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CUREe- Kajima Research Project Final Project Report
Analytical and Experimental Studies into the Identification and Control
of Intelligent Structural Systems
Mr. Yasuo Takenaka Mr. Norihide Koshika Mr. Hiroshi Ishida Mr. Kazuhiko Yamada Mr. 11asatoshi Ishida
By
Mr. Kazuhide Yoshikawa Mr. Yoshiki Ikeda Mr. Narito Kurata
Report No. CK 91-04 February 1991
California Universities for Research in Earthquake Engineering ( CUREe)
Prof. A. M. Abdel-Ghaffar Prof. Sami F. 11asri Prof. Richard K. Miller Mr. !sao Nishimura Prof. James L. Beck Prof. Thomas K. Caughey Prof. Wilfred D. I wan
Kajima Corporation
CUREe (California Universities for Research in Earthquake Engineering)
• California Institute of Technology • Stanford University • University of California, Berkeley • University of California, Davis • University of California, Irvine • University of California, Los Angeles • University of California, San Diego • University of Southern California
Kajima Corporation
• Kajima Institute of Construction Technology • Information Processing Center • Structural Department, Architectural Division • Civil Engineering Design Division • Kobori Research Complex
Final Report
ANALYTICAL AND EXPERIMENTAL STUDIES INTO THE IDENTIFICATION AND CONTROL
OF INTELLIGENT STRUCTURAL SYSTEMS
Submitted to
CUREe-Kajima Research Project cfo MC 104-44
California Institute of Technology Pasadena, California 91125
by
UNIVERSITY OF SOUTHERN CALIFbRNIA Department of Civil Engineering
Los Angeles, CA 90089-2531
A.M. Abdel-Ghaffar, S.F. Masri, R.K. Miller and I. Nishimura
and
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of Engineering and Applied Science
Pasadena, CA 91125
J.L. Beck, T.K. Caughey and W.D. Iwan
15 February 1991
SUMMARY The active control of large structural systems is a subject of growing worldwide interest. A
testament to the depth and breadth of this interest is the convening of, and the international participation in, the first "U.S. National Workshop on Structural Control Research" which was recently held on the campus of the University of Southern California under the auspices of the National Science Foundation. The proceedings of this Workshop furnish extensive information on the myriad research issues that need much more attention before the full potential of active control approaches can be fully exploited in the structural engineering field.
This report presents some of the results of an ongoing analytical and experimental study into the control of building-like structures subjected to nonstationary random excitations such as earthquakes. The structural model used resembles a 5-story building about 2.5 meters high. The building model was subjected to a variety of direct-force excitations. The control algorithm used employs an adaptive structural member at a pre-determined location in the model in order to attenuate the structural response relative to the moving building foundation. An electromagnetic actuator is used to generate the required control forces in the "smart" member. Among the key features of the algorithm under discussion are:
1. Only one active controller is required to attenuate the vibration response contributed by the first three modes; the damping factor is increased from virtually zero to about 20%.
2. Only two sensors are needed for this algorithm; this leads to simpler instrumentation and a more robust system.
3. Due to the optimization procedure used to select the controller location, a significant amount of damping augmentation is obtained from a relatively small amount of control energy.
4. The whole design procedure was demonstrated; special attention was devoted to the time lag problem of the active controller and the stability of the system.
As part of the design phase of this study, a system identification procedure was used to develop a suitable reduced-order mathematical model. The results of a simulation study using this identified model are compared to experimental measurements. Problems encountered in the experimental phase of the study are reported and discussed. It is shown that (1) the algorithm under discussion is capable of reliably controlling the motion of the test structure under arbitrary dynamic environments, and (2) the features of the algorithm make it a promising candidate for application to large civil structures.
A detailed discussion of some analytical issues related to active control algorithms is provided in Appendix I.
2
Contents
1 INTRODUCTION
1.1 Motivation . . .
1.1.1 Background
1.1.2 Research Objectives
1.2 Project Description ....
1.2.1 Research Overview
1.2.2 Technical Approach
1.3 Scope
2 ACTIVE BRACE CONTROL (ANALYTICAL PART)
2.1 Concept of Active Brace Control .......... .
2.2 State Space Representation of Active Brace Control
2.2.1 Assumption for modeling the problem ..
2.2.2 Equation of motion of the 3 DOF model .
2.2.3 State space representation . . . . . . . . .
2.3 Transfer Function Representation of Active Brace Control
2.3.1 Steady-state response ............ .
2.3.2 Control parameters and structure parameters
2.3.3 Transfer function from disturbance to response
2.4 Optimization of Feedback Gain and Control Position
2.4.1 Frequency response function ......... .
2.4.2 Steady points of frequency response functions
2.4.3 Definition of parameter optimization .....
2.4.4 Parametric study of frequency response functions .
2.5 Numerical Study; Earthquake Excitation ..... .
2.5.1 Analytical conditions and analytical model
2.5.2 Analytical result ........... .
2.5.3 Capacity requirement for active brace
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3 ACTIVE BRACE CONTROL (EXPERIMENTAL PART)
3.1 Specimen Structure and Controller
3.1.1 Specimen structure .....
3.1.2 Electromagnetic controller .
3.2 System Identification of Specimen Structure .
3.2.1 Modal frequencies and modal damping.
3.2.2 Modal vectors ....... .
3.2.3 Estimated stiffness matrix .
3.2.4 Rayleigh's quotient and estimated diagonal system matrices .
3.3 System Identification of Controller . . . . . . . . ..
3.3.1 Transfer function representation of controller
3.3.2 Open loop transfer function . . . . . .....
3.4 State Space Representation of the Comprehensive System
3.4.1 Feedback system in terms of control force .....
3.4.2 Open loop transfer function of a comprehensive system
3.4.3 Comprehensive closed loop system
3.4.4 Feedback gain matrix [H]
3.5 Stability of the Overall system
3.5.1 Characteristic equations
3.6 Controllability of High Frequency Mode Vibration
3.6.1 Controllability matrix .....
3.6.2 Energy response of the system
3.7 Experimental Investigation ..
3. 7.1 Set up of test structure
3.7.2 Results of excitation tests
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A FREE VIBRATION TEST OF SPECIMEN STRUCTURE {1) 63
B FREE VIBRATION TEST OF SPECIMEN STRUCTURE {2) 67
C FINITE ELEMENT ANALYSIS OF SPECIMEN STRUCTURE 69
D TEST OF OPEN LOOP TRANSFER FUNCTION OF CONTROLLER 72
4
Chapter 1
INTRODUCTION
1.1 Motivation
1.1.1 Background
Methods for the attenuation of structural response under seismic excitation, long an active area of
investigation in civil engineering, have been receiving an increasing amount of attention recently
due to several factors: advances in technology, successful demonstration projects, and availability
of new generations of microprocessors, sensors, devices, etc1 •
In recognition of the growing interest in, and importance of, seismic response attenuation, the
9WCEE Steering Committee organized a special theme session titled "Seismic Response Control
of Structural Systems." The session highlighted recent developments in seismic response reduction
and control methodologies, with emphasis on seismic load reduction, seismic load isolation and
seismic response control2 •
A detailed survey of research activities and technical issues associated with active vibration
control of civil structures are reported in two recent papers by Miller et al.3 and Soong4 • Space
limitations preclude a detailed listing, much less discussion, of the myriad research issues which
1 Kobori, T., Kanayama, H., and Kamagata, S., {1988), "A New Horizon in Design Philosophy of Anti-Seismic Structure - Dynamic Intelligent Building With Active Seismic Response Control System," Proc. Ninth World Conference on Earthquake Engineering, Tokyo-Kyoto, 2-9 August, Paper No. SE-11.
2 Masri, S.F., (1988), "Seismic Response Control of Structural Systems: Closure," Proc. Ninth World Conference on Earthquake Engineering, Tokyo-Kyoto, 2-9 August, Paper No. SE.
3 Miller, R.K., Masri, S.F., Dehghanyar, T.J., and Caughey, T.K., (1988), "Active Vibration Control of Large Civil Structures," Jnl. of Engineering Mechanics, Trans. ASCE, Vol 114, No. 9, September, pp. 1542-1570.
4 Soong, T. T., (1988), "State-of-the-Art Review: Active Structural Control in Civil Engineering," JournCll of Engineering Stri1ctur·cs, Vol. 10, April, pp. 74-84.
1
still await investigation. References (2] and (3] furnish an overview of the major research issues in
the dynamic response control of structural systems.
The U.S. National Science Foundation convened the first "U.S. National Workshop on Struc- ·
tural Control Research" at the University of Southern California on 25 and 26 October 1990.
Approximately one hundred participants with a variety of technical backgrounds participated in
the formal presentations and discussions covering current practices, research, and future needs. It
is clear from the Proceedings5 , which present the technical papers and summary statements of the
formal discussions, that numerous technical issues need to be investigated and resolved before the
potential of active control approaches can be fully exploited in the field of structural engineering.
1.1.2 Research Objectives
Even though considerable progress has been made in the field of seismic response control using
various passive and active approaches, additional studies are needed in virtually all aspects of this
field. Many serious technical challenges need to be overcome before wide-spread acceptance and
implementation of passive, active, semi-active and hybrid seismic response control approaches.
The overall research objective of the present ongoing study is to develop on-line identification
procedures suitable for use in active structural control and to develop practical active response
control concepts for large structures under earthquake excitations.
In order to speed the rate of progress in removing the serious impediments encountered in this
field, a multi-faceted joint project between USC and Caltech researchers is underway in order to
contribute to solving some of these problems, with particular application to tall buildings and cable-
supported long-span bridges. Among the specific research topics being addressed by the research
team are:
1. Development and evaluation of promising parameter control techniques related to active brace
control methods. 5 Housner, G.W. and Masri, S.F. (Editors), (1990), "Proceedings of the U.S. National Workshop on Structural
Control Research," 25-26 October 1990, University of Southern California, Los Angeles, CA, USC Publication No. CE-9013.
2
2. Development and evaluation of practical combined isolation/control methodologies suitable
for tall buildings.
3. Identification of the structural parameters of existing (built) structures which are needed
for designing the control actions and analyzing the controlled response, including explicit
treatment of the uncertainties arising from model error and measurement noise.
4. The combined usage of passive devices and active (hybrid) control systems to minimize costs.
5. Devising new control mechanisms for efficient vibration control in particular structures pos
sessing distinguishing configurations.
6. Active structural control of nonlinear systems such as cable-stayed bridges.
7. Development and use of computationally efficient procedures for optimizing the characteriza
tion and control algorithm of active control systems.
8. Evaluation of promising procedures for mitigating the dynamic response of long-span bridges
under arbitrary dynamic environments.
9. Experimental verification of the analytical results.
1.2 Project Description
1.2.1 Research Overview
As previously discussed, the vast number of topics needing research and development to advance
the state-of-the-art of active structural control is a major problem that will require considerable
effort directed to several areas of study by many investigators over a long period of time.
The present study was designed to identify several key issues and to formulate a project of
modest size which will produce results of immediate interest and lay sound foundations for future
developments. The detailed research plan presented in a subsequent section, while admittedly
broad in scope and complexity, is useful as a wide-perspective plan designed to investigate and
resolve some of the key issues that have a strong bearing on the development (eventually) of a
field-implementable methodology for the active control of smart structures.
3
The specific tasks which follow were carried out to the extent of the available resources in the
first year. The budgeted funds formed only a small part of the total resources needed to accomplish
the objectives of the planned two-phase project. The Principal Investigators viewed the budgeted
amount as representing seed money to initiate this project. Although useful results were obtained
from this small first-year effort, it is hoped that additional resources can be made available for
subsequent phases of the project.
Two different research topics consisting of an experimental study and a supporting analytical
component were investigated. The research topics of Phase-! are associated with a newly devel-
oped active control algorithm for large structures. System identification methods were used for
optimization of the control parameters specifically concerned with this control algorithm.
For robust control, it is important to insure stability and adequate performance of the control
system to high reliability, despite uncertainties in the dynamic model. For given maximum capa-
bilities of the control mechanism, the reduction in the uncertainties in the modeling that can be
achieved with system identification leads to a corresponding increase in the performance of the con-
trol system. There will always be residual uncertainties, however, and it is important to quantify
these and include their effects in the analysis of the stability and performance. In the future, this
will be achieved by using a probabilistic framework for system identification and control based on
an extension of earlier work6 •
Recently, one of the PI's has developed a new closed form analytical solution for the non-
stationary random response of general linear systems. This new solution provides a simple, direct,
and computationally efficient means of determining the statistics of the response of linear structures
subjected to earthquake-like excitation. This method works well for classically and non-classically
damped systems and is therefore ideally suited to structures with active control whether this control
is accomplished by external means or by internal parameter adjustments. Such an analytical tool
has not been available previously.
The availability of this new solution methodology has made it practical to consider the problem
6 Beck, J.L., (1989), "Statistical System Identification of Structures," Proc. Int. Conf. Struct. Safety and Reliability, San Francisco, August.
4
of optimizing both the design and control algorithm for actively controlled structures on a statistical
basis. Without the new solution, the optimization problem would require prohibitive computer time
even for the simplest structure and control models. With the new solution, more realistic models
can be analyzed without encountering unmanageable computational problems. By using a random
vibration approach, the uncertainty and random character of earthquake excitation can be taken
into account in the optimization. Deterministic studies have a much narrower range of application.
In view of the above, the promising analytical approach recently explored by I wan and Hou 7
was used to investigate system performance under stochastic earthquake-like excitation. The newly
developed analytical method was also used to study the optimization of control algorithms. The
results of this phase of the study are reported in Appendix I.
Most large civil engineering structures, such as tall buildings and long-span cable-supported
bridges, behave nonlinearly under the dynamic loads effects. The nonlinearity could be geometric
due to large curvature or mid-plane stretching for long-span restrained elements. It could also be due
to the self-excited forces due to wind or to structural material nonlinear behavior. The majority of
the previous applications of structural control technique were dedicated to linear elastic structures.
The USC investigators have begun a study on the nonlinear dynamic behavior of cable-supported
bridges.
In the future, the PI's wish to extend this study to include the control of nonlinear oscillations of
long-span cable-supported bridges, utilizing the newly acquired two shaking-table system at USC's
structural laboratory. To control such effects, one might need to use new control mechanisms.
Therefore, it is planned to compare the uses of traditional control mechanisms, such as tuned mass
dampers, pulse control technique, etc. and modify them for effective nonlinear control.
Among the main topics of the planned Phase-II of this study will be a study on a possible
combination of active and passive devices for structure control. The motivation for a hybrid system
of this type is that the constraints on active devices required for structural control can be reduced
and the control effectiveness, which is not attained by solely passive devices, may be achieved
71wan, W.D. and Hou, Z.K., {1988), "Explicit Solutions for the Response of Simple Systems Subjected to Nonstationary Random Excitation," Stochastic Structural Dynamics, Edited by Namachchivaya, Hilton and \Ven, Proc. Symposium held at the University of Illinois, November, pp. 255-268.
5
through this approach.
1.2.2 Technical Approach
The basic idea of active brace control is to change structural parameters such as stiffness and
damping at a certain location of a building structure by means of an actively controlled mechanism
so that the overall vibration response of the structure due to ground excitation will be effectively
suppressed. For the purpose of this preliminary study, the whole structure was divided into three
components: an upper portion, a lower portion and a controlled portion, depending on the location
of the active brace.
This unique control strategy has several advantages. First of all, constraints upon the active
mechanism, such as force capacity, velocity capacity, power, etc., are small compared to other
methods. One of the main reasons for this is that the inertia force of the upper portion is so
controlled as to act against the response vibration of the lower portion as a counter force. Another
advantage is that the active mechanism does not take much space because of the nature of the
control strategy, which is only local. However, this necessitates the solution of one optimization
problem concerning the optimum location of the active device.
There are, indeed, several optimization problems related to this control strategy. How much
damping and stiffness are most effective? Where is the optimum location of the control mechanism?
In order to answer these questions, system identification is indispensable to obtain a mathematical
model uniquely suitable for this active algorithm.
1.3 Scope
Chapter 2 of this report presents the studies conducted during the analytical phase of the investi
gation of active brace control approaches. The implementation procedure and related experimental
studies are reported in chapter 3. Further analytical and experimental studies are supplied in the
appendices.
6
Chapter 2
ACTIVE BRACE CONTROL (ANALYTICAL PART)
2.1 Concept of Active Brace Control
This report considers the problem of active seismic control for building structures. The proposed
problem is specifically associated with an active controller placed at a certain floor of a building
structure to generate inner force between the two adjacent floors in order to reduce the overall
vibration of the building. A schematic drawing of the system is shown in figure 2.1. Consequently,
the following topics need to be addressed:
• Most effective location of the controller
• Effective control law
• Cost of the controller (power, force, capacity)
• Stability of the entire system
In this chapter, the optimum location of the controller and the optimum feedback gain are
investigated. The control force and energy required for the active brace are also analyzed.
7
2.2 State Space Representation of Active Brace Control
2.2.1 Assumption for modeling the problem
It is assumed that there is an n-story building structure installed with an active brace at a certain
position. It is further assumed that the n-degree of freedom system can be reduced to a 3 DOF
system shown in figure 2.2.
The controller is supposed to adjust the stiffness and the damping at the local point. The
control stiffness and damping are kept constant while the structure is under control. Hence, the
proposed control law is linear, time-invariant feedback control.
In this section we derive the first order differential equation of the system that includes feedback
gain in terms of the control force rather than the control signal. Therefore, the feedback gain is
expressed in terms of stiffness and damping. Ground motion, or earthquake excitation, is regarded
as a disturbance input.
2.2.2 Equation of motion of the 3 DOF model
The controller is assumed to generate linear damping and stiffness at the control point. Hence, the
control force is given as:
(2.1)
where
• he : control damping
• kc : control stiffness
• u(t) : control force
• x2, X3 : response displacement (figure 2.2)
• x2, x3 : response velocity (figure 2.2)
The total story stiffness at the control floor is then given by:
(2.2)
8
/
where
• k 0 : original stiffness
• k2 : total stiffness
Hence, the equation of motion of the 3 DOF model is given by:
[M]{x} + [C]{:i:} + [K]{x} = -[M]{l}x9 = {!} (2.3)
where
[ m, 0
~,] [M] = ~ 0 0
[ k, -kl
0 l [K] = -~1 kl + k2 -k2 -k2 k2 + k3
(C] = [ ~ 0
-~, l he -he he
{x} = Ud {X}={::} {X}= on {/} = {
All notations are given in figure 2.2.
2.2.3 State space representation
After reducing x2, equation 2.3 can be written in a different manner, which is a first order differential
equation or a state space representation.
{i} = [A]{x} + [B]u(t) + [F]Ds(t) (2.4)
9
where
[ [<P] [I] l
[A]= -[M]-I[K] [<P]
[ [<P] l [F] = -[M]-t
[.M] = [ mt 0 l 0 m3
• u( t) : control force
• Ds(t) : disturbance (ground excitation x9
)
• [I] : identity matrix (2 X 2 matrix)
A block diagram representation of equation 2.3 is shown in figure 2.4. We try to control this
system according to the law given by equation 2.5.
u(t) = -[G]{x}
Substituting equation 2.5 into 2.4, we obtained the following equation:
{:i:} ([A]- [B][G]){x} + [F]Ds(t)
[Ac]{x} + [F]Ds(t)
10
(2.5)
(2.6)
Hence, this feedback control shifts the pole locations of state matrix [A] to that of [Ac]. All
we can do by changing the feedback gain is to relocate the pole location of the state matrix. As a
result, if we need to shift the zero location of the transfer function shown in equation 2.7, which is
directly derived from equation 2.6, we must change the [F] matrix.
{x(s)} = (s[I]- [Ac])-1[F]Ds(t) (2.7)
This is the primary reason for the necessity of the optimization of the controller location.
2.3 Transfer Function Representation of Active Brace Control
2.3.1 Steady-state response
Reconsidering the original equation of motion given by equation 2.3, we assume the following har-
monic ground excitation given by equation 2.8 and steady state response expressed in equation 2.10.
(2.8)
or
(2.9)
and
(2.10)
where w is the circular frequency of the harmonic excitation, and Xk is the complex amplitude
of the steady-state response. Substitution of equations 2.9 and 2.10 into equation 2.3 yields:
(2.11)
Equation 2.11 governs the steady state response of the 3 DOF system.
2.3.2 Control parameters and structure parameters
The following substitutions and normalizations are introduced to simplify equation 2.11.
11
Control parameter:
k2 a =
kt (2.12)
{3 he =
2m1Wt (2.13)
Structure parameter:
ml Jt =
ma (2.14)
~ Wt
wa (2.15)
where
2 kt 2 ka w Wt =- wa=- !=-
m1 ma wa (2.16)
Parameter a and {3 are subjected to the control gain, while Jt and ~ are not influenced by the
control force, although they are subjected to the conditions imposed by the location of the active
brace. J.l and~ are completely specified when the structure configuration and the controller location
are selected. Henceforth, they will be referred to as structure parameters. Parameter Jt represents
the ratio between the mass of the upper portion structure and that of the lower portion, while ~
is the ratio between the frequencies of the two. Hence, when the mass matrix and stiffness matrix
are given, the relation between Jt and ~ is specified. As a result, when the active brace location is
selected, J.l and ~ are determined completely.
On the other hand, a and {3 are determined when we select the feedback gain of active brace.
Henceforth, they will be referred to as control parameters.
2.3.3 Transfer function from disturbance to response
Substituting equations 2.13 through 2.16 into equation 2.11, we obtained equation 2.17.
(2.17)
The solution of equation 2.17 is given by equation 2.18, which is the transfer function represen-
12
tation from the disturbance input to the response displacement.
(2.18)
where
Ll = ~{ .!.(1 + o:)/4 - ( ae + .!.ae + .!.(1 + o:))/2 + .!.ae} +
JL JL JL JL
2f3J{.!.J4- <e + .!.e + .!.)!2 + .!.e}i (2.19)
JL JL JL JL
HtU) = ~J2{o:e(l +.!.) + .!.(1- / 2 )(1 +an+ 2/3f3{e(1 +.!.) + .!.(1- J2)}i (2.20) JL JL JL JL
H2U) = ~J2 {o:e(l +.!.) +.!.-.!. / 2(1 +an+ 2/3/3{e(l +.!.)-.!. / 2}i c2.21) JL JL JL JL JL
HaU) = ~/2 {ae(l + .!.) -.!. / 2(1 +a)}+ 2/3/3{e(l +.!.)-.!. / 2}i c2.22) JL JL JL JL
2.4 Optimization of Feedback Gain and Control Position
2.4.1 Frequency response function
From equations 2.19 through 2.22, we can define the following frequency response functions:
(2.23)
(2.24)
(2.25)
where
A = e J4{o:e(l +.!.) + .!.c1- / 2)(1 + o:)}2 JL JL
B = 416{e(l +.!.) + .!.(1- 12n2 J.L J.L
c = e ! 4 {ae(l +.!.)-.!. / 2(1 + o:) +.!. }2 J.L J.L J.L
D = 416{e(l +.!.) _.!. 12}2 J.L JL
E = e ! 4{ ae(l +.!.)-.!. / 2(1 + o:)}2
J.L JL
F = 4J6{eu +.!.)-.!. !2}2 Jl JL
13
P = e{ .!.c1 + a)f4- cae + !.ae + .!.(1 + a))P + .!.ae}2
J.l J.l J.l J.l
Q = 4f2{.!. 14- ce + !.e +.!. )12 + .!.e}2
J.l J.l J.l J.l
2.4.2 Steady points of frequency response functions
In this section we are discussing optimization of the frequency response functions. We require that
the three different response functions have the smallest maximum values over the entire region of
f > 0, under the same given parameter set (a,fJ,JL,(). However, the optimum parameters for
one function do not necessarily minimize the peak of the other functions. Hence, each function is
analyzed independently to compare each result of the optimization.
Equations 2.23 to 2.25 have a special feature in that each function has several points (!0 , y0 ) in
the f- y plane which are not subjected to parameter (3. For example, as shown in figure 2.3(a),
jG1(!)1 has four steady points (fo, Yo ) in the f- y plane that each function goes through regardless
of parameter {3. Hence, when the peak value of this function is equivalent to the largest constant
point, f3 is considered to be optimized. This is because the maximum of this function cannot be
less than the largest steady point that is determined by the remaining parameters. Therefore, the
parameter set (a, p, 0 should be selected in such a way that the largest steady point is minimized
as much as possible. To begin with, we study these steady points for function IG1(!)1. Steady
points must satisfy equations 2.26 and 2.27, if they exist.
2 . IG ( )!2 A(fo) Yo= 1~ 1 fo = P(fo) (2.26)
2 . I ( )12 B(fo) Yo=}!..~ G1 fo = Q(Jo) (2.27)
where A(fo), B(fo), P(fo), Q(fo) are evaluated by f = fo in equations 2.26 and 2.27. It can be
shown that equation y(f0 ) = y0 is always satisfied regardless of (3. In fact, we obtained equation 2.28
from equations 2.23, 2.26, and 2.27.
IG (fi W = A(Jo) + (32 B(Jo) = Y5{P(fo) + f32Q(Jo)} = 2
1 o P(Jo) + f32Q(Jo) P(fo) + f32Q(!o) Yo (2.28)
As shown below, there exists a point that satisfies both equation 2.26 and 2.27.
(2.29)
14
(2.30)
From equations 2.29 and 2.30, we obtained equation 2.31.
(2.31)
This equation can be solved numerically, if parameters a, f,, J.L are provided in advance. There-
fore, we have established that there exist such constant points where y = IG1(J)I is always satisfied
regardless of {3. If function y = IG1(J)I is maximized with respect to f at this steady point, it is
optimized with respect to {3.
The same argument is applied to function y = IG2(J)I and y = IGJ(/)1. This yields equa-
tions 2.32 and 2.33, respectively. These are the key equations that determine the location of the
steady points.
1 1 1 1 1 1 {e(1 +-)--J2 H -(1 + a)J4
- (ae + -ae + -(1 + a))J2 + -e} J.L J.L J.L J.L J.L J.L
{ 2 1) 1 ( 2 1 }2 { 1 4 ( 2 1 2 1 2 1 2 = ± a~ (1 +- -- 1 + a)f +- - f - ~ + -~ +- )! + -c; } J.L J.L J.L J.L J.L J.L Jl
(2.32)
1 1 1 1 1 1 {e(l +-)--t 2H-(1 + a)J4
- (ae + -ae + -(1 + a))J2 + -ae} J.L J.L J.L J.L J.L J.L
{ 2 1 1 2}{ 1 4 ( 2 1 2 1 2 1 2} = ± a~ (1 +-)- -(1 + a)f - f - ~ + -~ +- )! + -c; J.L J.L J.L J.L J.L J.L
(2.33)
2.4.3 Definition of param.eter optimization
As pointed out in the previous section, each response function has several steady points regardless of
{3. Hence, if we could find a parameter set (a, JL, ~) that makes each function take the smallest steady
value simultaneously, it could be an optimized parameter set. However, the optimum parameter
set for one function is not necessarily the optimum value for the other functions. As a result, we
must compromise in this case and come up with another definition of optimization. By definition,
parameter set (a, J.L, 0 is optimized when the following quantity Msum is minimized. Msum is the
summation of the largest steady value of each frequency response function y = IG1 (f) I, y = IG2(f)l,
15
2.4.4 Parametric study of frequency response functions
We restrict our parametric case study by the following equation:
(2.34)
The original building structure has a stiffness distribution and mass distribution in the vertical
direction. Hence, there is always one relation between parameter J.L and ~' even though it is not
similar to equation 2.34. If mass and stiffness are uniformly distributed in the vertical direction, J.L
and ~ should have the relation expressed in equation 2.34. Under various combinations of a and J.L,
each frequency response function was investigated and we finally obtained the following optimum
parameters:
{
aopt = 0.13 J.lopt = 0.90 ~opt = 1.11
(2.35)
Parameter optimization is not sensitive to the variation of a, J.L and ~; hence, the optimum
location of the controller is approximately at the center of the building structure. After several
case studies of parameter {3, we obtained the optimum value for {3.
f3opt = 0.28 (2.36)
Final results of these frequency response functions are shown in figure 2.3, where the dash-dot
line, the solid line and the dashed line indicate {3 = 0.10, {3 = 0.28, and {3 = 0.4, respectively.
As seen from these figures, the optimum value of {3 is approximately 0.28 and it is less sensitive
to frequency than are the other parameters. The expected peak value of each frequency response
function is less than 2.2. Therefore, the expected damping augmentation is approximately 20% or
more.
16
2.5 Numerical Study; Earthquak~ Excitation
2.5.1 Analytical conditions and analytical model
The objective of this section is to evaluate the optimization result obtained in the previous section.
The 3 DOF model shown in figure 2.5 is used for this study under an earthquake type excitation.
The natural.frequency of the lower portion of the 3 DOF system is set at 1.0 Hz, and the calculated
stiffness and mass are given in figure 2.5. Corrected acceleration data obtained at station No.157
during the San Fernando Earthquake of 9 February 1971 was used for input excitation. Time
history of the acceleration and velocity are shown in figure 2.6, as well as the Fourier spectrum of
the acceleration of this ground excitation. The average acceleration method was used for all the
numerical analyses with the time interval being 0.02 second.
2.5.2 Analytical result
The response vibrations of each mass point 1, 2, and 3, indicated in figure 2.5, are shown in
figures 2. 7 through 2.9. The transfer functions of the absolute acceleration at each mass point, with
respect to the input ground acceleration, are shown in figure 2.10. The peak of these functions are
nearly the same as those of the displacement-frequency response functions in figures 2.3
2.5.3 Capacity requirement for active brace
In this section, the capacity required for the active brace mechanism is investigated in the case
of earthquake type excitation. There are three fundamental conditions to be met: control dis
placement, control velocity, and control force. The control displacement is the relative response
displacement between mass points 2 and 3 (figure 2.2). If the story drift is larger than the device
capacity, the controller cannot generate adequate control force. The control velocity, which is the
relative response velocity between mass points 2 a.nd 3, is another important factor, since this
criterion is associated with the controller's necessary power limits. Obviously, the control force is a.
significant factor to take into account. In fact, the incentive to use active brace control comes from
the need to reduce the magnitude of the active control force. Taking a.dva.nta.ge of the counter force
associated with the inertia force of the upper portion of the structure against the lower portion of
17
the structure, we can reduce the contr.ol force.
The 3 DOF model shown in figure 2.5 is used for numerical analysis of the capacity requirement
for active brace control under random excitation. The input ground excitation is the same as that
used in the previous section. The original stiffness at control point k0 is assumed to be k2 • The
control force, control velocity, and control displacement are calculated as
{
fe = ~e(X2 .- X3) Ve = X2- X3
de= X2- X3
where fe is the control force, de is the control displacement, and Ve is the control velocity.
(2.37)
Response time histories of the control force, velocity, and displacement are shown in figure 2.11.
An interesting feature of the response control force or control velocity is its resemblance to the time
history of the ground velocity. From figures 2.6b and 2.11a, we recognize this similarity, although
there is some time-lag. The peak value of the required control force is as little as 3% of the total
weight of the building structure. The maximum control velocity obtainable from conventional
hydraulic actuator is estimated to be less than 50 em/sec. The peak control velocity calculated
here (18cm/sec) is far less than this limit. The maximum control displacement obtained here is
less than 5 em, which is also acceptable for the story drift in a building structure. However, these
control requirements are only for this special case study, where the natural frequency of the lower
portion of the structure is 1.0 Hz. Hence, it is necessary to know how these requirements change
as the stiffness of the structure varies. In other words, the response spectra of the control force,
control velocity, and control displacement should be obtained.
In this chapter, we have normalized the input harmonic excitation with respect to w3 ; conse-
quently, we will plot the control response spectrum with respect to w3 again. As w3 varies, peak
values of the control displacement, control velocity, and control force are calculated and plotted
in figure 2.12, where ko = k2 • Of course, these control spectra represent just one example under
a special ground excitation. Nevertheless, it is qualitatively seen from figure 2.12 that the control
velocity spectrum is almost fiat over the entire region between 0.3 and 2.0 seconds, while the control
force increases in proportion to w3 and the control displacement decreases in proportion to w3 •
As pointed out previously, there is a strong correlation between the control velocity and the
18
excitation velocity. Therefore, the peak of the control velocity response spectrum might be related
to the peak of the ground velocity. Hence, the following relation is implied by figure 2.12 for any
ground excitation.
maximum control force <X maximum ground excitation velocity
Therefore, we can roughly estimate the control force requirement and velocity requirement
according to the above relation.
19
Upper portion of structure
Controlled floor
Lower portion of structure
Floor of building structure
Column of building structure
controller
Force controller power supplier
Figure 2.1: Concept of active brace control
20
Upper portion
Active brace
Lower portion
~~~~~~ original structure reduced 3-DOF model
xg ground excitation
x1 displacement at the top floor of upper structure
x2 displacement at the control floor
x3 displacement at the top floor of lower structure
ml equivalent mass of upper structure
m3 equivalent mass of lower structure
k1 first mode equivalent stiffness of upper structure
"-3 first mode equivalent stiffness of lower structure
kz total stiffness
he control damping
Figure 2.2: Reduced model of the structure with an active brace
21
10~---r---,,---,----.----.----.----r----r---.----.
~ :~:~j~~~~.~~~2f~:;~i~~:~~f,~.:~·,:l~~~t;_: 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(a) Hz
1 1.2
(b) Hz
3~---r----.---~----~--~----~--~----,----.--~ ... ·.,; ; ~-. f ~ \. : :
2 ------------+·····-t···t····::><t:~-~-'-·;~:<·······--:_:_--------------+--------------+---------------r· -------------[,.------- ----: "' :
: .' :,' : : . ........... ]?~::>·'··············T······· ...... , .............. _....... --'-·-::·~:~~~:~-~.
OL-~~--~~--~--~----L----L----~--~--~--~
0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 (c) Hz
5~---r---.,---.----.----.----.----r---,----,----. ~ ... ,· ..... _
4 ··············+················~················}················f················h:,t ........ ~:";·~·················~················~················~·············· : : ; "1- : :
3 ............... _ ............. -~-' ················=_: .............. :_, .......... _ .• l_t·--· ········]· :-.,;········-~·-·--·········+ .. ···········_!······· .... . ''· ..... \
2 :: :: r :.:-t:::~~~;:::-:~-f::::::r::::::f· r : ;: OL---~~~~--~--~----L----L----~--~--_J--~
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(d) Hz
Figure 2.3: Frequency response (a) y =I Gl(f) I (example case) (b) y =I Gl(f) I (optimum case) (c) y =I G2(f) I (optimum case) (d) y =I G3(f) I (optimum case)
22
feed back gain
1 D(t) disturbance
l~l
"'>-{ .... x}--~
Figure 2.4. Block diagram of controlled structure
XI mass ml point-1
kl
mass point-2
mass point-3
.. X
~~
reduced 3-DOF model
Figure 2.5: 3-DOF model for numerical calculation
23
ml= 0.90
m3= 1.00
kl 43.86
k2 5.702
k3 39.438
he 3.515
200.-----.-----.-----.---~.----.,----.-----.-----.-----.-----,
. . 100 . . ·············· ··············· .... ········· ··············T··············· ··············T··············· ............... ··············· ··············
. . . . . .
-100 ··············l················j········ ·······!·········· ····r···············j················j················ ................ ! ............... "! ............. . : : : :
~00~--~----~----~--~----~----~--~----~----~--~
0 2 4 6 8 10 12 14 16 18 20 (a) sec
20~--~----~----~-----r----~----~----~----~--~-----,
10
0
•••••••••••••• -~- •••••••••••••••• ; • • • • • • • • 0 ••• ; ••••••••••••••• ; • • • • • • • • • • • • • • • ~... • ••••••••••• ~- •••••• 0 •••••••• -~- ................ ; •••••••••••••••• ; ••••••••••••••
: : : : . : : : : -10 : : : : : : : : : : : : : : : : : : : : : : :
-20L---~----~----~----~----~--~----~----~----~--~
0 2 4 6 8 10 12 14 16 18 20
(b) sec
100r-----~------~----~------~------r------r------~-----,
so ········ ·········-~-- ··················~·····················I·····················~·-···················;·····················!····················+·················· . . . . . . . . . . . . . . . . . . . 60 ................. : ................... ~ ................... + ................... j ..................... : .................... ; ..................... : .................. .
• • 0 • • . . . . . . . . . . . . . . . . . . 40 .......... ·······i·····················=·····················i·····················:·····················i·····················=··················· . . . . . .
: : : : : : : : : : : : . : : : : :
20 . . . . . .. . . .. . . .. ; .................... ~ ..................... ; .................... ~ .................... :· ................. . : : : : . . . . . . . . . . .
2 4 6 8 10 12 14 16
(c) Hz
Figure 2.6: (a) Acceleration of excitation (emf sec2 ) (b) Velocity of excitation (emf sec) (c) Fourier spectrum of acceleration (emf sec).
24
-5 ............................. ················~······· .. ····~················ ···············~················~················~··············· ·············· . . . .
-10~----~--------~--------~--------~----~--------~--------~--------~----~--------~
0 2 4 6 8 10 12 14 16 18 20 (a) sec
4r-----.---~.---~,---------~-----.-----.-----.-----.----~
-2 ···············=·················=·················>······ ....... , ................ , ......... ·····:·················=·················>······· ....... , ............. . : : : : : : : : : : : : : : : : : : . . . . . . . . . . . . . . . . . . . . . . . .
-4 .............. .: ................. :. ................ ~ .............. : ................ ; ................ ; ................ .=. ................ :. ................ :. ............. . : : : : : : : : : : : : : : : : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-6~--~----~----~----~----~----~--~~--~----~----~ 0 2 4 6 8 10 12 14 16 18 20
(b) sec
4.---~.----.-----.----~-----.-----.-----.-----.-----.----~
. . . . 2 ··············~·················:················=··········· . ··:················:················:················~·················:················;·············· . . : : : : : : . . . . . . . . . . . . . . . . . . . . . . . .
-2 ···············~········ .... ·····~······· . ······~ .......... ····!··· ·············!················~·················~·················~············· ····~···· .......... ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : : : . : : : : : : : . . . . . .
-4~--~~--~----~----~----~----~----~-----L-----L----~ 0 2 4 6 8 10 12 14 16 18 20
(c) sec
Figure 2.7: Displacement response (em) (a) at mass point-1 (b) at mass point-2 (c) at mass point-3
25
40.---~----~----~----.-----.----.----~----.-----.----.
20 ··············~·················~··············· ......... . . . . . . .
-20 ···············~·················~······· ········~····· . ········~················j--··············j················ ···············-~··············· ·············· : ; ~ : ;
: : : . . . ~~--~----~----~----~----~--~----~----~----~--~
0 2 4 6 8 10 12 14 16 18 20 (a) sec
30.---~-----r----.----.,----.----.----.-----.----.----.
. . . . . . . 20 ............... : ................. ; ................ ; ............... j .................................. , .........•...•... : ..............•.• ; ..•.•...•.•..... ~ ............. . : : : : : : : : : . . . . .
• • 0 • • . . . . . • • • 0 • . . . . .
10 ··············t···············-r················[·········· .... , ................ , ................ j················t···············+················!·············· . . . . . .
: : ......... ····-~-- ............ ··-~····· ........ -~ ............ --~- ........ ·······~········ ....... -~-- .............. ~ ............ ····~···· .......... --~ ............. . -10
: : : : : : : : : : : : : : : : : : : : :
-20~--~----~----~----~----~--~----~----~----~--~ 0 2 4 6 8 10 12 14 16 18 20
(b) sec
-10 ·············<······ ,.~ ...... , ........... ' ,, .............. , .............. , ....... , .......... .
: : : : : : : : : . . . . . . . . -20~--~----~----~----~--~----~----~----~--~----~
0 2 4 6 8 10 12 14 16 18 20
Figure 2.8: Velocity response (cmfsec) (a) at mass point-1 (b) at mass point-2 (c) at mass point-3
26
100.----.----.----.----.---~~--~----.----.----.----.
50
-50 ·············+···············+················}············ . . . . . . . . . . : : : : . .
-100~--~----~----~----~--~----~----~----~--~----~
0 2 4 6 8 10 12 14 16 18 20
(a) sec
60~--------~----.----.----~----.----.----.-----.---~
40 ··············-=·················=-················i········. ·····1················•················\················<\·················=-················i·············· ~ ~ ~ ~ ~ ~ ~ : ~ f ~ ~ ! ~ ~ ~ ~
20 ··············j·················~······· . ···~····· .... ····=················j······· . ······j················~················~················=·············· . . . . . 0 • • • • . . . . . . . . . . . . . . . . .
o~-~· . . .
-20 .............. : ................. :. .............. :....................... . . . ... : ................ : ................ ,;. ................ :. ................ : ..... ········· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -40~--~----~----~----~----~----~--~~--~-----L----~
0 2 4 6 8 10 12 14 16 18 20
(b) sec
150.----.----.-----.----.----.-----.----.----.-----.----,
100 . . . . . . . . . . . . .. . . . . . . . ·=· ............... ·:· ............... •,•..... . ........ : ................ ~ ................ ·:-- .............. ·=· ............... ·=· ................ : ... . . . . . . . . . . . . . . . . . . . . . : . : : . . . . .
2 4 6 8 10 12 14 16 18 20
(c) sec
Figure 2.9: Acceleration response (cmfsec2 ) (a) at mass point-1 (b) at mass point-2 (c) at mass point-3
27
2.5r----.-----.----~----~----.----.-----.----.-----.----.
2 ············· : .............. ......... : ..... ················~················~················j················ ················:·················f·············· . . .
1.5 ..... ·······~········· ······ ............... ···············-j-···············i···············+·············· ··············-r···············l·············· 1
0.5
0 0
............ ·j· .............. ··r . .. . . . ... .. . . . ............... ~- ............... !· ............... j·. .. .. .. . . .. . . . . . .............. ·r ............... ·~ ............. . : : : :
··············t···············r········· ··· ················l················\················l··············· ................ 1················1··············
0.5 1 1.5 2 2.5 3 3.5 4 4.5
(a) Hz
5
2~--~~--~----~----~----~----,-----~----r---~-----.
1.5 ······ ······1· ···············~················j················j················~················r··············r···············r··············!·············· . . . . . . . .
1 ············i······ ········t················i················1················j················J················r················[················j··············
. ~ . ~ : ~ ~ 0.5 ··············t·········· ····j················(·············~···············-r·············-r··············r·············r·············r············
Ol_~~--1L~====~===~:~:~~~~~~~ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
(b) Hz
2~----~----~--~----~----~----~----,-----~----r---~
. . . 1.5 ........... : ................ ; ................ , ................ ; ................ , ................. , ................ , ............. .
. . . . . . . . . . . . . . . . : ; : : . . . . . . . . . . . .
1 ....... ·······~·· ............... t············· ·~···· ........ ····~··· ·············j················j···· ············;······· .... ······t················t· ............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : : : : : : . . . . . . . . . . . . . . . .
0.5 ··············-~·-···············~·················}·········· ····~·-··············~················~·-··············-~················-:·················~·············· : : : : : : : : : : : : : : : : : : : :
OL_~'--_L'--~'--~--_c==~·~~,C=~~~= ~ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
(c) Hz
Figure 2.10: Transfer function of acceleration (a) at mass point-1 (b) at mass point-2 (c) at mass point-3
28
0.04 .------.----.------r---.-------,-----.---.-------.--.-----,
0.02
o~-----·
. . -0.02 ··············t··············r······· . ··---~---··· .. ······1·············-··t·············· ················j················ ···············-r·············
: : . . -0.04 L...----L---'----"'------'----'----.__ _ __._ __ _._ __ ..____---J 0 2 4 6 8 10 12 14 16 18 20
(a) sec
20.---~-----.----.-----.---~-----r----.-----.----.----~
10
o~-----·
. . . . . . . -10 .............. -:· ............... ·:- ............. -~ .............. : ................ ~- ............... -:· ............... -:· ............... -~ ................ ~ ............. .
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
-20~---'----'----.__--~-----'-----.__ __ __._ __ _.__ ____ .__ __ ~ 0 2 4 6 8 10 12 14 16 18 20
(b) sec
4.---~----~----~----~----~----~--~.---~----~----~
2 ···············~·· .......... ·····~······ . .
-2 ···············~·················~··········· . ···i ............ ··i··· ........ ·····~········· ·····:·········· ·······~··· ........ ······:············· .... , ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : : : : : : . . . . . . . . .
-4 ··············~·················=-················:········· .. ···=················:·-··············:················~················=-················=·············· : : : : : : : : : . . . . . . . . . . . . . . . . . . : : : : : : : : : . . . . . . . . .
-6~--~----~----~----~----~----~--~~--~----~----~ 0 2 4 6 8 10 12 14 16 18 20
(c) sec
Figure 2.11: (a) Control force (x total weight of structure) (b) Control velocity (cmfsec) (c) Control displacement (em)
29
20.----------.---------.----------.----------.---------.
15 ································t·································!••······························t .. ······························ ······························· . . . . . . . . . .
10 ································!·································~·································i····· ························· ······························· . .
5 ·································~·-·······························: ·····························-=································· .............................. .
0~--------~----------~----------~----------~--------~ 0 0.5 1 1.5 2 2.5
(a) sec
25~--------~-----------r----------.-----------r---------~
. . . . 20 ................................ , .................................. ;·····························:···r·······························r······························
15 ··················· ···········t·································~································ . . ............................ : .............................. .
10 ................................ ~ .................................. ; ................................. , .................................. , .............................. . : : : : : : : : . . . . 5 ................................ ; ................................. i ................................. , .................................. ; .............................. . : : : : : : : : :
OL---------~----------~----------~----------~--------~ 0 0.5 1 1.5 2 2.5
b
0.1~--------~----------~----------~----------~--------~
0.08
0.06
0.04
. . . ······················· ········=··································!··································=·······························-··=············· ·················· . . . . . . . . . . . .
: : ································=- ·········-·····················=·································:·-································=························"····· . . . . . . .
: . ; . . : : ............................... -~·.... . . . . . . . . . . . . . ............. ! ................................. ·: ................................. -~· .............................. . : . : : . . . . . . . . . . . .
0.02 ................................. f ................................. ~ .......................... . . . . . . . . . . . . .
OL---------~----------~----------~----------~--------~ 0 0.5 1 1.5
(c) sec
2 2.5
Figure 2.12: Response spectrum of (a) the control displacement (em) (b) the control velocity (em/sec) (c) the control force (xTotalweight)
30
Chapter 3
ACTIVE BRACE CONTROL (EXPERIMENTAL PART)
3.1 Specimen Structure and Controller
3.1.1 Specimen structure
Based on the parameteric study in chapter 2, a 5 DOF specimen structure was designed to eval-
uate the control efficiency experimentally. The final model is shown in figure 3.1. According to
the parametric study in chapter 2, the controller's position was selected at the center of the spec-
imen structure. A finite element analysis was conducted in advance (see Appendix C) and the
corresponding reduced stiffness matrix and mass matrix are given below:
[M] = 0 0 [
120 0 ~ l (kg) 0 0 130
[
33.5
[K] = ~33.5 -33.5 39.1 -5.6 ~5.6] (kgffcm)
39.1
Hence, structure parameters are:
{
J.l = 0.92 ~ = 0.96 Q = 0.167
(3.1)
(3.2)
(3.3)
The original stiffness of the control story is k0 ; thus, even if the control stiffness is 0, a=0.167
which is close to the optimum value for a. Therefore, the control force is to be designed proportional
only to velocity. The optimum value of f3 is 0.28, so he= 1.13 kgf/sec/cm.
31
3.1.2 Electromagnetic controller
The controller employed for this experimental study was an electromagnetic shaker. The controller's
basic principle is exactly the same as an audio speaker. It is composed of an inner armature coil
and a surrounding permanent magnet. The armature coil is capable of sliding unilaterally along
with a counter weight depending on the input voltage. Due to the uniformly distributed magnetic
field surrounding the armature coil, it moves according to the input current that is generated by
the power amplifier. This power amplifier generates a current which is proportional to the input
voltage. As a result, the generated control force is proportional to the input voltage. Figure 3.4
shows the dimension of the controller.
3.2 System Identification of Specimen Structure
3.2.1 Modal frequencies and modal damping
A hammer test was conducted to set the specimen structure under free vibration. Subsequently,
the acceleration of each mass point was recorded in digital form by using an A/D converter. The
results obtained from the experiments are given below. Details of the data processing and the
experiment are reported in Appendix A.
WI = 5. 7680radf sec w2 = 18.868radf sec w3 = 40.952rad/ sec w4 = 61.053radf sec ws = 71.790radfsec
hi= 0.5% h2 = 0.7% h3 = 0.3% h4 = 0.3% hs = 0.3%
where Wi and hi are the ith modal frequency and modal damping, respectively.
3.2.2 Modal vectors
(3.4)
A hammer test was conducted to determine the modal vectors as well. Details of the test are given
in Appendix B. The results of the test are given in equation 3.5.
32
1.000 -0.232 1.000 -0.053 -0.592 0.983 -0.178 0.048 0.043 1.000 0.955 -0.089 -0.807 0.015 -0.502 (3.5) 0.185 1.000 0.070 -0.271 0.066 0.098 0.571 0.067 1.000 -0.069
Experimentally obtained modal vectors are complex numbers under ordinary circumstances.
However, the absolute values of these complex numbers can be accurately estimated by neglecting
the damping matrix. In other words, the real modal vectors are obtained with high precision by
considering only the absolute values of complex modal vectors.
3.2.3 Estimated stiffness matrix
It is necessary to specify the modal masses which are associated with [U] matrix given by equa-
tion 3.5. The modal masses were obtained by utilizing Rayleigh's quotient given by equation 3.6.
2 { ui}t[K]{ u;} k; W·- --
' - { u;}t[M]{ u;} - m; (3.6)
Although we don't know exact [K] and [M] matrices, we can estimate the [K] matrix by means
of FEM analysis. The estimated stiffness matrix [Ke] is given by equation 3.7. Details of the FEM
model for this calculation are given in Appendix C.
66.268 -66.898 0.622 0.003 0.003 -66.898 134.299 -67.503 0.104 -0.001
[Ke] = 0.622 -67.503 72.396 -5.623 0.102 (kgf /em) (3.7) 0.003 0.104 -5.623 72.402 -67.465 0.003 -0.001 0.102 -67.465 129.148
3.2.4 Rayleigh's quotient and estimated diagonal system matrices
Rayleigh's quotient has a stationary value in the vicinity of an eigenvalue. If the estimated stiffness
matrix is close to the exact value, then equation 3.8 should yield a stationary value at (w£)2 which
is given by
(3.8)
where { ui} and w£ are the ith eigenvector and eigenvalue for the sy~tem of [Ke] and [M], respectively.
Here we expect that w£ is close tow;. As shown below, it is a good approximation.
33
It is assumed that the exact [K] matrix is composed of [Ke] and [b'K].
[K] = [Ke] + [oK] (3.9)
Substituting equation 3.9 into 3.8, we obtained the following equation:
(3.10)
or
(3.11)
The left hand side of equation 3.11 is supposed to have a stationary value in the vicinity of
eigenvalue Wi because of equation 3.8. But the second term of the right hand side of equation 3.11
does not have a stationary value around this eigenvalue, because the system of [M] and [b'K] does
not necessarily have an eigenvector { Ui}. Hence, it should be a small value, otherwise the left hand
side of equation 3.11 would not have a stationary value in the vicinity of this eigenvalue. As a result,
we can approximate wi by Wi· Finally, we obtain the following diagonal matrices by substituting
equations 3.4 and 3.5 into 3.8.
kl 0 0 k2
[Kd] = 0 0 0 0 0 0
where ki and mi are given as:
0 0 0 0 0 0 k3 0 0 0 k4 0 0 0 ks
kl = 4.3378 k2 = 39.1731 k3 = 112.4562 k4 = 171.8311 ks = 325.0559.
ml 0 0 m2
[A1d] = 0 0 0 0 0 0
m1 = 0.130400 m2 = 0.110000 m3 = 0.067040 m4 = 0.046099 ms = 0.063071
0 0 0 0 0 0 m3 0 0 (3.12) 0 m4 0 0 0 ms
A diagonal damping matrix is also obtained by substituting equation 3.4 and 3.12 into 3.13.
Ct 0 0 0 0 c1 = 2m1w1h1 0 c2 0 0 0 c2 = 2m2w2h2
[Cd] = 0 0 C3 0 0 [Cd] = c3 = 2m3w3h3 (3.13) 0 0 0 c4 0 c4 = 2m4w4h4 0 0 0 0 cs c5 = 2m5w5 hs
34
According to equations 3.12 through 3.16, we can retrieve the stiffness, mass, and damping
matrices given by equations 3.17, 3.18, and 3.19, respectively.
[K] = [U-1]t[Kd][u-t] (3.14)
[M] = [U-l]t[Md][u-1] (3.15)
[C)= (U-1]t[Cd)[U-1) (3.16)
62.289 -62.699 0.501 0.433 -0.247 -62.699 135.533 -72.203 -1.219 1.434
[K] = 0.501 -72.203 76.71 -5.308 -1.765 (kgf /em) (3.17) 0.433 -1.219 -5.308 69.529 -65.485
-0.247 1.434 -1.765 -65.485 130.315
34.190 1.168 -0.220 -0.224 -0.055 1.168 41.665 2.261 -2.027 0.242
[M] = -0.220 2.261 47.461 3.751 -0.517 (kg) (3.18) -0.224 -2.027 3.751 88.890 1.905 -0.055 0.242 -0.517 1.905 39.452
0.0077 -0.0042 -0.0012 -0.0012 -0.0004 -0.0042 0.0124 -0.0051 -0.0018 -0.0002
[C)= -0.0012 -0.0050 0.0097 -0.0004 -0.0003 (kgffcmfs) (3.19) -0.0012 -0.0018 -0.0004 0.0253 -0.0015 -0.0004 -0.0002 -0.0003 -0.0015 0.0142
Summation of every element of matrix [.M) to be less than the total mass of the structure, yet it
should be close to the total mass because there are some lateral vibration due to higher frequency
modes than the 5th mode, that should be negligible.
"' [m· ·] = 264.3kg ~i,j IJ
< mtotal = 266.9kg
The total weight of the structure, including the controller's mass, is 266.9kg ( 588.4lb ), with the
controller's weight being 36.5 kg.
Some of the mass matrix elements are negative. Summation of these negative values is about
2% of the total weight; hence, the presence of these terms could be regarded as an error due to the
experimental measurements and estimation of the trial stiffness matrix [Ke]·
35
3.3 System Identification of Controller
3.3.1 Transfer function representation of controller
It is imperative to identify the dynamic characteristics of the controller, because a comprehensive
system design needs a system model. Because the basic principle of the controller is exactly the
same as that of an audio speaker, the force generated by the controller is assumed to be proportional
to the input current. Therefore, it is basically a force controller rather than a position controller.
It is further assumed that the system is linear, so there exists a transfer function representation.
This transfer function is defined as the ratio of the output to the input: the input voltage to the
amplifier of controller is the system input, and the generated control force is the system output.
Based on the knowledge obtained from the experiment, the equation of motion which governs the
controller's behavior is expressed as
u(t) = mfi(t) + hy(t) = Ru8 (t) (3.20)
• u(t) : generated control force
• y(t) : relative displacement of armature coil
• if( t) : relative velocity of armature coil
• fi( t) : relative acceleration of armature coil
• u 8 (t) : input voltage
• m : mass of armature with weight
• h : damping coefficient
• R : static force coefficient
Equation 3.20 is expressed on the frequency domain as
Y(s) = R U(s) ms2 + hs
(3 .21)
36
where Y( s) and U(s) are the Laplace transforms of the displacement and input voltage, respectively.
The coefficients are given in equation 3.22. The following section gives the background for coefficient
m,R, and h.
{
m = 0.012 R = 0.060 h = 1.80
3.3.2 Open loop transfer function
(kgf fcmfsec2 )
(kgf /emf sec) (kgf fvol)
(3.22)
The open loop transfer function from input voltage to output acceleration, or s2 H(s), is shown in
figure 3.5a. The transfer function of H(s) is shown in figure 3.5c. Experimentally obtained results
are denoted by an'o' in those figures. From those magnitude transfer functions, the following
relations are obtained:
lim s2IH(s)l = R = 150 s-+oo m
(3.23)
lim siH(s)l = Rh = 30 S-+0
(3.24)
The weight of the armature coil is obtained from the equipment catalogue, while the other at-
tachment weights were actually measured, with the total weight being 11.77 kg or 0.0120 kgf/cmfsec2.
Finally, the transfer function of the acceleration is given by equation 3.25.
s2 H(s) = l50s s+5
(3.25)
The experimental results are denoted by an 'o' in figure 3.5, while the solid line is a calculated
result from equation 3.25. The static load coefficient was also checked by applying direct current
to the amplifier. The result was 1.77 kgfjvol that was well corresponding to R in equation 3.22.
3.4 State Space Representation of the Comprehensive System
3.4.1 Feedback system in terms of control force
The original specimen structure without the controller is expressed in a block diagram representa-
tion shown in figure 3.2, from which the equation of motion is given in equation 3.26.
{ {±} {Z}
= [A]{x} + [B]u(t) =[L]{x}
37
(3.26)
where
[ [<t>] [A]= -[Mt1[K)
[I) l -[Mt1[C]
(3.27)
[ [<t>] l [B] = -[M]-1[b] (3.28)
[L) = [ ~ 0 1 -1 0 0 0 0 0 ~ l 0 0 0 0 0 0 1 -1 (3.29)
{x} = {x1 x2 X3 x4 xs x1 x2 x3 x4 . }t xs (3.30)
{z} = {x3- X4 . . }t X3- X4 (3.31)
{b} = {o o 1 -1 o}t (3.32)
u(t) : control force
[M), [K), [C) : given by equations 3.17 through 3.19, respectively
[I) : identity matrix (5 X 5 matrix)
We wish to control this system according to the law given by
u(t) = -[G]{x(t)} (3.33)
where
[ G] = [0 0 91 - 91 0 0 0 92 - 92 0] (3.34)
This can also be written as
(3.35)
The optimum feedback gains, 91 and 92 , are determined according to the discussion in sec-
tion 3.1.1 and are given below. The feedback system block diagram, including the control law, is
shown in figure 3.3.
{ 91 = 0.0 92 = 1.13
(3.36)
If an active device such as an actuator is used as an open loop controller, the phase lag of this
system becomes a time lag which might cause an instability problem. Apart from this problem,
it is necessary to consider the dynamic effects of the controller itself. Hence, it is imperative to
38
construct a mathematical model that describes the dynamic behavior of the whole system, to obtain
the following equation:
(3.37)
where u_,(t) is the input control signal (not control force).
3.4.2 Open loop transfer function of a comprehensive system
Equation 3.37 contains the dynamic effect of the controller as well as that of the structure. Conse
quently, (A1] is different from (A], which is free from the controller's dynamics. The block diagram
representation of the open loop system is shown in figure 3.6, where the controller's dynamics are
considered. From figure 3.6, we can derive the following equation:
or
{x} = [A]{x} + [B]([Co]{z} + Rus(t))
{x} =([A]+ [B][Co][L]){x} + [B]Rus(t)
Equation 3.37 must be identical to equation 3.39, so we obtain
Hence, we obtain
[At] = [A]+ [B][Co][L]
[Bt] = [B]R
From figure 3.6, we know that { z} must satisfy the following relation:
{z} = [A0][L]{x} + [Bo]us(t) ( because [L]{x} = {z})
= [L]{x}
= [L][AJ]{x} + [L][Bt]us(t)
39
(3.38)
(3.39)
(3.40)
(3.41)
(3.42)
(3.43)
(3.44)
(3.'15)
Equations 3.43 and 3.45 must also be identical, so we obtain
(3.46)
(Eo] = (L](B1] (3.47)
From equations 3.40 and 3.41, we can construct the [A1] and [BI] matrices. We can then deter
mine the [Ao] and [Bo] matrices from equations 3.46 and 3.47. Next, we create the feedback signal
from { z} instead of { x}, because we have a relation between { x} and { z} given by equation 3.26
and 3.29.
3.4.3 Comprehensive closed loop system
From figures 3. 7 and 3.8, the feedback gain [ H] should satisfy the following equations:
[Ac] = [Ao]- [Bo][H]
[Cc] =[Co]- R[H]
where [Ac] and [Cc] are referred to in figure3.8
(3.48)
(3.49)
The final resulting feedback system is shown in figure 3.7, from which we know final system
dynamics are expressed as
Hence, we demand that
Consequently,
{x} = ([A]+ [B][Cc][L]){x}
([A]- [B][G]){x}
{z} = [Ac]{z} = [L]{±}
[Ac] = [L]([A]- [B][G])[L]t([L][L]t)- 1
40
(3.50)
(3.51)
(3.52)
(3.53)
(3.54)
The final feedback gain [H] can be derived from either equation 3.48 or 3.49. Both yield the
same result:
(3.55)
If the controller's dynamics are known, then [A1] and [B1] are available so that [H] is determined
by equation 3.55.
3.4.4 Feedback gain matrix (H]
From the result of the open loop controller test, we can obtain R and thus approximate matrix
[C0]. Because of the small amount of weight due to the armature coil (2.2 kg), the controller's
dynamics for the low frequency region can be approximated by equation 3.56, which is derived
from equation 3.20
u(t) = Ru5 (t)- hi;(t) (3.56)
where u(t) is the control force.
Considering the bracing geometry, we relate the controller's relative displacement y( t) to vector
{z(t)}.
(3.57)
Hence, [Co] matrix should be
[Co] = [0.0 0.80h] (3.58)
From equations 3.40, 3.41, 3.55, and 3.58 we obtained the feedback gain matrix [If].
[H] = [0.0 0.640] (3.59)
This is the final gain matrix based on the feedback gain [G] expressed in terms of voltage.
41
3.5 Stability of the Overall system
3.5.1 Characteristic equations
Refering to figure 3.6, the open loop transfer function of the controller is given by equation 3.60.
s[I]{z(s)} = [A0 ]{z(s)} + [Bo]u,(s) (3.60)
u(s) = [Co]{z(s)} + Ru,(s) (3.61)
Substituting equation 3.61 into 3.60, we obtain equation 3.62.
u(s) = ([Co](s[I]- [Ao])-1[Bo] + R)us(s) (3.62)
Hence, the transfer function of {x(s)} from Us(s) is given by
{x(s)} = (s[I]- [A])- 1 [B]([Co](s[I]- [Ao])-1[Bo] + R)us(s) (3.63)
The characteristic equation of the overall system is given by
F(s) = 1 + [G](s[I]- [A])- 1 [B]([Co](s[I]- [Ao])- 1 [Bo] + R) (3.64)
Refering to equations 3.40 and 3.41, we can obtain [Ao] [Bo] matrices from equations 3.46 and
3.47.
[Ao] = [ -1~31 -~.9] (3.65)
[Bo] = [ 6g.7] (3.66)
By substituting equations 3.65, 3.66, and 3.58 into 3.62, we obtained the open loop transfer
function of the controller which is installed inside the specimen (test) structure. This transfer
function is shown in figure 3.9 given by
( ) 1.8s2 + 0.5s + 2575 ( ) us = u s
s2 + 1.9s + 1431 s (3.67)
By substituting equations 3.65, 3.66, and 3.58 into 3.64, we obtained the characteristic equation
F(s) whose root locations are in the left hand side of the s-plane shown in figure 3.10, where the
feedback gain 92 varies from 0.0 to 1.13. Hence, this system is stable.
42
3.6 Controllability of High Frequency Mode Vibration
3.6.1 Controllability matrix
According to the algebraic controllability theorem, the time invariant system is controllable only if
the rank r(Q) of the controllability test matrix
(3.68)
is equal to kth order of the system. By substituting equation 3.27 and 3.28 into 3.68, we can check
that the rank of matrix Q is 10. Hence, this system is controllable. According to this definition, a
system is either controllable or it is not. In the real world, however, it is impossible to make such
sharp distinctions. Obviously, some modes are less controllable than other modes. As a result, we
should investigate further the controllability index.
3.6.2 Energy response of the system
The total energy input into the 5 DOF system, due to the control force u(t) during the limited
control time Tc, is expressed as
(3.69)
{Tc = lo {x}t{b}u(t)dt (3.70)
The control vector { b} can be expanded by using the modal vectors { Ui} such as
(3. 71)
where { ui} are given by equation 3.5.
Participation coefficients for input vector { b} are given by
fJl 5.905 (3.72)
(32 -9.900 (3.73)
(33 -13.09 (3.74)
(34 6.204 (3.7.5)
f3s - -9.002 (3.76)
43
Total energy is expressed as
(3.77)
where E(hi,wi) is the energy input into a single DOF system with unit mass and damping, hi, and
natural frequency, Wi, due to excitation, u(t), during the time Tc. If f3i = 0, the ith mode is not
controllable. All of the participation coefficients, f3i, are non-zero coefficients; hence, all modes are
controllable. However, some modes are more difficult to control than others.
H the system is easy to control, the total amount of energy input to the system due to the
random white noise during the same period of time should be large, because if the system is not
controllable, the system energy is kept constant or the energy input is zero. Hence, we can define
the controllabilty index as
ith mode controllability index:
m·f3? ' ' (3.78)
Using the definition given by equation 3. 78, we obtained the controllability index shown in
table 3.1.
m; f3i Controllability index
1st mode 0.1304 5.905 0.135 2nd mode 0.1100 -9.900 0.320 3rd mode 0.0670 -13.090 0.341 4th mode 0.0461 6.204 0.050 5th mode 0.0631 -9.002 0.152
Table 3.1: Controllability index.
Hence, the 4th mode is most difficult to control while the 1st, 2nd, 3rd, and 5th modes are
comparatively easy to control.
44
3. 7 Experimental Investigation
3.7.1 Set up of test structure
Details of the specimen (test) structure and controller have already been given in the previous
sections. The overall system has been identified and the optimum feedback gain in terms of the
control signal has so far been calculated. The final set up of specimen structure with the controller
is shown in figure 3.11, where the locations of the sensors are also indicated.
The relative displacement and relative velocity of the controlled story were measured and the
relative velocity was used to create the control signal. The signal taken from the velocity sensor
was amplified by means of a linear amplifier followed by a first order low pass filter whose corner
frequency was 60 Hz.
Response acceleration of each mass point was recorded by a servo-type accelerometer. Data
was taken from 5 different locations of the specimen structure, each corresponding to a mass point.
The data sampling rate used was 100Hz.
The control force was introduced to the specimen structure by means of a 1/8 inch wire cable
which was directly connected to the moving table of the controller and the next floor point.
Another APS shaker was placed on the top floor of the specimen structure to excite the whole
system. The excitation corresponds to the same earthquake type record which was used for the
numerical study in chapter 2. The shaker was set upside down so that the weight of its external
body (34.5 kg) was utilized as a counter weight against the generated excitation force.
There were two types of excitation tests carried out; one with control and the other without. In
the case of the no-control test, the tendon wire was released and taken off but the controller itself
was left because of weight compensation. The same excitation was used for both tests.
3. 7.2 Results of excitation tests
Response acceleration of each mass point in the case of control is shown in figure 3.12, while those
of the no-control case are shown in figure 3.13. Comparing both records, it is clear that the lower
4.5
frequency component of the response was reduced considerably in the controlled case.
Processing the acceleration data, we obtained the time history of the velocity at each mass
point. They are shown in figures 3.14 through 3.15. A low pass filter was used to remove the
drifting component from the acceleration data. Subsequently, the records were integrated in the
time domain to obtain the velocity response. The time history of the displacement was also obtained
by the same procedure and is plotted in figures 3.16 and 3.17.
The mathematical model which resulted from the identification task was used again here for
numerical calculations. The expected damping arising from control was 1.2 kgf em/sec, and the
control stiffness was zero. The average acceleration method was used with a time interval of 0.005.
The acceleration, velocity, and displacement associated with mass points 1, 2 and 3 are shown in
figures 3.18 through 3.19.
Judging from the experimental results, it is seen that the control algorithm proposed in chapter 1
was accomplished and the analytically predicted control effects were confirmed.
Had real earthquake type excitation been used instead of the sample excitation discussed above,
a drastic reduction in the response would have been observed.
4G
,.-j PL. 200 lt 1000 lt 19 0
675
0 \ 2-L 50lt50x5
2-SPL. 400ltl00lt3.2 675
/ PL.400 lt 2700 lt 3. 2 v 2,700 900
'V v 675
0 ~ .. -----.
~I PL.200ltl000ltl9 · ......... •
675 0
1 1,000 fllQQl 1 00
I PL. 200 lt 1000 lt 19 Uoit:[mm]
Figure 3.1: Specimen structure
47
u(t) {xf
-EJ----: {x} {z}
L
A
Figure 3.2: Block diagram of specimen structure
u(t) +
(xJ
-EJ----: +
control force
Active device
control signal
usC t)
{x} {z}
~~
A
G
Figure 3.3: Block diagram of feedback system with active device
48
\7
Input voltage
15.35
v 43.82
Unit [em]
t = =
Power amplifier
Model# 114 Model 113 Electro-seis shaker
Figure 3.4: Controller application
49
----0.0 Q)
"'0 '-"'
200r---~--,-~,-rrrnr----r--r-~rT~~----,-~~~-rrrn
:: -~i ::r ;:rrlllf~=-:1 ~~r:rrtJrfF:: :1:~r1:;:r~, ! 0 10-1
(a) Hz
101 102
100.----r--r-o-roTTor----r--r-ro-rrrn----.--.~~~rrn
Ill ! :I! , ,. ··············~=_i,: ········r,_·_······!····!··· ! ....... ( ................. ; ......... j .••••• i .... .; ... j .•. ; ..•.. j .. ) .•••.•.•...•.•..•..••.•... j .••••••.•••• i ...•... j •• ,( •• (
. I -II 1 if l I i i i 1·1 ~ I I II i l !I 0 --------- ···r·······r··· .. l .... :--n·rrr····---·--·--·-:·-------:--··--' .. :---··1·1-... ...... . ..... -r-----:----~--t·:--:·:
-50 lQ·l 1QO
(b) Hz
101 102
40r---~--,-~,-~rMr----r--.-~rT~~----,--.~~~~M
30 ............. ~ ........ " ... ..! ........ .I..LI .. U ............. , ........ L .. ..l .... l...l .. LU..l ................ , ......... , ...... i.. .. t .. .l ... Lu 20 ......... -----:--------~----·; ____ ;_ : .. l.t l.L ............. ! ...... ; ...... j .... ~.J ..... / ............................. 1 ... ) ... ; ... ; ...... :
i ;;:~~~~~ ~ ········ :::::: . . . . . . . . . : : : ::::: .. .
10 ··············-:-·········~······l····~····t···~··t••t·~···············o ·····!······t_._· ....... ,; ···• .. _: ... ___ i··f,>·f_._ •• :_._·················r_._· .......... _;_ ...... ~ ... ······.i_· .. t ••• ····!··-~.-. ·{··.
'··,: '··,:- '··:: : ~',, 1,:· '··:: l,,, =_·,,_ ; : : ; ; i ; ~ 1 ~ ~ i : ~ i oL---~~~LUiL __ _L_L~LL~~~--~-UUiD 10·1 100 101 1Q2
(c) Hz
0 ....
-20
-40
-60
-80
..... ~ .... ( .... i .. .L.L.LL ............... ) ......... j ...... l ..... t ... L . .i .. J .. ~ .. ~ ................ j ......... j ...... ~ ..... ; .•• ~ ••• j ••. : .• ~ ~~''"" ~-- ;;;;;;;; 1. \_;;;;:; . . . ; : ; : ; ~ : ; . . . .............. t ... ·--··t·····j···-~···r ; .. ; .. ~·t············--··~-·--·--·1· .... ·t····t--·~···t··~ .. [--~···--·--····--·-~·-·······~··--·t--··~- .. i···;··~· ~
.............. l ......... L ..... ; .... ~ .... L .. L.LL~ .............. L ........ L ..... L .. l. .. L.LLLL ............... L ........ L .... L .. .L .. L.LLI ............ l ........ _ ..... !---------H--~--~------- ......... ; ......... , ...... j '--! .. L ..... :-------~----..l ......... , ...... r .... j ....... J.._ ..
-100 '-----'-----'--''----'---'--'L....I....L-'------L---'--L..-.J.....L.JL...I....l.J._ __ ---L, __ ,__L..-.J.__.__,L....L.LJ lQ·I 100 101 102
(d) Hz
Figure 3.5: Open loop transfer function (a) s2 H(s) [Magnitude] (b) s2 H(s) [Phase] (c) sH(s) [Magnitude] (d) sH(s) [Phase]
50
control signal
ui t) {x}
controller's dynamics structure's dynamics
Figure 3.6: Block diagram of the system with controller
~~-------=
controller's dynamics structure's dynamics
Figure 3. 7: Block diagram of the system with controller
(z} [S}----4 B ~~~ +
A
compensator structure
Figure 3.8: Block diagram of the separated system
.51
2.-----,---~-.-.~~~------.--.--~.-ro~nr-----.---.-.-..-rrn
1 ·············· ·········=······
0 ·············· .............. .
rvr ················t········r···r··r·rrr ················ ··············· .... ; ... ·· ·(j-
• • ••• 0
: : : : : : • 0 ••• . . ... : : : : :
. ............... : ......... : ...... : .... .:. ... : ... : .. :.. ················ ··············· -~--~-; ; ; ~ : : : : : : : :
J.J. -lL-----~--~~~-L-L~L'~------~---L~~~LJ-L~------~---L--~~~~~
I0-1 IQO IQI 102
50 ...................... ····· ....... .
0 ............. .
-50 ·············· ········ ............ .
Transfer function of installed controler [kgf/vol]
• • 0 •
·:·:·r················:········:····••iJ®••J!•···············:·••••••••:••••••
. -~--~- t ............... ·f ........ -~ ... ···~ ... ··~· ... ~-··~.-~--~--~ ................ ~ ........ -~· .... . ::: : : : : : : : : ::: : : : : : : .
: . : : : . . . : . . . . .
-~·-:
·~··<-
-}QOL-----~--~-L-L-L~·~------L·--~~;~;-L~·-L~·~----~i---L~~~~~~-~~· IQ-1 IQO 10 I 1Q2
Transfer function of installed controler [degree]
Figure 3.9: Open loop transfer function of the controller.
52
80.---------.---------.---------.----------.--------, .. . . .
60 .............................. . ................................ ; ................................. 1' ........................ --r ............................. . 40 .............................................................. '1' ................ : ......... <:: _, .............................. .
20 .............................. . ······························-·:··········································
0 ··0 ····0 ·····O······<> ····
-20 ............................. ..
r <WlE ooooo~ .~-----------
... o ............. i; ......... o .. ~ .. o .. ·o·o·ooO<XXUUllOI--: __ ....................... ; .............................. .
ooooo ; 0' t'tiWF---~~--.. ~-.... ~ ............. -~ ............................... . ································:·····················. :
-40 ........................... . .................. i ........................... ~::":"::: ..... ~: ----..:-== ............................ .. -60 ............................. .. ................................ : ................................. ~ ......................... ..._, .............................. .
i ~ ... i : :
-80L-----------~-------------L------------~·------------~------------~ -20 -15 -10 -5 0 5
Figure 3.10: Root locus plate of the whole system (0.0 ~ 92 ~ 1.13, 91 = 0.0).
53
shaker (for excitation)
/ accelerometer 1
accelerometer 2
tendon accelerometer 3
table
accelerometer 4
controller accelerometer 5
Figure 3.11: Setup of the specimen structure.
54
200.--------.-------,--------.--------.--------.
100 ............................... .; ................................. L ............................... r·······························t·· .. ···························
0 io-"--WM.~ro./IIU'I'\1'1 '
. . -100 ·······························7································+································!··································1·······························
~ i ~ ~ -200 L..--------'----------'---------'----------'-------___J
0 5 w ~ w ~
(a) sec
200.--------.-------.--------,-------~--------.
100 .............................. +······ .. ······················+·······························+····· .. ·······················+······················· .. ··· ..
-1~ --- t·~~~=~=i···········--······:-·--········· ~ . ! ! -200L..--------'----------'---------'---------'-------___J
0 5 w 15 20 ~
(b) sec
200r--------.-------,--------.--------.--------.
100 ............................. :···· ............................ 1 ............................. ( ............................ ! ........................... .
or---~.A"''Jirt~,·,,~~~~~r i i -100 ................ r·····························:······ ........................ r··············· .............. ; ........................... .
-200 '---------'----------'---------'----------'-------___J 0 5 10 15
(c) sec
20 ~
200.-------.--------.--------.-------~--------.
1: ····························-············~--~····~····::.·································· .. ·································· .. ········ .. ·· If "~ 'i'(
-100
-200 0
............................ , .............................. : ........................... , ............................. +····· .. ·············· .. ···
5 10 15
(d) sec
20 25
Figure 3.12: (a) Ace. excitation of shaker (control) (emf sec2 ) (b) Ace. response at point-1 (control) (cmjsec2
) (c) Ace. response at point-3 (control) (cmfsec2 ) (d) Ace. response at point-4 (control) (cmfsec2 )
5.5
200.--------.--------~-------.------~,-------·
I 00 ·······························t················-············ .. i·················-··············r································t·······························
0 ~----~,rv..WIJ'l '
-IOO ······························T··········· ···················r···············-·············r······························-r······························
-200 0 s IO IS
(a) sec
20 25
200.-------~--------~-------.--------.-------~
-200~------~--------~------~--------~------~ 0 S IO IS 20 25
(b) sec
200.--------.--------~-------.--------.-------,
j ~ ,: ------- :~~1- ;~ijffN,~~ ---- r·····----······ -I 00 ... ························· ··;···························· ... ; .. ··················· .. ······j-························· ·····t·······························
~ ~ ~ -200~------~--------~------~---------"~------~
0 s 10 1S
(c) sec
20 25
200.--------.--------~-------.------~,-------,
100 ......................... ··-r···········::·:············:;. -~--~~-··········· ·····r······················ ... T ... ······················ ..
_,~ ... -- - T -~~--··:f·~:: ~·-- --··· :······························ -200
0 s 10 15
(d) sec
20 25
Figure 3.13: (a) Ace. excitation of shaker (no control) (cmfsec2 ) (b) Ace. response at point-1 (no control) (cmfsec2
) (c) Ace. response at point-3 (no control) (cmfsec2) (d) Ace. response at
point-4 (no control) (emf sec2 )
56
10~--------.---------~----------r---------.----------.
5 ································:································· ·································:··································:······························· . . . .
-5 ································~·································!·································i··································~································ : : : : : : : : : : : : : . . .
5 10 15
(a) sec
20 25
10.---------.----------.----------.---------.----------.
5 ································t·································j·································j··································; .............................. . . . . . . .
-5 ................................ , ................................. , ................................. , .................................. , .............................. . : : : : . . . . . . . . : : : : . . . . . . . . . . . . . . . .
-10~--------~--------~----------~--------~--------~ 0 5 10 15
(b) sec
20 25
10~--------.----------.----------r---------.----------.
. . . . 5 ································~·································:·································~··································:······················ . . . . . . .
0 f--------..,.J
-5 ································i··································:··································~··································:································ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -10~--------~--------~--------~~--------~--------~
0 5 10 15 20 25
Figure 3.14: Velocity response (emf sec): (a) at point-1 (b) at point-2 (c) at point-3 .
. 57
10~--------~---------.----------~---------.---------.
5 ............................... ·················· .......... ~ ..... ························· ·································j·······························
o~--............ ,....,_,.._,;
-5 ·····················:·································-:·······························
-10L---------~--------~--------~--------~~------~ 0 5 10 15 20 25
(a) sec
10~--------~--------~----------.----------r---------.
5
••••••••••••••••••••• ~- •••••••••••••••••••••••••••••••• "!' •••••••••••••••••••••••••••••••
: : -5 : : . . . . . . . . . . . .
-10~--------~--------~----------~--------~--------~ 0 5 10 15 20 25
(b) sec
10~--------~--------~~--------~---------r---------,
5 ································~·································i·································j·································+·······························
0
-5 ································~··································~·································~·································-~································ . . . . . . . . : : : : : : : : : : : : . . . .
-10~---------L--------~----------~--------~--------~ 0 5 10 15
(c) sec
20 25
Figure 3.15: Velocity response (em/sec): (a) at point-1 (b) at point-2 (c) at point-3.
58
2r----------.----------.----------.----------.---------~
. . 1 ······························· ·································j·································j································· .............................. .
. . . . . . . . : : . . . .
0
1 . . . - ............................... r································=································-r································ ·······························
-2L---------~----------~--------~----------~--------~ 0 5 10 15
(a) sec
20 25
2.----------.----------.-----------.----------.----------~
1 ································~·································j·································j··································t······························· : : : : : : : : . . . . . . . . . . . . • 0 • • . . .
o~------~v . ~~·------~--~
-1 ································:·································:·································:··································:······························· . . . . : : : : . . . . . . . . . . . . . . . . . . . . . . . .
-2L---------~----------~----------~--------~----------~ 0 5 10 15
(b) sec
20 25
2r----------.----------.-----------.----------.----------~
1 r································~·································~·································~··································~························ ······· : : . : . . . . . . . . . : : . . . . . . .
0 1------./. .. \;_;,,.,,...__,._~ ......... -..: . .. ··-~'-""""'·· ---..;.......----~------;
. . . . -l f-································:··································l·································1··································:································-
-2L---------~----------~----------~--------~----------~ 0 5 10 15
(c) sec
20 25
Figure 3.16: Displacement response (em) (a) at point-1 (control) (b) at point-3 (control) (c) at point-4 (control)
59
2.----------.-----------.----------.-----------.----------,
1
0
-1 ································~·-···························· . ~-··· ···························~··································!······················ ········· : : : : : : : : . .
-2L---------~----------~----------~----------L---------~ 0 5 10 15
(a) sec
20 25
2~---------.-----------r----------.-----------r---------~
1 ································:·································: ..... • ............ 0 •••••••••• ~- •••••••••••••••••• 0 ••••••••••••• -~ 0 •••••••••••••••••••••••••••••• . . . . . . . . . . . . . . . .
0 !-'"""--......_..-
-2L---------~~--------~----------~----------L---------~ 0 5 10 15
(b) sec
20 25
2.----------.-----------.----------.-----------.---------~
1 ................................ , ................................. ; ................................. ; .................................. ; ...................... . . . . . . . . . . . . . . . .
0 1-------.Y'
-1 ................... ······ .. ·····i····················· ············~····· ····························~···························· ...... ; ............................... . : : : : : : : : : : : : . . .
-2~--------~----------~----------~----------~--------~ 0 5 10 15
(c) sec
20 25
Figure 3.17: Displacement response (cm)(a) at point-1 (no control) (b) at point-3 (no control) (c) at point-4 (no control)
60
200.---------.---------~----------.---------.----------,
100 ·································································· ································· ································ ·······························
0~~~~~~~~~~ ~~RI1~~~~~~~~~~~~~~~~
-100 ································f·································i································+································ ······························· : :
-200~----------------~----------------~----------------~----------------~------------~ 0 5 10 15
(a) sec
20 25
200.----------.----------.----------.----------.---------~
. . . . ........................... ·····:-········· ...................... ··:· .......... ····················· -~·· ............................ ····:- .............................. . . . . . 100 . . . . . . . . : : : . . . . . .
-100 ............................... -~ ................................. ~ ................................. ~- ................................ -~ .............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : ;
-200~--------~--------~----------~--------~--------~ 0 5 10 15
(b) sec
20 25
200.----------.----------.----------.----------.---------~
100 . . ································:·································:·································-:··································:········· .. ·········· . . . . . . . . . . . . . . . . . . . . . .
0 1-------~' ""'-"~·VI.A\. v ..A. "'-A. :
-100 ................................ ;·································i··································:··································:································-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -200~----~-------~----~-------~--------~
0 5 10 15
(c) sec
20 25
Figure 3.18: Analyzed response of ace. (cmjsec2 )(a) at point-1 (control) (b) at point-3 (control) (c) at point-4 (control)
61
10~--------.-~-------.--------~---------.---------.
. . 5 ······························· ································l································r······························· .............................. .
0
-5 ................................ ~ ................................ + ................................ i···· .. ······ ................................................... . : : : : : : : : : : . .
-10L---------~----------~--------~----------~--------~ 0 . 5 10 15 20 25
(a) sec
10~--------~----------~---------.----------.----------.
5 ·····························r···················r· •···················· ·:····························
01-----.........,.;
-5 ................................ , ................................. , ................................. , .................................. , .............................. . : : : : : : : :
~ ~ ~ : : : :
-10~--------~----------~--------~----------~--------~ 0 5 10 15 20 25
(b) sec
10~--------~----------.---------~----------.----------.
5 f- ............................... -f ............................... + ................................ ~ .................................. \···· .. ··· .. ············· .. ····· . . . . . . : : : . . . . . . . . . . . . .... -,;_,
-5 1-·""""""""""''""''"+ ................................. f ................................. ; .................................. j ............................... . : : : : : : : : : : : : : : : : . . . .
-10L---------~----------~--------~----------~--------~ 0 5 10 15
(c) sec
20 25
Figure 3.19: Analyzed response of vel. (emf sec)( a) at point-1 (control) (b) at point-2 (control) (c) at point-3 (control)
62
Appendix A
FREE VIBRATION TEST OF SPECIMEN STRUCTURE (1)
Purpose : The modal vectors of the test structure should be obtained.
Test procedure : After an impulse shock was given to the specimen structure, the vibrating state
was measured by accelerometers attached to it. The data sampling rate and sampling duration are
given in Table A.l. These data collecting conditions were independently selected for each modal
frequency and the same test was then repeated three times for each mode.
Equipment and set up : Dimensions and details of the specimen structure are shown in fig
ure 3.1 in Chapter 3. The final setup is shown in figure A.l. Amplifiers were employed to increase
resolution. Their gain factors are given in Table A.2. The saturation range of the A/D converter
is 5.0 vol. The calibration factor of the accelerometer is 10.0 vol/G, where G = 981 emf sec2•
Data processing : After taking the complex Fourier coefficient of each time history, the peak
absolute value of each mode is picked up and shown in Table A.3, where the negative or positive
sign is given according to the phase angle of complex Fourier coefficient.
63
accelerometer 1
accelerometer 2
accelerometer 3
table (fixed)
accelerometer 4
controller accelerometer 5
Figure A.l: Set up of specimen and location of sensors.
Data sampling sampling total # of points rate (HZ) duration (sec)
1st mode 100 40 4000 2nd mode 400 10 4000 3rd mode 600 6.5 3900 4th mode 1000 4.0 4000 5th mode 1000 4.0 4000
Table A.l: Data sampling condition
64
model mode2 mode3 mode4 modeS Amp. 1 2.0 5.0 1.0 1.0 1.0 Amp. 2 2.0 5.0 2.0 1.0 1.0 Amp. 3 2.0 5.0 1.0 1.0 1.0 Amp. 4 10.0 1.0 4.0 1.0 1.0 Amp. 5 10.0 1.0 4.0 1.0 1.0
Table A.2: Gain of post amplifier
Test 1 Test 2 Test 3 Mean value 5592.6 5368.7 6243.7 1.000
1 st 5452.5 5381.0 6080.6 0.983 mode 5321.4 5202.6 5899.5 0.955
1028.4 996.6 1156.4 0.185 544.7 518.1 619.4 0.098 -328.7 -336.8 -383.5 -0.232
2nd -305.7 -249.7 -290.4 -0.178 mode -163.4 -120.1 -135.7 -0.089
1420.7 1559.4 1815.3 1.000 751.1 933.2 1066.1 0.571 1285.8 672.39 670.48 .1.000
3 rd 36.3 32.68 45.31 0.048 mode -1029.3 -542.0 -545.9 -0.807
106.9 42.16 43.21 0.070 100.4 41.01 41.83 0.067 46.02 47.01 36.76 -0.053
4th -37.76 -29.34 -46.69 0.048 mode -12.13 -8.74 -14.86 0.015
222.55 253.57 188.58 -0.271 -813.8 -895.84 -734.76 1.00
-1097.1 -938.01 -0.592 5th 1894.1 1553.38 1.000 mode -944.7 N.G. -782.35 -0.502
132.7 96.55 0.066 -120.4 -113.73 -0.069
Table A.3: Absolute value of meaured complex modal vectors (raw data)
6.5
Test 1 Test 2 Test 3 Mean value 0.0 0.0 0.0 0.0
1st -0.0 -0.5 -0.3 -0.3 mode -0.8 1.1 0.6 0.3
0.5 -0.4 0.2 0.1 -0.4 1.9 -0.8 0.2 0.0 0.0 0.0 0.0
2nd -1.3 2.6 1.1 0.8 mode 3.0 9.9 8.7 7.2
175.9 169.2 174.9 173.3 174.3 166.2 171.5 170.7 0.0 0.0 0.0 0.0
3rd 6.6 7.5 -29.0 -5.0 mode 180.8 181.3 181.1 181.1
1.0 -2.2 -0.8 -0.7 -0.2 -1.7 -4.0 -2.0 0.0 0.0 0.0 0.0
4th 170.6 160.2 154.6 161.8 mode 96.6 165.9 137.0 133.2
-5.0 0.9 10.7 2.2 182.1 177.7 163.4 174.4 0.0 0.0 0.0
5th 179.5 -182.4 178.6 mode 0.1 N.G. -1.5 0.7
176.6 -197.0 169.8 -4.3 3.4 -0.5
Table A.4: Angle of measured complex modal vectors (raw data)
GG
Appendix B
FREE VIBRATION TEST OF SPECIMEN STRUCTURE (2)
Purpose : Modal frequencies and modal dampings of the test structure should be obtained.
Test procedure: The test procedure is almost the same as test(l), except that only one accelerom-
eter at a suitable mass point of each mode was used to take a series of data. The data sampling
condition was changed because a high resolution was needed. The data sampling conditions are
given in table B.l. The same test is repeated twice for each mode test.
Data processing : After taking the FFT of each time history, a band pass filter was used to extract
the interesting component of a mode vibration. The inverse FFT of this processed data was then
used to obtain the time history of the data again.
Data sampling sampling total # of points rate (HZ) duration (sec)
1st mode 20 200 4000 2nd mode 50 80 4000 3rd mode 80 50 4000 4th mode 100 40 4000 5th mode 100 40 4000
Table ll.l: Data sampling condition
67
Test 1 Test 2 Mean 1st mode 0.918 0.918 0.918 2nd mode 3.003 3.003 3.003 3rd mode 6.519 6.494 6.507 4th mode 9.717 9.668 9.693 5th mode 11.426 11.426 11.426
Table B.2: Modal frquency (HZ)
Modal damping(%) 1st mode 0.5 2nd mode 0.7 3rd mode 0.3 4th mode 0.3 5th mode 0.3
Table B.3: Modal damping factor
68
Appendix C
FINITE ELEMENT ANALYSIS OF SPECIMEN STRUCTURE
Purpose: A reduced order (5 DOF model) stiffness matrix [K] of the specimen structure should be
obtained by the Finite Element Method. Resulting stiffness matrix is assumed to be an estimation
rather than the 'exact stiffness matrix'. It is supposed, however, that this estimated [K] matrix is
precise enough to retrieve the modal masses which are associated with the 'exact mass matrix' and
measured eigenvectors and eigenvalues. Further discussion is given in chapter 3.
Analytical condition : Bar elements provided by NASTRAN were used for the following analysis.
The dimension of the specimen structure along with node points are shown in Figure C.l The
material property and cross sectional data of each element are given in Table C.l. There are five
load conditions. In every analytical case, unit displacement in the lateral direction was given at
one of the reference points while the other four reference points were fixed in the lateral direction
(not in the vertical direction). The reaction forces obtained from the above analyses corresponded
to each element of the reduced stiffness matrix [K].
Result : A reduced 5 DOF stiffness matrix is given in Chapter 3. Errors are introduced into the
resulting [K] matrix because of the following reasons.
• The stiffness due to bolts and nuts are neglected. Also neglected is the looseness of the
assembled plates. The former contributes to the possible increase of stiffness, while the latter
corresponds to the reduction of it.
69
Member code Cross sectional area Moment of inertia A(cm2) I (cm4 )
C1 12.72 0.1072 C2 38.16 2.894 C3 22.72 1.252 B1 38.10 11.522 B2 58.10 40.859 B3 48.10 23.184
Table C.1: Member data for FEM
• Nominal value of Young's modulus (E = 2100 tfcm2) and Poisson's ratio(v = 0.3) is used,
which is probably the largest contribution to the error. However, this only introduces one
factor to the whole matrix.
• Linearity and small deflection are assumed as usual.
70
/B2 B1 /C3
/B3 B1
/C3
/C2
/B3 B1
C1
/C2
Bl /C3
/C2
Bl
II II 0 5 95 100
270 265
231 225 219 207.5
197.5 186 180 174
140
130
96 90 84 72.5
62.5 51 45 39
5
E: 2100 ton/cm2
v: 0.3
Unit [em]
Figure C.l: Finite element model and dimension.
71
Appendix D
TEST OF OPEN LOOP TRANSFER FUNCTION OF CONTROLLER
Purpose : The transfer function representation of the controller should be obtained.
Test procedure : The controller, from which the rubber band is removed, was excited by sine-
wave from a signal generator. After the system has reached a ste·ady oscillatory state, both the
acceleration and velocity of the moving auxiliary mass are recorded. The excitation voltage was
also recorded by using an A/D converter. The sampling rate and duration time were given in
table D.l. The final result of the transfer function is given in table D.2. For convenience, with no
special reason, 11.77 kg mass was used for this test.
Data processing : Even though the system was shaken by a sine-wave excitation, the analyses were
executed in the frequency domain, due to the presence of noise.
Frequency Data sampling sampling total # of points (HZ) rate (HZ) duration (sec)
0.3955 40 100 4000 0.5981 60 60 3600 0.8057 80 50 4000 1.0000 100 40 4000 2.0000 200 20 4000 6.1040 900 8 4000 10.010 2000 4 7200 14.063 2800 2.5 8000
Table D.l: Data sampling condition
72
Velocity Acceleration Phase Magnitude Phase Magnitude
0.3955 -29.9 25.68 61.4 70.290 0.5981 -40.8 23.59 50.2 96.120 0.8057 -50.3 19.34 41.0 110.23 1.0000 -57.6 17.74 33.0 124.15 2.0000 -74.8 9.93 15.0 139.44 6.1040 -83.5 2.86 7.90 148.12 10.010 -89.8 1.45 1.32 148.70 14.063 -90.0 1.20 -0.23 149.43
Table D.2: Test result of transfer function
73
APPENDIX I
SOME ISSUES RELATED TO ACTIVE
CONTROL ALGORITHMS
by
Wilfred D. Iwan and Zhikun Hou
Division of Engineering and Applied Sciences California Institute of Technology
Pasadena, CA 91125 U.S.A.
February, 1991
I-ii
Contents of Appendix I
Page
Contents of Appendix ........................................ :- .......... I-ii
CHAPTER 1-1 Introduction .......................................... I-1
CHAPTER 1-2 Background ........................................... I-5
2.1 Introduction ...................................................... I-5 2.2 Optimal Active Control ........................................... I-6 2.3 Modified Independent Modal Space Control ....................... I-8 2.4 Simplex Method for Optimization ................................ I-ll 2.5 Simplified State-variable Method for Nonstationary
Covariance ....................................................... I-12
CHAPTER 1-3 Optimal Location of Control Devices ............. I-15
3.1 Introduction ..................................................... 1-15 3.2 Formulation ...................................................... I-16 3.3 Example ......................................................... 1-18 3.4 Summary ............................................ · ............ 1-18
CHAPTER 1-4 Acceleration Control ............................... 1-21
4.1 Introduction ..................................................... I-21 4.2 Formulation ...................................................... I-22 4.3 Results and Conclusions .......................................... I-24
CHAPTER 1-5 Effect of Time Delay on Structural Control ...... 1-27
5.1 Introduction ..................................................... 1-27 5.2 Formulation ................................................... · ... I-28 5.3 Steady-state Solution ................. ~ .......................... I-29 5.4 Resonance and Critical Values of Delay Time ..................... 1-31 5.5 Effect of the Time Delay on Active Control ....................... I-34 5.6 Conclusions ...................................................... I-37
CHAPTER I-6 Summary ............................................ 1-46
Acknowledgement ....................................................... 1-47 References ............................................................... I-48
I-1
Chapter 1
Introduction
Structural control has recently become a focus of civil structural research. Us
ing structural control, the response of a structure to environmental loads is modified
or controlled in a desired way such that structural safety and/or functionality are
improved. A significant effort has been directed towards the development of devices,
algorithms and practical applications of structural control. For a detailed review,
the reader is referred to Soon (1988) and Kobori(1988).
If the undesirable motion of a structure is suppressed by introducing devices
such as base isolation systems, viscoelastic dampers, or tuned mass dampers, the
approach is referred to as passive control. If the motion is controlled by means
of devices such as tendons, active tuned mass dampers, or pulse generators, the
approach is referred to as active control. A combination of these two approaches is
referred to as hybrid control. This study is restricted to the active control.
As described in Yao (1972) and others, an active structural control system has
the basic configuration as shown in Figure 1-1 which consists of
(a) Sensers located at appropriate positions in or around the structure to measure
either external excitations and/ or structural response variables.
(b) Devices to process the measured information and to compute necessary control
forces needed based on a given control algorithm.
(c) Actuators, usually powered by external energy sources, to produce the required
forces.
This discussion is mainly focesed on active control algorithms.
There are three basic types of active control algorithm. When only the struc
tural response variables are measured, the control configuration is referred to as
I-2
closed-loop control since the structural response is continuously monitored and this
information is feedbacked to make continuous corrections to the applied forces. An
open-loop control results when the control forces are regulated only by the measured
excitations. In the case where the information on both the response qtntntities and
excitation are utilized for the control design, the term open-closed loop control is
used in the literature.
To see the effect of applying such control forces to a structure under ideal con
ditions, consider a building structure modeled by an n-degree-of-freedom lumped
mass-spring-dash pot system. The matrix equation of motion of the controlled struc
tural system can be written as
MX(t) + Cx(t) + Kx(t) = DF c(t) + GF e(t) (1 - 1)
where M, C, and K are respectively n x n mass, damping, and stiffness matrices of
the structure; x(t) is the n-dimensional displacement vector; F e(t) is the external
excitation vector; F c(t) is the applied control force vector from the actuators; D is
a n x m matrix specifying the location of actuators; and G is a n x r matrix defining
the location of the external excitations.
Assume the control force F c(t) is designed to be a linear function of the mea
sured displacement x(t), velocity x(t), and the excitation F e(t). That is
(1 - 2)
where K 1 , C 1 and E 1 are respective control gains which may be time-dependent.
Substituting Eq. (1-2) into Eq. (1-1) yields
MX(t) + (C-DC t)x(t) + (K- DKI)x(t) = (G + DEI)F e(t) (1 - 3)
Comparing Eq. (1-3) with the special case of Eq. (1-1) where the control force
is absent, it is concluded that the effect of the closed-loop control is to modify
the stiffness and damping properties of the structure so that it can respond more
favorably to the external excitation. The effect of the open-loop component is to
modify the external excitation. The choice of the control gain matrices K 1 , C 1 and
E1 is dictated by the control algorithm selected.
I-3
Various control algorithms have been developed based on different control de
sign criteria. Among them, representive examples are optimal control, pole assign
ment control, independent modal control (IMSC), instantaneous optimal control,
predictive control, and bounded state control. A comparison of these- algorithms
is given in the review paper by Soong(1988) where a list of their source references
is also given. Though this discussion is limited to optimal control combined with
the IMSC method, the concepts introduced may be extended to other control algo
rithms. Moreover, as pointed out by Yang(1987), both the open-loop control and
the open-closed loop control is not achievable for the optimal control of structures
subjected to earthquake ground motions since optimal control gains are obtained
by backwards solving the Riccati matrix equation and it can not be solved due to
the unpredictability of the seismic loads. Therefore, only the closed-loop control
algorithm is discussed in this investigation.
It is recognized that the previously mentioned control algorithms are developed
based on classical control theory as applied to highly idealized systems. Some
practical aspects must be considered in order to successfully apply these idealized
algorithms to real structures, which is exactly the focus of this investigation. Three
fundamental and practically important aspects, i.e. acceleration control, optimal
location of control devices, and the effect of time delays are discussed herein. In the
first two areas, stochastic control theory has been employed to render the results
useful not only for a particular sample of an environmental load, say a particular
past earthquake, but also for events which may be modelled by a stochastic process.
In following chapters, Chapter I-2 presents a necessary theoretical background
for this study; Chapter I-3 describes an algorithm for determining the optimal
location of the control devices in a structure subjected to nonstationary random
excitation; Chapter I-4 explores the possibility of directly using measured acceler
ation in the control force design and Chapter I-5 discusses the effect of time delay
on structural control. Finally, a summary is given in Chapter I-6.
I-4
EXTERNAL I STRUCTORE I STRUCTURAL EXCITATION RESPONSE
•• H I CONTROL FORCES t. H-
~~-
f SEJ.'lSORS ' I AC11JATORS -t f SENSORS J J~-
COMPUTATION -OF CONfROL FORCES
Figure 1-1 Schematic diagram of active control
I-5
Chapter I-2
Background
2.1 Introduction
This chapter presents the theoretical background for the analysis throughout
the report, which includes general concepts of the optimal control, the indepen
dant modal space control method for general nultiple-degree-of-freedom (MDOF)
systems, the simplex method for optimization, and the simplified state-variable
method for evaluating the nonstationary response of linear MDOF systems.
The optimal active control is to apply additional control forces from external
sources to counteract or control the effects of the originally-applied loads on the
system performance. The control forces should be designed such that the overall
structural performance is satisfied, or, mathematically, a predefined performance
index is minimized. Some basic concepts of the optimal control are introduced in
section 2.2.
To deal with the control problem of real structures, one may have to consider
a large-scale problem, especially when the dynamic performance of the structure
is not dominanted by a single mode. The size of the problem makes a complete
analysis difficult or even impossible. The independant modal space control method
provides a convenient way to cope with this problem. It can be shown that the
overall performance is governed by the independent performance of the system at
all of the modes. Therefore, the large-scale problem may be reduced to a set of
control problems for single-degree-of-freedom (SDOF) systems. The IMSC method
is presented in section 2-3 with a modification. The restriction that the number of
controllers used must be equal to the number of controlled modes is eliminated.
I-6
In order to minimize the performance index, the traditional control force is ob
tained from the classical Riccati matrix equation. The approach is widely accepted
in engineering practice. One criticism of this method is that the Riccati equation
is derived in the absence of applied loads and, therefore, the resuting control force
does not guarantee the minimization. In order to take effect of the external loads
into account, the simplex method is employed to directly search the optimizing
parameter set in a parameter space. A brief discription of the method is given in
Section 2.4.
The simplex method avoids differentiation of the performance index during
optimization, but requires repeated evaluation of the performance index for the
continuously modified controlled system. Therefore, an effecient method to calcu
late the performance index is needed. In this study, the main effort is devoted to the
structural control under stochastic excitation and the stochastic performance index
may be obtained in terms of the nonstationary covariance matrix of the response.
The simplified state-variable method is employed for this purpose since the nonsta
tionary structural response may be explicitly obtained by the method for a general
stochastic model of excitation. Section 2.5 introduces the basics of the method.
2.2 Optimal Active Control
The equation of motion of an actively controlled n-DOF system can be written
in the state variable space as
Z(t) = AZ(t) + BF c(t) + F e(t) (2- 1)
where Z(t) is a 2n-dimensional vector of state variables representing the displace
ment and velocity; A is a 2n x 2n system matrix obtained from the mass, damping,
and stiffness matrices of the system; F c(t) is an m-dimensional vector representing
control forces from the controllers; B is a 2n x m constant matrix indicating loca
tion of the controllers; and F e(t) is a 2n-dimensional vector specifying the external
excitations.
I-7
The standard quadratic performance index, J, is given by
t' J = Jo [zT(t)QZ(t) +F~(t)RFc(t)]dt (2- 2)
where Q is a 2n x 2n positively semi-definite matrix; R is am x m positive definite
matrix; and t 1 is the duration of the external excitation. Matrices Q and R are
referred to as weighting matrices, or penalty matrices, indicating the relative impor
tance between safety and economy. Though many other performance indices, such
as the maximum of the absolute structural response, can be defined, the definition
(2-2) is employed throughout this study.
The optimal active control is to choose appropriate control force F c(t) such
that the performance index J is minimized. The necessary condition can be shown
as follows. .\(t) = -AT .\(t)- 2QZ(t); .\(tl) = 0
F c(t) = _.!:_R -lBT .\(t) 2
(2- 3)
in which .\(t) is the 2n-dimensional costate vector of Lagrangian multipliers. If the
control force vector F c(t) or the costate vector .\(t) is regulated by the response
state vector and the external excitation, one has
.\(t) = P(t)Z(t) + q(t); .\(t1 ) = 0 (2 - 4)
where the first term on the right side indicates the closed-loop control and the
second term represents the open-loop control. Traditional control algorithms use
the assumption that q = 0 and Fe = 0, leading to the following Riccati matrix
equation for P (Yang, 1987).
. 1 P(t) + P(t)A- 2P(t)BR -lBTP(t) + ATP(t) + 2Q = 0; P(t1 ) = 0 (2- 5)
After solving P(t), the closed-loop feedback control force can be obtained by Eqs.
(2-3) and (2-4).
Strictly speaking, the Riccati matrix P(t) does not quarantee an optimal active
control, since it is derived in the absence of external excitation, which is an unreal
istic assumption. However, it is still frequently used in engineering practice mainly
I-8
due to its mathematical convenience. One great advantage of using the Riccati ma
trix is that it can be solved in advance since the control force is independent of the
external excitation. Also, it is found that the Riccati matrix is an approximately
constant matrix which can be solved by the steady-state version of Eq. f2-5) (Yang,
1987).
The external excitation vector F e(t) may be deterministic or stochastic. Con
sidering that most environmental loads such as earthquake, wind loads, wave forces,
etc. are random in nature, it is sometimes assumed that F e(t) is modeled as a ran
dom process, say a modulated white noise process. That is
F(t) = 7J(t)n(t)Fo (2- 6)
where 77(t) is a deterministic envelope function to describe nonstationarity of the
excitation, F 0 is a constant vector, and n(t) is a stationary white noise process with
properties E[n(t)] = 0
E[n(t!)n(t2)] = Soo(tl - t2) (2- 7)
in which 80 is the intensity of the white noise. If the stochastic excitation is applied,
the strategy of the active control is to design the control force F c(t) such that the
mean value of the performance index defined by Eq. (2-2) is minimized.
2.3 Modified Independent Modal Space Control
The governing equation of a controlled linear MDOF system may be written
as
Mx(t) + Cx(t) + Kx(t) = F c(t) + F e(t) (2 - 8)
where M, C, and K are the mass, damping, and stiffness matrix respectively, F c(t)
is the feedback control force vector, and F e(t) is the applied loads. Assume that
the damping matrix is classical and that
x(t) = ~y(t) (2 - 9)
I-9
where q> is the modal matrix of the system, and y(t) is the modal response vector,
Eq. (2-8) can be reduced to a set of modal equations.
y(t) + A~y(t) + Awy(t) = fc(t) + fe(t) (2- 10)
where A~ and Aw are n x n diagonal matrices whose ith diagonal components are
2~iWi and w'f respectively. ~i and Wi are the critical damping ratio and the natural
frequency of the ith mode. fc is the modal control force vector and fe denotes the
modal external load vector.
Typical governing equation for the ith mode is
yi(t) + 2~iWdJi(t) + w'fy(t) = /ci(t) + fei(t) (2- 11)
Note that Eq. (2-11) is not uncoupled since the modal control force lei generally
depends on the other modal responses implicitly. An important assumption that
the modal control force depends on only the modal response of the same mode has
been introduced to simplify the analysis. As a result, Eq. (2-11) becomes uncou
pled, which greatly facilitates solving the problem. It is seen from Eq. (2-9) that
the norm of the physical response, x( t), is controlled by the norm of the modal
response vector, y(t), and, therefore, the physical response may be effectively con
trolled by controlling the modal responses. The IMSC method is specially useful
in the control problem of large-scale structures since it reduces a large-scale prob
lem to a set of control problems for single-degree-of-freedom systems where many
analytical methods are available. Also, the modal truncation approach may apply
if the structural behavior is governed by some dominating modes, which makes a
complete analysis unnecessary. In the following, it is assumed that only the first l
modes are used in the analysis. That is
X= q>IYI (2 - 12)
where q>l is a n x l marix consisting of the first l columns of the modal matrix q>,
and Yt is a /-dimensional modal response vector.
Assume s controllers are used and FIJi is the control force executed by the ith
controller. Then
F" = RFc1 (2 - 13)
I-10
where F, is as-dimensional control force vector directly from the controllers, R is
as x l distribution matrix which assigns the control forces from the controllers to
each mode and F cl is the [-dimensional modal control force vector. Introduce then
x s location matrix U whith entries being 0 or 1, indicating the locations of these
controllers. the control force vector F c can be expressed as
Fe= UF, (2- 14)
Define
(2- 15)
and assume that its inverse exist. The distribution matrix R may be expressed as
(2- 16)
s =lis enforced in the old version of the method (Meirovitch and Silverberg, 1983),
implying a restriction that the number of controller is equal to the number of mode
controlled. In that case, Eq. (2-16) may be reduced to
(2- 17)
The restriction is lifted here in the sense that the control force from the controllers
is distributed such that the mean square error between the distributed and the
required modal control forces is minimized. A detailed discussion of the elimination
of the restriction is beyond the scope of this report.
Finally, the designed control algorithm for the modified IMSC method may be
summarized as following:
(a) Solve the eigenvalue problem of the uncontrolled system to obtain the modal
matrix, the natural frequencies, the critical damping ratios, and the paticipa
tion factors for the first l modes.
(b) Solve the control problem for a SDOF system for each mode to obtain the
required modal control force and the controlled modal response. The simplex
method is used for optimization and the simplfied state-variable method IS
employed to evaluate the nonstationary response of the modified system.
I-11
(c) Calculate the controlled total response and the required control forces from
each controller.
2.4 Simplex Method for Optimization
The simplex method has been effectively employed for optimization of the per
formance index, or objective function in this investigation. Generally speaking,
the objective function may be defined as norm of a vector F = (It, / 2 , ••• , f m.)
each of whose components /i, i = 1, 2, ... , m is a function of parameter set X =
(x~, x 2, ... , xn), i.e. /;. = /;.(X) = /;.(x~, x2, ... , Xn)· Both m and n are positive
integers and not necessary equal. The simplex method is used to find a special
parameter set X* such that IIF(X) II is minimized. Here II· II indicates any standard
norm of a vector. A brief description of the method is given in the following. The
fundamentals of the method may be found in textbooks on linear and nonlinear
programming including Himmeblau (1972).
Given an initial value of the parameter set, X 0 , and tollerance €, and assuming
that the initial length h = 0.3 for constructing vertices of the simplex, the following
procedure is performed.
(I) Construct n + 1 vertices of the simplex in n-dimensional X space. That is
x&O) = Xo
X~o) = (x~o), x~o), ... , xi~1 , xJ0> + h, xi~1 , ... , x~0))
(i=l,2, ... ,n)
(2- 18)
(2) Calculate Fi = jjF(X~0))II,i = O,l, ... ,n and determine their maximum Fb =
maxFi, sub-maximum Fn = maxFi, and minimum F~ = minF,. The corre-i i:;t.b i
sponding vertices are denoted by Xb,Xn, and X~. If Fi ::; E for all i = 0, 1, ... , n,
the algorithm is finished.
(3) Calculate the center of n vertices obtained by excluding the vertex Xb from the
original n+l vertices, i,e, Xc = ~ L Xi, and calculate the symmetric point XR i:;t.b
of the vertex xb about the center Xc in X space such that XR = 2Xc - xb. If
I-12
FR = IIF(XR)II 2 Fb, then go to step (4). Otherwise, calculate the symmetric
point Xr of Xc about XR such that Xr = 3Xc - 2Xb. Replace Fb by the
smaller value of FR and Fr, Xb by the corresponding vertex, and then goto
step (2).
( 4) Calculate the midpoint XA of XR and Xc such that XA = ~ (XR + Xc). If
FA = IIF(XA) II > Fn, then reduce the initial length h to its half and return to
step (1). Otherwise, replace Xb by XA and Fb by FA, and then return to step
(2).
In the simplex method, no calculation of the gradient of the objective function
is required. The only calculation needed is to evaluate the objective function itself,
which brings great convenience in this study, since the objective function can be
easily evaluated in terms of the covariance matrix of the response and the latter
can be explicitly solved by the simplified state-variable method.
2.5 Simplified State-variable Method for Nonstationary Covariance
A brief description of the simplified state-variable method for nonstationary
solution of linear MDOF systems is given in the following. A detailed derivation
and applications of the method can be found in Hou (1990).
The governing equation of motion of a linear MDOF system be written in a
state-variable form as
where
Y=
d - Y(t) = AY(t) + F(t) dt
( ':'(t)) y = (xo) x(t) 0 vo
Y(O) =Yo (2- 19)
F = F0 77(t)n(t)
in which M, C and K represent mass, damping and stiffness properties of the sys
tem, I is an identity matrix, F 0 is a constant vector, and n(t) is a stationary
zeromean white noise with intensity S0 modulated by a deterministic envelope 77(t).
I-13
The fundamental matrix of the system can be written as
~(t) = U.A.(t)u- 1 (2- 20)
where
(
.A.u(t) 0 0 .A.22(t)
.A.(t) = . . . . . . 0 0
(2- 21)
Note that N denotes the number of degree--of-freedom of the system. The column
components of U and the submatrice of .A.(t) are:
U2k-l = Re(v(2k-l))
k= 1, ... ,N
where -~kWk = Re(>.k)
Wdk = wky~1---~-f = Im().k) k=1, ... ,N (2- 22)
The general solution can be expressed as
Y(t) = ~(t)Y o +lot~ (t- r)F(r)dr
= ~(t)Yo + L lot P(t- r)ry(r)n(r)dr
(2- 23)
where
(2 - 24)
and the submatrice Pk(t) and Vk!k = 1, ... ,N are defined as
(2- 25)
where
(2- 26)
I-14
In the special case of SDOF systems, P(t) and L are given by
(2- 27)
For zero-mean response, the nonstationary covariance matrix of the response,
Q(tr,t2 ), can be expressed in a compact matrix form as
(2- 28)
For most envelopes used in engineering, Eq. (2-28) can be explicitly integrated.
I-15
Chapter 3
Optimal location of Control Devices
3.1 Introduction
In practice, only a limited number of control devices can be used to suppress
the structural response. With the limited set of control devices, consideration must
be given to the optimal location of these devices such that the structural response
can be reduced to the largest extent possible with limited power. The decision has
to be made before the installation of the devices because bulkness of the control
devices usually prohibits a trial and error placement. Since one does not actually
know in advance the nature of the excitation to which the structure will be sub
jected, a statistical approach is warranted in determining optimal device location.
In addition, the complex configuration of many civil structures may not allow the
determination of these positions intuitively.
An algorithm is developed in this chapter to determine the optimal locations
of control devices for a given number of devices and a given structural configura
tion. The external loads is modelled as a modulated white noise and the envelope
function may be chosen based on local characteristics of the environmental loads,
say earthquake ground motion, on the site. The optimal location of the devices is
determined by comparing the controlled performance indices with the same amount
of power for different plans of device placement and then selecting the one with min
imum performance index. A demonstration is given to a five-story building with
one actuator.
I-16
3.2 Formulation
The governing equation of the cotrolled structure can be written as
MX(t) + Cx(t) + Kx(t) = F c(t) + F e(t) (3- 1)
where M, C, and K are the mass, damping, and stiffness matrices respectively; x(t)
is the structural response. The external force takes the form
F e(t) = TJ(t)n(t)Fo (3 - 2)
where F 0 is ann-dimensional constant vector, 17(t) is a deterministic time-dependent
envelope function, and n(t) is the stationary modulated white noise with zeromean
and constant intensity 80 • 17(t) consists of terms expressed by
(3 - 3)
in which a is a positive number, f3 and 1 are nonnegative numbers, and H(t) is the
unit step function. The form of (3-3) includes most envelope functions currently
used in engineering analysis. The control force F c is written as
F c(t) = Ufc(t) (3 - 4)
where fc( t) is a s-dimentional vector representing the control force from the ac
tuators and U is the n x s constant location matrix with entries being 0 or 1,
indicating the preferred location of the control devices. Note that s is the number
of the devices used.
By the independent modal space control method, the problem is reduced to
a set of control problems of SDOF systems. For the ith mode, the modal control
force should be chosen such that the modal performance index
(3 - 5)
takes its minimum. In Eq. (3-5), t1 is the duration considered andRe and R1 are
weighting coefficients.
I-17
The algorithm may be summarized in the following steps.
(a) Solve the eigenvalue problem of the uncontrolled system to obtain the natural
frequencies, critical damping ratios, paticipation factors, and mode shapes for
all the modes. If necessary, the modal truncation technique may be employed.
(b) Choose a set of patterns of the device locations, or equivalently, give the loca
tion matrix U.
(c) For every given location matrix U, obtain the optimized structural response
and the control forces required. A curve of response or performance index
reduced versus control force required can be drawn by changing the weighting
matrices Re and R1.
(d) The optimum location is determined by comparing the optimal values of perfor
mance index for different placements of the devices, or equivalently, for different
assignment for the entries of matrix U.
The location matrix U may be given by considering all the possible plece
ments mathematically, or only a few favorable placements practically based on
some engineering judgement. The latter is useful for la,rge-scale structures due to
its computational efficiency.
For given U, the total control forces is contributed from the independent modal
control forces which is obtained by solving a set of control problems of uncoupled
SDOF systems. The simplex method is used for optimization and the simplified
state variable approach is employed to evaluate the nonstationary covariance matrix
of the response.
If the number of the control devices is equal to the number of the controlled
modes, the control forces supplied by actuators may be distributed to each mode
exactly as required. If these two number are not equal, the control forces will
be assigned to each mode in such a way that the mean square error between the
distributed and required control forces is minimized.
I-18
3.3 Example
The algorithm has been tested for a simple case of a five-story building with
equally distributed mass and stiffness properties. The critical damping ratios are
assumed the same for all the modes considered. One actuator is used for this
demonstration.
For a general n-story building with equally distributed mass and stiffness prop
erties, natural frequencues and corresponding mode shapes can be explicitly found.
The exact solution for the eigenvalue of the rth mode can be expressed as
{f . (2r- l)1r Wr =2 sm..:...,.--~
2(2n + 1) r = 1, 2, ... , n (3- 5)
where m is the lumped mass for each story and k is the inter-story stiffness. The
ith component of eigenvector v r corresponding to Wr is expressed as
(i) • (2r- l)i1r Vr =Sill
2n+ 1 i=1,2, ... ,n (3- 6)
For the special case of n = 5, the first five natural frequencies are found as w1 =
0.2ssfi,w2 = 0.831fi,w3 = 1.3lfi,w4 = 1.68fi, and w5 = 1.92fi.
The results are presented in Figure 3.1 for the Shinozuka-Sato type envelope.
Three curves represent the results for placing the actuator on 1st,. 3rd, and 5th
floor respectively. The roman numbers indicate the position of the actuator. It is
observed that a comparatively smaller mean-square control force is required for the
same amount of mean-square displacement of structural responce, or equivalently,
a relatively greater structural response can be suppressed for the same amount of
control force, by placing the device on the fifth floor. Therefore, the fifth floor is
the best location for the control device.
3.4 Summary
An algorithm is developed to determine the optimal position of the control de
vices such that the structural response may be reduced by less power supply. The
analysis is based on the concept of stochastic control due to uncertainty of envi
ronmental loads and the results so obtained may apply in a probabilistic sense. To
I-19
apply this algorithm to large-scale structures, the independent modal space control
approach is employed for the design of control forces and the modal truncation tech
nique can be incorporated into the analysis. In contrast to the IMSC method by
Meirovitch and Silverberg (1983), this algorithm does not require that-the number
of actuators be equal to the number of modes to be controlled. The simplex method
is used for optimization, which does not require to computing the gradient of the
performance index. The simplified state-variable method is used to efficiently and
accurately evaluate the nonstationary covariance and the performance index. The
algorithm is demonstrated for a five-story building with one actuator. The algo
rithm may be modified to apply to the control problems based on other definitions
of the performance indices.
I-20
200
150 I
100
III 50
v 0
0 50 100 (a)
J
300
100
30
10
3
(b)
Figure 3-1 Comparison of results for a Shinozuka-Sato type envelope function obtained by placing the actuator on the 1st, 3rd, and 5th floors respectively. (a) Maximum mean square response Sxx of the 5th floor; (b) Objective function J versus E[F;]. Roman numbers indicate the position of the actuator.
4.1 Introduction
I-21
Chapter 4
Acceleration Control
In the classical closed-loop feedback control, the feedback control force is nor
mally assumed to be a sum of terms proportional to the response displacement and
velocity. However, in most civil structures, the response variable which can be most
directly measured is acceleration. Therefore, the measured acceleration data must
be integrated in the classical approach to obtain the necessary control force. This
increases on-line calculation time and introduces additional system noise. The pos
sibility of employing a control force which is directly proportional to the measured
acceleration is explored for a simple control system in this chapter. The excitation
is assumed to be a modulated white noise process. A comparison of the results for
the acceleration control with those for the conventional control force is presented.
4.2 Formulation
Assume the differential equation of motion of a controlled SDOF system be
mx(t) + cx(t) + kx(t) + F(t) = -a(t) (4- 1)
where m, c, k are mass, damping, and stiffness of the uncontrolled system respec
tively; a( t) represents earthquake ground acceleration. The closed-loop feedback
I-22
control force F ( t) may be expressed in a general form as
F(t) = a(x(t) + a(t)) + f3:i;(t) + 1x(t) (4- 2)
where a,/3 and 1 are constant gain,s. Note that x(t) + a(t) denotes the absolute
acceleration of the structure.
Eq. (4-2) implies that the designed control force consists of terms proportional
to the absolute acceleration, relative velocity and displacement of the system re-
spectively. If a is zero, Eq. ( 4-2) reduces to the classical form of the feedback
control force which involves the velocity and displacement terms only. For the sake
of convenience, the case of nonzero a and zero {3 and 1 is refered to as acceleration
control or A- control. Similarily, velocity control (V-control), displacement control
(D-control), and their combination may be defined.
The ground acceleration a(t) is modelled as a modulated white noise process
expressed by
a(t) = 17(t)n(t) (4- 3)
where 17(t) is a determinent envelope function and n(t) is the stationary white noise
with properties E[n(t)] = 0
(4- 4)
in which S 0 is the intensity of the white noise and b"(·) denotes a 6-function. In this
study, two special forms of the envelope functions are employed, i.e. the unit step
envelope
7J(t) = { ~: if t 2: 0; otherwise.
and the Shinozuka-Sato envelope
{ a(e-bt _ e-ct)
7J(t) = ' 0,
if t 2: 0; otherwise.
(4- 5)
(4- 6)
I-23
It is assumed that a = 2.32, b = 0.09, and c = 1.49 to simulate the 1940 El Centro
earthquake ( Corotis and Marshall, 1977).
The control strategy is to choose appropriate a, f3, and 1 to minimi~e a stochas-
tic quadratic performance index defined by
(4- 7)
where t 1 is the duration considered; Re and R1 are weighting coefficients; and w0
is the natural frequency of the uncontrolled system.
In order to obtain more general conclusion, a nondimensional form is preferred.
Introduce the following nondimensional parameters:
where
a a=
m X
x=-L
- f3 f3 =-c
. X x=-
Lwo
- I 1=-k
- X x=-Lw2
0
a a= Lw5
f =two wo=lf
(4- 8)
(4- 9)
Note that the quantities with bar are differentiated with respect to f. The governing
equation of the control system becomes
d2 1 + fJ d 1 + i -x + --2~~x + --x = -a df2 1 + a dt 1 + a
( 4- 10)
where
a= ry(l)n(l) (4-11)
Note that n(l) is now a stationary zero-mean white noise process with unit intensity.
The nondimensional control force can be derived as
F(l) = F(t) mLw2
0
fJ-a . ;;y-&. = 2rx+ --x
1+&. ~ 1+&.
(4- 12)
I-24
The nondimensionl performance index can then be expressed by
(4- 13)
Eqs. (4-10)-(4-13) formulate an associated nondimensional optimal control
problem. The nondimensional control gains are chosen such that the nondimen
sional performance index J is minimized. J may be evaluated explicitly in terms of
the covariance matrix of the nondimensional response which is solved for Eqs. ( 4-10)
and (4-11) by the simplified state-variable method. The optimization is performed
by the simplex method. Both methods are briefly described in Chapter I-2. The
physical control force and the resulted response can be obtained by the relationship
(4-8), (4-9), and (4-12).
4.3 Results and Conclusions
Conclusions can be drawn from the representive results in Figure 4-1 for the
Shinozuka-Sato envelope function. Figure 4-l(a) gives a comparison of time histories
of the mean square displacement Bxx by using the acceleration control, velocity
control, displacement control, and velocity-displacement control respectively. The
dashed curve represents the result for the case without control. It is observed
that the acceleration control can be used effectively as the conventional velocity-
displacement control to suppress the structural response, but no on-line integration
is required to obtain the control force from the measured acceleration.
Figure 4-l{b) shows the results for the controlled maximum Bxx versus the
mean square of the control force required. It is seen that the acceleration control
may require a greater control force for the same amount of reduction of the response,
but the difference is probably reasonable.
I-25
Similar results can be obtained for the unit step envelope function. The con
clusions indicated here can be extended to more complicated systems subjected to
more general excitation.
Maximum S~c
4
3
2
I-26
1
(a)
-I - A-cOD&roi II - V -coD &rei III- D-coDnoi VI - VD-coD&roi
I - A-control II- VD-conuo.i
0~---------+.~--------~~------~~-------=~ EIF;r (b)
Figure 4-1 Comparison of results for a Shinozuka-Sato type envelope obtained by A-control, D-control, and VD-comrol. (a) Time history of the mean square response Sxx; (b) Objective function J versus E[F;]. ~ = 0.1. Dashed curves represent the results for the uncontrolled system.
I-27
Chapter 5
Effect of Time Delay on Structral Control
5.1 Introduction
To ensure that a control algorithm can be successfully applied to the control
of a real system, there are many factors that need to be taken into account. One is
the effect of time delays. In treating ideal systems, it is assumed that the control
force can be applied at a given time as desired. In reality, time is required in
acquisition of monitored data of the response, on-line calculation to obtain the
necessary feedback control force, and execution of the required force. Therefore,
there exists a time delay before the actual control force is applied. The time delay
causes unsynchronized applicaton of the control force. The time delay may be
reduced by developing new devices and equipment, or its effect may be partly taken
into account by introducing some finer considerations such as interaction of the
structure and actuator into the analysis. But it cann<:>t be eliminated with present
day technology.
In most applications of control algorithms to civil engineering structures, the
time delay effect has been neglected based on the argument that flexible structures
usually have long natural periods as compared with the time delay, which makes
the time delay effect small. Considerable effort has recently been devoted to the
study of the time delay effect on structural control. It has been found that the time
delay not only can render the control ineffective but also may cause instability in the
system. In order to take the time delay into account, compensation techniques have
been suggested such as the Taylor expansion approach by Abdel-Rohman (1985)
I-28
and the phase-shift technique by Chung, Reinhorn, and Soong (1988). The former
applies only if the time delay is small as compared with the natural periods of the
structure. The latter may fail for certain system where the delay time happens
to be close to some critical value which is determined by the system parameters,
excitation, and the feedback control force, as shown later in this chapter.
This chapter studies the dynamic behavior of a single-degree-of-freedom system
with a delayed closed-loop feedback control force. As a preliminary investigation,
the steady-state response is addressed. Explicit expressions for steady-state ampli
tude response and resonance frequency are obtained. The existence of critical values
of delay time is discussed for a given system with harmonic excitation and feedback
control force. The effect of the time delay on the optimal active control is investi
gated and illustrated by numerical examples.
5.2 Formulation
Consider a single-degree-of-freedom oscillator excited by an external force f( t)
and controlled by a control force F(t). The governing equation of the controlled
system can be written as
mx(t) + cx(t) + kx(t) + F(t) = f(t) (5-1)
where m, c and k describe the inertia, damping and stiffness properties of the system
and the control force is assumed to take the form of
F(t) = ax(t- r) + {Jx(t- r) + "fx(t- r) (5-2)
which implies that the feedback control force consists of terms proportional to the
acceleration, velocoty, and displacement with constant gains a, {3, "{, and the control
force is applied with a delay time r. In contrast to the conventional form of the
feedback force, Eq. (5-2) also includes an acceleration term as a part of the control
force. For the sake of simplicity, it is assumed that the delay time is the same for
the acceleration, velocity and displacement, but the analysis may be extended to
the case of different delay times.
I-29
5.3 Steady-state solution
As a preliminary study, the steady-state solution of Eq. (5-l) may be obtained.
Assume
x(t) = Xoeiw,t+<P (5-3)
where / 0 , X 0 are amplitudes of the excitation and response respectively, w1 is the
frequency of the excitation, and </> is the phase shift of the response. The phase
shift is caused by the system damping as well as the control force.
Substituting Eq. (5-3) into (5-l) yields
where
X _ fo/m o-y'A2+B2
B 4> = - arctan -
A
2 [ w J 2 I a ( w J ) 2) fJw J . J A= w0 1- (-) + (--- -- cosw1r + --smw1r wo k m w0 k
2 [ w f /3w f I a w J 2 . ] B = w0 2~- + -- coswJT- (-- -(-) ) smw1 r wo k k m wo
(5 - 4)
(5 - 5)
where wo is the natural frequency of the system and r is the critical damping ratio.
Introducing the following non-dimensional parameters:
0 = Wj
wo T
r=-To
). =I_ ~(Wf)2 k m wo
{J Wj J1- = 2~--
c w0
where !o is the natural period of the system and
(5-6)
(5-7)
I-30
the steady-state solution may be rewritten as
where
X _ fo/k 0- VA2+B2
B <P = - arctan -;:
A
A = 1 - 82 + >..cos 8 + p, sin 8
B = 2~8 + p,cosO- >..sinO
(5-8)
(5-9)
Note that in the special case where there is no the feedback control force, i.e.
a= f3 = 1 = 0, the parameters >.. and p, become zero and the solution (5-8)-(5-9) is
reduced to
(5- 10) 2~8
<P = - arctan 82 1-
which agrees with the classical results of the steady-state response.
Some numerical results for the steady-state amplitude response of a single
degree-of-freedom system with m = 1.0, k = 1.0, and ~ = 0.05 are illustrated in
Figure 5-1 for different relative delay times ;.0
= 0.0, 0.01, 0.25, and 0.5. The
feedback control force is assumed that o: = f3 = 0 and 1 = 0.5. Note that ;.0
= 0
implies a regular system without time delay. It is observed that all the curves for the
nonzero delay times show similar characteristics as that for the non-delayed control
force, but their magnitude and position of the peak of the amplitude response change
with the delay times. Small delay time does not necessarily mean that its effet on
the peak amplitude response will be correspondingly small. In Figure 5-1, in fact,
the amplitude response with smallest time delay ;.o has the highest peak.
5.4 Resonance and Critical values of delay time
For a given control system, the amplitude response changes for different delay
times, as seen from Figure 5-l. It is interesting to observe how the peak amplitude
response (peak-peak response) changes with the delay time. Figure 5-2 presents
I-31
such an example. The system has unit stiffness, unit mass, and the critical damping
ratio ~ = 0.05. The control force has constant gains o: = 0.0, f3 = 0.0, and 1 = 0.2.
The external excitation has a unit amplitude. It is clear that for the given control
system there exist certain delay times for which the peak-peak response becomes
unbounded, which implies that the active control of the system will fail. The number
of these critical values of the delay time seems infinite. Therefore, it is extremely
important for a successful structural control to determine the existance of such
critical values and, if they exist, what values they would have. That is the goal of
this section.
Resonance of the steady-state response is achieved when the denominator of
Xo in Eq. (5-4) becomes zero. Let this denominator be denoted by D where
D=A2 +iP
with
= [(1- 82)2 + (2~8) 2 + (.X2 + J.L2)] + 2[(1- 82 )-X + (2~8)J.L] cos 0+
2[(1- 82)J.L- (2~8)-X] sin 0
= [(1- 82)2 + (2~8) 2 + (.X2 + J.L2)] + Ccos(O- ¢)
(:2 = 4[(1- 82)2 + (2~8)2](_x2 + J.L2)
- (1- 82 )J.L- (2~8)-X 4> = arctan (1 _ P).X + (2~8)J.L
It can be proved that D = 0 if and only if
(5- 11)
(5- 12)
(5- 13)
Note that in the above equation, both A and J.L depend on 8, the ratio of the natural
frequency of the system to the frequecy of the harmonic external excitation.
Eq. (5-13) can be numerically solved for 8. For the standard form of the
feedback control force where o: = 0, the solution for 8 can be obtained explicitly.
The following discussion will be restricted to the standard control force, but the
conclusion may be extended to the general case where o: is not zero.
In the case of o: = 0, Eq. (5-13) may be simplified as
(5 - 14)
where
in which
I-32
p = 4~4 (1- ,8'2 ) -2
q = 1- "112
,8' = ,8 c
I "{
"' =-k
denote the relative control gains.
The solution for the quadratic equation (5-14) is given by
62 = _-_p_±_V....:..__P_2 _-_4_q 2
The existance of real 6 requires that
p2- 4q;::: 0
-p± Jp2- 4q __;;_ _ ____;....:....___----"- > 0 2 -
(5- 15)
(5- 16)
(5- 17)
The possible number of the solution for 6 may be zero, one, or two, depending on
system parameters and the feedback control gains, as shown later.
Assume the solved resonance frequency ratio, if it exists, is denoted by 6*. The
critical value(s) of the delay time may be found by
(5- 18)
or explicitly,
* 1 1 <Po T = -(k +- + -)
8* 2 271" (5- 19)
where k is any integer such that the so-obtained r* is non-negative. ¢0 is the
principal value of J satisfying Eq. (5-12). It is clear that if there exists one critical
value of the delay time, there will be an infinite number of critical values which
appear periodically with a period i· . A distribution map of the possible critical values of the time delay in the
"{1
- ,8' plane may be obtained from a detailed discussion of the inequality (5-17).
It is concluded that this map depends, to large extent, on the damping ratio ~·
I-33
For a heavily-damped system with ""32 < ~ < 1, the distribution map is shown
m Figure 5-3a. For "(12 > 1, there exists only one possible resonance frequency
and, in turn, there exists only one independant set of the critical delay time w~ich
contains an infinite number of critical values. These values can be obtained from
each other by adding or subtracting a multiple of l·, as observed from Eq. (5-19).
For 'Y' ~ 1, if the point in the "(1- {3' plane falls into the region enclosed by
(5- 20)
and
-1 < "(1 < 1 (5- 21)
there will be no critical values of the delay time. If the point (1',[3') lies outside
the enclosed region, there exist two independent sets of the critical delay time each
of which contains an infinite number of critical values but with different periods.
For a lightly-damped system with ~ < -32, the distribution map is shown in
Figure 5-3b. The map looks similar to that of the heavily-damped system except
that the closed region is shrunk such that it is enclosed by only two curves as
expressed by Eq. (5-20). The radius of that region along the "(1 axis is given by
(5- 22)
Note that 0 ~ r < 1 and r does not monotonically change with the damping ratio
~- For the special case of ~ = 0, i.e. an undamped system, the closed region is
shrunk to the origin, which implies that the critical delay times always exist for a
undamped system.
Three representative examples are illustrated in Figures 5-4, 5-5, and 5-6. In
these examples, all the parameters are chosen the same except for "(1• The system
has unit mass, unit stiffness, and the critical damping ratio ~ = 0.05. The feedback
control force is assumed such that a' = {3' = 0 and 1' = 0.08, 0.2, and 1.1. The
external excitation has unit amplitude. Since ~ = 0.05 is used, the map in Figure
5-3b applies and the three different values of 1' correspond three different regions
I-34
in Figure 5-3b where the number of the independant set of critical delay time may
be zero, one, and two respectively.
Figure 5-4 shows the result for "(1 = 0.08, the case that there is no critical delay
time exists. The peak-peak response is a harmonic-like function of the-delay time,
which is expected from Eq. (5-11).
Figure 5-5 illustrates the result for "(1 = 1.1 where only one independant set
of the critical delay times exists. By independant set, it is meant that within this
set an infinite number of critical delay times exists and they differ by a multiple of
the constant i· where 8* is the resonance frequency. In this example, 8* = 1.446
and the first six critical delay times ;.0
= 0.01451,0.7061, 1.3978, 2.0894, 2.7811, and
3.4727. The increament is approximately 0.6916. For these critical values, the peak
of the amplitude response becomes unbounded. It is noted that the critical delay
time may be very small, which challenges the conventional argument that the time
delay effect may be neglected if the ratio of the delay time to the natural period of
the system is small.
The result for "( 1 = 0.2 is presented in Figure 5-6. In this case, there exist two
independant sets of critical delay times which correspond to two different resonance
frequencies. Again, within each set there is an infinite number of critical values
which differ by a multiple of the inverse of the corresponding resonance frequency.
In this special case, 8; = 1.2186 corresponds the critical values of ;.0
=0.03215,
0.8528, 1.1673, 1.6734, 2.4940, 3.3147 etc. with the increament being 0.8206; 82 = 0.7107 corresponds the critical values of ;.
0 =0.6716, 2.0787, 3.4858, etc. with the
increament being 1.4071. Again, the critical delay time may be very small.
5.5 Effect of the time delay on active control
The governing equation of a single-degree-of-freedom control system may be
written as
mx(t) + cx(t) + kx(t) = F(t) + f(t) (5- 23)
I-35
where m, c, and k are system parameters, f(t) denotes an external excitation, and
the feedback control force F(t) is normally assumed to be
F(t) = f3x(t) + "fx(t) (5- 24)
{J and 'Y are chosen such that the performance index
(5- 25)
in minimized. In Eq. (5-25), t 1 is the duration considered, R1 is a weighting
coefficient measuring importance of the response and economy in the optimization.
In optimal control theory, the control gains {J and ')' can be obtained in terms
of a Riccati matrix P, as described in the literature. The Riccati matrix P is,
symmetric and positive-definite, and can be assumed to be time-independant, as
indicated by Yang et al. (1987). These properties of P have been used in many
optimal control algorithms. It should noted that the external excitation is assumed
to be zero in determining the matrix P for mathematical covenience. An explicit
expression for f3 and ')' is given by Meirovitch and Silverberg (1983) as
for an undamped system.
f3 = 2wo( -wo + Jw5 + Rj 1)
I= wo(Vw5 + R/ 1- wo)
(5- 26)
Though this approach has been employed by many investigators, it is doubtful
to use it in a control system with a delayed feedback control force. By this approach,
the constant control gains f3 and 1 are determined by solving a steady-state Riccati
matrix equation. However, from the analysis presented in Section 5.4, there may
exist critical values of delay time for the given system, harmonic external excitation,
and the control gains {3 and 1 found previously. If the control force is now applied
unsynchronously with a time delay and the delay time happens to be equal to or
close to one of these critical values, the control algorithm will fail. Figure 5-7
presents such an example. The system studied has unit mass and unit stiffness
and is originally undamped. The amplitude response of this uncontrolled system
I-36
is unbounded at the resonance frequency ~ = 1. The controlled system with wo
undelayed control force which is determined by solving the Riccati matrix equation
or directly given by Eq. (5-26) will have a bounded amplitude response curve which
is greatly reduced due to the application of the control force. However, if the control
force is now actualy applied with a time delay, the amplitude response curve will
change quite dramatically, as shown by the dashed lines in Figure 5-8. Critical delay
times exist and one of them is ;.0
= 0.11836 corresponding the resonance frequency
~ = 1.897. If the delay time is close to 1.1836, say 0.12, the amplitude response wo
becomes nearly unbounded. If the delay time is far away from this critical value,
say 0.10, the amplitude response becomes bounded but still comparatively greater,
implying that the control algorithm is somewhat ineffective.
In order to take the delay effect into account, some compensation techniques
have been proposed. Chung, Reinhorn, and Soong (1987) used a phase shift method
in their experimental studies on active control of seismic structures. By this ap
proach, the original feedback control force expressed by Eq. (5-2) is replaced by the
control force
(5- 27)
such that both the real system and the ideal system have the same active stiffness
and active damping. The new control gains g1 and g2 can be found by solving the
following matrix equation
(
COSWjTx
1 . --SlllWjTx
Wj
w1 sm rx
COSWJT:i; ) ( :~)- ( J) (5- 28)
where rx and T:i; denote the delay times associated with displacement and velocity
terms respectively. It is assumed herein that rx = T:i; without loss of generity.
While this compensation method has worked in experimental studies, the general
applicability of the method remains doubtful. For given 1 and f3, there may exist
an infinite number of critical delay times. And, if they exist, they remain in effect
for the modified control system with control gains g1 and g2 • If the delay time is
close to the critical value and the dominant frequency of the excitation is close to
the resonance frequency, the control algorithm will fail as seen from the previous
example.
I-37
The problem can be solved by directly optimizing the performance index of the
control system with the delayed control force. In the case of steady-state response,
the performance index may be written as
(5- 29)
For given delay time and harmonic excitation, control gains f3 and "( are obtained
such that J in Eq. (5-29) is minimized. The simplex method introduced in Chapter
2 is employed for the optimization. Numerical results for steady-state amplitude
response of a system with control delay time ;.0
= 0.12 are illustrated in Figure 5-9.
Note that ;.0
= 0.12 is close to one of the critical delay times. As a comparison,
the results for the uncontrolled system and the controlled system with undelayed
Riccati control force are also presented. As expected, this control algorithm remains
valid for all the frequency ratios and the amplitude response becomes bounded at
the frequency ratio ~ = 1.8 where the standard control algorithm fails. wo
5.6 Conclusion
The characteristics of the steady-state response of systems with delayed feed
back closed-loop control force is investigated and explicit solutions for the amplitude
response and the peak-to-peak response are presented. It has been found that there
may exist critical value of delay time for given system parameters and control gains.
The number of independant sets, or families, of these critical values may be zero,
one, or two. By an independant set, it is meant that within this set there exists
an infinite number of critical delay times which correspond to a unique resonance
frequency and these values differ a multiple of a constant which is the inverse of the
resonance frequency. Therefore, if one critical delay time can be found, then there
must be an infinite number of these values belonging to the same family. Distribu
tion maps for the number of independent sets are given in the f3, "( plane for both
heavily-damped and lightly damped systems. The bounbaries between regions with
different numbers of the independent sets are determined by the critical damping
ratio of the system and the control gains.
I-38
It has been found that unsynchronized application of the standard Riccati
control force has significant effect on active control for certain structures. The
existence of critical delay times may make the designed control algorithm fail if
the system is excited near the resonance frequency and the control force is applied
with one of the corresponding critical delay times. Since the critical delay times
may be very small, the above conclusion challenges the widely-accepted argument
that a time delay produces minor effects if the delay time is small compare to the
natural period of the system. While the phase-shift compensation method does not
eliminate the effects of these critical values, the proposed optimization procedure
provides an efficient approach even for these critical delay times.
The discussion in this section is basically for the steady-state response of simple
systems with a single delay parameter. It is expected that these results may be
extended to the transient response of large-scale systems, or systems with different
delay times.
I-39
A
r' = 0.01
9
"
6 r' = 0
r' - 0.25
3
0 0t=:::::~----4---~::~==~2~==========~3f===------~~~ wo
Figure 5-l Comparison of the steady-state amplitude responses of a linear SDOF system subjected to a harmonic excitation for different delay times. The natural frequency of the system wo = 1.0 and the critical damping ratio ~ = 0.05. The control gains a= f3 = 0 and 1 = 0.5. The delay time t' = ;.
0 = 0.0, 0.01, 0.25, 0.5. Dashed curve represents the result
for non-delayed control ( ;.0
=0).
40
30
20
10
0 0 2 3 r'
Figure 5-2 A demonstrative example for the steady-state peak-peak response versus delay time where the critical values of delay time are observed. The system and control parameters are the same as in Figure 5-l except that 1=0.2.
I-40
2 i I P'
II I I -
-f- v ~
-I
III
0 I'
I
"'--- ____./ 1- --1
II
-2 I ' -2 -1 0 2
Figure 5-3a Distribution map of critical values of delay time for heavily damped systems ( )2 < ~ < 1). Region I: one independent set; Region Ii: two
independent set; Region III: no critical delay time.
2 I
/3' I
1- I -
I III i I
0 I
!'
I l
r II -I
-1
-2 I I
-2 -1 0 1 2
Figure 5-3b Distribution map of critical values of delay time for lightly delayed systems ( ~ < ~). Region I: one independent set; Region Ii: two
independent set; Region III: no critical delay time.
I-41
20
15
10
5
00~----------~----------~2,_--------~3~--------~ r'
Figure 5-4 A representive curve for peak-peak response versus delay time for case I where only one independent set of the critical delay times exist. w0 = 1.0, ~ = 0.05; a: = {3 = 0.0, 1' = 1.1; The observed critical delay times are: 0.0145, 0. 706, 1.398, 2.089, 2.781, and 3.473. The corresponding resonance frequency is ~ = 1.446.
wo
I-42
75
50
25
Figure 5-5 A representive curve for peak-peak response versus delay time for case II where two independent sets of the critical delay times exist. wo = 1.0, ~ = 0.05; a = f3 = 0.0, 1' = 0.2; The observed critical delay times corresponding the resonance frequency !::..! = 1.219 are:
wo 0.0322, 0.853, 1.673, 2.494, and 3,315. Those corresponding the second resonance frequency ':!1.. = 0. 711 are: 0.672, 2.079, and 3.486. wo
l-43
9
6
3
0~--------~~---------k----------~--------~ 0 2 3 r'
Figure 5-6 A representive curve for peak-peak response versus delay time for case III where no critical delay time exists. wo = 1.0, ~ = 0.05; a = {3 = 0.0, 1' = 0.08
10
5
1 1 .5 r'
Figure 5-7 An illustration of existance of the critical delay times for the standard feedback control force obtained by solving the steady-state Riccati matrix equation. wo = 1.0, ~ = 0.0; a = 0.0, {3 and 1 are calculated from the solved Riccati matrix; Rs = R1 = 1.0.
I-44
Figure 5-8 Comparison of the steady-state response curves for controlled and uncontrolled system. Wo = 1.0, c; = 0.0; a: = 0.0, {3 and 1 are calculated from the solved Riccati matrix; Rs = R 1 = 1.0. One of the critical delay times is 0.1183. I - uncontrolled system; II -controled system with non-delayed steady-state Riccati control force; III- controlled system with delayed Riccati control force ( ;o =0.12};
N- controlled system with delayed Riccati control force ( ;0
=0.10).
I-45
A I I I I I I I I
I rn' I I I I I I \ I I I \
I \ I \
/ \ \
' ....... ---2 3 !!!.
wo
Figure 5-9 Comparison of the steady-state response curves for controlled and uncontrolled system. w0 = 1.0, ~ = 0.0; a = 0.0, f3 and 1 are calculated to monimize the performance index with R
8 = R 1 = 1.0.
I - uncontrolled system; II - controlled system with with non-delayed Riccati control force; III- controlled system with Riccati control force delayed by 0.12; IV - controlled system with a optimized delayed control force with ;.
0 =0.12.
I-46
Chapter 6
Summary
This study has focused on practical aspects of active control algorithms which
need to be considered in order to make the algorithms developed for idealized sys
tems applicable to the real structures. Three areas have been discussed, i.e. optimal
location of control devices, acceleration control, and the effect of time delays.
An algorithm has been developed to determine the optimal location of a limited
number of devices in order to effectively suppress the structural response with less
power. The algorithm is based on stochastic control theory to take the uncertainty
of the environmental loads, say earthquake ground motion, into account. The inde
pendent modal space control method is employed to convert the control problem of
a large-scale system into a set of control problems for SDOF systems. The method is
modified to eleminate the restriction that the number of the control devices should
be equal to the number of modes to be controlled. The simplex method is used to
optimize the modal performance indices and the simplified state-variable method is
used to evaluate the nonstationary convariance matrix of the structural response.
The algorithm has been demonstrated for a five-story building with one actuator.
While the standard feedback control force consists of terms propotional to the
measured velocity and displacement, the possibility to directly use the measured
acceleration data is explored. The results show that the acceleration control may
be used as effectively as conventional velocity-displacement control to reduce the
structural response, but no on-line integration is required to obtain the control
force. Acceleration may require a greater control force but the difference is probably
reasonable. The conclusion can be extended to more complicated systems subjected
to more general excitations.
I-47
The effect of a time delay on the active control algorithms has been extensively
discussed for the steady-state case. An explicit solution has been given for the
steady-state response of a controlled system with a delayed feedback control force.
The resonance relationship has been obtained. It has been found thai there may
exist resonance frequencies and corresponding independent sets of critical values of
the delay time for which the amplitude response becomes unbounded. The number
of the independent sets of the critical delay time may be zero, one, or two. Within
each independent set, there exists an infinite number of critical delay times which
appear periodically with a period equal to the inverse of the corresponding resonance
frequency. If one critical delay time can be found, there must exist an infinite
number of critical values belonging to the same family. The distribution maps
are drawn to show the regions with different numbers of independent sets. The
boundaries of these regions are determined by the critical damping ratio of the
uncontrolled system and the control gains.
The fact that the critical delay times may be very small challenges the argument
that the time delay effect is negligible if the ratio of the delay time to the natural
period of the system is small. Moreover, the conventional active control algorithms
using steady-state Riccati control force may fail in certain critical conditions, if
the control force is applied with a delay time. The critical conditions include case
thatthe ratio of the frequency of the harmonic excitation to the natural frequency
of the system is close the resonance frequancy ratio, and the delay time is close to
any one of the corresponding critical values. While the phase-shift method does not
change the nature of problem, a control algorithm based on directly optimizing the
controlled system with delayed control force is employed herein to efficiently reduce
the structural response even at these critical conditions. These preliminary results
suggest that further study on the effect of the time delay on the active control is
warranted.
Acknowledgement
Support from the CUREe-Kajima Research Project is gratefully acknowledged.
I-48
References
1 . Abdel-Rohman, Mohamed, "Structural Control Considering Time Delay Effect," Transaction of the CSME, Vol. 9, No. 4, 1985, pp. 224-227.
2 . Bryson, A.E., Jr., and Ho, Y.C., Applied Optimal Control, Wiley, New York, 1975.
3 . Chung. L.L., Reinhorn, A.M., and Soong, T.T., "Experimental Active Control of Seismic Structures," ASCE, Journal Engineering Mechanics, Vol. 114, No. 2, pp. 241-256, 1988.
4 . Corotis, R.B. and Marshall, T .A., "Oscillator Response to Modulated Random Excitation," ASCE Journal of Engrg. Mech. Die., Vol. 103, No. EM4, 1977, pp 501-513.
5 . Himmeblan, D.M., Applied Nonlinear Programming, McGraw-Hill, Inc., 1972.
6 . Hou, Z.K., "Nonstationary Response of Structures and Its Application to Earthquake Engineering," California Institute of Technology. EERL 90-01, 1990.
7 . Iwan, W.D. and Hou, Z.K., "Explicit Solutions for the Response of Simple Systems Subjected to Nonstationary Excitations," Structural Safety, 6(1989), pp. 77-86.
8 . Kobori, T., "State-of-the-Art Report, Active Seismic Response Control," Proceedings of Ninth World Conference on Earthquake Engineering, August 2-9, 1988, Tokyo-Kyoto, Japan. (Vol. VIII)
9 . Masri, S.F., Bekey, G.A., and Gaughey, T.K., "Optimal Pulse Control of Flexible Strucutres," ASME, Journal of Applied Mechanics, Vol. 48, pp. 619-626, 1981.
1 0. Meirovitch, L. and Silverberg, L.M., "Control of Structures Subjected to Seismic Excitations," ASCE, Journal of Engineering Mechanics, 109, pp. 604-618, 1983.
1 1. Soong, T.T., "State-of-the-Art Review: Acitve Control in Civil Engineering." Engineering Structures, Vol. 10, pp. 74-84, 1988.
1 2. Yang, J.N., Akbarpour, A., and Ghaernmaghami, Peiman, "New Optimal Control Algorithms for Structural Control," ASCE J. Engrg. Mech., Vol. 113, No. EM9, pp. 1369-1386, 1987.
1 3. Yao, J.T.P., "Concept of Structural Control," ASCE, J. Stru, Div., Vol. 98, pp. 1567-1574, 1972.
I-49
Publications
1 . Iwan, W.D. and Hou, Z.K., "Some Issues Related to Active Control Algorithms," Proceedings of U.S. National Workshop on Structural Control Research, University of Southern California, 25-26 October, 1990.
2 . Iwan, W.D. and Hou, Z.K., "The Effect of Time Delay on Active Control Algorithms." (in preparation)
l(ajilna-CUREe Project
Developtnent of Ot1-lit1e System Identification
for Structure
Final Project Report
.. Yasuo Takenaka Norihide Koshika Hiroshi Ishida Kazuhiko Yamada Masatoshi Ishida Kazuhide Yoshikawa Y oshiki Ikeda Narito Kurata
(Kajima Corporation)
Sami F. Masri Isao Nishimura (CUREe Team)
February 1991
SUMMARY
An on-line system identification procedure was developed based on extended
Kalman filter, which removes noise from a system and observation, and
estimates the best vector. This procedure can be classified as a parametric
method in the time domain and can be considered as one of the adaptive
filtering techniques. Kalman filter can make a scale of a system and observa
tion small, and it does not require a large computer.
This report describes the fundamental theory of this method, the numeri
cal tests of the newly developed code, and the application to the experimental
tests conducted by CUREe.
The conclusions lead from these tests and application are: (1) The proce
dure can remove noise from a system and observation, and estimate the outline
of the response, even under a nonstationary loading such as an earthquake. (2)
The accuracy of identification depends on excitations, initial conditions of
both parameters and Riccati equation, and covariance matrices for a system and
observation. (3) Unknown parameters can be identified under free vibration
better than a nonstationary disturbance. Furthermore, stiffness can be esti
mated better than damping factor. (4) Damping factor is more sensitive to the
values of covariance matrices than stiffness, especially under an earthquake.
(5) An increase of the number of observation sensors makes identification
stable and precise.
1. INTRODUCTION
2. ON-LINE SYSTEM IDENTIFICATION PROCEDURE ------------------
2.1 Extended Kalman filter
Page
1
2
2
2.2 Application to identification -------------------------- 5
2.3 U-D algorithm 7
3. NUMERICAL TEST 8
3. 1 Model -------------------------------------------------- 8
3.2 Initial condition of identified parameters ------------- 10
3.3 Identification result- Observation without noise- 11
(a) Free vibration ----------------------------------------- 11
(b) El Centro excitation ----------------------------------- 22
(c) Taft excitation ---------------------------------------- 34
3.4 Identification result- Observation with noise- ------- 44
(a) Free vibration ----------------------------------------- 44
(b) El Centro excitation ----------------------------------- 51
(c) Taft excitation ---------------------------------------- 59
4. APPLICATION TO EXPERIMENTAL TEST ------------------------- 66
4.1 Specimen structure ------------------------------------- 66
4.2 Initial condition of identified parameters ------------- 68
4.3 Identification result ---------------------------------- 69
5. CONCLUSIONS ---------------------------------------------- 82
References -------------------------------------------------- 84
1. INTRODUCTION
The system identification has become increasingly important in the area of
structural engineering, particularly in connection with structural response
control to adverse environmental loadings such as earthquakes and strong
winds. In this collaboration study, an on-line identification procedure was
researched for use in active response control.
In general, theie are three items in classifying system identification
methods. First, system identification methods can be classified as on-line
and off-line. In on-line methods, the identification and the observation of
response are conducted simultaneously. On the other hand, in off-line meth
ods, the identification is done after the completion of observation.
Secondly, these methods can also be classified as parametric and nonparamet
ric. Parametric identification involves estimation of system parameters,
while nonparametric identification determines the transfer function of the
system in terms of analytical representation. Lastly, identification methods
can be categorized as those in the time domain and the frequency domain. In
the time domain methods, system parameters are determined from observation
data sampled in time. On the other hand, in the frequency domain methods,
modal quantities such as natural frequencies, damping ratios and modal shapes
are identified using measurements in the frequency domain.
The developed on-line identification procedure is classified as an on
line, parametric and a method in the time domain, since the final goal of this
research is adaptive control for civil structures. And this procedure is
based on extended Kalman filter, which removes noise from a system and
observation and estimates the best state of a system.
Section 2 reviews extended Kalman filter and describes its application to
system identification. Section 3 describes some numerical test using the
-1-.
newly developed analytical code. The following section reports the applica
tion to the experimental tests conducted by CUREe. And the last section
concludes the numerical test result, the application to the experimental tests
and the fundamental characteristics of the developed on-line system identifi
cation procedure.
2. ON-LINE SYSTEM IDENTIFICATION PROCEDURE
2.1 Extended Kalman Filter
The developed on-line system identification procedure is based on extended
Kalman filter, which can remove noise from a system and observation and esti
mate the best state vector. Section 2.1 reviews extended Kalman filter before
applying it to identification of a structure.
A nonlinear stochastic system can be written in discrete-time space as
(1)
(2)
in which Xt n-dimensional state vector
Yt m-dimensional output vector
It n x n system matrix
Itt m x n output matrix
Wt n-dimensional noise vector for system
Vt m-dimensional noise vector for observation
-2-
Eq. (1) IS a state equation and Eq. (2) is an output equation. The system and
output matrices depend on time because of their nonlinearities.
These two equations are linearized, through expanding the system and
output matrices in Taylor's series and considering only the first-order dif-
ferential terms. This linearization is being done around the best estimations
Xe 1t and Xe 1t-1 as shown in Eqs. (3) and (4).
(3)
(4)
in which
Fe=(~) , OXt :r=iu,
He=(~) OXt :r=i,,,_, (5)
Under the assumption that noises are Gaussian and white, this linearization
leads to extended Kalman filter algorithm in the same way as Kalman filter.
This algorithm has three main steps as follows.
(1) Filtering equations
(6)
(7)
-3-
(2) Kalman gain
(8)
(3) Covariance equations
(9)
Initial conditions
,.. -Xot-1 = Xo , Pot-1 =l:o (11)
in which n x n covariance matrix for system noise
l?t m x m covariance matrix for observation noise
Pttt·: n x n solution of Ricatti equation
Xo n-dimensional initial state vector
l:o n x n initial solution of Ricatti equation
O) Ots, Rt (12)
(13)
-4-
Kalman filter algorithm is composed of two procedures, which are filtering and
prediction. The filtering procedure, which removes noise from the system and
the observation, corresponds to Eqs. (7) and (10). On the other hand, the
prediction procedure, which estimates the best state vector from the filtered
state vector, corresponds to Eqs. (6) and (9).
This algorithm shows that an only measured one-step-delayed vector is
required at every step for removal of noise and estimation of the state vec
tor. This advantage ·does not demand large volume of a computer.
2.2 Application to identification
This section presents an application of extended Kalman filter to an on-line
identification for structures.
The matrix equation of motion can be rewritten in discrete state-space form as
Xt+l =A (8) Xt+ B (8) Ut+ Wt (14)
1n which Xt is an n-dimensional state vector composed of velocities and
displacements relative to the base. Jt(8) and B(8) depend on identified param
eters. Under the assumption that an exact parameter vector 8 is invariable
on time, 8t+t=8t and 80 =8 can be lead. This assumption means a linear
structure. When a n + m extended state vector is considered as follow,
(15)
-5-
An extended state equation can be lead;
Zt+l = ( ::.<.O:J..~:.;;.~.~~~~.~:.~) + C6~·-)
=f(zt, t) +we (16)
(17)
Eqs. (16) and (17) are the same nonlinear stochastic state and outp~t equations
as Eqs. (1) and (2), respectively.
Extended Kalman filter algorithm can be applied to Eqs. (16) and (17) as
follows.
(18a)
(18b)
(19a)
(19b)
in which Ke and Le are the n x m and m x m Kalman gain matrices, respective-
Fe = ( .~.~~.:~~!. .. [ .... ~ .. :~.:~:~:.:.~.·.::.~.~.:~::.::.~~:: .. ) ( 2 0) 0 i I
(21)
-6-
In this method, both the state vector and the unknown parameter can be
estimated at every step simultaneously. This procedure is one of the adaptive
filtering.
The developed code can remove noise from measured responses and excita
tion, and identify stiffness and its corresponding damping coefficient simul
taneously.
2.3 U-D algorithm
The developed code utilizes U-D division algorithm for the purpose of avoiding
inverse-matrix calculation and raising stability of numerical analysis. The
algorithm can divide a nondiagonal matrix into a diagonal matrix and a unit
upper triangular matrix, and inverse-martrix calculation can be transformed
into division calculation.
Covariance matrices and a solution of Ricatti 'equation are also diagonal
ized by U-D algorithm.
-7-
3. NUMERICAL TEST
3. 1 Model
Some numerical-test was conducted for the purpose of grasping fundamental
characteristics of the newly developed identification procedure.
Before identification, an exact model of strupture was set up, and the
dynamic analysis was done to obtain its response observations. This process
corresponds to make an experimental model.
A five-story building model was chosen for the numerical test.
This model has five degrees of freedom, a stiffness and its corresponding
damping coefficient in each story. Internal viscous damping type was assumed
that its damping factor h1 was 0.01 at the first natural frequency.
Table 1 shows weights, stiffnesses and damping coefficients of an exact model.
The first natural frequency is 2. 77Hz, the second is 7.44 Hz, and the third
is 11.6 Hz. Five stiffnesses and five damping coefficients were selected as
parameters to be identified, so that their values were utilized to estimate
the validity of developed code.
-8-
Table 1 Exact building Model
Weight Stiffness Damping
( tf) coefficient
(cm/s) (tf·s/cm)
Roof level W5=50.0
K5=120.0 c5 =O. 1380
5th floor W4=50. 0
K4=150.0 c4=0. 1725
4th floor W3=50. 0
K3=120.0 c3=0.2070
3rd floor w2=5o.o
K2=210.o c2=0.2415
2nd floor W1=50.0
K1=210.0 c1=0.2415
G. L.
-9-
3.2 Initial condition of identified parameters
Initial condition of the unknown parameters is shown in Table 2. The initial
value of each stiffness has a decrease of 50 per cent in comparison with the
exact value in Table 1. On the other hand, the initial value of each damping
coefficient has an increase of 50 per cent.
Table 2 Initial values of identified parameters
Stiffness Damping
coefficient
(tf/cm) (tf·s/cm)
K5= 80.0 c 5=0.2070
K4=100.0 c 4=0.2888
K3=120.0 c 3=0.3105
K2=14o.o c 2=0.3623
K1=140.0 c 1=0.3623
-10-
3.3 Identification result- Observation without noise-
In numerical test, two cases of observation, i.e., without noise and with
noise, were studied. Section 3.3 reports identification result under obser
vation without noise.
The unknown parameters were identified under a free vibration and two
typical nonstationary loadings, i.e., El Centro and Taft earthquake excita
tions.
(a) Free vibration
In the free vibration test, the response of the objective model structure was
obtained by the initial displacements of the first mode shape, which has 1.0
em at the roof level. Fig. l(a) shows the response velocity relative to the
base at each floor. Fig.l(b) shows the response displacement relative to the
base at each floor. These responses were utilized as observation for identi
fication.
The covariance matrices Q and R were assumed to be diagonal ones and invaria
ble during identification, and the values of diagonal components are shown in
Table 3. An initial solution of Riccati equation P01 - 1 was assumed to be a
diagonal matrix and to be equivalent to the covariance matrix for system Q.
Three parametric cases were selected for study of observation scale. In the
first case, all relative velocities and displacements were measured for iden
tification of the structure. Fig.2(a) shows the identified stiffness at each
floor in comparison with the exact value in time. Fig.2(b) shows the identi-
fied damping coefficient in each story in the same representation. In these
-11-
figures, the solid lines are the identified values, while the dashed lines are
the exact values shown in Table 1.
In the second case, all relative velocities (five velocities) were meas
ured. Figs.3(a) and (b) show the identified stiffnesses and damping coeffi
cients in comparison with the exact values.
In the last case, only two relative velocities at the roof and the third
floors were measured. Figs.4(a) and (b) show the identification result.
Test Result
In the case' that the response velocity and displacement at each floor can be
measured, all stiffnesses and damping factors were identified very well. In
Figs.2(a) and (b), after 1.0 second, the estimated values correspond to the
exact values. Next, in the case that the response velocity at each floor can
be measured, all stiffness also were identified very well. In Fig.3(b), all
damping coefficients do not correspond to the exact values perfectly.
In the case that only two velocities can be measured, the damping coeffi
cients at the roof and third floors become negative as shown in Fig.4(b).
This means that the identification result is unstable under two-velocities
observation.
-12-
20.
:>. ..., ......... ...... C/l C) -0 a ....... C) Q) -......-
>
-20. o.o 10. 20. 30. 1!0.
(a) Velocity at roof level Time (s)
20.
:>. ..., ...... C/l C) -0 a
....... C) Q)
>
-20. o.o 10. 20. 30. 1!0.
(b) Velocity at 5th floor Time (s)
20.
:>. ..., -...... C/l C) -0 e;
....... C) Q)
>
-20. o.o 10. 20. 30. 1!0.
(c) Velocity at 4th floor Time (s)
20.
:>. ..., -...... C/l C) -0 a ....... C) Q)
>
-20. o.o 10. 20. 30. t!o.
(d) Velocity at 3rd floor Time (s)
20.
:>. ..., -C/l C) -0 a ....... C) .... ,_ Q)
>
-20. o.o 10. 20. 30. 1.!0 •
(e) Velocity at 2nd floor Time (s)
Fig. 1 (a) Observed response velocities under free vibration
-13-
1 .a ..., c Q) e Q) ,-... C,) e t'O C,)
....... '-" 0.. en .....
0
-1 .a a.a 1a. 2a. 3a. l!a.
(a) Displacement at roof level Time (s)
1 .a .... c:: Q) e Q) ,-... C,) e t'O C,)
....... 0. o:n .....
0
-1 .a a.a 1 a. 2a. 3a. l!a.
(b) Displacement at 5th floor Time (s)
1 . a .... c:: Q)
e Q) ,-... C,) e t'O C,)
....... '-" 0.. en .....
Q
-1 .a •• #_
a.a 1 a. 2a. 3a. l!a.
(c) Displacement at 4th floor Time (s)
1 . a ..., c:: Q) e Q) ,-... C,) e t'O C,)
....... 0. en .....
. Q
-1 .a a.a 1 a. 2a. 3a. l!a.
(d) Displacement at 3rd floor Time (s)
1 .o ... c:: Q)
e Q) ,-... C,) e t'O C,)
....... 0.. en .....
0
-1 .a a.a 1a. 2a. 3a. L!a.
(e) Displacement at 2nd floor Time (s)
Fig. 1 (b) Observed response displacements under free vibration
-14-
Table 3 Components in covariance matrices for system and observation noises
under free vibration
state Covariance mat ices state Covariance matrices
vector vector
system obser. system obser.
zt Q •• 11
R .. 11 zt Q ..
11 R ..
11
Vel. R 1. OE-2 1.0E-2 Disp.R 1. OE-4 1. OE-4
Vel. 5 1.0E-2 1. OE-2 Disp.5 1. OE-4 1. OE-4
Vel. 4 1. OE:-2 1. OE-2 Disp. 4 1.0E-4 1. OE-4
Vel. 3 1. OE-2 1.0E-2 Disp.3 1.0E-4 1. OE-4
Vel. 2 1. OE-2 1.0E-2 Disp.2 1.0E-4 1. OE-4
Stiff.5 1.0 -- Damp.5 1. OE-4 --
Stiff.4 1.0 -- Damp.4 1. OE-4 --
Stiff.3 1.0 -- Damp.3 1. OE-4 --
Stiff. 2 1.0 -- Damp.2 1. OE-4 --
Stiff. 1 1.0 -- Damp. 1 1. OE-4 --
Excit. -- -- --
-15-
200 tn
::.:: en rn ........ l1l a c: u ..... --..... ..... ... ...
tr.l
-200 o.o 10. 20· 30. 1,10.
(a) Story stiffness Ks Time (s)
200 ..,.
::.::
en
"' ........ l1l a c: u ..... --..... ..... ... ....
tr.l
-200 o.o 10. 20· 30. 1,10.
(b) Story stiffness K4 Time (s)
200 M
:><:
"' , ........ l1l a c: u ..... --..... ..... ... ....
tr.l
-200 o.o 10. 20. 30. 1,10.
(c) Story stiffness K3 Time (s)
300 N
::.::
rn en ........ l1l a c: u ..... --..... ..... ... .... tr.l
-300 o.o 10. 20· 30. 1,10.
(d) Story stiffness Kz Time (s)
300 ::.::
en
"' ........ Q) e c: u .... --.... ..... .... tr.l
-300 o.o 10. 20. 30. 1!0.
(e) Story stiffness Kl Time (s)
Fig. 2(a) Estimated stiffnesses and comparison to exact values ·-
-16-
If')
u ... 0-30 c: Cll .... ........ 0 e --------------------------------------------------------- ----------------------------------() .... --.... "' Cll 0 .... 0 ... 1>0 c:
a. e -0.30 "' o.o 10· 20. 30. 1.!0. Q
(a) Story damping coefficient c5 Time (s) '<2'
u ... 0-30 c: Cll .... ........ () e
0 .... --.... "' Cll 0 .... 0 ... 1>0 c: .... a. -0.30 e
"' Q o.o 10· 20. 30. LIO.
(b) Story damping (')
coefficient c4 Time (s) u ... O-LIO c: <I> .... ........ 0 e .... 0 .... --.... "' <I> 0 .... () ... 1>0 c:
a. -O.LIO e
"' o.o 10- 20. 30. LIO. Q
N (c) Story damping coefficient c3 Time (s) u ... O-LIO c: <I> .... ........ ----------- --() e
0 .... ._ .... "' <I> 0 .... 0 ... 1>0 c .... a. e -O.LIO "' Q o.o 10. 20. 30. 1.10.
- (d) Story damping coefficient Cz Time (s) u ... 0.1.10 c: <I>
........ () e .... 0 .... --...... "' <I> 0 .... () ... 1>0 c:
a. e -O.LIO "' o.o 10· 20. 30. 1.10. Q
(e) Story damping coefficient cl Time (s)
Fig, 2 (b) Estimated damping coefficients and comparison to exact values
-17-
200 .II)
~
"' "' ....._ Q) B c tJ ..... -..... ..... ..... ... ...
en
-200 o.o 10. 20· 30. 1!0.
(a) Story stiffness K5 Time (s)
200 ..,.
...:
"' "' ........ Q) B c tJ ..... -..... ..... ...... ..... ... en
-200 o.o 10. 20- 30. qo.
(b) Story stiffness K4 Time (s)
200 C')
b.:
"' "' ........ Q) e c tJ .... -..... ..... ...... ... ... (/)
-200 o.o 10. 20· 30. 1!0.
(c) Story stiffness K3 Time (s)
300 N
...:
"' "' ........ Q) B c tJ ..... -..... ...... ... .... en
-300 o.o 10. 20· 30. qo. (d) Story stiffness K2 Time (s)
300 0.::
"' "' ........ Q) e c tJ ..... -..... ..... ... ...
en
-300 o.o 10. 20· 30. 1!0.
(e) Story stiffness Kl Time (s)
Fig. 3 (a) Estimated stiffnesses and comparison to exact values
-18-
lt:l u .... c 0-30 Q) ..... ....... () a ..... () ..... ..._
...... "' Q)
0 ..... () .... bl) c::
a. a
"' -0.30 jQ o.o 10. 20. 30. 40.
""' u Time (s) (a) Story damping coefficient c5
.... c:: 0-30 Q) ..... ....... () a
()
...... ...... ...... "' Q)
0 ..... () .... bl) c
a. .. "' 0
-0.30 o.o 10. 20. 30.
M u
(b) Story damping coefficient c4 .... o.l!o c:: Q) ..... () a
()
...... ..._
..... "' Q)
0 ..... () .... bl) c::
a. .. -0.1!0 "' 0 o.o 10. 40. 20. 30.
N u
Time (s) (c) Story damping coefficient C3 .., 0-1!0 c:: Q) ..... ....... () e
() ..... ..._ .... "' Q)
0 ..... () .... bl) c:: ..... a. a
"' jQ
-0.1!0 o.o 10. 30. l!O. 20.
...... u
Time (s) (d) Story damping coefficient c2 .., 0-1!0 c:: Q) ---------=-=-=--=--=-"'""-~~-__;_ ________ _:.__ _________ _ ..... ....... () a
() .... ..._ .... "' Q)
0 ...... () .... bl) c ..... a. a
"' 0
-0.1!0 o.o 30. 40. 10. 20.
(e) Story damping coefficient c1 Time (s)
Fig. 3(b) Estimated damping coefficients and comparison to exact values
-19-
200 tn
~ ---------------------------------------------------------------------------------------------... ... ....... a> e c 0
...... -............ ... en
-200 o.o
200
10. 20. 30. YO.
(a) Story stiffness K5 Time (s)
~ --------------------------------------------------------------------------------------------~
... "' ....... a> e c 0
..... -..... ..... ... tl)
M ~
"' "' Q) c ..... ..... ... en
N .~
"' "' Q) c ..... ...... ... en
"'
....
....... e 0 -..... ...
....... e 0 -...... ...
"' ....... a> e c 0
..... -..... ..... ... en
-200
200
-200
300
-300
300
-300
o.o 10. 20. 30. YO.
(b) Story stiffness K4 Time (s)
o.o 10. 20. 30. 1!0.
(c) Story stiffness K3 Time (s)
o.o 10. 20. 30. YO.
(d) Story stiffness K2 Time (s)
o.o 10. 20. 30. 1!0.
(e) Story stiffness K1 Time (s)
Fig. 4(a) Estimated stiffnesses and comparison to exact values
-20-
tl')
u .... c:: 0.30 QJ ..... ........ () e ..... () ..... ........ ..... "' QJ 0 ..... () .... bO c::
0. e
"' -0.30 0 o.o 10. 20. 30. llO.
'<!' u
Time (s) (a) Story damping coefficient c5 ..... 0.70 c:: QJ ..... ........ () e
() ..... ........ ..... "' QJ 0 ..... () .... bO c::
0. e -0.70 "' 0 o.o 10. llO. 20. 30.
C')
u
Time (s) (b) Story damping coefficient c4
.... O.llO ·c:: QJ ..... ........ () e
() ..... ........ ..... "' QJ 0 ..... () .... bO c:: 0. e -O.llO "' 0 o.o 10. llO. 20. 30.
N u
Time (s) (c) Story damping coefficient C3 .... o.so c:: QJ ..... ........ () e
() ..... ........ ..... "' QJ 0 ..... () .... bO c::
0. e
"' 0
-o.so o.o 10. 20. 30. l!O.
-u (d) Story damping coefficient c2
Time (s)
.... 0.60 c:: QJ ..... ........ () e ..... () ..... ........ ..... "' QJ 0 ..... () .... bO c:: 0. e
"' 0
-0.60 o.o 30. llO. 10. 20.
(e) Story damping coefficient c1 Time (s)
Fig. 4(b) Estimated damping coefficients and comparison to exact values
-21-
(b) El Centro excitation
In one of the random vibration lest, the response of the objective model
structure was obtained under El Centro (N-S component) excitation. The exci
tation was normalized to be a maximum acceleration of 100 cm/s2, and was input
at the ground level, where the model structure was fixed. Fig.5 shows the
acceleration of El Centro excitation. Figs.6(a) and (b) show the response
relative velocity and displacement at each floor, respectively, The excita
tion and these responses were utilized as observation for identification.
The covariance matrices also were assumed to be diagonal ones and invariable
during identification, and the value of diagonal components are shown in
Table 4. An initial solution of Ricatti equationPot-talso was assumed to be a
diagonal matrix and to be equivalent to the covariance matrix for observation
Q. However, the covariance matrices Q and R are different from those under
the free vibration, particularly in the components for five damping coeffi
cients.
Under El Centro, three cases, which are the same cases as under the free
vibration, were selected for study of observation scale.
Figs, 7(a) and (b) show the identified stiffnesses and damping coeffi
cients in comparison with the exact values as shown in Table 1, under all
velocities-and-displacements observation. Figs.8(a) and (b) show the identi
fied parameters under all-velocities observation. And Figs,9(a) and (b) show
those under two-velocities observation.
-22-
Test Result
Identification under El Centro is less accurate than that under the free
vibration, particularly in the estimation of damping factors.
In both cases as shown in Figs. 7 and 8, the stiffnesses are estimated
more accurately than the damping factors. Nevertheless, the estimated stiff-
ness do not become constant during the first 30 seconds. During two seconds
from a identification start, the estimated values are the same as the initial
values, since the response level is very small as shown in Fig.6(b).
l
Fig.9 shows that measurement of only two velocities makes the identifica-
tion result unstable in the estimation of both stiffness and damping factor.
-23-
c: 150 0 ..... ~ ,....... ro· ('.'!
c... en Q) -....... e Q) tJ tJ tJ <
-150 o.o 10. 20. 30. l.lQ. so. 60.
Time (s)
Fig.5 Input acceleration of normalized El Centro excitation
-24-
30.
>. ....., ,..... ..... en (.) -0 = ....... (.) Q)
>
-30. o.o 10. 20. 30. 1!0. 50. so.
(a) Velocity at roof level Time (s)
30.
>. ....., ,..... ..... en (.) -0 = ....... (.)
Q)
>
-30. o.o 10. 20. 30. 1!0. 50. so.
(b) Velocity at 5th floor Time (s)
30.
>. ....., ,..... ..... en (.) -0 = ....... (.)
Q)
>
-30. o.o 10. 20~ 30. 1!0. so. so.
(c) Velocity 4th floor Time (s)
at 30.
>. ....., ..... en (.) -0 = ....... (.) Q)
>
-30. o.o 10. 20. 30. 1!0. so. so.
(d) Velocity at 3rd floor Time (s)
30.
>. ....., ,..... ..... en (.) - ~-······ .. ,. ·.v ' M'*•'-''•YifltVt.•u 0 =
y ,v .... , ¥C
........ (.)
Q)
>
-30. o.o 10. 20. 30. 1!0. so. so.
(e) Velocity at 2nd floor Time (s)
Fig. 6(a) Observed respons~ velocities under El Centro
-25-
nRc;I="RVOT T nM fl="l ri="MTnm
1 .5 .., 1::: Q) e Q) CJ e ro CJ ....... ..._, 0. tn ......
0
-1·5 o.o 10· 20. 30. qo. so. so.
(a) Displacement at roof level Time (s)
1.5 .., c:: Q) e Q)
CJ e ro CJ
....... 0. tn .....
0
-1 .s o.o 10. 20. 30. qo. so. so.
(b) Displacement at 5th floor Time (s)
1 • 5 .., 1::: Q) e Q) ......... CJ e ro CJ ....... 0. tn
0
-1 .s o.o 10. 20. 30. qo. so. so.
(c) Displacement at 4th floor Time (s)
1 • 5 .., 1::: Q) e Q) ......... CJ e
"' CJ ....... ..._, 0. tn
0
-1 .5 o.o 10. 20. 30. qo. so. so.
(d) Time (s)
Displacement at 3rd floor 1 .s
..... 1::: Q) e Q) ......... CJ e ro CJ
0. tn ......
0
-1 .s o.o 10. 20. 30· qo. so. so.
(e) Displacement at 2nd floor Time (s)
Fig. 6(b) Observed response displacements under El Centro
-26-
nocrtlltnT T nM
Table 4 Components in covariance matrices for system and observation noises
under earthquake excitations
state Covariance mat ices state Covariance matrices
vector vector
system obser. system obser.
zt Q .. 11
R .. 1 1 zt Qii Rii
Ve 1. R 1. OE-2 1. OE-2 Dis p. R 1. OE-4 1. OE-4
Ve 1. 5 1. OE-2 1. OE-2 Disp.5 1. OE-4 1. OE-4
Ve 1. 4 1.0E-2 1. OE-2 Disp.4 1. OE-4 1. OE-4
Vel. 3 1. OE-2 1. OE-2 Disp.3 1. OE-4 1. OE-4
Vel. 2 1. OE-2 1. OE-2 Disp.2 1. OE-4 1. OE-4
Stiff. 5 1.0 -- Damp.5 1. OE-6 --
Stiff. 4 1.0 -- Damp.4 1. OE-6 --
Stiff.3 1.0 -- Damp.3 1. OE-6 --
Stiff.2 1.0 -- Damp.2 1. OE-6 --
Stiff. 1 1.0 -- Damp. 1 1. OE-6 --
Excit. 1.0 1.0 --
-27-
200 .., ... -------- --"' "' ....... Q) e c:: CJ .... .._ .... .... ... ...
(/)
-200 o.o 10. 20. 30. qo. so. 60.
(a) Story stiffness K5 Time (s)
200 """' --------------------... "' "' Q) e c:: CJ .... .._ .... .... ... ...
(/)
-200 o.o 10. 20. 30- qo. so. 60.
(b) Story stiffness K4 Time (s)
300 C')
...:: -------------------"' "' ....... Q) e c:: CJ .... .._ .... .... ... ... (/)
-300 o.o 10. 20. 30- qo. so. 60.
(c) Story stiffness K3 Time (s)
300 N ----------------------... "' "' Q) e c:: CJ .... .._ .... ...... ...
(/)
-300 o.o 10. 20. 30- qo. so. 60.
(d) Story stiffness Kz Time (s)
300 ... ----------------------"' "' ....... Q) e c:: CJ .... .._
...... .... ... .... (/)
-300 o.o 10. 20. 30. qo. so. 60.
(e) Story stiffness Kl Time (s)
Fig. 7(a) Estimated s tiffnesses and comparison to exact values
-28-
lt:> u .... 0.30 c:: Cll .... ....... -----------------------------------------------------------------------------------() E! .... () .... -.... .... "' Cll 0 .... () .... bO c:: 0. -0.30 E!
"' o.o 10. 20. 30. liO. so. 60. 0
(a) Story damping coefficient Cs Time (s)
""" u 0.30 ....
c:: Cll ------------------------------------------------------·----------------------------.... ....... () E!
() ..... -.... ..... "' Cll 0 ..... () .... bO c::
0. -0.30 E!
"' o.o 10. 20. 30. liO. so. 60. 0
(b) Story damping coefficient c4 Time (s) M
u .... O-liO c:: Cll , ... () E!
() ..... -.... ..... "' Cll 0 .... () .... bO c:: 0. -o.llo E!
"' o.o 10. 20. 30. liO. so. 60. 0
N (c) Story damping coefficient c3 Time (s) u .... O-liO c:: Cll .... ....... () a
() .... -.... .... "' Cll 0 ..... () .... bO c:: .... 0. E! -O-liO "' 0 o.o 10. 20. 30. llO. so. 60.
(d) Story damping coefficient Cz Time (s) .....
u .... O-liO c:: Cll .... ....... -- -- -------() E!
() .... -.... .... "' Cll 0 ..... () .... bO c::
0. -o.llo E!
"' o.o 10. 20. 30. liO. so. 60. 0
(e) Story damping coefficient cl Time (s)
Fig. 7 (b) Estimated damping coefficients and comparison to exact values
-29-
200 \f)
~ --- -----------------"' "' ....... Q) e c u ..... -..... ..... ..., ..., til
-200 o.o 10. 20. 30- qo. 50. 60.
(a) Story stiffness K5 Time (s)
200
""' ~
"' "' Q) e c u ..... -..... ..... ..., ..., til
-200 o.o 10. 20. 30. qo. 50. 60.
(b) Story stiffness K4 Time (s)
300 M ~
"' "' ....... Q) e c u ..... -..... ..... ..... ..., ..., til
-300 o.o 10. 20. 30- qo. so. 60.
(c) Story stiffness K3 Time (s)
300 N ~
"' "' Q) e c u ..... -..... ..... ..... ..., ..., til
-300 o.o 10. 20. 30· qo. so. 60.
(d) Story stiffness Kz Time (s)
300 -~ "' "' ....... Q) e c u ..... -.... ..... ..., ... til
-300 o.o 10. 20. 30. qo. so. 60.
(e) Story stiffness Kl Time (s)
Fig, 8(a) Estimated stiffnesses and comparison to exact values
-30-
lJ')
L)
.... 0.30 c:: <!)
•-< ........ tJ B
tJ ..... ........ ..... "' <!)
0 ..... tJ .... bD c::
0. B
"' Q
-0.30 o.o to. 50. 20. 30. 1!0. 60.
"<!' L)
(a) Story damping coefficient c5 Time (s)
.... 0.30 c:: <!)
•-< tJ B
tJ ..... ........ ..... "' <!)
0 ..... tJ
bD c::
•-< 0. B
"' Q
-0.30 o.o to. 20. 30. qo. so. 60.
M (b) Story damping coefficient C4
Time {s)
L)
.... o.qo c:: <!)
•-< ........ tJ B
tJ ..... ........ ..... "' <!)
0 ..... tJ .... bD c::
0. B
"' Q
-0.1!0 o.o to. 20. 30. l!O. so. 60.
N (c) Story damping coefficient c3
Time (s)
L) O.l!O ....
c:: <!)
•-< ........ tJ B
•-< tJ ..... ........ ..... "' <!) 0 ...... tJ .... bD c::
0. B
"' Q
-0.1!0 o.o to. 20. 30. 1!0. so. 60.
- (d) Story damping coefficient c2 Time (s)
L)
.... o.qo c:: <!)
•-< ~
(.) a (.) ..... ........ ..... "' <!)
0 ..... (.) .... bD c::
0. B -0.1!0 "' Q o.o to. 20. 30. 1!0. so. 60.
(e) Story damping coefficient C1 Time (s)
Fig. 8(b) Estimated damping coefficients and comparison to exact values
-31-
200 II')
~ -----------------------------------------------------------------------------------"' "' --Q) e c tJ
..... --..... ..... .... en
..,. ~
"' "' Q) c ..... ..... ..... ....
en
"'
....
---e tJ --..... ....
"' --Q) e c tJ
..... --..... ..... .... en
N ~
"' "' Q) c ..... .... ..... .... en
-~ "' "' Q) c ..... ..... ... en
....
---B tJ --..... ....
....... e tJ --.... ....
-200 o.o 10. 20. 30. llO. so. 60.
(a) Story stiffness K5 Time (s)
1!00
-1!00 o.o 10. 20. 30. 1.10. so. 60.
(b) Story stiffness K4 Time (s)
300
--------------------------------------------------~--~---:::--~---~---~-~~-~--~---~--
-300 o.o 10. 20. 30. llO. so. 60.
(c) Story stiffness K3 Time (s)
700
,......,;J-_ _,..~r.:.-:::-:,..,_ __ --=-C---------------------------------------------------
-700 o.o 10. 20. 30. llO. so. 60.
(d) Story stiffness K2 Time (s)
. soo
-soo o.o 10. 20. 30. llO. so. 60.
(e) Story stiffness K1 Time (s)
Fig. 9(a) Estimated stiffnesses and comparison to exact values
-32-
1.0 u .... 0-30 c: Ql ..... ....... () e
() ..... ....... ..... "' Ql 0 ..... () .... bD c: .... a. a
"' Q
-0.30 o.o 10. 20. 50. 60. 30. 1!0.
..... Time (s) (a) Story damping coefficient c5
u 0-1!0 ....
c: Ql ..... ....... () a
() ..... ....... ..... "' Ql 0 ..... () .... bD c:
c. e
"' -0.1!0
60. o.o 10. 50. 20. 30. 1!0. Q
M
Time (s) (b) Story damping coefficient C4 u
1 .o .... c: Ql ..... ....... () a
() ..... ....... ..... "' Ql 0 ..... () .... bD c:
c. e
"' -1 .o
60. o.o 10. 20. 30. 1!0. 50. Q
N
Time (s) (c) Story damping coefficient c3 u .... 0-60 c: Ql ..... () e
() ..... ....... ..... "' Ql 0 .... () .... bD c:
a. e -0.60 "' Q
60. o.o 20. 1!0. 50. 10. 30.
- Time (s) (d) Story damping coefficient c2 u .... o.ao c: Ql ·- ....... () a
() ..... ....... ..... "' Ql 0 ..... () .... bD c:
a. e -0.80 "' Q
o.o 30. LJO. so. 10. 20. 60.
(e) Story damping coefficient c1 Time (s)
Fig. 9(b) Estimated damping coefficients and comparison to exact values
-33-
(c) Taft excitation
another random vibration test, the response of the objective model structure
was obtained under Taft (E-W component) excitation. The excitation also was
normalized to be a maximum acceleration of 100 cm/s2, and was input at the
ground level. Fig.lO shows the acceleration of Taft excitation. Figs.ll (a)
and (b) show the response relative velocity and displacement at each floor,
respectively.
The covariance matrices Q and R. and the initial solution of Riccati equation
Pot-l were the same matrices as those under El Centro.
Under Taft, three cases, which are the same cases as under the free vibration
and El Centro, were selected for study of observation scale.
Figs. 12(a) and (b) show the identified parameters in comparison with the
exact values as shown in Table 1, under all-velocities-and displacements
observation. figs. 13(a) and (b) shows those under all-velocities observation.
And Figs. 14(a) and (b) show those under two-velocities observation.
Test Uesult
Identification under Taft also 1s less accurate than that under the free
vibration, particularly in the estimation of damping factor.
In both cases as shown in Figs. 12 and 13, the identification are stable,
and stiffness is estimated more precisely than damping factor. On the other
hand, identification under two-velocities observation becomes unstable as
shown in Fig. 14.
-34-
150 c: 0 ...... ....., ........
"' N s:... til <1> -........ e <1> (.) (.) (.)
<
-150 o.o 10- 20. 30. 1!0. so. 60.
Time (s)
Fig. 10 Input acceleration of normalized Taft excitation
-35-
1!0.
» ~ ,.....
Vl tJ -0 e
tJ Q) ..._,
>
-1!0. o.o 10. 20. 30. 1.10. so. so.
(a) Velocity at roof level Time (s)
llO.
» ~ .... til tJ -0 e ...... tJ Q)
>
-1.10. o.o 10. 20. 30. 1.10. so. so.
(b) Velocity at 5th floor Time (s)
1.10.
» ~ ,..... .... Vl tJ -0 e ...... tJ Q)
>
-1!0. o.o 10. 20. 30. 1.10. so. so.
(c) Velocity 4th floor Time (s)
at 1.10.
» ~ ,..... .... til tJ -0 e ...... tJ Q) ..._, >
-1.10. o.o 10. 20. 30. 1.10. so. so.
(d) Velocity 3rd floor Time (s)
at 1!0.
» ~- ,..... .... til tJ -0 e ...... tJ Q) ..._, >
-1.10. o.o 10. 20. 30. 1.10. so. so.
(e) Velocity at 2nd floor Time (s)
Fig. 11 (a) Observed response velocities under Taft
-36-
3-0 ..., c:: (!) E (!) ,....... CJ E co CJ ...... 0.
"' .... Cl
-3.0 o.o 10. 20. 30. 1!0. 50. so.
(a) Displacement at roof level Time (s)
3.0 ..., c:: (!) E (!)
CJ E co CJ ...... 0.
"' Cl
-3.0 o.o 10. 20- 30. 1!0. 50. so.
(b) Displacement 5th floor Time ( s)
at 3.0 ...,
c:: (!)
E (!) ,....... CJ E co CJ ...... 0.
"' .... Cl
-3.0 o.o 10. 20. 30. 1!0. 50. so.
(c) Displacement at 4th floor. Time (s)
3-0 ..., c:: (!)
E (!) ,....... CJ E co CJ ...... 0.
"' .... Cl
-3.0 o.o 10. 20. 30. 1!0. 50. 60.
(d) Displacement at 3rd floor Time (s)
3-0 ..., c:: (!) E (!) ,....... CJ E co CJ
0.
"' .... Cl
-3.0 o.o 10. 20. 30. 1!0. 50. 60.
(e) Displacement at 2nd floor Time (s)
Fig, 11 (b) Observed response displacements ~nder Taft
-37-
200 tn ,.. "' "' Q) e c: (.) .... -... .... .... ..... .....
(I]
-200 o.o 10. 20. 30. qa. so. 60.
(a) Story stiffness K5 Time (s)
200 ..,. -------- -,..
"' "' ....... Q) e c: (.) .... -... .... .... ..... .....
en
-200 a.a 10. 20- 30. l!O. so. 60.
(b) Story stiffness K4 Time (s)
300 M ,.. -------- -"' "' ....... Q) e c: t)
...... -... .... .... ..... ... (/)
-300 a.o 10- 20. 30. 1!0. so. 60.
(c) Story stiffness K3 Time (s)
300 N ,.. -------- --"' "' ~
Q) e c: t) .... -... .... .... .... ..... en
-300 o.a 10. 20. 30. l!O. so. 60.
(d) Story stiffness K2 Time (s)
300 -,...:
"' "' ....... Q) e c: CJ ..... -... .... ......
..... ... (I]
-300 o.a 10. 20. 30. l!O. so. sa.
(e) Story stiffness Kl Time (s)
Fig. 12(a) Estimated stiffnesses and comparison to exact values
-38-
tn u ... 0-30 c Q) ..... _....... tJ e ---------------------------------------------------------------------..... tJ
...... ........ ...... "' Q)
0 ...... () ... 1:>0 c
0. e
"' Q
-0.30 o.o 10. 50. 20. 30. LIO. 60.
...:' u
(a) Story damping coefficient c5 Time (s)
... 0.30 c Q) ..... ---------------·----------------------------------------------·-------tJ e
tJ ...... ....... ...... "' Q)
0 ...... tJ ... 1:>0 c
0. e -0.30 "' Q
o.o 10. 20. 30. LIO. so. 60.
M u
(b) Story damping coefficient c4 Time (s)
... O-LIO c Q) ..... tJ e
() ...... ....... ...... "' Q)
0 ...... () ... 1:>0 c
0. e -O.LIO "' Q o.o 10. 20. 30. LIO. so. 60.
N u
(c) Story damping coefficient c3 Time (s)
... O-LIO c Q) ..... _....... tJ e
tJ .... ....... .... "' Ql 0 ..... () ... 1:>0 c ..... 0. e
"' Q
-O.LIO o.o 10. 20. 30. LIO. so. 60.
-u (d) Story damping coefficient c2
Time (s)
... c O-LIO Q) ..... () ..
tJ .... ....... .... "' Q)
0 ..... () ... 1:>0 c
0. e
"' 0
-O.LIO o.o 10. 30. LIO. so. 20. 60.
(e) Story damping coefficient c1 Time (s)
Fig. 12(b) Estimated damping coefficients and comparison to exact values
-39-
200 on
"" "' "' Q) e c 0 ...... --..... ..... ... ...
Ul
-200 o.o 10- 20. 30· qo. so. 60.
(a) Story stiffness K5 Time (s)
200 ""' "" "' "' ........ Q) E! c 0 ..... --..... ..... ... ...
Ul
-200 o.o 10. 20. 30. qo. so. 60.
(b) Story stiffness K4 Time (s)
300 C')
"" "' "'
,.__ Q) e c CJ ..... --..... ..... ... (/)
-300 o.o 10. 20. 30. 1!0. so. 60.
(c) Story stiffness K3 Time (s)
300 "" "" "' "' Q) a c 0 ..... --..... ..... ... ...
Ul
-300 o.o 10. 20. 30. qo. so. 60.
(d) Story stiffness Kz Time ( s)
300
"" ., ., ........ Q) e c 0 ..... --..... ..... ... ... Ul
-300 o.o 10. 20. 30· qo. so. 60.
(e) Story stiffness Kl Time (s)
Fig, 13 (a) Estimated stiffnesses and comparison to exact values
-40-
In u .... c 0-30 Q) .... ....... tJ e --------------- ------------------------------------------------------tJ ..... ....... ..... "' Q)
0 ..... tJ .... bO c Q.
e -0.30 "' Q o.o 10. 20. 30. 1!0. 50. 60.
(a) Story damping coefficient Cs Time (s) ..,. u .... 0.30 c Q) .... ....... tJ e
tJ ..... ....... ..... "' Q)
0 ..... tJ .... bO c Q. -0.30 e
"' o.o 10. 20. 30. 1!0. 50. 60. Q
(b) Story damping M
coefficient c4 Time (s) u ... 0-1!0 c Q) .... ....... tJ e .... tJ ..... ....... ..... "' Q)
0 ..... tJ .... bO c Q. -0.1!0 e
"' Q o.o 10. 20. 30. 1!0. 50. 60.
(c) Story damping coefficient c3 Time (s) N
u ... 0.1!0 c Q) .... ....... tJ e
tJ ..... ....... ..... "' Q)
0 ..... tJ
bO c Q. -0.1!0 e
"' o.o 10. 20. 30. 1!0. 50. 60. Q
- (d) Story damping coefficient c2 Time (s) u .... 0-1!0 c Q) ------------.... ....... ----------------------------------------------------------------------tJ e
tJ ..... ....... ..... "' Q)
0 ..... tJ .... bO c Q. -0.1!0 e
"' o.o 10. 20. 30. 1!0. 50. 60. Q
(e) Story damping coefficient cl Time (s)
Fig. 13 (b) Estimated damping coefficients and comparison to exact values
-41-
100 lf)
"" "' "' Q) a c t) .... -... .... .... ..... ..... ....
til
-100 o.o 10. 20. 30. 1!0. so. 60.
(a) Story stiffness K5 Time (s)
1!00 ...,.
"" "' "' ,.... Q) e c t) .... -... .... .... ... ..... til
-1!00 o.o 10. 20. 30. 1!0. so. 60.
(b) Story stiffness K4 Time (s)
200 (")
"" "' "'
,.... Q) e c t) .... -... .... .... ..... ..... .....
til
-200 o.o 10. 20. 30. 1!0. so. so. (c) Story stiffness K3
Time (s)
700 N
"" "' "' ,-... Q) e c t) .... -... .... .... ..... ..... til
-700 o.o 10. 20. 30. 1!0. so. so.
(d) Story stiffness K2 Time (s)
600
"" "' "' ,-... -------------------------Q) e c t) ..... -... ...... ...... ..... ..... ...... ........ til
-600 o.o 10. 20. 30. 110. so. so.
(e) Story stiffness K1 . Time (s)
Fig. 14(a) Estimated stiffnesses and comparison to exact values
-42-
In u ... 0-30 c: Q) ..... .-.. t) e =....-~-----:.:-:.:::-:::._-·-----------------------------
t) .... ........ .... "' Q)
0 ..... t) ... t>ll c:
0. e -0.30 "' Q o.o 10. 60. 30. 1!0. so. 20.
.;,. u
Time (s) (a) Story damping coefficient C5
... o.I!O c: Q) ..... .-.. t) e
t) ..... --.... "' Q)
0 ..... t) ... t>ll c:
0. e -0.1!0 "' Q
o.o 10. 60. 30. so. 20. 1!0.
M u
(b) Story damping coefficient c4 Time (s)
... o.so c: Q) ..... ,..... t) e
t) ..... ........ ..... "' Q)
0 ..... t) ... t>ll c:
0. e -o.so "' Q o.o 10. so. 60. 30. LIO. 20.
N u
Time (s) (c) Story damping coefficient c3
... o.so c: Q) ..... t) e
t) ..... ........ ..... "' Q)
0 ..... t)
t>ll c:
0. e -0.90 "' Q o.o 10. so. 60. 30. 20. 1!0.
u
Time (s) (d) Story damping coefficient c2 ... 0-60 c: Q) ..... ,..... t) e
t) .... .._ ..... "' Q)
0 ..... t) ... t>ll c:
0. e -0.60 "' Q o.o so. 60. 10. 30. LIO. 20.
(e) Story damping coefficient Cl Time (s)
Fig. l4(b) Estimated damping coefficients and comparison to exact values
3.4 Identification result- Observation with noise-
Section 3.4 reports identification results under the observation with noise.
The unknown parameters were identified under the free vibration and two earth
quake loadings, i.e., El Centro and Taft. However, the observation included
white noise. In the noise-included cases, only the observation case that
relative velocity was measured at each floor was studied.
(a) Free vibration
The response of the objective model structure was obtained by the initial
displacements of the first mode shape, which has 1.0 em at the top level. The
relative velocities are the same as those shown in Fig. 1(a) in Section 3.3.
However, Fig. 15 shows that white noises were added to the obtained all rela
tive velocities responses. These noise-included responses were utilized as
observation for identification.
The covariance matrices Q and R were assumed to be diagonal ones and invaria
ble during identification, and the values of diagonal components are shown in
Table 5. An initial solution of Riccati equation P01 - 1 was assumed to be a
diagonal matrix and to be equivalent to the covariance matrix for system Q.
Figs. 16(a) and (b) show the estimated parameters 1n comparison with the exact
values in time under all-velocities observation. Fig.l7 shows the estimated
response velocities in which noises were removed.
-44-
Test Result
During the first ten seconds, the stiffnesses and damping coefficients corre
spond to the exact values. On the other hand, after len seconds, identifica
tion result become worse, because the observation includes only noise as shown
in Fig.l5. In the noise-included observation, stiffness also is identified
more accurately than damping factor.
The estimated response can be acquired during identification, since
Kalman filter can remove noise from observation. Comparing Fig.l7 with
Fig. l(a), the response is estimated accurately.
l 1
20·
»
1
..., ,....... ·~ ..... en
I
t) - 1
0 a ...... tJ Q)
__, >
-20·
1
o.o 10· 20·
30· 110·
(a) VelocitY at roof level Time (s) 1
1 20·
» ..., - I ...... "' -t) -0 a ...... C) 1
(!) ......,
>
-20· o.o 10· 20·
30· 1!0·
(b) VelocitY at 5th floor
Time (s)
20.
:» ..., ........ ...... "' ~~~ t) -0 e
....... C)
Q) .....,
>
-20· o.o 10. 20·
30. 1!0·
(c) VelocitY at 4th floor Time (s)
20·
» ..., ,-.. .... en -~~~~'ti*N
tJ -0 a ...... tJ Q) ·->
-20· o.o 10. 20· 30·
llO.
(d) VelocitY at 3rd floor . Time (s) 1
20.
'1-Jti-,J~ » ..., ,...... ..... en t) -0 a
....... tJ Q)
..._, >
-20. o.o 10. 20.
30· 1!0· I
(e) VelocitY at 2nd floor 'Time (s)
Fig. 15 Observed response velocities under free vibration
--16-
Table 5 Components in covariance matrices for system and observation noises
under free vibration
state Covariance malices state Covariance matrices
vector vector
system obser. system obser.
zt Q •• 1 1
R •. 1 1 zt Q ••
11 R ..
11
Vel. R 1. OE-2 1. OE+1 Disp. R 1. OE-4 --
Vel. 5 1. OE-2 1. OE+ 1 Disp.5 1. OE-4 --
Vel. 4 1. OE-2 1. OE+1 Disp.4 1. OE-4 --
Vel. 3 1.0E-2 1. OE+ 1 Disp.3 1. OE-4 --
Vel. 2 1. OE-2 1. OE+1 Disp.2 1. OE-4 --
Stiff. 5 1.0 -- Damp.5 1. OE-4 --
Stiff. 4 1.0 -- Damp.4 1. OE-4 --
Stiff.3 1.0 -- Damp.3 1. OE-4 --
Stiff. 2 1.0 -- Damp.2 1. OE-4 --
Stiff. 1 1.0 -- Damp. 1 1. OE-4 --
Exci t. -- -- --
-47-
200 tn
"" ---------------------------------------------------------------------------------------------"' "' Q) e c u ..... ......... ...... ..... ..,
tl)
-200 o.o 10. 20- 30. 40.
(a) Story stiffness Ks Time (s)
200 ..... ----------------------------------------------------
"" "' "' Q) e c u ...... ......... ..... ..... .., .... en
-200 o.o 10. 20. 30. 40.
(b) Story stiffness K4 Time (s)
200 --- ---C')
"" "' "' Q) e c tJ ..... ......... ...... ...... ..., .... tl)
-200 o.o 10. 20. 30. 40.
(c) Story stiffness K3 Time (s)
300 N
"" "' "' Q) El c tJ ...... -...... ...... .... ..... tl)
-300 o.o 10. 20. 30. 40.
(d) Story stiffness Kz Time (s)
300 -"" -- ---
"' "' ......... Q) El c tJ ...... ......... ..... ..... ..., ..... tl)
-300 o.o 10. 20. 30. 40.
(e) Story stiffness Kl Time (s)
Fig. 16(a) Estimated s ti ffnesses and comparison to exact values
-48-
tn u .... o.ao c Ql ..... ,..... tJ e
tJ ..... ....... ..... "' Ql 0 .... tJ .... bD c
Q. e
"' Q
-o.ao o.o 20. 30. qo. 10·
«:' u
Time (s) (a) Story damping coefficient c5 .... o.l!o c Q) ..... ,..... tJ e
tJ ..... ....... ..... "' Ql 0 .... tJ .... bD c Q. e -0.1!0 "' Q
o.o 30. 1!0. 10. 20.
M
Time (s) (b) Story damping coefficient c4 u .... 0-1!0 c Ql
tJ e ..... tJ ..... -... ..... "' Ql 0 .... tJ .... bD c
Q. e
"' -0.1!0
o.o 30. 1!0. 10. 20. Q
N
Time (s) (c) Story damping coefficient C3 u .... o.so c Ql ..... ,..... tJ e ..... tJ ...... ....... ..... "' Q)
0 .... tJ .... bD c Q. e
"' Q
-o.so o.o 30. 1!0. 10. 20.
- Time (s) (d) Story damping coefficient c2 u o.so .... c Ql ..... tJ e
tJ .... ....... .... "' Ql 0 .... tJ .... bD c
Q. e
"' -o.so
o.o 1!0. 10. 20. 30. Q
(e) Story damping coefficient c1 Time (s)
Fig. 16(b) Estimated damping coefficients and comparison to exact values
-49-
20.
» .... ,...... ..... en C) -0 e ..... C) (I)
>
-20. o.o 10. 20. 30. YO.
(a) Velocity at roof 1 evel Time (s)
20.
» .... ,...... ..... en C) -0 e ...... C)
(I)
>
-20. o.o 10. 20. 30. YO.
(b) Velocity at 5th floor Time (s)
20.
» .... ,...... ..... en C) -0 e ..... C)
(I)
>
-20. o.o 10. 20· 30. YO.
(c) Velocity at 4th floor Time (s)
20.
» .... ,...... ..... en C) -0 e ..... C)
<» >
-20. o.o 10. 20. 30. YO.
(d) Velocity at 3rd floor Time (s)
20.
» .... ,...... ..... en C) -0 e ..... C) (I) '-"
>
-20. o.o 10. 20. 30. YO.
(e) Velocity at 2nd floor Time (s)
Fig. 17 Estimated response velocities under free vibration
-50-
(b) El Centro excitation
The response velocities were obtained under El Centro (N-S component) excita
tion, which was normalized to be a maximum acceleration of 100cm/s2 and was
input at the ground level of the structure. White noise was added to the
excitation and the exact response velocities as measurements. Figs.18 and 19
show the noise-included excitation and response, respectively.
The covariance matrices were assumed to be diagonal ones and invariable during
estimation, and the value of diagonal components are shown in Table 6. An
initial solution of Hiccati equation P 01-1was assumed to be a diagonal matrix
and to be equivalent to the covariance matrices for observation Q.
Figs.20(a) and (b) show the estimated parameters in time, under all velocities
observation. Figs.21 and 22 show the estimated excitation and the response
velocities, respectively.
Test Hesult
Stiffness is identified well, while damping is not done. Comparing the exact
excitation in Fig.5 and the estimated excitation in Fig.2l, the outline of the
acceleration is estimated well. The high-frequency waves of the exact accel
eration are removed as no1se. On the other hand, there is a goorl correspond
ence between Fig.6 and Fig.22, since the exact response velocities have lower
frequencies than the ground acceleration.
-51-
150 c:: 0 ..... ..., -"' N (... ., Q) -..... e Q) CJ CJ ......... CJ
<
-150 o.o 10. 20. 30. 1!0. so. 60.
Time (s)
Fig.18 Observed noise-included acceleration of El Centro excitation
-52-
30.
» ..... ,.-.... ....... "' C) -0 e ...... C)
Q)
>
-30. o.o 10. 20. 30. 1!0. so. 60.
(a) Velocity at roof level Time (s)
30.
» ..... ....... "' C) -0 e ...... C)
Q)
>
-30. o.o 10. 20. 30. 1!0. so. 60.
(b) Velocity at 5th floor Time (s)
30.
» ..... ,.-.... ....... "' C) -,0 e
C)
Q)
>
-30. o.o 10. 20. 30. 1!0. so. 60.
(c) Velocity at 4th floor Time (s)
30.
» ..... ,.-.... ....... "' C) -0 e
C)
Q)
>
-30. o.o 10. 20. 30. 1!0. so. 60.
(d) Velocity at 3rd floor Time (s)
30.
» ..... ,.-.... ....... "' C) - ~/fA~~ w,•,-,_ .. ,t\'li~NltVIt"(NNNN..·n;Att~#''• '/.: c: •I ..... q ,., ... 0 e ,., ' 4 4 •u;
...... C) Q)
>
-30. o.o 10. 20. 30. qo. so. 60.
(e) Velocity at 2nd floor Time (s)
Fig. 19 Observed response velocities under El Centro
-53-
Table 6 Components in covariance matrices for system and observation noises
under earthquake excitations
state Covariance mat ices state Covariance matrices
vector vector
system obser. system obser.
zt Q •• 11
R .. 11 zt Q ••
11 R ..
11
Vel. R 1. OE-2 1. OE+ 1 Disp.R 1. OE-4 --
Vel. 5 1. OE-2 1. OE+ 1 Disp.5 1. OE-4 --
Vel. 4 1. OE-2 1. OE+1 Disp.4 1. OE-4 --
Vel. 3 1. OE-2 1. OE+ 1 Disp.3 1. OE-4 --
Vel. 2 1. OE-2 1. OE+ 1 Disp.2 1.0E-4 --
Stiff.5 1.0 -- Damp.5 1. OE-6 --
Stiff.4 1.0 -- Damp.4 1. OE-6 --
Stiff. 3 1.0 -- Damp.3 1. OE-6 --
Stiff. 2 1.0 -- Damp.2 1. OE-6 --
Stiff.1 1.0 -- Damp. 1 1. OE-6 --
Exc it. 1. OE+2 1. OE+2 --
200 In
l>d --------- -------------------------------------------------------------------"' "' ,.... Q) B c u ..... ....... ..... ..... ..., ...,
o:n
-200 o.o to. 20. 30. 1!0. so. so. (a) Story stiffness Ks Time (s)
200 .... l>d ----------------------------------------"' "' ,.... Q) e c u ..... ....... ..... ..... .... ..., ...,
o:n
-200 o.o 10. 20. 30. 1!0. so. so.
(b) Story stiffness K4 Time (s)
300 (")
l>d --- ------- -- ------ --- --------
"' "' ,.... Q) e c u ..... ....... ..... ..... ..., ...,
o:n
-300 o.o 10. 20. 30. 1!0. so. so. (c) Story stiffness K3 Time (s)
300 <'I
l>d --------------"' "'
,.... Q) e c u ..... ....... ..... ..... ..., ...,
o:n
-300 o.o 10. 20. 30. 1!0. so. so.
(d) Story stiffness K2 Time (s)
300 ..... ---------------- -------------------- --------------l>d
"' "' ,.... Q) e c u ..... ....... ..... ..... .... ..., ...,
o:n
-300 o.o 10. 20. 30. 1!0. so. so. (e) Story stiffness Kl Time (s)
Fig. 20 (a) Estimated stiffnesses and comparison to exact values
-55-
In u u 0-30 c Ql ..... ,..... t) El ..... t) ..... ....... ..... rn Ql . 0 ..... t) u
t>C c
0. e -0.30 "' Q o.o 10. 30. so. llO. 20. 60.
(a) Story damping coefficient c5 Time (s) ....
u 0-30 u c Ql ..... t) E!
t) ..... ....... ..... rn Ql . 0 ..... t) ... t>C c Q. -0.30 e
"' o.o 20. 30. 1!0. so. 10. 60.
Q
(b) Story damping coefficient c4 Time (s) M
u 0-llO u c Ql ..... ,..... -----------------------------~-----------------------------~-----------------------t) El
t) ..... ....... ..... "' Ql . 0 .... t) u
t>C c:
Q. -0.1!0 E!
"' o.o llO. so. 10. 20. 30. 60.
Q
N (c) Story damplng coefficient c3
Time (s)
u 0-llO u c Ql ..... ,..... t) E!
t) ..... ....... ..... rn Ql 0 .... t) ... t>C c Q. E!
"' -0.1!0
o.o 20. 30. 1!0. so. 10. GO. Q
- (d) Story damping coefficient c2 Time (s)
u 0-llO ... c Ql ..... ,._ 0 E!
t) .... ....... .... rn Ql 0 .... t) ... t>C c
Q. E!
"' -0.1!0
o.o llO. so. 10. 20. 30. 60. Q
(e) Story damping coefficient Cl Time (s)
Fig. 20(b) Estimated damping coefficients and comparison to exact values
-56-
150 c:: 0 ....., ,-....
"' N
"" tl)
Q) -...... Ei Q) (..) (..) (..)
< -150 o.o 10- 20. 30. 1!0. so. so.
Time (s)
Fig,21 Esimated acceleration of El Centro excitation
-!i7-
30.
» ...., ,..... ...... en (.) -0 e ....... (.)
(!) ......., >
-30. o.o 10. 20. 30. qo. so. so.
(a) Velocity at roof level Time (s)
30.
» ...., ...... en
(.) -0 9 ....... (.)
(!)
>
-30. o.o 10. 20. 30. qo. so. so.
(b) Velocity at 5th floor Time (s)
30.
» ...., ...... en (.) -0 e ....... (.)
(!)
>
-30. o.o 10. 20. 30. qo. so. so.
(c) Velocity at 4th floor Time (s)
30.
» ...., ,..... ...... en (.) -0 e
....... (.) (!) .......,
>
-30. o.o 10. 20. 30. qo. so. so.
(d) Velocity at 3rd floor Time (s)
30.
» ...., ,..... ...... en (.) -0 e
....... (.) (!)
>
-30. o.o 10. 20. 30. qo. so. so.
(e) Velocity at 2nd floor Time (s)
Fig. 22 Estimated response velocities under El Centro
.-58-
(c) Taft excitation
The response velocities were obtained under Taft (E-W component) excitation,
which also was normalized to be a maximum acceleration of 100cm/s2 and was
input at the ground level. White noi•e was added to the excitation and the
obtained response velocities. Figs.23 and 24 show the noise-included excita
tion and response, respectively.
The covariance matrices for system and observation, and the initial solut'ion
of the Riccati equation were the same as those under El Centro.
Figs.25(a) and (b) show the estimated parameters in time, under all-velocities
observation. Figs.26 and 27 show the noise-removed excitation and response
velocities, respectively.
Test Result
Stiffness is identified more precisely than damping, even under Taft. Compar
ing Fig.lO with Fig.26 and Fig.ll with Fig.27, the noise is removed and the
outline of the response is estimated in the excitation and the response veloc
ities very well.
-59-
150 r:: 0 ...... ...., ....... "' CN ~ , Q) -....... e Q) CJ CJ CJ <
-150 o.o 10. 20. 30. 1!0. so. 60.
Time (s)
Fig.23 Observed noise-included acceleration of Taft excitation
-60-
L!O.
;::., ..., ..... "' CJ ......_ 0 e ...... CJ Q)
>
-1.!0. o.o 10. 20. 30. L!O. so. so.
(a) Velocity at roof level Time (s)
L!O.
;::., ..., ..... "' CJ ......_ 0 e ...... CJ Q)
>
-1.!0. o.o 10. 20. 30. l!Q. so. so.
(b) Velocity at 5th floor Time (s)
L!O.
;::., ..., ..... "' CJ ......_ 0 e ...... CJ Q)
>
-1.!0. o.o 10. 20. 30. L!O. so. so.
(c) Velocity at 4th floor Time ( s)
L!O.
;::., ..., ,...... ..... "' CJ ......_ 0 e
...... CJ Q)
>
-1.!0. o.o 10. 20. 30. L!O. so. so.
(d) Velocity at 3rd floor Time (s)
L!O.
;::., ..., ,...... ..... "' CJ ......_ 0 e ...... CJ Q)
>
-1.10. o.o 10. 20. 30. 1.10. so. so.
(e) Velocity at 2nd floor Time (s)
Fig. 24 Observed response velocities under Taft
-61-
200 tn
:X: -------------------------- -----------------------"' "' ........ Q) a c t) .... ....... .... .... .... ....
til
-200 o.o 10. 20. 30- 1!0. so. 60.
(a) Story stiffness Ks Time (s)
200
""' -------------- -------------------------:X:
"' "' Q) a c t) .... ....... .... .... .... .... ....
til
-200 o.o 10. 20. 30. 1!0. so. 60.
(b) Story stiffness K4 Time (s)
300 M
:X: ----------------------------"' "' Q) B c tJ .... ....... .... .... .... .... til
-300 o.o 10. 20. 30. 1!0. so. so.
(c) Story stiffness K3 Time (s)
300 N
:X: ------------------"' "' Q) a c tJ .... ....... .... .... .... .... til
-300 o.o 10. 20. 30. 1!0. so. 60-
(d) Story stiffness K2 Time (s)
300 :X: -------------- -----------------"' "' ........ Q) e c tJ .... ....... .... .... .... .... V)
-300 o.o 10. 20. 30- 1!0. so. 6Q ..
(e) Story stiffness Kl Time (s)
Fig. 25 (a) Estimated s tiffnesses and comparison to exact values
-62-
If)
u .... 0.30 c: Q) ..... ......_ tJ E! ..... tJ ..... ........ ..... 1/)
Q)
0 ..... tJ .... bO c:
c. E! -0.30 "' 0 o.o 10. 20. so. 30. 1!0. so.
""' Time (s) (a) Story damping coefficient c5
u .... 0.30 c: Q) ..... ......_ --------------- -----------------------------------------------------·-------------tJ E!
tJ ..... ........ ..... 1/)
Q)
0 ..... tJ .... bO c: ..... c. E!
"' Q
-0.30 o.o 10. 20. 30. 1!0. so. so.
M
Time (s) (b) Story damping coefficient c4 u .... 0-1!0 c: Q) ..... ,...._ --------------------------- -·---------------------------------------tJ E!
tJ ..... ........ ..... 1/)
Q)
0 ..... tJ
bO c:
c. E!
"' 0
-0.1!0 o.o 10. 20. so. so. 30. 1!0.
N Time (s) (c) Story damping coefficient c3 u
.... 0-1!0 c: Q) ..... ,...._ -------------------------------------------------------------------tJ E!
tJ .... ........ .... 1/)
Q)
0 ..... tJ .... bO c:
c. E! -0.1!0 "' Q
o.o 10. so. so. 20. 30. 1!0.
- Time (s) (d) Story damping coefficient c2 u .... 0-1!0 c: Q) ..... ,...._ tJ E! ..... tJ ..... ........ ..... 1/)
'1)
0 .... tJ
bO c:
c. E!
"' Q
-0.1!0 o.o 10. 20. so. so. 30. 1!0.
(e) Story damping coefficient c1 Time (s)
Fig. 25(b) Estimated damping coefficients and comparison to exact values
-63-
150 c:: 0 ...... ~ ........
"' N f., "' Q) -..... e Q) (.) (.) (.)
<
-150 o.o 10. 20. 30. l.lO. so. GO.
Time (s)
Fig.26 Esimated acceleration of Taft excitation
-6,1-
LIO.
:>. ...., ""' ...... en
tJ -0 s -< tJ Q) ...._.
>
-LIO. o.o 10. 20. 30. LIO. so. so.
(a) Velocity at roof level Time (s)
LIO.
:>. ...., ...... en tJ -0 s
tJ Q)
>
-LIO. o.o 10. 20. 30. LIO. so. so.
(b) Velocity at 5th floor Time (s)
LIO.
:>. ...., ""' ...... en
tJ - ~'MW•'Ht<'" Ut.'''"'"' 0 s ....... tJ Q)
>
-LID. o.o 10. 20. 30. LIO. so. so.
(c) Velocity at 4th floor Time (s)
LIO.
:>. ...., ,-.. ...... en tJ - .liNN"· • • ,ININ._ ., ~ttv.J,'t'"'ln'IJNfNw'IIM" 0 s .. ....
....... tJ Q)
>
-LIO. o.o 10. 20. 30. LIO. so. so.
(d) Velocity at 3rd floor Time (s)
LID.
:>. ...., ,-.. ...... en
~W< tJ - tY" •>lt•vY/h\ 0 s ....... tJ
Q)
>
-Lio. o.o 10. 20. 30. LIO. so. so.
(e) Velocity at 2nd floor Time (s)
Fig. 27 Estimated response velocities under Taft
-65-
4. APPLICATION TO EXPERIMENTAL TEST
After the numerical test, the newly developed system identification procedure
was applied to an experimental test conducted by CUREe. Section 4 describes
an experimental specimen structure used for identification, and report the
application result to the structure.
4.1 Specimen structure
A specimen structure for identification test is 2. 7 meters high and a steel
framed model which has five stories, as shown in Fig.28. The third story is
the highest than other stories. The each floor is made of a steel plate,
which has thickness of 1.9 em. Total weight of the structure and each story
weight had already known before identification test. For identification, the
structure was model as a mathematical model, which 1s a five-degree-of-free
doms lumped mass-spring-dashpot model. Thus, five stiffnesses of stories and
five corresponding damping coefficient are selected as parameters to be iden
tified.
-66-
3 6. 3 kg
If I PL. 200 X 1000 X 19 ~ 45 0
It 44. 8kg
4 r\ 2-L 50x50x5
4 ~ 0 ~. 45
f·:::.:·:::::::.J 675
lo. 48. 4kg j
so-
2-SPL. 400x100x3.2 675
2,700 90 4 /
~ 0 ~ PL.400 X 2700 X 3.2 4 /
v 84. 8kg v ...: 675
f .,
45 0 4 ~ 4 ~ '
44. 8kg ...: -r-; .. l
- PL. 200 X 1000 X 19 r.·.·.·.·.·.·.·.·.·.·.J
675 45 0
....l IL JIL
1 1,000 1 w /
7 PL. 200 X 1000 X 19 Unit: [mm]
Fig. 28 Specimen structure
-67-
4.2 Initial condition of identified parameters
Initial condition of the identified parameters is shown in Table 7. All
initial values of stiffness are the same, and all damping coefficients are the
same.
Table 7 Initial values of identified parameters
Stiffness Damping
coefficient
(kgf/cm) (kgf·s/cm)
K5=10.0 C5=0.0010
K4=10.0 c4=0.0010
K3=10.0 C3=0.0010
K2=10.0 C2=0.0010
K1=10.0 C1=0.0010
-68-
4.3 Identification result
The unknown parameters, i.e., stiffness and damping coefficient, were identi-
fied under two free vibration tests. In one of them, the specimen structure
was vibrated by a hammer which shocked at the top floor level. In another
·test, the structure was vibrated by a hammer which shocked at the fourth floor
level. As the results, the former test included the first natural frequency
mainly, while the latter the second natural frequency mainly.
The observed acceleration was integrated into velocity before identifica
tion, because the developed procedure can work under the measurements of
velocity and/or displacement in theory. Fig.29 shows the observed response
accelerations and Fig.30 shows the integrated velocities used directly for
identification almost under first mode free vibration. Figs.31 and 32 show
the observed response accelerations and the integrated velocities, respective
ly, under the second mode free vibration.
The covariance matrices for system and for observation R were assumed to be
diagonal ones and invariable during identification, and the value of diagonal
components are shown in Table 8. An initial solution of Riccati equation also
was assumed to be a diagonal matrix and to be equivalent to the covariance
matrix for observation.
Fig.33(a) and Fig.33(b) show the identified stiffnesses and damping coeffi
cients, respectively, in time, under the first mode free vibration. On the
other hand, Fig.34(a) and Fig.34(b) show the estimated parameters under the
second mode free vibration.
-69-
Test Result
Table 9 shows values of the parameters estimated just at the 40th second from
identification start. Table 10 shows the modal quantities calculated from the
identified parameters.
Comparing the first mode test result with the second mode one, identifi
cation in the estimation of stiffness is better than of damping coefficient.
That is, stiffness keeps almost constant in both tests, while damping does
not. In other words, the natural periods is identified better than the modal
damping factors. This tendency is the same as one in the numerical test in
Section 3.
-70-
c:: 0 ......
c:: 0 ...... ..., .....__ ~ C\:J ~ en (l) .........
........ e (l) tJ tJ '-"' tJ
....::
.... .....__ ~ C\:J ~ en (l) .........
1
0
TEST2R.I01 (a) Acceleration at roof level
1
--. e (l) tJ tJ '-" u
<
TEST2A.102
(U ~ ~.UU.Ii.iJ -~ l ' I V t1 I ' I JJV"~'fV1'
. ...,
c:: 0
.., ....... ~ C\:J ~ en (l) .........
........ e (l) u tJ ...._ tJ
....::
1
0
TEST2R .J 03
TEST2R ·1 0'-l
(b) Acceleration at 5th floor
(c) Acceleration at 4th floor
(d) Acceleration at 3rd floor
..., .........
~ ~ Q 1-"-.J·-U'-11 (l) ......... ....... e; (l) u tJ '-' tJ
<
TEST2R.J05 (e) Acceleration at 2nd floor
FIG.u.s.c EXPERIMENT Fig. 29
Observed response accelerations under lst mode free Vibration -71-
'-lO SEC
o.so
:>. ..... ..... tn (.) -·o 5 ...... (.) (I)
> -o.so
o.o 10. 20· 30. 1.!0.
(a) Velocity at roof level
o.so
>o ....., ......... ...... en (.) -0 a ...... (.)
(I)
>
-o.so o.o 10. 20· 30. 1.!0 •
(b) Velocity at 5th floor o.so
:>. ....., -..... ., (.) -0 a ...... (.)
(I) .......... >
-o.so o.o 10 0 20· 30. l!O.
(c) Velocity at 4th floor o.so
:>. ....., ......... ..... en (.) -0 e ...... (.) (I)
>
-o.so o.o 10. 20. 30. 1.!0-
(d) Velocity at 3rd floor o.so
:>. ....., ......... ..... en (.) -0 a ...... (.)
(I) ->
-o.so o.o 10. 20. 30. 1.!0-
(e) Velocity at 2nd floor Time (s)
Fig. 30 Integrated velocities under 1st mode free vibration
-72-
c:: 0 ..... ...., ,......, «l C'l 1-< tn
Q) - e Q) () () '-' ()
<
c:: 0 ..... ...., ,......, «l C'l 1-< tn
Q) -...... e Q) () () ()
<
c:: 0 ..... ...., ,......, «l C'l 1-< tn
Q) -...... e Q) () () ()
<
o.s
0
0-2
2
0
c:: 1 0 ...... ...., ,......, ~ C'JC/) 0 Q) -...... e Q) () () '-' ()
<
TEST2B.101 (a) Acceleration at roof level .
TEST2B.102 (b) Acceleration at 5th floor
TEST2B .1 03 (c) Acceleration at 4th floor
TEST2B .lOLl (d) Acceleration at 3rd floor
TEST2B-105 (e) Acceleration at 2nd floor
0
FIG.u.s.c EXPERIMENT Fig. 31 Observed response accelerations under 2nd mode free vibration
-73-
10 SEC
;:.. +-' ,...., ..... tn
tJ -o e ...... tJ
Q) ..........
>
» ............ ..... tn
tJ -o e ...... tJ
Q)
>
;:.. ............ ....... tn
tJ -o e ...... tJ
Q) .........
>
..... tn
tJ -o e ....... tJ
Q)
>
;:.. ...., ,...., ..... tn
tJ -o e ...... tJ
Q)
>
0-10
-0-10
0-10
-0-10
0-10
-0-10
0-10
-0-10
0-10
-0-10
o.o
o.o
o.o
o.o
o.o
Fig. 32
2-0 IJ.O 6-0 a.o (a) Velocity at roof level
IJ.O 6-0 a.o (b) Velocity at 5th floor
IJ.O 6-0 a.o (c) Velocity at 4th floor
IJ.O 6-0 a.o
(d) Velocity at 3rd floor
IJ.O 6-0 a.o
(e) Velocity at 2nd floor
Integrated velocities under 2nd mode free vibration
-H-
10-
10.
10.
10.
10.
Time (s)
Table 8 Components in covariance matrices for system and observation noises
under 1st and 2nd mode free vibrations
state Covariance matices state Covariance matrices
vector vector
system obser. system obser.
zt Q .. 11
R .. 11 zt Q ••
11 R ..
11
Vel. R 1. OE-8 1. OE-8 Disp,R 1. OE-10 --
Vel. 5 1. OE-8 1. OE-8 Disp.5 1. OE-10 --
Vel. 4 1. OE-8 1. OE-8 Disp.4 l.OE-10 --
Vel. 3 1. OE-8 1. OE-8 Disp,3 l.OE-10 --
Vel. 2 1. OE-8 1. OE-8 Disp,2 l.OE-10 --
Stiff.5 1. OE-10 -- Damp,5 1. OE-15 --
Stiff.4 1. OE-10 -- Damp,4 l.OE-15 --
Stiff.3 1. OE-10 -- Damp.3 1. OE-15 --
Stiff.2 1. OE-10 -- Damp.2 1.0E-15 --
Stiff. 1 l.OE-10 -- Damp, 1 1. OE-15 --
Exci t. -- -- -- -- --
-75-
IJ')
u .. c Q) ...... ..... E! tJ tJ
....... ... ., ... . Q) ... 0 1>0 tJ .... 1>0 c
0.. E!
"' 0
..,. u .. c Q) ..... ...... tJ e
tJ ... ....... .... ., Q) . 0 .... tJ 1>0 .... 1>0 c ..... 0. E!
"' 0
<') u .. c Q) ..... ...... tJ E!
tJ .... ....... .... ., Q)
0 .... tJ 1>0 .... 1>0 c
0.. e
"' 0
N u .. c Q) ...... ..... e tJ tJ ..... ....... .... ., .... Q) .... 0 1>0 tJ .... 1>0 c
0. E!
"' 0
.... u .. c Q) ..... ..... 0 E!
tJ ... ....... .... ., Q)
0 ... 0 1>0 .... bO c 0. e
"' 0
o.oso
-o.oso o.o 10. 20. 30. llO.
Story damping coefficient c5 o.oso
-o.oso o.o 10. 20. 30. L!O.
Story damping coefficient c4
o.oso
--~~~
-o.oso o.o
0-020
-0.020 o.o
0.020
-0.020 o.o
Fig. 33(b)
10. 20. 30. L!O.
Story damping coefficient c3
10. 20- 30. l!O.
Story damping coefficient c2
10. 20. 30. llO.
Story damping coefficient cl Time (s)
Estimated damping coefficients(lst mode free vibration)
-77-
60. lr.l
::.0::
en ,_.... en e Q) (.),
c:: -...... ...... ...... bO ..... .><: +-" -U)
-60. o.o 2-0 li.O 6.0 a.o 10.
Story stiffness K5 60.
~ ::.0::
en ,_.... en e Q) (.)
c:: -..... ..... ..... bO ..... .><: +-" -U)
-60. o.o 2-0 li.O 6-0 a.o 10.
Story stiffness K4 60.
M ::.0::
en ........ en e Q) (.)
c:: -..... ..... ..... bO ..... ~ +-" U)
-60. o.o 2-0 li.O 6-0 a.o 10.
Story stiffness K3
N 60.
::.0::
en ........ "' e Q) (.)
c:: -..... ..... ...... bO ..... ~ .., U)
-60. o.o 2-0 li.O 6.0 a.o 10.
Story stiffness K2 60 . .....
::.0::
"' ........ "' e Q) (.)
c:: -...... ..... ...... bO ..... ~ .., U)
-60. o.o 2-0 ll-0 6-0 a.o 10.
Story stiffness Kl Time (s)
Fig. 34(a) Estimated stiffnesses(2nd mode free vibration)
-78-
tt> u 0.020 ... = Ql ....... .... e t) t)
........ ..... "' ..... Ql .... 0 toO t) "" toO = .... c.
-0.020 e
"' o.o Q
.... u ... 0-020 = Ql .... ....... t) e
t) .... ........ .... "' Ql 0 .... t) toO
"" toO
= c. e -0.020 "' Q o.o
C') u ... 0-020 = Ql .... ....... t) E
t) .... ........ .... "' Ql 0 .... t) toO
"" toO = c. e -0.020 "' Q o.o
N 0-020 u ...
= Ql ....... .... e t) t)
........ .... "' .... Ql ..... 0 toO t) "" toO = c. -0.020 e
"' o.o Q
..... u ... 0-020 = Ql .... ....... t) a
t) .... ........ .... "' Ql 0 ..... t) toO
"" toO
= .... c. e -0.020 "' Q o.o
Fig. 34(b)
2.0 ij.Q 6.0 a.o 10.
Story damping coefficient Cs
2-0 1.!.0 6-0 a.o 10.
Story damping coefficient c4
2-0 1.!.0 6.0 a.o 10.
Story damping coefficient c3
0
2-0 1.!.0 6.0 a.o 10.
Story damping coefficient cz
2-0 1.!-0 6.0 a.o 10.
Story damping coefficient ci Time (s)
Estimated damping coefficients(2nd mode free vibration) -79-
Table 9
Parameter
Stiff. 5
Stiff. 4
Stiff. 3
Stiff. 2
Stiff. I
Damp.5
Damp.4
Damp.3
Damp.2
Damp. 1
Identified parameters
Identified values
1st mode 2nd mode
test test
27.4 29.8
43. 4 38.3
6. 6 7.3
46. 3 51.8
49.6 58.3
0.0307 0.0019
0.0265 0.0042
0.0467 0.0187
0.0064 0.0117
o. 0092 0.0154
-80-
Unit:
kgf/cm
Unit:
kgf·s/cm
Table 10 Identified natural peripods and modal damping factors
Natural periods (s) Damping factors (%)
Mode 1st mode 2nd mode 1st mode 2nd mode
test test test test
1 1. 051 1. 004 1. 46 0.56
2 0.356 0.335 1. 48 0.64
3 o. 215 0.213 2. 12 0.50
4 0. 129 0. 131 0.92 0.30
5 o. 128 o. 120 1. 71 0.66
-81-
5. CONCLUSIONS
An on-line system identification procedure was developed based on extended
Kalman filter, which removes noise from a system and observation, and
estimates the best vector. This procedure can be classified as a parametric
method in the time domain and can be considered as one of the adaptive
filtering techniques.
Section 2 describes the fundamental theory of the method. Section 3
reports the numerical tests of the newly developed code. And Section 4
reports the application to the experimental tests conducted by CUREe.
The following conclusions are introduced from the numerical tests and the
application to the experimental tests, and these are the fundamental charac-.
teristics of the developed procedure.
1. The procedure can remove no1se from a system and observation, and can
estimate the outline of the response, even under a nonstationary loading such
as an earthquake.
2. The accuracy of identification depends on excitations, initial values of
the estimated parameters, covariance matrices for a system and observation,
and an initial solution of Ricatti equation.
3. Unknown parameters can be identified under free vibration better than under·
a nonstationary disturbance such as an earthquake. Furthermore, stiffness can
be estimated better than damping factor. Because an observation under an
earthquake is very transient and the identification is conducted from only
one-step measurement.
-82-
4. Damping coefficient is more sensitive to the values of covariance matrices
than stiffness, especially under an earthquake.
5. An increase of the number of observation sensors makes identification
stable and precise.
In this collaboration study, an identification procedure for use in active
response control was researched. This purpose demands a small scale of an
observation and computer system. In addition, it requires a method in the
time domain and a high speed algorithm for a computer. These are why Kalman
filter technique was applied. However, only one-step measurement is not
enough particularly for the estimation of damping factor, and nonlinearity in
a system matrix demands much computional time in the estimation of a large
multi-degree-of-freedoms structure. These characteristics might be the dis
advantages for use in active response control, and further investigation and
improvement would be expected.
-83-
References
1. T.Katayama, 'Application of Kalman filter', Asakura-syoten, 1983
(in Japanese)
2. M.Hoshiya and E.Saito, 'Structural identification by extended Kalman
filter', Journal of engineering mechanics, ASCE Vol.llO No. 12(1984),
pp. 1757-1770
3. K. Toki, T.Sato and J.Kiyono, 'Identification of structural parameters
and input ground motion from response time histories', Structural
engineering/Earthquake engineering, Proceeding of JSCE(Japan Society
of Civil Engineers) Vol.6 No.2(1989), pp.413-421
4. E.~afak, 'Adaptive modeling, identification, and control of dynamic
structural systems !:Theory', Journal of engineering mechanics,
ASCE Vol.ll5 No. 11(1989), pp.2386-2405
5. E.~afak, 'Adaptive modeling, identification, and control of dynamic
structural systems 2:Applications', Journal of engineering mechanics,
ASCE Vol. 115.No.ll(l989), pp.2406-2425
-84-
