which is then solved. The simple organization and approximations made in qdes
make it practical only for small problems. The paper by Oakley and Barratt OB90]
describes the use of qdes to design a controller for a exible mechanical structure.
16.3.4
FIR Filter Design via Convex Optimization
A relevant parallel development took place in the area of digital signal processing.
In about 1969, several researchers observed that many nite impulse response (FIR)
lter design problems could be cast as linear programs see, for example, the arti-
cles CRR69, Rab72] or the books by Oppenheim and Schaefer OS70, 5.6] and
x
Rabiner and Gold RG75, ch.3]. In RG75, 3.39] we even nd designs subject to
x
both time and frequency domain speci cations:
Quite often one would like to impose simultaneous restrictions on both
the time and frequency response of the lter. For example, in the design
of lowpass lters, one would often like to limit the step response over-
shoot or ripple, at the same time maintaining some reasonable control
over the frequency response of the lter. Since the step response is a
linear function of the impulse response coe cients, a linear program is
capable of setting up constraints of the type discussed above.
A recent article on this topic is OKU88].
Like the early work on the closed-loop design method, convexity is not recognized
as the property of the FIR lter design problem that allows e cient solution. Nor is
it noted that the method actually computes the global optimum, i.e., if the method
fails to design an FIR lter that meets some set of convex speci cations, then the
speci cations cannot be achieved by any FIR lter (of that order).
16.4
Some Extensions
16.4.1
Discrete-Time Plants
Essentially all of the material in this book applies to single-rate discrete-time plants
and controllers, provided the obvious changes are made (e.g., rede ning stability to
mean no poles on or outside the unit disk). For a discrete-time development, there
is a natural choice of stable transfer matrices that can be used to form the (analog
of the) Ritz sequence (15.8) described in section 15.1:
Qijk(z) = Eijz (k 1)
1 i n
;
;
u 1 j ny k = 1 2 ...
(Eij is the matrix with a unit i j entry, and all other entries zero), which corresponds
to a delay of k 1 time steps, from the jth input of Q to its ith output. Thus in the
;
Ritz approximation, the entries of the transfer matrix Q are polynomials in z 1,
;
i.e., FIR lters. This approach is taken in the program qdes BBB88].
16.4 SOME EXTENSIONS
381
Many of the results can be extended to multi-rate plant and controllers, i.e.,
a plant in which di erent sensor signals are sampled at di erent rates, or di er-
ent actuator signals are updated at di erent rates. A parametrization of stabi-
lizing multi-rate controllers has recently been developed by Meyer Mey90] this
parametrization uses a transfer matrix Q(z) that ranges over all stable transfer
matrices that satisfy some additional convex constraints.
16.4.2
Nonlinear Plants
There are several heuristic methods for designing a nonlinear controller for a non-
linear plant, based on the design of an LTI controller for an LTI plant (or a family
of LTI controllers for a family of LTI plants) see the Notes and References for chap-
ter 2. In the Notes and References for chapter 10, we saw a method of designing a
nonlinear controller for a plant that has saturating actuators. These methods often
work well in practice, but do not qualify as extensions of the methods and ideas
described in this book, since they do not consider all possible closed-loop systems
that can be achieved. In a few cases, however, stronger results have been obtained.
In DL82], Desoer and Liu have shown that for stable nonlinear plants, there is
a parametrization of stabilizing controllers that is similar to the one described in
section 7.2.4, provided a technical condition on P holds (incremental stability).
For unstable nonlinear plants, however, only partial results have been obtained.
In DL83] and AD84], it is shown how a family of stabilizing controllers can be
obtained by rst nding one stabilizing controller, and then applying the results
of Desoer and Liu mentioned above. But even in the case of an LTI plant and
controller, this \two-step compensation" approach can fail to yield all controllers
that stabilize the plant. This approach is discussed further in the articles DL84a,
DL84b, DL85].
In a series of papers, Hammer has investigated an extension of the stable factor-
ization theory (see the Notes and References for chapter 7) to nonlinear systems
see Ham88] and the references therein. Stable factorizations of nonlinear systems
are also discussed in Verma Ver88].
382