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where 8 t (i = 1 , 2 , . .. , n] denote the sliding margins. These controllers will have the state move from an
arbitrary initial state to the sliding mode and then move along the user-specified sliding mode to origin.
And the control input at boundary layer is assumed as follows equation.
u c - -Kx = -(A^ + & 2 jc 2 + .. . + k n x n ) (13)
+,!-/; xinf a, -£,) £ f >o
The boundary layer is a hypothetical band centred on the sliding mode cr = 0 , The width of the boundary layer,
A represents the tolerance or allowable deviation from the ideal sliding motion due to parameter uncertainty and
desturbances. The contol input where the boundary layer equivalence is considered can be reprezented by
where (f> represents parameter uncertainty and other disturabances. w is the classical sliding mode given by
disturbance-accommodating.
In the above treatment, the semi-active force is supposed to be realized by variable dampers. Actually, the
damping coefficients should be positive, and remain within a certain range for practical use. Therefore, the
damping coefficients of semi-active dampers are limited by the inequality. By supposing that the control force
generated by the semi-active dampers under the above constraint is as close as possible to the control force of
active control, the damping coefficient of semi-active damper is expressed as follows:
C =
NUMERICAL CALCULATION
In order to investigate the feasibility and the effectiveness of the semi-active control using variable dampers
which is designed by the method proposed in this study, the numerical calculation was carried out.
And the control characteristics of the proposed semi-active base isolation control are examined in both time
and frequency domains. Since the semi-active control is essentially nonlinear, the vibration transmissibilityunder
semi-active control is obtained as a ratio in frequency range of the spectral densities of input and output. And a
white noise is used as an input.
For the semi-active contol based upon the disturbance accommodating sliding mode using the bilinear optimal
switching hyperplane, an observer used VSS observer was, since the measured physical values are sup-