Civil Engineering Reference
In-Depth Information
denoting
224
T
C
0
e
e
T
f
1
()
203
K
S
224
f
2
()
--------------------
f
1
()
(
C
0
)
(3.31)
Substituting (3.31), into (3.30) yields
zf
1
()
z
1
.
G
PC
G
TWP
()
z
f
2
()
-------------------------------------------------
(3.32)
(
)
(
zC
0
)
The discrete PI-type controller can be rewritten as follows:
K
PROP
1
z
1
(
1
)
.
G
FC
Z
()
--------------
-------------------------------------
(3.33)
z
1
Thus if
varies sufficiently slowly or the controller is sufficiently fast,
can be chosen such that
1
f
1
()
------------
1
(3.34)
then complete open-loop discrete transfer function is defined as
K
PROP
1
1
zC
0
G
FC
G
PC
G
TWP
z
()
--------------
f
2
()
----------------
(3.35)
with an overall effective gain controller of
K
PROP
1
K
EFFECTIVE
--------------
f
2
()
(3.36)
Equations (3.31) through (3.36) are used to determine the PI force controller gains in the controller
design update of the self-tuning regulator. The effectiveness of the pole-zero cancellation is related to the
process variation frequency spectrum and the gain update rate.
The implementation of this controller uses the windowed data handling scheme as previously dis-
cussed. Thus, the size of the data window, is the key parameter to the success of this approach. The
smaller the window, the faster the gain update. However, as the sample window gets smaller, the variance
of the parameter estimation increases. This reduces the confidence of the estimate and ultimately reduces
the probability of an effective pole-zero cancellation.
Controller Stability and Bounds
An important issue in the successful application of adaptive control to nonlinear systems is the stability
of the combined plant and controller. Several approaches have been developed by authors in this regard.
The most common approach is to ignore the stability analysis, letting a successful implementation stand
on its own merit without a rigorous stability analysis. This is dangerous and should be avoided.
The most notable stability analysis method, and the most popular employed, is assessing stability of
a controlled system in the sense of Lyapunov (Slotine and Li, 1991). Lyapunov's direct method requires
the existence of an energy-type, scalar function of the plant state that is positive definite, with continuous
