Truxal cites his Ph.D. thesis and a 1951 article by Aaron
].
Aar51
On the di erence between classical controller synthesis and the method he pro-
poses, he states (
]):
Tr
u55,
p278{279
The word synthesis rigorously implies a logical procedure for the transi-
tion from speci cations to system. In pure synthesis, the designer is able
to take the speci cations and in a straightforward path proceed to the
nal system. In this sense, neither the conventional methods of servo
design nor the root locus method is pure synthesis, for in each case the
designer attempts to modify and to build up the open-loop system until
he has reached a point where the system, after the loop is closed, will
be satisfactory.
... The closed-loop design] approach to the synthesis of closed-loop sys-
tems represents a complete change in basic thinking. No longer is the
designer working inside the loop and trying to splice things up so that
the overall system will do the job required. On the contrary, he is now
saying, \I have a certain job that has to be done. I will force the system
to do it."
So Truxal views his closed-loop controller design method as a \more logical synthesis
pattern" ( 278) than classical methods. He does not extensively justify this view,
p
except to point out the simple relation between the classical error constants and the
closed-loop transfer function ( 281). (In chapter 8 we saw that the classical error
p
constants are a ne functionals of the closed-loop transfer matrix.)
The closed-loop design method is described in the books
],
NGK57
RF58,
],
], and
] (see also the Notes and References from
ch7
Hor63,
x5.12
FPW90
,
x5.7
chapter 7 on the interpolation conditions).
16.3.2
Fegley’s Linear and Quadratic Programming Approach
The observation that some controller design problems can be solved by numerical
optimization that involves closed-loop transfer functions is made in a series of papers
starting in 1964 by Fegley and colleagues. In
] and
], Fegley applies
Feg64
FH65
linear programming to the closed-loop controller design approach, incorporating
such speci cations as asymptotic tracking of a speci c command signal and an
overshoot limit. This method is extended to use quadratic programming in PF66,
]. In
] and
], speci cations on RMS values of signals are included.
BF68
CF68
MF71
A summary of most of the results of Fegley and his colleagues appears in
],
FBB71
which includes examples such as a minimum variance design with a step response
envelope constraint. This paper has the summary:
Linear and quadratic programming are applicable ... to the design of
control systems. The use of linear and quadratic programming fre-
quently represents the easiest approach to an optimal solution and often
16.3 SOME HISTORY OF THE MAIN IDEAS
379
makes it possible to impose constraints that could not be imposed in
other methods of solution.
So, several important ideas in this book appear in this series of papers by Fegley
and colleagues: designing the closed-loop system directly, noting the restrictions
placed by the plant on the achievable closed-loop system expressing performance
speci cations as closed-loop convex constraints and using numerical optimization
to solve problems that do not have an analytical solution (see the quote above).
Several other important ideas, however, do not appear in this series of papers.
Convexity is never mentioned as the property of the problems that makes e ective
solution possible linear and quadratic programming are treated as useful \tools"
which they \apply" to the controller design problem. The casual reader might
conclude that an extension of the method to inde nite (nonconvex) quadratic pro-
gramming is straightforward, and might allow the designer to incorporate some
other useful speci cations. This is not the case: numerically solving nonconvex
QP's is vastly more di cult than solving convex QP's (see section 14.6.1).
Another important idea that does not appear in the early literature on the
closed-loop design method is that it can potentially search over all possible LTI
controllers, whereas a classical design method (or indeed, a modern state-space
method) searches over a restricted (but often adequate) set of LTI controllers. Fi-
nally, this early form of the closed-loop design method is restricted to the design
of one closed-loop transfer function, for example, from command input to system
output.
16.3.3
-Parameter Design
Q
The closed-loop design method was rst extended to MAMS control systems (i.e., by
considering closed-loop transfer matrices instead of a particular transfer function),
in a series of papers by Desoer and Chen DC81a, DC81b, CD82b, CD83] and
Gustafson and Desoer GD83, DG84b, DG84a, GD85]. These papers emphasize
the design of controllers, and not the determination that a set of design speci cations
cannot be achieved by any controller.
In his 1986 Ph. D. thesis, Salcudean Sal86] uses the parametrization of achiev-
able closed-loop transfer matrices described in chapter 7 to formulate the controller
design problem as a constrained convex optimization problem. He describes many of
the closed-loop convex speci cations we encountered in chapters 8{10, and discusses
the importance of convexity. See also the article by Polak and Salcudean PS89].
The authors of this book and colleagues have developed a program called qdes,
which is described in the article BBB88]. The program accepts input written
in a control specication language that allows the user to describe a discrete-time
controller design problem in terms of many of the closed-loop convex speci cations
presented in this book. A simple method is used to approximate the controller
design problem as a nite-dimensional linear or quadratic programming problem,
380