2 1 2 3
A
S
S
S
A
S
S
S
1 1 2
1 1 3
1 2 3
2 1 2
2 1 3
2 2 3
A
S
S
A
S
S
A
S
S
A
S
S
A
S
S
A
S
S
1 1
1 2
1 3
2 1
2 2
2 3
A
S
A
S
A
S
A
S
A
S
A
S
The possible actuator and sensor congurations, with the
Figure
16.4
partial ordering induced by achievability of a set of specications.
1 1 2 3
2 1 2 3
A
S
S
S
A
S
S
S
achievable
D
1 1 2
1 1 3
1 2 3
2 1 2
2 1 3
2 2 3
A
S
S
A
S
S
A
S
S
A
S
S
A
S
S
A
S
S
1 1
1 2
1 3
2 1
2 2
2 3
A
S
A
S
A
S
A
S
A
S
A
S
unachievable
D
The actuator and sensor congurations that can meet the
Figure
16.5
specication .
D
Modifying the Plant Model and Specifications
After choosing an achievable set of design speci cations, the design is veri ed: does
the controller, designed on the basis of the LTI model and the design speci ca-
P
tions , achieve the original goals when connected in the real closed-loop system?
D
If the answer is no, the plant and design speci cations have failed to accu-
P
D
rately represent the original system and goals, and must be modi ed, as shown in
gure 16.6.
Perhaps some unstated goals were not included in the design speci cations. For
example, if some critical signal is too big in the closed-loop system, it should be
added to the regulated variables signal, and suitable speci cations added to , to
D
constrain the its size.
16.3 SOME HISTORY OF THE MAIN IDEAS
377
System
Plant
Feas. yes
yes
Prob.
OK?
no
no
Goals
Specs.
If the outcome of the feasibility problem is inconsistent with
Figure
16.6
designer's criteria then the plant and specications must be modied to
capture the designer's intent.
As a speci c example, the controller designed in the rise time versus undershoot
tradeo example in section 12.4, would probably be unsatisfactory, since our design
speci cations did not constrain actuator e ort. This unsatisfactory aspect of the
design would not be apparent from the speci cations|indeed, our design cannot
be greatly improved in terms of rise time or undershoot. The excessive actuator
e ort would become apparent during design veri cation, however. The solution, of
course, is to add an appropriate speci cation that limits actuator e ort.
An achievable design might also be unsatisfactory because the LTI plant is
P
not a su ciently good model of the system to be controlled. Constraining various
signals to be smaller may improve the accuracy with which the system can be
modeled by an LTI
adding appropriate robustness speci cations (chapter 10)
P
may also help.
16.3
Some History of the Main Ideas
16.3.1
Truxal’s Closed-Loop Design Method
The idea of rst designing the closed-loop system and then determining the con-
troller required to achieve this closed-loop system is at least forty years old. An
explicit presentation of a such a method appears in Truxal's 1950 Ph.D. the-
sis
], and chapter 5 of Truxal's 1955 book, Automatic Feedback Control
Tr
u50
System Synthesis, in which we nd
]:
Tr
u55,
p279
Guillemin in 1947 proposed that the synthesis of feedback control sys-
tems take the form ...
1. The closed-loop transfer function is determined from the speci ca-
tions.
2. The corresponding open-loop transfer function is found.
3. The appropriate compensation networks are synthesized.
378