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,I III '
PB88-163738
NATIONAL ~ENTER FOR EARTHQUAKEENGINEERING RESEARCH
State University of New York at Buffalo
PRACTICAL CONSIDERATIONS FORSTRUCTURAL CONTROL: SYSTEM
UNCERTAINTr: SYSTEM TIME DELA~
AND TRUNCATION OFSMALL CONTROL FORCES
by
J.N. Yang and A. AkbarpourDepartment of Civil, Mechanical and Environmental Engineering
The George Washington UniversityWashington, D.C. 20052
Technical Report NCEER-87-0018
August 10, 1987
This research was conducted at George Washington University and was partiallysupported by the National Science Foundation under Grant No. ECE 86-07591.
REPRODUCED BYU.S. DEPARTMENT OF COMMERCE
NATIONAL TECHNICALINFORMATiON SERVICESPRINGFIELD. VA. 22161
NOTICEThis report was prepared by The George Washington University as a result of research sponsored by the National Center forEarthquake Engineering Research (NCEER) and the NationalScience Foundation. Neither NCEER, associates of NCEER, itssponsors, The George Washington University, nor any personacting on their behalf:
a. makes any warranty, express or implied, with respect to theuse of any information, apparatus, method, or process disclosed in this report or that such use may not infringe uponprivately owned rights; or
b. assumes any liabilities of whatsoever kind with respect to theuse of, or the damages resulting from the use of, any information, apparatus, method or process disclosed in this report.
III III II
PRACTICAL CONSIDERATIONS FOR STRUCTURAL CONTROL:SYSTEM UNCERTAINTY, SYSTEM TIME DELAY, AND
TRUNCATION OF SMALL CONTROL FORCES
by
IN. Yangl and A. Akbarpour2
August 10, 1987
Technical Report NCEER-87-0018
NCEER Contract Number 86-3021
NSF Master Contract Number ECE-86-07591
and
NSF Grant No. ECE-85-21496
Professor, Dept. of Civil, Mechanical and Environmental Engineering, George WashingtonUniversity
2 Graduate Research Assistant, Dept. of Civil, Mechanical and Environmental Engineering,
George Washington University
NATIONAL CENTER FOR EARTHQUAKE ENGINEERING RESEARCH
State University of New York at BuffaloRed Jacket Quadrangle, Buffalo, NY 14261
REPRODUCED BYU.S. DEPARTMENT OF COMMERCE
NATIONAL TECHNICALINFORMATION SERVICESPRINGFIELD, VA 22161
50272-10
4. Title and SubtitlePractical Considerations for Structural Control:System Uncertainty, System Time Delay, and Truncationof Small Control Forces.
7. Author(s)
J.N. Yang and A. Akbarpour9. Performing Organization Name and Address
National Center for Earthquake Engineering ResearchSUNY/BuffaloRed Jacket QuadrangleBuffalo, NY 14261
12. Sponsoring Organization Name and Address
Same as box 9.
3. Recipient's Accession No.
P888 1 6 3 7' 3 BIAS5. Report Date
August 10, 1987
6.
8. Performing Organization Rept. No:
10. Project/Task/Work Unit No.
11. Contract(C) or Grant(G) No.
£ECE-85-21496(C) <:.. 86-3021
~13. Type of Report & Period Covered
Technical Report
14.
15. Supplementary Notes
This research was conducted at George Washington University and was partiallysupported by the National Science Foundation under Grant No. ECE 86-07591.
16. Abstract (Limit: 200 words)
For practical application of structural control system, various importantfactors should be considered, including the system time delay, uncertaintyin structural identification, truncation of small control forces, etc. Asensitivity study is conducted to investigate the effects of these factors onthe control system of structures. THe influence of system time delay, estimationerrors for structural parameters and elimination of small control forces on theefficiency of the control system depends upon the particular control algorithmused. Four control algorithms practical for applications to earthquake-excitedstructures have been studied. These include the Riccati closed-loop controlalgorithm and three instantaneous optimal control algorithms recently pro-posed. The methodologies for the sensitivity analysis are presented. Boththe active tendon control system and the active mass damper have been considered.Numerical examples are worked out to demonstrate the criticality and toleranceof these factors for practical implementation of active control systems.
17. Document Analysis a. Descriptors
b. Identlfiers/Open·Ended Terms
20. Security Class (This Page)unclassified
19. Security Class (This Report)unclassified
STRUCTURAL CONTROLCONTROL ALGORITHMSSEISMIC RELIABILITY ANALYSIS
c. COSATI Field/Group
18. Availability Statement
Release unlimited
EARTHQUAKE ENGINEERINGSTRUCTURAL ANALYSIS
I
I21. No. of Pages
159l--------------+---~._---
(See ANSI-Z39.18) See Instructions on Reverse OPTIONAL FORM 27Z (4-77)(Formerly NTIS-35)Department of Commerce
ABSTRACT
For practical application of structural control system, various
important factors should be considered, including the system time delay,
uncertainty in structural identification, truncation of small control
forces, etc. A sensitivity study is conducted to investigate the effects of
these factors on the control system of structures. The influence of system
time delay, estimation errors for structural parameters and elimination of
small control forces on the efficiency of the control system depends upon
the particular control algorithm used. Four control algorithms practical
for applications to earthquake-excited structures have been studied. These
include the Riccati closed-loop control algorithm and three instantaneous
optimal control algorithms recently proposed. The methodologies for the
sensitivity analysis are presented. Both the active tendon control system
and the active mass damper have been considered. Numerical examples are
worked out to demonstrate the criticality and tolerance of these factors for
practical implementation of active control systems .
. Pr1'" 'f I
SECTION TITLE
TABLE OF CONTENTS
PAGE
1 INTRODUCTION 1-1
2 CONTROL ALGORITHMS IN IDEAL ENVIRONMENTS 2-12.1 Riccati Closed-Loop Control 2-42.2 Instantaneous Optimal Closed-Loop Control 2-62.3 Instantaneous Optimal Open-Loop Control 2-62.4 Instantaneous Optimal Closed-Open-Loop Control 2-8
3 STRUCTURAL CONTROL WITH SYSTEM UNCERTAINTY 3-13.1 Riccati Closed-Loop Control 3-13.2 Instantaneous Optimal Closed-Loop Control 3-23.3 Instantaneous Optimal Open-Loop Control 3-43.4 Instantaneous Optimal Closed-Open-Loop Control 3-7
4 STRUCTURAL CONTROL WITH SYSTEM TIME DELAy 4-14.1 Riccati Closed-Loop Control 4-14.2 Instantaneous Optimal Closed-Loop Control 4-24.3 Instantaneous Optimal Open-Loop Control 4-44.4 Instantaneous Optimal Closed-Open-Loop Control 4-8
5 TRUNCATION OF SMALL CONTROL FORCES 5-15.1 Riccati Closed-Loop Control '" 5-15.2 Instantaneous Optimal Closed-Loop Control 5-45.3 Instantaneous Optimal Open-Loop Control 5-55.4 Instantaneous Optimal Closed-Open-Loop Control 5-7
6 NUMERICAL EXAMPLES 6-16.1 Ideal Control Environments '" 6-16.2 Structural Control With System Uncertainty 6-46.3 Structural Control With System Time Delay 6-86.4 Truncation of Small Control Forces 6-11
7 CONCLUSIONS 7-1
8 ACKNOWLEDGEMENT AND REFERENCES 8-1
ii
FIGURE TITLE
LIST OF ILLUSTRATIONS
PAGE
1 Structural Model of a Multi-Story Building WithActive Control System , 2-2
2 Block Diagram For Riccati Closed-Loop Control 2-53 Block Diagram For Instantaneous optimal Closed-Loop
Control 2-74 Block Diagram For Instantaneous Optimal
Open-Loop Control 2-95 Block Diagram For Instantaneous Optimal Closed-Open-Loop
Control 2-11
6 Block Diagram For Riccati Closed-Loop Control WithUncertainties in Structural Identification 3-3
7 Block Diagram For Instantaneous Optimal Open-Loop ControlWith Uncertainties in Structural Identification , 3-6
8 Block Diagram For Instantaneous Optimal Closed-Open-LoopControl With Uncertainties in Structural Identification ... 3-9
9 Block Diagram For Riccati Closed-Loop Control WithTime Delay 4-3
10 Block Diagram For Instantaneous Optimal Closed-LoopControl With Time Delay 4-5
11 Block Diagram For Instantaneous Optimal Open-Loop ControlWith Time Delay 4-6
12 Block Diagram For Instantaneous optimal Closed-Open-LoopControl With Time Delay 4-10
13 Block Diagram For Riccati Closed-Loop Control WithElimination of Small Control Forces , 5-3
14 Block Diagram For Instantaneous Optimal Closed-LoopControl With Elimination of Small Control Forces 5-6
15 Block Diagram For Instantaneous optimal Open-Loop ControlWith Elimination of Small Control Forces 5-8
16 Block Diagram For Instantaneous Optimal Closed-Open-LoopControl With Elimination of Small Control Forces 5-10
17 Simulated Ground Acceleration 6-2318 Uncontrolled Response Quantities 6-2419 Response Quantities and Active Control Force Using Tendon
Control System and Riccati Closed-Loop Control Algorithm.. 6-2520 Response Quantities and Active Control Force Using Tendon
Control System and Instantaneous Optimal ControlAlgorithm , 6-26
21 Response Quantities and Active Control Force Using anActive Mass Damper and Riccati Closed-Loop ControlAlgorithm 6-27
22 Response Quantities and Active Control Force Using AnActive Mass Damper and Instantaneous Optimal Control
Algorithm 6-28
iii
FIGURE TITLE
LIST OF ILLUSTRATIONS (Continued)
PAGE
23 Top Floor Relative Displacement For an 8-Story BuildingUsing Tendon Control System and Riccati Closed-LoopControl Algorithm 6-29
24 Base Shear Force For an 8-Story Building Using Tendon ControlSystem and Riccati Closed-Loop Control Algorithm 6-30
25 Active Control Force From the First Controller For an8-Story Building Using Tendon Control System and Riccati
Closed-Loop Control Algorithm '" '" 6-3126 Top Floor Relative Displacement for an 8-Story Building
Using Tendon Control System and Instantaneous OptimalClosed-Loop Control Algorithm 6-32
27 Base Shear Force For an 8-Story Building using TendonControl System and Instantaneous Optimal Closed-LoopControl Algorithm 6-33
28 Active Control Force From the First Controller For an8-Story Building Using Tendon Control System andInstantaneous Optimal Closed-Loop Control Algorithm 6-34
29 Top Floor Relative Displacement For an 8-Story BuildingUsing Tendon Control System and Instantaneous optimalOpen-Loop Control Algorithm 6-35
30 Base Shear Force For an 8-Story Building Using TendonControl System and Instantaneous Optimal Open-LoopControl Algorithm 6-36
31 Active Control Force From the First Controller For an8-Story Building Using Tendon Control System andInstantaneous Optimal Open-Loop Control Algorithm 6-37
32 Top Floor Relative Displacement For an 8-Story BuildingUsing Tendon Control System and Instantaneous Optimal
Closed-Open-Loop Control Algorithm 6-3833 Base Shear Force For an 8-Story Building Using Tendon
Control System and Instantaneous Optimal Closed-Open-LoopControl Algorithm 6-39
34 Active Control Force From the First Controller For an8-Story Building Using Tendon Control System andInstantaneous Optimal Closed-Open-Loop Control Algorithm .. 6-40
35 Top Floor Relative Displacement For an 8-Story BuildingUsing Active Mass Damper and Riccati Closed-LoopControl Algorithm 6-41
36 Base Shear Force For an 8-Story Building Using Active MassDamper and Riccati Closed-Loop Control Algorithm 6-42
37 Active Control Force For an 8-Story Building Using ActiveMass Damper and Riccati Closed-Loop Control Algorithm..... 6-43
38 Top Floor Relative Displacement For an 8-Story Building UsingActive Mass Damper and Instantaneous Optimal Open-LoopControl Algorithm 6-44
39 Base Shear Force For an 8-Story Building Using Active MassDamper and Instantaneous Optimal Open-Loop ControlAlgorithm 6-45
iv
FIGURE TITLE
LIST OF ILLUSTRATIONS (Continued)
PAGE
40 Active Control Force For an 8-Story Building Using ActiveMass Damper and Instantaneous Optimal Open-Loop ControlAlgorithm , 6-46
41 Top Floor Relative Displacement For an 8-Story BuildingUsing Active Mass Damper and Instantaneous OptimalOpen-Loop Control Algorithm '" .. , '" ., 6-47
42 Base Shear Force For an 8-Story Building Using Active MassDamper and Instantaneous Optimal Open-Loop ControlAlgorithm 6-48
43 Active Control Force For an 8-Story Building Using ActiveMass Damper and Instantaneous Optimal Open-Loop ControlAlgorithm 6-49
44 Top Floor Relative Displacement For an 8-Story BuildingUsing Active Mass Damper and Instantaneous Closed-Open-Loop Control Algorithm 6-50
45 Base Shear Force For an 8-Story Building Using Active MassDamper and Instantaneous Closed-Open-Loop ControlAlgorithm 6-51
46 Active Control Force For an 8-Story Building Using ActiveMass Damper and Instantaneous Closed-Open-Loop ControlAlgorithm 6-52
47, 48 Effect of Time Delay, ~, on Maximum Top Floor RelativeDisplacement and Control Force For Building With TendonControl System 6-53
49 Effect of Time Delay, ~, on Top Floor Relative DisplacementFor an 8-Story Building Using Tendon Control System andRiccati Closed- Loop Control Algorithm 6-55
50 Effect of Time Delay, ~, on Base Shear Force For an 8-StoryBuilding Using Tendon Control System and Riccati Closed-Loop Control Algorithm 6-56
51 Effect of Time Delay, ~, on Control Force From the FirstController For an 8-Story Building Using Tendon ControlSystem and Riccati Closed-Loop Control Algorithm 6-57
52 Effect of Time Delay, ~, on Top Floor Relative DisplacementFor an 8-Story Building Using Tendon Control System andInstantaneous Optimal Closed-Loop Control Algorithm 6-58
53 Effect of Time Delay, ~, on Base Shear Force For an 8-StoryBuilding Using Tendon Control System and InstantaneousOptimal Closed-Loop Control Algorithm 6-59
54 Effect of Time Delay, ~, on Control Force From the FirstController For an 8-Story Building Using Tendon ControlSystem and Instantaneous Optimal Closed-Loop ControlAlgorithm 6-60
55 Effect of Time Delay, ~, on Top Floor Relative DisplacementFor an 8-Story Building Using Tendon Control System andInstantaneous Optimal Open-Loop Control Algorithm 6-61
56 Effect of Time Delay, t, on Base Shear Force For an 8-StoryBuilding using Tendon Control System and InstantaneousOptimal Open-Loop Control Algorithm 6-62
v
FIGURE TITLE
LIST OF ILLUSTRATIONS (Continued)
PAGE
57 Effect of Time Delay, ~, on Control Force From the FirstController For an 8-Story Building Using Tendon ControlSystem and Instantaneous Optimal Open-Loop ControlAlgorithm 6-63
58 Effect of Time Delay, ~, on Top Floor Relative DisplacementFor an a-Story Building Using Tendon Control System andInstantaneous Optimal Closed-Open-Loop Control Algorithm.. 6-64
59 Effect of Time Delay, ~, on Base Shear Force For an 8-StoryBuilding Using Tendon Control System and InstantaneousOptimal Closed-Open-Loop Control Algorithm 6-65
60 Effect of Time Delay, ~, on Control Force From the FirstController For an 8-Story Building Using Tendon ControlSystem and Instantaneous Optimal Closed-Open-LoopControl Algorithm 6-66
61, 62 Effect of Time Delay, ~, on Maximum Top Floor RelativeDisplacement and Control Force For Building With ActiveMass Damper 6-67
63 Effect of Time Delay, ~, on Top Floor Relative DisplacementFor an a-Story Building Using an Active Mass Damper andRiccati Closed-Loop Control Algorithm 6-69
64 Effect of Time Delay, ~, on Base Shear Force For an 8-StoryBuilding Using an Active Mass Damper and Riccati Closed-Loop Control Algorithm............•....................... 6-70
65 Effect of Time Delay, ~, on Control Force For an 8-StoryBuilding Using an Active Mass Damper and Riccati Closed-Loop Control Algorithm 6-71
66 Effect of Time Delay, ~, on Top Floor Relative DisplacementFor an a-Story Building Using an Active Mass Damper andInstantaneous Optimal Closed-Loop Control Algorithm 6-72
67 Effect of Time Delay, ~, on Base Shear Force For an 8-StoryBuilding Using an Active Mass Damper and InstantaneousOptimal Closed-Loop Control Algorithm 6-73
68 Effect of Time Delay, ~, on Control Force For an 8-StoryBuilding Using an Active Mass Damper and InstantaneousOptimal Closed-Loop Control Algorithm 6-74
69 Effect of Time Delay, ~, on Top Floor Relative DisplacementFor an a-Story Building Using an Active Mass Damper andInstantaneous Optimal Open-Loop Control Algorithm 6-75
70 Effect of Time Delay, ~, on Base Shear Force For an 8-StoryBuilding Using an Active Mass Damper and InstantaneousOptimal Open-Loop Control Algorithm 6-76
71 Effect of Time Delay, ~, on Control Force For an 8-StoryBuilding Using an Active Mass Damper and InstantaneousOptimal Open-Loop Control Algorithm 6-77
72, 73 Effect of Time Delay, ~, on Top Floor Relative DisplacementFor an 8-Story Building Using an Active Mass Damper andInstantaneous Optimal Closed-Open-Loop Control Algorithm .. 6-78
vi
FIGURE TITLE
LIST OF ILLUSTRATIONS (Continued)
PAGE
74 Effect of Time Delay, ~, on Control Force For an 8-StoryBuilding using an Active Mass Damper and InstantaneousOptimal Closed-Open-Loop Control Algorithm 6-80
75 Effect of Truncation of Small Control Forces on Top FloorRelative Displacement For an 8-Story Building With TendonControl System Using Riccati Closed-Loop ControlAlgorithm For Different Truncation Levels c 6-81
76 Effect of Truncation of Small Control Forces on Base ShearForce For an 8-Story Building With Tendon Control SystemUsing Riccati Closed-Loop Control Algorithm For DifferentTruncation Levels c 6-82
77 Effect of Truncation of Small Control Forces on ControlForce From First Controller For an 8-Story Building WithTendon Control System Using Riccati Closed-Loop ControlAlgorithm For Different Truncation Levels 6-83
78 Effect of Truncation of Small Control Forces on Top FloorRelative Displacement For an 8-Story Building WithTendon Control System Using Instantaneous OptimalClosed-Loop Control Algorithm For Different Truncation
Levels c 6-8479 Effect of Truncation of Small Control Forces on Base Shear
Force For an 8-Story Building With Tendon Control SystemUsing Instantaneous Optimal Closed-Loop ControlAlgorithm For Different Truncation Levels c 6-85
80 Effect of Truncation of Small Control Forces on ControlForce From First Controller For an 8-Story Buildingwith Tendon Control System Using Instantaneous OptimalClosed-Loop Control Algorithm For Different TruncationLevels E 6-86
81 Effect of Truncation of Small Control Forces on Top FloorRelative Displacement For an 8-Story Building With TendonControl System Using Instantaneous Optimal Open-LoopControl Algorithm For Different Truncation Levels E 6-87
82 Effect of Truncation of Small Control Forces on BaseShear Force For an 8-Story Building With Tendon ControlSystem Using Instantaneous Optimal Open-Loop ControlAlgorithm For Different Truncation Levels c 6-88
83 Effect of Truncation of Small Control Forces on ControlForce From First Controller For an 8-Story Building WithTendon Control System Using Instantaneous OptimalOpen-Loop Control Algorithm For Different TruncationLevels E 6-89
84 Effect of Truncation of Small Control Forces on Top FloorRelative Displacement For an 8-Story Building With TendonControl System Using Instantaneous Optimal ClosedOpen-Loop Control Algorithm For Different TruncationLevels c 6-90
vii
FIGURE TITLE
LIST OF ILLUSTRATIONS (Continued)
PAGE
85 Effect of Truncation of Small Control Forces on Base ShearForce For an 8-Story Building With Tendon Control SystemUsing Instantaneous Optimal Closed-Open-Loop ControlAlgorithm For Different Truncation Levels E 6-91
86 Effect of Truncation of Small Control Forces on ControlForce From First Controller For an 8-Story Building WithTendon Control System Using Instantaneous OptimalClosed-Open-Loop Control Algorithm For DifferentTruncation Levels E 6-92
87 Effect of Truncation of Small Control Forces on Top FloorRelative Displacement For an 8-Story Building With anActive Mass Damper Using Riccati Closed-Loop ControlAlgorithm For Different Truncation Levels E 6-93
88 Effect of Truncation of Small Control Forces on Base ShearForce For an 8-Story Building With an Active Mass DamperUsing Riccati Closed-Loop Control Algorithm For DifferentTruncation Levels E 6-94
89 Effect of Truncation of Small Control Forces For an 8-StoryBuilding With Active Mass Damper Using RiccatiClosed-Loop Control Algorithm For Different TruncationLevels E 6-95
90 Effect of Truncation of Small Control Forces on Top FloorRelative Displacement For an 8-Story Building With ActiveMass Damper Using Instantaneous Optimal Closed-LoopControl Algorithm For Different Truncation Levels E 6-96
91 Effect of Truncation of Small Control Forces on Base ShearForce For an 8-Story Building With Active Mass DamperUsing Instantaneous Optimal Closed-Loop Control AlgorithmFor Different Truncation Levels E '" .6-97
92 Effect of Truncation of Small Control Forces For an 8-StoryBuilding With Active Mass Damper Using InstantaneousOptimal Closed-Loop Control Algorithm For Different
Truncation Levels E 6-9893 Effect of Truncation of Small Control Forces on Top Floor
Relative Displacement For an 8-Story Building WithActive Mass Damper Using Instantaneous Optimal Open-LoopControl Algorithm For Different Truncation Levels E 6-99
94 Effect of Truncation of Small Control Forces on Base ShearForce For an 8-Story Building With Active Mass DamperUsing Instantaneous Optimal Open-Loop Control AlgorithmFor Different Truncation Levels E '" .6-100
95 Effect of Truncation of Small Control Forces for an8-Story Building With Active Mass Damper UsingInstantaneous Optimal Open-Loop Control Algorithm ForDifferent Truncation Levels E 6-101
96 Effect of Small Control Forces on Top Floor RelativeDisplacement For an 8-Story Building With Active MassDamper Using Instantaneous Optimal Closed-Open-LoopControl Algorithm For Different Truncation Levels E 6-102
viii
FIGURE TITLE
LIST OF ILLUSTRATIONS (Continued)
PAGE
97 Effect of Truncation of Small Control Forces on Base ShearForce For an 8-Story Building With Active Mass DamperUsing Instantaneous Optimal Closed-Open Loop ControlAlgorithm For Different Truncation Levels E '" 6-103
98 Effect of Truncation of Small Control Forces For an 8-StoryBuilding With Active Mass Damper Using Instantaneous optimalClosed-Open- Loop Control Control Algorithm For DifferentTruncation Levels E 6-104
ix
TABLE TITLE
LIST OF TABLES
PAGE
1
2
3
4
5
6
7
8
9
10
11
12
13
Maximum Structural Responses and Control Force For an8-Story Building With Active Tendon Control System 6-14
Maximum Structural Responses and Control Force For an8-Story Building With an Active Mass Damper 6-14
Maximum Structural Responses and Control Force For an8-Story Building Using Riccati Closed-Loop ControlAlgorithm 6-15
Maximum Structural Responses and Control Force For an8-Story Building Using Instantaneous Optimal Open-LoopControl Algorithm 6-16
Maximum Structural Responses and Control Force For an8-Story Building Using Instantaneous Optimal Closed-Open-Loop Control Algorithm 6-17
Maximum Response Quantities and Control Force For an8-Story Building With Tendon Control System Using RiccatiClosed-Loop Control Algorithm 6-18
Maximum Response Quantities and Control Force For an8-Story Building with Tendon Control System UsingInstantaneous Optimal Closed-Loop Control Algorithm 6-19
Maximum Response Quantities and Control Force For an8-Story Building With Tendon Control System UsingInstantaneous Optimal Open-Loop Control Algorithm 6-20
Maximum Response Quantities and Control Force For an8-Story Building With Tendon Control System UsingInstantaneous Optimal Closed-Open-Loop Control Algorithm .. 6-20
Maximum Response Quantities and Control Force For an8-Story Building With an Active Mass Damper UsingRiccati Closed-Loop Control Algorithm 6-21
Maximum Response Quantities and Control Force For an8-Story Building With an Active Mass Damper UsingInstantaneous Optimal Closed-Loop Control Algorithm 6-21
Maximum Response Quantities and Control Force For an8-Story Building Using with an Active Mass DamperInstantaneous Optimal Open-Loop Control Algorithm 6-22
Maximum Response Quantities and Control Force For an8-Story Building with an Active Mass Damper UsingInstantaneous Optimal Closed-Open-Loop ControlAlgorithm £: Maximum Control Force Umax = 232 KN 6-22
x
I. INTRODUCTION
The use of protective systems, such as passive or active devices, to
improve the reliability and safety of structures subjected to severe
environmental loads, such as strong earthquakes, wind turbulences, etc. has
received increasing attention recently. One potentially promising
protective system is the active control device [e.g., Refs. 1-24] In this
regard, considerable research efforts have been made for active control of
seismic-excited building structures both theoretically and experimentally
[e.g., 1-24]. In particular, new optimal control algorithms practical for
applications to earthquake-excited structures have been proposed [17-18] and
verified experimentally [4, 5]. Most studies in active control of civil
engineering structures, however, are based on ideal control environments in
the sense that the structural system can be identified precisely and the
system time delay is negligible.
Even under well-controlled laboratory environments, the experimental
results differ from those computed theoretically, and the experimental
control efficiency is lower than the theoretical one [4, 5]. This can be
attributed to several factors, such as system time delay, uncertainty in
structural identification, software and hardware control devices, etc. For
practical implementation of active control systems to full-scale structures,
such important problems as system time delay, identification of structural
parameters, etc., should be studied [24]. The investigation of these
problems can be made theoretically by conducting a sensitivity study to
determine their criticality and tolerance for the design of a control
system.
The knowledge of a structural model and its parameters, such as
stiffnesses, damping coefficients, and natural frequencies, should be given
in the design of a control system. These parameters should be obtained via
1-1
a set of measurements for an as built structure. In practice, however,
these parameters are estimated and their estimations involve errors and
uncertainties. Thus, the efficiency of a control system depends on how
closely these estimated parameter values approximate the actual ones. Like
wise, the structural characteristics are affected by the service
environments and they may change with service time. Any change in
structural parameters may cause a change in controlled quantities. Thus,
the uncertainties involved in the estimation of structural parameters may
result in an adverse effect on the efficiency of the active control system.
Time delay exists within the control system, because of the following
operations: (i) taking measurements of the response vector and/or the
earthquake base accelerations and processing them, (ii) computing the
required active control forces, (iii) generating signals to activate control
devices, and (iv) generating the required magnitude of control force (i.e.,
the reaction time for the controller). Whether system time delay is
detrimental to the control system and to what extent a time delay is
tolerable for a particular control algorithm should be examined.
Preliminary investigations in this regard have been made in Ref. 1.
When a structure is exposed to an earthquake, the time history of the
active control force contains many cycles of small amplitudes. Because of
limitations of actuators, it may be desirable to eliminate those control
forces with magnitude smaller than a certain value. The effect of
truncating small control forces on the structural response and the
efficiency of the control system should be investigated.
The objective of this report is to investigate the effect of (i)
uncertainty in structural identification, (ii) system time delay, and (iii)
truncation of small control forces, on the efficiency of the active control
system. A sensitivity study is conducted to study the effect and
1-2
criticality of these factors on the control system with respect to various
control algorithms. Four control algorithms that have been demonstrated to
be feasible and practical for earthquake-excited structures are investigated
herein. These include the Riccati closed-loop control algorithm and three
instantaneous optimal control algorithms recently proposed [17-18]. The
methodology for the sensitivity analysis associated with each control
algorithm is presented. Both the active tendon control system and the
active mass damper have been considered. Numerical examples are worked out
to demonstrate the tolerance of a control system for system time delay,
estimation error for structural parameters, and truncation of small control
forces.
1-3
II _ CONTROL ALGORITHMS IN IDEAL ENVIRONMENTS
The method of analysis for investigating the influence of various
factors described above on the control system varies with respect to the
particular control algorithm used. Hence, the following control algorithms
practical for applications to seismic-excited structures will be
investigated separately. These include the Riccati closed-loop control
algorithm and three instantaneous optimal control algorithms proposed
recently [17-18].
For simplicity, consider a shear beam type building structure
implemented by an active tendon control system as shown in Fig. 1. The
structure is idealized by an n degrees of freedom system and subjected to a
--one-dimensional earthquake ground acceleration XO(t). The matrix equation
of motion can be expressed as [e.g., 17-19]
Z (t) (2.1)
with the initial condition ~(O) = O. In Eq. (2.1), ~(t) = 2n state vector,
~( t) r dimensional control vector, ~ = (2nx2n) system matrix, ~ (2nxr)
matrix specifying the location of active controllers, and ~l is an
appropriate 2n vector [see Ref. 17 for these matrices]. In what follows, an
under bar denotes a vector or matrix and a prime denotes the transpose of a
vector or matrix.
Let B be a (2nx2n) diagonal matrix consisting of complex eigenvalues B.]
(j = 1, 2, ... , 2n) of matrix ~, and T be a (2nx2n) modal matrix consisting
of the corresponding eigenvectors of A. Then, the solution of the equation
of motion, Eq. (2.1), can be obtained numerically as [17)
~(t) (2.2)
2-1
(a) (b)
Fig. 1: Structural Model of a Multi-Story Building withActive Control System; (a) Active Tendon ControlSystem; (b) Active Mass Damper
2-2
in which ~t is the integration step,
(2.3)
8~tis a vector containing all elements evaluated at t-~t, and e- is a (2nx2n)
diagonal matrix with the jth diagonal element being exp[8.~t].J
A time dependent performance index J(t) has been proposed recently by
Yang, et a1 for optimizing the control system [17-19]
J(t) ~'(t) 9 ~(t) + ~'(t) ~ ~(t) (2.4)
in which 9 is a (2nx2n) positive semi-definite weighting matrix, and ~ is a
(rxr) positive definite weighting matrix. The three instantaneous optimal
control algorithms to be investigated later are obtained by minimizing Eq.
(2.4) subjected to the constraint of the equations of motion, Eq. (2.2),
[17-18).
On the other hand, the classical performance index J is defined as the
integral of J(t) over the time duration t f longer than that of the earth-
quake excitation, i.e.,
Jtf f J(t) dt
o(2.5)
As mentioned previously, the sensitivity of the control system with
respect to various factors described above depends on the individual control
algorithm. Further, the method of sensitivity analysis differs for each
control algorithm. To facilitate the derivation of the solutions for each
control algorithm in different environments, the four control algorithms in
2-3
ideal control environments, i.e., without system time delay, structural
uncertainty, and truncation of control forces, will be summarized in the
following. For detail derivations, the reader is referred to Refs. 17-18.
2.1 Riccati Closed-Loop Control: When the classical performance index J
defined above, Eq. (2.5), is minimized subjected to the equation of motion,
Eq. (2.1), and the external excitation XO(t) is disregarded, the so-called
Riccati closed-loop control has been derived [Ref. l7-18J. For Riccati
closed-loop control in ideal environments, the control vector ~(t) is
regulated by the measured response state vector ~(t),
~(t) _1 R- l B' P Z(t)2 - - -
(2.6)
in which ~ is a (2nx2n) Riccati matrix computed from the following matrix
Riccati equation
(2.7)
Although the Riccati matrix is time dependent, it has been shown in Refs.
17-18 that the constant Riccati matrix is an excellent approximation for
earthquake excitations. The response state vector ~(t) is computed
numerically from the equation of motion
~(t) A ~(t) + B ~(t) + ~1 XO(t) (2.8)
A block diagram for the Riccati closed-loop control algorithm is displayed
in Fig. 2.
2-4
N I Ul
u<t>
• Z<t>
z<t>
Fig
.2
1-
1"
--
RB
P2
__
_
Blo
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O.tl
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ol-
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ol.
2.2. Instantaneous Optimal Closed-Loop Control: The so-called
instantaneous optimal closed-loop control algorithm is obtained by
minimizing the time dependent performance index J(t), Eq. (2.4), subjected
to the constraint of the equations of motion, Eq. (2.2), and assuming that
the control vector ~(t) is regulated by the measured response state vector
[17-18]. For such a control algorithm, the control vector ~(t) is related
to the feedback state vector ~(t) as
~(t) 2 ~(t) (2.9)
in which the gain matrix G is given by
G -(~t/2) R-l
B' 9 (2.10)
The response state vector ~(t) is computed numerically using the following
discretized equation of motion
~(t) (2.11)
where ~(t-~t) is the vector denoting the system's boundary conditions at
time t-~t given by Eq. (2.3). A block diagram describing such a control
algorithm is shown in Fig. 3.
2.3 Instantaneous Optimal Open-Loop Control: Based on the instantaneous
optimal open-loop control algorithm, the control vector ~(t) is computed on
line, real-time, using the sensed base acceleration, XO(t), at time t as
follows [17-18]
2-6
N I --.J
.,~Wll
2- ¢
Z<t>
HeM
T-1 •
_c
I-I.
.-
-,..
e-_
~B
I..
..2
---r-
_~tR-IB'Q
2-
--
• u<t)
-
Fig
.3
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in which
~ ~(t)
L = [ (~t/2)2 B' 9 B + R ]-1
(2.12)
(2.13)
~(t) (2.14)
where ~(t-~t) is defined by Eq. (2.3). The response state vector ~(t) is
given by
~(t) ! ~(t-~t) + (~t/2) B L ~(t) + (~t/2) ~1 XO(t) (2.15)
A block diagram for the instantaneous optimal open-loop control
algorithm in ideal environments is displayed in Fig. 4.
2.4 Instantaneous Optimal Closed-Open-Loop Control: For the instantaneous
optimal closed-open-loop control algorithm, the control vector ~(t) is
regulated by both the measured base acceleration, XO(t), and the feedback
state vector, ~(t), as follows [17-18]
(~t/4)~-1 ~' [ ~ ~(t) + get) ] (2.16)
in which Ais a (2nx2n) constant gain matrix representing the closed-loop
portion of the control force, and get) is a 2n vector denoting the open-loop
portion of the control force,
2-8
N I \..0
~L::
J;:t
.....CB
Q'W
I.<
"2
/-
--1
i
'"<
.~t
BQ
I,--
2--
At u
. 1..
.~
"12~1
-,.
.--
@A
t,,-
ll..
<e
_
Fig
.4
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A _ [ (~~)2 9 ~ ~-1 B' + rJ-1 9
The response state vector ~(t) is calculated numerically as
(2.17)
(2.18)
~(t) (2.19)
A block diagram describing the instantaneous optimal c1osed-open-1oop
control algorithm in ideal environments is shown in Fig. 5.
2-10
(A
t7\
"L..
._
2-
-1i
At
I_
,---
__-,1
-'J
1')
-2
-,
N I I-'
I-'
- ~(t)
ILlt
-1+~
y<t>
I~tI~~(trl
---r
RB
--B
.._
_2
_
At
R-1 B
A,_
..I....
I
4-
--
e~~tfll
-(
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.5
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III. STRUCTURAL CONTROL VIlli SYSTEM UNCERTAINTY
When the structural parameters, such as stiffnesses, natural
frequencies, damping ratios, etc., involve uncertainties, the response state
vector ~(t) and the control vector ~(t) deviate from the solutions derived
above in ideal environments. The solutions for ~(t) and ~(t) for a control
system with estimation errors in structural parameters will be derived in
this section.
The system matrix ~ represents the structural characteristics and the
elements of matrix A are functions of structural parameters, including
masses, stiffnesses, and dampings. In reality, structural (system)
parameters can not be identified precisely and they invotve considerable
statistical variabilities. As a result, the actual A matrix is unknown.
* *Let A be the best estimate of the system matrix~. Of course, ~ deviates
from A and the extent of deviation with respect to the degradation of the
control efficiency will be studied.
3.1 Riccati Closed-Loop Control: *With the estimated system matrix A the
*estimated Riccati matrix, denoted by £ ' is computed from Eq. (2.7) in which
*~ replaced by ~ , i.e.,
* *P A 1 * B R- l * *, *2 £ B' P + A P + 2 9 o (3.1)
*The control vector ~(t) is computed using P and the measured state
vector ~(t) as,
~(t) 1 R- 1 B' p* _Z(t)2 -
3-1
(3.2)
The equations of motion for the entire structural system is obtained by
substituting Eq. (3.2) into Eq. (2.1) as follows
~(t) (3.3)
in which Ais a (2n x 2n) matrix given by
A - 1 B R- l B' p*2 - -
(3.4)
Thus, the state vector ~(t) can be computed numerically using Eq. (3.3)
as follows
Z(t) (3.5)
in which exp(E~t) is a (2nx2n) diagonal matrix with the jth diagonal element
being exp(O.~t), where O. is the jth eigenvalue of A. In Eq. (3.5), T isJ J -
the modal matrix consisting of eigenvectors of Amatrix.
A block diagram for simulating the entire control process with
uncertainties in structural parameters is displayed in Fig. 6.
3.2 Instantaneous Optimal Closed-Loop Control: For the instantaneous
optimal closed-loop control algorithm, the control vector is related to the
feedback state vector ~(t) through a gain matrix ~, Eq. (2.9). The gain
matrix G given by Eq. (2.10) is, however, not a function of the system
matrix A. Hence, both the control vector £(t) and the gain matrix ~ are not
functions of structural parameters. Therefore, instantaneous optimal
closed-loop control is independent of the uncertainties involved in
3-2
w I w
U(t
)
,+
'+
z<t)
z<t)
1-1
/•
--
RB
P2
__
_
Fig
,6
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,
determining structural parameters, such as stiffnesses, dampings, natural
frequencies, etc. In other words, whether the structural parameters are
estimated accurately or not the control efficiency is not affected.
3.3 Instantaneous Optimal Open-Loop Control: Based on the instantaneous
optimal open-loop control algorithm without system uncertainty, the control
vector ~(t) is computed from the sensed ground acceleration, Xo(t) , at time
t and the calculated vector Q(t-~t), see Eqs. (2.12)-(2.14) and Eq. (2.3).
The vector Q(t-~t) is a function of the state vector ~(t-~t), where ~(t-~t)
is estimated rather than measured for the open-loop control algorithm.
Further, the response state vector ~(t) is estimated numerically from Eq.
(2.15) rather than being measured.
When the structural parameters involve uncertainties, the estimated
*system matrix ~ , rather than the true matrix~, is used for computing the
* * * *state vectors, ~ (t-~t) and ~ (t), as well as vectors Q (t-~t) and ~ (t).
Thus, the estimated equation of motion is given as follows
.*~ (t) * * *A ~ (t) + ~ ~ (t) + ~l XO(t) o (3.6)
*Let f be a (2nx2n) diagonal matrix consisting of complex eigenvalues
* * *B. (j = 1, 2, ... , 2n) of matrix ~ , and! be a (2nx2n) modal matrixJ
*consisting of eigenvectors of A
obtained numerically as
Then, the solution of Eq. (3.6), can be
*Z (t)
in which
(3.7)
3-4
*!2 (t-l-.t)*8 l-.t * -1 { * [ *e- [!] ~ (t-l-.t) + (l-.t/2) ~ ~ (t-l-.t)
*and the estimated control vector ~ (t) applied to the structure is
(3.8)
*~ (t) *!: ~ (t) (3.10)
where L is given by Eq. (2.13) and
*G (t) (3.11)
Thus, the control vector ~*(t) and the estimated response state vector
*~ (t) can be computed numerically from Eqs. (3.7)-(3.11). A block diagram
for such a control operation is shown on the left hand side of Fig. 7.
*Note that the estimated control vector ~ (t) is actually applied to the
*structure, whereas the estimated state vector, ~ (t), that is computed for
the purpose of estimating the control vector ~*(t), is not the actual
response state vector. The actual response state vector ~(t), can be
synthesized using the actual system matrix ~ as well as the control vector
*~ (t) that is applied to the structure as follows:
or
~(t) *~ ~(t) + B U (t) + ~l XO(t) ~(O) a (3.12)
~(t) ! !2(t-l-.t) + (l-.t/2) [ B U*(t) + ~l XO(t) ]
3-5
(3.13)
Syn
the
sIs
Of
Actu
Ql
Syste
MR
esp
on
seV
ecto
r~(t)
,-
C:-<
t)rL
l~<t)
lI
Ibt~
Z<t>
b--I!
!At
-1L
-r"
:ri
..e
TI
II
IW I 0
\
IA
tB
"QTII
I1
t:=1
II
-r---
I
I~(
t)1
L_
--
--
---
Fig
.7
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where
(3.14)
It should be mentioned that the system matrix ~ is unknown. However,
for the purpose of investigating the effect of uncertainty in structural
identification, the actual response state vector ~(t) is synthesized using
the A matrix. A block diagram for the synthesis of the actual response
state vector ~(t) is shown on the right hand side of Fig. 7.
3.4 Instantaneous Optimal Closed-Open-Loop Control: For instantaneous
optimal closed-open-loop control without system uncertainty, the control
vector ~(t) is regulated by the response state vector ~(t) and the measured
earthquake base acceleration given by Eqs. (2.16)-(2.18).
*With system uncertainties, the best estimate, ~ , for the system matrix
A should be used in computing the control vector ~(t). Thus, the estimated
*control vector to be applied to the structure, U (t), is given by
*U (t) (3.15)
in which the response state vector ~(t) is measured. -*In Eq. (3.15), g (t)
is a 2n vector representing the contribution from open-loop control, which
*is computed based on the estimated system matrices ~ and its modal matrix
*T , i.e.,
-*g (t) (3.16)
*in which Ais given by Eq. (2.18) and D (t-~t) is estimated as follows
3-7
(3.17)
The actual response state vector ~(t) that is measured during the
control operation can be synthesized using the true system matrix ~ as well
*as the control vector £ (t) computed above, i.e.,
~(t) (3.18)
and the numerical solution of Eq. (3.18) can be written as
~(t) 0.19)
in which Q(t-~t) is synthesized again using the true system matrix~, i.e.,
First, ~ (t) is computed by
*Second, £ (t) is obtained by
above into Eq. (3.15). A block
The computational procedures for determining ~*(t) and for synthesizing
~(t) are summarized in the following.
substituting Eq. (3.15) into Eq. (3.19).
substituting the computed ~(t) obtained
diagram for the synthesis of the entire control operation is presented in
Fig. 8, when the system identification involves uncertainties.
3-8
e!.
Atfl
I..
.<
~.At
.-1.e
T....
...-..~
<
z<t)
.6..t
-1.l
~R
IS"."
.1J"
J
---
~---
--""
''''
I'~
t\J
1~
~i-
_..
~(t1~B1Bh
!!(t)·+~~~1
-{>~(tT4
..jw I \..
0
Fig
.8
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IV. STRUCTURAL CONTROL WITH SYSTEM TIME DEIAY
Four control algorithms without system time delay have been summarized
in Section II. With a system time delay, the response state vector ~(t) and
the control vector ~(t) will deviate from those given in Section II. The
solution for ~(t) and ~(t) using each control algorithm with a system time
delay will be derived in the following.
Let r be the time delay for the control vector ~(t). In other words,
the control vector ~(t-r) is applied to the structural system at time t,
such that the equation of motion becomes
~(t) (4.1)
4.1 Riccati Closed-Loop Control: For Riccati closed-loop control, the gain
matrix, denoted by 2, is constant, i.e.,
~(t) 2 ~(t) _1 R- 1 B'P ~(t)2 - - (4.2)
in which G - (1/2) R-1
B' P. Hence, the equation of motion is given by
~(t) ~ ~(t) + B G ~(t-r) + ~1 XO(t) (4.3)
Eq. (4.3) can be integrated step-by-step with an integration inteval ~t if
the response state vector, ~(t-r), at t-r is known. For simplicity, the
integration step size ~t is chosen to be equal to the time delay r, i.e., ~t
= r. Thus, the numerical solution for Eq. (4.3) can be expressed similar to
Eqs. (2.2) and (2.3) as
4-1
~(t) T efr r- l {~(t-r) + (r/2) [~~ ~(t-2r) + ~l xo(t-r)]}
+ (r/2) [~G ~(t-r) + ~l Xo(t)] (4.4)
The response state vector, ~(t), for a control system with a time delay r
can be computed step-by-step using Eq. (4.4). A block diagram for such
computations is shown in Fig. 9.
4.2 Instantaneous Optimal Closed-Loop Control: For instantaneous optimal
closed-loop control, the gain matrix, denoted by ~, is constant, i.e.,
in which
!:!:(t)
s
~ ~(t)
-(~t/2) R- l B' g
(4.5)
(4.6)
(4.7)
For a time delay r, the equation of motion is given by Eq. (4.1), where
!:!:(t-r) is obtained from Eq. (4.5) with t being replaced by t-T. If the
numerical integration step ~t is chosen to be equal to the time delay r for
simplicity, then the response state vector ~(t) for a control system with a
time delay T can be computed numerically similar to Eq. (4.4) as follows
~(t) T efr T-l
{ ~(t-r) + (r/2) [~ ~ ~(t-2r) + ~l XO(t-T) ] }
+ (T/2) [~~ ~(t-T) + ~l XO(t) ]
4-2
>l:> I w
1-1
/G
=--R
BP
-2
---
U<
t-T
)•
+1
+Z
<t)
•I
Z<
t)
Fig
.9
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Eq. (4.7) is similar to Eq. (4.4) as expected, because both control
algorithms described above are closed-loop control. A block diagram for the
instantaneous optimal closed-loop control algorithm with a time delay T is
shown in Fig. 10.
4.3 Instantaneous Optimal Open-Loop Control: Based on the instantaneous
optimal open-loop control algorithm the control vector ~(t) is computed from
the sensed ground acceleration, XO(t), at time t, and the calculated vector
Q(t-~t), Eq. (2.12), at t-~t. Unlike closed-loop control, the response
state vector ~(t) is not measured in open-loop control operation. Instead,
~(t) is estimated without considering system time delay. This is because,
in reality, the magnitUde of time delay is unknown.
With a system time delay T, let the estimated response state vector be
*denoted by ~ (t), and the corresponding estimated quantity at the previous
* * *time step t-~t be denoted by Q (t-~t). Of course ~ (t) and Q (t-~t) are
different from the actual ~(t) and Q(t-~t). In actual operation, the con-
* * *trol vector, denoted by ~ (t), is computed from ~ (t) and D (t-~t), whereas
. *the est~mated response state vector ~ (t) is computed as
*~ (t) (4.8)
The block diagram shown on the left hand side of Fig. 11 illustrates the
operation described above at t-~t.
For simplicity of computation, the integration time step ~t is chosen
to be identical to the time delay T, i.e., ~t = T. Because of time delay,
*the control vector at t-T, ~ (t-T), is applied to the structure at time t,
4-4
,e, I
()l
...I..
.'WI'
2-
z<t)
-.
...'.!.
.B.L------t_
2- U
(t-
T)
~t
-1,
>..
,S
=--R
BQ
-2
---
aT
-1..
.-..
.<-
Te
_
Fig
.10
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ith
TiM
eD
eln
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T\J
I_
,
2"-
1----
~
• Z(t-
T)
I I
~~t-T
):
--1e~T!I~:
I I I L
CO
Mp
ute
rE
stll'
lo.t
lon
s...
ISyn
th:S
ISD
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ctu
al
Sys
teM
Beh
o.vlo
r(
\tIl
tho
ut
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elo.
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r-
/'
--
--
--
Ito XJt~
•--
G(t
-T
)-
..
At
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!2
-
(A
t)1J
'"Q
'J~
2-
--1
.t:> I <5\
Fig
.11
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and the actual response state vector ~(t) can be synthesized as shown on the
right hand side of the block diagrams depicted in Fig. 11.
*The estimated control vector ~ (t~r), that is applied to the structure
at time t, is given by [see Fig. 11]
*~ (t-r) (4.9)
in which L is given by Eq. (2.13) and
*~ (t-r)
where
-(~t/2) B' g T D*(t-2r) ~ (~t/2)2 B' g ~l XO(t-r) (4.10)
*D (t-2r) (4.11)
involves quantities estimated at t-2r or t~2~t. The response state vector
*~ (t-~t) estimated at t-~t or t-r is computed by substituting Eq. (4.9) into
Eq. (4.8) with ~t = r
*~ (t-r) * * .! ~ (t-27) + (r/2) ~ ~ ~ (t-r) + (7/2) ~1 XO(t-r) (4.12)
The estimated control vector ~*(t-7) with a time delay 7 is actually
applied to the structure at time t. The actual response of the structure
can be synthesized numerically using the equation of motion, Eq. (4.1) as
~(t) ! ~(t-7) + (7/2) ~ ~*(t-7) + (r/2) ~1 XO(t)
4-7
(4.13)
where
~(t-1") (4.14)
A step-by-step numerical computation with the step size ~t = 1" can be
carried out to determine the actual response state vector ~(t). A block
*diagram showing the computer estimation of the control vector ~ (t-1") and
the synthesis of the actual system behavior is displayed in Fig. 11.
4.4 Instantaneous Optimal Closed-Open-Loop Control: For instantaneous
optimal closed-open-loop control without system time delay, the control
vector ~(t) is regulated by the response state vector ~(t) and the measured
earthquake base acceleration as shown in Eqs. (2.16)-(2.19). The response
state vector ~(t) and the earthquake ground acceleration Xo(t) are measured
from which the control vector ~(t) is computed. This is different from
open-loop control in which the response state vector ~(t) and the control
vector ~(t) are estimated. As a result, the problem of time delay for the
closed-open-loop control is, in general, expected to be less serious than
the open-loop control.
With a time delay 1", the control vector ~(t-1") is actually applied to
the structure at time t. Choosing the integration interval ~t to be equal
to 1", i.e., ~t = 1", the response vector is computed as
~(t) T ~ ( t - 1") + (1" / 2 ) [~~ ( t - 1") + ~1 Xo ( t) ] (4.15)
in which the control vector ~(t-1") at t-r is given by
4-8
where
!!(t-r)
~(t-r)
(b.t/4) R- 1 B' [~~ ( t - r) + S (t - r) ] (4.16)
(4.17)
(4.18)
Q(t-2r) Or -1 { [ ..]}e-! ~(t-2r) + (r/2) ~ !!(t-3r) + ~1 XO(t-2r) (4.19)
A step-by-step numerical computation can be carried out easily with a
step size r for the determination of ~(t) and !!(t). A block diagram for the
entire operation of the control system using the instantaneous optimal
closed-open-loop control algorithm with a system time delay is shown in Fig.
12.
4-9
At7
\wT
-:.
t
<&>
II.
Z~,(
t",)I
I9
T_I
I•
(~eI
Mff
lJfA
4-
--
---:
----
,1_
~q
(t-
T)I~~II~
-•
..G
- A!
..t:. I I-'
o
Fig
.12
IB
lock
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lny.
V. TRUNCATION OF SMAIJ.. CONTROL FORCES
Under earthquake excitations, the time history of active control forces
involve many cycles of small amplitude. Because of limitations of
actuators, it may be desirable to eliminate all cycles of control forces
with amplitudes smaller than a certain value, thus simplifying the control
operation. When small control forces are truncated, not only the behavior
of the structure under control will be different, but also the control
forces will deviate from the results presented in section II. In this
section, the response state vector ~(t) and the required control vector ~(t)
will be derived, when control forces smaller than a certain value are
eliminated.
5.1 Riccati Closed-Loop Control: Based on Riccati closed-loop control, the
response state vector ~(t) is measured and the control vector ~(t) is
computed from ~(t), i.e.,
in which
~(t)
G
G ~(t)
1 -1- - R B' P2 -
(5.1)
(5.2)
Let E be a preselected value such that any control force smaller than E
will be truncated, i.e., will be set to be zero. This is equivalent to pass
the control vector ~(t) through a filter, such that any element of ~(t)
becomes zero if it is smaller than E, and it is unaffected if larger than E.
*Then, the resulting control vector from the filter, denoted by ~ (t), is
5-1
applied to the structure. The process is repeated for every time instant t
throughout the entire episode of the earthquake.
For the analytical/numerical investigation, the entire operations
described above can be simulated to determine the response state vector ~(t)
*and the control vector ~ (t). A block diagram for such a simulation the
entire control operation is shown in Fig. 13.
The response state vector ~(t) is obtained from the equations of
motion,
Z (t) *~ ~(t) + B U (t) + ~l XO(t) (5.3)
. *in wh~ch ~ (t) is the truncated control vector actually applied to the
structure,
*~ (t) £ ~(t) (5.4)
where ~(t) is the untruncated control vector computed from the response
vector ~(t) given by Eqs. (5.1) and (5.2), and £ is a nonlinear operator
indicating the truncation effect.
The nonlinear truncation operator £ can be represented by a (rxr)
diagonal matrix with the diagonal elements given as follows
1
a
if U.(t) > €~
(5.5)
in which Ui(t) is the ith element of ~(t).
*The response vector ~(t) and the control vector ~ (t) applied to the
5-2
Ul I VJ
•+
'+
Z<t
)i
iZ
<t)
tf<
t)8
u<t)
1-
1/
--
RB
P2
__
_
Fig
.13
IB
lock
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.gro
.MF
or
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l
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orc
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structure can be solved using the system of equations in Eqs. (5.1)-(5.5).
However, the system of equations is nonlinear because the operator £ is
nonlinear. As a result, the following iterative procedures are used to
*determine the state vector ~(t) and the control vector ~ (t).
(i) The response state vector ~(t) is computed from Eq. (5.3) by use of
*the untruncated control vector ~(t), i.e., ~ (t) = ~(t). In other words,
*~ (t) in Eq. (5.3) is replaced by ~ ~(t), Eq. (5.1),
(ii) ~(t) is computed from ~(t) obtained in step (i) using Eq. (5.1),
*(iii) ~ (t) is obtained from ~(t) using Eqs. (5.4)-(5.5), and
(iv) The response state vector ~(t) is computed from Eq. (5.3) using
*~ (t) obtained in step (iii).
*The iterative procedure is repeated until both vectors ~(t) and ~ (t)
* .converge. In general, ~(t) and ~ (t) converge rap~dly and few cycles of
iteration are sufficient. Such an interative procedure is repeated for
*every time instant t to obtain the time histories of ~(t) and ~ (t).
5.2 Instantaneous Optimal Closed-Loop Control: The control operation using
the instantaneous optimal closed-loop control algorithm is essentially
identical to that of the Riccati closed-loop control, except that the gain
matrix, ~' is different, i.e.,
where
~(t)
s
~ ~(t)
-(~t/2) R- 1 B' 9
5-4
(5.6)
(5.7)
In simulating the entire control operation using the instantaneous optimal
closed-loop control algorithm, the following equation for computing ~(t) is
employed, Eq. (2.2),
~(t) (5.8)
*in which ~ (t) is given by Eqs. (5.4) and (5.5), and
(5.9)
Thus, instead of Eq. (5.3), Eqs. (5.8) and (5.9) are used in
conjunction with Eqs. (5.4)-(5.7) for iteration and simulation. Again, the
iterative procedure converges very rapidly. A block diagram for simulating
*the response state vector ~(t) and the control vector ~ (t) is shown in Fig.
14.
5.3 Instantaneous Optimal Open-Loop Control: For instantaneous optimal
open-loop control, the operation for the truncation of small control forces
is identical to that described previously. The difference between the
present control algorithm and the closed-loop control algorithm is that the
response state vector ~(t) is computed rather than measured. Because of
such a difference, the iterative procedure is not necessary in simulating
*the response state vector ~(t) and the control vector ~ (t). The following
*equations can be used directly to compute ~ (t) and ~(t) at each time
instant t.
~(t)
5-5
(5.10)
....~t"ll
2-
.-o
r-
AtB
2- •
z<t)
~B
2-
--r-
I..n
~(t)
_.
...A_le~At!-1~
T
U1 I CJ'.
-A
t R-1B'Q
2-
__
, U<t>
8
• U(t)
Fig
.1
4I
Blo
ck
Dln
grn
MF
or
Instn
ntn
ne
ou
sO
ptl
Mn
lC
lose
d-L
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ntr
ol
\.11t
hE
llMln
ntl
on
Of
SM
nll
Co
ntr
ol
Fo
ree
s.
*~ (t) ~ ~(t)
(5.11)
(5.12)
(5.13)
(5.14)
A block diagram for simulating the control operation is shown in Fig.
15.
5.4 Instantaneous Optimal Closed-Open-Loop Control: By use of the
instantaneous optimal closed-open-loop control algorithm, the response state
vector ~(t) is measured,
measured ~(t). In other
*whereas the control vector ~ (t) depends on the
*words, both ~(t) and ~ (t) are coupled in the simu-
lation process. Hence, the iterative procedure described previously is
*needed. The equations used for the simulation of ~(t) and ~ (t) are given
in the following
*~ (t) = ~ ~(t)
(5.15)
(5.16)
~(t) (~t/4) R-l ~' [ ~ ~(t) + g(t) ]
5-7
(5.17)
(5.18)
/At'
\.cr
rQW
I."
<:"
2/
---
•
At
~-IT
'!!..I
.-..
--y--
U1 I Cf)
le!!At!
-ll•<
€9>
<.6
tB"
Q!
.L..
_2
--
T -
Fig.
15I
Blo
ck
Dlo
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.MF
or
Inst
o.n
to.n
to.n
eo
us
Opt
lMo.
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pe
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ntr
ol
'-11t
hE
lIMln
o.tl
on
Of
SM
o.ll
Co
ntr
ol
Fo
rce
s.
(5.19)
*The iterative procedures for determining ~(t) and 2 (t) are described in the
following
(i) The initial solutions for ~(t) and 2(t) are obtained by setting
*2 (t) = 2(t) and using Eqs. (5.15) and (5.17)
*(ii) With 2(t) determined in (i), 2 (t) is computed from Eq. (5.16)
*(iii) ~(t) is obtained from Eq. (5.15) using 2 (t) determined in (ii)
and
(iv) 2(t) is computed from ~(t) using Eq. (5.17).
*The iterative procedure is repeated until ~(t) and 2 (t) converge.
*Again, numerical results indicate that ~(t) and ~ (t) converge rapidly. A
block diagram for such a simulation procedure is shown in Fig. 16.
5-9
L-----
c~t
AW
lI..
.2
--
)~...
.A
t\J
I..
.2
-1i
(Jl I I--' o
-.
~(1;1~Bllih
.\!(1
;)-W
.\!(1;1~l!1
_{>~(1;~ z<
t)
...
1A
tR-1
Jl-
,--
__/\
r-t-
I---
---.
I.,
..
;'A
tfl l
..<
Fig
.16
rB
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rce
s.
VI. NUMERICAL EXAMPLES
To demonstrate the sensitivity and criticality of the active control
system with respect to the variant from ideal control environments,
extensive numerical examples are worked out in this section. In particular,
two configurations of the active control system will be considered,
including the active tendon control system and active mass damper. A
simulated earthquake ground acceleration shown in Fig. 17 is considered as
the input excitation.
An eight-story building in which every story unit is identically
constructed is considered for illustrative purposes. The structural
properties of each story are: m floor mass = 345.6 tons; k = elastic
stiffness 53.404 x 10 KN/m; and c internal damping coefficient = 2,937
tons/sec. that corresponds to a 2% damping for the first vibrational mode of
the entire building. The external damping is assumed to be zero. The
computed natural frequencies are 5.79, 17.18, 27,98, 37.82, 46.38, 53.36,
58.53, and 61.69 rad/sec.
Without control system, the top floor relative displacement with
respect to the ground and the base shear force of the building are shown in
Fig 18.
6.1 Ideal Control Environments: Two examples, one with an active tendon
control system and another with an active mass damper, will be illustrated
in the following under ideal control environments. Ideal control
environments refer to the control situation without system time delay and
free of estimation errors for structural parameters. The structural
response quantities and required active control forces under ideal control
environments will be presented in this subsection. These will serve as
bases for comparing the corresponding results when control environments
6-1
deviate from ideal ones, including system time delay, uncertainty in
structural identification, and truncation of small control forces.
Example 1: Active Tendon Control System
Suppose four active tendon controllers are installed in the lowest four
story units and the angle of inclination of the tendons with respect to the
floor is 25 0. The dimension of the weighting matrices 9 and ~ are (16x16)
and (4x4) , respectively. For Riccati closed-loop control, both 9 and R
*matrices are chosen to be diagonal matrices with elements Q.. = Q 1.3 x~~
105 (for i = 1, 2, ... 8), Qjj=O (for j = 9, 10, ... 16), and Rll = R22 R33-4
R44 = 10
With the application of instantaneous optimal control algorithms, the
weighting matrix ~ is identical to the one given above. However, the
(16x16) weighting matrix 9 will be partitioned more efficiently as follows
9 (6.1)
in which 921 and 9 22 are (8x8) matrices and a is a constant. Note that 911
and 9 12 do not contribute to the active control force and, hence, they are
chosen to be zero [17, 18]. The choice of 921 and 922 requires some consid
erations as discussed in Ref. 17. For simplicity, 921 and 922 are chosen to
b l · * h 1 f * b * ..e equa , ~.e., 921 = g22 - Q Tee ements 0 9 , denoted y Q (~,J), are
* *given in the following; Q (i,j) = j for i ~ 4 and Q (i,j) = 0 for i > 4.
For a 68% reduction of the building response, a value of 5,000 is used for
Q.
The time histories of (i) the top floor relative displacement to the
ground, (ii) the base shear force, and (iii) the control force from the
6-2
lowest controller, are shown in Fig. 19, when the Riccati closed-loop
control algorithm is used.
Under ideal control environments, it has been shown in Refs. 17-18 that
the control efficiencies for three instantaneous optimal control algorithms
are identical. In other words, the resulting structural response quantities
and required active control forces for these three control algorithms are
identical. The time histories of the response quantities and the required
active control force from the lowest controller are shown in Fig. 20, when
one of the instantaneous optimal control algorithms is used. The maximum
values of the time histories shown in Figs. 18 to 20 are tabulated in Table
1.
Example 2: Active Mass Damper
The same example as the previous one is considered except that, instead
of the active tendon control system an active mass damper is installed on
the top floor of the building as shown in Fig. l(b). The properties of the
active mass damper are md = mass of the damper 29.63 tons, cd = damping of
the damper = 25 tons/sec., k d = stiffness of the damper = 957.2 KN/m. Note
that the mass md
is 2% of the generalized mass associated with the first
vibrational mode, the frequency of the damper is 98% of the first natural
frequency of the building, and the damping coefficient of the damper is
approximately 7.3%. In the present situation, the weighting matrix R
consists of only one element, i.e., R = R, whereas the dimension of 9 matrix
is (18x18).
For Riccati closed-loop control, the weighting matrix 9 is considered
5as a diagonal matrix with Q
ii= 1.3 x 10 (for i=l, 2, ... , 8), and Qjj = 0
(for j = 9, 10, .,., 18). The element of R is 10-3
.
-3In applying instantaneous optimal control algorithms, g = 10 is used,
and the 9 matrix is partitioned as shown by Eq. (6.1) in which 921 and 922
6-3
are (2x9) matrices. The following values are assigned to elements of these
two matricies for illustrative purposes:
[-33.5
-33.5
-67.
-67.
-100.5
-100.5
-134.
-134.
-167.5
-167.5
-201.
-201.
-234.5
-234.5
-268.
-268.-375.6 ]
32.2
[67.5
5.8
135.0
11.6
202.5
17.6
270.0
23.2
338.5
29.0
405.0
34.7
472.5
40.5
540.0
46.332.2 ]5.7
A value of 67.0 is chosen for a, Eq. (6.1), such that the top floor relative
displacement is reduced approximately by 60%
Using the Riccati closed-loop control algorithm, the building response
quantities and required active control force are displayed in Fig. 21. With
the application of anyone of three instantaneous optimal control
algorithms, the corresponding quantities are displayed in Fig. 22. The
maximum values of the time histories shown in Fig. 21 and 22 are tabulated
in Table 2.
Finally, it should be mentioned that the control efficiency for each of
three instantaneous optimal control algorithms is quite different when a
system time delay or an estimation error for structural parameters is
introduced. Numerical examples will be presented in the next subsections.
6.2 Structural Control Yith System Uncertainty: When the system identifi-
cation involves uncertainty, the actual system matrix A is unknown and the
*estimated system matrix A is used for control operation. Various degrees
of estimation error in stiffness and damping will be introduced to
illustrate the effect of system uncertainty on the control system. The
error in stiffness estimation for every story unit, denoted by ~k, is
6-4
represented by the percentage of the true stiffness. The corresponding
estimation error in the fundamental frequency wf ' denoted by 6.w, is
expressed in terms of the percentage of wf
. The estimation error in damping
coefficient for every story unit, denoted by 6.c, is similarly defined. The
estimation error for the damping ratio, €, in the first mode is expressed in
terms of the percentage of e, as denoted by 6.e.
Example 3: Active Tendon Control System
Example 1 is reconsidered and various estimation errors for structural
parameters are introduced. Using the Riccati closed-loop control algorithm,
the time histories of structural response quantities and active control
force from the first controller are presented in Figs. 23-25. In these
figures, ±40% estimation errors in stiffness or damping have been
introduced. The maximum response quantities and control force in the entire
time histories of 30 seconds are summarized in the upper part of Table 3.
The control force shown in the table is the one from the controller in the
lowest story unit. It is observed from Table 3 that in comparison with the
results under ideal control environments, the response quantities may
increase or decrease depending on whether the structural properties are
under or over estimated. However, as the response quantities increase, the
corresponding active control force always decreases and vice versa. This
indicates that the degradation of the control efficiency is not quite
significant. The control system is moderately sensitive to the stiffness
estimation error (or natural frequency), whereas it is not sensitive the
estimation error of damping coefficient. In general, an estimation error
within 20% for stifffness and 50% for damping is quite acceptable, when the
Riccati closed-loop control algorithm is used.
With the application of the instantaneous optimal-open-loop control
algorithm, the time histories of the structural response quantities and the
6-5
active control force from the first controller for various degrees of
stiffness estimation error are shown in Figs. 26-28. The maximum response
quantities and the maximum active control force are summarized in the upper
part of Table 4. It is observed from Table 4 that the control system is
quite sensitive to the estimation error for the stiffness as compared to the
Riccati closed-loop control algorithm. An estimation error for stiffness
may result in a considerable degradation of the control effectiveness.
Further, it is observed from Fig. 26 that the response quantities beyond 17
seconds do not die down as rapidly as that of the ideal system without
uncertainty. On the other hand, the control system is not sensitive to the
estimation error for damping as shown in Fig. 29-31 and Table 4.
It is concluded that the instantaneous optimal open-loop control
algorithm is quite sensitive to the system uncertainty in stiffness or
natural frequency, and the control efficiency can deteriorate considerably
if the stiffness estimation error is more than 10%. This situation may have
been expected, because in open-loop control no feedback state vector is
measured. Note that the feedback state vector reflects to some extent the
behavior of true system parameters.
Using the instantaneous optimal closed-open-loop control algorithm, the
time histories of the structural response quantities and active control
force are depicted in Figs. 32-34. In these figures, estimation errors of
±40% for stiffness and damping are introduced. The maximum response
quantities and active control force in 30 seconds of the time history are
summarized in the upper part of Table 5. Table 5 and Figs. 32-34 indicate
that the instantaneous optimal closed-open-loop control algorithm is
practically insensitive to system uncertainties.
6-6
Finally, it has been shown in Section III that the instantaneous
optimal closed-loop control algorithm is independent of structural
parameters or system uncertainty.
Example 4: Active Mass Damper
Example 2 is reconsidered and various estimation errors of structural
parameters are taken into account. With the Riccati closed-loop control
algorithm, time histories of the response quantities and control force are
shown in Figs. 35-37 for different degrees of estimation errors. The
maximum values in these time histories are summarized in the lower part of
Table 3. A comparison between the upper part and lower part of Table 3
shows that the active mass damper is less sensitive to the uncertainties (or
errors) in system identification. An estimation error of 40% for stiffness
is acceptable when the Riccati closed-loop control algorithm is used.
Again, the error in damping estimation has a negligible effect on the
efficiency of the control system.
With the application of the instantaneous optimal open-loop control
algorithm, the time histories of the response quantities and active control
force are depicted in Figs. 38-43. The corresponding maximum response
quantities and control force are summarized in the lower part of Table 4.
As observed from Table 4, the control system is sensitive to the uncertainty
in stiffness estimation, in particular, when the stiffness is over-
estimated. An over-estimation of the stiffness for more than 10% will
result in a significant degradation of the control efficiency. Again, the
control system is not sensitive to the estimation error for damping.
With the application of instantaneous optimal closed-open-loop control,
the time histories of response quantities and control force are shown in
Figs. 44-46. The corresponding maximum values are displayed in the lower
part of Table 5. It is observed from the table that the instantaneous
6-7
optimal closed-open-loop control algorithm is not sensitive to the
estimation errors for structural parameters.
6.3 Structural Control Yith System Time Delay: To demonstrate the effect
of system time delay on the control system, the same examples worked out in
Section 6.1 under ideal control environments will be considered, in which
various degrees of time delay T will be introduced. The system time delay,
-3T, is expressed in 10 seconds and also in percentage of the fundamental
period of the structure Tf
= 1.OS5 sec.
Example 5: Active Tendon Control System
Example 1 will be reconsidered herein except that a time delay T is
taken into account. The solutions for the structural response quantities
and active control forces have been derived in Section V.
By use of the Riccati closed-loop control algorithm, time histories of
the response quantities and active control forces are computed for various
degrees of time delay T. Some results are displayed in Figs. 49 to 51. The
absolute maximum value within the entire time history of 30 seconds of
earthquake episode for the response and control force is considered as a
measure for the effect of system time delay. In particular, the deviation
of the absolute maximum value from that obtained without system delay. Let
YS be the maximum top floor relative displacement with respect to the ground
in the time history of 30 seconds with a system time delay, and YS
be the
corresponding quantity under ideal control environments without a time
delay. Then, the deviation, YS Ys ' measured in terms of the percentage of
Ys ' is plotted in Fig. 47(a) as a solid curve. In other words, the solid
curve represents the deviation, (Yg - YS)/ YS ' as a function of time delay
6-8
As observed from the figure, the deviation increases as the time delay
T increases. With a time delay T, the maximum control force from the
controller installed in the lowest story unit is denoted by U1 , whereas the
corresponding maximum control force without a time delay is denoted by U1 .
The deviation of the maximum control force measured in terms of the
percentage of U1
, i.e., (U1 - Ul )/ Ul , is plotted in Fig. 47(a) as a dashed
curve. Again, the deviation increases as the magnitude of time delay T
increases.
From Fig. 47(a), two observations made in the following are
significant. (1) As the time delay T increases, the maximum response
quantities increase whereas the required maximum control forces are always
larger than that without a system time delay. This indicates that the
control efficiency degrades monotonically with respect to the time delay T.
(2) The absolute maximum values of response quantities increase drastically
as the time delay T is over 5% of the fundamental structural period Tf
. In
other words, the slope of the solid curve increases rapidly for T > 5% Tf
.
The structural response quantities and required active control forces
have been computed for various degrees of time delay T using three
instantaneous optimal control algorithms. The deviations of the absolute
maximum of the top floor relative displacement and control force are
displayed in Figs. 47(b) and 48, when three instantaneous optimal control
algorithms are used. Further, the time histories of response quantities and
active control force are displayed in Figs. 52 to 60. Again, the two
conclusions described previously for Riccati closed-loop control hold for
instantaneous optimal control algorithms.
It is observed that time delay is most serious for instantaneous
optimal open-loop control. Riccati closed-loop control and instantaneous
6-9
optimal closed-loop control are less sensitive to system time delay than
instantaneous optimal closed-open-Ioop control. A time delay within 3% of
the fundamental natural frequency, Tf = 1.085 seconds, is
tolerable for the instantaneous optimal open-loop control algorithm, whereas
a time delay of 4.5% of Tf is acceptable for other control algorithms.
Beyond these limits described above, the degradation of the control
efficiency is quite significant.
The fact that open-loop control may be susceptible to system time delay
can be explained in the following. The control force is regulated by the
measured input excitation rather than the feedback state vector in the case
of open-loop control. The predominant frequency of the input excitation is
usually quite different from that of the structure. Theoretically, the
frequency content of the control force is expected to be close to that of
the structural response. Since the frequency of the structural response is
contributed by the frequencies of the input excitation and the structure, a
time delay may result in a significant phase shift for the control force
with respect to the response. This is particularly true when the magnitude
of time delay is large.
Example 6· Active Mass Damper
Example 2 is reconsidered in which a system time delay r is introduced.
The deviation of the absolute maximum top floor relative displacement and
that of the absolute maximum control force are presented in Figs. 61 and 62
as functions of time delay T. For different control algorithms, the time
histories of the structural response quantities and control force are
displayed in Figs. 63 to 74.
From the results presented above, the effect of time delay on the
active tendon control system and active mass damper is comparable, although
6-10
the active mass damper is slightly less sensitive.
previously hold for the active mass damper also.
The observations made
6.4 Truncation of Small Control forces: The sensitivity and criticality of
truncating small control forces on the control system will be illustrated
using the following numerical examples. The truncation level € is expressed
in term of KN and also in term of the percentage of maximum control force
from the first controller under ideal control environments obtained in
Section 6.1.
Example 7: Active Tendon Control System
Example 1 will be reconsidered in which different truncation levels, €,
for active control forces will be made. Numerical results for the time
histories of the top floor relative displacement, the base shear force, and
the active control force from the first controller are presented in Figs. 75
to 77 when the Riccati closed-loop control algorithm is used. With the
application of three instantaneous optimal control algorithms, the
corresponding results are displayed in Figs. 78 to 86.
For the particular earthquake ground acceleration input considered, the
structural response and the active control force have a most intense segment
roughly from 3 to 17 seconds. The active control forces outside this region
are quite small. Thus, a majority of small control forces truncated are
outside this region. As such, the structural response quantities do not die
down after 17 seconds as rapidly when the small control forces are
truncated. However, the truncation effect on the maximum structural
response that occurs in the most intense segment is extremely limited as
evidenced by Figs. 75 to 86. The truncation effect may become significant
when the truncation level is high enough such that some control forces in
the most intense segment are eliminated.
6-11
The maximum structural response is of great concern, whereas the rate
of decay of the response beyond the most intense segment may not be
important. Consequently, the maximum response quantities and the maximum
control force within the time period of 30 seconds are tabulated in Tables
6-9 for different truncation levels, E, and different control algorithms.
The first column in the table indicates the actual truncation level E in KN,
and the second column shows the truncation level expressed in terms of the
percentage of the maximum control force under ideal control environments as
given in the first row of the table. For different truncation levels, the
maximum response quantities and maximum control force are also expressed in
terms of the percentage of the corresponding quantity obtained without a
truncation, i.e., E = O.
It is observed from Tables 7 to 9 and Figs. 78 to 86 that a truncation
of all control forces smaller than 20% of the maximum control force
practically does not affect the maximum response quantities. However, the
response quantities do not die down as rapidly as the results obtained
without a truncation of small control forces. The conclusion holds for all
control algorithms considered.
Example 8: Active Mass Damper
Example 2 is reconsidered in which different levels of truncation for
control forces have been made. The time histories of the top floor relative
displacement, the base shear force, and the control force are presented in
Figs. 87 to 98 for different control algorithms. The maximum response
quantities and control force are tabulated in Tables 10-13. As observed
from Figs. 87 to 98 and Tables 10 to 13, a truncation of all control forces
smaller than 20% of the maximum control force has an insignificant effect of
the active mass damper system.
6-12
Unlike the system uncertainty and time delay, the effect of truncation
of small control forces is not sensitive to different control algorithms.
Likewise, the sensitivity for active tendon control system is almost
identical to that of the active mass damper.
6-13
Table 1: Maximum Structural Responses and Control Force for an 8-StoryBuilding with Active Tendon Control System
TOP FLOOR BASE SHEAR CONTROL FORCE FROMCONTROL LAW DISPLACEMENT FORCE FIRST CONTROLLER
(cm) (KN) (KN)
Uncontrolled 4.10 2,506 ------
Riccati Closed- 1.36 853 437Loop Control
Instantaneous 1. 34 847 421Optimal Control
Table 2: Maximum Structural Responses and Control Force for an 8-StoryBuilding with an Active Mass Damper
TOP FLOOR BASE SHEAR CONTROLCONTROL LAW DISPLACEMENT FORCE FORCE
(cm) (RN) (RN)
Uncontrolled 4.10 2,506 ------
Riccati Closed- 1. 61 1,075 250Loop Control
Instantaneous 1. 54 1,045 232Optimal Control
6-14
Table 3: Maximum Structural Responses and Control Force for an 8-Story BuildingUsing Riccati Closed-Loop Control Algorithm.
I ACTIVE TENDON CONTROL SYSTEM II ESTIMATION ERROR I I I BASE 1 1 I 11 I 1 I I TOP FLOOR IDIFFERENCE ISHEAR IDIFFERENCE 1CONTROL IDIFFERENCE I1 ~kl ~w I ~cl ~rIDISPLACEMENTI IN % OF IFORCE I IN % OF I FORCE I IN % OF II %\ % 1 %1 % (CM) I 1. 36 CM I (KN) I 853 KN I (KN) I 437 KN II I I 1 I I I I 11 I 1 I I I 1 1 I1 ° I ° I °1 ° 1. 36 I 853 I I 437 I I1 I I I I I I II 401 18.31 ° ° 1.53 I 12.5 965 I 13.1 I 389 I -11.0 11 I 1 1 I I I 1I 2°1 9.6l ° ° 1.43 I 5.1 899 I 5.4 I 413 I -5.5 I1 I I 1 I I I I1-201-10.51 0 0 1. 32 1 -2.9 792 I -7.1 I 460 I 5.3 II I I I 1 I I 1I-40 I -22 . 6 I ° ° 1.27 I -8.4 698 1 -18.1 I 502 1 14.9 I1 I I I I I I I1 ° I 0 I 40 40 1.43 I 5.4 901 I 5.6 I 416 I -4.8 II 1 1 1 I 1 1 I1 ° I 0 1-40 -40 1. 33 I -2.2 812 I -4.8 I 443 I 1.4 I1 I I I I 1 I I II ACTIVE MASS DAMPER 1
ESTIMATION ERROR I I I BASE I I I 1I I I I TOP FLOOR IDIFFERENCEISHEAR IDIFFERENCE ICONTROL IDIFFERENCE 1
~kl ~w I ~cl ~rIDISPLACEMENTI IN % OF IFORCE I IN % OF I FORCE I IN % OF I%1 % 1 %\ %1 (CM) 1 1.61 CM I (KN) 1 1,070 KN 1 (KN) 1 250 KN \
1 I I I 1 I I I I II I I I I I I I I
° I ° I °1 ° 1. 61 I \1,070 I I 250 I II I I I I I I
401 18.31 °1 ° 1.71 6.2 1,047 I -2.1 I 254 I 1.6 II I I I I 1 I
2°1 9.61 °1 ° 1.65 2.5 1,052 I -1. 7 I 255 I 2.0 1I I I I I I I
-201-10.51 °1 ° 1. 59 -1. 2 1,096 I 2.4 I 253 1 1.2 II I I I I I I
-4°1-22.61 °1 ° 1. 58 -1. 8 1,096 I 2.4 I 278 1 11.2 1I I 1 I I I 1
° I ° I 401 40 1.64 1.8 1,085 I 1.4 I 240 I -4.0 II I I 1 I 1 I
° I 0 1-4°1- 40 1.59 -1. 2 1,054 I -1.4 I 259 1 3.6 II I 1 I I I I
6-15
Table 4: Maximum Structural Responses and Control Force for an 8-Story BuildingUsing Instantaneous Optimal Open-Loop Control Algorithm.
ACTIVE TENDON CONTROL SYSTEM IESTIMATION ERROR I I 1BASE I I I I
I I I I TOP FLOOR IDIFFERENCEISHEAR jDIFFERENCEjCONTROLIDIFFERENCEJ~kl ~w 1 ~cl ~rIDISPLACEMENTI IN % OF IFORCE I IN % OF I FORCE I IN % OF I
%1 % I %1 %1 (CM) I 1.34 CM I (KN) I 847 KN I (KN) I 421 KN I
I I I I I I I I I II I I I I I I I I I
o I 0 I 0 I 0 I 1.34 I 1 847 1 1 421 I 1I I I I I I 1 I I I
20 I 9.61 0 I 0 I 1. 64 I 22.4 I 996 I 17.6 I 415 I -1. 4 I1 I I I 1 I 1 I
101 4.9 01 01 1.51 I 12.7 I 866 I 2.2 I 403 -4.3 II I I I I I I I
-10 I - 5.1 °1 °I 1. 43 I 6.7 I 910 I 7.4 I 439 4.3 II 1 I I I I I I
-2°1-10.5 01 0\ 1.71 1 27.6 \1,022 \ 20.7 I 457 8.6 I1 I I I J 1 I I° I ° 401 401 1.37 1 2.2 I 884 1 4.4 I 409 -2.8 II I I 1 I I I I
° 1 ° -401-401 1.33 I -0.7 1 799 I -5.6 I 441 4.7 II I! I I I I I
ACTIVE MASS DAMPER 1ESTIMATION ERROR I I I BASE I I I I
I I I I TOP FLOOR IDIFFERENCEISHEAR I DIFFERENCE ICONTROL DIFFERENCE I~kl ~w I ~cl ~rIDISPLACEMENTI IN % OF IFORCE I IN % OF I FORCE IN % OF I
%1 % I %1 %1 (CM) 1 1.54 CM I (KN) 11,045 KN I (KN) 232 KN II I I I I I I I 1I I I I I I I I° 1 ° I 0 °1 1. 54 I 11 ,045 I I 232 II I I I 1 1 I I
201 9.61 ° 01 1.96 I 27.3 11,202 I 15.0 I 217 -6.5 II I I I 1 I I I
101 4.91 ° 01 1.77 1 14.9 11,094 I 4.7 I 223 -3.9 II I I I I I I I I1-1°1 -5.11 ° 01 1.52 I -1.3 11,031 I -1.3 I 246 6.0 II I I I I I I I I1-201-10.51 0 01 1.55 I 0.6 11,082 I 3.5 1 274 18.1 IJ I I 1 I I I I II ° I 0 \ 40 401 1.56 I 1.3 11,045 1 I 230 -0.9 1I 1 I I I I I 1 I1 0 1 ° 1-40 -40\ 1.54 I 11,047 I 0.2 I 234 0.9 I
I I I I I I I I I
6-16
o
Table 5: Maximum Structural Responses and Control Force for an 8-StoryBuilding Using Instantaneous Optimal C1osed-Open-Loop ControlAlgorithm.
ACTIVE TENDON CONTROL SYSTEM 1ESTIMATION ERROR I I I BASE I 1 I I
1 1 I I TOP FLOOR DIFFERENCEISHEAR 1DIFFERENCE 1CONTROL IDIFFERENCE 1~k ~w 1 ~cl ~rIDISPLACEMENT IN % OF IFORCE I IN % OF 1 FORCE 1 IN % OF I
% % I %1 %1 (CM) 1.34 CM I (kn) I 847 KN I (KN) I 421 KN I
1 I I I 1 I II 1 I I I 1 1
o 1 0 0 I 1. 34 1 847 1 1 421 I II 1 I I I 1 I
40 18.31 0 01 1.39 3.7 1 829 1 -2.1 438 I 4.0 II 1 1 1 1 1
20 9.61 0 01 1.37 2.2 I 838 1 -1.1 428 1 1.6 I1 1 I 1 1 1
- 20 -10.51 0 0 I 1. 34 I 856 I 1. 1 414 I -1. 6 I1 I I 1 I 1 I1-40 -22.61 0 01 1.38 3.0 I 878 I 3.6 413 1 -1.9 11 1 1 1 1 I 1I 0 0 I 40 40 I 1. 34 I 852 I 0.5 418 I -0.7 I1 1 I 1 I 1 I 11 0 1 0 1-40 -401 1.35 0.7 1 843 1 -0.4 425 1 1.4 I1_4-1_--1-1-+-----ll ~ -l-I__II---__--1-__---lll--- 11 ..:-!A~CT~I~V_!=E'--'MAf..!£=..!:S~S~DAM~P_!=E~R 11 ESTIMATION ERROR 1 1 1 BASE 1 1 I 1I \ I I I TOP FLOOR \DIFFERENCE\SHEAR \DIFFERENCE 1CONTROL 1DIFFERENCE I1 ~kl ~w I ~c ~rIDISPLACEMENTI IN % OF IFORCE 1 IN % OF I FORCE 1 IN % OF 1
I %I % 1 % %I (CM) I 1. 54 CM 1 (KN) I 1,045 KN 1 (KN) I 232 KN II I I I I I I I I I1 I I 1 I I1 0 0 1 0 01 1.54 11,045 232 1I I I I II 40 18.31 0 01 1. 55 0.7 11,046 0.1 232 11 1 I I 11 20 9.61 0 01 1.54 11,045 232 I1 1 1 1 I1-20 -10.51 0 01 1.54 11,045 232 11 I 1 1 I1-40 -22.61 0 01 1.54 11,046 0.1 232 1I 1 I I II 0 0 1 40 401 1.54 11,046 0.1 232 I1 1 1 I 110 0 1-40 -401 1.54 11,046 0.1 232 I1_4--_-1-1_I---~I- +--__--+I__~__-l-__-I-- I
6-17
Table 6: Maximum Response Quantities and Control Force for an 8-Story Buildingwith Tendon Control System Using Riccati Closed-Loop ControlAlgorithm.
I I I I€ € IN TOP FLOOR CHANGE IBASE SHEAR I CHANGE ICONTROL I CHANGEIN % of DISPLACEMENT IN % OF I FORCE I IN % OF I FORCE I IN % OFKN 437KN (CM) 1. 36 CM I (KN) I 853 KN I (KN) I 437 KN
I I I II I I I
a 0.0 1. 36 I 853 I I 437 II I I I
50 11.44 1.36 I 844 I -1.0 I 439 I 0.4I I I I
75 17.16 1. 36 I 847 I -0.7 I 442 I 1.1I I I I
100 22.88 1. 37 0.7 I 848 I -0.6 I 444 I 1.6I I I I
6-18
Table 7: Maximum Response Quantities and Control Force for an 8-Story Buildingwith Tendon Control System Using Instantaneous Optimal Closed-LoopControl Algorithm.
I I I I€ I € INI TOP FLOOR CHANGE BASE SHEAR CHANGE CONTROL I CHANGE IIN I % OFIDISPLACEMENT IN % OF FORCE IN % OF FORCE I IN % OF IKN 1421KNI (CM) 1. 34 CM (KN) 847 KN (KN) I 421 KN I
1__ 1 I 1I I I I
0 I 0.0 I 1.34 847 421 I II I I I
50 111.871 1. 34 848 0.1 421 I II I I I
75 117. 81 1 1. 33 -0.7 850 0.3 424 1 0.7 I1 I I I
100 123 . 75 1 1. 32 -1. 5 853 0.7 426 1 1.1 II I I I
6-19
Table 8: Maximum Response Quantities and Control Force for an 8-Story Buildingwith Tendon Control System Using Instantaneous Optimal Open-LoopControl Algorithm.
I I€ I € IN TOP FLOOR CHANGE BASE SHEAR CHANGE CONTROL I CHANGEIN I % OF DISPLACEMENT IN % OF FORCE IN % OF FORCE 1 IN % OFKN 1421KN (CM) 1. 34 CM (KN) 847 KN (KN) I 421 KN
I_- II I
0 I 0.0 1.34 847 421 II I
50 111 . 87 1.33 -0.7 849 0.2 422 I 0.2I I
75 117.81 1. 33 -0.7 852 0.6 424 I 0.71 I
100 123.75 1. 33 -0.7 853 0.7 426 I 1.1I I
Table 9: Maximum Response Quantities and Control Force for an 8-Story Buildingwith Tendon Control System Using Instantaneous Optimal C1osed-OpenLoop Control Algorithm.
I I 1€ € INI TOP FLOOR I CHANGE IBASE SHEAR CHANGE CONTROL CHANGEIN % OFIDISPLACEMENT IN % OF I FORCE IN % OF FORCE IN % OFKN 421KNI (CM) 1. 34 CM 1 (KN) 847 KN (KN) 421 KN
__I II I
0 0.0 I 1. 34 1 847 4211 I
50 11. 87 1 1. 34 I 848 0.1 421I I
75 17. 81 1 1. 33 -0.7 I 852 0.6 424 0.7I I
100 23.75 1 1. 33 -0.7 I 853 0.7 426 1.1I 1
6-20
Table 10: Maximum Response Quantities and Control Force for an 8-Story Buildingwith an Active Mass Damper Using Riccati Closed-Loop ControlAlgorithm.
I I 1 I€ € INI TOP FLOOR I CHANGE IBASE SHEAR I CHANGE ICONTROL I CHANGEIN 1 % OFI DISPLACEMENT , IN % OF FORCE I IN % OF 1 FORCE I IN % OFKN I 250KNI (CM) I 1. 61 CM (KN) 11,070 KN 1 (KN) 1 250 KN
1__1 1 I I II I I I I I
° I ° 1 1. 61 , 1,070 I 1 250 II I I 1 1 1
50 1 20 , 1.66 I 3.1 1,073 I 0.2 I 249 1 -0.4\ \ I I I I
75 1 30 1 1. 74 , 8.1 1,088 , 1.7 1 237 I -5.2I I , ,
1 1
Table 11: Maximum Response Quantities and Control Force for an 8- StoryBuilding with an Active Mass Damper Using Instantaneous OptimalClosed-Loop Control Algorithm.
1 1 1 I€ € INI TOP FLOOR I CHANGE IBASE SHEAR I CHANGE ICONTROL CHANGEIN I % OFI DISPLACEMENT I IN % OF FORCE 1 IN % OF I FORCE IN % OFKN 1232KNI (CM) I 1. 54 CM (KN) 11,047 KN 1 (KN) 232 !<N
1__1 1 1 II / 1 / 1
0 1 ° 1 1. 54 I 1,047 1 I 232I I I I 1
50 121.551 1.56 I 1.2 1,036 1 -1. 0 I 225 -3.0I I I I I
75 /32.321 1. 60 1 3.9 977 I -6.7 1 203 -12.51 1 I 1 I
6-21
Table 12: Maximum Response quantities and Control Force for an 8-Story Buildingwith an Active Mass Damper Using Instantaneous Optimal Open-LoopControl Algorithm.
I I I I€ € INI TOP FLOOR I CHANGE IBASE SHEAR I CHANGE CONTROL I CHANGEIN I % OF IDISPLACEMENT I IN % OF I FORCE I IN % OF FORCE I IN % OFKN 1232KN I (CM) I 1. 54 CM I (KN) 11 ,047 KN (KN) I 232 KN
1__1 I I I I1 I I I 1 I
0 I 0 I 1. 54 I I 1,047 I 232 II I I I I I
50 121.551 1. 56 I 1.2 I 1,035 I -1.1 224 I -3.4I I I I I I
75 132. 32 1 1.62 I 5.2 I 977 I -6.7 205 I -11.6I I I I I I
Table 13: Maximum Response Quantities and Control Force for an 8-Story Buildingwith an Active Mass Damper Using Instantaneous Optimal Closed-Open-Loop Control Algorithm €: Maximum Control Force U 232 KN.max
€ € IN TOP FLOOR CHANGE BASE SHEAR CHANGE CONTROL CHANGEIN % OF DISPLACEMENT IN % OF FORCE IN % OF FORCE IN % OFKN 232KN (CM) 1. 54 CM (KN) 1,047 KN (KN) 232 KN
0 0 1.54 1,047 232
50 21. 55 1.55 0.6 1,035 -1.1 224 -3.4
75 32.32 1.61 4.5 976 -6.8 204 -12.0
6-22
..zo 0.8-~ffi C\I 0.4...J oW WU(JJ 0U""c(~
c -0.4z::Jo -0.8c:(!) o 5 10 15 20
TIME IN SECONDS25 30
Fig. 17: Simulated Ground Acceleration
6-23
4,...----------::----------
-2~
(a)
25 3010 15 20
TIME IN SECONDS
5
OM
-4 ------.1---.1..-'_~I__..1..'__...I..'_--J
a
w> ~-~ c.:-oJ ~
LU Zc: wc: ~o wo ()...J <~ ~c. tJ')
o 25~
(b)
1
o
3,.----------------......,
2
-2- 3 '--_.......-~---'------'-__.l......__ __J
o 5 10 15 20 25 30TIME IN SECONDS
w()a:ou.a: Z« ~w Cf):r: 0C/) ~ -1wC/)
«CD
Fig. 18: Uncontrolled Response Quantities: (a) Top Floor RelativeDisplacement, (b) Base Shear Force
6-24
(a)
(e)
2-------------------,
(b)
o
o '-"-''''V'' l
- 2 L..-_----J~_ __..:.L____L.___..t____L.__ _____...J
1.5 ,....-------------------,
o
- 1.5 L..-__L--._--J:......-_---1__---J..__---L.__.....J
500 r----------------__--..
-500----~-_~__.L___.L___~_--i
o 5 10 15 20 25 30;
.IZw2w(.)
<..leel)-Q
w()a:olJ...Joa:...zo()
w>-~..IW=a:oo..IU.
c..o~
TIME IN SECONDS
Fig. 19: Response Quantities and Active Control Force Using Tendon ControlSystem and Riccati Closed-Loop Control Algorithm: (a) Top FloorRe la tive Displacement. (b) Base Shear Force. (c) Active ControlForce.
6-25
-500 OL---.1..5--.....110-------:-1::-5----:2::0~~2~5;--:;310
TIME IN SECONDS
(a)
o
2
o
-2 L--..-L__...L...-_--.l.__--l.-__l..-.-_---J
1.5---------------(-b-)I
o
-1.5=====~====~====~====~====~===~500 (c)
.....zw~w(.)<....c..en-QZ~
~o~.w(.)a:ou..
w(.)a:ou.-oJoa:~zo()
a:<w:tC/)
wC/)
<CD
w>-~-'wa:a:oo...Ja.a.oto-
Fig. 20: Response Quantities and Active Control Force Using Tendon ControlSystem and Instantaneous Optimal Control Algorithm: (a) Top FloorRelative Displacement, (b) Base Shear Force, (c) Active ControlForce.
6-26
w> ~-t< (.) 2.
(a)..J .-W Zc: wc: ~
0 w 0(.)0 <..J ..J... a.a- U)
0 - -2Qt- 1.5c: ~ (b)< ('f)w 0:t - 0en .w wen (.)
< c:CD 0... -1.5w 350() (c)c:0
~1J.
...J 00 -c: ::Jt-Z0 -350()
0 5 10 15 20 25 30
TIME IN SECONDS
Fig. 21: Response Quantities and Active Control Force Using an Active MassDamper and Riccati Closed-Loop Control Algorithm: (a) Top FloorRelative Displacement, (b) Base Shear Force, (c) Active ControlForce.
6-27
2
(b)
25 302015105
(e)
(a)
o
o
- 2 '--__-'-- --'-__---" .l...--.__....l-..__---J
1.5
o
-1.5 '--__-!.-__~__._.I_____L_____J___'
350 ,...........------------------,
-350 '--__---ll..-.----L----'---~--~---Jo
w(.)a:oI.L.
.~zw~w(.)c:-lQ.Cf)-QZ~
Mo,..
w(.)a:oI.L....Joa:~zou
w>-~-lWct£too-l~
Q.o...a:<w:x:(I)
w(I)
<a:1
TIME IN SECONDS
Fig. 22: Response Quantities and Active Control Force Using an Active MassDamper and Instantaneous Optimal Control Algorithm: (a) Top FloorRelative Displacement, (b) Base Shear Force, (c) Active ControlForce.
6-28
o
2r----------------.(a)
(b)
(d)
(c)
(e)o
o
o
o- 2 '--_--I.-_---l.__....l-_-.....__.J.....-_--J
2
- 2 ~_-..-_~_ __l.__.l__._.....l.__--I
2
- 2 L..--_-J...-.-_~_ __l.__.J.....-_ __L__ _J
2
-2 ~_.....L.-_......L-_--L_-J__l..----J
2
·~zw~w(.)4(...Q.en-QW>-~...wc:a:oo......Q.o...
305 10 15 20 25TIME IN SECONDS
- 2 ':---...L...--_...J-_--l..-_~_ __L...__
o
Fig. 23 Top Floor Relative Displacement for an 8-Story Bui Iding using Tendon
Control System and Riccati Closed-Loop Control Algorithm:(a) No System
Uncertainty, (b) llK = 40%, (c) lIK = -40%, (d) lIC = 40%, (e) lIC = -40%.
6-29
1.5
o _"'l"lJ
(a)
(e)
(b)
o
o
o "'---V"lJ
o
-1.5 '---_.l--_...I--_....J.-_-L.-_--L.._--'
1.5
-1.5 ""'---_....L.-_-"--_~_ __'___ _1.___
1.5r---------
C-d
----.)
-1.5 L....-.._...L-._...L.--_....J.-_.-I--_~----J1.5r------------------,
(0)
-1. 5 ~_.l..-_.l..--_..l--_...l.-.-_..I---J
1.5
..w()Ctou.a::<w~UJWUJ<m
5 10 15 20 25 30TIME IN SECONDS
Fig. 24 Base Shear Force for an 8-Story Bu i I ding Us i ng Tendon Contra I
System and Rlccati Closed-Loop Control Algorithm: (a) No System
Uncertainty, (b) llK = 40%, (c) LlK = -40%, (d) LlC = 40%, (e) LlC =
-40%.
6-30
500
a(a)
Cd)
(b)
(e )
(c)
5 10 15 20 25 30TIME IN SECONDS
-500500
az~
-500.... 500::>..w a()a:0 -500u..-J 5000a: at-z0() -500
500
a
-5000
Fig. 25 Active Control Force From the First Controller tor an 8-Story Bui Iding
Using Tendon Control System and Riccati Closed-Loop Control Algorithm:
(a) No System Uncertainty, (b) ~K = 40%, (c) 6K = -40%, 6C = 40%, (e)
6,C = -40%.
6-31
2.....----------~__:I
(d)
(c)
(b)
(e)
o
o
o
o
o
- 2 L------l_-..l._--L.._--L_---l-_---J
2
- 2 L...-----:L.-------l_---l._---J.._---I...---J
2
- 2 L-----L_----L_--I-_--L---....L--
2
- 2 L-.._..L..-_....l..-_-L-_-.L...._----l..-_--.J
2
•...zw~w(.)<-JQ.en-QW>-~..JWa:a:oo...JL&-a..oI-
- 20L----L-----..L---I----L--..L--
30
5 10 15 20 25TIME IN SECONDS
~ig. 26: Top Floor Relative Displacement for an 8-Story Bui Iding Using Tendon
Control System and Instantaneous Optimal Closed-Loop Control Algorithm
: (a) No System Uncertainty, (b) ~K = 40%, (c) ~K = -40%, (d) ~C = 40%,(e) 6.C = -40%.
6-32
(b)
(c)
(a)
(d)
(e)
5 10 15 20 25 30TIME IN SECONDS
1.51
0
-1.51.5
z~ 0
(¥)0~ -1.5..w 1.5ua:0 0u.a:< -1.5w:I: 1.5enw 0en<CD -1.5
1.5
0
-1.50
r:- i g. 27: Base Shear Force for an 8-Story Bu i I ding Us i ng Tendon Cont ro I System
and Instantaneous Optimal Closed-Loop Control Algorithm: (a) No System
Uncertainty, (b) 6K = 40%, (c) 6K = -40%, (d) 6C = 40%, (e) 6C = -40%.
6-33
o
500 r------------_
(a) i
(b)o
- 500 L.-------'-_----....-._---i-_....!------L_--J
500
(c)
(e)
(d)o
o
o
- 500 L.-.-_.L...-..-_..L....--_..L....--_..L...-_.l------J
500
-500 L...-~_--J-_-'------l...-_--J......-_
500
-500 L..--_..L...-_...l--_...l--_..J...-_..J...-_
500..
..w()a:ou...Joa:tzo()
5 10 15 20 25 30TIME IN SECONDS
- 500 '"--_"'O""-_.....l--_......L.-_--l--_-l-__
o
rig. 28: Active Control Force from the First Controller for an 8-Story Sui Iding
Using Tendon Control System and Instantaneous Optimal Closed-Loop
Control Algorithm: (a) No System Uncert~inty, (b) 6K = 40%, 6K = -40%,
LC = 40%. LC = -40%.
6-34
(a)
(b)
(c)
25 305 10 15 20
TIME IN SECONDS
2
o
o
o
- 2 L--_---I.__---'-__--'--__...I..-__~_ __'
2
- 2 '---_---l.__---'-__.....!-__~__:..._.__ __'
2
- 2 '--------'------'----"-----'------1.-------'
o
•t-ZW~W(.)
<-JQ.Cf)-cw>-~.JW~
c:roo-'LA..0ot-
Fig. 29: Top Floor Relative Displacement for an 8-Story Building UsingTendon Control System and Instantaneous Optimal Open-Loop GontrolAlgorithm: (a) No System Uncertainty, (b) ~C - 40%, (c) ~G = -40%.
6-35
1.5(a)
z 0~
M0..- -1.5..w 1.5 (b)ua:0
0L&-
a:4(w -1.5:tC/) 1.5 (c)wC/)
<m 0 ~....-
305 10 15 20 25
TIME IN SECONDS
-1.5 '-------'------'----'----'--------'-----'o
Fig. 30: Base Shear Force for an 8-Story Building Using Tendon ControlSystem and Instantaneous Optimal Open-Loop Control Algorithm: (a)No System Uncertainty, (b) ~c - 40%, (c) ~c - -40%.
6-36
- 500 o\--_~_---I-__l..-...-_-l.-_--l..-_--!
5 10 15 20 25 30
TIME IN SECONDS
500(a)
az::.::~ -500~.. 500
(b)1w() I
a:: i
0I
0 w-v./ll.,j'.rv-.,--.r _JI
U.
...J0 -500a:.... 500
(c)z0()
0
Fig. 31: Active Control Force from the First Controller for an 8-StoryBuilding Using Tendon Control System and Instantaneous OptimalOpen-Loop Control Algorithm: (a) No System Uncertainty, (b) ~C 40%, (c) ~C - 40%.
6-37
2r------------
(d)
(e)
(a)
(b)
(e)
"'"'-'-------.o
o
o
o
o- 2 I..--_---I.-_---.:.__-l.-_--J....__.J--_--I
2
- 2 L..--_....l-_---J......_---..J...__L-_--L.-_-.I
2
- 2 L..--_....l-_---J......_----L__L-_--L.-_--l
2
- 2 I..--_--'--_---L__...l--_----L...__.l...--_--J
2
•....zw~wc.:><..Ja.en-cw>-te..JW0=a:oo...~
CO...
- 2 L-_-l-.-_--l...-_---l..__.l..--_.....L.-_-J
o 5 10 15 20 25 30
TIME IN SECONDS
Fig. 32: Top Floor Relative Displacement for an 8-Story Building UsingTendon Control System and Instantaneous Optimal Closed-Open-LoopControl Algorithm: (a) No System Uncertainty, (b) tJ( - 40%, (c)tJ( - -40%, (d) ~C - 40%, (e) ~C - -40%.
6-38
1.5
o(8)
(b)
(c)
(d)
(e)~--'-1o
o
o
O~·L'
-1.5 l.--_L.-_.L--_..L--_..l.....-_.J...------J
1.5
-1.5 L-.--.l_---L._--l-_---L..._---l..-_---J
1.5
-1.5 l.--_l.-.-_..L--_..L-_....L--_....l----.J
1.5
-1.5 L-----JL.------l_---J.._--'-_---l-_-'
1.5
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305 10 15 20 25TIME IN SECONDS
-1. 5 ~-..J.....-_ __L..__---J...__'____-.....-__
o
Fig. 33: Base Shear Force for an 8-Story Building Using Tendon ControlSystem and Instantaneous Optimal Closed-Open-Loop ControlAlgorithm: (a) No System Uncertainty, (b) ~K = 40%, (c) ~K =-40%, (d) ~C - 40%, (e) ~C - -40%.
6-39
500~---------------,(a)
a
5 10 15 20 25 30TIME IN SECONDS
(d)
(e)
(c)
(b)-500
500
az~ -500~ 500::)
..w a()a::0 -500LL.
~ 5000a::
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Fig. 34: Active Control Force from the First Controller for an 8-StoryBuilding Using Tendon Control System and Instantaneous OptimalClosed-Open-Loop Control Algorithm: (a) No System Uncertainty,(b) 6K - 40%, (c) 6K - -40%, (d) ~C - 40%, (e) 6K - -40%.
6-40
a
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(d)
(e)
(e)
(b)
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o
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·to-ZW~W(.)
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-2a 5 10 15 20 25
TIME IN SECONDS30
Fig. 35: Top Floor Relative Displacement for an a-Story Building UsingActive Mass Damper and Riccati Closed-Loop Control Algorithm: (a)No System Uncertainty, (b) ~ - 40%, (c) ~ - -40%, (d) ~c = 40%,(e) ~c - -40%.
6-41
(c)
(d)
(a)
(e)
(b)
5 10 15 20 25 30TIME IN SECONDS
1.5
0
-1.51.5
z~ 0
M0..... -1.5.w 1.5(.)a:0 0u.a:
-1.5«w
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Fig. 36: Base Shear Force for an 8-Story Building Using Active MassDamper and Riccati Closed-Loop Control Algorithm: (a) No SystemUncertainty, (b) ~K - 40%, (c) ~K - -40%, (d) ~C - 40%, (e) ~C = 40%.
6-42
(c)
(d )
(b)
(e)
(a)
5 10 15 20 25 30TIME IN SECONDS
350
0
-350350
z 0~
~ -350~
350..w()
0a:0u..
-350...J0 350a:~z 00<.:>
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0
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Fig. 37: Active Control Force for an 8-Story Building Using Active MassDamper and Riccati Closed-Loop Control Algorithm: (a) No SystemUncertainty, (b) lll( - 40%, (c) lll( - -40%, (d) t::.C - 40%, (e) t::.C-40%.
6-43
(b)
(e)
(d)
(c)
o
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TIME IN SECONDS
Fig. 38: Top Floor Relative Displace~ent for an 8~Story »uilding UsingActive Mass Damper and Instantaneous Optimal Open-Loop ControlAlgorithm: (a) No System Uncertainty, (b) ~ - 20%, (c) 6K ~ 10%,(d) 6K - -10%, (e) bK - -20%.
6-44
(e)
(d)
(e)
(b)
o
1.5
o
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305 10 15 20 25TIME IN SECONDS
-1.5 L..-_....L--_---l.-_-.L_--L__L.--I
o
Fig. 39: Base Shear Force an 8-Story Building Using Active Mass Damper andInstantaneous Optimal Open-Loop Control Algorithm: (a) No SystemUncertainty, (b) 6K - 20%, (c) 6K - 10%, (d) 6K - -10%, (e) ~K - 20%.
6-45
o
350 r-------------.-,.~. (a)
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5 10 1S 20 2S 30TIME IN SECONDS
-350 ~---'--_ _J...__"---~_ _J...__
o
Fig. 40: Active Control Force an 8-Story Building Using Active Mass Damperand Instantaneous Optimal Open-Loop Control Algorithm: (a) NoSystem Uncertainty, (b) ~K - 20%, (c) ~ = 10%, (d) ~K - -10%, (e)~ - -20%.
6-46
30
(b)
25
~-_.-I
10 15 20
TIME IN SECONDS
5
o
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o
2~----------------
(a)
- 2 I.--_~__---L.__---l.-__....I....-__.l..--_-..!
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Fig. 41: Top Floor Relative Displacement for an 8-Story Building UsingActive Mass Damper and Instantaneous Optimal Open-Loop ControlAlgorithm: (a) No System Uncertainty, (b) l:.C - 40%, (c) l:.C =
-40%.
6-47
1.5,..------------------,
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30255 10 15 20
TIME IN SECONDS
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o
Fig. 42: Base Shear Force for an 8-Story Building Using Active Mass Damperand Instantaneous Optimal Open-Loop Control Algorithm: (a) NoSystem Uncertainty, (b) ~c - 40%, (c) ~C - -40%.
6-48
(c)
(b)
o
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5 10 15 20 25
TIME IN SECONDS
30
Fig. 43: Active Control Force for an 8-Story Building Using Active MassDamper and Instantaneous Optimal Open-Loop Control Algorithm: (a)No System Uncertainty, (b) ~C - 40%, (c) ~C - -40%.
6-49
2
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(d)
(e)
(c)
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o
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o 5 10 15 20 25 30TIME IN SECONOS
- 2 L..--_..l..-_...l--_-J------J----J--
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Fig. 44: Top Floor Relative Displacement for an 8-Story Building UsingActive Mass Damper and Instantaneous Closed-Open-Loop ControlAlgorithm: (a) No System Uncertainty, (b) ll.K - 40%, (c) ~K - 40%, (d) ~C - 40%, (e) ~C - -40%.
6-50
(d)
(a)
(b)
(e)
(e)
\/'-------
5 10 15 20 25 30TIME IN SECONDS
1.5
0
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0
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Fig. 45: Base Shear Force for an 8-Story Building Using Active Mass Damperand Instantaneous Closed-Open-Loop Control Algorithm: (a) NoSystem Uncertainty, (b) ~ - 40%, (c) 6K - -40%, (d) 6C - 40%, (e)6C - -40%.
6-51
(d)
(c)
(a)
(e)
~ ---4
5 10 15 20 25 30TIME IN SECONDS
350
0
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0
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Fig. 46: Active Control Force for an 8-Story Building using Active MassDamper and Instantaneous Closed-Open-Loop Control Algorithm: (a)No System Uncertainty, (b) ~K ~ 40%, (c) ~ - -40%, (d) ~C = 40%,(e) ~C - -40%.
6-52
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25 305 10 15 20
TIME IN SECONDS
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-3Loop Control Algorithm: (a) r -0 (No Delay), (b) T - 45 x 10
sec., (c) T - 60 x 10- 3 sec.
6-55
1.5(a)
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30255 10 15 20
TIME IN SECONDS
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o
Fig. 50: Effect of Time Delay, r, on Base Shear Force for an 8-StoryBuilding Using Tendon Control System and Riccati Closed-Loop
-3Control Algorithm: (a) r -0 (No Delay), (b) r - 45 x 10 sec.,
(c) r - 60 x 10- 3 sec.
6-56
500 (a)
0z~
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(b)w(.)a: a0u...J0 -500a:... 500
(c)z0(.) a
305 10 15 20 25
TIME IN SECONDS
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Fig. 51: Effect of Time Delay, T, on Control Force from the FirstController for an 8-Story Building Using Tendon Control System andRiccati Closed-Loop Control Algorithm: (a) T -0 (No Delay), (b) T
- 3 - 3- 45 x 10 sec., (c) T - 60 x 10 sec.
6-57
(a)
(b)
(c)
25 30
o
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o 5 10 15 20
TIME IN SECONDS
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Fig. 52: Effect of Time Delay, r, on Top Floor Relative Displacement for an8-Story Building Using Tendon Control System and InstantaneousOptimal Closed-Loop Control Algorithm: (a) r -0 (No Delay), (b) r
_ 45 x 10- 3 sec., (c) r - 60 x 10-3
sec.
6-58
1.5(a)
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5 10 15 20
TIME IN SECONDS
25 30
Fig. 53: Effect of Time Delay, r, on Base Shear Force for an 8-StoryBuilding Using Tendon Control System and Instantaneous OptimalClosed-Loop Control Algorithm: (a) r -0 (No Delay), (b) r ~ 45 x
10- 3 sec., (c) r - 60 x 10-3
sec.
6-59
500(a)
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TIME IN SECONDS
Fig. 54: Effect of Time Delay, 'T, on Control Force from the FirstController for an 8-Story Building Using Tendon Control System andInstantaneous Optimal Closed-Loop Control Algorithm: (a) 'T -0 (No
_3 - 3Delay), (b) 'T - 45 x 10 sec., (c) 'T - 60 x 10 sec.
6-60
25 305 10 15 20
TIME IN SECONDS
2r-----------------,(a)
o
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2r------------------,(b)
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2.---------------------,(c)
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Fig. 55: Effect of Time Delay, r, on Top Floor Relative Displacement for an8-Story Building Using Tendon Control System and InstantaneousOptimal Open-Loop Control Algorithm: (a) r -0 (No Delay), (b) r =
- 3 - 330 x 10 sec., (c) r - 45 x 10 sec.
6-61
1.5 (a)
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TIME IN SECONDS
Fig. 56: Effect of Time Delay, r, on Base Shear Force for an 8-StoryBuilding Using Tendon Control System and Instantaneous OptimalOpen-Loop Control Algorithm: (a) r -0 (No Delay), (b) r - 30 x
10- 3 sec., (c) T - 45 x 10- 3 sec.
6-62
500 (a)
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TIME IN SECONDS
Fig. 57: Effect of Time Delay, T, on Control Force from the FirstController for an 8-Story Building Using Tendon Control System andInstantaneous Optimal Open-Loop Control Algorithm: (a) T =0 (No
Delay), (b) T - 30 x 10. 3 sec., (c) T - 45 x 10-3
sec.
6-63
(a)
(b)
(c)
25 30
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o 5 10 15 20
TIME IN SECONDS
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Fig. 58: Effect of Time Delay, r, on Top Floor Relative Displacement for an8-Story Building Using Tendon Control System and InstantaneousOptimal Closed~Open-Loop Control Algorithm: (a) T =0 (No Delay),
(b) T _ 45 x 10- 3 sec., (c) T - 60 x 10.3
sec.
6-64
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TIME IN SECONDS
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Fig. 59: Effect of Time Delay, 1', on Base Shear Force for an 8-StoryBuilding Using Tendon Control System and Instantaneous OptimalClosed-Open-Loop Control Algorithm: (a) l' -0 (No Delay), (b) T =
- 3 - 345 x 10 sec., (c) l' - 60 x 10 sec.
6-65
500(a)
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5 10 15 20
TIME IN SECONDS
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Fig. 60: Effect of Time Delay, T, on Control Force from the FirstController for an 8-Story Building Using Tendon Control System andInstantaneous Optimal Closed-Open-Loop Control Algorithm: (a) T
- 3 - 3-0 (No Delay), (b) r - 45 x 10 sec., (c) r - 60 x 10 sec.
6-66
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TIME IN SECONDS
2r-------------------,(a)
o
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2
o
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•.-zw:2w(.)<..Ie>el)-QW>-~..IWa:ttoo..I~
CO...
Fig. 63: Effect of Time Delay, T, on Top Floor Relative Displacement foran 8-Story Building Using an Active Mass Damper and RiccatiClosed-Loop Control Algorithm: (a) T - 0 (No Delay), (b) T =
- 3 - 345xlO sec., (c) r = 60xlO sec.
6-69
1.5(a)
z 0~
(¥)
0.,... -1.5.w 1.5(.) (b)a:0 0LL.
c:<w -1.5J:(J) 1.5w (c)(J)
<CD 0
5 10 15 20
TIME IN SECONDS
25 30
Fig. 64: Effect of Time Delay, T, on Base Shear Force for an 8-StoryBuilding Using an Active Mass Damper and Riccati Clos~-LoopContro 1 Al gori thm: (a) T - 0 (No Delay). (b) T = 45xlO sec. •
(c) T - 60xlO- 3 sec.
6-70
350(a)
z 0~
..... -350~
350...(b)w
()a:: 00U.
...I0 -350a:
350t- (c)z00
0
-350ol---s...L----,...J..o-----L,S---l2-0--2l.--5---'~
TIME IN SECONDS
Fig. 65: Effect of Time Delay, T, on Control Force for an 8-Story BUildingUsing an Active Mass Damper and Riccati Closed-Loop Control
Algorithm: (a) T - 0 (No Delay), (b) T - 4SxlO- 3 sec., (c) T =
-360xlO sec.
6-71
(c)
(b)
25 305 10 15 20
TIME IN SECONDS
o
o
o
2,--------------------.(a)
- 2 "--_---'-__---l.-__--L..-__L.-_---L__-.J
2
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2
•...zw~wc.J<..JaU)-QW>-te-'wa:a:oo...JL&.
a.o...
Fig. 66: Effect of Time Delay. r, on Top Floor Relative Displacement foran 8-Story Building Using an Active Mass Damper and Instantaneous Optimal Closed-Loop Control Algorithm: (a) r = 0 (No
Delay), (b) r - 45xlO- 3 sec., (c) r - 60xlO-3
sec.
6-72
1.5(a)
z 0~
M0.... -1.5..w 1.5 (b)()a:0 0u.a:<w -1.5J:(J') 1.5w (c)(J')
<m 0
30255 10 15 20
TIME IN SECONDS
-1.5 '-_-.l.-__..l--_---L.__-'--_----l..-_----l
o
Fig. 67: Effect of Time Delay, r, on Base Shear Force for an 8-StoryBuilding Using an Active Mass Damper and Instantaneous OptimalClosed-Loop Control Algorithm: (a) r - 0 (No Delay), (b) r =
-3 -345xlO sec., (c) r - 60xlO sec.
6-73
30255 10 15 20
TIME IN SECONDS
- 350 I.--.----l..-_---I.-__I.--_-J..-.-_---L_---J
o
Fig. 68: Effect of Time Delay, T, on Control Force for an 8-Story BuildingUsing an Active Mass Damper and Instantaneous Optimal Closed
-3Loop Control Algor i thm: (a) T - 0 (No Delay), (b) T = 45xlO
sec., (c) T - 60xlO- 3 sec.
6-74
(b)
25 305 10 15 20
TIME IN SECONDS
2~-------------------.
(a)
o
o
o
- 2 l--_---.1.__----J-__--'--__.1--_---'-__--l
2
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(c)
•~
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Fig. 69: Effect of Time Delay, r, on Top Floor Relative Displacement foran 8-Story Building Using an Active Mass Damper and Instantaneous Optimal Open-Loop Control Algorithm: (a) r - 0 (No
Delay), (b) r - 45xlO- 3 sec., (c) r - 60xlO- 3 sec.
6-75
(b)
(c)
o
o
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1.5,----------------
-1. 5 l-.--_-l..-__.l-_~__..I...___ _.L._ _____.J
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1.5
·w(Ja:oLJ..
a:<wJ:(J)
wen<m
30255 10 15 20
TIME IN SECONDS
-1.5 L--_--!..-_-----I.__.....l-._--L..__..l.-_--l
o
Fig. 70: Effect of Time Delay, T, on Base Shear Force an 8-Story BuildingUsing an Active Mass Damper and Instantaneous Optimal Open-Loop
-3Control Algorithm: (a) T - 0 (No Delay), (b) T - 45xlO sec.,
(c) T - 60xlO- 3 sec.
6-76
350(a)
0z~
... -350:::>.. 350(b)w
ua: 00LI..-J0 -350a:
350f- (c)z0()
0
-3500l---....l.------L...---L-------J---2---LS--3---'O5 10 1S 20
TIME IN SECONDS
Fig. 71: Effect of Time Delay, T, on Control Force for an 8-Story BuildingUsing an Active Mass Damper and Instantaneous Optimal Open-Loop
·3Control Algorithm: (a) T - 0 (No Delay), (b) T - 45xlO sec.,
(c) T - 60xlO- 3 sec.
6-77
(c)
(a)
25 30
o
o
o
- 2'---_~_ ___I.__..L__ ___L__..L__ _._.J
2r----------------.(b)
-2 ~-=--...L----L----L----L---.Jo 5 10 15 20
TIME IN SECONDS
- 2 -J..--_-----L__....L.-_---L.__..l..-._-J
2r------r---------
~() 2r----------------.•...
zw~w(.)
<-'0-m-QW>-te-'wa:Ctoo...L\.Q.o...
Fig. 72: Effect of Time Delay, r, on Top Floor Relative Displacement foran 8-Story Building Using an Active Mass Damper and InstantaneousOptimal Closed-Open-Loop Control Algorithm: (a).,. = 0 (No
Delay), (b) T - 45xlO- 3 sec" (c) .,. - 60xlO-3
sec.
6-78
1.5(a)
z 0~
C")
0,... -1.5..w 1.5(.) (b)a:0
0LL.
a:<w -1.5:I:UJ 1.5w (c)UJ<m 0
30255 10 15 20
TIME IN SECONDS
-1.5 '--_--l..-._----I.__--l....-_-----L__...L.-_---.1
o
Fig. 73: Effect of Time Delay, T, on Top Floor Relative Displacement foran 8-Story Building Using an Active Mass Damper and Instantaneous Optimal Closed-Open-Loop Control Algorithm: (a) T = 0
(No Delay), (b) T - 45xlO- 3 sec., (c) T = 60xlO-3
sec.
6-79
350(a)
z 0~... -350:)
350..w (b)ua: 00IJ...J0 -350a:~ 350z (c)0u
0
302S-350 ;;:--~_.l..-.-_.1..--_..L..-_..L.-~
o 5 10 15 20
TIME IN SECONDS
Fig. 74: Effect of Time Delay, ~, on Control Force for an 8-Story BuildingUsing an Active Mass Damper and Instantaneous Optimal ClosedOpen-Loop Control Algorithm: (a) T - 0 (No Delay) I (b) T -
45xlO- 3 sec., (c) T - 60xlO-3
sec.
6-80
(c)
(b)
o
o
o
2,..---------------~(a)
-2 L-_.....J..-_---L._--.J__...1.-.._----L-_~
o 5 10 15 20 25 30
TIME IN SECONDS
-2L.-_~_~__...1___ ___i___.L..___ _...J
2
- 2 L-_---L.__....L-_--L.__.....l--_---'__--I
2
•....zw:Ew(.)
<...IC.CI)-QW>-~.JWa:a::oo...L&-
a.o...
Fig. 75: Effect of Truncation of Small Control Forces on Top Floor Relative Displacement for an 8-Story Building with Tendon ControlSystem Using Riccati Closed-Loop Control Algorithm for DifferentTruncation Levels e: (a) No Truncation e - 0, (b) e - 75 KN(17.l6i), (c) e - 100 KN (22.88%).
6-81
1.5 (a)
z 0~
(f)
0-1.5't-
..1.5 (b)w
()a:0LL.
a:<w -1.5:I:(/) 1.5
(C)w(J)
<0CD
-1.5 L--...L-.--L----l--~-~=___:3~Oo 5 10 16 20 25
TIME IN SECONDS
Fig. 76: Effect of Truncation of Small Control Forces on Base Shear Forcefor an 8-Story Building with Tendon Control System Using RiccatiClosed-Loop Control Algorithm for Different Truncation Levels E:(a) No Truncation E - 0, (b) E - 75 KN (17.16%), (c) E = 100 KN(22.88%).
6-82
500(a)
az~..... -500~ 500.. (b)wuCt a0LI....J0 -500a: 500....
(c)z0u
0
30255 10 15 20
TIME IN SECONDS
- 500 l---_--'--_----I.__......L-_---J.-__....L.--__
o
Fig. 77: Effect of Truncation of Small Control Forces on Control Forcefrom First Controller for an 8-Story Building with Tendon ControlSystem Using Riccati Closed-Loop Control Algorithm for DifferentTruncation Levels: (a) No Truncation € - 0, (b) € = 75 KN(17.16%), (c) € - 100 KN (22.88%).
6-83
(a)
(b)
(c)
2
o
o
o
- 2 L.-_-l.-_-L_-.1.__..I..--_--"-- ____
2
- 2 '---_-L-_--L-_--'-_----L.__"'---_
2
-2 '--..l----..L----L--2~O=----=:2~5~3;;-;Oo 5 10 15
TIME IN SECONDS
•to-Zw:iw(..)
<..JC.CIJ-QW>-!C-'wa:a:oo-'LL.CO...
Fig. 78: Effect of Truncation of Small Control Forces on Top Floor Relative Displacement for an 8-Story BUilding with Tendon ControlSystem Using Instantaneous Optimal Closed-Loop Control Algorithmfor Different Truncation Levels E: (a) No Truncation E = 0, (b)E - 75 KN (17.81%), (c) E - 100 KN (23.85%).
6-84
1.5(a)
z 0~
M0
-1.5....•
1.5w (b)(.)a:0u.a:<w -1.5:z:U) 1.5 (c)wU)
<0m
-1.50L..--.....1-----L----J-----.L---
2--'-S--3--'O
5 10 15 20TIME IN SECONDS
Fig. 79: Effect of Truncation of Small Control Forces on Base Shear Forcefor an 8~Story BUilding with Tendon Control System UsingInstantaneous Optimal Closed-Loop Control Algorithm for DifferentTruncation Levels €: (a) No Truncation € - 0, (b) e - 75 KN(17.81%), (c) E:" 100 KN (23.75%).
6-85
500(a)
az~..~ -500::>
500 (b)..w(.)a: 00u....J0 -500a: 500....
(e)z0(.)
0
- 500 0L -._.1..s---,....Lo--'-l..S--2~O=----=2-::-S---;3~OTIME IN SECONDS
Fig. 80: Effect of Truncation of Small Control Forces on Control Forcefrom First Controller for an 8-Story Building with Tendon ControlSystem Using Instantaneous Optimal Closed-Loop Control Algorithmfor Different Truncation Levels e: (a) No Truncation e - 0, (b)f - 75 KN (17.81%), (c) e - 100 KN (23.75%).
6-86
(c)
(b)
(a)2
o
o
o
- 2 L-_---l-_--L__..L.-_-L..-_---I-_----'
2
- 2 l.--_--L.__.L-_---l....__..l...--_-.L..._----'
2
-2 L....-_--L__.L-_---l....__..l...--_-.L..._--:-::
o 5 10 15 20 25 30
TIME IN SECONDS
•t-ZW~W(.)
<...a.CI'J-QW>-t<...wa:e:too..JLA.CO...
Fig. 81: Effect of Truncation of Small Control Forces on Top Floor Relative Displacement for an 8-Story Building with Tendon ControlSystem Using Instantaneous Optimal Open-Loop Control Algorithmfor Different Truncation Levels e: (a) No Truncation e - 0, (b)e - 75 KN (17.81%), (c) e - 100 KN (23.75%).
6-87
1.5(a)
z 0~
C'?0
-1.5.,...•
1.5(b)
w0a:0LL
a:<w -1.5::t(J) 1.5
(c)wCJ)
<0m
-1.5 0L_--L.s --1-LO--1.....J.5-~20-::----=2~5-~30
TIME IN SECONDS
Fig. 82: Effect of Truncation of Small Control Forces on Base Shear Forcefor an 8-Story BUilding with Tendon Control System UsingInstantaneous Optimal Open-Loop Control Algorithm for DifferentTruncation Levels E: (a) No Truncation E .. 0, (b) E = 75 KN(17.81%), (c) E .. 100 KN (23.75%).
6-88
500(a)
0z~
•~ -500:::>
500.. (b)w(Ja: 0 (n.l\r0LL
...J0 -500a:to- 500
(c)z0(.)
0
- 500 '--0----L-----'---.l..---...I.---2--l..~--3.-JOS 10 15 20 ~
TIME IN SECONDS
Fig. 83: Effect of Truncation of Small Control Forces on Control Forcefrom First Controller for an 8-Story Building with Tendon ControlSystem Using Instantaneous Optimal Open-Loop Control Algorithmfor Different Truncation Levels f: (a) No Truncation f - 0, (b)f - 75 KN (17.81%), (c) f - 100 KN (25.75%).
6-89
(b)
(e)
25 305 10 15 20
TIME IN SECONDS
o
2,-----------------.,(a)
o
o
- 2 '-----...L..---1--_---I__----I..-__...1--_---l
2
- 2 ~_--t.__--J.-__._........___.L...___----l-___.J
2
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o
•.-zw~w(.)
<-JC.CfJ-QW>-~-JWtta:oo..JL&.Q.o...
Fig. 84: Effect of Truncation of Small Control Forces on Top Floor Relative Displacement for an 8-Story Building with Tendon ControlSystem Using Instantaneous Optimal Closed-Open-Loop ControlAlgorithm for Different Truncation Levels e: (a) No Truncation e- 0, (b) € - 75 KN (17.81%), (c) e - 100 KN (23.75%).
6-90
1.5(a)
z 0~
C")
0-1.5~
•w 1.5 (b)ua:0
0u.a:<w -1.5J:(f) 1.5 (c)wen<m 0
30255 10 15 20
TIME IN SECONDS
-1.5 '--_--'-__-'---_---I__~--~-----'
o
Fig. 85: Effect of Truncation of Small Control Forces on Base Shear Forcefor an 8-Story Building with Tendon Control System UsingInstantaneous Optimal Closed-Open-Loop Control Algorithm forDifferent Truncation Levels €: (a) No Truncation € = 0, (b) e 75 KN (17.81%), (c) € - 100 KN (23.75%).
6-91
500(a)
0z~
~ -500~.. 500 (b)w ~(.)
~ • h j ~ ~l'a: 0~0 1 ~ ~
LL.
-I0 -500 I I I I I
a:.... 500 (c)z0(.)
0
-500 0L --5..L-.-1..LO--1--LS--2~O=---~2-=-5~3~O
TIME IN SECONDS
Fig. 86: Effect of Truncation of Small Control Forces on Control Forcefrom First Controller an 8-Story Building with Tendon ControlSystem Using Instantaneous Optimal Closed-Open-Loop ControlAlgorithm for Different Truncation Levels f: (a) No Truncation f
- 0, (b) f - 75 KN (17.81%), (c) f - 100 KN (23.75%).
6-92
(a)
(c)
(b)
25 305 10 15 20
TIME IN SECONDS
2
o
o
o
- 2 L-._---l-__.l.--_----l-__...L---_--L-_----'
2
- 2 L-_---L-_-.L__--L.-_----L..__...I.--_--J
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o
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Fig. 87: Effect of Truncation of Small Control Forces on Top Floor Relative Displacement for an 8-Story Building with an Active MassDamper Using Riccati Closed-Loop Control Algorithm for DifferentTruncation Levels €: (a) No Truncation € - 0, (b) € - 50 KN(20%), (c) € - 75 KN (30%).
6-93
(a)1.5
z 0~
M0
-1.5.,...• 1.5 (b)UJ
"a:0u... 0a:<UJ -1.5~en 1.5 (C)w(f.)
<m 0
-1.5 OL--...L...---1.----L..-----L....--~2-=-5-3~O5 10 15 20
TIME IN SECONDS
Fig. 88: Effect of Truncation of Small Control Forces on Base Shear Forcefor an B~Story BUilding with an Active Kass Damper Using RiccatiClosed-Loop Control Algorithm for Different Truncation Levels €:(a) No Truncation e - 0, (b) £ - 50 KN (20%), (c) € _ 75 KN(30%) .
6-94
350(a)
0z~
~ -350~
350..w (b)uCt 00u..-J0 -350 1
Ct350....z (c)
0u
0
30255 10 15 20
TIME IN SECONDS
-350 '---_--J---~--.1.-----I...------l...-----I
o
Fig. 89: Effect of Truncation of Small Control Forces for an 8-StoryBuilding with Active Mass Damper Using Riccati Closed-LoopControl Algorithm for Different Truncation Levels €: (a) NoTruncation € - 0, (b) € - 50 KN (20%), (c) € - 75 KN (30%).
6-95
(a)
(b)
25 30S 10 15 20
TIME IN SECONDS
2
o
o
o
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2
- 2 l.--_---l--__.L--_---L-__.l--_-..L._----.J
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2r---------------.........(c)
•....zw:Ew(.)
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Fig. 90: Effect of Truncation of Small Control Forces on Top Floor Relative Displacement for an 8-Story Building with Active Mass DamperUsing Instantaneous Optimal Closed-Loop Control Algorithm forDifferent Truncation Levels E: (a) No Truncation E = 0, (b) e 50 KN (21.55%), (c) E - 75 KN (32.32%).
6-96
1.5 (a)
~-~z 0~
('l)
0-1.5,...
• 1.5w (b)()a:0
0LL.
a:<w -1.5:t:fJ) 1.5 (c)wfJ)
<0m
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o 5 10 15 20 2530
TIME IN SECONDS
Fig. 91: Effect of Truncation of Small Control Forces on Base Shear Forcefor an 8-Story Building with Active Mass Damper UsingInstantaneous Optimal Closed-Loop Control Algorithm for DifferentTruncation Levels £: (a) No Truncation £ - 0, (b) £ - 50 KN(21. 55%), (c) £ - 75 KN (32.32%).
6-97
350(a)
0z~
.,..-350:J
350 (b)..w(,)a: 00u..-J0 -350a:
350t-(c)z
00
0
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TIME IN SECONDS
Fig. 92: Effect of Truncation of Small Control Forces for an 8-StoryBuilding with Active Mass Damper Using Instantaneous OptimalClosed-Loop Control Algorithm for Different Truncation Levels E:(a) No Truncation E - 0, (b) E - 50 KN (21. 55%), (c) E _ 75 KN(32.32%).
6-98
(b)
(a)
(c)
2
o
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- 2L-_-l.-_--l__-l-_--l...__-'---_
2
- 2 L-_-L-_----..l__-l-_---L..__....L.--
2
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TIME IN SECONDS
•....zw~w(.)
<-'a.(fJ-QW>-~..JW£:~oo-'u.CO....
Fig. 93: Effect of Truncation of Small Control Forces on Top Floor Relative Displacement for an 8-Story Building with Active Mass DamperUsing Instantaneous Optimal Open-Loop Control Algorithm forDifferent Truncation Levels e: (a) No Truncation e - 0, (b) e _50 KN (21.55%), (c) e - 75 KN (32.32%).
6-99
1.5(a)
z 0 ~~-~~
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a:0
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(c)wUJ<
0£D
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o 5 10 15 20 25 30
TIME IN SECONDS
Fig. 94: Effect of Truncation of Small Control Forces on Base Shear Forcefor an 8-St o ry BUilding with Active Mass Damper UsingInstantaneous Optimal Open-Loop Control Algorithm for DifferentTruncation Leve Is e: (a) No Truncation e - 0, (b) e - 50 KN(21.55%), (c) e - 75 KN (32.32%).
6-100
350 (a)
0 f\/VV-----Z~
..... -350:::>.. 350 (b)w(.)a: 00LL...J0 -350a:
350...(c)z
0(.)
0
-3500L.--....L---t---L-----l.---
2--L.
S--
3--'0
5 10 1S 20
TIME IN SECONDS
Fig. 95: Effect of Truncation of Small Control Forces for an 8-StoryBuilding with Active Mass Damper Using Instantaneous OptimalOpen-Loop Control Algorithm for Different Truncation Levels €:(a) No Truncation € - 0, (b) € - 50 KN (21. 55%), (c) € - 75 KN(32.32%) .
6-101
(b)
25 305 10 15 20
TIME IN SECONDS
o
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2r--------------(a)
o
- 2 ~_---1....-_ __l____L__ __L__l.___ __.J
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•t-Zw:Ew(.)
<..Ja.CfJ-QW>-=c-Iwa:a:oo...LL.Q..o...
Fig. 96: Effect of Truncation of Small Control Forces on Top Floor Relative Displacement for an 8-Story Building with Active Mass DamperUsing Instantaneous Optimal Closed-Open-Loop Control Algorithmfor Different Truncation Levels E: (a) No Truncation E - 0, (b)E - 50 KN (21. 55%), (c) E - 75 KN (32.32%).
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-1. 5 l-_-L..._---L__l...--_--l.-_--I.-_~
o 5 10 15 20 25 30
TIME IN SECONDS
Fig. 97: Effect of Truncation of Small Control Forces on Base Shear Forcefor an 8-Story Building with Active Mass Damper UsingInstantaneous Optimal Closed-Open-Loop Control Algorithm forDifferent Truncation Levels f: (a) No Truncation f = 0, (b) f =
50 KN (21.55%), (c) f - 75 KN (32.32%).
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-350 OL--.l.----!.--....L----I..-----L2-5-3~O5 10 15 20
TIME IN SECONDS
Fig. 98: Effect of Truncation of Small Control Forces for an 8-StoryBuilding with Active Mass Damper Using Instantaneous OptimalClosed-Open-Loop Control Algorithm for Different TruncationLevels c (a) No Truncation e - 0, (b) e - 50 K.N (21.55%), (c) €
- 75 KN (32.32%).
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VII. CONCLUSIONS
Under seismic excitations, the effect of system uncertainty, time
delay, and truncation of small control forces on the structural control
system have been investigated. The sensitivity, criticality and tolerance
for system identification errors, time delay, and elimination of small
control forces depend on the particular control algorithm used. Four
control algorithms, which have been verified experimentally to be useful for
earthquake-excited structures [4,5], have been studied. These include the
Riccati closed-loop control algorithm, and three instantaneous optimal
control algorithms recently proposed [17-18]. The method of sensitivity
analysis for each control algorithm has been presented. Both active tendon
control system and active mass damper have been investigated. Conclusions
are summarized in the following.
(A) For uncertainty in system identification, the following conclusions
are observed.
(i) The instantaneous optimal closed-loop control algorithm is
independent of the system uncertainty. In other words, any estimation error
in structural properties, such as damping, stiffness and natural frequency,
does not affect the efficiency of the control system.
(ii) The instantaneous optimal closed-open-loop control algorithm is
not sensitive to system uncertainties at all. A substantial estimation
error for various structural parameters has an insignificant effect on the
control system.
(iii) The instantaneous optimal open-loop control algorithm is quite
sensitive to the system uncertainty in stiffness (or natrual frequency). An
estimation error over 10% for the stiffness (or 5% for the first natural
frequency) may result in a serious degradation for the control system.
However, the control algorithm is not sensitive to the damping estimation.
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In other words, a large estimation error for the structural damping does not
affect the control system.
(iv) For the Riccati closed-loop control algorithm, the uncertainty in
stiffness (or frequency) estimation has a moderate effect on the structural
response and control force; however, its effect on the control efficiency is
less prominent. A 20% estimation error for the stiffness (or 10% for the
fundamental natrual frequency) is clearly acceptable. Again, the Riccati
closed~loop control algorithm is not sensitive to the uncertainty in damping
estimation. A 40% estimation error for damping is still acceptable.
From numerous sensitivity analysis results, it can be concluded, in
general, that the control system is not sensitive at all to the statistical
uncertainty involved in determining the dampings of the structure. The
effect of damping estimation error on the control system is negligible.
This is a very important and beneficial conclusion for the application of
active control system, because the accurate identification of the damping
coefficient for a structure is rather difficult. For some control
algorithms, however, care should be taken in estimating the stiffness or
natural frequencies in order to avoid excessive degradation of the control
system.
(B) For system time delay, the following conclusions are obtained.
The instantaneous optimal open-loop control algorithm is most sensitive
and critical to system time delay. It is followed by the instantaneous
optimal closed-open-Ioop control algorithm. Two closed-loop control
algorithms investigated are less sensitive to time delay because they depend
exclusively on the feedback response state vector. In general, open-loop
control is expected to be more critical and sensitive to system time delay.
A system time delay always results in a degradation of control efficiency in
the sense that both the response quantities and required active control
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forces are larger than the corresponding results associated with no system
time delay.
One important observation is that the tolerance for time delay is
rather stringent, with 3% of the fundamental structural period for the
instantaneous optimal open-loop control algorithm, and 4.5% of the
fundamental structural period for other control algorithms. For the
structure with a fundamental period, Tf , of 1.085 seconds considered in this
3 -3report, the tolerances are 33 x 10- and 49 x 10 seconds, respectively.
The problem becomes more critical when the period of the structure is
shorter (or the fundamental frequency is higher). As a result, the system
time delay may be an important issue for practical implementation of active
control systems. In fact, such a conclusion has also been revealed
experimentally [4, 5]. The experimental results using a scale structural
model and excited by an earthquake record on a shaking table [4, 5] indicate
that the response quantities and control force are always larger than those
computed theoretically without a system time delay. In this regard, future
research efforts are needed to establish methodologies for compensating
system time delay, such as th: preliminary study conducted in Ref. 9.
(C) For the truncation of small control forces, the following
conclusions are obtained from extensive numerical results.
i) The active control systems investigated are not sensitive to the
elimination of control forces that are smaller than 20% of the maximum
control force.
(ii) With the truncation of small control forces, the structural
response quantities do not die down as rapidly as in the case without
truncation. However, the maximum response quantities remain practically
unchanged, if the truncation level is within 20% of the maximum control
force.
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(iii) The conclusions obtained above hold for all control algorithms
studied in this report. It is expected that these conclusions will hold for
other control algorithms not investigated.
It should be emphasized that conclusions derived in this report are
restricted to seismic-excited structures. Further studies are needed for
other types of environmental loads.
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ACKNOWLEDGEMENT
This research is supported in part by the National Science Foundation
Grant No. ECE-85-2l496 and National Center for Earthquake Engineering
Research Grant No. NCEER-86-302l.
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