In order to prevent structural damages, it is more important to bound the vibration amplitude than to reduce the vibration energy. However, in the performance index for linear quadratic regulator (LQR), the instantaneous amplitude of vibration is not minimized. An ordinary LQR may have an unacceptable amplitude at some time instant but still have a good performance. In this paper, we have developed an LQR with adjustable gains to guarantee bounds on the vibration amplitude. For scalar systems, the weighting for control is switched between two values which give a low-gain control when the amplitude is inside the bound and a high-gain control when the amplitude is going to violate the given bound. For multivariable systems, by assuming a matching condition, a similar controller structure has been obtained. This controller is favored for application since the main structure of a common LQR is not changed; the additional high-gain control is required only if the vibration amplitude fails to stay inside the bound. We have applied this controller to a five-story building with active tendon controllers. The results show that the largest oscillation at the first story stays within a given bound when the building is subject to earthquake excitation.

1.
Anderson, B. D. O., and Moore, J. B., 1970, Linear Optimal Control, Prentice Hall.
2.
Bryson, A. E., and Ho, Y.-C., 1975, Applied Optimal Control, Hemisphere.
3.
Harvey
C. A.
, and
Stein
G.
,
1978
, “
Quadratic Weights for Asymptotic Regulation Properties
,”
IEEE Trans. Automatic Control
, Vol.
AC-23
, pp.
378
387
.
4.
Khalil, H. K., 1992, Nonlinear Systems, Macmillan Publishing Company.
5.
Moore
B. C.
,
1976
,“
On the Flexibility Offered by State Feedback in Multivariable Systems Beyond Closed-Loop Eigenvalue Assignment
,”
IEEE Trans. Automatic Control
, Vol.
21
, pp.
689
691
.
6.
Pruca
Z.
,
Soong
T. T.
and
Reinhorn
A. M.
,
1985
, “
An Analysis of Pulse Control for Simple Mechanical Systems
,”
ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL
, Vol.
107
, pp.
123
131
.
7.
Soong, T. T., 1990, Active Structural Control: Theory & Practice, Longman.
8.
Van Loan
C. F.
,
1977
, “
The Sensitivity of the Matrix Exponential
,”
SIAM J. Numerical Analysis
, Vol.
14
, No.
6
, pp.
971
981
.
9.
Yang
J. N.
,
Akbarpour
A.
, and
Ghaemmaghami
P.
,
1987
, “
New Optimal Control Algorithms for Structure Control
,”
ASCE J. of Engineering Mechanics
, Vol.
113
, pp.
1367
1386
.
This content is only available via PDF.
You do not currently have access to this content.