We can express the design speci cation of internal stability in terms of the
interpolation conditions (section 7.2.5) for T, the I/O transfer function: T is stable
and satis es
T(1) = T( ) = 0:
(12.34)
1
This in turn can be expressed in terms of the step response s13(t): s13 is the step
response of a stable transfer function and satis es
Z
1
s13(t)e t dt
;
= 0
(12.35)
0
s13(0) = 0:
(12.36)
Now if (12.33) holds then
Z
Z
1
s
t dt 0:8 1 e t dt = 0:8e Tmax:
(12.37)
;
;
T
13(t)e;
max
Tmax
If (12.32) holds then
T
T
Z
max s
Z
max
13(t)e t dt
U
e t dt = U
e Tmax : (12.38)
;
max
;
max ;1
;
0
;
0
;
;
Adding (12.37) and (12.38) we have
Z
0 = 1 s13(t)e t dt
Tmax U
e Tmax :
;
0:8e;
max ;1
;
0
;
;
Hence if the design speci cations with Umax and Tmax are feasible,
T
U
0:8e max
;
max
1 e T :
max
;
;
This relation is shown in gure 12.3. We have shown that every achievable under-
shoot, rise-time speci cation must lie in the shaded region of gure 12.3 in other
words, the shaded region in gure 12.3 includes the region of achievable speci ca-
tions in performance space.
In fact, the speci cations with limits Umax and Tmax are achievable if and only
if
T
U
max
max > 0:8e;
1 e T
(12.39)
max
;
;
so that the shaded region in gure 12.3 is exactly the region of achievable speci -
cations for our family of design speci cations.
We will brie y explain why this is true. Suppose that Umax and Tmax sat-
isfy (12.39). We can then nd a step response s13(t) of a stable rational transfer
function, that satis es the interpolation conditions (12.35{12.36) and the overshoot
12.4 A SIMPLE RISE TIME, UNDERSHOOT EXAMPLE
285
2
1:8
1:6
1:4
1:2
Umax
1
X
0:8
;
q
;
0:6
0:4
0:2
0
0
0:2
0:4
0:6
0:8
T 1 1:2 1:4 1:6 1:8 2
max
The tradeo between achievable undershoot and rise-time
Figure
12.3
specications.
and undershoot limits. If Umax and Tmax are near the boundary of the region of
achievable speci cations, this step response will have to \hug" (but not violate)
the two constraints. For Umax = 0:70 and Tmax = 1:0 (marked \X" in gure 12.3)
a suitable step response is shown in gure 12.4 it is the step response of a 20th
order transfer function (and corresponds to a controller K of order 22). (A detailed
justi cation that we can always design such a step response is quite cumbersome
we have tried to give the general idea. See the Notes and References at the end of
this chapter for more detail about this particular transfer function.)
The rapid changes near t = 0 and t = 1 of the step response shown in gure 12.4
suggest very large actuator signals, and this can be veri ed. It should be clear that
for speci cations Umax, Tmax that are nearly Pareto optimal, such rapid changes in
the step response, and hence large actuator signals, will be necessary. So controllers
that achieve speci cations near the tradeo curve are probably not reasonable from
a practical point of view but we point out that this \side information" that the
actuator signal should be limited was not included in our design speci cations. The
fact that our speci cations do not limit actuator e ort, and therefore are probably
not sensible, is re ected in the fact that the Pareto optimal speci cations, which
satisfy
T
U
max
max = 0:8e;
1 e Tmax
;
;
are not achievable (see the comments at the end of section 3.5).
286