Adaptive control of base-isolated structures against near-field earthquakes using variable friction dampers

Transcription

1 Adaptive control of base-isolated structures against near-field earthquakes using variable friction dampers Osman E. Ozbulut 1, Maryam Bitaraf 2 and Stefan Hurlebaus 3 Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU College Station, TX Abstract This paper investigates the effectiveness of two adaptive control strategies for modulating control force of variable friction dampers (VFDs) that are employed as semi-active devices in combination with laminated rubber bearings for seismic protection of buildings. The first controller developed in this study is an adaptive fuzzy neural controller (AFNC). It consists of a direct fuzzy controller with self-tuning scaling factors based on neural networks. A simple neural network is implemented to adjust input and output scaling factors such that the fuzzy controller effectively determine command voltage of the damper according to current level of ground motion. A multi-objective genetic algorithm is used to learn the shape of the activation functions of the network. The second controller is based on the simple adaptive control (SAC) method, which is a type of direct adaptive control approach. The objective of the SAC method is to make the plant, the controlled system, track the behavior of the structure with the optimum performance. Here, SAC methodology is employed to obtain the required control force which results in the optimum performance of the structure. For comparison purposes, an optimal LQG controller is also developed and considered in the simulations together with maximum passive operation of the friction damper. Results reveal that developed adaptive controllers can successfully improve seismic response of base-isolated buildings against various types of earthquakes. Keywords: variable friction damper, semi-active control, adaptive control, base isolation, near-field earthquake 1. Introduction One of the main considerations of civil engineers when designing a structure is to reduce the risks of damage and injuries caused by dynamic loads such as earthquakes and strong winds. Over past decades, base isolation is found to be an effective means to protect structures and their contents from destructive effects of dynamic excitations. However, recent studies have shown that near-field earthquakes characterized by long-duration pulses with peak velocities result in significant relative displacements at the isolation level of a seismically isolated structure [1-3]. As a result of large isolator displacements, the size of the isolation device significantly increases. This may require very large seismic gaps between buildings or large bridge expansion joints. Therefore, these requirements, in return, increase the cost of the construction, which contradicts the primary goal of 1 Corresponding Author, Post Doctoral Research Associate, ozbulute@neo.tamu.edu, phone Ph.D. Student, mbitaraf@tamu.edu 3 Assistant Professor, shurlebaus@civil.tamu.edu

2 seismic isolation to design structures more efficiently and economically by reducing the earthquake forces transferred to the superstructure. In order to enhance the performance of base-isolated structures against near-field excitations, passive, active and semi-active control devices have been proposed [4-1]. Passive systems can reduce the deformations of the isolation bearings during strong ground motions, yet, they can cause an increase in superstructure response due to large damper forces applied to the structure [11-13]. Furthermore, in case of a moderate or weak excitation, the passive devices can have detrimental effects on an isolated structure since the desired isolation characteristics may be different for these ground motions and the passive devices cannot be adapted online. On the other hand, the active devices are generally able to control the seismic response of the isolation system for a wide range of loading conditions. However, active devices use an external energy source to produce the control forces imparted on the structure and require a considerable size of power source, which makes them vulnerable to power failure. Also, an active control system has the potential to destabilize the structural system. Semi-active control devices have received considerable attention in recent years because of their great adaptability with low power requirements [14-17]. These devices only absorb or store the vibratory energy and they do not input the energy to the system. Several researchers studied the use of a semi-active device in a base isolation system in order to reduce the displacement response of isolation system without an increase in superstructure response. Yoshioka et al. [18] performed experimental tests to demonstrate the effectiveness of a base isolation system that employs an MR damper. Symans and Kelly [19] investigated the performance of a variable viscous damper modulated by a fuzzy controller for seismic protection of an isolated bridge. In another study, Nagarajaiah and Sahasrabudhe [2] proposed a variable stiffness device that is used in a sliding isolation system to reduce seismic response of a base-isolated building and the effectiveness of the proposed semi-active device was shown by performing analytical and experimental studies. Madhekar and Jangid [21] evaluated the dynamic response of a seismically isolated benchmark bridge equipped with viscous and variable dampers and assessed the performance of such dampers. One semi-active device that appears to be promising for structural control applications is variable friction damper. Over the past few years, friction dampers have been widely investigated as an energy dissipation system to reduce the structural response due to dynamic loading [22-25]. A typical passive friction damper consists of two solid bodies that slide over each other to develop friction, which absorbs energy. As it is the case for other passive devices, the passive friction dampers cannot adjust the slip force in real-time according to structural response. When the contact force is very large, the damper will not dissipate energy for moderate and weak ground motions since it may not slide. On the other hand, when the contact force is reduced, the damper will have small energy dissipation capacity under strong earthquakes due to its small sliding friction force. Therefore, a controllable normal force is favorable in order to ensure the required amount of energy dissipation for various levels of ground motions. A variable friction damper enables the adaptation of the contact force by an actuator and can effectively suppress the vibrations of a structure. Several variable friction dampers have been conceptually investigated and some have been implemented [26-3]. In order to utilize full capabilities of a variable friction damper employed in a smart isolation system, an effective control algorithm that is practically viable is needed. However, the task of developing an optimal controller is challenging due to uncertainties in the nature of ground motions and in the characteristics of isolation system. For instance, a controller designed for near-field

3 ground motions that cause significant deformations in the isolation system might develop large damper forces during a far-field earthquake of generally more moderate excitation. As a result, the isolation system may not perform as expected and significant increase in acceleration response of superstructure can be observed. Alternatively, if the controller is designed for an earthquake with far-field characteristics, the damper force may not be large enough to effectively damp out the motion during a pulse-like ground motion. In the past, several control algorithms have been proposed to modulate the contact force of variable friction dampers. Inaudi et al. [31] developed a modulated homogenous friction (MHF) strategy that assumes the slip force of the damper is proportional to the absolute value of the prior local peak of the damper deformation. He et al. [32] improved the MHF controller to enable the slippage in the damper at different levels of seismic loadings. Chen and Chen [29,33] suggested the viscous and Reid friction (VRF) control strategy where the control force is computed as a function of the displacement and velocity of the damper. Xu and Chen [34] presented a modified VRF controller with a Kalman filter that uses acceleration response as feedback signal. Lu [35] proposed a control algorithm named predictive control that takes into account the features of friction dampers and predicts a slip force such that the damper remains in its slip state throughout the excitation. Ng and Xu [36] developed non-sticking friction controller that uses the actuated velocity in the current state of damper to ensure a continuous slipping and energy dissipation. In this study, two adaptive control strategies are developed in order to adjust the contact force of variable friction dampers that are used in a smart isolation system. The first controller is based on an intelligent control strategy. In particular, fuzzy logic control theory is used to design an adaptive controller whose input and output scaling factors are tuned by a simple neural network according to current level of ground motion. The adaptive fuzzy neural controller (AFNC) determines the command voltage by using isolation displacement and velocity as the two input variables. The second control method that is used to improve the performance of a base-isolated structure is the simple adaptive control (SAC) method. The goal of SAC is to force the controlled system to track the behavior of the model which exhibits the desired behavior. In other words, the error, or difference between the output of the controlled system and model, is reduced using SAC method. This method does not require explicit system identification because in the direct method the identification and control functions are merged into one scheme where control gains are computed directly. SAC is suitable for control systems with uncertainty, disturbance, or any change in the characteristics of the system such as damage. Here, the required force to optimize the performance of a base-isolated structure against different earthquakes is obtained using SAC method and then, the command voltage of the VFD is determined to generate the required force. Also, a linear quadratic Gaussian (LQG) controller is designed and considered in the simulations together with maximum passive operation of the variable friction damper for comparison purposes. A five-story building isolated by laminated rubber bearings is modeled together with variable friction dampers that are installed to the base of the building. Numerical simulations of the base-isolated building are performed and various response quantities are evaluated in order to assess the performance of the controllers.

4 2. Modeling of a Base-Isolated Structure with Variable Friction Dampers Consider an n DOF linear base-isolated structure subject to earthquake ground acceleration x& g. The equation governing dynamic response of the structural system equipped with variable friction dampers located at certain levels of the structure is given by Mx& () t + Cx& () t + Kx() t = MIx& () t + Du() t, (1) where M, C, and K denote n n mass, damping, and stiffness matrices, respectively; D is n 1 damper location vector; I is an n dimensional identity matrix; x(t), x& () t, x& () t n 1 displacement, velocity, and acceleration vectors, respectively; and u(t) is n 1 control force vector. Rewriting equation (1) in state-space form gives g z& () t = Az() t + Bu() t + Hx& () t, (2) where z() t = [ x(), t x& ()] t T denotes the state vector of the system. Also, the system matrix A and force location matrices B and H are defined as follows: I A 2n 2n =, B 2n m =, H 2 n r = M K -M C M D -I (3) In this study, the variable friction damper investigated experimentally and analytically by Lu and Lin [37] is adopted in numerical simulations. This damper employs a piezoelectric actuator to provide a controllable contact force as shown in Fig. 1. The friction force of the variable damper is proportional to normal contact force and friction coefficient between the friction pad and friction bar of the damper. The contact force of the damper is given as g N( t) = N C V ( t), (4) pre + pz where N(t) is the total normal contact force, N pre represents constant preload, C pz is the piezoelectric coefficient of the piezoelectric actuator, and V(t) is the applied voltage on the stack actuator. Then, the friction force of the semi-active damper, f(t), is given by f ( t) = µ N( t)sgn( x! ) if x! (5) µ N ( t) f ( t) µ N( t) if x! = (6) where μ is friction coefficient of the damper and ) sgn(x! denotes the sign of the slip rate of the damper. The parameters N pre, C pz, and μ are given as 1 N, 1.1 N/V and.2, respectively. Also, the maximum driving voltage is 1 V.

5 (a) (b) Fig. 1. Schematic illustration of components of variable friction damper: (a) side view, (b) top view (Lu and Lin [37]). During the motion of a structure, a friction damper has two possible states: sticking and slipping phases. A combination of stick and slip phases describes the complete state of the friction force. In the slip phase, the sliding velocity is nonzero and the friction force can be computed by equation (5). In the stick phase, two friction plates are stuck together, i.e., the sliding velocity equals to zero. The absolute value of the friction force in the sticking phase, f s, can be approximated for numerical analyses as follows f + s = fi fr when fs f and x! = (7) where f i is the inertial force applied on the mass, and f r is the restoring force provided by structural stiffness. For a base-isolated building where variable friction dampers are installed between the first floor and ground, the approximate values of f i and f r are given as f i = m u! t g (t) and f = r kb xb (t), where m t is the total mass of the structure, k b is the isolator stiffness and x b is the relative displacement of the base isolators [32]. A total of six historical earthquake records that are commonly used in structural control research are employed as external excitation in the numerical simulations. The characteristics of selected ground motions are given in Table 1. Fig. 2 illustrates 5% damped elastic acceleration and velocity response spectra for the selected earthquakes. Record (Station & Direction) Table 1. Characteristics of the ground motions used in the analyses. Earthquake Magnitude (M w ) Peak Ground Acceleration (g) Peak Ground Velocity (cm/s) Source of Data* El Centro Array # El Centro USGS Pacoima Dam San Fernando CDMG Hollister - South & Pine 1989 Loma Prieta CDMG Sylmar - Olive View Med FF Northridge DWP H KOB E-W 1995 Kobe JMA TCU68 N-S 1999 Chi-Chi CWB * USGS= CDMG= DWP= JMA= CWB =

6 3 3 Spectral acceleration (g) El Centro San Fernando Loma Prieta Northridge Kobe Chi-Chi Spectral velocity (cm/s) Period (s) Period (s) Fig. 2. Acceleration and velocity response spectra for the selected earthquakes. 3. Adaptive Control Strategies Adaptive control is a method by which a controller is designed to have the capability of tuning its parameters automatically. The primary goal of using adaptive controllers is to improve the performance of the controller online in the face of changing/uncertain parameters. This section introduces two adaptive controllers that are designed to adjust the contact force of variable friction dampers according to current level of external excitation. 3.1 Adaptive fuzzy neural control strategy One of the effective methods to deal with complex nonlinear systems is the fuzzy logic approach. Fuzzy logic enables to describe relationships between inputs and outputs of a controller using simple verbose statements instead of complicated mathematical terms. Due to its inherent robustness and simplicity, several researchers have used fuzzy logic theory to develop controllers for semi-active devices [38-4]. The design of a fuzzy logic controller (FLC) involves four main steps: (i) fuzzification of variables, which includes the transformation of the crisp inputs to fuzzy variables by defining membership functions to each input, (ii) definition of a rule base that relates the inputs to output by means of if-then rules, (iii) definition of an inference engine that evaluates the rules to produce the system output and (iv) defuzzification, where the output variable that is a fuzzy quantity is transformed to a non-fuzzy discrete value. The performance of a conventional FLC depends on various controller parameters such as the scaling factors, the membership functions and the rule base. In order to improve the effectiveness of the controller, an adaptive FLC can be designed by altering each of these parameters. Here, neural networks, trained with Non-dominated Sorting Genetic Algorithm version II (NSGA-II), are employed to modify the input and output scaling factors of a fuzzy controller to ensure satisfactory controller performance for both near-field and far-field ground motions. By tuning the scaling factors of input and output variables, the corresponding universe of discourse of the variable will

7 enlarge or reduce, resulting in better specification of fuzzy parameters. The block diagram of the adaptive fuzzy neural controller is shown in Fig. 3. Ground motion Structural response Control force Base Isolated Building Isolation displacement and velocity Variable Friction Damper Voltage S volt Fuzzy Controller S d S v Neural Networks NSGA-II Fig. 3. Block diagram of adaptive fuzzy neural controller. Fuzzy logic controller developed in this study employs isolation system displacement and velocity as two input variables and provides the command voltage of the damper. Seven triangular membership functions are defined for each input variable as shown in Fig. 4. The fuzzy sets for input variables are NL = negative large, NM = negative medium, NS = negative small, ZE = zero, PS = positive small, PM = positive medium, and PL = positive large. Note that universe of discourse for each input variable is defined from -1 to 1. Scaling factors are used to keep the input variables in the range of universe of discourse. For each input, a reasonable scaling factor should be selected because if the inputs are scaled to the values such that they become too small, then the innermost membership functions will be used frequently. On the other hand, if they become too big after being scaled, then the outermost membership function will be mostly employed, and this limits the effectiveness of the controller. Since the amplitudes of isolation deformation and velocity differ greatly for near-field and far-field ground motions, the decision on the scaling factors is done by a simple feedforward neural network introduced later in this section.

8 Degree of membership Degree of membership 1 NL NM NS ZE PS PM PL Isolation displacement 1 NL NM NS ZE PS PM PL Isolation velocity Fig. 4. Input membership functions for sub-level FLC. Five triangular membership functions are defined to cover the universe of discourse of the output variable voltage. The maximum driving voltage for the VFD is 1V. Yet, if the damper is operated at its full capacity during far-field earthquake, the normal contact force of the damper will be too large and the damper will not dissipate much energy since it will not slide as expected. Therefore, a lower command voltage, i.e. a normal contact force, is desirable for moderate and weak earthquakes for effective energy dissipation. On the other hand, when the contact force is reduced too much, there will not be enough energy dissipation for strong earthquakes due to small sliding friction force of the damper. To overcome this difficulty, an output scaling factor that varies in the range of to 1 is used to scale the command voltage of the damper. The output membership functions are equally-spaced over output domain as shown in Fig. 5. The fuzzy sets for output variables are VL = very large, L = large, M = medium, S = small, and ZE = zero. Degree of membership 1 ZE S M L VL Voltage (V) Fig. 5. Output membership functions for sub-level FLC. After the fuzzification of input and output variables, a fuzzy rule base is defined for the FLC. The rule bases adopted for the developed fuzzy controller are given in Table 2. The control rules are in the form of if-then statements and map the link between the input and output membership functions. Since the rules are words instead of mathematical equations, it is easy to interpret and

9 modify the rules. For example, a rule in the Table 2 can be read as if the isolation displacement is negative large and isolation velocity is negative large, then the voltage is very large. The rationale used to form to rule bases is as follows: if the displacement and velocity of the isolation system is of opposite sign (i.e., the isolation system returns to its original position), then the output voltage is small, and if the isolation displacement and velocity have the same sign, then the output voltage is large. The magnitude of the output (the degree of being small or large ) is linearly proportional with the magnitude of the input variables. When the displacement and velocity are almost zero or small, the command voltage is about zero; that means variable friction damper acts against motion as a passive Coulomb damper. The center of area method is used as a defuzzification method for the FLC to get a crisp output value. Isolation Velocity Table 2. Fuzzy rule base for sub-level FLC. Isolation Displacement Voltage NL NM NS ZE PS PM PL NL VL VL L L M S ZE NM VL L L M S ZE S NS L L M S ZE S M ZE L M S ZE S M L PS M S ZE S M L L PM S ZE S M L L VL PL ZE S M L L VL VL Neural networks have been used widely used for adaptive control of uncertain systems [41]. An artificial neural network consists of a number of simple artificial neurons that are usually organized into three layers, namely, an input layer, a hidden layer and an output layer, with random connections between the layers. Fig. 6 shows the schematic representation of an artificial neuron. The input signals, represented by x 1, x 2,, x n, are modified by synaptic weights. The output of a neuron is specified by an activation function whose input is the sum of weighted inputs. x 1 x 2. x i w 1j w ij w nj w j Sum of weighted inputs Activation function Output path y j x n Fig. 6. Schematic representation of an artificial neuron. A simple neural network that consists of three artificial neurons as shown in Fig. 7 is constructed to specify the input and output scaling factors of the fuzzy controller. As discussed earlier, near-field earthquakes usually possess long duration pulses with peak velocities. Therefore, the ground velocity x! g is selected as input of each neuron in order to determine the characteristics of

10 the ground motion. The range of the input x! g is defined to be [-1, 1] cm/s, where the upper limit of 1 cm/s is set as saturation point and seismic records with ground velocities beyond this value are directly classified as near-field earthquakes. The outputs of the network are scaling factors for isolation displacement S d, isolation velocity S v, and command voltage S volt. The activation functions for artificial neurons that produce scaling factors S d and S v, are selected to be triangular basis functions, while a bipolar sigmoid function is chosen for S volt. These functions are defined as F x) = c ( c c )! (8) 1( x g 2 ( x) c3 ( c4 c3) x g F =! (9) F 3 ( 1 exp( αx! g ) x) =, 1+ exp( αx! (1) ) where c 1, c 2, c 3, c 4 and α are the constants that define the shape of the activation functions. The shapes of these functions are illustrated in Fig. 8. Note that the activation functions F 1 and F 2 produce a maximum value of c 1 and c 3 for scaling factors S d and S v, respectively and the magnitude of the scaling factors decreases when absolute value of ground velocity increases. On the other hand, the activation function F 3 yields an output scaling factor S volt that increases with the magnitude of ground velocity and changes in the range of [-1, 1]. However, the absolute value of S volt is used as scaling factor and, therefore, the output of fuzzy controller, i.e. the command voltage of the damper, is scaled by a number between and 1. The variation of the output activation function F 3 with the constant α is shown in Fig. 8. Note that this function is highly nonlinear and the value of α significantly affects the output of the function. g F 1 (x) S d Ground velocity w 1 w 2 F 2 (x) S v w 3 F 3 (x) S volt Fig. 7. Neural network architecture.

11 c 1 or c 3 α = 5 α = 1 α = 2 1 c 2 or c 4-1 Input Input - 1 Fig. 8. Shapes of activation functions. Genetic algorithm (GA) is a search technique that emulates biological evolutionary theories for optimization and search. The method has been successfully applied to solving a wide range of optimization problems. In recent years, the use of genetic algorithms for training neural networks has been explored in several studies [42,43]. During training of a neural network, the genetic algorithm can be used to determine the network topology and/or weights and/or transfer functions. In this study, a multi-objective genetic algorithm optimizer, namely NSGA-II, is implemented to search the optimum activation function parameters of the neural network that generate the input and output scaling factor of the fuzzy controller. NSGA-II is a computationally fast and elitist evolutionary algorithm based on a non-dominated sorting approach. Among a pool of initial random candidate values that reside within a user-defined range, NSGA-II generates a set of Pareto-optimal solutions through an iterative process. In particular, it compares each solution with every other solution in the population to determine if it is dominated, and then evaluates the solutions in accordance with given performance objectives. The detailed description of NSGA-II algorithm can be found in [44]. In order to determine optimal activation functions using NSGA-II, five parameters, namely, c 1, c 2, c 3, c 4 and α are encoded into GA chromosome as shown in Fig. 9. From preliminary simulations, it is found that the parameters c 1 (maximum value) and c 2 (minimum value) of input scaling factor S d are within the ranges [7., 12.] and [1.5, 3.], respectively. Similarly, the ranges for the parameters c 3 and c 4 of input scaling factor S v are within the ranges [1.5, 4.] and [.5, 1.5], respectively and the parameter α for output scaling factor S volt is varied in the range of [3., 2.]. c 1 c 2 c 3 c 4 α Fig. 9. Chromosome representation of neural network. For evaluation of candidate solutions during NSGA-II optimization, a seismic excitation is required. Since here the goal is to have a single controller that is effective against both far-field and near-field earthquakes, 194 El Centro and 1999 Chi-Chi earthquakes are used to represent far-field and near-field excitations respectively during GA simulations. In order to evaluate each chromosome, four objective functions are defined and computed as follows

12 J 1 = max t(ff ) x b (t) max!! t( FF ), f x f ( t) J 2 = max t(ff ) ˆx b (t) max!! xˆ ( t) J 3 max = max t( NF) t( NF) x ( t) b xˆ ( t) b J 4 max = max t( FF ), f t( NF), f t( NF), f where x b and x ˆb denote uncontrolled and controlled base displacement, respectively; f!! x f!! xˆ f ( t) ( t) x& f and (11)! xˆ! are uncontrolled and controlled absolute floor acceleration, respectively; t is time; and f represents the story that is considered. The objective functions J 1 and J 3 evaluate peak base displacement in the controlled structure normalized by corresponding displacement in the uncontrolled structure during far-field (FF) and near-field (NF) earthquakes, respectively; the objective functions J 2 and J 4 compute peak absolute floor acceleration in the controlled structure normalized by corresponding acceleration in the uncontrolled structure during far-field and near-field earthquakes, respectively. Here, the term uncontrolled refers to the isolated building without any semi-active dampers. With above-described settings, a population of 5 chromosomes is initialized for a total of 1 GA runs. 3.2 Simple adaptive control strategy The simple adaptive control technique was introduced by Sobel et al. [45] and developed further by Wen and Balas [46] and Barkana and Kaufman [47]. In the simple adaptive control method, the desired behavior is defined by the behavior of the model. This method evaluates the system behavior by comparing the system output and the model output, and its goal is to match the response of the controlled system (plant) with the model that exhibits the desired behavior. The governing equations [4] for the plant (the controlled system) and the model (the system with the desired performance) are x& () t = A x () t + B u () t + d () t (12) p p p p p i yp() t = Cpxp() t + d () t (13) x& () t = A x () t + B u () t m m m m m f (14) ym() t = Cmx m() t, (15) where A p and A m are state matrices, B p and B m are input matrices and C p and C m are the output matrices for the plant and model, respectively. The vectors x p and x m are the n 1 plant state vector and n m 1 model state vector. The plant output vector is y p, y m is the model output vector, u p is the m 1 input control vector, and u m is the m 1 input command vector. The order of y p is m. The dimension of the model state may be less than the dimension of the plant state but it should be large enough to create the desired command for the plant. However, the order of model outputs is equal to the order of the plant outputs because y p is supposed to track y m. The variables d i (t) and d (t) represent input and output disturbances. Fig. 1 shows the block diagram of the adaptive control system.

13 Fig. 1. Block diagram of adaptive control system. where The control command can be calculated by T T T T u = K()[ t y y x () t u ()] t = K() t r () t (16) p m p m m K() t = [ K (), t K (), t K ()] t = K () t + K () t (17) e x u I P The overall gain K(t) is the sum of the integral gain K I (t) and proportional gain K p (t) which can be calculated from K& () t = ( y () t y ()) t r T () t T σ K () t I m p I T p() t = ( m() t p()) t () t (18) K y y r T (19) The rate of adaptation is controlled by the positive definite scale matrices T and T. The proportional gain is used to increase the rate of convergence of the plant performance to the desired performance, and the integral gain is required for the stability and tracking of the system. The coefficient σ in equation (18) can be very small and is used to prevent the integral gain from reaching very high values or diverging in the presence of the disturbance [48]. In order to define the model with the desired behavior to control the performance of the structure with VFDs, the behavior of the model should always be better than the behavior of the controlled system and should not depend on the external loads applied to the structure. Therefore, here, the model in SAC is designed such that it always exhibit better performance than the plant. In particular, it is defined as a structure having the output equal to zero. In this study, the output is the velocity of each floor of the structure, and the velocity equal to zero results in the minimum displacement and drift for the controlled structure. 3.3 Optimal controller For comparison purposes, a linear quadratic Gaussian control algorithm is also adopted to tune the contact normal force of the semi-active friction dampers. In LQG control design, the control

14 force is computed by minimizing a given performance index using isolation velocity and displacement as feedback signals. Here, ground acceleration is considered as stationary white noise, and a quadratic performance index with infinite horizon is selected as r 1 T T J = lim E { ( t) ( t) + ( t) ( t) } dt r τ z Qz u Ru, (2) where the matrices R and 11 Q are referred to as weighting parameters, whose magnitudes are 2n 2n assigned according to relative importance attached to the state variables and to the control force in minimization procedure. In this study, weighting matrix Q is selected as follows M Q = K, (21) where M and K are the mass and stiffness matrices, respectively. The above form of Q matrix represents the sum of the kinetic and strain energies, and it is commonly adopted in structural control applications. The selection of the parameter R is based on the goal of balancing the isolation deformation and superstructure acceleration responses, and through a trial and error process, R =.1 is chosen for the simulations. The control force obtained from optimal control theory is termed as active control force and cannot always be achieved by semi-active devices. A variable friction damper can only provide a resistant force whose direction is always opposite to that of the slip velocity. Also, the desired control force can be realized when the damper is in slipping state. Hence, a modified clipped control strategy is used to achieve the desired force as close as possible [36]. In this approach, the following criteria are used to adjust the desired contact force: u N( t) = µ ACTIVE if if where u ACTIVE is the control force obtained from the LQG control method, and μ is the Coulomb friction coefficient. 4. Numerical Study In this section, a five-story base-isolated building, as studied in Johnson et al. [49] is selected to investigate the performance of the developed adaptive controllers. A lumped-mass structure model with one degree of freedom per floor is used in the numerical simulations. The model of the smart base-isolated structure with variable friction dampers is shown in Fig. 11. The fundamental period of the building is.3 s, with a damping of 2% in the first mode. The mass of each floor is equal to 5,897 kg. The damping coefficients for the five floors are 67, 58, 57, 5 and 38 kns/m, and the stiffnesses of the five floors are 33,732, 29,93, 28,621, 24,954 and 19,59 kn/m, respectively. The isolation system consists of low-damping rubber bearings with a base mass of 6,8 kg and assumed to have linear force deformation behavior with viscous damping. The total damping coefficient and the stiffness of the rubber bearings are 7.45 kns/m and 232 kn, respectively. For the given isolation parameters, the base-isolated building has a period of 2.5 s and a damping ratio of 4% for the fundamental mode. In order to improve the performance of the base-isolated building against u u ACTIVE ACTIVE u! b u! b < (22)

15 different earthquakes, variable friction dampers with a total force capacity of 16.8 kn are installed to the base of the structure. x 5 m 5 k 5 c 5 x 4 m 4 k 4 c 4 x 3 m 3 k 3 c 3 Fixed Reference x 2 m 2 k 2 c 2 x 1 Variable Friction Damper m 1 k 1 c 1 Rubber Bearing VFD x b k b m b c b Fig. 11. Model of smart base-isolated building as six degrees of freedom system. Time history analyses of the base-isolated building are performed in MATLAB/Simulink [5] for the six historical earthquakes. For adaptive fuzzy neural controller, the optimum values of the parameters c 1, c 2, c 3, c 4 and α are found to be 9.3, 2.1, 3.3, 1.1 and 3.4, respectively. For SAC method, the value of σ, T and T are chosen to be.1, 1 I 4 4 and 1 I 4 4, respectively. The results for the applied ground motions are presented in Table 3. The evaluated response quantities are maximum isolator displacement x b,max, maximum interstory drift d s,max, maximum floor acceleration s,max and maximum damper force F d,max. The results for the base-isolated building without any damper (uncontrolled structure), for the maximum passive operation of the variable friction dampers, and for the modified clipped-optimal controller described above, are also given in Table 3 for comparison purposes. The minimum value of each response quantity for various controller cases is indicated in bold in the table. It can be seen that passive operation of VFDs with maximum voltage significantly reduces the peak isolation deformation for all considered excitations without an increase in interstory drifts, except in the case of the San Fernando earthquake. Specifically, the reduction of the maximum base displacement is of the order of 32% to 76%. However, there is a significant amplification in the maximum floor acceleration for most of the cases. In particular, the maximum floor acceleration increases 36%, 156%, 131% and 23% for San Fernando, Kobe, El Centro and Loma Prieta earthquakes, respectively. The semi-active control of VFDs notably improves the performance of base-isolated building concerning the peak acceleration response at the cost of slight deterioration in the peak isolation deformation. When the performances of three control strategies are compared, it can be seen that developed adaptive control methods are more effective than clipped-optimal controller, especially in controlling the acceleration response of the base-isolated building. For example, there are 8%, 95%

16 and 19% increases in the peak floor accelerations compared to the uncontrolled cases for El Centro, Kobe, and San Fernando earthquakes respectively, when VFDs are modulated by a clipped-optimal controller, while the same increases are only 3%, 47%, and 58% for the AFNC case and 3%, 28%, and 78% for the SAC method. Also, when the performances of adaptive controllers are compared, it is observed that the results of the AFNC and SAC method are mostly close to each other; however, the SAC method produces slightly better results for the structural response of the base-isolated building while the AFNC commands smaller control forces for most of the excitation cases. Table 3. Maximum responses of base-isolated building to various seismic excitations. Earthquake Response Uncontrolled Passive-max Optimal AFNC SAC El Centro x b,max (cm) d s,max (cm) s,max (m/s 2 ) F d,max (kn) San Fernando x b,max (cm) d s,max (cm) s,max (m/s 2 ) F d,max (kn) Loma Prieta x b,max (cm) d s,max (cm) s,max (m/s 2 ) F d,max (kn) Northridge x b,max (cm) d s,max (cm) s,max (m/s 2 ) F d,max (kn) Kobe x b,max (cm) d s,max (cm) s,max (m/s 2 ) F d,max (kn) Chi-Chi x b,max (cm) d s,max (cm) s,max (m/s 2 ) F d,max (kn) Figs. 12 and 13 shows the time histories of the isolator displacement and top floor acceleration for the semi-active control of the base-isolated building with AFNC and SAC method subjected to El Centro and Chi-Chi earthquakes, respectively. The results of the uncontrolled structure are also provided as a measure of performance evaluation of semi-active controllers. Also, forcedisplacement diagram of VFDs and time history of command voltage for the AFNC and SAC methods are presented in Fig. 14 and 15 for the same excitation cases. As shown in 12, in case of a far-field earthquake such as the El Centro excitation, both adaptive controllers operate the VFDs as

17 passive Coulomb dampers with nearly zero voltage. Therefore, the structural responses for both controllers are very close as illustrated in Fig. 12. (a) Isolator displacement (cm) Uncontrolled AFNC SAC Passive Time (s) Acceleration (m/s 2 ) Time (s) Fig. 12. Time histories of (a) isolator displacement and (b) top floor acceleration of uncontrolled and semiactive systems subjected to El Centro earthquake. (a) 4 6 (b) Isolator displacement (cm) Uncontrolled AFNC SAC Passive Time (s) Acceleration (m/s 2 ) Time (s) Fig. 13. Time histories of (a) isolator displacement and (b) top floor acceleration of uncontrolled and semiactive systems subjected to Chi-Chi earthquake. When the responses of base-isolated building subjected to Chi-Chi earthquake (which has nearfield characteristics) are compared for developed adaptive controllers, it can be seen that the peak isolator displacement and top floor acceleration are close to each other. However, the structural response ceases earlier when the dampers are controlled with SAC method. Also, as it is shown in Fig. 15, the controller developed using SAC method is more aggressive and commands higher voltage values. Overall, both controllers reduce both the displacement and acceleration response of the base isolated building when it is subjected to the Chi-Chi earthquake. (b)

18 (a) (b) Damper force (kn) Damper force (kn) Isolator displacement (cm) Isolator displacement (cm) Voltage (V) Voltage (V) AFNC Time (s) 5 SAC Time (s) Fig. 14. Force-displacement diagram of VFDs and time history of command voltage for (a) AFNC and (b) SAC method subjected to El Centro earthquake. (a) (b) Damper force (kn) Damper force (kn) Isolator displacement (cm) Isolator displacement (cm) Voltage (V) Voltage (V) AFNC Time (s) 5 SAC Time (s) Fig. 15. Force-displacement diagram of VFDs and time history of command voltage for (a) AFNC and (b) SAC method subjected to Chi-Chi earthquake. In order to further assess the performance of the developed controller, the energy balance equations of the based-isolated structure are established. Energy concepts can be used to evaluate the performance of seismically-excited structures since vibrations of structures due to an earthquake can be described as an energy transferring process. For a linear base-isolated structure with an installed variable friction damper, the seismic input energy is the sum of the kinetic energy, the

19 strain energy, the energy dissipated by structural damping, and the energy dissipated by VFD. The energy balance equations of the base-isolated structure can be expressed as E + E + E + E = E (23) K ξ S H I where each term in equation (23) can be obtained by integrating individual force terms in equation (1) over the entire relative displacement history and are given as follows 1 T E K = x& t Mx& (24) t 2 t E = T ξ dx = T xc & xcx & & dt (25) t ES t T = xk dx (26) EH t T T = ud dx (27) EI t = x& gmidx (28) In above equations, E K is the absolute kinetic energy, E ξ is the damping energy, E S is the elastic strain energy, E H is the hysteretic energy provided by the VFD, and E I represents the absolute input energy. Note that E H is zero for the base-isolated building without any damper. Fig. 16 shows the time histories of the input energy E I, the viscous damped energy of the building and rubber bearings E ξ, and the energy dissipated by the VFD E H, if it is present, for the uncontrolled base-isolated building and the semi-active control of the base-isolated building with AFNC and SAC method for El Centro earthquake. It can be seen that although input energy somewhat increases when the VFD is installed to the structure, the VFD successfully dissipates the most of the input energy. Also, Fig. 17 illustrates the sum of the kinetic energy and strain energy (E K + E S ) that is known as the total energy of the system or the damage energy. It should be also noted that the maximum damage energy of base-isolated building with the VFD is 6% smaller than without any damper.

20 Input energy Energy dissipated by isolated building Energy dissipated by VFD Energy (kn-m) 1 5 Uncontrolled 1 5 Passive Energy (kn-m) 1 5 AFNC 1 5 SAC Time (s) Time (s) Fig. 16. Energy time histories for uncontrolled and controlled base-isolated buildings subjected to El Centro earthquake. Recoverable energy (kn-m) Uncontrolled AFNC SAC Passive Time (s) Fig. 17. Time histories of damage energy for uncontrolled and controlled base-isolated buildings subjected to El Centro earthquake.

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