B^{*} (q^{- 1}) \underline{u} (t) + \tilde{G} (q^{- 1}) \underline{y} (t)]

(28)

where

B^{*} (q^{- 1}) = q B (q^{- 1})

Using equations (27) and (28), it is shown that the control strategy is given by

H (q^{- 1}) \underline{u} (t) = \tilde{C} (q^{- 1}) R_{w} (q^{- 1}) \underline{w} (t + d + 1) - \tilde{G} (q^{- 1}) \underline{y} (t)

(29)

where

H (q^{- 1}) = \tilde{F} (q^{- 1}) B^{*} (q^{- 1}) + \tilde{C} (q^{- 1}) Q_{m} (q^{- 1})

We have shown the efficiency of this control strategy, but the major problem of this method is the huge calculation matrix of Diophantine equation. There is a problem of implementation, control calculation and also problem of the coupling terms that are taken into consideration.

Empirical GMV control algorithm for the structure

The aim of this work is the calculation of one model (the empirical model) that permits us the generalization of the SISO GMV algorithm to the MIMO case without any further modification. So, we calculate the decoupled model by ignoring the interaction term (ϕ_i*). So we can obtain a mono-variable linear time varying system for each story for the structure.

To derive the GMV algorithm, the ARMAX model of the system is used⁷

a (q^{- 1}) y (t) = q^{- d} b (q^{- 1}) u (t) + c (q^{- 1}) e (t)

(30)

where

a (q^{- 1}) = 1 + a_{1} q^{- 1} + \dots + a_{n_{a}} q^{- n_{a}}

b (q^{- 1}) = b_{1} q^{- 1} + \dots + b_{n_{b}} q^{- n_{b}}

c (q^{- 1}) = 1 + c_{1} q^{- 1} + \dots + c_{n_{c}} q^{- n_{c}}

$d \geq 0$ is the time delay of the system y(t) system output u(t) control input e(t) white noise with zero mean and of variance σ².

The performance index to be minimized is

J = E [{(p y (t + d + 1) - r_{w} w (t + d + 1))}^{2} + {(Q' u (t))}^{2}]

(31)

where E is the expected value and w(t + d + 1) is the reference signal p(q⁻¹) r_w(q⁻¹) and Q′(q⁻¹) weighing polynomials. With

p (q^{- 1}) = \frac{p_{N} (q^{- 1})}{p_{D} (q^{- 1})} Q' (q^{- 1}) = \frac{Q'_{N} (q^{- 1})}{Q'_{D} (q^{- 1})}

The degrees of P and r_w can be chosen arbitrarily. We remark in this criterion that w(t + d + 1) is a disposable information, but y(t + d + 1) is not. It is the future information that we must predict. After some calculations, the prediction of py(t + d + 1) = ψ(t + d + 1) is (see Appendix 1)

ψ^{*} (t + d + 1) = \frac{p_{N} b^{*}}{p_{D} a} u (t) + \frac{r}{p_{D} a} e (t)

(32)

where

b^{*} = q b

and R are the solution of the Diophantine equation defined as

a p_{D} s' + q^{- (d + 1)} r = p_{N} c

(33)

With

s^{'} (q^{- 1}) = {s^{'}}_{0} + {s^{'}}_{1} q^{- 1} + \dots + {s^{'}}_{d} q^{- d}

r (q^{- 1}) = r_{0} + r_{1} q^{- 1} + \dots + r_{n_{r}} q^{- n_{r}}

n_{r} = n + n_{d} - 1

n_d is the degree of the polynomial p_D

The minimization of J leads to the GMV control law given by (see Appendix 2)

u (t) = \frac{p_{D} c r_{w} w (t + d + 1) - r y (t)}{p_{D} (s + Q c)}

(34)

where

Q (q^{- 1}) = \frac{{q^{'}}_{N_{0}} p_{D_{0}}}{{q^{'}}_{D_{0}} p_{N_{0}} b_{1}} Q^{'} (q^{- 1})

The controller described by equation (34) is written on the RST form^2,21

u (t) = \frac{T (q^{- 1})}{R (q^{- 1})} w (t + d + 1) + \frac{S (q^{- 1})}{R (q^{- 1})} x (k)

(35)

The closed loop transfer function is deduced from equation (35)

H_{C L} = \frac{x (k)}{x_{r} (k)} = \frac{T (q^{- 1}) B^{*} (q^{- 1}) q^{- d}}{A (q^{- 1}) S (q^{- 1}) + R (q^{- 1}) B^{*} (q^{- 1}) q^{- (d + 1)}}

(36)

The resulting closed-loop poles associated to an RST controller are the poles of the polynomial C(q⁻¹) (with: P_N =1) according to the Diophantine equation described by equation (33). We can remark that similar tracking behavior with the pole placement can be obtained. The polynomial C(q⁻¹) defines the tracking trajectory regulation, and the closed-loop dynamic at the same time¹⁸

P o l e s {C (q^{- 1})} = P o l e s {A (q^{- 1}) S (q^{- 1}) + R (q^{- 1}) B^{*} (q^{- 1}) q^{- (d + 1)}}

So the polynomial parameters C(q⁻¹) lead to an optimal prediction. Moreover, the closed loop characteristic polynomial is C(q⁻¹) = T(q⁻¹), and it also defines the closed loop stability condition, where the poles must be in the unit circles.¹⁹ The empirical model derived previously is used to implement the monovariable case of the GMV to each story independently. Figure 8 is a schematic representation of this control approach. Our contribution is to calculate the GMV controller that has the charge to compensate the perturbations, represented by the term $φ_{i}^{*}$ , due to the interconnection with other subsystems.

Figure 7.

Generalized minimum variance control architecture.

Figure 8.

Schematic representation of the empirical GMV approach for a 3DOF structure. GMV: generalized minimum variance.

Mathematical model of earthquake ground motion

The earthquake ground acceleration is modeled as a uniformly modulated non-stationary random process.²³

{\ddot{x}}_{g} (t) = ψ (t) {\ddot{x}}_{s} (t)

(37)

where ψ(t) is a deterministic nonnegative envelope function and

{\ddot{x}}_{s} (t)

is a stationary random process with zero mean and a Kanai-Tajimi power spectral density

φ_{g} (ω) = [\frac{1 + 4 ξ_{g}^{2} {(\frac{ω}{ω_{g}})}^{2}}{{(1 - {(\frac{ω}{ω_{g}})}^{2})}^{2} + 4 ξ_{g}^{2} {(\frac{ω}{ω_{g}})}^{2}}] S_{0}^{2}

(38)

where ξ_g_, ω_g are the filter parameters and S₀ is the constant spectral density of the white noise. However, it can be shown that the velocity and displacement spectra, which are derived from the acceleration spectra described by equation (38), have strong singularities at zero frequency. These singularities can be removed by using high-pass filter, as suggested by Clough–Penzien.²⁴

Using such a second high-pass filter, the Kanai-Tajimi spectrum is modified as follows to obtain the Clough–Penzien spectrum

φ_{c} (ω) = [\frac{1 + 4 ξ_{g}^{2} {(\frac{ω}{ω_{g}})}^{2}}{{(1 - {(\frac{ω}{ω_{g}})}^{2})}^{2} + 4 ξ_{g}^{2} {(\frac{ω}{ω_{g}})}^{2}}] [\frac{{(\frac{ω}{ω_{c}})}^{4}}{{(1 - {(\frac{ω}{ω_{c}})}^{2})}^{2} + 4 ξ_{c}^{2} {(\frac{ω}{ω_{c}})}^{2}}] S_{0}^{2}

(39)

A particular envelope function ψ(t) given in the following will be used

ψ (t) = {\begin{cases} 0 for t < 0 \\ {(\frac{t}{t_{1}})}^{2} for 0 \leq t \leq t_{1} \\ 1 for t_{1} \leq t \leq t_{2} \\ \exp [- a (t - t_{2})] for t \geq t_{2} \end{cases}

(40)

where t₁, t₂ and a are parameters that should be selected appropriately to reflect the shape and duration of the earthquake ground acceleration. Numerical values of parameters are t₁=3 s, t₂=13 s, a = 0.26, ξ_g=0.65, ω_g=19 rad/s, ξ_c = 0.6, ω_c = 2 rad/s, S₀ = 0.8 × 10⁻² m/s.

The Kanai-Tajimi and Clough–Penzien ground accelerations have been simulated and are presented in Figure 9.

Figure 9.

Simulated ground accelerations.

Simulation results

Simulation tests are performed using a 3DOF structure with the following structural parameters¹¹

m₁=2100 Kg, k₁=1,262,450 N/m, c₁=3675 Ns/m

m₂=2100 Kg, k₂ = 2,607,500 N/m, c₂=10500 Ns/m

m₃=2100 Kg, k₃=2,607,500 N/m, c₃=10500 Ns/m

An active tendon controller is installed in every story unit and the angle of incline of the tendons with respect to the floor is 25°. Thus, the control force vector from the controllers is u/cos25°. Thus, we can suppose that the force is applied at the top of each story and assumed to be activated externally by an independent power supply.

To implement the empirical approach, Table 1 gives the structural parameters, and the corresponding ARMAX model for each story.

Table 1.

Structural parameters and ARMAX model parameters.

	1st story	2nd story	3rd story
m_i	2100 kg	2100 kg	2100 kg
c_i	14,175 Ns/m	21,000 Ns/m	10,500 Ns/m
k_i	3,869,950 N/m	5,215,000 N/m	2,607,500 N/m
	na1=2	na2=2	na3=2
	nb1=2	nb2=2	nb3=2
	nc1=1	nc2=1	nc3=1
a ₀	1	1	1
a ₁	−1.225	−0.9905	−1.451
a ₂	0.8737	0.8187	0.9048
b ₁	8.568 × 10⁻⁸	8.215 × 10⁻⁸	8.842 × 10⁻⁸
b ₂	8.182 × 10⁻⁸	7,667 × 10⁻⁸	8.547 × 10⁻⁸
c ₀	1	1	1
c ₁	0.9549	0.9333	0.9666

ARMAX: autoregressive moving average exogenous

Ponderation polynomials used in the GMV algorithm are

Q_{i} (q^{- 1}) = {2.10}^{- 7}, P_{i} (q^{- 1}) = 1 - 0.5 q^{- 1}, i = \bar{1, 3}

In order to test the control strategy performances under real strong earthquakes, the structure has been subjected to EL Cento earthquake as presented in Figure 10. The Empirical GMV control algorithm has demonstrated good performances.

Figure 10.

El Centro Earthquake.

Responses of the structure to Kanai-Tajimi and Clough–Penzien excitation models are shown in Figures 11 to 22.

In order to demonstrate the effectiveness of our approach, several simulation comparisons have been made.

First, the LQR algorithm has been applied to mitigate the structural vibration under Kanai-Tajimi and Clough–Penzien earthquake. This controller has been calculated using the MIMO structural model. Then, the LQR controllers, using the empirical model, have been calculated to each story of the structure. In this case, we calculate the decoupled model by ignoring the interaction term (ϕ_i*). So we can obtain a mono-variable linear time varying system for each story for the structure.

We have compared all the controller performances with the Empirical GMV control algorithm proposed in this paper. The LQR algorithm proposed (see section LQR algorithm)has presented weak performances compared to our approach. In fact, the results show the superiority of the empirical approach as presented in Tables 2 to 7 and Figures 11 to 20.

Figure 11.

LQR approach: structural response to Clough–Penzien ground acceleration. LQR: linear quadratic regulator.

Figure 12.

LQR approach: structural response to Kanai-Tajimi ground acceleration. LQR: linear quadratic regulator.

Figure 13.

Empirical LQR approach: structural response to Kanai-Tajimi ground acceleration. LQR: linear quadratic regulator.

Figure 14.

Empirical LQR approach: structural response to Clough–Penzien ground acceleration. LQR: linear quadratic regulator.

Figure 15.

GMV approach: structural response to Kanai-Tajimi ground acceleration. GMV: generalized minimum variance.

Figure 16.

GMV approach: structural response to Clough–Penzien ground acceleration.

Figure 17.

Empirical GMV approach: structural response to Kanai-Tajimi ground acceleration. GMV: generalized minimum variance.

Figure 18.

Empirical GMV approach: structural response to Clough–Penzien ground acceleration. GMV: generalized minimum variance.

Figure 19.

LQR approach: structural response to EL Centro Earthquake.

Figure 20.

Empirical LQR approach: structural response to El Centro Earthquake. LQR: linear quadratic regulator.

Figure 21.

GMV approach: structural response to EL Centro Earthquake. GMV: generalized minimum variance.

Figure 22.

Empirical GMV approach: structural response to EL Centro Earthquake. GMV: generalized minimum variance.

Table 2.

Output variance.

	Controller	Output variance
	Controller	1st floor	2nd floor	3rd floor
Kanai-Tajimi	Without control	8.479210^− 7	1.5669 × 10⁻⁶	2.0394 × 10⁻⁶
	LQR	9.4824 × 10⁻⁸	1.7702 × 10⁻⁷	2.3151 × 10⁻⁷
	Empirical LQR	1.3158 × 10⁻⁷	2.3740 × 10⁻⁷	3.1014 × 10⁻⁷
	GMV	5.1373 × 10⁻⁹	8.5847 × 10⁻⁹	8.2635 × 10⁻⁹
	Empirical GMV	1.4868 × 10⁻⁸	1.4473 × 10⁻⁸	7.9022 × 10⁻⁹
Clough-Penzien	Without control	8.5882 × 10⁻⁷	1.5872 × 10⁻⁶	2.0660 × 10⁻⁶
	LQR	9.5233 × 10⁻⁸	1.7788 × 10⁻⁷	2.3269 × 10⁻⁷
	Empirical LQR	1.3249 × 10⁻⁷	2.3913 × 10⁻⁷	3.1245 × 10⁻⁷
	GMV	4.9138 × 10⁻⁹	8.2192 × 10⁻⁹	7.9106 × 10⁻⁹
	Empirical GMV	1.4698 × 10⁻⁸	1.4287 × 10⁻⁸	7.7709 × 10⁻⁹
El Centro	Without control	1.1717 × 10⁻⁶	2.1639 × 10⁻⁶	2.8151 × 10⁻⁶
	LQR	2.1761 × 10⁻⁷	4.0450 × 10⁻⁷	5.2720 × 10⁻⁷
	Empirical LQR	3.0564 × 10⁻⁷	5.5314 × 10⁻⁷	7.2166 × 10⁻⁷
	GMV	4.7789 × 10⁻⁹	1.3361 × 10⁻⁸	1.9143 × 10⁻⁸
	Empirical GMV	5.9258 × 10⁻⁹	1.6835 × 10⁻⁸	2.3354 × 10⁻⁸

LQR: linear quadratic regulator; GMV: generalized minimum variance.

Table 3.

Maximum displacement.

Earthquakes	Controllers	Drift (cm)
		1st floor		2nd floor		3rd floor
		RMS	Peak	RMS	Peak	RMS	Peak
Kanai-Tajimi	Without control	0.6219	1.9152	0.8474	2.6131	0.9605	3.0058
	LQR	0.3125	1.0854	0.4278	1.5023	0.4864	1.7379
	Empirical LQR	0.3663	1.2865	0.4975	1.7512	0.5645	1.9495
	GMV	0.0501	0.0165	0.0682	0.0221	0.0786	0.0251
	Empirical GMV	0.0620	0.0217	0.0737	0.0289	0.0921	0.0309
Clough-Penzien	Without control	0.6236	2.0154	0.8482	2.7502	0.9638	3.1196
	LQR	0.3125	1.0761	0.4236	1.4727	0.4808	1.6833
	Empirical LQR	0.3605	1.3375	0.4915	1.7765	0.5671	1.9583
	GMV	0.0508	0.0166	0.0685	0.0266	0.0789	0.0258
	Empirical GMV	0.0626	0.0224	0.0743	0.0302	0.0929	0.0314
El Centro	Without control	1.0854	5.0291	1.4768	6.8324	1.6896	7.7829
	LQR	0.4756	3.2483	0.6410	4.3809	0.7361	4.9832
	Empirical LQR	0.5563	3.5702	0.7428	4.7593	0.8557	5.3341
	GMV	0.0069	0.0442	0.0113	0.0763	0.0135	0.0925
	Empirical GMV	0.0093	0.0523	0.0141	0.0812	0.0208	0.0961

LQR: linear quadratic regulator; GMV: generalized minimum variance.

Table 4.

Maximum acceleration.

Earthquakes	Controllers	Acceleration (g)
		1st floor		2nd floor		3rd floor
		RMS	Peak	RMS	Peak	RMS	Peak
Kanai-Tajimi	Without control	0.1019	0.3471	0.1369	0.4768	0.1554	0.5478
	LQR	0.0543	0.2562	0.0740	0.3406	0.0931	0.3676
	Empirical LQR	0.0821	0.3873	0.1054	0.5163	0.1052	0.4182
	GMV	0.0024	0.0105	0.0024	0.0102	0.0026	0.0123
	Empirical GMV	0.0030	0.0124	0.0030	0.0124	0.0083	0.0146
Clough–Penzien	Without control	0.1034	0.3356	0.1398	0.4668	0.1576	0.5489
	LQR	0.0583	0.2691	0.0773	0.3443	0.103	0.03726
	Empirical LQR	0.0914	0.3936	0.1094	0.5238	0.1098	0.4246
	GMV	0.0024	0.0108	0.0025	0.0110	0.0026	0.0126
	Empirical GMV	0.0030	0.0127	0.0031	0.0126	0.0083	0.0149
El Centro	Without control	0.1654	0.9524	0.2276	1.2381	0.2564	1.3428
	LQR	0.0714	0.6174	0.1045	0.8091	0.1136	0.8951
	Empirical LQR	0.1026	0.7059	0.1249	0.8942	0.1375	1.0024
	GMV	0.0033	0.0381	0.0055	0.0583	0.0067	0.0672
	Empirical GMV	0.0042	0.0408	0.0078	0.0624	0.0091	0.0712

LQR: linear quadratic regulator; GMV: generalized minimum variance.

Table 5.

Maximum control effort.

	Controller	Max control force U_max (N)
	Controller	1st floor	2nd floor	3rd floor
Kanai-Tajimi	LQR	2704.7	989.3	523
	Empirical LQR	583.7	1677	10596
	GMV	1474.3	1398.6	1053.5
	Empirical GMV	1567.4	1217.7	738.6
Clough-Penzien	LQR	2713.6	1000	528.5
	Empirical LQR	604.3	1637.6	10,578
	GMV	1462.8	1384.4	1012.3
	Empirical GMV	1492.7	1021.3	721.8
El Centro	LQR	8561	3254.3	1796.8
	Empirical LQR	1485.3	3912.3	30,598
	GMV	2210.6	1324.5	761
	Empirical GMV	2834.2	1573.8	943.4

LQR: linear quadratic regulator; GMV: generalized minimum variance.

Table 6.

Responses and accelerations ratios (evaluation criteria).

	Controller	J1			J2
	Controller	1st floor	2nd floor	3rd floor	1st floor	2nd floor	3rd floor
Kanai-Tajimi	LQR	0.5654	0.5747	0.5767	0.7353	0.7234	0.6667
	Empirical LQR	0.6702	0.6705	0.6467	1.1176	1.0851	0.7778
	GMV	0.0086	0.0085	0.0084	0.0294	0.0213	0.0222
	Empirical GMV	0.0113	0.0110	0.0103	0.0364	0.0263	0.0273
Clough-Penzien	LQR	0.5323	0.5345	0.5402	0.7879	0.7391	0.6852
	Empirical LQR	0.6617	0.6436	0.6270	1.1818	1.1304	0.7593
	GMV	0.0083	0.0083	0.0083	0.0303	0.0239	0.0222
	Empirical GMV	0.0111	0.0109	0.0100	0.0384	0.0273	0.0275
El Centro	LQR	0.6454	0.6413	0.6467	0.6421	0.6504	0.6642
	Empirical LQR	0.7112	0.6955	0.6851	0.7368	0.7236	0.7463
	GMV	0.0088	0.0112	0.0119	0.0400	0.0472	0.0500
	Empirical GMV	0.0104	0.0118	0.0123	0.0421	0.0507	0.0531

LQR: linear quadratic regulator; GMV: generalized minimum variance.

Table 7.

RMS responses and acceleration ratios (evaluation criteria).

	Controller	J3			J4
	Controller	1st floor	2nd floor	3rd floor	1st floor	2nd floor	3rd floor
Kanai-Tajimi	LQR	0.4989	0.5013	0.5024	0.5647	0.5712	0.5783
	Empirical LQR	0.5877	0.5806	0.5815	0.8669	0.7717	0.6816
	GMV	0.0816	0.0819	0.0823	0.0244	0.1882	0.1724
	Empirical GMV	0.0826	0.0845	0.0857	0.0278	0.2084	0.1954
Clough-Penzien	LQR	0.4970	0.4996	0.5007	0.5639	0.5703	0.5775
	Empirical LQR	0.5863	0.5793	0.5802	0.8643	0.7697	0.6905
	GMV	0.0818	0.0821	0.0825	0.0244	0.1883	0.1725
	Empirical GMV	0.0827	0.0849	0.0864	0.0278	0.2086	0.1961
El Centro	LQR	0.4310	0.4324	0.4327	0.4448	0.4477	0.4515
	Empirical LQR	0.5107	0.5056	0.5063	0.6021	0.5564	0.5387
	GMV	0.0542	0.0739	0.0754	0.0573	0.0751	0.0759
	Empirical GMV	0.0568	0.0783	0.0791	0.0607	0.0796	0.0801

LQR: linear quadratic regulator; GMV: generalized minimum variance ; RMS : root mean square.

On the other hand, we have proposed the GMV control algorithm for MDOF structure under earthquake. This controller was developed based on the ARMAX model calculation using the MIMO structural model. The GMV controller has shown its efficiency.

The Empirical GMV method has been compared to the MDOF GMV algorithm. The results have shown the robustness of the proposed method as presented in Tables 2 to 7 and Figures 11 to 22.

Simulation results show the effectiveness of the control approach. Structural responses are significantly reduced with an acceptable control effort. Indeed, the deep comparison made in this paper has shown a small advantage for the multivariable case over the empirical approach. This result is logical because the MIMO case takes the interconnection terms into account in the control law calculation which brings more suitable performances. On the other hand, even if the interconnection terms are considered as external perturbations, the empirical approach has shown very satisfying results (very close to MIMO GMV results) using a simplified structural modal with less complicated control law calculations. On the other hand, the GMV controller has shown good robustness for compensating the external perturbations

For comparison purposes, Table 2 gives the output variance for each story.

To highlight the controllers performances, the displacement Peak, the RMS and the control effort peak of each controller of each story of the structures under different earthquakes are presented in Tables 3 to 5.

To evaluate the control algorithm performances, the following evaluation criteria are adopted from literature.^28–31 are considered.

Story displacement ratio

J_{1} = \frac{\max | x_{c} |}{\max | x_{u} |}

where x_c is the controlled story displacement x_u is the uncontrolled story displacement.

Story acceleration ratio

J_{2} = \frac{\max | {\ddot{x}}_{c} |}{\max | {\ddot{x}}_{u} |}

where

${\ddot{x}}_{c}$ is the controlled story acceleration.

${\ddot{x}}_{u}$ is the uncontrolled story acceleration.

Root mean square (RMS) story displacement ratio

J_{3} = \frac{{\vec{x}}_{c}}{{\vec{x}}_{u}}

is calculated from

\vec{x} = s t d (x) * \sqrt{T_{s} / T_{f}}

where

T_s is the sampling time.

T_f is the total excitation duration.std is the standard deviation.

RMS story acceleration ratio

J_{4} = \frac{{\vec{\ddot{x}}}_{c}}{{\vec{\ddot{x}}}_{u}}

$\vec{\ddot{x}}$ is calculated from

\vec{\ddot{x}} = s t d (\ddot{x}) * \sqrt{T_{s} / T_{f}}

Tables 6 and 7 show the reduction in the peak story displacement without control compared with the proposed control algorithms.

Conclusions

This paper investigated the GMV control algorithm applied to earthquake excited MDOF structures. The control approach was developed and presented based on an empirical model. Our approach has given good performances and the structural responses are significantly reduced. This approach is an extension to the mono-variable case of the classical GMV algorithm. It is designed for the structure as a multivariable system based on local GMV controllers.

Simulation results have shown that our method (the empirical approach) is efficient, despite the fact that this approach does not consider the interaction effects between story units.

It has been shown that the multivariable model is complicated with a well-established theoretical development. In fact, the MIMO case has the disadvantage of the great amount of calculations and matrix manipulation especially in the case of tall buildings with a large number of degrees of freedom. However, our approach has the advantage of presenting a decentralized system. Each local GMV controller can be implemented in local computer hardware which reduces significantly time calculations and memory requirements.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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