benchmark base isolated building with controlled ...

16th ASCE Engineering Mechanics Conference July 16-18, 2003, University of Washington, Seattle

BENCHMARK BASE ISOLATED BUILDING WITH CONTROLLED BILINEAR ISOLATION Baris Erkus1, Student Member ASCE and Erik A. Johnson2, Assoc. Member ASCE ABSTRACT This paper presents a sample control design for the base isolated benchmark building with bilinear hysteretic bearings (e.g., lead-rubber bearings). Designing a standard linear optimal controller (e.g., LQR) requires a linear model of the structure. An appropriate linearized model of the nonlinear structure requires knowledge of the response characteristics of the structure. However, the response depends on the controller, leading to a set of circular dependencies. Thus, this paper addresses the coupled problem of finding a good linear model for the controlled system and designing a linear optimal control for the structure. This is intended primarily as a tutorial and not as a competitive control strategy for the Benchmark Problem. First, the dynamics of the benchmark structure are reviewed and developed in a control-oriented manner. Second, an initial linear model, based on small-motion behavior, is developed. This model is then used to determine a reasonable cost function, trading off drifts and absolute accelerations in the structure. Then an iterative and a parametric study are described for developing the linearized model and corresponding linear control design. A Kalman filter is used as an estimator, and a Kanai-Tajimi filter is used in the control design to model typical ground motion frequency content. Finally, the responses of the controlled equivalent linear system and the controlled nonlinear system are compared for a specific ground motion data. Keywords: structural control, base isolation, benchmark problems, linearization

INTRODUCTION Base isolated structures are one of the most widely applied structural control strategies. Various researchers have studied the potential of improving base isolation performance using hybrid control strategies, such as employing passive, active and/or semiactive control devices in addition to the isolation (e.g., Reinhorn et al., 1987; Kelly et al., 1987; Nagarajaiah et al., 1993; Schmitendorf et al., 1994; Reinhorn and Riley, 1994; Yoshida et al., 1994; Nagarajaiah, 1994; Yang et al., 1996; Taylor and Constantinou, 1996; Symans and Kelly, 1999; Yoshida et al., 1999; Johnson et al., 1999; Spencer et al., 2000; Ramallo et al., 2002). Accommodating the large base drifts demanded by current design code has driven some of this work to make isolation more cost effective. Further, hybrid isolation can make further improvements on isolation performance. The Base Isolation Benchmark Problem (Narasimhan et al., 2003a,b) has been developed by the

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Grad Rsrch Asst, Dept Civil & Env Engrg., Univ Southern California, Los Angeles, CA 90089-2531; [email protected] Asst Prof, Dept Civil & Env Engrg., Univ Southern California, Los Angeles, CA 90089-2531; [email protected]

ASCE Technical Committee on Structural Control to facilitate comparisons between these hybrid control strategies. The Benchmark definition paper (Narasimhan et al., 2003b) gives the formal description of the benchmark problem. This paper gives a sample hybrid isolation design to serve as an example that participants in the benchmark study can use and adapt to their own control design methodologies. To use linear optimal control theory to design a controller for the benchmark structure, the model of the structure must be linear. In this Benchmark problem, the nonlinearities are only in the isolation layer, governed by the combined behavior of the nonlinear isolation bearings. Herein, the isolation bearings are chosen to be 31 rubber bearings and 61 lead-rubber bearings (LRBs). The rubber bearings are modeled with linear stiffness and damping properties. The LRBs are modeled similarly but with the addition of a lead plug that acts as an elastic-perfectly-plastic hysteretic element. Thus, the aim is to find a linear model of the combined effects of the 61 lead plugs that give a response approximately that of the nonlinear model. The difficulty in developing a linearized model is that it is dependent on the actual response of the nonlinear structure. The response is influenced by both the excitation and the control force. The excitation is not known a priori; herein, a stochastic definition based on the Kanai-Tajimi spectrum (Soong and Grigoriu, 1993) is assumed. The control forces are specified by the optimal control design. However, design of the control requires a linear model. Thus, there is a circular dependency. In this paper, a parametric study of linear behavior is developed to determine a good linear model for control design. This linear model is given, and a sample linear quadratic Gaussian (LQG) controller for the system is developed. The resulting controller is used in the simulation of the nonlinear system and the various performance indices, as described in the benchmark problem definition, are reported. A REVIEW OF THE BENCHMARK STRUCTURE In this section, some major characteristics of the benchmark structure and its mathematical model are given briefly, and a control-oriented state-space equation of motion is presented. The benchmark superstructure, as stated in the definition paper (Narasimhan et al., 2003b), is an eight-story steel-braced framed building. Stories one through six have an L-shape plan while the higher floors have a rectangular plan. The superstructure rests on a rigid concrete base, which is isolated from the ground by the isolation system. The superstructure has beam, column and bracing elements and rigid slabs above the base. Below the base, the isolation system consists of a variety of 92 isolation bearings. In the nominal benchmark model (to which other designs are compared), 31 of the bearings are linear elastomeric bearings and the remaining 61 are sliding friction bearings. The former have linear stiffness and damping behavior, whereas the latter have some stiffness but then slide with a given friction coefficient. Benchmark participants are allowed to modify the properties of the isolation bearings, as well as add passive, active or semiactive devices in the isolation layer between the ground and the base. The mathematical model of the benchmark structure is complex and is not directly convenient for many control design methodologies. Therefore, the model is reviewed and developed here in a manner that is somewhat more amenable for control design. The isolated building is modeled in two parts: (1) the superstructure, which consists of the eight-floor structure above the base; and (2) the base, isolation bearings and any additional control devices.

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The superstructure is modeled as a linear system and the slabs are assumed to be rigid. Therefore, the response of each story can be characterized by three degrees-of-freedom (DOF), two horizontal DOF and one rotational DOF, located at the center of mass of the corresponding floor. Thus, the superstructure finite element model is condensed to a 24 DOF model where only the horizontal slab responses and rotation are considered. In modal form, the superstructure equation of motion can be written as abs % η& b + K % ηb = −Φ T M R && && ηsb + C s s s s s s s xb

(1)

where ηbs is the modal responses of the superstructure with respect to the base and it is related to the floor responses through xbs = Φs ηbs , where Φs is the superstructure-mass normalized eigenmatrix of the superstructure and xbs = [ x8b y8b θ8b L θ1b ]T where xib , yib and θib are the displacements in the x- and y-directions and the rotation of the ith floor with respect to % s = ΦsT Cs Φs = diag (2ζ iωi ) are the modal stiffness % s = ΦTs K s Φs = diag (ωi2 ) and C the base, K and damping matrices where ωi and ζ i are the frequency and damping ratio of the ith mode, and M s , Cs and K s are the superstructure mass, damping and stiffness, R s is the influence is the absolute base acceleration. matrix and &x&abs b The isolation layer is also modeled by the three degrees-of-freedom of the rigid base about its center of mass. The isolator and control device forces are transferred to the base center of the mass, resulting in the base equation of motion M b && xb + Cb x& + K b x b = −M b R b && x gabs + S′c u + S ′f f + Fs

(2)

where xb = [ xb yb θ b ]T and xb , yb , and θ b are the base displacements and rotation with respect to the ground, M b is the mass matrix of the base, C b and K b are the linear damping and stiffness matrices associated with the linear elements between the base and the ground, u is a vector of control forces, f = f (x,x& ) is a vector of nonlinear isolator forces, Fs is the shear force applied by the superstructure to the base, R b , S′c and S′f are influence vectors of appropriate dimensions for the inertial forces, control forces and nonlinear isolator forces, is the absolute ground acceleration vector. The base shear force is respectively, and &x&abs g Fs = R Ts M s ( && xbs + R Ts && xabs b ).

(3)

An equation of motion for the whole structure can be obtained by combining equations (1), (2) and (3) as && + Cx& + Kx = Rx &&g + S cu + S f f (x,x& ) Mx

(4)

where x = [( ηsb )T (x b ) T ]T , or by the state &&g + Ff (q) q& = Aq + Bu + Ex

(5)

CONTROL DESIGN AND LINEAR MODEL FOR THE BENCHMARK STRUCTURE

The goal here is to develop a good linear model of the controlled benchmark structure and the corresponding sample linear controller. As the linear control strategy, a linear quadratic (LQ) controller is chosen. The 31 rubber bearings in the nominal isolation system (Narasimhan et al.,

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2003b) are used, and the 61 sliding friction bearings in the nominal isolation are replaced by bilinear elastomeric bearings, such as lead-rubber bearings, that have hysteretic stiffness and linear viscous damping. In LRBs, the rubber provides some linear stiffness and viscous damping whereas the lead plug provides hysteretic elastic-perfectly-plastic behavior. While the approach herein for developing a good linear model and corresponding control law is explained in the context of the bilinear nonlinearity, the procedure is applicable to other types of nonlinearities as well with some modification in the steps to determine the equivalent linear system. It is not intended that the resulting control strategy necessarily be a strongly competitive design, but rather intended as a tutorial for developing control laws for this nonlinear system. Since the “plant” (i.e., system to be controlled) in LQG theory should be a linear time invariant system, the model given by equation (5) cannot be used for the control design. Hence, a linear time-invariant model, which approximates the controlled nonlinear structure behavior, should be developed as the controller plant. However, this requires knowledge of response characteristics  such as peak and mean square values  of the nonlinear structure with the LQ controller, which requires the LQ controller itself. Therefore, development of a LQ controller for the nonlinear system turns out to have significant interdependencies. 1. The iterative procedure starts with an initial guess of the equivalent linear time-invariant structure. The approach that is used to create an equivalent linear system depends on the type of nonlinearity. For the benchmark structure with nonlinear bearings, which have bilinear hysteretic stiffness and linear damping, these nonlinear elements are simply replaced with fictitious elements with linear stiffness and linear damping. The linear stiffness and the linear damping of these new elements are set to the pre-yield stiffness, as shown in FIG. 1 and linear damping of the nonlinear bearings. Finally, an initial guess of the equivalent linear system is obtained. 2. The second step in the design process is the design of a linear controller. In the benchmark problem an LQ controller for the equivalent linear system is designed. For the current problem, a Kalman estimator is employed to estimate the states. Absolute accelerations of the center of masses of each floor, ground accelerations and controller drifts are assumed to be available by sensors. The index to be minimized is based on the drifts of the columns at each story and absolute accelerations of the floor mass centers. Also included is a Kanai-Tajimi filter (Soong and Grigoriu, 1993) to represent the ground acceleration frequency spectrum in the controller design for better performance. 3. The third step is the closed-loop numerical analysis of the original nonlinear structure and equivalent linear structure obtained in Step 1, both controlled by the linear control designed in Step 2. For this purpose, the two models are excited by some well-known historical ground motion data, and response characteristics are obtained. 4. The next step in the design is to compare the response characteristics of the original nonlinear structure and the equivalent linear structure, and update the equivalent linear structure guess. For example, considering the current problem, if the response levels of the nonlinear structure are higher than the linear structure response, one can suggest reducing the linear stiffness of the fictitious elements representing the nonlinear bearings and vice versa. By this comparison, a new guess for the equivalent linear system is made. 5. The iterative procedure continues with Step 2 until convergence.

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As can be seen easily, the criteria that is used to compare the responses of the nonlinear and the equivalent system is quite subjective and depends on the nonlinearity, the earthquake ground motion used and control designs goals.

FIG. 1 Selection of the linear stiffness

Details of the LQ controller and Kalman estimator

Consider a linear time-invariant system and an output to be minimized as follows q& = Aq + Bu + Ew

(6)

z = Cz q + Dz u + E z w

The stochastic LQ problem is defined as to find a control gain K that satisfies the following optimization problem (Stengel, 1994) % + uR % T u + 2z T Nu % ] , subject to (5) and u = −Kq min J , where J = E[z T Qz K

(7)

% =Q %T ≥0 , R % =R % T > 0 and N % are weighting matrices. The outputs to be where Q minimized are chosen to be absolute floor accelerations and corner drifts. Here, the corresponding weight matrices are chosen to be diagonal

(

)

α1b = diag (α xj,i ) , (α yj,i ) , 2

2

(

)

β = diag ( β x ,i ω x2 ) , ( β y ,i ω y2 ) , 2

2

α = diag ( diag(α1b ,L, α bN ),L,diag(α18 ,L, α 8N ) ) , β = diag(β b ,L, β8 ),

(8)

% =  aα 0  , R % = rI , N % =0 Q  0 bβ 

where α xj,i and α yj,i are the relative importance of the drifts of the j th corner of the i th floor in the x- and y-directions, respectively, β x ,i and β y ,i are the relative importance of the floor absolute accelerations of the i th floor in the x- and y-directions, respectively. The inclusion of ω x and ω y , the natural frequencies of the fundamental x- and y-dominant modes, is an aid in

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normalizing drifts and accelerations. With the control design parameters given above, the control design problem reduces to a choice of parameters, a and b , which determines the relative importance of the corner drifts and absolute floor accelerations.

FIG. 2 Final controller and structure model used in control design and analysis

The states are estimated using a Kalman filter for a measurement output given by y v = Cyq + Dyu + E y w + v

(9)

where v is the noise with zero mean and y v is the noisy output. Given that E[ ww T ] = Q , E[ vv T ] = R and E[ wv T ] = N , Kalman filter estimates the states qˆ that minimizes covariance of the steady state error in the states given by lim E[(q − qˆ )(q − qˆ )T ] . t →∞

(10)

The transfer function for the Kanai-Tajimi filter (Soong and Grigoriu, 1993) is given by GKT ( s ) =

2ζ gωg s + ωg2 s 2 + 2ζ gωg s + ωg2

(11)

Due to a previous study by Ramallo et al., (2002) on scaling of the Kanai-Tajimi filter using some historical ground motion data, the filter frequency and the damping ratio are selected as ζ g = 0.3 and ωg = 17 rad/sec . A schematic representation of the final controlled system is given by FIG. 2. It should be noted that the combined Kanai-Tajimi filter and structure model is used in LQG and Kalman filter design. NUMERICAL SIMULATIONS

In this section, the iterative procedure is applied to the benchmark structure. As noted previously, the sliding bearings in the nominal benchmark structure are replaced with LRBs. LRBs are modeled as combination of two different elements: The first element, that corresponds to the rubber, has a linear stiffness and linear damping while the second element, that corresponds to the lead, has bilinear stiffness with zero post-yield stiffness, and it has zero damping as shown in FIG. 3. Therefore the final isolation layer is modeled as 91 linear elements and 61 bilinear

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elements. 91 linear elements are assumed to have stiffness 919.422 kN/m and damping 101.439 kN ⋅ s/m . The 61 bilinear elements are assumed to have 5546.677 kN/m pre-yield stiffness, zero post-yield stiffness, 132.469 kN yielding force and zero damping. The values for the other structural parameters are set to the original values of the nominal structure. The iterative procedure starts with an initial guess of the equivalent linear structure. As the initial guess, the bilinear elements are assumed to have a linear stiffness the same as the pre-yield stiffness of the bilinear elements. Then a linear controller is designed using some numerical simulations of the controlled linear structure. The LQG parameters are selected as a = 1 × 10−2 , b = 1 × 10−4 and r = 1 × 10−12 .

FIG. 3. Mathematical modeling of the bearings

In the second step of the iteration, the nonlinear system and corresponding equivalent linear system, both with controller, are simulated for the 1940 El Centro ground motion record. The force at the center of mass due to the lead plugs is considered as the iteration criteria. The goal is to achieve same lead plug force levels in both controlled nonlinear and equivalent linear models. Therefore, the entire lead plug forces are transferred to the center of mass and the following value is computed for each iteration step:

γf =

RMS[ Fbnonlin ] RMS[Fblin ]

(12)

where Fbnonlin and Fblin are the force of the lead plugs (transferred to the center of mass of the base) in the nonlinear and linear controlled systems. The physical meaning of this parameter is that if γ f < 1 , the forces due to lead plugs of the linear system should be decreased so that force levels of the lead plugs of the new linear system will be similar to the nonlinear lead plug forces. Then, in the equivalent linear model, the stiffness of each of the lead plugs are updated as K ilinear = γ fi K ilinear +1

(13)

until it converges. The linear system is updated using this new stiffness value. Redesigning the LQG controller (i.e., finding new values of the design parameters a , b and r ) may be desirable in each iteration, though it requires interaction by the designer in each iteration. Rather, in this study, the initial controller design is not changed and is used through all iterations.

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TABLE 1. Results of the iterative procedure for several earthquakes

Ground Motion Newhall Sylmar El Centro Rinaldi Kobe

K final / K 0

0.33 0.28 0.32 0.25 0.35

This iterative procedure is performed using the criteria given by Eqs. (12) and (13) for five different historical ground motion records, and the results are presented in TABLE 1. The iterations are continued until reasonable accuracy is achieved. To understand the iteration path better, the variation of the γ f is investigated for several K linear / K preyield values where K linear is the stiffness of the lead plugs in the equivalent linear model and K preyield is the pre-yield stiffness of the lead plug of the original nonlinear structure, and the results are presented in FIG. 4. According to results and plots given, the iterative procedure will converge to γ f = 1 as the function γ f is monotonically decreasing.

RMS[ Fbnonlin ] γf = RMS[Fblin ]

FIG. 4. γ f vs. K linear / K preyield values for five historical earthquakes

Considering both the iteration and the parametric results, the equivalent linear structure is formed by substituting the bilinear lead plugs by linear elements with stiffness 0.3K preyield . The corresponding LQG controller parameters are a = 1 × 10−2 , b = 1 × 10−4 and r = 1 × 10−12 . Comparison of the controlled equivalent linear system with the controlled nonlinear system is

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given by comparing the responses both structures for the historical earthquake data used before. FIG. 5 shows the displacement of the center of mass of the base and the total lead plug force acting on the center of mass and hysteretic behavior of the base for the Newhall earthquake. FIG. 6 shows the absolute acceleration of the base, 5th and 8th floor. One can immediately observe that, for these specific response and force quantities, the controlled equivalent linear system shows behavior very similar to the controlled nonlinear structure even though yielding and considerable nonlinear behavior occurs.

Linear Sys. Nonlinear Sys.

FIG. 5. Comparison of the controlled equivalent linear and nonlinear responses: base displacements and the lead plug forces at the center of mass of the base in the x- and y-directions, hysteretic behavior of the base

There are alternate criteria that could be used in this iterative procedure. For example, the ratio given by Eq. (12) does not include any superstructure response. Neither does it account for the effect of force of the linear behavior of the isolation layer (i.e., the rubber) nor the energy dissipated by the bilinear hysteresis as can be seen in FIG. 5. Moreover, the entire set of bearings is represented in one index, which is not very informative. Also peak values are not included in the index. Similar discussion can be extended to other criteria as well. The choice herein is intended to be simple so to aid benchmark participants in understanding the control design approach presented in this study. Therefore, if this method is chosen for the design of the nonlinear benchmark structure, an extensive study should be carried out for better performance.

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Linear Sys. Nonlinear Sys.

FIG. 6 Absolute accelerations of the center of mass of the base, 5th and 8th floors

CONCLUSIONS

In this paper, a sample linear control design is developed for the recently introduced Base Isolated Benchmark structure with bilinear hysteresis. An LQG controller is designed for a fictitious linear system such that the chosen linear system and the nonlinear benchmark system have similar response characteristics when controlled. An iterative procedure is developed to satisfy a criterion that is based on the similarity of the forces associated with the nonlinear elements in the linear and nonlinear systems. Results for the iterative procedure are given for several earthquakes and, based on these results, the final LQG controller and the corresponding equivalent linear structure parameters are given. It is also shown for the Newhall earthquake ground motion that the selected linear structure shows behavior very similar to the nonlinear structure when they are controlled. The procedures presented in this study are intended for aiding to the benchmark problem participants; it is expected that results can be improved further by a comprehensive study. ACKNOWLEDGMENTS

The authors gratefully acknowledge the partial support of this research by the National Science Foundation, under CAREER grant 00-94030, and by the METRANS National Center for Metropolitan Transportation Research.

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