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Active control of a tall structure excited by wind
Journal of Wind Engineering and Industrial Aerodynamics 83 (1999) 209}223
Active control of a tall structure excited by wind Jin Zhang, Paul N. Roschke* Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, USA
Abstract A control strategy is developed for application to a #exible laboratory structure excited by simulated wind forces for the purpose of minimizing along-wind accelerations. Static and dynamic characteristics of the structure are identi"ed through a modal analysis method that formulates a linear model of the system. Actual wind speed data is used to produce a simulated wind loading by means of drag forces. An LQG/LTR control strategy based on acceleration feedback is used in conjunction with a magnetorheological (MR) damper to reduce structural response. When a strong wind loading is applied to the structure, the control force notably reduces simulated peak #oor accelerations. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Acceleration; Active control; Building; Magnetorheological damper; System identi"cation; Vibration; Wind
1. Introduction As modern materials and construction methods lead to taller buildings that are increasingly #exible, environmental loads such as strong wind gusts and earthquakes can be expected to increase building response. Whereas safety is a major concern for a civil engineering structure subjected to an earthquake, it is not a consideration for most building structures in strong wind environments. The main concern for high-rise buildings in strong wind events is discomfort to the occupants, such as physical symptoms due to motion sickness or psychological responses like anxiety. One approach to mitigate undesirable motions due to hazardous wind or earthquake loads is to alter the dynamic characteristics of a building with respect to a given loading. This idea has developed into the concept of structural control, which was "rst
* Corresponding author. Fax: #1-409-845-6554. E-mail address: [email protected] (P.N. Roschke) 0167-6105/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 9 ) 0 0 0 7 3 - 2
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presented by Yao [1]. Over the past several decades, a number of physical devices have been investigated for the control of buildings. Such devices include the active bracing system (ABS) and magnetorheological (MR) dampers. An ABS (see Fig. 1) typically consists of a set of pre-stressed tendons or braces that can be attached to a frame of the structure in the plane for which motion is to be controlled. These braces are connected with an actuator that sti!ens or relaxes the system according to a control algorithm. An MR damping system (see Fig. 2) consists of a damper that is rigidly connected between the ground and the "rst #oor or between two neighboring #oors. Viscous properties of the damper are changed according to the voltage from a control algorithm. An algorithm (termed as a `controllera) is used to determine the control force or damping that is to be applied to a structure through an ABS or MR system, respectively. Virtually, all current control methods utilize optimum strategies that minimize one or more performance indices [2]. The most basic optimal controller is the linear quadratic regulator (LQR). This control strategy minimizes a function that relates the response (or states) of the structure and the control input. If the needed states of the structure are not directly measured, an estimator such as a Kalman "lter can be employed to approximate these states. The addition of a Kalman "lter to an LQR strategy leads to what is termed as the linear quadratic Gaussian (LQG) problem. The Kalman "lter characterizes statistical distribution of the noise processes
Fig. 1. Active tendon control system.
Fig. 2. MR damper control system.
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inherent in the system as they are related to the sensors [2]. However, the LQG approach has been shown to be overly sensitive to changes in structural parameters. This motivates the use of a variation of the LQG strategy that takes into account frequency domain characteristics and feedback properties. This variation is termed as the loop transfer recovery (LQG/LTR) problem. In what follows, system identi"cation is carried out on a #exible four-storey model of a tall building in a laboratory to determine the salient dynamic characteristics of the structure. A simulated wind loading is applied to a numerical model of the building that is equipped with an MR damper. Finally, a control algorithm that uses acceleration feedback and a time delay is shown to provide signi"cant reduction of acceleration.
2. Experimental setup A tall, slender, four-storey model of a #exible tall building structure was constructed in a laboratory (see Fig. 3) in order to demonstrate the performance of a semi-active control system with an MR damper. Dynamic parameters are collected through system identi"cation as described in the following section, and controller performance
Fig. 3. Laboratory structure.
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is simulated through MATLAB. The main structural members are graphite epoxy tubes donated to the study from NASA Johnson Space Center. Vertical members are continuous throughout the height of the structure. Horizontal tubes are joined to the vertical members at each #oor level. Joints that connect vertical and horizontal tubes are custom-made from polyvinyl chloride (PVC) pipe. Flat, plastic plates with additional mass span the horizontal beam members in a slab-like fashion at each level. The bottom of each column is secured by a PVC pipe encased in a wooden sub-assemblage that is bolted to the concrete #oor. In order to ensure that very little motion occurs normal to the along-wind direction, the structure is also restrained by a light X-bracing cable. Nominal dimensions of the model are 0.77 m]1.25 m]6.52 m. Height of the 1st, 2nd, 3rd, and 4th #oors above the semi-rigid base are 1.73 m, 3.30 m, 4.93 m, and 6.52 m, respectively. To measure the response of the structure to loading, an accelerometer is placed on each #oor. An MR damper purchased from Lord Corporation is intended to be used to provide a controllable force. MR #uids typically consist of micron-sized, magnetically polarizable particles dispersed in a carrier medium such as mineral or silicone oil. Normally, MR #uids are free-#owing liquids having a consistency similar to that of motor oil. However, when a magnetic "eld is applied within several ms their consistency changes and the #uid becomes semi-solid with a yield strength of up to 100 kPa. The degree of change is proportional to the magnitude of the applied magnetic "eld. MR dampers are also stable over a broad temperature range, from !40}1503C.
3. System identi5cation 3.1. Introduction The knowledge about the dynamic characteristics of a system is one of the most important aspects of control design. An accurate mathematical model of the system determines whether a controller works properly or becomes unstable. The most popular way to represent a model is through a complex frequency response function that relates the input and output characteristics of the experimental system. For the identi"cation of many civil engineering structures, inputs to the system are applied forces, while outputs include displacements, velocities and accelerations. The complex frequency response function can be thought of as a series of transfer functions that transform disturbance inputs to acceleration outputs. Natural frequencies, mode shapes and equivalent viscous damping characteristics are determined from analysis of the input and output data and are used in the formulation of a state-space representation of the structure. 3.2. Identixcation procedure For this investigation, a complex frequency response function that relates input forces to #oor accelerations is used to determine natural frequencies, mode shapes and
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equivalent viscous damping characteristics of the structure. These functions are obtained as follows: f The laboratory model is excited with an impact hammer and a time history of the impact force and accelerations of each #oor are read into a data "le. f Transfer functions are calculated using the acceleration and force time histories. f Natural frequencies and corresponding amplitudes of the structure are identi"ed (see Table 1). f Amplitude ratios of frequency domain response and the corresponding phase angles are used to determine the mode shapes. f Equivalent viscous damping characteristics are determined by the half-power bandwidth method: f i !f i 1, m"2 (1) i f i #f i 2 1 where f and f are frequencies at which the transfer function amplitudes equal 0.707 1 2 times the transfer function peak at the ith natural frequency. 3.3. State space representation The general equation of motion for a multi-degree-of-freedom system excited by a forcing function is: Mx$ (t)#CHx5 (t)#Kx(t)"d(t),
(2)
where M, CH, and K are the mass, damping and sti!ness matrices, respectively, of the structure; d(t) is a time-dependent vector of disturbance forces; and xK (t), x5 (t) and x(t) are time histories of the acceleration, velocity and displacement vectors, respectively. De"ning the state vector
CD
z"
x i x5 i
(3)
Table 1 Structural parameters Parameter (1)
Mode 1 (2)
Natural frequency (Hz) Damping ratio (%) Normalized mode shape Floor 1 Floor 2 Floor 3 Floor 4
1.053 9.64 0.3364 0.6358 0.7257 1
Mode 2 (3)
Mode 3 (4)
3.835 3.86
8.108 2.55
!0.8542 !1.0588 !0.2354 1
1.9110 !0.1887 !1.6763 1
Mode 4 (5) 13.662 1.22 !3.6626 4.1205 !3.0009 1
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and requiring that the desired output vector y consist of the accelerations of each #oor xK allows Eq. (2) to be recast into the general state space form: z5 "Az#Bd, (4)
y"Cz#Dd, where
C
A"
0 4 !M~1K
D
I 4 !M~1CH
C"[!M~1K !M~1CH]
C D
B"
0
4 , M~1
D"[M~1].
(5)
Note that for the structure at hand 0 is a 4]4 zero matrix, I is a 4]4 identity 4 4 matrix, M is determined through a lumped mass method, and matrices K and CH are determined as follows [3]: (6) K"M/ X/TM CH"M/ W/TM, n n n n where X is the diagonal matrix [u2 , u2 , 2,u2], W is the diagonal matrix n 1 2 [2m u , 2m u ,2,2m u ], u is the ith natural frequency, and m is the damping ratio 1 1 2 2 n n i i of the ith mode. In summary, all of the required parameters necessary to obtain state space representation matrices A, B, C, and D are available from the system identi"cation described earlier. 4. Wind disturbance 4.1. Wind speed data Wind speed data to be used in simulation of a strong wind event on the laboratory structure was obtained from the Wind Engineering Research Field Laboratory (WERFL) at Texas Tech University [4]. Data were collected from a free-standing tower at heights of 4, 10, 21 and 49 m above the surface of the earth in #at, open terrain. A selected portion of the total time history is presented in Fig. 4. The sampling rate was 10 Hz and low-pass "ltering was conducted at 8 Hz.
Fig. 4. Wind speed set M15N571.
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4.2. Characterization of wind Vibration of tall structures is elicited by the turbulent component of the wind velocity. Wind turbulence in the atmospheric boundary layer is typi"ed by random #uctuations of velocity and pressure. Kaimal et al. [5] have proposed the following normalized power spectral density of wind velocity for structural design purposes: u ) S(z, u) 200f " , u2 (1#50f )5@3 H where uz f" U(z)
(7)
(8)
z is the height above the surface, ;(z) is the mean wind speed at height z, and u is the frequency in Hz. S(z, u) is the power spectral density (PSD) at height z for the frequency u. The parameter u is the shear #ow velocity that is determined by: H kU(z) u " , (9) H z ln z 0 where k is von Karman's constant (generally assumed to be approximately 0.4), and z is the roughness length, a variable characterizing the terrain [5]. Solving for S(z, u) 0 in Eq. (7) gives the wind PSD
A B
u2 200f H S(z, u)" . u(1#50f )5@3
(10)
The force that a #uid produces on a body is of the following form [6]: 1 F "C o<2A, D D2
(11)
where C is the drag coe$cient, o is the density of the #uid, < is the #ow velocity, and D A is the area exposed to the #ow. The drag coe$cient of the rectangular laboratory model is taken to be 1.48 [6]. The total along-wind response, such as de#ection, velocity or acceleration, may be viewed as a sum of two parts: X(z)"xN (z)#x(z),
(12)
where x6 (z) is the mean response, and x(z) is the #uctuating response that is induced by the wind gustiness [5]. In what follows, only the latter component is considered. 4.3. Application of wind speed data The force to be applied to each #oor of the laboratory model from the Texas Tech wind data set is calculated using Eq. (11). The tributary areas that relate wind pressure
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to an equivalent force at a #oor level are taken to be the surface areas on the building model that are normal to the wind #ow. Also, according to Eq. (12), only the #uctuating component is considered. The main purpose of this study is to design and test an active control algorithm that can reliably reduce the vibration of a wind-excited building. Within reasonable bounds, an acceptable controller should function properly for any wind load. Therefore, relationships of similitude are not emphasized in this research work. Here, the wind loading from each of the four anemometer readings is directly applied to the corresponding #oor of the structure, although the heights of the #oors in the laboratory model do not directly scale to the heights of the "eld readings. Another simpli"cation made is that the coe$cient of drag is applied uniformly to each #oor. Alternatively, pressure coe$cients for the windward and leeward walls could be determined for a building that has aspect ratios that match those of the laboratory model. Resultant pressures could then be approximated by discrete forces.
5. Control design 5.1. Control algorithm After salient dynamic characteristics of the experimental structure have been determined and a time history of wind disturbance for each of the four stories has been speci"ed, a control strategy is formulated that is aimed at amelioration of the vibration of the structure. The control force is assumed to be capable of being produced by an active control device such as an active tendon system or an MR damper acting between the ground and the "rst #oor. Line of action of the control force is taken to be parallel to the direction of the wind (see Figs. 1 and 2). A diagram of relationships for this system is presented in Fig. 5 [2], where d denotes the disturbance input and G represents the four-storey laboratory structure. Acceleration response of the structure is denoted by y. Measurement of the output is taken to be a noise n that is included in the feedback. H denotes the controller that produces a force vector u that, if designed correctly, reduces the output and stabilizes the closed-loop system. A linear quadratic regulator is used to optimize a quadratic function of the state and control vectors. Given the linear system shown in Fig. 5, a function J that relates
Fig. 5. Control block representation.
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the state vector and control force is as follows:
P
1 = (z@Qz#u@Ru) dt, (13) J" 2 0 where Q and R are weighting matrices of the state and control forces, respectively. The integrand has a quadratic form and Q and R are usually symmetric. It is assumed that Q is positive de"nite (i.e., it is symmetric and has positive eigenvalues) and R is positive semi-de"nite (i.e., it is symmetric and has non-negative eigenvalues). These assumptions imply that the cost is non-negative and, therefore, its minimum value is zero. Minimization of Eq. (13) leads to the necessity of solving the algebraic Riccati equation (ARE) for the Riccati matrix P [2]: A@P#PA#Q!PBR~1B@P"0.
(14)
Once the Riccati matrix is known from Eq. (14), the gain matrix K and target control force vector u are calculated by the following relations: K"R~1B@P,
u"!Kz.
(15)
If the system is both controllable and observable, and if an appropriate Q has been selected, the positive-de"nite solution of the ARE results in an asymptotically stable closed-loop system. However, if not all of the state variables can be measured, optimal control cannot be applied directly and a state estimator is needed. The solution to this problem was "rst provided by Kalman and Bucy [2]: z(5 "Az( #Bu#L(y!y( ),
(16)
where z( and y( are estimates of the actual states z and y, respectively, and L is the observer gain. This gain is computed from the following set of equations so as to minimize the size of the estimation error intensity matrix E: AE#EA@#Q !EC@R~1CE"0, 0 0 (17) L"EC@R~1 0 where matrices Q and R represent the intensity of the process and sensor noise 0 0 inputs, respectively. The LQG algorithm combines a Kalman "lter and an LQR algorithm in a single controller. This controller may be constructed by substituting Eqs. (4) and (15) into Eq. (16) and simplifying z(5 "(A!BK!LC#LDK)z( #Ly, u"!Kz( .
(18)
Experience has shown that although LQG is an improvement over LQR, it demonstrates low robustness [2]. Moreover, it is overly #exible due to the arbitrariness of the noise weighting matrices. To alleviate these inadequacies, the LQG/LTR technique modi"es the design procedure to recover the advantageous stability
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margins of the LQR algorithm while maintaining the basic mathematics of LQG. The open-loop transfer function for an LQR controller at the input to the structure is given by L(s) "K(sI!A)~1B LQR and the LQG loop transfer function may be expressed as
(19)
L(s) "K(sI!A#BK#LC!LDK)~1L(D#C(sI!A)~1B). (20) LQG Two criteria for the LQG/LTR approach are that the structure has a minimum phase (i.e., it has no zeros in the right-hand plane) and the noise matrices are given by R "1 0 and Q "q2BB@, where q is a tuning parameter. Through these assumptions it can be 0 shown that [2] lim L(s) "L(s) . (21) LQG LQR q?= The tuning parameter q can be increased until the open-loop transfer function is close to the LQR loop transfer function. That is, the feedback loop transfer function for the structure is recovered as the noise goes to in"nity. 5.2. Time delay The previous discussion is based on the assumption that all operations in the control loop can be performed instantaneously. However, in reality, each operation experiences a delay between sending the command signal and measuring the response. This delay is caused mostly by signal processing, on-line computation and response time of the actuator system [7]. It may lead to instability in the controlled system. Thus, the degree to which a controller can tolerate a certain amount of time delay while not signi"cantly degrading the controlled response, is an important index to evaluate a controller. Provisions for time delay are made in the controller described in the following section. 5.3. Controller design Design of an LQG/LTR controller that is to specify a control force to be applied at the "rst #oor of the laboratory structure is implemented through MATLAB and SIMULINK [8]. Fig. 6 is a SIMULINK model of the controller. A 0.05-s time delay is taken into account. Width of the connecting lines represents the dimension of the data vector that is being transmitted. The &plant' block represents the state-space model of the laboratory building with the input as the external force and the output as displacement, velocity and acceleration of each #oor. In order to simulate motion of an uncontrolled building, wind disturbance is the only external force. For the controlled building, the force speci"ed by the controller that is to be generated by the MR damper is added to the force of the wind. The &LQG/LTR controller' block is the state-space form of Eq. (18), with the input being accelerations of each #oor and the
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Fig. 6. LQG/LTR controller system model.
output being the control force that is applied by the MR damper. Measurement noise is added to the acceleration time history of each sensor. The magnitude of this white noise is determined by an analysis of a stationary accelerometer. 5.4. Feedback properties A properly designed controller should be able to operate e!ectively and maintain stability of the structure in a working environment. This environment may cause system parameters to change or disturbances to this system might vary. Even if these conditions were not considered, modeling of the system by a mathematical representation has uncertainty. For instance, in this study uncertainty of the natural frequencies, masses and mode shapes of the structure needs to be considered. Also of importance is the noise of the measurement devices themselves. In order for the feedback/control system to be e!ective and stable, it should satisfy certain performance speci"cations and allow for uncertainties in the model. These issues may be studied through the frequency domain. The block diagram for Fig. 5 may be written in equation form as follows: Y(s)5
GH 1 GH R(s)1 D(s)! N(s). 11GH 11GH 11GH
(22)
The relationships between the output y, disturbance d, and noise n are de"ned by the sensitivity S and transmissibility T functions as follows: 1 S" , 1#GH
(23)
GH . T" 1#GH
(24)
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Therefore, small values of the sensitivity function reduce the e!ect that disturbances have on the response. Small values in the transmissibility function will likewise reduce the e!ect that noise has on the response. Note that the sum of the sensitivity and the transmissibility is unity. Also note that the disturbances are low-frequency signals while measured noise is a high-frequency signal. Therefore, both objectives can be met by keeping S small in the low-frequency range and T small in high frequencies. Weighting matrices Q and R of Eq. (13) are determined so that the resulting controller provides desirable frequency characteristics. 5.5. Simulation results For a 20-s interval of numerical simulation the resulting uncontrolled and controlled accelerations of the fourth #oor are shown in Fig. 7. Table 2 summarizes the peak accelerations of all four #oors for both controlled and uncontrolled vibrations; also, the percent reduction of the peak value is given. Table 3 lists the uncontrolled and controlled root mean squared (RMS) accelerations of all #oors along with the percent reduction of the peak value. RMS values are decreased by 63%, 45%, 36% and 37% for the "rst through the four #oor levels, respectively.
Fig. 7. Fourth #oor accelerations from wind speed set M15N571.
Table 2 Peak accelerations due to wind M15N571 Floor (1)
Uncontrolled peak (g) (2)
Controlled peak (g) (3)
Peak reduction (%) (4)
1 2 3 4
0.0195 0.0355 0.0351 0.0486
0.0075 0.0201 0.0252 0.0311
61.4 43.5 28.1 36.0
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Table 3 RMS accelerations due to wind M15N571 Floor (1)
Uncontrolled RMS (g) (2)
Controlled RMS (g) (3)
RMS reduction (%) (4)
1 2 3 4
0.0050 0.0088 0.0099 0.0130
0.0018 0.0048 0.0063 0.0082
63.4 45.5 36.0 37.1
Fig. 8. Control force.
Fig. 9. Voltage required for MR damper.
A time history of the control force required to minimize building response for this strong wind event is shown in Fig. 8. Fig. 9 shows a portion of the time history of the voltage required to operate the MR damper for this same period of time.
6. Summary and conclusion A #exible laboratory structure was constructed of graphite epoxy members for the purpose of demonstrating semi-active control of its #oor accelerations when
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the structure is excited by wind. This structure is considered to be a discrete linear system with one degree of freedom per #oor in the horizontal direction. It is braced out-of-plane to reduce the transverse and torsional response. Masses are assumed to be lumped at each #oor. An experimental method of system identi"cation that uses modal analysis is carried out to determine the dynamic and static characteristics of the four-storey system. These characteristics are used to formulate a state-space representation of the structure. Actual wind speed data taken at various heights in a free "eld environment were converted to numerically simulated disturbance forces on the structure using a drag force assumption in the along-wind direction. An LQG/LTR controller was designed to simulate disturbance, noise, and time delay rejection in an e!ort to reduce #oor accelerations for this structural system. The controller speci"es a control force that is to be applied to the "rst #oor. Numerical simulations of this controller show that when the structure is excited by the wind disturbance, the controller reduces peak accelerations of the #oors of the structure by approximately 35%}60%. Even allowing for a period of time delay, these results indicate decidedly positive prospects for this controller. The controller is independent of the system producing the control force. Thus, the calculated control force required is the same for the active tendon system as for the MR damper system. However, since the semi-active MR damper uses a relatively small amount of external voltage (2}25 VDC [9]), and an active tendon system requires much more power to operate (115/230 VAC plus 0}10 VDC position command signal [10]), it is concluded that an MR damper may possess distinct advantages over active systems of control. Several caveats need to be mentioned in closing. It is known that system identi"cation of the dynamic physical structure with an active or semi-active control device installed possesses di!erent dynamic characteristics than those determined from testing when the device was not present. That is, it is expected that the installation of an MR damper or an active tendon will increase natural frequencies of the system, thereby changing the system model. In addition, the state estimator uses the noise weighting matrices and has poles or frequencies, that are relatively high; therefore, the estimator might erroneously track noise and degrade system stability.
References [1] J.T.P. Yao, Concept of structural control, J. Struct. Division ASCE 98 (ST7) (1972) 1567}1574. [2] R.T. Stefani et al., Design of Feedback Control Systems, Saunders, Boston, MA, 1994. [3] A.W. Miller, Active control of a tall structure excited by wind, Master of Engineering Report, Texas A&M University, 1996. [4] Texas Tech Field Experiment Data Package, Wind Engineering Research Field Laboratory, Texas Tech University, Lubbock, TX, 1995. [5] F. Simiu, R.H. Scanlan, Wind E!ects on Structures, 2nd Edition, Wiley, New York, 1986. [6] V. Kolousek et al., Wind E!ects on Civil Engineering Structures, Elsevier, Amsterdam, 1984. [7] T.T. Soong et al., Control. I: design and simulation, J. Struct. Eng. 117 (11) (1991) 3516}3536.
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[8] MATLAB, Reference Guide, The MathWorks, Inc., Natick, MA, 1998. [9] J.D. Carlson et al., Magneto-rheological #uid dampers for semi-active seismic control, Proceedings of the 3rd International Conference on Motion and Vibration Control, Vol. III, Chiba, Japan, 1996, pp. 35}40. [10] Linear and Rotary Positioning Systems & Controls, Industrial Devices Corporation, Novato, CA, 1998.
Active control of a tall structure excited by wind Jin Zhang, Paul N. Roschke* Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, USA
Abstract A control strategy is developed for application to a #exible laboratory structure excited by simulated wind forces for the purpose of minimizing along-wind accelerations. Static and dynamic characteristics of the structure are identi"ed through a modal analysis method that formulates a linear model of the system. Actual wind speed data is used to produce a simulated wind loading by means of drag forces. An LQG/LTR control strategy based on acceleration feedback is used in conjunction with a magnetorheological (MR) damper to reduce structural response. When a strong wind loading is applied to the structure, the control force notably reduces simulated peak #oor accelerations. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Acceleration; Active control; Building; Magnetorheological damper; System identi"cation; Vibration; Wind
1. Introduction As modern materials and construction methods lead to taller buildings that are increasingly #exible, environmental loads such as strong wind gusts and earthquakes can be expected to increase building response. Whereas safety is a major concern for a civil engineering structure subjected to an earthquake, it is not a consideration for most building structures in strong wind environments. The main concern for high-rise buildings in strong wind events is discomfort to the occupants, such as physical symptoms due to motion sickness or psychological responses like anxiety. One approach to mitigate undesirable motions due to hazardous wind or earthquake loads is to alter the dynamic characteristics of a building with respect to a given loading. This idea has developed into the concept of structural control, which was "rst
* Corresponding author. Fax: #1-409-845-6554. E-mail address: [email protected] (P.N. Roschke) 0167-6105/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 9 ) 0 0 0 7 3 - 2
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presented by Yao [1]. Over the past several decades, a number of physical devices have been investigated for the control of buildings. Such devices include the active bracing system (ABS) and magnetorheological (MR) dampers. An ABS (see Fig. 1) typically consists of a set of pre-stressed tendons or braces that can be attached to a frame of the structure in the plane for which motion is to be controlled. These braces are connected with an actuator that sti!ens or relaxes the system according to a control algorithm. An MR damping system (see Fig. 2) consists of a damper that is rigidly connected between the ground and the "rst #oor or between two neighboring #oors. Viscous properties of the damper are changed according to the voltage from a control algorithm. An algorithm (termed as a `controllera) is used to determine the control force or damping that is to be applied to a structure through an ABS or MR system, respectively. Virtually, all current control methods utilize optimum strategies that minimize one or more performance indices [2]. The most basic optimal controller is the linear quadratic regulator (LQR). This control strategy minimizes a function that relates the response (or states) of the structure and the control input. If the needed states of the structure are not directly measured, an estimator such as a Kalman "lter can be employed to approximate these states. The addition of a Kalman "lter to an LQR strategy leads to what is termed as the linear quadratic Gaussian (LQG) problem. The Kalman "lter characterizes statistical distribution of the noise processes
Fig. 1. Active tendon control system.
Fig. 2. MR damper control system.
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inherent in the system as they are related to the sensors [2]. However, the LQG approach has been shown to be overly sensitive to changes in structural parameters. This motivates the use of a variation of the LQG strategy that takes into account frequency domain characteristics and feedback properties. This variation is termed as the loop transfer recovery (LQG/LTR) problem. In what follows, system identi"cation is carried out on a #exible four-storey model of a tall building in a laboratory to determine the salient dynamic characteristics of the structure. A simulated wind loading is applied to a numerical model of the building that is equipped with an MR damper. Finally, a control algorithm that uses acceleration feedback and a time delay is shown to provide signi"cant reduction of acceleration.
2. Experimental setup A tall, slender, four-storey model of a #exible tall building structure was constructed in a laboratory (see Fig. 3) in order to demonstrate the performance of a semi-active control system with an MR damper. Dynamic parameters are collected through system identi"cation as described in the following section, and controller performance
Fig. 3. Laboratory structure.
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is simulated through MATLAB. The main structural members are graphite epoxy tubes donated to the study from NASA Johnson Space Center. Vertical members are continuous throughout the height of the structure. Horizontal tubes are joined to the vertical members at each #oor level. Joints that connect vertical and horizontal tubes are custom-made from polyvinyl chloride (PVC) pipe. Flat, plastic plates with additional mass span the horizontal beam members in a slab-like fashion at each level. The bottom of each column is secured by a PVC pipe encased in a wooden sub-assemblage that is bolted to the concrete #oor. In order to ensure that very little motion occurs normal to the along-wind direction, the structure is also restrained by a light X-bracing cable. Nominal dimensions of the model are 0.77 m]1.25 m]6.52 m. Height of the 1st, 2nd, 3rd, and 4th #oors above the semi-rigid base are 1.73 m, 3.30 m, 4.93 m, and 6.52 m, respectively. To measure the response of the structure to loading, an accelerometer is placed on each #oor. An MR damper purchased from Lord Corporation is intended to be used to provide a controllable force. MR #uids typically consist of micron-sized, magnetically polarizable particles dispersed in a carrier medium such as mineral or silicone oil. Normally, MR #uids are free-#owing liquids having a consistency similar to that of motor oil. However, when a magnetic "eld is applied within several ms their consistency changes and the #uid becomes semi-solid with a yield strength of up to 100 kPa. The degree of change is proportional to the magnitude of the applied magnetic "eld. MR dampers are also stable over a broad temperature range, from !40}1503C.
3. System identi5cation 3.1. Introduction The knowledge about the dynamic characteristics of a system is one of the most important aspects of control design. An accurate mathematical model of the system determines whether a controller works properly or becomes unstable. The most popular way to represent a model is through a complex frequency response function that relates the input and output characteristics of the experimental system. For the identi"cation of many civil engineering structures, inputs to the system are applied forces, while outputs include displacements, velocities and accelerations. The complex frequency response function can be thought of as a series of transfer functions that transform disturbance inputs to acceleration outputs. Natural frequencies, mode shapes and equivalent viscous damping characteristics are determined from analysis of the input and output data and are used in the formulation of a state-space representation of the structure. 3.2. Identixcation procedure For this investigation, a complex frequency response function that relates input forces to #oor accelerations is used to determine natural frequencies, mode shapes and
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equivalent viscous damping characteristics of the structure. These functions are obtained as follows: f The laboratory model is excited with an impact hammer and a time history of the impact force and accelerations of each #oor are read into a data "le. f Transfer functions are calculated using the acceleration and force time histories. f Natural frequencies and corresponding amplitudes of the structure are identi"ed (see Table 1). f Amplitude ratios of frequency domain response and the corresponding phase angles are used to determine the mode shapes. f Equivalent viscous damping characteristics are determined by the half-power bandwidth method: f i !f i 1, m"2 (1) i f i #f i 2 1 where f and f are frequencies at which the transfer function amplitudes equal 0.707 1 2 times the transfer function peak at the ith natural frequency. 3.3. State space representation The general equation of motion for a multi-degree-of-freedom system excited by a forcing function is: Mx$ (t)#CHx5 (t)#Kx(t)"d(t),
(2)
where M, CH, and K are the mass, damping and sti!ness matrices, respectively, of the structure; d(t) is a time-dependent vector of disturbance forces; and xK (t), x5 (t) and x(t) are time histories of the acceleration, velocity and displacement vectors, respectively. De"ning the state vector
CD
z"
x i x5 i
(3)
Table 1 Structural parameters Parameter (1)
Mode 1 (2)
Natural frequency (Hz) Damping ratio (%) Normalized mode shape Floor 1 Floor 2 Floor 3 Floor 4
1.053 9.64 0.3364 0.6358 0.7257 1
Mode 2 (3)
Mode 3 (4)
3.835 3.86
8.108 2.55
!0.8542 !1.0588 !0.2354 1
1.9110 !0.1887 !1.6763 1
Mode 4 (5) 13.662 1.22 !3.6626 4.1205 !3.0009 1
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and requiring that the desired output vector y consist of the accelerations of each #oor xK allows Eq. (2) to be recast into the general state space form: z5 "Az#Bd, (4)
y"Cz#Dd, where
C
A"
0 4 !M~1K
D
I 4 !M~1CH
C"[!M~1K !M~1CH]
C D
B"
0
4 , M~1
D"[M~1].
(5)
Note that for the structure at hand 0 is a 4]4 zero matrix, I is a 4]4 identity 4 4 matrix, M is determined through a lumped mass method, and matrices K and CH are determined as follows [3]: (6) K"M/ X/TM CH"M/ W/TM, n n n n where X is the diagonal matrix [u2 , u2 , 2,u2], W is the diagonal matrix n 1 2 [2m u , 2m u ,2,2m u ], u is the ith natural frequency, and m is the damping ratio 1 1 2 2 n n i i of the ith mode. In summary, all of the required parameters necessary to obtain state space representation matrices A, B, C, and D are available from the system identi"cation described earlier. 4. Wind disturbance 4.1. Wind speed data Wind speed data to be used in simulation of a strong wind event on the laboratory structure was obtained from the Wind Engineering Research Field Laboratory (WERFL) at Texas Tech University [4]. Data were collected from a free-standing tower at heights of 4, 10, 21 and 49 m above the surface of the earth in #at, open terrain. A selected portion of the total time history is presented in Fig. 4. The sampling rate was 10 Hz and low-pass "ltering was conducted at 8 Hz.
Fig. 4. Wind speed set M15N571.
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4.2. Characterization of wind Vibration of tall structures is elicited by the turbulent component of the wind velocity. Wind turbulence in the atmospheric boundary layer is typi"ed by random #uctuations of velocity and pressure. Kaimal et al. [5] have proposed the following normalized power spectral density of wind velocity for structural design purposes: u ) S(z, u) 200f " , u2 (1#50f )5@3 H where uz f" U(z)
(7)
(8)
z is the height above the surface, ;(z) is the mean wind speed at height z, and u is the frequency in Hz. S(z, u) is the power spectral density (PSD) at height z for the frequency u. The parameter u is the shear #ow velocity that is determined by: H kU(z) u " , (9) H z ln z 0 where k is von Karman's constant (generally assumed to be approximately 0.4), and z is the roughness length, a variable characterizing the terrain [5]. Solving for S(z, u) 0 in Eq. (7) gives the wind PSD
A B
u2 200f H S(z, u)" . u(1#50f )5@3
(10)
The force that a #uid produces on a body is of the following form [6]: 1 F "C o<2A, D D2
(11)
where C is the drag coe$cient, o is the density of the #uid, < is the #ow velocity, and D A is the area exposed to the #ow. The drag coe$cient of the rectangular laboratory model is taken to be 1.48 [6]. The total along-wind response, such as de#ection, velocity or acceleration, may be viewed as a sum of two parts: X(z)"xN (z)#x(z),
(12)
where x6 (z) is the mean response, and x(z) is the #uctuating response that is induced by the wind gustiness [5]. In what follows, only the latter component is considered. 4.3. Application of wind speed data The force to be applied to each #oor of the laboratory model from the Texas Tech wind data set is calculated using Eq. (11). The tributary areas that relate wind pressure
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to an equivalent force at a #oor level are taken to be the surface areas on the building model that are normal to the wind #ow. Also, according to Eq. (12), only the #uctuating component is considered. The main purpose of this study is to design and test an active control algorithm that can reliably reduce the vibration of a wind-excited building. Within reasonable bounds, an acceptable controller should function properly for any wind load. Therefore, relationships of similitude are not emphasized in this research work. Here, the wind loading from each of the four anemometer readings is directly applied to the corresponding #oor of the structure, although the heights of the #oors in the laboratory model do not directly scale to the heights of the "eld readings. Another simpli"cation made is that the coe$cient of drag is applied uniformly to each #oor. Alternatively, pressure coe$cients for the windward and leeward walls could be determined for a building that has aspect ratios that match those of the laboratory model. Resultant pressures could then be approximated by discrete forces.
5. Control design 5.1. Control algorithm After salient dynamic characteristics of the experimental structure have been determined and a time history of wind disturbance for each of the four stories has been speci"ed, a control strategy is formulated that is aimed at amelioration of the vibration of the structure. The control force is assumed to be capable of being produced by an active control device such as an active tendon system or an MR damper acting between the ground and the "rst #oor. Line of action of the control force is taken to be parallel to the direction of the wind (see Figs. 1 and 2). A diagram of relationships for this system is presented in Fig. 5 [2], where d denotes the disturbance input and G represents the four-storey laboratory structure. Acceleration response of the structure is denoted by y. Measurement of the output is taken to be a noise n that is included in the feedback. H denotes the controller that produces a force vector u that, if designed correctly, reduces the output and stabilizes the closed-loop system. A linear quadratic regulator is used to optimize a quadratic function of the state and control vectors. Given the linear system shown in Fig. 5, a function J that relates
Fig. 5. Control block representation.
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the state vector and control force is as follows:
P
1 = (z@Qz#u@Ru) dt, (13) J" 2 0 where Q and R are weighting matrices of the state and control forces, respectively. The integrand has a quadratic form and Q and R are usually symmetric. It is assumed that Q is positive de"nite (i.e., it is symmetric and has positive eigenvalues) and R is positive semi-de"nite (i.e., it is symmetric and has non-negative eigenvalues). These assumptions imply that the cost is non-negative and, therefore, its minimum value is zero. Minimization of Eq. (13) leads to the necessity of solving the algebraic Riccati equation (ARE) for the Riccati matrix P [2]: A@P#PA#Q!PBR~1B@P"0.
(14)
Once the Riccati matrix is known from Eq. (14), the gain matrix K and target control force vector u are calculated by the following relations: K"R~1B@P,
u"!Kz.
(15)
If the system is both controllable and observable, and if an appropriate Q has been selected, the positive-de"nite solution of the ARE results in an asymptotically stable closed-loop system. However, if not all of the state variables can be measured, optimal control cannot be applied directly and a state estimator is needed. The solution to this problem was "rst provided by Kalman and Bucy [2]: z(5 "Az( #Bu#L(y!y( ),
(16)
where z( and y( are estimates of the actual states z and y, respectively, and L is the observer gain. This gain is computed from the following set of equations so as to minimize the size of the estimation error intensity matrix E: AE#EA@#Q !EC@R~1CE"0, 0 0 (17) L"EC@R~1 0 where matrices Q and R represent the intensity of the process and sensor noise 0 0 inputs, respectively. The LQG algorithm combines a Kalman "lter and an LQR algorithm in a single controller. This controller may be constructed by substituting Eqs. (4) and (15) into Eq. (16) and simplifying z(5 "(A!BK!LC#LDK)z( #Ly, u"!Kz( .
(18)
Experience has shown that although LQG is an improvement over LQR, it demonstrates low robustness [2]. Moreover, it is overly #exible due to the arbitrariness of the noise weighting matrices. To alleviate these inadequacies, the LQG/LTR technique modi"es the design procedure to recover the advantageous stability
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margins of the LQR algorithm while maintaining the basic mathematics of LQG. The open-loop transfer function for an LQR controller at the input to the structure is given by L(s) "K(sI!A)~1B LQR and the LQG loop transfer function may be expressed as
(19)
L(s) "K(sI!A#BK#LC!LDK)~1L(D#C(sI!A)~1B). (20) LQG Two criteria for the LQG/LTR approach are that the structure has a minimum phase (i.e., it has no zeros in the right-hand plane) and the noise matrices are given by R "1 0 and Q "q2BB@, where q is a tuning parameter. Through these assumptions it can be 0 shown that [2] lim L(s) "L(s) . (21) LQG LQR q?= The tuning parameter q can be increased until the open-loop transfer function is close to the LQR loop transfer function. That is, the feedback loop transfer function for the structure is recovered as the noise goes to in"nity. 5.2. Time delay The previous discussion is based on the assumption that all operations in the control loop can be performed instantaneously. However, in reality, each operation experiences a delay between sending the command signal and measuring the response. This delay is caused mostly by signal processing, on-line computation and response time of the actuator system [7]. It may lead to instability in the controlled system. Thus, the degree to which a controller can tolerate a certain amount of time delay while not signi"cantly degrading the controlled response, is an important index to evaluate a controller. Provisions for time delay are made in the controller described in the following section. 5.3. Controller design Design of an LQG/LTR controller that is to specify a control force to be applied at the "rst #oor of the laboratory structure is implemented through MATLAB and SIMULINK [8]. Fig. 6 is a SIMULINK model of the controller. A 0.05-s time delay is taken into account. Width of the connecting lines represents the dimension of the data vector that is being transmitted. The &plant' block represents the state-space model of the laboratory building with the input as the external force and the output as displacement, velocity and acceleration of each #oor. In order to simulate motion of an uncontrolled building, wind disturbance is the only external force. For the controlled building, the force speci"ed by the controller that is to be generated by the MR damper is added to the force of the wind. The &LQG/LTR controller' block is the state-space form of Eq. (18), with the input being accelerations of each #oor and the
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Fig. 6. LQG/LTR controller system model.
output being the control force that is applied by the MR damper. Measurement noise is added to the acceleration time history of each sensor. The magnitude of this white noise is determined by an analysis of a stationary accelerometer. 5.4. Feedback properties A properly designed controller should be able to operate e!ectively and maintain stability of the structure in a working environment. This environment may cause system parameters to change or disturbances to this system might vary. Even if these conditions were not considered, modeling of the system by a mathematical representation has uncertainty. For instance, in this study uncertainty of the natural frequencies, masses and mode shapes of the structure needs to be considered. Also of importance is the noise of the measurement devices themselves. In order for the feedback/control system to be e!ective and stable, it should satisfy certain performance speci"cations and allow for uncertainties in the model. These issues may be studied through the frequency domain. The block diagram for Fig. 5 may be written in equation form as follows: Y(s)5
GH 1 GH R(s)1 D(s)! N(s). 11GH 11GH 11GH
(22)
The relationships between the output y, disturbance d, and noise n are de"ned by the sensitivity S and transmissibility T functions as follows: 1 S" , 1#GH
(23)
GH . T" 1#GH
(24)
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Therefore, small values of the sensitivity function reduce the e!ect that disturbances have on the response. Small values in the transmissibility function will likewise reduce the e!ect that noise has on the response. Note that the sum of the sensitivity and the transmissibility is unity. Also note that the disturbances are low-frequency signals while measured noise is a high-frequency signal. Therefore, both objectives can be met by keeping S small in the low-frequency range and T small in high frequencies. Weighting matrices Q and R of Eq. (13) are determined so that the resulting controller provides desirable frequency characteristics. 5.5. Simulation results For a 20-s interval of numerical simulation the resulting uncontrolled and controlled accelerations of the fourth #oor are shown in Fig. 7. Table 2 summarizes the peak accelerations of all four #oors for both controlled and uncontrolled vibrations; also, the percent reduction of the peak value is given. Table 3 lists the uncontrolled and controlled root mean squared (RMS) accelerations of all #oors along with the percent reduction of the peak value. RMS values are decreased by 63%, 45%, 36% and 37% for the "rst through the four #oor levels, respectively.
Fig. 7. Fourth #oor accelerations from wind speed set M15N571.
Table 2 Peak accelerations due to wind M15N571 Floor (1)
Uncontrolled peak (g) (2)
Controlled peak (g) (3)
Peak reduction (%) (4)
1 2 3 4
0.0195 0.0355 0.0351 0.0486
0.0075 0.0201 0.0252 0.0311
61.4 43.5 28.1 36.0
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Table 3 RMS accelerations due to wind M15N571 Floor (1)
Uncontrolled RMS (g) (2)
Controlled RMS (g) (3)
RMS reduction (%) (4)
1 2 3 4
0.0050 0.0088 0.0099 0.0130
0.0018 0.0048 0.0063 0.0082
63.4 45.5 36.0 37.1
Fig. 8. Control force.
Fig. 9. Voltage required for MR damper.
A time history of the control force required to minimize building response for this strong wind event is shown in Fig. 8. Fig. 9 shows a portion of the time history of the voltage required to operate the MR damper for this same period of time.
6. Summary and conclusion A #exible laboratory structure was constructed of graphite epoxy members for the purpose of demonstrating semi-active control of its #oor accelerations when
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the structure is excited by wind. This structure is considered to be a discrete linear system with one degree of freedom per #oor in the horizontal direction. It is braced out-of-plane to reduce the transverse and torsional response. Masses are assumed to be lumped at each #oor. An experimental method of system identi"cation that uses modal analysis is carried out to determine the dynamic and static characteristics of the four-storey system. These characteristics are used to formulate a state-space representation of the structure. Actual wind speed data taken at various heights in a free "eld environment were converted to numerically simulated disturbance forces on the structure using a drag force assumption in the along-wind direction. An LQG/LTR controller was designed to simulate disturbance, noise, and time delay rejection in an e!ort to reduce #oor accelerations for this structural system. The controller speci"es a control force that is to be applied to the "rst #oor. Numerical simulations of this controller show that when the structure is excited by the wind disturbance, the controller reduces peak accelerations of the #oors of the structure by approximately 35%}60%. Even allowing for a period of time delay, these results indicate decidedly positive prospects for this controller. The controller is independent of the system producing the control force. Thus, the calculated control force required is the same for the active tendon system as for the MR damper system. However, since the semi-active MR damper uses a relatively small amount of external voltage (2}25 VDC [9]), and an active tendon system requires much more power to operate (115/230 VAC plus 0}10 VDC position command signal [10]), it is concluded that an MR damper may possess distinct advantages over active systems of control. Several caveats need to be mentioned in closing. It is known that system identi"cation of the dynamic physical structure with an active or semi-active control device installed possesses di!erent dynamic characteristics than those determined from testing when the device was not present. That is, it is expected that the installation of an MR damper or an active tendon will increase natural frequencies of the system, thereby changing the system model. In addition, the state estimator uses the noise weighting matrices and has poles or frequencies, that are relatively high; therefore, the estimator might erroneously track noise and degrade system stability.
References [1] J.T.P. Yao, Concept of structural control, J. Struct. Division ASCE 98 (ST7) (1972) 1567}1574. [2] R.T. Stefani et al., Design of Feedback Control Systems, Saunders, Boston, MA, 1994. [3] A.W. Miller, Active control of a tall structure excited by wind, Master of Engineering Report, Texas A&M University, 1996. [4] Texas Tech Field Experiment Data Package, Wind Engineering Research Field Laboratory, Texas Tech University, Lubbock, TX, 1995. [5] F. Simiu, R.H. Scanlan, Wind E!ects on Structures, 2nd Edition, Wiley, New York, 1986. [6] V. Kolousek et al., Wind E!ects on Civil Engineering Structures, Elsevier, Amsterdam, 1984. [7] T.T. Soong et al., Control. I: design and simulation, J. Struct. Eng. 117 (11) (1991) 3516}3536.
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[8] MATLAB, Reference Guide, The MathWorks, Inc., Natick, MA, 1998. [9] J.D. Carlson et al., Magneto-rheological #uid dampers for semi-active seismic control, Proceedings of the 3rd International Conference on Motion and Vibration Control, Vol. III, Chiba, Japan, 1996, pp. 35}40. [10] Linear and Rotary Positioning Systems & Controls, Industrial Devices Corporation, Novato, CA, 1998.