1
0:8
0:6
0:4
(t)s13 0:2
0
;0:2
;0:4
max = 0 466
U
:
;
;
;0:6
us = 0 70
:
;
;
;0:8
0
0:5
1
1:5
t2
2:5
3
3:5
4
A step response with an undershoot of 0.70 and a rise time of
Figure
12.4
1.0, which achieves the specications marked \X" in gure 12.3. Undershoot
as small as 0 466 with a rise time of 1 0 are also achievable.
:
:
The tradeo curve in gure 12.3 is valuable even though the design speci cations
do not limit actuator e ort. If we add to our design speci cations an appropriate
limit on actuator e ort, the new tradeo curve will lie above the one we have found.
Thus, our tradeo curve identi es design speci cations that are not achievable, e.g.,
Umax = 0:4, Tmax = 1:0, when no limit on actuator e ort is made a fortiori these
design speci cations are not achievable when a limit on actuator e ort is included.
We remark that the tradeo for this example is considerably more general than
the reader might suspect. (12.39) holds for
any LTI plant with P0(1) = 0,
the 2-DOF controller con guration,
any nonlinear or time-varying controller.
This is because, no matter how the plant input u is generated, the output of P0,
yp, must satisfy conditions of the form (12.35{12.36).
12.5
A Weighted Peak Tracking Error Example
In this section we present a less trivial example of a plant and family of design
speci cations for which we can explicitly solve the feasibility problem.
12.5 A WEIGHTED PEAK TRACKING ERROR EXAMPLE
287
We consider the classical 1-DOF system of section 2.3.2 with
P
2
0(s) = s ;
s2 1:
;
Designing a controller for this plant is quite demanding, since it has an unstable
zero at s = 2 along with an unstable pole only an octave lower, at s = 1.
Our design speci cations will be internal stability and a limit on a weighted
peak gain of the closed-loop tracking error transfer function:
WS pk gn Emax
(12.40)
k
k
where
W(s) =
1
1 + sTtrk
and S is the closed-loop transfer function from the reference input r to the error
;
e = r +yp (see sections 5.2.5 and 8.1.2). Thus we have a two-parameter family of
;
design speci cations, indexed by Emax and Ttrk.
Roughly speaking, Emax is an approximate limit on the worst case peak mis-
tracking that can occur with reference inputs that are bounded by one and have a
bandwidth 1=Ttrk. Therefore, 1=Ttrk represents a sort of tracking bandwidth for the
system. It seems intuitively clear, and turns out to be correct, that small Emax can
only be achieved at the cost of large Ttrk.
These design speci cations are simple enough that we can explicitly solve the
feasibility problem for each Emax and Ttrk. As in the previous section, however, these
design speci cations are not complete enough to guarantee reasonable controller
designs, so the comments made in the previous section hold here as well.
As we did for the previous example, we express internal stability in terms of the
interpolation conditions: S is stable and satis es
S(1) = 0 S(2) = S( ) = 1:
1
Equivalently, WS is stable, and satis es
WS(1) = 0
(12.41)
WS(2) = (1 + 2Ttrk) 1
;
(12.42)
lim
1
s
sWS(s) = T;trk:
(12.43)
!1
Let h be the impulse response of WS, so that
WS
Z
1
pk gn =
h(t) dt
k
k
0 j
j
288