Optimal design of a hybrid control system for the mitigation of the wind-induced torsional response of tall buildings Ilaria Venanzi, Filippo Ubertini, Annibale Luigi Materazzi Department of Civil and Environmental Engineering, University of Perugia, Via G. Duranti 93, 06125 Perugia, Italy email: [email protected], [email protected] , [email protected] ABSTRACT: In this paper a general procedure is developed for the optimization of a hybrid control system for tall buildings subjected to wind-induced vibrations. The control system is conceived to mitigate the flexural and torsional response and satisfy serviceability limits. The hybrid control system is made of one active tuned mass damper with two longitudinal and one torsional degrees of freedom. The passive device, that is, the active tuned mass damper without active control actuators, is preliminarily designed to minimize the response of the flexural and torsional modes. The introduction of the control actuators improves the performances of the device especially for the torsional response. The feedback information necessary to compute the control forces is provided by a limited number of accelerometers arranged over the building’s height. The Kalman observer provides the estimate of the states of the system known the measured accelerations and the covariances of the wind forces. The reduction of the top corner’s accelerations and the control forces are chosen as competing targets of the multi-objective optimization problem. The design variables are the accelerometers’ positions over the building height and the weighting matrices of the LQR performance index. To reduce the computational effort, subsequent optimization sub-procedures are employed that take advantage of the genetic algorithm to find the solution of the nonlinear, constrained optimization problems. As an illustrative example, a control system is optimized for the response mitigation of a square tall building 180 m high. Wind forces are obtained by wind tunnel tests on a rigid model of the building. Results highlight that the procedure is effective in finding the optimal balance between the need of energy saving and the structural response reduction. KEY WORDS: Hybrid control system; Optimization; Tall buildings; Wind-excited vibrations. 1 INTRODUCTION Many recent studies demonstrated that the wind-induced rotational motions in high-rise buildings can be relevant in some cases . In particular, the increased accelerations near the building perimeter may cause inhabitants discomfort also compounded by the increased awareness of the motion for the occupants, provided by the visual cue of the rotating horizon . Moreover, the current tendency towards the increase of the buildings’ height and towards the realization of structures with irregular shapes makes the serviceability conditions a crucial problem for the design . In order to satisfy serviceability limits, i.e. to reduce structural accelerations, many high-rise building are equipped with passive, semi-active or hybrid control systems [4-6]. The most common ones are passive devices, such as tuned mass dampers (TMDs) , that do not require power supply. However, these systems are well-known to be subjected to frequency mistuning which might strongly weaken their effectiveness in practical applications. Multiple tuned mass dampers can partially circumvent this drawback , but their application in tall buildings is not free from a degree of complexity. A quite promising approach, able to effectively control both flexural and torsional motions in tall buildings, is to use hybrid control systems , such as active tuned mass dampers (ATMDs), which share the advantages of the active control, needing a lower actuation power with respect to the purely active systems, with the capability of working as passive systems when power supply is missing. Although it is true that semi-active and hybrid control  allow, in principle, to solve most of the drawbacks that have limited the applications of active systems until today, some aspects still need to be investigated. On this respect, one main concern in the field is the overall reliability of the control systems which must properly operate even in the case of extreme events. In this viewpoint, the role of physical limits, such as actuators saturations and limited stroke extensions of inertial actuators, cannot be neglected, as they were usually in the past. Two are the most common approaches in the literature to deal with these issues: (i) to account for physical limitations directly in the control algorithms; (ii) to design the control systems in some evolved way considering the physical limits as design constraints. The former approach, followed for instance in [11-15] using the theory of nonlinear control, is probably more suited for earthquake engineering applications where the input severity is hardly predictable and there is the need of conceiving a system which is required to be effective also when physical limitations are reached in order to prevent the structure from attaining an ultimate limit state. The latter approach , on the contrary, is probably preferable when the control system is designed to prevent serviceability issues, which is the case, for instance, of wind-excited vibration mitigation in tall structures. In this case, the optimal design of the control system plays a crucial role, while the control algorithm can be chosen with some freedom among the classic ones of linear control theory . In this paper, a methodology for the optimal design of a hybrid control system using a three degrees of freedom (DOFs) ATMD for the flexural/torsional response mitigation of tall buildings is presented. The reduction of the top corner’s accelerations and the control forces are the competing targets of the multi-objective optimization problem. In a general framework of incomplete number of measurements , the design variables are the accelerometers’ positions over the building height. Also the LQR performance index is optimized to balance the need of energy saving and structural response reduction. To reduce the computational effort, subsequent optimization sub-procedures are employed that take advantage of the genetic algorithm to find the solution of the nonlinear, constrained, multi-objective optimization problem. 2 THE HYBRID CONTROL SYSTEM The hybrid control system considered, without loss of generality, in this paper, is made of an ATMD with 3 DOFs (Figures 1,2). The device is composed by a rigid body with translational and rotational mass, placed on the top floor of a tall building. The structure is schematized as a simplified dynamic system having 3 DOFs for each floor. The total number of DOFs, including those of the ATMD, is 3n+3 where n is the storeys number. The control forces produced by the four actuators in the x and y directions, named as u1x , u1y , u2 x , u2 y , result in two translational forces Fx, Fy and a torsional moment Mt applied at the elastic center, G, of the ATMD which coincide with the elastic center of the structural storey. Figure 2. Schematic representation of the ATMD location at the top floor of the tall building. The control forces are: u1x Fx 2 M t e u2 x Fx 2 M t e u u F 2 2y y 1y (1) where e is the distance between each control device acting along the x direction and the elastic center G of both the structural storey and the ATMD. Under the hypothesis of neglecting aeroelastic effects, the classic equations of motion of the structure-ATMD system are written in second-order form as: M s q Cs q Ks q f B0u (2) where q is the vector of generalized displacements of the structural-TMD system, having dimension 3n+3, M s , Cs and K s are the mass, damping and stiffness matrices, respectively, f is the vector of wind loads, u is the vector of control forces which, in the present case, is defined as: u Fx M t T Fy (3) B0 is a convenient collocation matrix and a dot denotes time derivative. The state space formulation of the equation of motion of the actively controlled system is stated from Equation (2) as follows: z Az Bu Hf (4) where z q q is the state vector, A is the system matrix, B T Figure 1. Schematic representation of the ATMD’ location over the tall building’s height. and H are the location matrices for the vectors u and f , respectively. Owing to the common availability of accelerometers as monitoring sensors, tracking of the state by means of a state observer using only acceleration measurements is here considered. In particular, a limited number of storeys are instrumented with 3 sensors per floor in order to measure the alongwind, acrosswind and torsional accelerations. The output, y , thus results in a linear combination of generalized nodal accelerations, as: y Ca q (5) where Ca is a convenient matrix that selects the monitored DOFs. After well-known computations, Equation (5) can be rewritten in terms of state vector and control forces as: y Cz Du Hf (6) where: C Ca M s 1 K s M s 1Cs D Ca M s 1 B0 (7) and v is the vector of measurement noise. The linear optimal control algorithm is used for the problem at hand. The linear quadratic performance index can be written as: J zT Qz uT Ru t 0 histories and from the real-time measurement of the mean wind velocity and direction at the reference height. In this way, the wind process is correctly reproduced in the numerical calculations, which allows an indirect check of the effectiveness of the Kalman observer for tracking the state in wind excited vibrations of tall buildings. 3 OPTIMAL DESIGN OF THE CONTROL SYSTEM 3.1 Statement of the problem In a general framework of incomplete number of measurements and limited power supply, the optimization problem is aimed at designing the hybrid control system which gives the best trade off between structural response reduction and energy saving, with due account given to the physical limitations of the actuators. In order to minimize the computational effort, this goal is achieved through subsequent optimization steps (Figure 3): 1) Optimal TMD tuning; 2) Optimal calibration of the LQR performance index; 3) Optimal choice of the sensors’ location. (8) where Q and R are the weighting matrices of the state vector and the control forces vector respectively. By application of the classic LQR algorithm the optimal gain matrix K, which allows minimizing the performance index J in Equation (8), is computed and the feedback is calculated as u Kz . To provide an estimate, zˆ , of the state from the incomplete measurement set, a classic Kalman filter is used. Accordingly, the equation of the state observer reads as: zˆ Azˆ BKzˆ L y Czˆ DKzˆ (9) where L is the optimal Kalman gain matrix. In Equation (9), calculation of the feedback using the state estimate should be noticed. For fast time integration, Equations (4), (6) and (9) are readily converted in compact form using the augmented state z zˆ as: T z A zˆ LC BK z Hf A BK LC zˆ 0 (10) The computation of the Kalman filter gain matrix L , requires the hypothesis that both measurement and process vectors are realization of white Gaussian stochastic processes. Here, the measurement noise, v , is assumed to satisfy such hypothesis and its covariance matrix Rn E vvt is directly assigned. The wind process, on the contrary, is non-Gaussian and non-white but, for the purpose of computing matrix L in Equation (9), the hypothesis of a white Gaussian wind process is retained. Given the practical difficulty of directly measuring the cross-correlation of the wind forces f, due to the high number of anemometers required, the covariance matrix Qn E f f t may be obtained from simultaneous wind tunnel measurements of the pressure coefficients time Figure 3. Outline of the optimization procedure. 3.2 Optimal TMD tuning The parameters of the passive TMD are adjusted to minimize the response of the controlled modes. In particular, according to the solution proposed by Warburton  for i random excitation, the optimal tuning ratio, opt , of the i-th mode is: i opt i TMD Si 1 i / 2 (11) 1 i and the corresponding damping ratio is: i opt i 1 3i / 4 1 i 1 i / 2 (12) i where TMD is the i-th TMD’s angular frequency, si is the i- i th angular frequency of the structure, i mTMD M Si is the i mass ratio, mTMD is the i-th generalized mass of the TMD and M Si is the participating mass of the selected mode. In order to control the flexural mode, the stiffness kTMD and the damping 1 1 and opt for a fixed cTMD of the TMD are computed from opt value of the mass ratio 1 . To control the structural rotations, the TMD must also be tuned to the torsional mode with an iterative procedure as follows: - a trial value of the rotational mass ratio 3j is fixed; - 3 the tuning ratio opt , j is computed according to - Warburton (11); 3 the angular frequency TMD , j and the mass moment j of inertia JTMD kTMD 2 3 TMD are computed; i 1, 2 j x, y (15) where ui j are the control forces, umax is the upper bound of the control forces, qTMD, ij are the strokes of the TMD in the directions x and y, qTMD,max is the upper bound of the TMD strokes. The upper bounds umax and qTMD ,max depend on the technical characteristics of the selected control devices. To keep into account the constraints, a penalty function P q, f is added to the objective function when the constraints are violated so that the solution may be discarded. - the mass ratio - the procedure is repeated with the new value 3j 1 3.4 until 3j 1 3j tol , with tol assigned tolerance; The third phase of the optimization procedure leads to the optimization of the sensors locations along the height of the building. In this optimization sub-problem the multi-objective function, f 2 p , to be minimized is: - j 1 3 JTMD M 3 S is computed; 3 the damping ratio opt is computed from Equation (12) in correspondence of the final value 3j 1 . 3.3 ui j umax qTMD ,i j qTMD ,max Optimal choice of the sensors locations Optimal calibration of the LQR performance index The optimal calibration of the weight matrices R and Q applied to the state vector and the control forces in the LQR performance index, Equation (3), is achieved through an optimization procedure. In this optimization sub-problem the multi-objective function, f1 k , to be minimized is: q u f2 p q i x, y i observed q i x, y i P q, u (16) active where the design variable p is a (m x 1) location vector with m number of instrumented floors, i x , y qi observed is the sum P q, u (13) of the standard deviations of the top corner accelerations of the actively controlled system instrumented with a limited number of sensors, i x , y qi active is the sum of the where C1, C2 are the weighting coefficient of the objective function, i x , y qi active is the sum of the standard standard deviations of the top corner accelerations of the ideal actively controlled system. The location vector p contains numbers from 1 to n, with n equal to the total number of storeys, to identify which are the instrumented floors: f1 k C1 i i x, y q i x, y i active passive C2 i x, y j 1,2 ij e i x, y ij deviations of the top corner accelerations of the actively controlled system, i x , y qi passive is the sum of the standard deviations of the top corner accelerations of the passively controlled system, i x , y uij is the sum of the j 1,2 standard deviations of the control forces, i x, y eij is the sum of the standard deviations of the elastic restoring forces of the TMD and P q, f is the penalty function. The matrix Q is assumed to be equal to the identity matrix while the matrix R is the product between the identity matrix and a coefficient k that is the design variable of this subproblem. In order to reduce the computational effort, a lower and an upper bounds are set to the design variable: kmin k kmax (14) where the bounds kmin and kmax are assigned on the basis of a preliminary sensitivity analysis. The non-linear constraints to the sub-problem are the following: 1 pi n i 1,..., m (17) Three accelerometers are arranged over each instrumented floor to measure accelerations along the 3 DOFs. During the optimization process, the trial components of the vector p which are real numbers are rounded to the closer integer. The non-linear constraints to the sub-problem are expressed by Equation (15). 3.5 Optimization algorithm The problem at hand is a nonlinear constrained optimization problem for which the objective function cannot be written as an explicit function of the design variables. For this reason, it is not possible to analytically compute the first and second derivatives of the function to be minimized and gradient based methods cannot be used. The proposed procedure is based on the use of the genetic algorithm that does not require any information about the gradient of the objective function but is based on random generation of trial solutions [20, 21]. In particular, the genetic algorithm is an optimization method that is based on natural selection, the process that drives biological evolution. It repeatedly modifies a population of individual solutions selecting at each step individuals at random from the current population to be parents and using them to produce the children for the next generation. Over successive generations, the population evolves toward an optimal solution using techniques inspired by natural evolution such as selection, crossover and mutation. 4 4.1 NUMERICAL EXAMPLE Description of the structure The proposed optimization procedure was applied to a square tall building 180 m high with an aspect ratio of 6. The structure is made of steel with central cores and systems of bracings in both the principal directions. Floors are reinforced concrete slabs capable of warranting a rigid inplane behavior. The structure was modeled as a simplified dynamic system having 3 DOFs for each floor obtained by static condensation from a finite element model of the structure (Figure 4). profiles and turbulence intensity corresponding to suburban terrain conditions. To make the pressure time histories measured in the wind tunnel representative of the real phenomenon, it was chosen to respect the similitude criterion on the reduced frequency, as it is usually done for tall buildings. 4.3 Results In the present section the following cases will be analyzed: uncontrolled case, passively controlled case, ideally actively controlled case and control-with observer case. In the ideal active case, the entire state is assumed to be known, while in the control-with observer case it is estimated by means of the state observer presented in Section 2. The first step of the optimization process was the optimal tuning of the passive control system to the first flexural and torsional modes as explained in Section 3.2. The optimal values of the mass, tuning and damping ratio are reported in table 1. Table 1. TMD optimal parameters. Mode No. 1 (Flexural x) 2 (Flexural y) 3 (Torsional) Mass ratio Tuning ratio opt Damping opt 0.020 0.019 0.001 0.985 0.986 0.999 0.070 0.069 0.866 The modal characteristics of the uncontrolled structure and the passively controlled structure are summarized in Table 2. Table 2. Modal characteristics of the uncontrolled structure and the structure with TMD. Mode 1 2 3 Figure 4. Structural model of the analyzed system. 4.2 Wind load modeling The forcing functions representing the wind load were obtained from synchronous pressure measurements carried out in the wind tunnel. Experimental tests were carried out in the boundary-layer wind tunnel operated by CRIACIV (Interuniversity Research Center on Buildings Aerodynamic and Wind Engineering) in Prato, Italy. The rigid 1/500 scale model of the building was instrumented with 120 pressure taps, 30 for each side. Tests were carried out with wind speed Frequency (Hz) uncontrolled 0.208 0.215 0.287 Frequency (Hz) with TMD 0.192 0.195 0.222 The second step of the optimization procedure was the calibration of the LQR weighting coefficients, that is, matrices Q and R in Equation (8). In this case, the choice of the control actuators and the maximum extensions of the strokes of the ATMD are the main control constraints to be satisfied. Indeed, in principle, there would be essentially no upper limit to the control effectiveness of the system if these constraints were not accounted for. Nevertheless, the obvious counterparts of a large control effectiveness are large strokes extensions and large control forces. Hence, the aforementioned physical limitations dictate, in practice, the maximum achievable control effectiveness. The control constraints considered in this work are summarized in Table 3. Table 3. Constraints of the hybrid control system. Maximum control forces [kN] Maximum strokes [m] u1x , u2 x u1 y , u2 y qxATMD q yATMD 800 800 3.0 3.0 It was decided to consider unit weights for all generalized structural displacements and their time derivatives. Moreover, a preliminary analysis showed the convenience of applying small penalties also to the displacements of the ATMD, in order to satisfy the stroke limitation. The weighting coefficients of the objective function C1 and C2 expressed by Equation (13) are assumed equal to 1 and 0.2 respectively, in order to privilege the response reduction with respect to the control force limitation. Then, by applying the optimization procedure described in Section 3.3, the optimal value of the coefficient k appearing in Equation (14) was obtained. The results are summarized in Table 4. J qcon quncon quncon where qcon is the standard deviation of the top corner acceleration obtained with the controlled system and quncon is the standard deviation of the top corner acceleration obtained with the uncontrolled system. The performance indexes are reported in Table 5 with reference to the top corner’s accelerations in x and y directions and to: - the passively controlled system; - the ideal actively controlled system; - the actively controlled system with state observer. Table 4. Results of the calibration of the LQR coefficients. qxATMD q yATMD qrATMD k 0.6 0.7 0.6 10-12 1 qx [m/s2] 0.5 0 : -0.5 -1 200 300 t [s] 400 500 600 1 0.5 qy [m/s2] Acceleration x (%) - 26.5 - 38.3 - 37.7 Passive Active ideal Active with observer In Figure 5 are shown the time histories of the x and y components of the acceleration at the corner of the top floor for the cases of uncontrolled structure, passively controlled structure and ideal actively controlled structure. 100 Table 5. Performance indexes J . Control penalty State penalties 0 (18) 0 Acceleration y (%) - 28.9 - 48.4 - 47.9 It can be noticed that the passive solution proves to be quite effective in reducing the structural response. However, as it is well-known, such a result is not robust in the sense that the almost unavoidable mistuning of the device would strongly impair its control effectiveness. Hence, the need for an upgrading towards the hybrid approach clearly points out. The slightly improved performance of the hybrid system with respect to the passive one should therefore be interpreted in this viewpoint. It should be also observed that the technical difficulties in making the TMD devices effective both in translation and in torsion may lead to the design of a control system in which the torsional response is mitigated only by the active forces. A minor remark concerns the effectiveness of the active system using acceleration information from an incomplete measurement set. Indeed, it was observed that, already with a relatively small number of sensors, the effectiveness of the control system was essentially similar to the ideal case. On this respect, Figure 6 shows a comparison between the ideal active response and the one with state observer considering an instrumented storey every four storeys. : -1 0 100 200 300 t [s] 400 500 600 Figure 5. Accelerations in the x and y directions at the corner of the top floor: uncontrolled (gray), passively controlled (black), actively controlled (red). In order to quantitatively represent the effectiveness of the control system in reducing the structural accelerations, a performance index J was defined as follows: qy [m/s2] -0.5 0.5 0 -0.5 0 100 200 300 t [s] 400 500 600 Figure 6. Comparison between ideal active response (red) and response with state observer (black). It should be noticed that the overall optimization of the control parameters pursued in this example depends upon the severity of the external input. In other words, different weight coefficients would be obtained under different levels of external excitation. This aspect could be solved by means of a gain scheduling approach, as proposed, for instance, in  for earthquake engineering applications: a similar approach goes beyond the purposes of the present investigation but shall represent its natural future development. Table 6. Optimal position of the accelerometers along the building’s height. Sensor no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 ATMD x stroke [m] 3 2 1 0 -1 -2 -3 0 100 200 300 t [s] 400 500 600 ATMD y stroke [m] 3 2 1 0 -1 -2 -3 0 100 200 300 t [s] 400 500 600 100 200 300 t [s] 400 500 600 Actuator x force [kN] 300 200 100 0 -100 -200 -300 0 Actuator y force [kN] 200 100 0 -100 -200 -300 0 100 200 300 t [s] 400 500 600 Figure 7. ATMD strokes along the x and y directions (top) and corresponding actuators forces (bottom). Heigth (m) 6 18 21 42 51 72 75 87 108 111 120 135 144 147 153 156 165 168 171 180 183 In Figure 7 are shown the strokes of the ATMD along the principal x and y directions and the corresponding forces of one actuator. It can be noted that the limits summarized in Table 3 are never exceeded although stroke limitation appeared to be a rather more severe constraint than actuator force saturation. The final step of the proposed design methodology is to optimize the position of the monitoring sensors by applying the procedure presented in Section 3.4. This analysis, in the present case, led to the results presented in Table 6. It should be noticed that, in the optimal solution, sensors are placed almost uniformly along the height to represent the motion of the first three modal shapes which are, for the symmetric building chosen as case study, nearly linear along the height. The genetic algorithm showed to be effective and fast in providing the optimal set of design variables and the solution proved to be quite stable with respect to the initial guess population. 5 300 Floor no. 2 6 7 14 17 24 25 29 36 37 40 45 48 49 51 52 55 56 57 60 61 CONCLUSIONS A general procedure for the optimal design of a hybrid control system for the mitigation of the wind-induced flexural/torsional response of tall buildings has been presented in this paper. The considered control actuator is represented by an ATMD with 2 longitudinal and 1 torsional DOFs. The chosen control algorithm is the classic LQR complemented with a Kalman observer for state tracking using only acceleration measurements. The proposed formulation allows to optimize the weight parameters of the controller as well as the position of the monitoring sensors, under appropriate constraints represented by the physical limitations of the control actuators. Indeed, the choice of handling these limitations offline, i.e. as “a priori” optimization constraints, was found to be quite appropriate for applications where the structure must be preserved from serviceability issues. The proposed methodology has been also applied to a numerical example in order to show its effectiveness. The following are the main observations of this work: - The adopted 3 DOFs passive TMD, if properly tuned, is seen able to strongly reduce the wind excited structural response; - The improvement of control effectiveness achieved by means of the hybrid approach, for realistic values of the control constraints, with respect to the optimal passive case is significant although not extraordinary. However, this improvement should be regarded as substantial in the sense that the hybrid control solution is more robust and does not suffer from mistuning issues; - In the optimal design procedure, stroke limitation appeared to be a rather more severe constraint than actuator force saturation; - Although the wind forces acting on the structure are neither Gaussian nor white (in the present case they were extracted from wind tunnel tests), the classic Kalman state observer is seen to be quite effective for the chosen application; - The proposed procedure is seen to be a quite effective tool for the optimal design of the control system. Particularly, its ability and in yielding the optimal positions of the sensors is noteworthy; - As the main future developments of this work, different levels of input severity shall be incorporated in the formulation using a gain scheduling approach. Also, the robustness against random variations of structural parameters shall be investigated. 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