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To examine the effect of the control forces on the uncontrolled modes, residual modes can be written
^R(^= : A R w R (t) J rB R f(t) + E R x g (12)
where W R is a residual state vector by uncontrolled modes. Substituting Eq. (10) into Eq. (4) and
considering Eq. (12), we obtain
w c (r) = A c w c (t)-B c K c w c (t) + E c x g} w R (t)-A R w R (t)-B R K c w c (f) + E R x (13)
Moreover, substituting Eqs. (10) and (11) into Eq. (8), we can write the observer equation in the form
* c (t) = (Ac - B C K C )>v c (r) + LC C (w c (t) - w c (r)) + LC R W R (r) + E c x g (14)
The error vector is defined such as e c (t) = w c (t) - w c (t). Then the Equations can be written in the
matrix form
A C -B C K C 0
0
~B C K C
— BffK r
(15)
LC R A C -LC C
Note that the term -B^Kc in Eq. (15) is responsible for the excitation of the residual modes by the
control forces and is known as control spilloveralas. If CR is zeros, which means the sensor signal only
include controlled modes, the term -B&Kc has no effect on the eigenvalues of the closed-loop system.
Hence, we conclude that control spillover cannot destabilize the system, although it can cause some
degradation in the system performance. Normally, however, the above system can not satisfy the
separate principle because the term LC# affects eigenvalues of the controlled system by the observer.
This effect is known as observation spillover and can produce instability in the residual modes.
However, a small amount of damping inherent in the structure is often sufficient to overcome the
observation spillover effect. At any rate, observation spillover can be eliminated if the sensor signals
are prefiltered so as to screen out the contribution of the uncontrolled modes (Meirovitch, 1990)
Numerical Example
To evaluate the proposed modal control scheme for use with the MR damper, a numerical example is
considered in which a model of a six-story building is controlled with four MR dampers (Fig. 1). This
numerical example is the same with that of Jansen and Dyke (2000) and is adopted for direct
comparisons between the proposed modal control scheme and other control algorithms. In simulation,
the model of the structure is subjected to the NS component of the 1940 El Centro earthquake. Because