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

Empirical multi-degree-of-freedom-generalized minimum variance control for buildings during... 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 1,2 important in the last few decades. Researchers are very interested in control against the external disturbances 2–8 (vehicles, buildings, sensors disturbances, etc.). One of the important missions of structural control is to ensure 9–13 14 the safety of structures and cities during large earthquakes. In fact, Weng et al. 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. Gudarzi has presented a robust l-synthesis output-feedback controller design for seismic alleviation of multi-structural buildings with parametric uncertain- ties and to tackle the instabilities and performance declines due to these uncertainties. Tınkır et al. 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 LSEI Laboratory, University of Science and Technology Houari Boumediene (USTHB), Algiers, Algeria Corresponding author: Mohamed Azira, LSEI Laboratory, University of Science and Technology Houari Boumediene (USTHB), BP 32 Alia, Algiers 16111, Algeria. Email: mohamed.azira@yahoo.fr Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 4 Journal of Low Frequency Noise, Vibration and Active Control 37(1) 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. 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 perspec- tive. In this case, the system is characterized as stochastic process described by a linear model subjected to external disturbances. Ying et al. 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. 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. 19,20 The GMV algorithm has been widely studied in literature. It was introduced by Clarke as an extended 21–23 version of the minimum variance (MV) algorithm initially developed by Astrom. 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 ) 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 conclu- sions 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 (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 as f ðtÞþ f ðtÞþ f ðtÞ f ðtÞ f ðtÞ¼ u ðtÞ u ðtÞ (1) I A E A E i iþ1 i i i iþ1 iþ1 where f ðtÞ is the inertial force of the mass m I i f ðtÞ¼ m x ðtÞ (2) I i i i Azira and Guenfaf 5 (a) (b) Figure 1. Schematic representation of a multi-degree-of-freedom structure under seismic excitation (a) motion of the structure th (b) forces equilibrium of the i story unit. th f ðtÞ is the damping force of the i story _ _ f ðtÞ¼ c ðx ðtÞ x ðtÞÞ (3) A i i i1 th f ðtÞ is the elastic force of the i story f ðtÞ¼ k ðx ðtÞ x ðtÞÞ (4) E i i i1 th where relative displacement x (t).u (t) is the control force from the controller installed between the (i–1) story and i i th the i story. Absolute displacement x ðtÞ is defined as x ðtÞ¼ x ðtÞþ x ðtÞ (5) i g where x (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 ðx þ x Þþ c ðx  x Þþ k ðx  x Þ c ðx  x Þ k ðx  x Þ¼ u  u (6) i i g i i i1 i i i1 iþ1 iþ1 i iþ1 iþ1 i i iþ1 with i ¼ 1; n where c ¼ 0, k ¼ 0, x ¼ 0, x ¼ 0, u ¼ 0, and x ðtÞ represent the ground acceleration. nþ1 nþ1 0 nþ1 nþ1 g 6 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Equation (6) can be written in the matrix form as € _ € MXðtÞþ CXðtÞþ KXðtÞ¼ LUðtÞ MI x ðtÞ (7) v g where X ðtÞ¼ ½ x ðtÞ x ðtÞ  x ðtÞ  is the structural displacement vector 1 2 n U ðtÞ¼½ u ðtÞ u ðtÞ  u ðtÞ control vector 1 2 n 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 =[1 1.... . .1] 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 hi T T T T Z ðtÞ¼bZ ðtÞ Z ðtÞc ¼ X ðtÞ X ðtÞ as a state vector. _ € ZðtÞ¼ AZðtÞþ BUðtÞþ Ex ðtÞ (8) YðtÞ¼ DZðtÞ "# "# 0 I 0 where A ¼ of dimension (2n  2n), B ¼ of dimension (2n  n) 1 1 1 M K M C M L 2 3 "# I 0 4 5 E ¼ of dimension (2n  1), D ¼ of dimension (2n  2n) 0 0 I is the identity matrix of dimension (n  n) 0 is the null matrix of dimension (n  n) 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 24–26 model, described by equation (8), can be written in the following form 2 3 2 3 2 3 1 1 1 1 1 1 A ðq Þ  A ðq Þ B ðq Þ B ðq Þ C ðq Þ  C ðq Þ 11 1n 11 1n 11 1n 6 7 6 7 6 7 6 7 6 7 6 7 . . . . . . . . . . . . XðtÞ¼ . . . UðtÞþ . . . eðtÞ (9) 6 7 6 7 6 7 . . . . . . . . . 4 5 4 5 4 5 1 1 1 1 1 1 A ðq Þ  A ðq Þ B ðq Þ B ðq Þ C ðq Þ  C ðq Þ n1 nn n1 nn n1 nn where XðtÞ¼½ x ðtÞ x ðtÞ is the n  1 output vector 1 n UðtÞ¼½ u ðtÞ u ðtÞ is the n  1 input vector 1 n eðtÞ¼½ e ðtÞ  e ðtÞ is the n  1 zero-mean white noise vector with covariance matrix 1 n EeðtÞe ðtÞ¼ r  I n m n ij ij ii X X X 1 l 1 l 1 d l ij A ðq Þ¼ 1 þ a q ; A ðq Þ¼ a q ; B ðq Þ¼ q b q ii ii ij ij ij ij l l l l¼1 l¼1 l¼1 Azira and Guenfaf 7 nc nc ij ii X X 1 l 1 l C ðq Þ¼ 1 þ c q ; C ðq Þ¼ c q ii ii ij ij l l l¼1 l¼1 for i ¼ 1;nj ¼ 1;nd  0 ij q shift operator is defined as q xðÞ t þ 1 ¼ x ðÞ t i i 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. th Considering the dynamical equation of motion of the i story € _ _ _ _ € m x þ c ðx  x Þþ k ðx  x Þ c ðx  x Þ k ðx  x Þ¼ u  u  m x (10) i i i i i1 i i i1 iþ1 iþ1 i iþ1 iþ1 i i iþ1 i g Equation (10) can be written as € _ € _ _ m x þðc þ c Þx þðk þ k Þx ¼ u  u  m x þ c x þ c x þ k x þ k x (11) i i i iþ1 i i iþ1 i i iþ1 i g i i1 iþ1 iþ1 i i1 iþ1 iþ1 Introducing new notations, equation (11) has the form € _ € m x þ c x þ k x ¼ u  m x þ / (12) i i i g i i i i i i where m ¼ m c ¼ c þ c i iþ1 k ¼ k þ k i iþ1 th u ¼ u  u is the effective control force applied to the i story. i iþ1 _ _ / ¼ c x þ c x þ k x þ k x contains all the coupling terms from the other story units. i i1 iþ1 iþ1 i i1 iþ1 iþ1 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). k c i i By neglecting the term / and introducing the following notations: x ¼ ,n ¼ then applying Laplace i 0i m i 2m x 0i i i transform to equation (12), we obtain m 1 X ðÞ s ¼ U ðÞ s  X ðÞ s (13) i g 2 2 2 2 s þ 2n x s þ x s þ 2n x s þ x 0i 0i i i 0i 0i € € where X ðÞ s , X ðÞ s and U ðÞ s are the Laplace transform of x ðÞ t , x ðÞ t and u ðÞ t respectively; s is the Laplace i g i g i i operator. th Figure 2 shows a bloc diagram of the i 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 H ðÞ s H ðÞ s 1Ni 2Ni X ðÞ s ¼ U ðÞ s þ X ðÞ s (14) i g H ðÞ s H ðÞ s 1Di 2Di 8 Journal of Low Frequency Noise, Vibration and Active Control 37(1) th Figure 2. Bloc diagram of the i story model. where 2 2 > H ðÞ s ¼ H ðsÞ¼ s þ 2n x s þ x 1Di 2Di 0i > 0i H ðÞ s ¼ 1Ni > m > i H ðÞ s ¼ 1 2Ni The ARMAX model of the structure is obtained by the discretization of equation (14) 1 1 B q C q i i x ðÞ t ¼ u ðÞ t þ x ðÞ t (15) i g 1 1 AðÞ q AðÞ q i i where 1 i1i11 2 A ðq Þ¼ 1 þ a q þ a q i i2 i1 1 1 2 B ðq Þ¼ b q þ b q i i1 i2 1 1 2 C ðq Þ¼ c q þ c q i i1 i2 By analytical discretization of equation (14) using the Z-transform, the polynomial parameters are given by 8 8 8 1 n x i 0i > > 1 n x > i 0i > > > c ¼ 1  a b þ c > > > i1 i b ¼ 1  a b þ c i i > > > i1 i i i > >  > x x 0i i < < m x x < 0i i a ¼2a b i i1 i ; ; > > > > a ¼ a > > i2 i 1 n x > > > i 0i 2 1 n x > > > i 0i > > b ¼ a þ a c  b > i2 i : : i i i : c ¼ a þ a c  b i2 i i i i m x x 0i i i x x 0i i qffiffiffiffiffiffiffiffiffiffiffiffiffi with x ¼ x 1  n , a ¼ expðn x T Þ, b ¼ cosðx T Þ, c ¼ sinðx T Þ and T is the sampling period. i 0i i i 0i e i e i e e i i i Case 2. The ground acceleration is described by the Kanai-Tajimi model, i.e. X ðÞ s ¼ G ðsÞEsðÞ (16) g 1 2n x sþx G ðsÞ g g 1N g where G ðÞ s ¼ ¼ 1 2 2 G ðsÞ s þ2n x sþx 1D g g E(s) is the Laplace transform of white noise. Substituting equation (16) into equation (13), we obtain 2n x s þ x m 1 g g X ðÞ s ¼ U ðÞ s  EsðÞ (17) 2 2 2 2 2 2 s þ 2n x s þ x s þ 2n x s þ x s þ 2n x s þ x i 0i i 0i g g 0i 0i g Azira and Guenfaf 9 th Figure 3. Block diagram of i story model under Kanai-Tajimi seismic excitation. which can also be written as H ðÞ s H ðÞ s G ðsÞ 1Ni 2Ni 1N X ðÞ s ¼ U ðÞ s þ EsðÞ (18) H ðÞ s H ðÞ s G ðÞ s 1Di 2Di 1D By reducing the second member of equation (18) to a common denominator, we obtain F ðÞ s F ðÞ s 1Ni 2Ni X ðÞ s ¼ U ðÞ s þ EsðÞ (19) F ðÞ s F ðÞ s Di Di 2 2 2 2 F ðÞ s ¼ H ðÞ s G ðÞ s ¼ s þ 2n x s þ x s þ 2n x s þ x > Di 1Di 1Di 0i g i 0i g g 2 2 where F ðÞ s ¼ H ðÞ s G ðÞ s ¼ s þ 2n x s þ x 1Ni 1Ni 1Di g g g > m : 2 F ðÞ s ¼ H ðsÞG ðsÞ¼ 2n x s þ x 2Ni 2Ni 1Ni g g g Block diagrams are shown in Figure 3 to illustrate the previous calculations. 10 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Discretization of equation (19) gives the discrete ARMAX model of the structure under the Kanai-Tajimi ground acceleration 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. LQR for MDOF structure The LQR is applied for the system represented in equation (8). In this case, the control gain matrix is determined as 1 T G ¼ R B P (20) where R is the weighting matrix for the control force vector. P is the solution of Riccati equation given as T 1 T A P þ PA  PBR B P þ 2Q ¼ 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 intercon- nection with other subsystems. € _ € m x þ c x þ k x ¼ u  m x (22) i i i g i i i i i Using the state space concept, equation (22) can be written as _ € Z ðtÞ¼ A Z ðtÞþ B U ðtÞþ E x ðtÞ (23) i i i i i i g Figure 4. Schematic representation of the LQR approach for a 3DOF structure. LQR: linear quadratic regulator. Azira and Guenfaf 11 Figure 5. Schematic representation of the empirical LQR approach for a 3DOF structure. LQR: linear quadratic regulator. 2 3 2 3 "# "# 01 0 x ðtÞ 0 4 5 4 5 where Z ¼ ðtÞ ; A ¼ k c ; B ¼ 1 ; E ¼ i i i i i i x 1 m m m i i i In this case, the control gain matrix is 1 T G ¼ R B P (24) i i i where R is the weighting matrix for the control force vector. P is the solution of Riccati equation given as T 1 T A P þ P A  P B R B P þ 2Q ¼ 0 (25) i i i i i i i i i where Q is the weighting matrix for the state vector. Generalized minimum variance control algorithm 20,21 The generalized minimum variance (GMV) algorithm was introduced by Clarke to control the non-minimum phase systems. It is an extension of the MV algorithm 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 hi 2 0 2 J ¼ E kPytðÞ þ d þ 1 R wtðÞ þ d þ 1k þkQ utðÞk (26) where E is the expected value w(t þ d þ 1) is the n  1 vector defining the reference signal. 12 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Figure 6. Schematic representation of the GMV approach for a 3DOF structure. GMV: generalized minimum variance. P, R ,Q are the weighing polynomial matrices of dimension n  n each. w m 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 1 1 1 Aðq ÞXðtÞ¼ Bðq ÞUðtÞþ Cðq ÞI x ðtÞ (27) v g where 2 3 1 1 A ðq Þ A ðq Þ 0 11 12 6 7 6 7 1 1 1 6 7 A ðq Þ A ðq Þ A ðq Þ 21 22 23 6 7 6 7 ðq Þ¼ A 6 7 6 . 7 6 7 6 7 4 5 1 1 0 A ðq Þ A ðq Þ nðn1Þ nn 2 3 1 1 B ðq Þ B ðq Þ 0 11 12 6 7 6 7 1 1 6 7 B ðq Þ B ðq Þ 22 23 6 7 6 7 Bðq Þ¼ 6 7 6 . 7 6 7 6 7 4 5 1 1 0 B ðq Þ B ðq Þ nðn1Þ nn 2 3 C ðq Þ 0 6 7 6 7 6 7 1 . Cðq Þ¼ . 6 7 6 7 4 5 0 C ðq Þ nn Azira and Guenfaf 13 With this model, the MIMO GMV controller is calculated. As in the mono-variable case, we first have to derive the optimal predictor u ðt þ d þ 1Þ of uðt þ d þ 1Þ¼ P yðt þ d þ 1Þ and since it is a future information, it is given by hi 1 1  1 1 ~ ~ ~ u ðt þ d þ 1ÞC ðq Þ Fðq ÞB ðq ÞuðtÞþ Gðq ÞyðtÞ (28) 1 1 where B ðq Þ¼ qBðq Þ Using equations (27) and (28), it is shown that the control strategy is given by 1 1 1 1 ~ ~ Hðq ÞuðtÞ¼ Cðq ÞR ðq Þwðt þ d þ 1Þ Gðq ÞyðtÞ (29) 1 1  1 1 1 ~ ~ where Hðq Þ¼ Fðq ÞB ðq Þþ Cðq ÞQ ðq Þ 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 (u *). 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 used 1 d 1 1 aðq ÞytðÞ ¼ q bðq ÞutðÞ þ cðq ÞetðÞ (30) where 1 1 n aðq Þ¼ 1 þ a q þ þ a q 1 n 1 1 n bðq Þ¼ b q þ  þ b q 1 n 1 1 n cðq Þ¼ 1 þ c q þ þ c q 1 n d  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 r . The performance index to be minimized is hi 2 2 J ¼EpðÞ yðÞ t þ d þ 1 r wtðÞ þ d þ 1 þðÞ Q’utðÞ (31) 1 1 1 where E is the expected value and w(t þ d þ 1) is the reference signal p(q ) r (q ) and Q (q ) weighing polynomials. With 1 0 1 p ðq Þ Q ðq Þ 1 0 1 N pðq Þ¼ Q ðq Þ¼ 1 0 1 p ðq Þ Q ðq Þ The degrees of P and r can be chosen arbitrarily. We remark in this criterion that w(t þ d þ 1) is a disposable information, but y(t þ d þ 1) is not. It is the future information that we must predict. After some calculations, the prediction of py(t þ d þ 1) ¼ w(t þ d þ 1) is (see Appendix 1) 14 Journal of Low Frequency Noise, Vibration and Active Control 37(1) p b r wðÞ t þ d þ 1¼ utðÞ þ etðÞ (32) p a p a D D where b ¼ qb and R are the solution of the Diophantine equation defined as 0 ðdþ1Þ ap s þ q r ¼ p c (33) D N With 0 1 0 0 1 0 d s ðq Þ¼ s þ s q þ þ s q 0 1 d 1 1 n rðq Þ¼ r þ r q þ þ r q 0 1 n n ¼ n þ n  1 r d n is the degree of the polynomial p d D The minimization of J leads to the GMV control law given by (see Appendix 2) p cr wtðÞ þ d þ 1 ryðÞ t D w utðÞ ¼ (34) pðÞ s þ Qc where q p N 0 1 0 0 1 Qðq Þ¼ Q ðq Þ q p b N 1 D 0 2,21 The controller described by equation (34) is written on the RST form 1 1 Tðq Þ Sðq Þ uðtÞ¼ wðt þ d þ 1Þþ xðkÞ (35) 1 1 Rðq Þ Rðq Þ The closed loop transfer function is deduced from equation (35) 1  1 d xðkÞ Tðq Þ B ðq Þ q H ¼ ¼ (36) CL 1 1 1  1 ðdþ1Þ x ðkÞ Aðq Þ Sðq Þþ Rðq Þ B ðq Þ q Figure 7. Generalized minimum variance control architecture. Azira and Guenfaf 15 Figure 8. Schematic representation of the empirical GMV approach for a 3DOF structure. GMV: generalized minimum variance. The resulting closed-loop poles associated to an RST controller are the poles of the polynomial C(q ) (with: P =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 ) defines the tracking trajectory regu- lation, and the closed-loop dynamic at the same time no 1 1 1 1  1 ðÞ dþ1 Poles C q ¼ Poles A q Sq þRq B q q So the polynomial parameters C(q ) lead to an optimal prediction. Moreover, the closed loop characteristic 1 1 polynomial is C(q ) ¼ T(q ), and it also defines the closed loop stability condition, where the poles must be in the unit circles. 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 / , due to the interconnection with other subsystems. Mathematical model of earthquake ground motion The earthquake ground acceleration is modeled as a uniformly modulated non-stationary random process. € € x ðÞ t ¼ wðÞ t x ðÞ t (37) g s where w(t) is a deterministic nonnegative envelope function and x ðÞ t is a stationary random process with zero mean and a Kanai-Tajimi power spectral density 2 3 6 7 1 þ 4n 6 g 7 / ðÞ x ¼ S (38) 6 7 g 0 4 2 2 5 x x 1  þ 4n x g x g g where n x are the filter parameters and S is the constant spectral density of the white noise. However, it can be g, g 0 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. 16 Journal of Low Frequency Noise, Vibration and Active Control 37(1) (a) (b) Figure 9. Simulated ground accelerations. Using such a second high-pass filter, the Kanai-Tajimi spectrum is modified as follows to obtain the Clough– Penzien spectrum 2 32 3 2 4 x x 6 76 7 1 þ 4n g x x g c 6 76 7 / ðÞ x ¼ S (39) 6 76 7 4 2 2 54 2 2 5 2 2 x x x x 1  þ 4n 1  þ 4n g c x x x x g g c c A particular envelope function w(t) given in the following will be used 0 for t < 0 for 0  t  t wðÞ t ¼ (40) 1 for t  t  t 1 2 exp½ atðÞ  t for t  t 2 2 where t , t and a are parameters that should be selected appropriately to reflect the shape and duration of the 1 2 earthquake ground acceleration. Numerical values of parameters are t =3 s, t =13 s, a ¼ 0.26, n =0.65, x =19 1 2 g g rad/s, n ¼ 0.6, x ¼ 2 rad/s, S ¼ 0.8  10 m/s. c c 0 The Kanai-Tajimi and Clough–Penzien ground accelerations have been simulated and are presented in Figure 9. Simulation results Simulation tests are performed using a 3DOF structure with the following structural parameters m =2100 Kg, k =1,262,450 N/m, c =3675 Ns/m 1 1 1 m =2100 Kg, k = 2,607,500 N/m, c =10500 Ns/m 2 2 2 m =2100 Kg, k =2,607,500 N/m, c =10500 Ns/m 3 3 3 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. Azira and Guenfaf 17 Table 1. Structural parameters and ARMAX model parameters. 1st story 2nd story 3rd story m 2100 kg 2100 kg 2100 kg c 14,175 Ns/m 21,000 Ns/m 10,500 Ns/m k 3,869,950 N/m 5,215,000 N/m 2,607,500 N/m na1¼2 na2¼2 na3¼2 nb1¼2 nb2¼2 nb3¼2 nc1¼1 nc2¼1 nc3¼1 a 11 1 a 1.225 0.9905 1.451 a 0.8737 0.8187 0.9048 8 8 8 b 8.568  10 8.215  10 8.842  10 8 8 8 b 8.182  10 7,667  10 8.547  10 c 11 1 c 0.9549 0.9333 0.9666 ARMAX: autoregressive moving average exogenous Figure 10. El Centro Earthquake. Ponderation polynomials used in the GMV algorithm are 1 7 1 1 Q ðq Þ¼ 2:10 ; P ðq Þ¼ 1  0:5q ; i ¼ 1; 3 i i 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 dem- onstrated good performances. Responses of the structure to Kanai-Tajimi and Clough–Penzien excitation models are shown in Figures 11 to 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 (u *). 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. 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. 18 Journal of Low Frequency Noise, Vibration and Active Control 37(1) (a) (b) (c) Figure 11. LQR approach: structural response to Clough–Penzien ground acceleration. LQR: linear quadratic regulator. (a) (b) (c) Figure 12. LQR approach: structural response to Kanai-Tajimi ground acceleration. LQR: linear quadratic regulator. Azira and Guenfaf 19 (a) (b) (c) Figure 13. Empirical LQR approach: structural response to Kanai-Tajimi ground acceleration. LQR: linear quadratic regulator. (a) (b) (c) Figure 14. Empirical LQR approach: structural response to Clough–Penzien ground acceleration. LQR: linear quadratic regulator. 20 Journal of Low Frequency Noise, Vibration and Active Control 37(1) (a) (b) (c) Figure 15. GMV approach: structural response to Kanai-Tajimi ground acceleration. GMV: generalized minimum variance. (a) (b) (c) Figure 16. GMV approach: structural response to Clough–Penzien ground acceleration. Azira and Guenfaf 21 (a) (b) (c) Figure 17. Empirical GMV approach: structural response to Kanai-Tajimi ground acceleration. GMV: generalized minimum variance. (a) (b) (c) Figure 18. Empirical GMV approach: structural response to Clough–Penzien ground acceleration. GMV: generalized minimum variance. 22 Journal of Low Frequency Noise, Vibration and Active Control 37(1) (a) (b) (c) Figure 19. LQR approach: structural response to EL Centro Earthquake. (a) (b) (c) Figure 20. Empirical LQR approach: structural response to El Centro Earthquake. LQR: linear quadratic regulator. Azira and Guenfaf 23 (a) (b) (c) Figure 21. GMV approach: structural response to EL Centro Earthquake. GMV: generalized minimum variance. (a) (b) (c) Figure 22. Empirical GMV approach: structural response to EL Centro Earthquake. GMV: generalized minimum variance. 24 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Table 2. Output variance. Output variance Controller 1st floor 2nd floor 3rd floor 7 6 6 Kanai-Tajimi Without control 8.479210 1.5669  10 2.0394  10 8 7 7 LQR 9.4824  10 1.7702  10 2.3151  10 7 7 7 Empirical LQR 1.3158  10 2.3740  10 3.1014  10 9 9 9 GMV 5.1373  10 8.5847  10 8.2635  10 8 8 9 Empirical GMV 1.4868  10 1.4473  10 7.9022  10 7 6 6 Clough-Penzien Without control 8.5882  10 1.5872  10 2.0660  10 8 7 7 LQR 9.5233  10 1.7788  10 2.3269  10 7 7 7 Empirical LQR 1.3249  10 2.3913  10 3.1245  10 9 9 9 GMV 4.9138  10 8.2192  10 7.9106  10 8 8 9 Empirical GMV 1.4698  10 1.4287  10 7.7709  10 6 6 6 El Centro Without control 1.1717  10 2.1639  10 2.8151  10 7 7 7 LQR 2.1761  10 4.0450  10 5.2720  10 7 7 7 Empirical LQR 3.0564  10 5.5314  10 7.2166  10 9 8 8 GMV 4.7789  10 1.3361  10 1.9143  10 9 8 8 Empirical GMV 5.9258  10 1.6835  10 2.3354  10 LQR: linear quadratic regulator; GMV: generalized minimum variance. Table 3. Maximum displacement. Drift (cm) 1st floor 2nd floor 3rd floor Earthquakes Controllers 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. 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 Azira and Guenfaf 25 Table 4. Maximum acceleration. Acceleration (g) 1st floor 2nd floor 3rd floor Earthquakes Controllers 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. Max control force U (N) max 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. 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 litera- 28–31 ture. are considered. Story displacement ratio max jx j J ¼ max jx j u 26 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Table 6. Responses and accelerations ratios (evaluation criteria). 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). 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. where x is the controlled story displacement x is the uncontrolled story displacement. c u Story acceleration ratio max jx j J ¼ max jx j where x is the controlled story acceleration. x is the uncontrolled story acceleration. Root mean square (RMS) story displacement ratio J ¼ is calculated from qffiffiffiffiffiffiffiffiffiffiffiffi x ¼ stdðxÞ T =T s f Azira and Guenfaf 27 where T is the sampling time. T is the total excitation duration.std is the standard deviation. RMS story acceleration ratio J ¼ x is calculated from qffiffiffiffiffiffiffiffiffiffiffiffi € € x ¼ stdðxÞ T =T s 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. 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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Integration of decentralized structural control and the identification of unknown inputs for tall shear building models under unknown earthquake excitation. Eng Struct 2013; 52: 306–316. 19. Athans M. The role and use of the stochastic linear-quadratic-Gaussian problem in control system design. IEEE Trans Automat Contr 1971; 16: 529–552. 20. Clarke DW, Mohtadi C and Tuffs PS. Generalized predictive control – part I. The basic algorithm. Automatica 1987; 23: 137–148. 21. Clarke DW. Self tuning control of non-minimum phase systems. Automatica 1984; 20: 501–517. 22. Astrom € KJ. Introduction to stochastic control theory. New York: Academic Press, 1970. 23. Borison U. Self-tuning regulators for a class of multivariable systems. Automatica 1979; 15: 209–215. 24. Koivo HN. A multivariable self-tuning controller. Automatica 1980; 16: 351–366. 25. Liu M-H and Lin W. Multivariable self-tuning control with decoupling for robotic manipulator. IEE Proc 1988; 135: 43–48. 26. Dugard L, Goodwin GC and Xianya X. The role of the interactor matrix in multivariable stochastic adaptive control. Automatica 1984; 20: 701–709. 27. Liu M-H and Lin W. Pole assignment self-tuning controller for robotic. Int J Control 1987; 46: 1307–1317. 28. Clough RW and Penzien J. Dynamics of structures. New York: McGraw-Hill, 1993. 29. Ohtori Y, Christenson R, Spencer B, Jr, et al. Benchmark control problems forseismically excited nonlinear buildings. J Eng Mech 2004; 130: 366–385. 30. Spencer B, Dyke S and Deoskar H. Benchmark problems in structural control: part II – active tendon system. Earthq Eng Struct Dyn 1998; 27: 1141–1148. 31. Spencer B, Dyke S and Deoskar H. Benchmark problems instructural control: part I – active mass driver system. Earthq Eng Struct Dyn 1998; 27: 1127–1140. Appendix 1 Derivation of the optimal predictor for SISO systems We need to find the best prediction w ðt þ d þ 1Þ of wðt þ d þ 1Þ in the least square sense. This prediction will 2,29 minimize the following criterion J ¼ E½ wðt þ d þ 1Þ w ðt þ d þ 1Þ (41) Using equation (32), we have pb pc wðt þ d þ 1Þ¼ pyðt þ d þ 1Þ¼ uðtÞþ eðt þ d þ 1Þ (42) a a Azira and Guenfaf 29 Taking into account the Diophantine equation (36), equation (42) can be written as p b r wðt þ d þ 1Þ¼ uðtÞþ eðtÞþ s eðt þ d þ 1Þ (43) p a r a D D Now substituting equation (43) into equation (41) 2 d p b r 0 2 2 J ¼ E uðtÞþ eðtÞ w ðt þ d þ 1Þ þ s r 1 i p a p a D D i¼0 0 0 p b s rs þ 2E uðtÞeðt þ d þ 1Þþ eðtÞeðt þ d þ 1Þ s w ðt þ d þ 1Þeðt þ d þ 1Þ p a p a D D (44) But e(t) is a white noise with the following characteristics 2 2 Ee ðtÞ¼ r Ee½ ðtÞeðt þ sÞ¼ 0; for s ¼ 0 (auto-correlation function) Ee½ ðtÞxðt þ sÞ¼ 0; 8s  0 (inter-correlation function) Thus, equation (44) becomes p b r N 2 0 2 2 J ¼ E uðtÞþ eðtÞ w ðt þ d þ 1Þ þ s r (45) 1 i p a p a D D i¼0 J is minimal if the first term is null, thus p b r wðÞ t þ d þ 1¼ utðÞ þ etðÞ (46) p a p a D D Appendix 2 Derivation of the GMV control for SISO systems Noting that pyðt þ d þ 1Þ¼ wðÞ t þ d þ 1þ fðt þ d þ 1Þ (47) where fðt þ d þ 1Þ¼ s eðt þ d þ 1Þ. The criterion J defined by equation (31) becomes hi 2 2 0 0 J ¼ EðÞ wðÞ t þ d þ 1þ s eðt þ d þ 1Þ r wtðÞ þ d þ 1 þðÞ Q uðtÞ (48) Applying the proprieties of e(t) reported in Appendix 1, we have hi d 2 2 0 0 2 2 J ¼ EðÞ wðÞ t þ d þ 1 r wtðÞ þ d þ 1 þðÞ Q uðtÞ þ s r (49) w i i¼0 30 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Minimization of J with respect to u(t) can be written as @J @wðÞ t þ d þ 1 @Q uðtÞ ¼ 2EðÞ wðÞ t þ d þ 1 r wtðÞ þ d þ 1 þðÞ Q uðtÞ ¼ 0 (50) @uðtÞ @uðtÞ @uðtÞ p b @wðÞ tþdþ1 N 1 From equation (40), we have ¼ , where p and p are the first coefficients of the polynomials N D 0 0 @uðtÞ p D 0 0 q @Q utðÞ N 1 1 p ðq Þ and p ðq Þ, respectively, and ¼ . Thus, equation (49) becomes N D 0 @uðtÞ q wðÞ t þ d þ 1  r wtðÞ þ d þ 1 þ QuðtÞ¼ 0 (51) q p N D 1 0 0 1 where Qðq Þ¼ Q ðq Þ q p b D N 1 From equations (43) and (46), we have r s b w ðt þ d þ 1Þ¼ yðtÞþ uðtÞ (52) p c c Now substituting equation (52) into equation (51), we can deduce the control law u(t) p cr wtðÞ þ d þ 1 ryðÞ t D w utðÞ ¼ (53) p ðs þ QcÞ where s ¼ s b . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE

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

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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 1,2 important in the last few decades. Researchers are very interested in control against the external disturbances 2–8 (vehicles, buildings, sensors disturbances, etc.). One of the important missions of structural control is to ensure 9–13 14 the safety of structures and cities during large earthquakes. In fact, Weng et al. 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. Gudarzi has presented a robust l-synthesis output-feedback controller design for seismic alleviation of multi-structural buildings with parametric uncertain- ties and to tackle the instabilities and performance declines due to these uncertainties. Tınkır et al. 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 LSEI Laboratory, University of Science and Technology Houari Boumediene (USTHB), Algiers, Algeria Corresponding author: Mohamed Azira, LSEI Laboratory, University of Science and Technology Houari Boumediene (USTHB), BP 32 Alia, Algiers 16111, Algeria. Email: mohamed.azira@yahoo.fr Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 4 Journal of Low Frequency Noise, Vibration and Active Control 37(1) 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. 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 perspec- tive. In this case, the system is characterized as stochastic process described by a linear model subjected to external disturbances. Ying et al. 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. 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. 19,20 The GMV algorithm has been widely studied in literature. It was introduced by Clarke as an extended 21–23 version of the minimum variance (MV) algorithm initially developed by Astrom. 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 ) 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 conclu- sions 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 (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 as f ðtÞþ f ðtÞþ f ðtÞ f ðtÞ f ðtÞ¼ u ðtÞ u ðtÞ (1) I A E A E i iþ1 i i i iþ1 iþ1 where f ðtÞ is the inertial force of the mass m I i f ðtÞ¼ m x ðtÞ (2) I i i i Azira and Guenfaf 5 (a) (b) Figure 1. Schematic representation of a multi-degree-of-freedom structure under seismic excitation (a) motion of the structure th (b) forces equilibrium of the i story unit. th f ðtÞ is the damping force of the i story _ _ f ðtÞ¼ c ðx ðtÞ x ðtÞÞ (3) A i i i1 th f ðtÞ is the elastic force of the i story f ðtÞ¼ k ðx ðtÞ x ðtÞÞ (4) E i i i1 th where relative displacement x (t).u (t) is the control force from the controller installed between the (i–1) story and i i th the i story. Absolute displacement x ðtÞ is defined as x ðtÞ¼ x ðtÞþ x ðtÞ (5) i g where x (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 ðx þ x Þþ c ðx  x Þþ k ðx  x Þ c ðx  x Þ k ðx  x Þ¼ u  u (6) i i g i i i1 i i i1 iþ1 iþ1 i iþ1 iþ1 i i iþ1 with i ¼ 1; n where c ¼ 0, k ¼ 0, x ¼ 0, x ¼ 0, u ¼ 0, and x ðtÞ represent the ground acceleration. nþ1 nþ1 0 nþ1 nþ1 g 6 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Equation (6) can be written in the matrix form as € _ € MXðtÞþ CXðtÞþ KXðtÞ¼ LUðtÞ MI x ðtÞ (7) v g where X ðtÞ¼ ½ x ðtÞ x ðtÞ  x ðtÞ  is the structural displacement vector 1 2 n U ðtÞ¼½ u ðtÞ u ðtÞ  u ðtÞ control vector 1 2 n 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 =[1 1.... . .1] 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 hi T T T T Z ðtÞ¼bZ ðtÞ Z ðtÞc ¼ X ðtÞ X ðtÞ as a state vector. _ € ZðtÞ¼ AZðtÞþ BUðtÞþ Ex ðtÞ (8) YðtÞ¼ DZðtÞ "# "# 0 I 0 where A ¼ of dimension (2n  2n), B ¼ of dimension (2n  n) 1 1 1 M K M C M L 2 3 "# I 0 4 5 E ¼ of dimension (2n  1), D ¼ of dimension (2n  2n) 0 0 I is the identity matrix of dimension (n  n) 0 is the null matrix of dimension (n  n) 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 24–26 model, described by equation (8), can be written in the following form 2 3 2 3 2 3 1 1 1 1 1 1 A ðq Þ  A ðq Þ B ðq Þ B ðq Þ C ðq Þ  C ðq Þ 11 1n 11 1n 11 1n 6 7 6 7 6 7 6 7 6 7 6 7 . . . . . . . . . . . . XðtÞ¼ . . . UðtÞþ . . . eðtÞ (9) 6 7 6 7 6 7 . . . . . . . . . 4 5 4 5 4 5 1 1 1 1 1 1 A ðq Þ  A ðq Þ B ðq Þ B ðq Þ C ðq Þ  C ðq Þ n1 nn n1 nn n1 nn where XðtÞ¼½ x ðtÞ x ðtÞ is the n  1 output vector 1 n UðtÞ¼½ u ðtÞ u ðtÞ is the n  1 input vector 1 n eðtÞ¼½ e ðtÞ  e ðtÞ is the n  1 zero-mean white noise vector with covariance matrix 1 n EeðtÞe ðtÞ¼ r  I n m n ij ij ii X X X 1 l 1 l 1 d l ij A ðq Þ¼ 1 þ a q ; A ðq Þ¼ a q ; B ðq Þ¼ q b q ii ii ij ij ij ij l l l l¼1 l¼1 l¼1 Azira and Guenfaf 7 nc nc ij ii X X 1 l 1 l C ðq Þ¼ 1 þ c q ; C ðq Þ¼ c q ii ii ij ij l l l¼1 l¼1 for i ¼ 1;nj ¼ 1;nd  0 ij q shift operator is defined as q xðÞ t þ 1 ¼ x ðÞ t i i 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. th Considering the dynamical equation of motion of the i story € _ _ _ _ € m x þ c ðx  x Þþ k ðx  x Þ c ðx  x Þ k ðx  x Þ¼ u  u  m x (10) i i i i i1 i i i1 iþ1 iþ1 i iþ1 iþ1 i i iþ1 i g Equation (10) can be written as € _ € _ _ m x þðc þ c Þx þðk þ k Þx ¼ u  u  m x þ c x þ c x þ k x þ k x (11) i i i iþ1 i i iþ1 i i iþ1 i g i i1 iþ1 iþ1 i i1 iþ1 iþ1 Introducing new notations, equation (11) has the form € _ € m x þ c x þ k x ¼ u  m x þ / (12) i i i g i i i i i i where m ¼ m c ¼ c þ c i iþ1 k ¼ k þ k i iþ1 th u ¼ u  u is the effective control force applied to the i story. i iþ1 _ _ / ¼ c x þ c x þ k x þ k x contains all the coupling terms from the other story units. i i1 iþ1 iþ1 i i1 iþ1 iþ1 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). k c i i By neglecting the term / and introducing the following notations: x ¼ ,n ¼ then applying Laplace i 0i m i 2m x 0i i i transform to equation (12), we obtain m 1 X ðÞ s ¼ U ðÞ s  X ðÞ s (13) i g 2 2 2 2 s þ 2n x s þ x s þ 2n x s þ x 0i 0i i i 0i 0i € € where X ðÞ s , X ðÞ s and U ðÞ s are the Laplace transform of x ðÞ t , x ðÞ t and u ðÞ t respectively; s is the Laplace i g i g i i operator. th Figure 2 shows a bloc diagram of the i 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 H ðÞ s H ðÞ s 1Ni 2Ni X ðÞ s ¼ U ðÞ s þ X ðÞ s (14) i g H ðÞ s H ðÞ s 1Di 2Di 8 Journal of Low Frequency Noise, Vibration and Active Control 37(1) th Figure 2. Bloc diagram of the i story model. where 2 2 > H ðÞ s ¼ H ðsÞ¼ s þ 2n x s þ x 1Di 2Di 0i > 0i H ðÞ s ¼ 1Ni > m > i H ðÞ s ¼ 1 2Ni The ARMAX model of the structure is obtained by the discretization of equation (14) 1 1 B q C q i i x ðÞ t ¼ u ðÞ t þ x ðÞ t (15) i g 1 1 AðÞ q AðÞ q i i where 1 i1i11 2 A ðq Þ¼ 1 þ a q þ a q i i2 i1 1 1 2 B ðq Þ¼ b q þ b q i i1 i2 1 1 2 C ðq Þ¼ c q þ c q i i1 i2 By analytical discretization of equation (14) using the Z-transform, the polynomial parameters are given by 8 8 8 1 n x i 0i > > 1 n x > i 0i > > > c ¼ 1  a b þ c > > > i1 i b ¼ 1  a b þ c i i > > > i1 i i i > >  > x x 0i i < < m x x < 0i i a ¼2a b i i1 i ; ; > > > > a ¼ a > > i2 i 1 n x > > > i 0i 2 1 n x > > > i 0i > > b ¼ a þ a c  b > i2 i : : i i i : c ¼ a þ a c  b i2 i i i i m x x 0i i i x x 0i i qffiffiffiffiffiffiffiffiffiffiffiffiffi with x ¼ x 1  n , a ¼ expðn x T Þ, b ¼ cosðx T Þ, c ¼ sinðx T Þ and T is the sampling period. i 0i i i 0i e i e i e e i i i Case 2. The ground acceleration is described by the Kanai-Tajimi model, i.e. X ðÞ s ¼ G ðsÞEsðÞ (16) g 1 2n x sþx G ðsÞ g g 1N g where G ðÞ s ¼ ¼ 1 2 2 G ðsÞ s þ2n x sþx 1D g g E(s) is the Laplace transform of white noise. Substituting equation (16) into equation (13), we obtain 2n x s þ x m 1 g g X ðÞ s ¼ U ðÞ s  EsðÞ (17) 2 2 2 2 2 2 s þ 2n x s þ x s þ 2n x s þ x s þ 2n x s þ x i 0i i 0i g g 0i 0i g Azira and Guenfaf 9 th Figure 3. Block diagram of i story model under Kanai-Tajimi seismic excitation. which can also be written as H ðÞ s H ðÞ s G ðsÞ 1Ni 2Ni 1N X ðÞ s ¼ U ðÞ s þ EsðÞ (18) H ðÞ s H ðÞ s G ðÞ s 1Di 2Di 1D By reducing the second member of equation (18) to a common denominator, we obtain F ðÞ s F ðÞ s 1Ni 2Ni X ðÞ s ¼ U ðÞ s þ EsðÞ (19) F ðÞ s F ðÞ s Di Di 2 2 2 2 F ðÞ s ¼ H ðÞ s G ðÞ s ¼ s þ 2n x s þ x s þ 2n x s þ x > Di 1Di 1Di 0i g i 0i g g 2 2 where F ðÞ s ¼ H ðÞ s G ðÞ s ¼ s þ 2n x s þ x 1Ni 1Ni 1Di g g g > m : 2 F ðÞ s ¼ H ðsÞG ðsÞ¼ 2n x s þ x 2Ni 2Ni 1Ni g g g Block diagrams are shown in Figure 3 to illustrate the previous calculations. 10 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Discretization of equation (19) gives the discrete ARMAX model of the structure under the Kanai-Tajimi ground acceleration 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. LQR for MDOF structure The LQR is applied for the system represented in equation (8). In this case, the control gain matrix is determined as 1 T G ¼ R B P (20) where R is the weighting matrix for the control force vector. P is the solution of Riccati equation given as T 1 T A P þ PA  PBR B P þ 2Q ¼ 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 intercon- nection with other subsystems. € _ € m x þ c x þ k x ¼ u  m x (22) i i i g i i i i i Using the state space concept, equation (22) can be written as _ € Z ðtÞ¼ A Z ðtÞþ B U ðtÞþ E x ðtÞ (23) i i i i i i g Figure 4. Schematic representation of the LQR approach for a 3DOF structure. LQR: linear quadratic regulator. Azira and Guenfaf 11 Figure 5. Schematic representation of the empirical LQR approach for a 3DOF structure. LQR: linear quadratic regulator. 2 3 2 3 "# "# 01 0 x ðtÞ 0 4 5 4 5 where Z ¼ ðtÞ ; A ¼ k c ; B ¼ 1 ; E ¼ i i i i i i x 1 m m m i i i In this case, the control gain matrix is 1 T G ¼ R B P (24) i i i where R is the weighting matrix for the control force vector. P is the solution of Riccati equation given as T 1 T A P þ P A  P B R B P þ 2Q ¼ 0 (25) i i i i i i i i i where Q is the weighting matrix for the state vector. Generalized minimum variance control algorithm 20,21 The generalized minimum variance (GMV) algorithm was introduced by Clarke to control the non-minimum phase systems. It is an extension of the MV algorithm 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 hi 2 0 2 J ¼ E kPytðÞ þ d þ 1 R wtðÞ þ d þ 1k þkQ utðÞk (26) where E is the expected value w(t þ d þ 1) is the n  1 vector defining the reference signal. 12 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Figure 6. Schematic representation of the GMV approach for a 3DOF structure. GMV: generalized minimum variance. P, R ,Q are the weighing polynomial matrices of dimension n  n each. w m 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 1 1 1 Aðq ÞXðtÞ¼ Bðq ÞUðtÞþ Cðq ÞI x ðtÞ (27) v g where 2 3 1 1 A ðq Þ A ðq Þ 0 11 12 6 7 6 7 1 1 1 6 7 A ðq Þ A ðq Þ A ðq Þ 21 22 23 6 7 6 7 ðq Þ¼ A 6 7 6 . 7 6 7 6 7 4 5 1 1 0 A ðq Þ A ðq Þ nðn1Þ nn 2 3 1 1 B ðq Þ B ðq Þ 0 11 12 6 7 6 7 1 1 6 7 B ðq Þ B ðq Þ 22 23 6 7 6 7 Bðq Þ¼ 6 7 6 . 7 6 7 6 7 4 5 1 1 0 B ðq Þ B ðq Þ nðn1Þ nn 2 3 C ðq Þ 0 6 7 6 7 6 7 1 . Cðq Þ¼ . 6 7 6 7 4 5 0 C ðq Þ nn Azira and Guenfaf 13 With this model, the MIMO GMV controller is calculated. As in the mono-variable case, we first have to derive the optimal predictor u ðt þ d þ 1Þ of uðt þ d þ 1Þ¼ P yðt þ d þ 1Þ and since it is a future information, it is given by hi 1 1  1 1 ~ ~ ~ u ðt þ d þ 1ÞC ðq Þ Fðq ÞB ðq ÞuðtÞþ Gðq ÞyðtÞ (28) 1 1 where B ðq Þ¼ qBðq Þ Using equations (27) and (28), it is shown that the control strategy is given by 1 1 1 1 ~ ~ Hðq ÞuðtÞ¼ Cðq ÞR ðq Þwðt þ d þ 1Þ Gðq ÞyðtÞ (29) 1 1  1 1 1 ~ ~ where Hðq Þ¼ Fðq ÞB ðq Þþ Cðq ÞQ ðq Þ 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 (u *). 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 used 1 d 1 1 aðq ÞytðÞ ¼ q bðq ÞutðÞ þ cðq ÞetðÞ (30) where 1 1 n aðq Þ¼ 1 þ a q þ þ a q 1 n 1 1 n bðq Þ¼ b q þ  þ b q 1 n 1 1 n cðq Þ¼ 1 þ c q þ þ c q 1 n d  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 r . The performance index to be minimized is hi 2 2 J ¼EpðÞ yðÞ t þ d þ 1 r wtðÞ þ d þ 1 þðÞ Q’utðÞ (31) 1 1 1 where E is the expected value and w(t þ d þ 1) is the reference signal p(q ) r (q ) and Q (q ) weighing polynomials. With 1 0 1 p ðq Þ Q ðq Þ 1 0 1 N pðq Þ¼ Q ðq Þ¼ 1 0 1 p ðq Þ Q ðq Þ The degrees of P and r can be chosen arbitrarily. We remark in this criterion that w(t þ d þ 1) is a disposable information, but y(t þ d þ 1) is not. It is the future information that we must predict. After some calculations, the prediction of py(t þ d þ 1) ¼ w(t þ d þ 1) is (see Appendix 1) 14 Journal of Low Frequency Noise, Vibration and Active Control 37(1) p b r wðÞ t þ d þ 1¼ utðÞ þ etðÞ (32) p a p a D D where b ¼ qb and R are the solution of the Diophantine equation defined as 0 ðdþ1Þ ap s þ q r ¼ p c (33) D N With 0 1 0 0 1 0 d s ðq Þ¼ s þ s q þ þ s q 0 1 d 1 1 n rðq Þ¼ r þ r q þ þ r q 0 1 n n ¼ n þ n  1 r d n is the degree of the polynomial p d D The minimization of J leads to the GMV control law given by (see Appendix 2) p cr wtðÞ þ d þ 1 ryðÞ t D w utðÞ ¼ (34) pðÞ s þ Qc where q p N 0 1 0 0 1 Qðq Þ¼ Q ðq Þ q p b N 1 D 0 2,21 The controller described by equation (34) is written on the RST form 1 1 Tðq Þ Sðq Þ uðtÞ¼ wðt þ d þ 1Þþ xðkÞ (35) 1 1 Rðq Þ Rðq Þ The closed loop transfer function is deduced from equation (35) 1  1 d xðkÞ Tðq Þ B ðq Þ q H ¼ ¼ (36) CL 1 1 1  1 ðdþ1Þ x ðkÞ Aðq Þ Sðq Þþ Rðq Þ B ðq Þ q Figure 7. Generalized minimum variance control architecture. Azira and Guenfaf 15 Figure 8. Schematic representation of the empirical GMV approach for a 3DOF structure. GMV: generalized minimum variance. The resulting closed-loop poles associated to an RST controller are the poles of the polynomial C(q ) (with: P =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 ) defines the tracking trajectory regu- lation, and the closed-loop dynamic at the same time no 1 1 1 1  1 ðÞ dþ1 Poles C q ¼ Poles A q Sq þRq B q q So the polynomial parameters C(q ) lead to an optimal prediction. Moreover, the closed loop characteristic 1 1 polynomial is C(q ) ¼ T(q ), and it also defines the closed loop stability condition, where the poles must be in the unit circles. 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 / , due to the interconnection with other subsystems. Mathematical model of earthquake ground motion The earthquake ground acceleration is modeled as a uniformly modulated non-stationary random process. € € x ðÞ t ¼ wðÞ t x ðÞ t (37) g s where w(t) is a deterministic nonnegative envelope function and x ðÞ t is a stationary random process with zero mean and a Kanai-Tajimi power spectral density 2 3 6 7 1 þ 4n 6 g 7 / ðÞ x ¼ S (38) 6 7 g 0 4 2 2 5 x x 1  þ 4n x g x g g where n x are the filter parameters and S is the constant spectral density of the white noise. However, it can be g, g 0 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. 16 Journal of Low Frequency Noise, Vibration and Active Control 37(1) (a) (b) Figure 9. Simulated ground accelerations. Using such a second high-pass filter, the Kanai-Tajimi spectrum is modified as follows to obtain the Clough– Penzien spectrum 2 32 3 2 4 x x 6 76 7 1 þ 4n g x x g c 6 76 7 / ðÞ x ¼ S (39) 6 76 7 4 2 2 54 2 2 5 2 2 x x x x 1  þ 4n 1  þ 4n g c x x x x g g c c A particular envelope function w(t) given in the following will be used 0 for t < 0 for 0  t  t wðÞ t ¼ (40) 1 for t  t  t 1 2 exp½ atðÞ  t for t  t 2 2 where t , t and a are parameters that should be selected appropriately to reflect the shape and duration of the 1 2 earthquake ground acceleration. Numerical values of parameters are t =3 s, t =13 s, a ¼ 0.26, n =0.65, x =19 1 2 g g rad/s, n ¼ 0.6, x ¼ 2 rad/s, S ¼ 0.8  10 m/s. c c 0 The Kanai-Tajimi and Clough–Penzien ground accelerations have been simulated and are presented in Figure 9. Simulation results Simulation tests are performed using a 3DOF structure with the following structural parameters m =2100 Kg, k =1,262,450 N/m, c =3675 Ns/m 1 1 1 m =2100 Kg, k = 2,607,500 N/m, c =10500 Ns/m 2 2 2 m =2100 Kg, k =2,607,500 N/m, c =10500 Ns/m 3 3 3 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. Azira and Guenfaf 17 Table 1. Structural parameters and ARMAX model parameters. 1st story 2nd story 3rd story m 2100 kg 2100 kg 2100 kg c 14,175 Ns/m 21,000 Ns/m 10,500 Ns/m k 3,869,950 N/m 5,215,000 N/m 2,607,500 N/m na1¼2 na2¼2 na3¼2 nb1¼2 nb2¼2 nb3¼2 nc1¼1 nc2¼1 nc3¼1 a 11 1 a 1.225 0.9905 1.451 a 0.8737 0.8187 0.9048 8 8 8 b 8.568  10 8.215  10 8.842  10 8 8 8 b 8.182  10 7,667  10 8.547  10 c 11 1 c 0.9549 0.9333 0.9666 ARMAX: autoregressive moving average exogenous Figure 10. El Centro Earthquake. Ponderation polynomials used in the GMV algorithm are 1 7 1 1 Q ðq Þ¼ 2:10 ; P ðq Þ¼ 1  0:5q ; i ¼ 1; 3 i i 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 dem- onstrated good performances. Responses of the structure to Kanai-Tajimi and Clough–Penzien excitation models are shown in Figures 11 to 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 (u *). 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. 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. 18 Journal of Low Frequency Noise, Vibration and Active Control 37(1) (a) (b) (c) Figure 11. LQR approach: structural response to Clough–Penzien ground acceleration. LQR: linear quadratic regulator. (a) (b) (c) Figure 12. LQR approach: structural response to Kanai-Tajimi ground acceleration. LQR: linear quadratic regulator. Azira and Guenfaf 19 (a) (b) (c) Figure 13. Empirical LQR approach: structural response to Kanai-Tajimi ground acceleration. LQR: linear quadratic regulator. (a) (b) (c) Figure 14. Empirical LQR approach: structural response to Clough–Penzien ground acceleration. LQR: linear quadratic regulator. 20 Journal of Low Frequency Noise, Vibration and Active Control 37(1) (a) (b) (c) Figure 15. GMV approach: structural response to Kanai-Tajimi ground acceleration. GMV: generalized minimum variance. (a) (b) (c) Figure 16. GMV approach: structural response to Clough–Penzien ground acceleration. Azira and Guenfaf 21 (a) (b) (c) Figure 17. Empirical GMV approach: structural response to Kanai-Tajimi ground acceleration. GMV: generalized minimum variance. (a) (b) (c) Figure 18. Empirical GMV approach: structural response to Clough–Penzien ground acceleration. GMV: generalized minimum variance. 22 Journal of Low Frequency Noise, Vibration and Active Control 37(1) (a) (b) (c) Figure 19. LQR approach: structural response to EL Centro Earthquake. (a) (b) (c) Figure 20. Empirical LQR approach: structural response to El Centro Earthquake. LQR: linear quadratic regulator. Azira and Guenfaf 23 (a) (b) (c) Figure 21. GMV approach: structural response to EL Centro Earthquake. GMV: generalized minimum variance. (a) (b) (c) Figure 22. Empirical GMV approach: structural response to EL Centro Earthquake. GMV: generalized minimum variance. 24 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Table 2. Output variance. Output variance Controller 1st floor 2nd floor 3rd floor 7 6 6 Kanai-Tajimi Without control 8.479210 1.5669  10 2.0394  10 8 7 7 LQR 9.4824  10 1.7702  10 2.3151  10 7 7 7 Empirical LQR 1.3158  10 2.3740  10 3.1014  10 9 9 9 GMV 5.1373  10 8.5847  10 8.2635  10 8 8 9 Empirical GMV 1.4868  10 1.4473  10 7.9022  10 7 6 6 Clough-Penzien Without control 8.5882  10 1.5872  10 2.0660  10 8 7 7 LQR 9.5233  10 1.7788  10 2.3269  10 7 7 7 Empirical LQR 1.3249  10 2.3913  10 3.1245  10 9 9 9 GMV 4.9138  10 8.2192  10 7.9106  10 8 8 9 Empirical GMV 1.4698  10 1.4287  10 7.7709  10 6 6 6 El Centro Without control 1.1717  10 2.1639  10 2.8151  10 7 7 7 LQR 2.1761  10 4.0450  10 5.2720  10 7 7 7 Empirical LQR 3.0564  10 5.5314  10 7.2166  10 9 8 8 GMV 4.7789  10 1.3361  10 1.9143  10 9 8 8 Empirical GMV 5.9258  10 1.6835  10 2.3354  10 LQR: linear quadratic regulator; GMV: generalized minimum variance. Table 3. Maximum displacement. Drift (cm) 1st floor 2nd floor 3rd floor Earthquakes Controllers 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. 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 Azira and Guenfaf 25 Table 4. Maximum acceleration. Acceleration (g) 1st floor 2nd floor 3rd floor Earthquakes Controllers 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. Max control force U (N) max 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. 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 litera- 28–31 ture. are considered. Story displacement ratio max jx j J ¼ max jx j u 26 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Table 6. Responses and accelerations ratios (evaluation criteria). 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). 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. where x is the controlled story displacement x is the uncontrolled story displacement. c u Story acceleration ratio max jx j J ¼ max jx j where x is the controlled story acceleration. x is the uncontrolled story acceleration. Root mean square (RMS) story displacement ratio J ¼ is calculated from qffiffiffiffiffiffiffiffiffiffiffiffi x ¼ stdðxÞ T =T s f Azira and Guenfaf 27 where T is the sampling time. T is the total excitation duration.std is the standard deviation. RMS story acceleration ratio J ¼ x is calculated from qffiffiffiffiffiffiffiffiffiffiffiffi € € x ¼ stdðxÞ T =T s 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. 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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This prediction will 2,29 minimize the following criterion J ¼ E½ wðt þ d þ 1Þ w ðt þ d þ 1Þ (41) Using equation (32), we have pb pc wðt þ d þ 1Þ¼ pyðt þ d þ 1Þ¼ uðtÞþ eðt þ d þ 1Þ (42) a a Azira and Guenfaf 29 Taking into account the Diophantine equation (36), equation (42) can be written as p b r wðt þ d þ 1Þ¼ uðtÞþ eðtÞþ s eðt þ d þ 1Þ (43) p a r a D D Now substituting equation (43) into equation (41) 2 d p b r 0 2 2 J ¼ E uðtÞþ eðtÞ w ðt þ d þ 1Þ þ s r 1 i p a p a D D i¼0 0 0 p b s rs þ 2E uðtÞeðt þ d þ 1Þþ eðtÞeðt þ d þ 1Þ s w ðt þ d þ 1Þeðt þ d þ 1Þ p a p a D D (44) But e(t) is a white noise with the following characteristics 2 2 Ee ðtÞ¼ r Ee½ ðtÞeðt þ sÞ¼ 0; for s ¼ 0 (auto-correlation function) Ee½ ðtÞxðt þ sÞ¼ 0; 8s  0 (inter-correlation function) Thus, equation (44) becomes p b r N 2 0 2 2 J ¼ E uðtÞþ eðtÞ w ðt þ d þ 1Þ þ s r (45) 1 i p a p a D D i¼0 J is minimal if the first term is null, thus p b r wðÞ t þ d þ 1¼ utðÞ þ etðÞ (46) p a p a D D Appendix 2 Derivation of the GMV control for SISO systems Noting that pyðt þ d þ 1Þ¼ wðÞ t þ d þ 1þ fðt þ d þ 1Þ (47) where fðt þ d þ 1Þ¼ s eðt þ d þ 1Þ. The criterion J defined by equation (31) becomes hi 2 2 0 0 J ¼ EðÞ wðÞ t þ d þ 1þ s eðt þ d þ 1Þ r wtðÞ þ d þ 1 þðÞ Q uðtÞ (48) Applying the proprieties of e(t) reported in Appendix 1, we have hi d 2 2 0 0 2 2 J ¼ EðÞ wðÞ t þ d þ 1 r wtðÞ þ d þ 1 þðÞ Q uðtÞ þ s r (49) w i i¼0 30 Journal of Low Frequency Noise, Vibration and Active Control 37(1) Minimization of J with respect to u(t) can be written as @J @wðÞ t þ d þ 1 @Q uðtÞ ¼ 2EðÞ wðÞ t þ d þ 1 r wtðÞ þ d þ 1 þðÞ Q uðtÞ ¼ 0 (50) @uðtÞ @uðtÞ @uðtÞ p b @wðÞ tþdþ1 N 1 From equation (40), we have ¼ , where p and p are the first coefficients of the polynomials N D 0 0 @uðtÞ p D 0 0 q @Q utðÞ N 1 1 p ðq Þ and p ðq Þ, respectively, and ¼ . Thus, equation (49) becomes N D 0 @uðtÞ q wðÞ t þ d þ 1  r wtðÞ þ d þ 1 þ QuðtÞ¼ 0 (51) q p N D 1 0 0 1 where Qðq Þ¼ Q ðq Þ q p b D N 1 From equations (43) and (46), we have r s b w ðt þ d þ 1Þ¼ yðtÞþ uðtÞ (52) p c c Now substituting equation (52) into equation (51), we can deduce the control law u(t) p cr wtðÞ þ d þ 1 ryðÞ t D w utðÞ ¼ (53) p ðs þ QcÞ where s ¼ s b .

Journal

"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Mar 21, 2018

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

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