National Instruments Xmath Interactive Control Design Module ICDM - Manual (Page 66)

Chapter 7 LQG Synthesis
© National Instruments Corporation 7-5 Xmath Interactive Control Design Module
Integral Action
When integral action is enabled, the controller minimizes a variation on the
LQG cost:
where
As in the standard mode, the sensor noise parameter
ν is the ratio of the
sensor noise intensity to the input-referred process noise intensity.
Penalizing the “running integral” of the plant output forces the power
spectral density of the plant output to vanish at zero frequency.
In classical control terms, this forces a pole at S = 0 in the loop transfer
function, that is, integral control. As with PID design, the parameter T
int
gives the time scale over which the effects of the integral action will take
place.
Exponential Time Weighting
When this feature is enabled, the plant is first changed to P(s – a), where a
is the Decay Rate parameter. In other words, the plant is made less stable;
its poles (and zeros) are shifted to the right by the value a. Then, the LQG
controller for this “destabilized” plant is computed. Finally, the poles and
zeros of this controller are shifted left by the Decay Rate parameter a.
One effect of this shifting is that the closed-loop poles are guaranteed to
have real part less than the Decay Rate parameter a. In other words, the
closed-loop time domain responses are guaranteed to decay at least as fast
as exp(–at). This is why the parameter is called Decay Rate.
J E
t ∞ →
lim ρu t ( )
2
y t ( )
2
z t ( )
2
+ +
( )
=
z t ( ) 1 T
int
⁄ ( ) y τ ( ) τ d
0
t
∫
=