Evaluation of an active variable-damping structure

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 23, 1259-1274 (1994)

EVALUATION OF AN ACTIVE VARIABLE-DAMPING STRUCTURE*

E. POLAK AND G. MEEKER Department of Electrical Engineering and Computer Sciences. University of California, Berkeley, CA 94720, U.S.A.

AND

K. YAMADA AND N. KURATA Kobori Research Complex. Kajima Corporation, KI Building, 6-5-30, Akasaka, Minato-ku, Tokyo 107, Japan

SUMMARY

We present a numerical evaluation of the potential improvements in seismic disturbance rejection to be obtained by using active variable damping control in a structure. Using the responses to seismic excitation of an optimally controlled variable structure and of a minimax-optimal designed fixed structure, we obtain an upper bound on the achievable performance and a lower bound on the acceptable performance of a control system for a variable-damping structure. Both of these bounds are relative to an energy-function criterion. Our numerical experiments lead to the following conclusions:

(i) The gap between the upper and lower bounds is rather small, which makes designing a feedback law, that results in performance superior to that of a minimax-optimal designed structure, very difficult. The best choice for a feedback law appears to be continuous moving horizon control, whose implementation requires ground motion prediction up to 0.2 sec ahead, possibly using sensors located a small distance away from the site.

(ii) A minimax-optimal designed structure gives very good seismic disturbance suppression, not only for the earthquakes used in its design, but also for other earthquakes of similar intensity. Controlled variable structures are likely to offer advantages when earthquakes are moderate to severe, particularly at sites, such as landfills and dry lake beds, where resonances can be expected, but the resonance frequency cannot be estimated in advance.

1. INTRODUCTION

The problem of controlling seismically excited vibrations in a terrestrial structure has some unique features. First, seismic disturbances are of short duration, they are potentially of destructive intensity, and their occurrences are separated by long periods of quiescence. Second, by comparison to most other applications, the required control forces and associated energy requirements can be very large. Third, for reasons of reliability, the energy, for operating the control system of a seismic resistant structure, ought to be stored locally. As a result, a considerable literature dealing specifically with a great variety of passive/active control schemes for the reduction of the damage caused by earthquakes to buildings and other structures has developed: see Reference 1 for a list of references.

The actuator choices have a considerable impact both on the energy consumption and on the potential effectiveness of the resulting control system. For many active control schemes, such as active base isolation and active tendons, because of the enormous weight of a multistoried structure, the energy requirements can

*The research reported herein was sponsored by the Kajima Corporation, the Air Force Office of Sclentific Research contract AFSOR-90-0068, and the National Science Foundation grant ECS-8916168.

CCC OO98-8847/94/ 11 1259-16 0 1994 by John Wiley & Sons, Ltd.

Received 15 June 1993 Revised 4 October 1993

1260 E. POLAK ET AL.

be very large, and hence the development of safe energy storage methods for these types of actuators remains an open problem. The actuators proposed in References 2 and 3 are distinguished by the fact that they use very little energy, but they raise the question as to how effectively one can control a structure without applying considerable external forces.

The function of variable structure control system is to produce an impedance mismatch between the ground motion and the structure, so as to minimize the amount of ground motion energy that gets pumped into the structure. The success of such a control law, therefore, is bound to depend considerably on the spectral properties of the ground motion. The purpose of this study was to establish, by means of optimization techniques, an upper bound on the achievable performance and a lower bound on the acceptable performance of a control law for a variable damping structure, with respect to an energy-function performance criterion and a set of scaled ground motions. The selection of the criterion was guided by the fact that similar quadratic criteria are used in feedback laws. In our numerical experiments we used only five scaled ground motions, since these sufficed to establish that the gap between the upper and lower bounds can be fairly narrow.

In Section 2, we describe a model of the planar variable damping structure that we wish to study. In Section 3, we describe the numerical experiments and the optimization algorithms that we used in establishing achievable/acceptable performance bounds. For each of the selected earthquakes, the upper bound on achievable performance was obtained by computing the responses of our variable damping structure, subjected to these earthquakes, and controlled using open-loop optimal damping modulation, computed under the unrealistic assumption that the ground motion is known in advance. We obtained a lower bound on the acceptable performance of a control law, by designing a structure with $xed damping coeflcients that was minimax-optimal (i.e. worst-case optimal), with respect to four of the earthquakes considered and our energy criterion, and used its performance as the lower bound. The fifth earthquake (El Centro 1940) was not included in the minimax design, because we wanted to use it as a control case, to determine whether a structure that has been minimax-optimal designed for a given set of earthquakes performs well on other earthquakes.

Because we found the performance gap to be rather narrow, we explored, in Section 4, the performance of two moving horizon control laws (see e.g. Reference 4) since they have the best chance of giving behaviour that falls in the gap between our upper and lower performance bounds. The first moving horizon control law was of the sampled-data type, while the second one was a continuous control law.

In Section 5, we evaluate our numerical results, which show that our minimax-optimal designed structure performs surprisingly well compared to open-loop optimal control. Its excellent performance on the scaled El Centro earthquake, which was not used in its design, leads us to speculate that one can get very good design results (in the elastic range) by performing a minimax-optimal design using a fairly small number of earthquake samples. We also found that the performance of our continuous moving horizon control law falls between our upper and lower bounds.

2. SYSTEM MODELS FOR NUMERICAL EXPERIMENTATION

We will consider the control of a planar variable three-storey structure consisting of a main structure and an auxiliary structure that are linked through variable damping elements, as shown in Figure l(a). This configuration is the variable stiffness and damping structure model proposed in Reference 3, in which an auxiliary structure is connected to a main structure by means of variable dampers.

2. I . Vuriuhle structure system model We will assume that the mass of the main structure is concentrated in its floors, and that the mass of the

auxiliary structure is concentrated at the points of connection of the auxiliary structure to the variable dampers. Hence, our mathematical model corresponds to the situation in Figure l(b).

ACTIVE VARIABLE-DAMPING STRUCTURE 1261

(a) Phy.*.l Configuntbn C) l d r b l t b n

Figure 1. Configuration of variable damping-stiffness structure

Table I. Parameter values for variable structure

M,' 2 . 0 ~ lo5 kg M: 1 . 2 ~ lo5 kg Ma 1 . 2 ~ 105kg

Ks' 1.96 x lo7 N/m &* 1.96 x lo7 N/m K3 2.45 x lo' N/m & 0.004 Kf N s/m

K!i 9.8 x lo7 N/m fl: 0.004Ki N s/m

M: 1 . o ~ 104kg

We will model the variable structure, as a planar structure with horizontal displacements only, in terms of absolute coordinates, as follows:+

where the vector x E R6 is given by

x(t) = (x; ( t ) , xf(t), X3(t), xb ( t ) ,X ,2@), X3(t)) (2) where the xf(t), i = 1,2,3, are the absolute displacements of the floors of the main structure, the x;(t) , i = 1,2,3, are the absolute displacements of the masses of the auxiliary structure, M is the mass matrix, C is the internal viscous damping matrix, K is the stiffness matrix, pi@), i = 1,2,3, are the variable damping coefficients, Ui, i = 1,2,3, are location matrices for the variable dampers, J, is a 6 x 1 vector with each element equal to 1, and x,( t ) and ie(t) are ground motion and ground velocity obtained from an earthquake record. We assume that at rest the positions of the floors and of the auxiliary masses are zero, in absolute co-ordinates. The parameter values for our model are given in Table I where i = 1,2,3.

'Since only ground awleration records are available, in our numerical experiments we used a model in relative co-ordinates, for which the disturbance is the ground acceleration, and use ground velocity records, obtained by numerical integration, to obtain absolute velocities for kinetic energy calculations. Thus, we avoided errors that would have been caused by the computation of ground displacements by double integration of the acceleration records.


2.2. Sampled-data damper modulation Because one of the control laws that we propose to explore is a moving horizon sampled-data control law,

we need to develop a model for the response of the variable damper to 'on-off' commands. Since in sampled-data mode, the command to each clamp can be either 'on' or 'off, at each point of time, the behaviour of the structure in Figure 1 is determined by one of the eight possible combinations of the command signals. To keep track of this situation, we introduce the command vector c ( t ) E R3, t 2 0, with the components ~ ' ( t ) , i = 1, 2,3, assuming only the values of 0 or 1, i.e., c( t ) E C, where

When c'(t) = 1, the command to the ith damper is 'on', and when c'(t) = 0, the command is 'off'. We will assume that c( t ) can only change at the sampling times kT, k = 0, 1,2,3, . . . , where T is the sampling period.

We assume that the damping coefficients of the variable dampers can be driven from 0 0 N sec/m to ,urnax = 1.96 x 108Nsec/m, in T, = 018 sec, which leads us to choose a minimum sampling period of T = 0.2 sec. We will model the damper response to switching inputs as follows. Let pL(t), i = 1,2,3, denote the response of the ith damper to the command signal ci(t). Then we assume that, with r = 4,

pmaX((t - kT)/T')' for t E [kT, kT + T,] for t E ( ~ T + T,,(k + 1)T) { pmax

d ( t ) = (4)

if c'(kT - E ) = 0 and c'(kT) = 1 (with E E (0, T ) ) ;

(5) pmax(l - (t - kT)/T')' for t E [ k T , k T + T']

for t E (kT + Tc,(k + 1)T)

if c'(kT ~ F ) = 1 and c'(kT) = 0;

pk(t) = 0 for t E [kT, ( k + 1) T )

if c'(kT - E ) = 0 and c'(kT) = 0 and, finally,

&(t) = pma, for t E [kT, (k + 1)T)

if c ' (kT- E ) = 1 and c'(kT) = 1.

3. ESTIMATION OF PERFORMANCE BOUNDS

Since the control system for an actively controlled variable structure cannot impart energy to the structure, it is not clear, a priori, what is its potential for earthquake damage mitigation. Hence, before attempting to develop control laws, we will establish upper and lower bounds on its performance. The upper bounds will be established using optimal control laws under the assumption that the earthquake ground motion is known in advance. A lower bound on acceptable performance will be obtained by carrying out a minimax optimal design that assigns fixed values to the variable dampers, on the basis of the histories of a fixed number of earthquakes.

3.1. Development of a performance criterion Minimal performance requirements for a structure subjected to earthquakes are usually expressed in terms

of maximum allowable interstory drifts, shear forces and forces on the active elements. In addition, the accelerations of the floor masses must be kept as small as possible, so as to help protect the contents of the structure during an earthquake. The above requirements are best captured by a minimax type criterion. However, there are no commonly used feedback laws based on a minimax criterion and in addition it would


be rather cumbersome to implement a moving horizon control law based on a minimax criterion. Hence, we will use a weighted energy criterion that reflects these requirements indirectly, but which relates naturally to common feedback laws. The weights will be tuned to make the energy criterion represent the performance requirements as well as possible.

We define the state vector z E R12 by z ( t ) = ( i ( t ) , x(t)) , and e(t) E R2 by e(t) = (ig(t), xg( t ) ) and denote by z( t ;zo,To,p,e) the response of the structure, determined by equation (l), to a ground motion e(t) and damping coefficient modulation function p(t) , from an initial state zo and at a time To 2 0. At time t 2 To, we will associate with each storey of the structure two energy functions. Let E:,K(t; zo, To; p,e) = t M i ( t ) 2 be the kinetic energy in the ith storey of the main structure, EfSp( t ; zO , To,p, e) = *KS(x.(t) - xf"(t))', the potential energy in the ith storey of the main structure. We will use as our performance criterion a weighted energy function of the form

3 3

E(t;zO, TO,We) = 1 Wi,KEb,K(t;ZO* TO,p,e) + 1 Wi,pE6,P(t;Z0,T0,p,e) (8) i = 1 i = l

where the weights w ~ , ~ , w ~ , ~ , i = 1,2,3 are all positive. The choice of the weights w = ( w ~ , ~ , w ~ , ~ , w ~ , ~ , w ~ , ~ , w ~ ~ ~ , w ~ , ~ ) affects the response of the controlled

structure. Increasing the potential energy weights wi,p tends to decrease corresponding interstorey drifts, while increasing the kinetic energy weights w ~ , ~ tends to reduce the velocity of the ith storey mass. After considerable numerical experimentation, using a command signal determined by the sampled-data moving horizon control law presented in Section 4, and the scaled earthquake records described below, we found that the choice of w = (3,2.5,4,1,1,1) gave the best results.

For the purpose of establishing design performance bounds, we chose as a benchmark five earthquake records, scaled so that their peak velocities were equal to that of the El Centro 1940 (comp SOOE) earthquake. In addition to the El Centro record, we used the Kern County (Taft) 1952 (comp S69E), the Puget Sound 1965 (comp S86W), the San Fernando (Pacoima) 1971 (comp S74W), and the Western Washington 1949 (comp N86E), records. The scaled ground acceleration for each of the earthquakes is shown in Figure 2(a)-(e).

3.2. Upper performance bounds: optimal controls

each of the five scaled earthquake records described in Section 3.1 To establish upper performance bounds, we will compute optimal damping modulation coefficients for

The optimal control problem. Let 0 < To < < cg be given initial and final times, e(t), To < t < Tf be a given ground motion, and let

M = { P E L: [To, Tcl I 0 < pi(t) < pmax, i = 1,2,3, t E CO, 51) (9) Finally, let the cost function$ M + R be defined by

T,

TO fbu) E ( t ; zot TO, P, e) dt (10)

where E(t; zo, To, p, e) is defined as in equation (8). To compute an optimal damping coefficient modulation function (control) &t), we propose to solve the following optimal control problem:

To solve P, we used a constrained Armijo gradient method (see References 1 and 5-7). Optimal control problems require considerable computing time to solve, with the solution time depending strongly on the choice of the initial control. We therefore solved the optimal control problems in two stages. The first consisted of solving a simplified problem, in which the control is required to be constant for all times, whose solution was used to initialize the optimal control algorithm in the second stage. Referring to equation (1 l),


300

-200 ' ,. 0 0.5 1 1.5 2 2.5 3 35 4 4.5 5

(4 Time (aeccmds)

I 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-300 1

(4 Time (seconds)

-300 I 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(b) Time (seconds) -250' 0 0.5 1 1.5 2 2.5 3 3.5 4 I 4.5 5 I

Time(recads) (el

Figure 2. Scaled ground accelerations

ACTIVE VARIABLE-DAMPING STRUCTURE

we see that the simplified problem has the form

PO minf(u) OSV

1265

(12)

wheref(o) A f ( p u ) , with pL,(t) = o for all k, and V A ( o E R3)0 < u j < pmax, j = 1,2,3}. The constrained Armijo gradient algorithm'$ 5-7 was also used to solve problem Po.

3.3. Lower performance bounds: minimax-optimal fixed structure To justify the additional cost, a controlled variable structure must perform better than any well designed,

comparable fixed structure. We therefore computed minimax-optimal fixed values for the damping coefficients with respect to the Kern County, Puget Sound, San Fernando, and Western Washington earthquakes. The El Centro earthquake was not included in our minimax design, to give us an opportunity to evaluate the response of the minimax-optimal fixed structure to an earthquake that was not included in its design.

Formulation of the minimax-optimal design problems. For j = 1,2,3,4, let p(v) be defined as in equation (12), with the subscript j denoting the particular earthquake exciting the dynamic equations (I), as follows: j = 1 corresponds to Kern County, j = 2 corresponds to Puget Sound, j = 3 corresponds to San Fernando, and j = 4 corresponds to Western Washington. The minimax problem that we solved to obtain fixed values for the damping coefficients was

MMP min max { j j ( u ) } u e V jen

where m = (1,2,3,4}. The problem MMP was solved by means of a constrained minimax algorithm (see References 1 and 5-7).

4. MOVING HORIZON FEEDBACK CONTROL LAWS

Two moving horizon control laws were evaluated with respect to our performance bounds. The first is a sampled-data relay type law that controls the dampers in on-off fashion, while the second is continuous modulation type law. Both of these moving horizon control laws must be considered to be conceptual, since they require knowledge of the ground motion one horizon ahead of the present time, which may not be available in a real-time control system.

Sampled-data mooing horizon feedback control law. Our sampled-data moving horizon control law is based on a sampling period T. At time t = kT, k E N (i.e. at the beginning of the kth sampling interval), the control law chooses an element ck E C that determines via equations (4)-(7) the action of the dampers over the kth sampling interval, so as to minimize the cost function

E((k + 1)T; z ( W , k T , p , , e ) (14) Note that the control remains constant between the sampling times.

Continuous time mooing horizon feedback control law. Our continuous time moving horizon control law is based on a horizon of Th sec. At time t = k Th, k E N, the control law uses the optimal control algorithm (as in Section 3.2) to determine an optimal control Pk(t) which is then used over the time interval [kT,,(k + l)Th).

A detailed description of both moving horizon feedback control algorithms is given in Reference 1. In our numerical experiments, we used again the weight w = (3,2-5,4,1, l), the dynamics given by equation (l), our five scaled ground motion records, and the horizon length Th of 0.2sec. A complete evaluation of the performance of the continuous time moving horizon control law is given in Section 5, where for the El Centro earthquake, a full set of plots will be given in Figure 5.


5. EVALUATION OF NUMERICAL RESULTS

We will now discuss the results of our numerical experiments. Because of the need to perform a very large number of numerical experiments, to reduce the total computation time, we performed experiments using only the first 5 sec of each earthquake. In addition, to establish the validity of our conclusions based on these ‘short-time’ experiments, we performed one set of experiments using the ground motion supplied by the first 20 sec of El Centro earthquake.

5.1. Numerical results In the interest of keeping the size of this paper reasonable, we have included a full set of plots only for the

computations that used the El Centro ground motion record. Our complete numerical results are summarized in Tables 11-VIII. In these tables, EC denotes the El Centro 1940 (SOOE) earthquake, KC denotes the Kern County (Taft) 1952 (comp S69E) earthquake, PS denotes the Puget Sound 1965 (comp S86W)

Table 11. Comparison of optimal cost function values (Nms x 10)

Optimal Minimax Fixed Moving Sampled-data control design design horizon moving horizon

~~

Earth- quake f ( a ) f ( o M M ) T(fi,) f ( ~ ) f (W EC 2.77 4.46 4.4 1 3.26 4.93 KC 1.65 2.10 2.09 1.83 4.92 PS 1.93 2.5 1 2.51 2.65 3.56 SF 1.58 2.22 2.07 2.1 1 2.89 ww 1.21 1.57 1.49 1.27 2.08

Table 111. Comparison of peak interstorey drifts, and peak average accelerations

Earth- quake Optimal control Minimax design Fixed design

(a) Comparison of peak interstorey drifts (cm) (xt, i = 1,2,3)

EC 0.80 1.45 1.10 1.30 1.90 1.75 1.20 1.90 1.60 KC 0.65 1-45 1.00 0.75 1.25 1.10 0.90 1.30 1.00 PS 0.75 1.18 0.70 065 095 1.00 0.60 1.00 1.00 SF 0.62 1.12 1.08 0-85 1.30 1.15 0.75 1.15 1-15 WW 0.75 1.30 0.80 0.65 1.05 0.90 0.60 0.85 0.80

(b) Comparison of peak accelerations (cm/sec2)’(ji.besk, i = 1,2,3)

EC 290 260 280 500 410 360 460 375 340 KC 320 275 290 350 280 280 340 275 280 PS 260 220 330 235 255 320 230 250 325 SF 310 300 360 315 280 210 350 310 225 WW 280 260 290 265 220 225 310 260 240

(c) Comparison of average accelerations (cm/sec2) (xkvg, i = 1,2,3)

EC 59.6 51,5 79.8 98.5 80.2 72.8 91.6 75.4 72-1 KC 38.0 36.5 50.3 50.4 43.4 44.1 45.1 42.6 44.2 PS 43.8 50.5 77.9 64.0 58.6 70.6 64.0 58.6 70.7 SF 46.1 47.2 59.8 544 47.6 49.9 62.7 52.2 50.5 WW 35.8 35.3 45.3 35.8 34.2 41.3 41.8 39.3 43.7


Table IV. Comparison of damper energy dissipation, and peak shear and damping forces

Earth- quake Optimal control Minimax design Fixed design

(a) Comparison of total energy dissipation in dampers (Nm x lo4) (Ei , i = 1,2,3)

EC 1.20 3.75 3.10 2.60 6.20 460 2.60 6-44 4.53 KC 0.70 2-10 1.20 1-00 2.40 1.80 1.06 2.51 1.62 PS 0.61 1.50 1.40 0.90 2.1 1.70 093 2-10 1.66 SF 041 1.52 0.81 1.00 2.35 1.80 0.73 1.72 1.24 WW 0.33 0.87 0.50 0.46 1-12 0.92 042 0.96 0-67

(b) Comparison of peak shear force (N x lo5) ( F i , i = 1, 2, 3)

EC 1.7 2 8 2.6 2-6 3.8 4.3 2.5 3.7 4.0 KC 1.4 2,4 2.9 1.7 2.5 2 6 1.8 2.6 2.4 PS 1.5 2.3 1.7 1.2 1.9 2.5 1.2 1.9 2.5 SF 1.2 2.2 2.7 1.7 2.6 2.8 1.5 2.3 2-9

1.7 2 0 WW 1.4 2.5 2.0 1.3 2.0 2.2 1.2

(c) Comparison of peak damping force (N x 10') ( F i , i = 1,2,3)

EC 5.3 8.5 9.6 7.5 11.5 14.0 6.9 104 12.8 KC 5.2 9.0 10.0 5.7 8.3 10.0 5.4 8.0 9.8 PS 4-6 6.7 7.4 3.6 5.2 6.8 3.6 5.2 6 8 SF 5.1 9.1 100 4.9 7.7 9.6 5-7 8.8 105 WW 5.2 7.9 7.7 4.3 6.3 7.5 5.1 7.5 8.8

Table V. Comparison of peak interstorey drifts, peak and average accelerations

Earth- quake S-D moving horizon Moving horizon

(a) Peak interstorey drifts (cm) xi, i = 1,2,3) 1.59 1.31 1.50 1.18 1.47 1.20 1.27 097 1.58 0.87

EC 1.34 1.46 2-00 KC 1-74 2.86 3.17 PS 1.56 1.89 1.49 SF 0.76 1.65 1.72 WW 1.93 2.27 1.19

(b) Peak accelerations (cm/sec2

EC 579 385 392 KC 460 395 272 PS 452 363 364 SF 417 342 453 WW 283 230 229

0.78 0.89 077 0.7 1 075

(%k,

35 1 390 258 369 276

= 1,2,3)

279 329 317 321 221 329 341 359 226 423

(c) Average accelerations (cm/sec2) (&, i = 1,2,3)

EC 1029 91-3 94.3 704 64.5 74.8 KC 70.2 55-3 46.0 41.3 36.5 45.0 PS 61.6 65.8 80.2 55.3 53.2 722 SF 59.3 54.1 808 57.3 504 59.4 WW 38.1 33.9 40.5 32.7 34.5 44.7

1268 E. POLAK E T AL.

Table VI. Comparison of damper energy dissipation, and peak shear and damping forces

Earth- quake S-D moving horizon Moving horiLon

(a) Comparison of total energy dissipation in dampers (Nm x 1 04)

EC 083 1.59 2.25 1.32 4.06 4.23 KC 0.65 1.46 1.62 0.69 2 10 1.68 PS 0.25 0.72 1.15 0.63 2.00 1.34 SF 0.27 0.81 0.84 0.74 2.80 1.39 WW 0.50 089 0.76 0.32 1.04 0.57

(b) Comparison of peak shear force (N x lo5) ( F k , i = 1,2,3)

EC 2.7 2.9 4.9 1.5 3.1 3.2 KC 3.4 5.6 7.8 I .8 2.9 2.9 PS 3.1 3.7 3.7 1 5 2.9 3.0 SF 1.5 3.2 4.2 1.4 2.5 2.4 WW 3.8 4.5 2.9 1.5 3.1 2.1

( E i , i = 1,2,3)

(c) Comparison of peak damping force (N x lo5) (F;z, i = 1,2,3) EC 9.8 13.7 13-6 5-9 8.9 9.3 KC 9.5 14.9 183 7.4 11.5 12.7 PS 7.4 12.2 13.3 4.8 7.8 10.0 SF 7-3 9.7 12.0 6.6 10.7 12.0 WW 7-4 10.4 10-0 5.3 7.9 8.3

Table VII. Results of 20 sec simulations

(a) Comparison of peak interstorey drifts (cm) (xi, i = 1,2,3)

Optimal control 0.74 1.33 0.99 Minimax design 1.34 1.91 1.77 Moving horizon 0.83 1.61 1.17

(b) Comparison of peak accelerations (cm/sec2 (xbeak, i = 1,2,3)

Optimal control 283 253 273 Minimax design 498 417 365 Moving horizon 375 373 337

(c) Comparison of average accelerations (cm/sec2) (xivg, i = I, 2,3)

Optimal control 40.4 37.5 47.2 Minimax design 61.0 502 46.3 Moving horizon 48.5 42.9 45%

earthquake, SF denotes the San Fernando (Pacoima) 1971 (comp S74W) earthquake, and WW denotes the Western Washington 1949 (comp N86E) earthquake.

In our tables we will use the following definitions and notation: In Table IIf(tz), y(~MM),f(i?FD),f(tiMH) and f(a SD) denote the cost function values for optimal control, minimax design, fixed design, the continuous moving horizon control law and the sampled-data moving horizon control law, for a particular earthquake, respectively. In Tables IV(a) and VI(a) E6 denotes the energy dissipated in the variable damper connected to


Table VIII. Initial design: optimal fixed damper values

Earthquake P‘ P2 P 3

El Centro 7.5 x 106 6.8 x 10“ 1.45 x lo7 Kern County 7.9 x 106 7.6 x 10“ 1.80 x lo7 Puget Sound 9.1 x lo6 8.7 x 10“ 1.71 x lo7 San Fernando 1.72 x 107 1.63 x 107 3.15 x lo7 Western Washington 1.43 x 1 0 7 1.44 x 107 3.20 x 107

the ith floor of the structure, up to time G = 5 sec, in Tables IV(b) and VI(b), F: denotes the peak shear force at the ith floor of the main structure, in Tables IV(c) and VI(c), FL denotes the peak force on the variable damper connecting the ith level of the main and auxiliary structures.

We define the peak acceleration magnitude vector %peak by

the average acceleration vector xavg by

and the maximum interstorey drift vector xd by

Fixed optimal design experiments. As part of initializing the optimal control algorithm for each of our five scaled earthquake ground motions, we computed optimal fixed damping coefficient values that are shown in Table VIII.

Optimal control experiments. The responses of the optimally controlled structure subjected to the El Centro ground motions are shown in Figures 3(a)-(d).

Minimax-optimal design experiments. The minimax-optimal damping coefficient values were found to be

5 = (0.90 x lo7, 0-87 x lo7, 1-70 x lo7) N sec/m

which happen to be nearly identical to the fixed optimal values computed for the Puget Sound earthquake. Figures 4(a)-(e) show responses of the minimax-designed structure to the El Centro earthquake.

Sampled-data moving horizon control experiments. With the sampling time set at T = 0.2 sec., and the damper modulation law given by equations 4-7, we simulated the response of the structure, controlled by the sampled-data moving horizon control law, with co = (0, 0, 0), subject to the ground motions specified by our five scaled earthquake records.

Continuous moving horizon control experiments. With the horizon time set to T = 0.2 sec., we simulated the response of the structure, controlled by the continuous moving horizon control law, subject to our five scaled ground motion records. For the El Centro earthquake, we have included a full set of plots of the responses in Figures 5 (a)-(d).

5.2. Evaluation of numerical results We will now try to interpret the results summarized in Tables 11-VIII.


Interstomy Drilis Optipd Control (EC) I5

I 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-1.5‘

(4 r i c (w) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -m‘

@I r= (SOCOW

4

-10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(4 TEmc(sad.)

Figure 3. Responses of the optimally controlled structure. Solid lines: first floor; dashed lines: second floor; dotted lines: third floor

Available performance gap. Referring to the data for the El Centro earthquake in Table I1 we see that there is a large gap between the cost function values corresponding to optimal control and minimax design, respectively; for the other test earthquakes, the gap is smaller, but still significant. However, referring to Tables IT1 and IV, we see that, with the exception of the El Centro earthquake, the cost function gap does not clearly translate into a corresponding gap in structural response performance. Overall, the performance gap between the realistic, minimax designed structure and the conceptual, optimally controlled structure is fairly small, and thus it may be quite difficult to obtain a feedback control law that does significantly better than the minimax design.

Sampled-data mooing horizon control law. Referring to Tables 11, V and VI, we see that in general, the performance of the sampled-data moving horizon control law is significantly worse than that of the continuous moving horizon control law, and from Tables I11 and IV, we see that it is somewhat worse than that of the minimax design.

Continuous moving horizon control law. We see from Table I1 that, with the exception of the Puget Sound earthquake, the use of the continuous moving horizon control law resulted in cost function values that are roughly half-way between those of the optimally controlled structure and those of the minimax designed structure. However, Tables I11 and V show that the use of continuous moving horizon control law results in


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interstorey drifts and peak floor accelerations that are sometimes smaller than those in the minimax designed structure, but always larger than those resulting from the use of the optimal control law.

From Tables IJ1 and V, we see that on the top two floors, the continuous moving horizon control tends to produce higher average accelerations than optimal control, but lower average accelerations than in the minimax designed structure. On the bottom floor, continuous moving horizon control tends to produce lower average accelerations than optimal control, but higher than in the minimax designed structure.

Referring to Tables IV and VI, we see that the continuous moving horizon control results in energy dissipation in the variable dampers that is higher than when optimal control is used, but lower than in the minimax designed structure.

Referring to Tables IV and V1, we see that for all earthquakes, the structure with the continuous moving horizon control law exhibited larger peak shear and damper forces than the optimally controlled structure, but, for some earthquakes, smaller shear and damper forces than the minimax designed structure.

5.3. Evaluation of minimax design Referring to Tables I1 and IV, we see that the minimax design exhibits only slightly larger optimal cost

function values, peak interstorey drifts, peak accelerations, shear forces, and more energy dissipation in the dampers than the optimal fixed designs, including the El Centro earthquake, which was not used in the minimax design. Thus, our results show that, although the minimax design was based on a very small number

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of earthquakes, it appears likely to result in excellent performance when subjected to earthquakes not used in the minimax optimization.

5.4. Validation of ‘short-time’ experiments To establish confidence in our experimental results that examined only the first 5 sec of the response of

a structure to a ground motion excitation, we performed an optimal control computation, a simulation of the response of the minimax designed structure, and a simulation of the response of the structure controlled by the continuous moving horizon control law, using the first 20 sec of the El Centro earthquake, see Figure 6. Referring to Tables 111, V and VII, we conclude that the 5 sec experiments give an adequate indication of the performance of the various designs over a horizon of 20 sec, and most likely on a longer horizon as well.

6. CONCLUSION

Using the responses to seismic excitation of an optimally controlled variable structure and of a minimax optimally designed fixed structure, we have established an upper bound on the achievable performance and a lower bound on the acceptability of a control system for a variable damping structure. The gap between the upper and lower bounds is rather small, which indicates that to design a control system that results in a variable structure that is clearly superior to a minimax optimal designed fixed structure is a very difficult task indeed. Our numerical results indicate that a controlled variable structure is likely to perform better than a fixed structure in the case of moderate to severe earthquakes. Further reflection leads us to believe that controlled variable structures are likely to perform best at sites, such as landfills and dry lake beds, where resonances can be expected, but the resonance frequency cannot be estimated in advance; hence the design of a reasonable fixed structure would be difficult.

In addition, we have found that even fairly simple minimax formulations of the design of a fixed structure, based on a small number of ground motions, produce very good results, in the sense that the resulting structure meets specifications on interstorey drifts, shear forces, and floor accelerations not only for the earthquakes considered in the design process, but also for other earthquakes scaled to comparable intensity. Sophisticated minimax design techniques (see e.g. References 6 and 7), using direct measures of performance requirements in the form of bounds on instantaneous interstorey drifts, shear forces, accelerations, etc. are likely to produce even better results.

Finally, we have found that a continuous moving horizon control law, using a horizon of only 0.2 sec, produces quite acceptable performance in a variable structure. However, the implementation of this control law will require the resolution of two issues. The first is that of speed of computation, and will require finding suitable hardware as well as developing particularly efficient software. The second issue is that of ground motion prediction over an interval of 0.2 sec. It will be necessary to determine whether such a prediction can be made by using ground motion monitoring sensors located a small distance away from the site.

7. REFERENCES

1. E. Polak, G. Meeker, N. Kurata and K. Yamada, ‘Evaluation of an active variable-damping structure’, Earthquake Engineering

2. T. J. Dehghanyar, S. F. Masri, R. K. Miller and T. K. Caughey, ‘On-line parameter control of nonlinear flexible structures’. in

3. T. Kobori et al., ’Research on active seismic response control system with variable structure characteristics’, J . struct. eng. J S C E 37B,

4. E. Polak and T. H. Yang, ‘Moving horizon control of linear systems with input saturation, disturbances, and plant uncertainty, Part I:

5. T. E. Baker and E. Polak, ‘On the optimal control of systems described by evolution equations’, Memorundurn No: UCBIERL

6. E. Polak, ‘Minimax algorithms for structural optimization’, in Proc. IUTAM symp. on structural optimization, Melbourne, Australia,

7. E. Polak, ‘Nonsmooth optimization algorithms for the design of controlled flexible structures’, in J. E. Marsden, P. S. Krishnaprasad

Research Center Report No. UCB/EERC-93/02, April 1993.

Liepholr H.H.E. (ed.), Structural Conmd, Martinus Nijhoff, Dordrecht, 1987.

133-149 (1991).

robustness’, Int. j . control, 68, 613-638 (1993).

M89/113, Electronics Research Laboratory, University of California, Berkeley, CA, 1989.

1988.

and J. C. Simo (eds), Contemporary Mathematics, Vol. 97, American Math. SOC., Providence RI, 1989, pp. 337-371.

Evaluation of an active variable-damping structure

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