Neuro-Fuzzy Control of Structures Using Acceleration ... - CiteSeerX

Neuro-Fuzzy Control of Structures Using Acceleration Feedback KYLE C. SCHURTER1 AND PAUL N. R OSCHKE 2 This paper describes a new approach for reduction of environmentally induced vibration in constructed facilities by way of a neuro- fuzzy technique.

The new control technique is

presented and tested in a numerical study that involves two types of building models. Energy of each building is dissipated through magnetorheological (MR) dampers whose damping properties are continuously updated by a fuzzy controller. This semi-active control scheme relies on development of a correlation between accelerations of the building (controller input) and voltage applied to the MR damper (controller output). This correlation forms the basis for development of an intelligent neuro- fuzzy control strategy. To establish a context for assessing effectiveness of the semi-active control scheme, responses to earthquake excitation are compared with passive strategies that have similar authority for control. According to numeric al simulation, MR dampers are less effective control mechanisms than passive dampers with respect to a single degree of freedom (DOF) building model. On the other hand, MR dampers are predicted to be superior when used with multiple DOF structures for reduction of lateral acceleration. KEYWORDS ACCELERATION F EEDBACK, MAGNETORHEOLOGICAL DAMPER, NEURO-FUZZY, SEMI-ACTIVE CONTROL, STRUCTURAL DYNAMICS, VIBRATION

INTRODUCTION Protection of civil engineering structures from excessive vibration due to uncontrollable events—environmental and otherwise—is important for the purpose of increasing survivability of the constructed facility and protection of its human occupants. Vibration control of buildings is accomplished primarily through reduction of interstory drift, lateral floor acceleration, and column base shear. In most cases, control of building vibration is employed through modification of stiffness, damping, or mass of a structure. The amount of 1 Research Asst., Dept. of Civ. Engrg., Texas A&M Univ., College Station, TX 77843-3136.

Telephone: (979) 845-1985 FAX: (979) 845-6554 E-mail: [email protected] 2 Prof., Dept. of Civ. Engrg., Texas A&M Univ., College Station, TX 77843-3136.

Telephone: (979) 845-1985 FAX: (979) 845-6554 E-mail: [email protected]

energy required to change stiffness and mass of a full- size structure is typically large enough to be both economically and technologically prohibitive. However, damping of a structure can be modified with a relatively small energy requirement.

Thus, semi-active control

schemes that involve variable damping devices are ideal for use with civil engineering applications [see, for example, Dyke et al. (1996b); Patton (1997); Kurata et al. (1999); Symans and Kelly (1999)]. A magnetorheological (MR) damper is a semi-active device that shows great potential for use in vibration control of full-scale structures. The device resembles an ordinary viscous damper, is filled with MR fluid, and has one or more electromagnetic coils wrapped around the piston head. The fluid contains very small magnetically polarizable particles that allow properties of the fluid to change dramatically according to the strength of an accompanying magnetic field (Dyke et al. 1996b). Thus, when no current is supplied to the coil wrapped around the piston head, the MR damper behaves as an ordinary viscous damper. On the other hand, when current is sent through the coil and produces a magnetic field, the MR fluid becomes semi-solid (Dyke et al. 1996b; Carlson and Spencer 1996; Spencer et al. 1997). This phenomenon occurs as the result of the particles being suspended in the fluid and aligning themselves parallel to the direction of the magnetic field.

Yield strength of a

magnetorheological damper is directly proportional to the strength of the electromagnetic field applied to the MR fluid. In turn, the electromagnetic field is directly proportional to the voltage applied to the MR damper. The proper magnitude of voltage supplied to the coils as mandated by a control algorithm produces an appropriate resisting force for the damper to mitigate vibration of a structure. Application of MR dampers to control of vibration has been applied to reduction of bridge motion by Hansen et al. (1994) and to mitigation of wind and seismic effects on tall buildings by Zhang and Roschke (1999) and Jansen and Dyke (1999), respectively. One important benefit of an MR damper is its capacity to operate from a remote power source such as a chemical battery, thus increasing its viability during destructive environmental events. A sensory device that exhibits similar bene fits in terms of reliability is the accelerometer.

It is a durable measuring device with a small power requirement.

Accelerometers are widely available and are relatively inexpensive.

Use of acceleration

feedback control in structural applications has been shown to be a feasible and successful

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method to reduce vibration of civil engineering structures (Dyke et al. 1996a; Chung et al. 1998). This paper accomplishes two objectives: first, a fuzzy controller based on acceleration feedback is developed to reduce vibration of seismically excited buildings equipped with MR dampers.

Second, a numerical study is presented that compares performance of the

introduced semi-active control scheme with passive control strategies of comparable control force. The first section of this paper describes the theory and structure of the proposed fuzzy controller based on acceleration feedback. The second section applies the control technique to a single degree of freedom (DOF) building model and evaluates its performance via numerical simulation. Similarly, the third section applies the control method to the more complex and realistic case of a multiple DOF building model.

The concluding section

comments on significance of these results for control of undesirable vibratio n in constructed facilities designed by civil engineers.

THEORY AND STRUCTURE OF FUZZY CONTROLLER Effectiveness of the proposed fuzzy controller is dependent on defining a correlation between acceleration of a vibrating building and a control signal (voltage) applied to an installed MR damper(s). This correlation is defined by means of a fuzzy- mapping that is implemented according to the neural network architecture of ANFIS (adaptive neuro- fuzzy inference system) that was developed and introduced by Jang (1993, 1996) and Jang et al., (1997). The following steps outline the design procedure used for design of the fuzzy controller: 1. Create a target controller for use with a specified structure. 2. Integrate the target controller with a model of the building and damper. 3. Collect time histories of building acceleration and voltage produced by the control system during a representative disturbance. 4. Use ANFIS to create a fuzzy controller that is defined by training data collected in Step 3. 5. Replace the target controller with the newly developed fuzzy controller. 6. Test the new acceleration feedback controller by means of copious computer simulations.

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A more detailed description of the design procedure follows. TARGET CONTROLLER The acceleration feedback fuzzy controller outlined here is based on a strategy of mimicking an expert according to a design method known as expert control. Traditionally, an expert is defined as a human operator. Through a process of knowledge acquisition involving lengthy interviews with human operators and numerical observation, a set of linguistic descriptors (membership functions) and if-then rules are defined to construct a fuzzy inference system (Jang et al. 1997; Yen and Langari 1999). The goal of this process is to create a fuzzy controller that closely emulates, and can therefore replace, a human operator (Mamdani and Assilian 1975; Takagi and Sugeno 1983). With recent advances of computer technology and development of neuro-fuzzy tools such as ANFIS, experts of greater complexity and higher dimensionality are now within the scope of emulation. One such set of “experts” whose structure and behavior is well understood is the vast collection of classical and modern control algorithms. Techniques such as PID control, quadratic optimizations (e.g. LQR), and various classes of robust control schemes are now viable models for creation of a fuzzy-based expert controller.

This new concept of expert control forms the basis for

development of the proposed fuzzy controller. DESCRIPTION OF CONTROL SYSTEM The block diagram in Fig. 1 illustrates how the concept of a target controller is implemented. The diagram consists of four components; first is a linear and time invariant, state space model of a building structure (Soong 1990) that has the following form:

E +

State Space Model of Structure

?? –

Umr

Forward Damper Model

V

Inverse Damper Model

x, x?

Ud

Target Controller

x, x?

ASAC

FIG. 1. Block Diagram for Control Feedback Loop Containing Target Controller

4

x? ? Ax ? Bu y ? Cx ? Du

(1)

where x is the state vector of the system, u is the input vector of the structure. In this case, u is taken to be the sum of an earthquake force, E, and force from the MR damper(s), Umr. y is the output vector. A and B are matrices that define characteristics of the building, while C and D dictate the form of output produced by the solution of (1). The second component of the block diagram is a target controller that defines a desired control force, Ud, based on dynamics of the vibrating structure. The remaining two components of Fig. 1 function as a unit and are referred to as an ASAC (Active to Semi- Active Converter). The function of the ASAC is to convert a desired control force—produced by the target controller—to an actual control force resulting from a magnetorheological damper (Schurter 2000). This conversion is necessary because, in the field of structural vibration control, most classical and modern control algorithms manifest themselves as active control strategies. The target controller is created without regard to semi-active control requirements of a structure equipped with magnetorheological dampers. An ASAC remedies this incompatibility of control theories. As a result of the conversion process, only portions of an active control signal can be used to specify operation of a semiactive device. Usable portions of the active control signal are consistent with the dependence on force and velocity that is characteristic of classical damper behavior. The first element of the ASAC is an inverse damper model that is based on a modified version of the Bingham viscoplastic model (Stanway et al. 1985, 1987; Shames and Cozzarelli 1992). This model is specific to an MR damper and serves to predict voltage, V, required by the MR device to meet the force commanded by the target controller. Input to the inverse damper model is x , x? , and Ud. Output of the model is voltage. The second element of an ASAC is the forward damper model that predicts the damping force produced by the MR damper given the inputs of x , x? , and V. Many options exist when defining the structure of the forward damper model. Spencer et al. (1997) present several mechanical models for emulation of the complex behavior of MR dampers. For this paper, however, the forward damper model is based on a fuzzy representation developed by Schurter and Roschke (2000) that is both accurate and computationally efficient. Thus the inverse and forward models act together as an ASAC to transform an active control signal produced by the target controller 5

into a signal that is consistent with behavior of a magnetorheological device. For a more detailed investigation of the ASAC, refer to Schurter (2000). TRAINING OF THE FUZZY CONTROLLER The system represented in Fig. 1 is used to collect data for training of the fuzzy controller. Dynamics of the controlled system are evaluated through numerical simulation. For the duration of the disturbance, E, histories are recorded of building acceleration and voltage specified by the inverse damper model. The neuro-fuzzy tool ANFIS is used to establish a relationship between the two parameters. ANFIS uses a hybrid learning technique to construct an input-output mapping based on the acceleration-voltage data pairs gathered from simulation. ANFIS is structured as an adaptive neural network that is functionally equivalent to a first-order Sugeno fuzzy model. Such a model is composed of r rules of the form Rule i: IF x 1 is Ai1 and x 2 is Ai2 … and x n is Ain THEN yi = bi0 + bi1 x1 + bi2x 2 + … + binxn where x 1 , x 2 , …, x n are antecedent variables, and yi is the consequent variable of rule i. Ai1 , Ai2 , …, Ain are fuzzy sets defined over the domains of the respective antecedents. bi0 , bi1 , …, bin are constant coefficients that characterize the linear relationship defined by the ith rule in the rule set, i = 1, 2, …, r. Total output y of a Sugeno model is determined from the following equation: r

y?

?

i ?1 r

?

r

wi yi

i? 1

?

?

i? 1

wi ?bi 0 ? bi1 x1 ? bi 2 x 2 ? ? ? bin x n ? (2) r

?

wi

i? 1

wi

where wi is the matching degree of the ith rule defined as wi ?

n

?

i ?1

? Ai ?xi ?.

(3)

Furthermore, ? Ai(x i) is the membership value of x i with respect to the fuzzy set Ai. By a combination of the steepest descent method and least squares estimation (hybrid learning), ANFIS adjusts antecedent and consequent parameters of the Sugeno model until an error measure between the fuzzy model and the acceleration–voltage data pairs is sufficiently small or, alternatively, for a predefined number of epochs. 6

Two error measures are used to

determine efficacy of a model: training error and checking error. The metric used for both types is root mean squared (RMS) error. Training error is defined as

TE RMS ?

?

1 n i v ? v ifuzzy n ?i?1 train

?

2

(4)

where v itrain is voltage from the training data at the ith time step, v ifuzzy is voltage predicted by the fuzzy model at the ith time step, and n is the total number of data points. Training error measures how closely the fuzzy model matches the data used for training. Checking error is defined as

CERMS ?

?

1 n i v ? v ifuzzy n ?i? 1 check

?

2

(5)

where v icheck is voltage according to the checking data (similar but unique from training data). Checking error is used to measure ability of the fuzzy model to generalize its behavior on untrained (checking) data. A situation where training error is much smaller than checking error is undesirable. This would indicate a model with poor generalization abilities. An ideal scenario yields both low training and checking error. Thus, when both errors are satisfactorily minimized, a correlation between acceleration of the building and voltage applied to the MR damper(s) is established and design of the fuzzy controller is complete. The control system presented in Fig. 1 can now be greatly simplified by replacing the target controller and inverse damper models with the newly established fuzzy controller thus creating a direct link between dynamic response of the building and voltage applied to the MR damper. A revised version of the block diagram is shown in Fig. 2. Note that ?x? represents an acceleration vector of the building.

E

State Space Model of Structure

?? + –

Umr

Next, a set of indices is introduced that allows

Forward Damper Model

V

x, x?

x, x?, ?x?

Fuzzy Controller

?x?

FIG. 2. Block Diagram for Control Feedback Loop Containing Fuzzy Controller 7

performance of the fuzzy controller to be assessed. EVALUATION OF CONTROLLER PERFORMANCE Six arithmetic expressions are defined to quantify performance of the fuzzy controller. These expressions are based on those defined by Ohtori et al. (1999). They measure peak and normed response of the building with respect to interstory drift, lateral acceleration, and base shear. Small values of each index are generally considered more desirable. Each expression is presented in Table 1 in a general form and is appropriate for use with multiple DOF building models. The expressions can be specialized for use with single DOF models. Note that the base shear indices for the single DOF case are not shown since they are synonymous with the acceleration indices. The first three performance indices address peak interstory drift (J1 ), peak acceleration (J2 ), and peak base shear (J3 ) of the building where i is building story, di(t) is interstory drift for the duration of the earthquake, and hi is the height of each associated story. ? max is maximum interstory drift of the uncontrolled building calculated by the formula max max d i ?t ?/ hi . ?x?ai ?t ? and ? x?ai are absolute acceleration at the ith building story with and t, i

without control devices, respectively. mi is mass of the ith level and Fbmax is maximum base shear of the uncontrolled structure. The final three performance indices define normed values for interstory drift (J4 ), acceleration (J5 ), and base shear (J6 ) of the building where norm, ? , is determined according to the following equation

?? and t f is duration of the disturbance.

1 tf

tf

???? dt 2

(6)

0

d max ? max d i ?t ?/ hi , i

?x?a max , and

Fb max

are

maximum normed interstory drift, acceleration, and base shear force of the uncontrolled structure. For indices that describe behavior of the single DOF model, x(t) is controlled lateral displacement of the building, and x max is the maximum uncontrolled lateral displacement of the building.

x c ?t ? and x u ?t ? are normed displacements of the controlled and uncontrolled

building, respectively.

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TABLE 1. Performance Indices for Fuzzy Controller Description (1)

Peak Drift

Peak Acceleration

Performance Indices for Single DOF Model (2)

J1 ?

J2 ?

Peak Base Shear

max x ?t ? t

x max

max ?x?a ?t ? t

?x?a max

—

Normed Drift

J4 ?

Normed Acceleration

J5 ?

Normed Base Shear

Performance Indices for Multiple DOF Model (3)

? d ?t ? ? max ? i ? t ,i ? hi ? J1 ? d max max ?x?ai ?t ? J 2 ? t, i max ?x?a J3 ?

max ? mi ?x?ai ?t ? t

i

Fb max

? d ?t ? ? max ? i ? i ? hi ? J4 ? d max

xc ?t ?

x u ?t ?

x??ac ?t ? x??au ?t ?

J5 ?

—

J6 ?

max ?x?ai ?t ? i

?x?a max

?

mi ? x?ai ?t ?

i

Fb

max

EXAMPLE 1: SINGLE DOF BUILDING For purposes of illustration, an idealized single DOF structure with an installed MR damper and an attached accelerometer is subjected to a seismic excitation as shown in Fig. 3. The MR damper model has a peak force capacity of approximately 2.5 kN and a voltage input range of 0–2.25 V. A target controller is chosen in the form of a linear quadratic statefeedback regulator (LQR). Relevant parameters of the control system are listed in Table 2. COLLECTION OF TRAINING DATA According to LQR theory, an optimal gain matrix ? is obtained when control force u(t) = -?? x(t). ? is determined such that the quadratic cost function

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MR Damper Accelerometer

Rigid Chevron Brace

Earthquake

FIG. 3. Arrangement of Damper, Brace, and Accelerometer on Building Model

J?

?

??x ?t ?Qx?t ?? u ?t ?Ru?t ??dt T

T

(7)

0

is minimized for the state space model of the building. Generally, Q is a 2n ? 2n symmetric, positive semi-definite, weighting matrix for the structural responses where n is the number of DOFs of the building. R is a p ? p symmetric, positive definite, weighting matrix for the input control forces where p represents the number of control force locations. ? for the single DOF building model is determined with the use of the MATLAB Control Toolbox (1999). The system loop shown in Fig. 1 is employed for production of training data. The quality of data used for training the fuzzy controller is of utmost importance. The data must be representative of situations expected during operation of the controller. Also, there must be enough data to minimize the amount of interpolation required by the fuzzy controller. The seismic disturbance, E, is used to define the type and range of excitation that the fuzzy controller is suited to handle. The disturbance used for training of the fuzzy controller is

TABLE 2. Single DOF System Parameters System Parameter (1)

Parameter Value (2)

Mass, m Stiffness, k Damping, c Feedback Gain Matrix, ? ?

4.43494 ? 104 kg 1.75084 ? 106 N/m 1.1146 ? 104 N-s/m [0.010 0.040]

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created from band- limited, Gaussian white noise with a frequency content and magnitude representative of a typical design-seismic event.

Frequency content of the artificial

earthquake data is primarily from 0–8 Hz and has a duration of 120 seconds. Simulation is performed using a Dormand-Prince solver with a time step of 0.001 seconds to maintain convergence. TRAINING OF FUZZY CONTROLLER WITH ANFIS Chung et al. (1998) show that while acceleration feedback control is typically unsuccessful when only data for the current time-step are used, it “becomes feasible when the feedback data is extended to cover accelerations of the previous time-steps.” The fuzzy controller, therefore, uses input acceleration feedback data from the current time-step, ? x??t ?, and one past time-step, ? x??t ? ? ?, where ? is a selected delay time of the acceleration signal. By an iterative optimization process, the optimal delay time, ?opt , is determined to be 0.230 seconds. A process of trial and error determines the optimal number of membership functions (MFs) used with each input variable. If too few MFs are used, training will not sufficiently capture important relationships within the data. On the other hand, overfitting occurs if too many MFs are used.

Two and three MFs are used for inputs ? x??t ? and x? ??t ? 0.230 ?,

respectively. The input space is divided according to the grid partition method resulting in six rules and six output equations. A minimum checking error of 0.3721 V is reached after 40 training epochs for the voltage to be applied to the MR damper. The corresponding training error is 0.3724 V. Close proximity of the two values indicates good generalization properties of the controller to unforeseen inputs. Fig. 4 shows the control surface of the fully trained fuzzy controller.

x??t ? and ?

??t ? 0.230 ? are represented along the horizontal axes and voltage is shown on the vertical x? axis. x? ??t ? 0.230 ? is seen to be more influential for determination of voltage than ? x??t ?. Fig. 5 compares a portion of the voltage signal used for training with the voltage commanded by the fuzzy controller. While voltage of the training data is smoother than the signal produced by the fuzzy controller, similarities between the two signals are unmistakable. Hence it is seen that even though determination of the control signal from the target controller is based on

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Voltage (V)

2

1 0 0.4 0

??t ? (g) x?

-0.4 -0.8

-0.4

0

x??t ? 0. 230? (g) ?

0.4

FIG. 4. Control Surface for Fuzzy Controller

Voltage (V)

2.5

Target Voltage ANFIS Prediction

2 1.5 1 0.5 0 30

31

32

33

34

35

36

37

38

39

40

Time (s)

FIG. 5. Ten Second Comparison of Target Voltage and Voltage Predicted by the Single DOF Fuzzy Controller

displacement and velocity (state) of the structure, a strong correlation exists between voltage and acceleration of the building. EVALUATION OF CONTROLLER PERFORMANCE After the fuzzy controller is fully trained, it is integrated with the building and damper models (as shown in Fig. 2) and tested for its ability to reduce building vibration. The building is subjected to one artificial earthquake (similar to that used for training of the fuzzy controller) and the following four historic earthquakes of variable intensity and duration:

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1. El Centro, May 18, 1940: North-South (N-S) component recorded in El Centro, California, at the Imperial Valley Irrigation District substation.

Absolute peak

acceleration of this far-field disturbance is 3.417 m/s2 (0.348 g). 2. Hachinohe , May 16, 1968: N-S component of the Tokachi-oki earthquake recorded in Hachinohe City, Japan. This is also a far- field earthquake with a recorded absolute peak acceleration of 2.250 m/s2 (0.229 g). 3. Northridge, January 17, 1994: N-S component recorded in Sylmar, California, at the parking lot of the Sylmar County Hospital. Absolute peak acceleration of this nearfield quake is 8.276 m/s2 (0.844 g). 4. Kobe, January 17, 1995: N-S component recorded during the Hyogo-ken Nanbu earthquake at the Kobe Japanese Meteorological Agency (JMA). Absolute peak acceleration of this far-field earthquake is 8.1782 m/s2 (0.834 g). A signal delay of 30 milliseconds and sensor noise with a magnitude of 0.07 RMS gs is included to increase realism of the simulation. For purposes of comparison, response of the building to four types of control strategies is considered: 1. No Control. MR dampers removed from the building. 2. Passive-Off. Dampers present with no voltage applied. 3. Passive-On. Saturation voltage applied to dampers. 4. Semi- Active Control. Voltage to dampers defined by fuzzy controller. Through examination of performance indices from the four scenarios, comparison of semiactive and passive control strategies are made with respect to a single DOF structure. Response of the building is examined when the devices are used in a passive mode by maintaining a constant voltage to the dampers. When the dampers function in a passive-off capacity, voltage is set to zero for all time. This situation corresponds with a period of total power loss. Results show how MR dampers can be effective control mechanisms even during unforeseen times of zero power supply. Control of the dampers when input voltage is set to maximum (2.25 V) is tested in the passive-on configuration. In this way, strength of the dampers is maximized for the entire duration of the test.

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DISCUSSION OF RESULTS From Table 3 it is observed that all of the tested control schemes are successful in reducing dynamic response of the building for each earthquake. Thus it is shown that by increasing the damping of a single DOF system, both peak and normed responses of displacement and acceleration of the system are diminished. According to Table 3, control of the building by increasing its damping is most pronounced during the artificial and Hachinohe earthquakes. These are the least intense of the records. Effectiveness of increased damping is lowest during the Northridge and Kobe earthquakes. The two near- field records exhibit the most intense acceleration signals of the earthquakes considered. The information contained in Table 3 is especially interesting when performance of the three control schemes is considered. For every earthquake and every performance index, greatest control is attained when the dampers are set to the passive-on configuration. In this scenario, the dampers produce their maximum possible force. Displacement and acceleration responses of the building are reduced more using the higher control force than when the passive-off or semi- active control strategies are enforced. The least effective form of control is the passive-off configuration, while performance of the semi-active scheme is intermediate

TABLE 3. Evaluation Criteria for Single DOF Building Criteria: Passive and Semi-Active Cases Disturbance (1)

Artificial Earthquake El Centro

Hachinohe

Northridge

Kobe

Type of Control (2)

Passive-Off Passive-On Semi-Active Passive-Off Passive-On Semi-Active Passive-Off Passive-On Semi-Active Passive-Off Passive-On Semi-Active Passive-Off Passive-On Semi-Active

J1 (3)

J2 (4)

J4 (5)

J5 (6)

0.6574 0.3698 0.5289 0.8090 0.5268 0.6392 0.6326 0.5080 0.6034 0.9321 0.8077 0.8699 0.9692 0.8943 0.9143

0.7684 0.4946 0.6331 0.8676 0.6000 0.6965 0.7392 0.6162 0.7038 0.9670 0.8866 0.9340 0.9773 0.9215 0.9352

0.5903 0.3281 0.4780 0.7216 0.4860 0.5853 0.5693 0.2472 0.4563 0.7840 0.5842 0.6622 0.7743 0.5575 0.6053

0.6579 0.4588 0.5693 0.7775 0.5835 0.6692 0.5898 0.2899 0.4795 0.8116 0.6425 0.7082 0.7998 0.6131 0.6529

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to the two passive systems. Results from the table indicate that for this single DOF system a direct correlation exists between the amount of damping force and effectiveness of control. According to the definition of control performance used here, it is concluded that semiactive control is not an optimal control strategy for vibration reduction of a single DOF building. In a situation where no limitation exists on magnitude of the damping force, a passive system surpasses performance of a semi-active system when the upper limit control force of the two controllers is equal. Validity of this statement is based on two primary assumptions: ?? The building is modeled as a linear, time invariant, single DOF structure. ?? Evaluation criteria for control are based solely on reduction of peak and normed measurements of lateral displacement and acceleration of the building. Variation from these key assumptions may nullify the above conclusion. Based on this finding, the best device for control of a single DOF system is not a magnetorheological damper. MR dampers require development of a semi-active control algorithm and an electric power source. When an MR damper remains in the passive-on configuration (with voltage set to a level of saturation) for an extended amount of time, large amounts of heat are produced by the constant current passing through the device. Long-term operation under such extreme circumstances would severely shorten the life of the damper. In practice, voltage to an MR damper is zero during a majority of its operation. It is designed to easily accommodate short periods of activity such as during a seismic event. A typical passive damper is a more appropriate device for vibration control of a single DOF structure. Mechanical properties of a passive damper are constant and do not require an external power source making it ideal for long-term use when a constant level of damping is desired. Comparison of passive and semi- active control is made again in the following example that involves a multiple DOF building model.

EXAMPLE 2: MULTIPLE DOF BUILDING The fuzzy controller developed in this example determines the voltage signal sent to a single MR damper that is installed in a multiple DOF building. The voltage is a function of the acceleration of all of the floors. A simplified rendition of the laboratory-building model, damper, chevron brace, four accelerometers, and direction of the seismic disturbance is

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Floor 4

Accelerometers (four total)

Floor 3

Floor 2 MR Damper Floor 1

Rigid Chevron Brace

Earthquake

FIG. 6. Arrangement of Damper, Brace, and Accelerometers on Building Model

presented in Fig. 6. The MR damper is located between the ground and first floor. In this location, it has been found to be most effective in reducing overall dynamic response of the structure (Subramaniam 1994; Jansen and Dyke 1999). Parameters of the 4-DOF building model are specified to resemble a scaled model of a tall, slender building with natural frequencies of 1.08, 3.91, 8.34, and 13.65 Hz for the first four modes of vibration. As with the previous example, the target controller is chosen to be a linear quadratic state- feedback regulator (LQR). Relevant parameters of the control system are listed in Table 4. Data for training of the fuzzy controller are acquired in a manner identical to that used in the previous example. TRAINING OF FUZZY CONTROLLER WITH ANFIS Based on the input selection of the previously designed fuzzy controller, an analogous set of inputs for the multiple DOF building is an instantaneous and time-delayed acceleration reading from each of the four accelerometers for a total of eight input variables. By an

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TABLE 4. Multiple DOF System Parameters System Parameter (1)

Parameter Value (2)

0 0 0 ? ?70.83 ? 0 78.60 0 0 ?? kg ? ? 0 0 77.71 0 ? ? ? 0 0 73.63? ? 0

Mass, m

Stiffness, k

66,430 ? 10,700 ? ? 258 ,000 ? 198,100 ?? 198,100 272 ,880 ? 168,060 37,050 ? ? ? N/m ? 66,430 ? 168,060 212 ,390 ? 88,760 ? ? ? 37 ,050 ? 88,760 54,510 ? ? ? 10,700

Damping, c

? 168.13 ? 14.13 ? 20 .75 10 .05 ? ? ? 14.13 152.5 ? 6.369 ? 10.71 ? ? ? N-s/m ?? 20 .75 ? 6.369 164.0 ? 34.00 ? ? ? ? 10.05 ? 10.71 ? 34.00 120 .4 ?

Feedback Gain Matrix, ? ?

?0.1651 ? 0 ? ? 0 ? ? 0

- 0.1339 0 0 0

0.0561 0 0 0

- 0.0136 0 0 0

0.0684 0 0 0

- 0.0539 0 0 0

0.0220 0 0 0

- 0.0053 ? 0 ? ? 0 ? ? 0 ?

iterative optimization method, the optimal time delay ?opt is found to be approximately 0.20 seconds.

Each of these eight variables provides added information about the dynamic

response of the building and individually contributes to the accuracy of the fuzzy approximation. However, in the interest of maintaining minimal computational complexity during training of the fuzzy controller, the number of input variables is limited to five. It has be shown that the addition of one input variable to a variable set increases computation time required by ANFIS approximately at a logarithmic (base 10) rate.

At the same time,

improvement in checking error associated with each added input variable decreases approximately by the power of one-half (Schurter 2000). Thus, as more input variables are added the law of diminishing return becomes quite apparent. Selection of the five most “useful” input variables is accomplished through use of the best subset method and forward selection procedure (Yen and Langari 1999). The optimal input combination is achieved with instantaneous acceleration readings from floors 1, 2, and 3 in conjunction with delayed readings from floors 1 and 4. After 30 epochs, training and

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checking errors are 0.4688 V and 0.4687 V, respectively. Complete training of the 5–input controller requires approximately four hours and 37 minutes of CPU time with a Pentium III (450 MHz) machine. (Addition of a sixth input variable is projected to require approximately 40 hours.) A graphical gauge of how well the fuzzy controller emulates the LQR target controller is presented in Fig. 7. For two different segments of time, the voltage signal predicted by the fuzzy controller is plotted simultaneously with the voltage signal used as training data. The two signals vary significantly for most instants of time. Virtually no similarities can be discerned between the two signals in part (a) of the figure. During portions of the histories shown in part (b), an approximate correlation is detected.

These two samples are

representative of the quality of training for the entire 120 seconds of data. The lack of correlation between voltage signals in Fig. 7 reveals less than encouraging training results. However, a more enlightening view of the capacity of the fuzzy controller to reduce vibration of the building is gained through numerical simulation. The fuzzy controller is tested through a series of numerical simulations of five earthquakes that are similar to those used in the previous example. Again, both sensor noise and a signal delay are introduced to the system, and the damper is tested in the passive-off, passive-on, and semi-active configurations. Effectiveness of the controller is measured according to the six multiple DOF

Voltage (V)

performance indices presented in Table 1. Results of the simulations are shown in Table 5.

Voltage (V)

(a)

(b)

2

Target Voltage ANFIS Prediction

Saturation Voltage

1 0 70

70.5

71

115.5

116

71.5

72

72.5

73

116.5

117

117.5

118

2 1 0 115

Time (s)

FIG. 7. Two Comparisons of Target Voltage and Voltage Predicted by the Fuzzy Controller: (a) Seconds 70-73; (b) Seconds 115-118

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TABLE 5. Evaluation Criteria for Multiple DOF Building: Passive and SemiActive Cases Disturbance (1) Artificial Earthquake El Centro

Hachinohe

Northridge

Kobe

Type of Control (2)

J1 (3)

J2 (4)

J3 (5)

J4 (6)

J5 (7)

J6 (8)

Passive-Off Passive-On Semi -Active Passive-Off Passive-On Semi -Active Passive-Off Passive-On Semi -Active Passive-Off Passive-On Semi -Active Passive-Off Passive-On Semi -Active

0.7269 0.5859 0.6844 0.7700 0.9042 0.7010 0.7506 0.5628 0.6589 0.8568 0.7781 0.8018 0.8373 0.5764 0.7857

0.8410 0.7079 1.0290 0.8358 1.2702 0.7855 0.7950 0.7101 0.7151 0.9518 0.9556 0.9103 0.8866 0.7145 1.2147

0.6613 0.5149 0.6384 0.7130 0.8530 0.7657 0.8139 0.5854 0.6947 1.0697 1.2107 1.1114 0.7743 0.6955 0.8540

0.7654 0.5750 0.6976 0.7082 0.6550 0.6275 0.6079 0.4470 0.5050 0.8125 0.6158 0.7508 0.7500 0.5849 0.6649

0.7629 0.8411 0.7501 0.7428 0.8359 0.7194 0.6867 0.6871 0.6426 0.8581 0.8724 0.8265 0.8227 0.7633 0.7933

0.8140 0.7009 0.7749 0.7552 0.9679 0.7032 0.6467 0.5423 0.5689 0.8535 0.7883 0.8205 0.8141 0.6678 0.7668

DISCUSSION OF RESULTS For most cases presented in Table 5, each of the three damper configurations improves control of the building according to the six performance criterion. Improvement is indicated by a performance index that is less unity. Only six of the 90 performance indices shown in the table are greater than unity.

These are associated with peak responses of absolute

acceleration and base shear of the structure. The six anomalous indices do not seem to be associated with any particular control scheme or earthquake. Generally, in comparison with response of the uncontrolled building the table reflects an overall improvement of structural response for all excitation and control cases. First, differences between the passive-off and passive-on control schemes are examined. Generally, the passive-on strategy results in improved control of the building. However, for 30% of the cases, performance of the passive-off configuration is superior. (Note that a given earthquake and a performance index define a case.) These typically occur for measures of acceleration and base shear of the structure indicating that in some cases, an increase of damping force does not improve dynamics of the building. This fact is contrary to the

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conclusion made with respect to the single DOF building where increased damping force always results in improved vibration control. A more interesting comparison occurs between the semi-active strategy of the fuzzy controller and the two passive strategies. 30% of the cases (signified by bold type in Table 5) show that the semi-active control scheme is better at reducing vibration of the building than either of the corresponding passive approaches. Four of the five cases associated with the J5 index demonstrate this superiority of the variable damping strategy. Recall that J5 provides a measure of improvement for the normed acceleration response of the building. Five of the six cases associated with the El Centro disturbance show optimum control to be exhibited by the semi-active configuration. The cause of this apparent trend is no t clear. It might be a reflection of the earthquake type (near field) or might be related to its magnitude and frequency content. The fact that very few strong trends exist in the data of Table 5 makes the statement of broad generalizations imprudent. Nevertheless, some conclusions can be drawn: ?? Overall, the passive-on strategy is the most successful form of control when all aspects of performance are considered. ?? The semi- active scheme dictated by the fuzzy controller is optimum if reduction of normed absolute acceleration is of most interest. Effectiveness of the semi-active control scheme with respect to reduction of acceleration is not unexpected and is supported through the consideration of extreme circumstances. Suppose the damper located at the bottom floor of the building is passive in nature and exhibits very high damping with respect to the structure. On a global scale, the building behaves in a more rigid manner consequently decreasing the amount of interstory drift of the structure. Concomitantly, an increase of rigidity results in higher floor accelerations and therefore higher base shears within the building. Thus, a tradeoff is established between various control objectives.

A well-designed semi- active control scheme should balance

benefits of the different objectives within the requirements of the specific design scenario. Reduction of building acceleration is important in situations where occupant comfort is a high priority. A circumstance where control of acceleration takes precedent over control of drift and base shear often arises with the design of very tall and slender buildings. Referring

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to Table 5, it is seen that optimum results are linked with the semi-active scheme for two of the five cases associated with peak acceleration. Hence, the argument is further reinforced for use of a variable damping strategy when acceleration control of a tall building is the greatest concern. According to these results, one of the most promising opportunities for research in the field of magnetorheological dampers lies in their application to tall building design.

CONCLUSIONS Recent temblors in Turkey and Taiwan have again shown the urgency of need for practical and effective solutions to ameliorate building vibration. The inherent stability and low power requirements of variable damping devices have served to promote the use of such semi-active control strategies by researchers and designers worldwide. Continued work in this area is expected to play a key role in the evolution of safe and reliable structures. Practical benefits of the proposed control scheme are numerous. Use of neuro- fuzzy methods produce a computationally efficient control law based on well-established techniques exhibited through a target controller. Neural networks provide a straightforward way to quantify complex nonlinear relationships without the use of complex mathematical modeling practices. Durability and energy efficiency of MR dampers and accelerometers make them ideal for use in civil engineering structures during severe excitation. Use of semi-active devices contributes very little energy to the structure hence guaranteeing stability of the control system. Such qualities increase the potential for future full-scale implementation of the proposed control strategy. With respect to a single DOF building model, the fuzzy controller based on acceleration feedback is found to be less effective than a simple passive damping scheme. Effectiveness of the controller is measured through consideration of interstory drift, lateral acceleration, and column base shear. Testing of a semi-active controller in the context of a multiple DOF building model shows acceleration to be notably reduced when compared to passive damping schemes of similar authority.

ACKNOWLEDGMENT This research was supported by a grant entitled “Robust Semi-Active Control of Structural Vibrations” from the Texas Advanced Research Program.

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