Earthquake Engineering Research - HKU Libraries - The University ...

233
g ^ w (4)
Notice that the stability condition depends only on the natural frequency rather than the damping ratio of
the structures. In order to achieve maximum damping effect in the PPF control, the natural frequency
should be closely matched to that of the structure.
2.2 Strain Rate Feedback Control
Strain rate feedback control (SRF) is implemented by feeding the velocity co-ordinate to the compensator.
The position co-ordinate of the compensator is then fed back with a negative gain to the structure. When a
smart structure is involved using a collocated PZT actuator and sensor, this control is achieved by feeding
the derivative of the voltage from the sensor, which is proportional to the strain rate, to the input of the
compensator and applying the negative compensator output voltage to the actuator. The scalar equations
governing the vibration of the structure in a single mode and the SRF controller are given as:
-gQ} 2 7l (5)
= a; t 2 f (6)
The variables above are the same as those defined in the PPF control. The stability condition requires that
the frequency in the second order compensator be greater than the controlled mode frequency of the
structure. As compared with the PPF, SRF has a much wider active damping frequency region, which
gives a designer some flexibility. Selecting a precise compensator frequency for SELF is not as critical as
for PPF. As long as the compensator satisfies the stability condition, a certain amount of damping will be
provided. SRF can also stabilize more than one mode given a sufficient bandwidth. A limitation of SRF is
that the magnitude of the transfer function in the active damping region becomes extremely small very
quickly. Therefore, the amount of damping provided SRF over a certain frequency range is limited.
2.3 Sliding-Mode Based Robust Controller
Sliding mode controller (SMC) technique is naturally robust with respect to the uncertainty in the
structural parameters and external disturbances. In principle, SMC consists of the control law that
switches with infinite speed to drive the system on a specified state trajectory, called the sliding surface,
and has capability to keep the state on this surface. Clearly, this controller belongs to the non-linear robust
controller. Among the design issues, the most important is to design the sliding surface and reduce the
chatter phenomenon. The governing equation of the structure is the same as described in equation (1). The
control force in the SMC framework is obtained by following the equivalent control method, which
requires that the sliding surface r = £ + A£ and r = 0 , and the control force is called as the equivalent
force. The parameter A is a positive constant. The control law can be expressed as
u = -K D r~-pszt(ar) (7)
where K D and r are positive constants. The parameter p called robust gain is maximum value of the
nonlinear switching control action to account for the disturbance. And the sat function is a nonlinear
saturation function imposing upper and lower bounds on a signal. The function is defined as
Isgn(ar)
otherwise
(8)