Stanford, CA Keywords: H-infinity control, feedback structural control, decentralized control, smart structures.

Transcription

1 Decentralize H Controller Design for Large-scale Civil Structures Yang Wang a, Jerome P. Lynch b, Kincho H. Law c a School of Civil an Environmental Engineering, eorgia Institute of echnology Atlanta, A 3033 b Department of Civil an Environmental Engineering, University of Michigan Ann Arbor, MI 4809 c Department of Civil an Environmental Engineering, Stanfor University Stanfor, CA ABSRAC Complexities inherent to large-scale moern civil structures pose many challenges in the esign of feeback structural control systems for ynamic response mitigation. With the emergence of low-cost sensors an control evices creating technologies from which largescale structural control systems can eploy, a future control system may contain hunres, or even thousans, of such evices. Key issues in such large-scale structural control systems inclue reuce system reliability, increasing communication requirements, an longer latencies in the feeback loop. o effectively aress these issues, ecentralize control strategies provie promising solutions that allow control systems to operate at high noal counts. his paper examines the feasibility of esigning a ecentralize controller that minimizes the H norm of the close-loop system. H control is a natural choice for ecentralization because imposition of ecentralize architectures is easy to achieve when posing the controller esign using linear matrix inequalities. Decentralize control solutions are investigate for both continuous-time an iscrete-time H formulations. Numerical simulation results using a 3-story an a 0-story structure illustrate the feasibility of the ifferent ecentralize control strategies. he results also emonstrate that when realistic semi-active control evices are use in combination with the ecentralize H control solution, better performance can be gaine over the passive control cases. It is shown that ecentralize control strategies may provie equivalent or better control performance, given that their centralize counterparts coul suffer from longer sampling perios ue to communication an computation constraints. Keywors: H-infinity control, feeback structural control, ecentralize control, smart structures.. INRODUCION Real-time feeback control has been a topic of great interest to the structural engineering community over the last few ecaes [-4]. A feeback structural control system inclues an integrate network of sensors, controllers, an control evices that are installe in the civil structure to mitigate unesire vibrations uring external excitations, such as

2 earthquakes or typhoons. Uner an external excitation, the ynamic response of the structure is measure by sensors. Sensor ata is communicate to a centralize controller that uses the ata to calculate an optimal control solution. he optimal solution is then ispatche by the controller to control evices which irectly (i.e. active evices) or inirectly (i.e. semi-active evices) apply forces to the structure. his process repeats continuously in real time to mitigate, or even eliminate, unesire structural vibrations. It was recently reporte that more than 50 builings an towers have been successfully instrumente with various types of structural control systems from 989 to 003 [5]. In practice, semi-active control is usually preferre over active control because it can achieve at least an equivalent level of performance, consumes orers of magnitue less power, an provies higher level of reliability. Examples of semi-active control evices inclue active variable stiffness (AVS) evices, semi-active hyraulic ampers (SHD), electrorheological (ER) ampers, an magnetorheological (MR) ampers [6]. Aitional avantages associate with semi-active control inclue aaptability to real-time excitation, inherent Boune Input/Boune Output (BIBO) stability, an invulnerability against power failure. raitional feeback structural control systems employ centralize architectures. In such an architecture, one central controller is responsible for collecting ata from all the sensors in the structure, making control ecisions, an ispatching these control ecisions to control evices. Hence, the requirements on communication range an ata transmission banwith increase with the size of the structure an with the number of sensors an control evices being eploye. he communication requirements coul impose economical an technical ifficulties for the implementation of feeback control systems in increasingly larger civil structures. he centralize controller itself represents a single point of potential failure; failure of the controller may paralyze the entire control system. In orer to overcome these inherent challenges, ecentralize control architectures coul be alternatively aopte [7-9]. For example, a structural control system consisting of 88 fully ecentralize semi-active oil ampers has been installe in the 70m-tall Shioome ower in okyo, Japan [0; ]. his paper examines both fully ecentralize scheme, where the controller on a floor only has local sensor ata from that floor, an partially ecentralize scheme, where the controller also receives sensor ata from neighboring floors (or substructures). In a ecentralize control system architecture, multiple controllers are istribute throughout the structure. Acquiring ata from a local subnet of sensors, each controller commans control evices in its vicinity. he benefits of localizing a subset of sensors an control evices to each controller inclue shorter communication ranges an reuce ata transmission rates in the control system. Decentralization also eliminates the risk of global control system failure if one of the controllers shoul fail. For large-scale structures, occasional failure of ecentralize controllers may only cause minor egraation to the control performance. Decentralize control esign base on the linear quaratic regulator () optimization criteria has been previously explore by the authors to stuy the feasibility of utilizing wireless sensors as controllers for feeback structural control [; 3]. his paper investigates a ifferent approach to the esign of a ecentralize control system base on

3 H control theory, which is known to offer excellent control performance when worstcase external isturbances are encountere. Due to the multiplicative property of the H norm [4], H control esign can also consier moeling uncertainties (as is typical in most civil structures). Centralize H controller implementation in the continuous-time omain for civil structural control has been extensively stuie [5-]. When compare with traitional linear quaratic aussian (LQ) controllers, H controllers can achieve either comparable or superior performance [; 3]. For example, it has been shown that H control esign may achieve better performance in attenuating transient vibrations of the structure [4]. However, ecentralize H controller esign, either in the continuous time omain or iscrete time omain, has rarely been explore in structural control. he H control solution can be reaily formulate as an optimization problem with constraints expresse in terms of linear matrix inequalities (LMI) [5]. For such problems, sparsity patterns can be easily applie to the controller matrix variables. his property offers significant convenience for esigning ecentralize controllers, where certain sparsity patterns can be applie to the gain matrices consistent with certain esire feeback architecture. his paper presents pilot stuies investigating the feasibility of ecentralize H control that may be employe in large-scale structural control systems. More specifically, ecentralize H controller esign is presente in both the continuoustime an iscrete-time omains. Using properties of LMI, the ecentralize H control problem is converte into a convex optimization problem that can be conveniently solve using available mathematical packages. Numerical simulations are conucte to valiate the performance of the ecentralize H controller esign. In the first example, a 3-story structure is use to emonstrate the etaile proceure for the esign of the ecentralize H controller. he control performance of ecentralize H controllers is then compare with the performance of ecentralize -base controllers [; 3]. In the secon example, simulations of a 0- story benchmark structure are conucte to illustrate the efficacy of the ecentralize H control solution for large-scale civil structures. Different information feeback architectures an control sampling rates are employe so as to provie an in-epth stuy of the propose approaches. Control performance using ieal actuators an large-capacity semi-active hyraulic (SHD) ampers are presente for the 0-story structure. Performance of the ecentralize control system is compare with passive control cases where the SHD ampers are fixe at minimum or maximum amping settings.. FORMULAION OF DECENRALIZED H CONROL his section first iscusses the esign of a ecentralize H controller for structural control in the continuous-time omain. he controller s counterpart in the iscrete-time 3

4 omain is then erive. In both erivations, properties of linear matrix inequalities are utilize to convert the formulation of the ecentralize control esign problem into a convex optimization problem... Continuous-time ecentralize H control For a lumpe-mass structural moel with n egrees-of-freeom (DOF) subjecte to m external excitations, an controlle by m control evices, the equations of motion can be formulate as: Mqɺɺ t + Cqɺ t + Kq t = u t + w t () ( ) ( ) ( ) ( ) ( ) n where q(t) R is the isplacement vector relative to the groun; M, C, K R n n are the mass, amping, an stiffness matrices, respectively; u(t) R m an w(t) R m are the n control force an external excitation vectors, respectively; an m n u R an m w R are the control an excitation location matrices, respectively. For simplicity, the iscussion is base on a -D shear-frame structure subjecte to uniirectional groun excitation. In the example structure shown in Figure, it is assume that the external excitation, w(t), is a scalar function (m = ) containing the groun q t M. acceleration time history ɺɺ g ( ) ; the spatial loa pattern w is then equal to { } n Entries in u(t) are efine as the control forces between neighboring floors. For the 3- story structure, if a positive control force is efine to be moving the floor above the evice towars the left irection, an moving the floor below the evice towars the right irection (as shown in Figure ), the control force location matrix u is efine as: u w 0 0 u = () 0 0 he secon-orer orinary ifferential equation (ODE), Eq. (), can be converte to a first orer ODE by the state-space formulation as follows: xɺ t = A x t + B u t + E w t (3) ( ) ( ) ( ) ( ) I I I I I u 3 q 3 u 3 q u u q u u qɺɺ ( t) Figure. A three-story controlle structure excite by uniirectional groun motion. g 4

5 where x n I = q ( t) ; qɺ ( t) R is the state vector; A I n n n R, B I m n R, an E I m R are the system, control, an excitation matrices, respectively: [ 0] [ I] n n n n A I = M K M C, [ 0] n m BI =, { 0} n E M u { } I = (4) n In this stuy, it is assume that inter-story rifts an velocities are measurable. he isplacement an velocity variables in x I, which are relative to the groun, are first transforme into inter-story rifts an velocities (i.e. rifts an velocities between neighboring floors). he inter-story rifts an velocities at each story are then groupe together as: x = [q qɺ q q qɺ qɺ qn qn qɺ n qɺ n ] (5) n n A linear transformation matrix Γ R can be efine such that x = Γx I. Substituting x I = Γ x into Eq. (3), an left-multiplying the equation with Γ, the state space representation with the transforme (inter-story) state vector becomes: xɺ t = Ax t + Bu t + Ew t (6) ( ) ( ) ( ) ( ) where A = ΓA IΓ, = I B ΓB, E = ΓE I (7) p he system output z(t) R is efine as the sum of linear transformations to the state vector x(t) an the control force vector u(t): z t = C x t + D u t (8) ( ) ( ) ( ) z p n p where C z R an D m z R are the output matrices for the state an control force vectors, respectively. Assuming static state feeback, the control force u(t) is etermine m n by u(t) = x(t), where R is terme the control gain matrix. Substituting x(t) for u(t) in Eq. (6) an Eq. (8), the state-space equations of the close-loop system can be written as: xɺ ( t) = ACLx ( t) + Ew( t) (9) z( t) = CCLx ( t) z where ACL = A + B C = C + D CL z z (0) In frequency-omain, the system ynamics can be represente by the transfer function p m H zw (s) C from isturbance w(t) to output z(t) as [6]: H s = C si A E () zw ( ) ( ) CL where s is the complex Laplacian variable. he objective of H control is to minimize the H -norm of the close-loop system, which in the frequency omain is efine as: CL 5

6 ( s) supσ ( jω) H = zw H zw () ω where ω represents angular frequency, j is the imaginary unit, σ [ i ] enotes the largest singular value of a matrix, an sup enotes the supremum (least upper boun) of a set of real numbers. he efinition shows that in the frequency omain, the H -norm of the system is equal to the peak of the largest singular value of the transfer function H ( s) along the imaginary axis (where s = jω). he H -norm also has an equivalent interpretation in the time omain, as the supremum of the -norm amplification from the isturbance to the output: ( ) ( ( ) ( ) ) Hzw s = sup z t w t (3) w, w( t) 0 t=+ where the -norm of a signal f(t) is efine as ( ) ( ) ( ) f t f t f t t, which represents = t = the energy level of a signal. In this stuy, the H -norm can be viewe as the upper limit of the amplification factor from the isturbance (i.e. seismic groun motion) energy to the output (i.e. structural response) energy. he isturbance is calle a worst-case isturbance when this upper limit is reache. By minimizing the H -norm, the system output (which inclues structural response measures) can be greatly reuce when a worstcase isturbance (which is the earthquake excitation) is applie. Accoring to the Boune Real Lemma, the following two statements are equivalent for a H controller that minimizes the smallest upper boun of the H norm of a continuoustime system [5]:. H γ an A CL is stable in the continuous-time sense (i.e. the real parts of all zw < the eigenvalues of A CL are negative);. here exists a symmetric positive-efinite matrix inequality hols: A Θ + ΘA + EE γ ΘC * I CL CL CL < 0 n n Θ R such that following where * enotes the symmetric entry (in this case, CCLΘ ), an < 0 means that the matrix at the left sie of the inequality is negative efinite. Using the close-loop matrix efinitions in Eq. (0), Eq. (4) becomes: AΘ + ΘA + BΘ + Θ B + EE γ ΘC + Θ D * I z z < 0 he above nonlinear matrix inequality can be converte into linear matrix inequality (LMI) by introucing a new variable Y R m n where Y = Θ : AΘ + ΘA + BY + Y B + EE γ ΘC + Y D * I z z < 0 zw (4) (5) (6) 6

7 In summary, the continuous-time H control problem is now transforme into a convex optimization problem: minimize γ (7) subject to Θ > 0 an the LMI expresse in Eq. (6) Here Y, Θ, an γ are the optimization variables. Numerical solutions to this optimization problem can be compute, for example, using the Matlab LMI oolbox [7] or the convex optimization package CVX [8]. After the optimization problem is solve, the control gain matrix is compute as: = YΘ (8) In general, the algorithm fins a gain matrix without any sparsity constraints; in other wors, it represents a control scheme consistent with a centralize state feeback architecture. o compute gain matrices for ecentralize state feeback control, appropriate sparsity constraints can be applie to the optimization variables Y an Θ while solving the optimization problem of Eq. (7). For most available software packages, the sparsity constraints can be conveniently efine by assigning corresponing zero entries to the Y an Θ optimization variables. For example, gain matrices of the following sparsity patterns may be employe for a 3-story structure: I = 0 0, an II = (9) Note that each entry in the above matrices represents a block. Accoring to the linear feeback control law u( t) = x ( t), when the sparsity pattern in I is use, only the inter-story rift an velocity at the i-th story are neee to etermine the control force u i at the same story. When the sparsity pattern in II is aopte, the inter-story rifts an velocities from both the i-th story an the neighboring stories (story) are neee in orer to etermine the control force u i at the i-th story. Consiering the relationship between an Y as specifie in Eq. (8), in orer to fin the control gain matrices satisfying the shape constraints in I, the following shape constraints may be applie to the optimization variables Y an Θ : YI = 0 0, an ΘI = 0 0 (0) Similarly, to compute control gain matrices satisfying the shape constraints of II, the following shape constraints may be applie to the optimization variables: YII =, an ΘII = 0 0 () It is important to realize that ue to the constraints impose on the Y an Θ variables, the presente ecentralize H controller preclues the possibility that a ecentralize gain 7

8 matrix may exist with Y an Θ variables not satisfying the corresponing shape constraints. For example, it is possible that a gain matrix may satisfy the sparsity pattern in I while the corresponing Y an Θ variables o not conform to the sparsity patterns shown in Eq. (0). he application of sparsity patterns to Y an Θ variables makes the gain matrix easily computable using existing software packages, although the approach may not be able to explore the complete solution space of ecentralize gain matrices. hat is, the approach for ecentralize H controller esign may not guarantee that a minimum H -norm is obtaine over the complete solution space; rather, only a minimum H -norm is obtaine for the solution space containe within the bounary impose by the shape constraints on Y an Θ... Discrete-time ecentralize H control For implementation in typical igital control systems, the ecentralize H control esign in iscrete-time omain is neee. Using zero-orer hol (ZOH) equivalents, the continuous-time system in Eq. (9) can be transforme into an equivalent iscrete-time system [9]: x [ k + ] = A CLx [ k] + Ew [ k] () z [ k] = C CLx [ k] where the subscript inicates that the variables are expresse in the iscrete-time omain, an the close-loop system matrices A CL an C CL are efine accoringly: ACL = A + B (3) CCL = Cz + Dz For linear state feeback, the control force u [ k] is etermine as [ k] = [ k] u x. Accoring to the Boune Real Lemma, the following two statements are equivalent for iscrete-time systems [6]:. he H -norm of the close-loop system in Eq. () is less than γ, an A CL is stable in the iscrete-time sense (i.e. all of the eigenvalues of A CL fall in the unit circle on the complex plane);. here exists a symmetric matrix Θ > 0 such that the following inequality hols: A CL C CL γ Θ 0 A CL E Θ 0 0 < CL γ E 0 0 I C 0 0 I Replacing Θ with Θ ɶ γ an using the Schur complement [5] an congruence transformation, it can be shown that the matrix inequality in Eq. (4) is equivalent to: Θɶ 0 A CL Θɶ CCL * γ I E Θɶ 0 > 0 * * Θɶ (5) 0 * * * I (4) 8

9 Left-multiplying an right-multiplying the matrix above with a positive efinite blockiagonal matrix iag( Θɶ, IΘ, ɶ, I ), an letting Θ = Θ ɶ, the following matrix inequality is obtaine: Θ 0 ΘACL ΘC CL * γ I E 0 > 0 (6) * * Θ 0 * * * I Similar to the continuous-time system, by replacing the close-loop matrices A CL an C CL in Eq. (6) with their efinitions in Eq. (3), an letting Y = Θ, the above matrix inequality can be converte into: Θ 0 ΘA + Y B ΘCz + Y Dz * γ I E 0 > 0 (7) * * Θ 0 * * * I herefore, the iscrete-time H control problem can be converte to a convex optimization problem with LMI constraints: minimize γ (8) subject to Θ > 0 an the LMI expresse in Eq. (7) Here again, Y, Θ, an γ are the optimization variables. After the optimization problem is solve, the control gain matrix is compute as: = YΘ (9) Furthermore, the sparsity pattern of the gain matrix can be obtaine by specifying appropriate zero entries to the LMI variables Y an Θ, following the same proceure as escribe in the continuous-time case. 3. NUMERICAL SIMULAIONS Since the iscrete-time formulation is suitable for implementation in moern igital controllers, numerical simulations are conucte to evaluate the performance of the iscrete time ecentralize H control schemes escribe in Section.. In Section 3., the proceure for esigning the ecentralize H controller is illustrate in etails using a 3- story structure. Performance of the H controllers is compare with the performance of controllers base on the optimization criteria. In Section 3., simulations using a 0- story benchmark structure are conucte to illustrate the efficacy of the ecentralize H control solution for large-scale civil structures. Results using both ieal actuators an large-capacity semi-active hyraulic (SHD) ampers are presente for the 0-story structure. 3.. Numerical simulation of a 3-story structure 9

10 3... Decentralize H control Simulations of a 3-story shear-frame structure are first presente to illustrate the proceure employe in ecentralize H control esign. he frame structure is moele as an inplane lumpe-mass shear structure with one actuator allocate between every two neighboring floors. It is assume that both the inter-story rifts an inter-story velocities between every two neighboring floors are measurable. Such an assumption is reasonable consiering that moern semi-active hyraulic ampers contain internal stroke sensors an loa cells that measure real-time amper isplacements an forces, respectively []. Assuming V-brace elements are use as shown in Figure, the isplacement an force measurements can be use to estimate inter-story rifts an velocities. he following mass, stiffness, an amping matrices are aopte for the example structure: M = 6 0 kg, K = N/m, (30) C = N/(m/s) For uniirectional groun excitation, the continuous-time system matrices A, B an E can be obtaine via Eq. (7) as: A = , (3) B = , E = Note that the state-space vector corresponing to these matrices no longer contains isplacements an velocities relative to the groun. Instea, the vector has been formulate to contain inter-story rifts an velocities that are groupe by floors as given in Eq. (5). he system matrices in Eq. (3) can be reaily converte into their iscrete-time equivalents for a given sampling frequency [8]. For the results presente here, a sampling frequency of 00 Hz is employe. he output matrices C z an D z shown in Eq. (3) are efine as: C z = , D z = (3)

11 he above assignments for C z an D z make the -norm of the output vector [ k] quaratic function of the inter-story rifts an control forces: [ k] = ( + ) [ k] z C D x z z = C x + [ k] D u [ k] z z q [ k] ( q [ k] q [ k] ) ( q3 [ k] q [ k] ) ( u [ k] u [ k] u3 [ k] ) = z a he ecentralize H controller esign aims to minimize the close-loop H -norm, which is efine as the system norm from the excitation input to the output z. he relative weighting between the structural response an the control effort is reflecte by the magnitue of the output matrices, C z an D z. If higher attenuation of structural response is neee, larger magnitue shoul be assigne to C z ; on the contrary, if less control effort is available, larger magnitue shoul be assigne to D z. Using the matrices efine in Eqs. (3) an (3), the convex optimization problem as shown in Eq. (8) can be pose with ifferent sparsity patterns impose on the Y an Θ matrices. For the two sparsity patterns I an II efine in Eq. (0) an (), the solutions to the optimization problem result in the following two ecentralize control gain matrices, respectively: I = (34) II = (35) When no sparsity pattern is applie to Y an Θ, a full gain matrix representing centralize feeback is generate: III = (36) he open-loop H -norm of the uncontrolle structure an the close-loop H -norms of the controlle structure using the above gain matrices are liste in able. he H -norm of the uncontrolle structure is compute using the iscrete-time system efine in Eq. () with the gain matrix set as a zero matrix. Comparing the four cases in able, the uncontrolle structure has the highest H -norm (8.4366), which inicates the largest worst-case amplification from the excitation input w to the output z. Among the three controlle cases, because the centralize case with gain matrix III assumes that complete state information is available for control ecisions, the lowest H -norm (0.7045) is achieve (which means best control performance). he fully ecentralize case with gain matrix I has the largest norm (0.85) among the three H controllers; this is somewhat expecte because the fully ecentralize controller has the least amount of information available for calculating control ecisions for each control evice. (33)

12 able. H an H -norms of the open-loop transfer function H zw an the closeloop norms using both the Open-loop (Uncontrolle) I H controllers an controllers Close-loop Fully Partially Centralize ecentralize I ( H ) ecentralize II ( H ) II III III ( H ) H norm H norm he 940 El Centro NS (Imperial Valley Irrigation District Station) earthquake recor with its peak acceleration scale to m/s is use as the groun excitation. hree ieal actuators that generate any esire control force are eploye at the three stories. In contrast to a realistic semi-active or active control evice, an ieal actuator offers unlimite force capacity, an has zero time elay while elivering the force. Maximum inter-story rifts an control forces uring the ynamic response are plotte in Figure. he inter-story rift plots in Figure (a) inclue the results for the uncontrolle structure an the structure controlle using the three ifferent gain matrices. Using ieal actuators, all three controlle cases achieve significant reuction in inter-story rifts compare with the uncontrolle case. Among the three controlle cases, the fully ecentralize case using gain matrix I achieves the smallest reuction in inter-story rifts, which is consistent with the performance comparison inicate by the H -norms in able. he ifference between the cases using gain matrices II an III is minor, with III achieving slightly better performance. Figure (b) presents the peak control forces for the three controlle cases. he fully ecentralize controller imposes the lowest requirements on the control force capacity. 3 Maximum Inter-story Drifts 3 Maximum Actuator Force ain I ain II ain III No Control ain I ain II ain III Story Story Drift (m) Force (N).5 x 0 4 (a) (b) Figure. Simulation results when ieal actuators are eploye on the 3-story structure.

13 he peak control forces are similar between the partially ecentralize case II an the centralize case III. he largest sum of the three peak actuator forces is about 35kN, which represents the total capacity of all the actuators. his amount of total actuator capacity is about 9% of the builing weight (80kN), a realistic ratio as suggeste by most references [30-3]. Further illustration of the trae-off between structural response attenuation an control effort is shown in able. he table first lists the RMS (root-of-mean-square) values of the inter-story rifts at all three floors when the structure is uncontrolle, i.e..9mm,. mm, an.4mm, respectively. For the three controlle cases, changes in RMS rifts are liste in percentages relative to the uncontrolle values, i.e. negative numbers represent reuction from the uncontrolle value. For example, case III offers 96% reuction to the RMS rift at the first story. RMS values of three actuator forces are also liste for the three controlle cases. For each controlle case, the bottom row of able provies average rift changes an RMS forces across the three stories. It is again shown that cases II an III offer greater reuction to structural response through larger control effort Comparison with ecentralize control It coul be instructive to compare the ecentralize H controller esign with the ecentralize controller esign that was previously stuie []. he control algorithm aims to select the optimal control force trajectory u by minimizing the expecte value of a quaratic cost function, J : ( [ ] [ ] [ ] [ ]) n n m m k = J = x k Qx k + u k Ru k, Q 0 an R > 0 (37) Using the same efinition of the output matrices as escribe in Eq. (3), the following weighting matrices are employe for the controller esign: Q C C (38) = z z R = D z D z (39) As a result, the optimization inex J is proportional to the signal -norm of the able. RMS values of control forces an the changes in RMS inter-story rifts ( H control cases) RMS of Controlle uncontrolle I II III inter-story rift (mm) Drift change Force (kn) Drift change Force (kn) Drift change Force (kn) Story-.9-85%.06-95% % 3.09 Story-. -78%.35-95%.08-96%.06 Story-3.4-8% %.05-94%.09 Average N/A -8%.37-95%.07-95%.08 3

14 system output: K k = K ( x [ ] Cz Czx [ ] u [ ] Dz Dzu [ ]) J = k k + k k = C x + = [ k] [ k] D u [ k] k = z z K k = = z t z (40) where t is the sampling perio, an note that Cz Dz = 0 an Dz Cz = 0 using the efinitions in Eq. (3). he esign of the controller iteratively searches for an optimal control gain matrix by traversing along the optimization graient. Sparsity shape constraints are iteratively applie to the search graient in orer to compute the ecentralize gain matrices. he following three ecentralize/centralize gain matrices are obtaine: I = (4) II = (4) III = (43) able also lists the H an close-loop norms using both the H zw an the H -norms of the open-loop transfer function H controllers an the controllers. Since the control approach is equivalent to an H control esign that minimizes the close-loop H - norm, controllers are expecte to perform well in reucing the close-loop H -norm [4; 33]. Similar to the H norm, efinition of the system H -norm can also be written in terms of the singular values of the transfer function matrix: + H ( s) = σ ( j ) i ω ω π zw H zw (44) i As expecte, able shows that the controllers, no matter ecentralize or centralize, consistently perform better than their H counterparts in reucing the H - norm, while the H controllers consistently perform better than their counterparts in terms of reucing the H -norm. In this example, the secon imension of the transfer function matrix ( jω) H zw is one, because the isturbance w is a scalar that represents the groun excitation. herefore, H zw ( jω) has only one singular value at each frequency ω, which is the largest singular value. Figure 3 plots the singular value of the close-loop system transfer function H zw ( jω) using the ecentralize H controller I an the ecentralize controller 4

15 .4. ain I (H ) ain I Singular Value I. he efinition of the system H norm in Eq. () shows that the H norm shoul be equal to the peak of the largest singular value over the frequency span. Corresponingly, Figure 3 shows that the peak of the singular value using the H controller I is about 0.85, while the peak for the controller Frequency (ra/secon) Figure 3. Singular values of the close-loop system transfer function ( jω) the ecentralize H controller I an ecentralize controller H using is about.8; both of which are consistent with the H norms liste in able. Figure 3 also illustrates that the ecentralize H controller excels at pushing own the peak of the largest singular value. In comparison, the ecentralize controller is shown to excel in reucing all singular values over the entire frequency span, which agrees with the objective of minimizing the H norm (as efine in Eq. (44)). Simulations are conucte using the controllers, with the same 940 El Centro NS earthquake excitation scale to m/s. hree ieal actuators are again eploye at the three stories. Maximum inter-story rifts an control forces uring the ynamic response are zw I I 3 Maximum Inter-story Drifts ain I 3 Maximum Actuator Force ain I ain II ain II ain III ain III No Control Story Story Drift (m) Force (N).5 x 0 4 (a) (b) Figure 4. Simulation results when ieal actuators are eploye on the 3-story structure. 5

16 plotte in Figure 4. Comparison between Figure 4 an Figure shows that controllers generally achieve less reuction to peak inter-story rifts. On the other han, the avantage of the controllers in this example is that they impose lower requirements to the force capacity of the structural control evices. For the control cases, the total actuator capacity is approximately 0kN, i.e. about % of the builing weight. his ratio is much lower than the ratio of 9% for the H control cases. Similar to able for the H control cases, further illustration of the trae-off between structural response attenuation an control effort is shown in able 3 for control. For the controlle cases, changes in RMS rifts are again liste in percentages relative to the uncontrolle values. he table shows that as compare with case I, cases II an III offer greater reuction to structural response at the expense of larger control effort. Comparing the bottom rows of able 3 an able, it is again illustrate that the cases achieve less attenuation to structural response, while emaning less control effort. 3.. Numerical simulation of a 0-story benchmark structure 3... Simulation using ieal actuators o explore the performance of ecentralize H control for a larger-scale structure, a 0- story benchmark builing esigne for the Structural Engineers Association of California (SAC) project is selecte [34]. Same as the 3-story example, iscrete-time controllers are aopte in the simulation. he builing is moele as an in-plane lumpe-mass shear structure with control evices allocate between every set of neighboring floors. Figure 5(a) shows the mass, stiffness, an amping parameters of the structure. In the numerical simulations, it is assume that both the inter-story rifts an inter-story velocities between every two neighboring floors are measurable. As shown in Eq. (5), the state-space equations are formulate such that the state-space vector contains inter-story rifts an velocities. Simulations are conucte for ifferent ecentralization schemes as shown in Figure 5(b). he egree-of-centralization (DC) reflects the ifferent communication architectures, with each communication subnet (as enote by channels Ch, Ch, etc.) able 3. RMS values of control forces an the changes in RMS inter-story rifts ( control cases) RMS of uncontrolle inter-story rift (mm) I Controlle II III Drift change Force (kn) Drift change Force (kn) Drift change Force (kn) Story-.9-66%.0-7%.3-73%.4 Story-. -65% %.04-7%.0 Story % 0.4-7% % 0.59 Average N/A -66% % %.0 6

17 Seismic Mass F.6 x 0 6 kg F F9.00 x 0 6 kg F0.70 x 0 6 kg Inter-story Stiffness F-F x 0 3 kn/m F6-F x 0 3 kn/m F-F x 0 3 kn/m F5-F7 9.3 x 0 3 kn/m F8-F x 0 3 kn/m F x 0 3 kn/m F0 F5 F0 F5 Ch4 Ch3 Ch Ch F0 F5 Ch F5 Ch3 F0 Ch F0 F5 Ch F0 F5 Ch F0 F5 F0 F5 Ch Damping 5% Natural Damping (a) DC = DC = DC = 3 DC = 4 Figure 5. wenty-story SAC builing for numerical simulations: (a) moel parameters of the lumpe mass structure; (b) communication subnet partitioning for ifferent egreesof-centralization (DC). covering a limite number of stories. he controllers covere by a subnet are allowe to access the sensor ata within that subnet. For example, the case where DC = has each subnet covering only five stories with a total of four subnets utilize. For DC =, each subnet covers ten stories an a total of three subnets are utilize; meanwhile, overlaps exist between subnets for DC =. For stories covere by multiple overlapping subnets, each controller at these stories shoul have communication access to ata within all the overlapping subnets. Although each controller may comman multiple control evices, in this example, a control evice can only be commane by one controller. he gain matrices for the ecentralize information structures with DC = an DC = have following sparsity patterns: (b) = when DC = 0 40 = when DC = 0 40 ; (45) Each entry in the above matrices represents a 5 0 block submatrix. o achieve the sparsity patterns in gain matrix, the matrix variable Y in Eq. (8) is efine to have the same sparsity pattern as, an Θ is efine to be always block-iagonal. For the cases where DC = 3 an DC = 4, the number of stories covere by each communication subnet increases accoringly, which result in fewer zero blocks in. Clearly, the case where DC = 4 correspons to a centralize feeback structure with all evices in the same subnet (i.e. Ch ). o investigate the effectiveness of the propose ecentralize control esign, we first assume the 0-story structure is instrumente with ieal actuators that can prouce any esire force. Output matrices C z an D z in Eq. (3) are efine as: 7

18 C z 0 I , Dz 0 0 I = = 0 (46) Simulations are performe for ifferent egrees-of-centralization (DC =,,4) an sampling perios (ranging from 0.0 s to 0.06 s at a resolution of 0.0 s). Aitionally, three groun motion recors all scale to a peak groun acceleration (PA) of m/s are use for the simulation: 940 El Centro NS (Imperial Valley Irrigation District Station), 995 Kobe NS (JMA Station), an 999 Chi-Chi NS (CU-076 Station). wo representative performance inices, J an J, as propose by Spencer, et al. [34] are aopte: [ ] ˆ { i i [ ]} k i k i J = max max k max k (47) Earthquakes,, { ˆ } J = max z z (48) Earthquakes Here J an J correspon to maximum inter-story rifts an output vector z, respectively. k represents the inter-story rift between floor i (i =,, n) an its lower In Eq. (47), i [ ] floor at time step k, an [ ] max i k, i k is the maximum inter-story rift over the entire time history an among all floors. he maximum inter-story rift is normalize by its counterpart max ˆ i [ k ], which is the maximum response of the uncontrolle structure. he k, i largest normalize ratio among the simulations for the three ifferent earthquake recors is efine as the performance inex J. Similarly, the performance inex J is efine in Eq. (48) base on the -norm of the output vector z ; i.e. = K t [ k] [ k] k = z z z, with K being the last time step of the simulation. When computing the two inices, a uniform k sampling perio of 0.00 s is use to collect the structural response ata points for i [ ] an z [ k], regarless of the sampling perio of the feeback control scheme. Because these inices have been normalize against the performance of the uncontrolle structure, values less than one inicate that the close-loop control solution is effective with smaller inex values inicating better overall control performance. Figure 6 shows the control performance inices for ifferent egrees-of-centralization an sampling rates. enerally speaking, control performance is better for higher egrees-ofcentralization an shorter sampling perios. he plots show that all control schemes achieve obvious reuction in structural response when compare to the uncontrolle case, i.e. the normalize performance inices are much less than one. o better review the simulation results, the performance inices for the four ifferent control schemes are replotte as a function of sampling perio in Figure 6(c) an (). Figure 6() clearly illustrates the expecte comparison among the four control cases, i.e. for each sampling time, the achieve output norm generally ecreases as the egree of centralization increases. 8

19 Maximum Drift Among All Stories Z Performance Inex J Sampling Perio(s) 0.0 (a) 4 3 Degree-of-Centralization Performance Inex J Sampling Perio(s) 0.0 (b) 4 3 Degree-of-Centralization Performance Inex J Maximum Drift Among All Stories DC DC DC 3 DC 4 Performance Inex J DC DC DC 3 DC 4 Z Sampling Perio(s) (c) Sampling Perio(s) Figure 6. Simulation results for the 0-story SAC builing instrumente with ieal actuators. he plots illustrate performance inices for ifferent sampling steps an egrees-of-centralization (DC): (a) 3D plot for performance inex J ; (b) 3D plot for performance inex J ; (c) conense D plot for J ; () conense D plot for J. () While it may appear from Figure 6 that a centralize control architecture always performs better than ecentralize ones operating at the same sampling frequency, such a centralize system with high noal counts might be economically an technically ifficult to implement in large-scale civil structures. For example, significant communication an computation resources are usually require to implement a large-scale centralize control system. As a result, longer sampling perios nee to be aopte, which in turn, reuces the effectiveness of the centralize solution. In contrast, if a ecentralize architecture is implemente, the control system woul be capable of shorter sampling perios that lea to potential improvement in the control performance. It can be observe from Figure 6 that if shorter sampling perios are aopte in partially ecentralize control systems (DC or DC3), smaller performance inices can be achieve when compare with a centralize system (DC4) that aopts a longer sampling perio. he trae-off between centralization an sampling perio will be further explore in the next simulation analysis Simulation using semi-active hyraulic ampers (SHD) 9

20 o investigate the performance of ecentralize H control using realistic structural control evices, semi-active hyraulic ampers (SHD) are employe in the simulations for the 0-story structure. he arrangement of SHD ampers in the builing is shown in Figure 7(a). From lower to higher floors, the number of instrumente SHD ampers ecreases graually from 4 to. Figure 7(b) shows the installation of a SHD amper between two floors using a V-brace, together with key parameters of the amper. o accurately moel the amping force, the Maxwell element propose by Hataa, et al. [35] is employe. In a Maxwell element, a ashpot an a stiffness spring are connecte in series, which result in a amping force escribe by the following ifferential equation: keff uɺ ( t) + u( t) = keff qɺ ( t) (49) c ( t) SHD where u(t) an ɺ q( t) enote the amping force an the inter-story velocity, respectively, k eff represents the effective stiffness of the amper in series with the V-brace, an cshd ( t ) is the ajustable amping coefficient of the SHD amper. When the SHD amper is eploye in a feeback control system, if the esire amping force u(t) is in an opposite irection to the inter-story velocity ɺ q( t), as shown in Figure 7(b), the amping coefficient cshd ( t ) is ajuste so that the amper generates a force closest to the esire force. If the esire force is in the same irection to the inter-story velocity, the amping coefficient is set to its minimum value at,000 kn s/m. An important criterion to consier in evaluating a feeback control system, whether being centralize or ecentralize, is that the feeback control system shoul perform better than a passive control system. When the SHD ampers are employe, fixing the amping Floors F ~ F5 Floors F6 ~ F5 Floors F ~ F5 Upper floor moves towars right relative to the lower floor ɺ q ( t) SHD u(t) Floors F6 ~ F0 Force generate by the SHD amper attempts to move upper floor towars left Floors F ~ F5 Floors F6 ~ F5 Floors F ~ F5 Maximum Control Force Maximum Displacement Stiffness of the SHD Maximum Damping Coefficient Minimum Damping Coefficient Maximum Shaft Velocity Power Consumption,000 kn +/- 6 cm 400,000 kn/m 00,000 kn s/m,000 kn s/m 5 cm/s 70 Watts (a) (b) Figure 7. Instrumentation of semi-active hyraulic ampers (SHD) in the 0-story structure: (a) layout of ampers on the floor plans; (b) key parameters of the ampers. 0

21 coefficients of all ampers at either minimum (,000 kn s/m) or maximum (00,000 kn s/m) values constitutes a passive control system. Figure 8 presents the simulate maximum inter-story rifts when the structure is excite using the three groun motions with the PA (peak groun acceleration) scale to m/s : 940 El Centro NS, 995 Kobe NS, an 999 Chi-Chi NS. Four cases are plotte for each earthquake: the case without control, the passive case with minimum amping, the passive case with maximum amping, an a ecentralize semi-active control case. For the ecentralize semi-active control case, the DC (egrees-of-centralization) is (Figure 5) an the sampling frequency is 00 Hz. It is foun that the ecentralize H controller with the following output matrices, C z an D z : 5 0 I Cz =, Dz = (50) I0 0 achieves satisfactory results. As shown in Figure 8, all three control schemes, incluing two passive an one semi-active, reuce the maximum inter-story rifts compare with the uncontrolle case. he passive control case with maximum amping generally results in less inter-story rifts than the passive case with minimum amping, except at a few higher floors for the El Centro an Kobe earthquakes. he ecentralize semi-active control case not only effectively reuces rifts at lower floors, but also achieves greater mitigation of rifts at the higher floors compare to the two passive cases. Better performance of the ecentralize semi-active control case is observe for all three earthquake recors. For the Kobe earthquake, ecentralize semi-active control reuces the rift at the 8 th story by about 75% compare with the uncontrolle an the two passive control cases. his shows that in the passive case with maximum amping, ampers at each story may only attempt to reuce local responses an results in conflict among amper efforts at ifferent stories. 0 El Centro, PA = m/s 0 Kobe, PA = m/s 0 Chi-Chi, PA = m/s Story 0 Story 0 Story No Control Min Damp Max Damp Semi-active Maximum Drift (m) 4 No Control Min Damp Max Damp Semi-active Maximum Drift (m) 4 No Control Min Damp Max Damp Semi-active Maximum Drift (m) Figure 8. Maximum inter-story rifts for cases without control, with passive control (amping coefficients of all SHD ampers fixe at minimum or maximum), an with ecentralize semi-active control (DC = an 00 Hz sampling frequency).

22 Maximum Drift Among All Stories Performance Inex J DC (00Hz) DC (00Hz) DC3 (66.7Hz) DC4 (50Hz) Max. Damping Min. Damping (a) Z Performance Inex J DC (00Hz) DC (00Hz) DC3 (66.7Hz) DC4 (50Hz) Max. Damping Min. Damping (b) Figure 9. Simulation results for the 0-story SAC builing instrumente with semi-active hyraulic ampers (SHD). he plots illustrate performance inices for passive control cases an semi-active feeback control cases with ifferent egrees-of-centralization (DC) an sampling frequencies: (a) performance inex J ; (b) performance inex J. While in the semi-active control case that aims to minimize the overall H norm of the global structural system, efforts from ampers at ifferent stories can be better coorinate to reuce overall structural response. Figure 9 compares the performance inices for the 0-story structure instrumente with semi-active hyraulic ampers, when ifferent control schemes are aopte. he two passive control schemes inclue the maximum an minimum amping cases. o illustrate the effect of faster sampling frequency (i.e. shorter sampling perios) in ecentralize feeback control, feeback control cases with ifferent centralization egrees (DC =,,4) are associate with ifferent sampling frequencies. For each centralization egree, the sampling frequency is selecte in reverse proportion to the number of stories containe in one communication subnet (shown in Figure 5). For example, a sampling frequency of 00 Hz is associate with case DC, while a sampling frequency of 50 Hz is associate with the centralize case DC4 ue to larger communication an computation burens. he same three groun motion recors scale to a peak acceleration of m/s are use in the simulation: 940 El Centro NS, 995 Kobe NS (JMA Station), an 999 Chi-Chi. As shown in Figure 9, the feeback control cases generally achieve better performance when compare with the two passive control cases. Furthermore, the figure illustrates that although ecentralize feeback control cases o not have complete sensor ata available when calculating control ecisions, they may outperform the centralize case ue to the faster sampling frequencies that are available through ecentralization. For example, compare with the centralize scheme DC4 (at 50Hz), the partially ecentralize scheme

23 DC (at 00Hz) can provie larger reuction to both maximum inter-story rift an the - norm of the output vector z. 4. SUMMARY AND CONCLUSION his paper presents pilot stuies in exploring ecentralize structural control esign that minimizes the close-loop H norm. he ecentralize control esign offers promising solutions to large-scale structural sensing an control systems. Solutions are evelope for both continuous-time an iscrete-time formulations. he properties of linear matrix inequalities are utilize to convert the complicate ecentralize H control problem into a simple convex optimization problem. For such a convex optimization problem, ecentralize architectures can be easily incorporate to yiel ecentralize H control solutions. Such solutions are necessary to provie control systems with the ability to scale with the number of sensors an actuators implemente in the system. Nevertheless, it shoul be note that the shape constraining approach for ecentralize H controller esign is heuristic. he approach may not guarantee the minimum H -norm over the complete solution space. Numerical simulation results using a 3-story an a 0-story structure illustrate the feasibility of the ifferent ecentralize control architectures. Comparison between the performance of the ecentralize H controllers an the performance of ecentralize -base controllers illustrates that both controllers eliver expecte performance. Using the simulation results for the 3-story structure, the trae-off between structural response attenuation an control effort is emonstrate for both the H controllers an controllers. he ratio of the require total actuator capacity over the builing weight appears realistic for both types of controllers. For the 0-story structure, the simulation results emonstrate that when realistic semi-active control evices (such as the SHD ampers) are use in combination with the ecentralize H control algorithm, better performance can be gaine over the passive control cases. It is also illustrate that ecentralize control strategies may provie equivalent or even superior control performance, given that their centralize counterparts coul suffer longer sampling perios ue to communication an computation constraints. On the other han, since the propose control esign is base on the assumption of system linearity, further investigation on how to improve the control performance with non-linear semi-active control evices is neee. he rawbacks of the presente ecentralize H control esign inclue the inability to consier the effect of time elay in the feeback loop an the requirement for inter-story rift an velocity ata for feeback. Future research in ecentralize H control may consier time elay effects in the control algorithm an utilize system output feeback. Furthermore, ynamic output feeback will be explore instea of static feeback, to capitalize on a much larger controller parametric space. Comparative stuies will then be conucte between the ecentralize H an ecentralize control esigns with consieration of time elay effects []. Future investigation may also inclue eveloping 3