fH
j
H
w
2
Z
g
is a convex subset of .
H
6.3 CLOSED-LOOP CONVEX DESIGN SPECIFICATIONS
135
6.3
Closed-Loop Convex Design Specifications
Many design speci cations have the property that the set of closed-loop transfer
matrices that satisfy the design speci cation is convex. We call such design speci-
cations closed-loop convex:
A design specication is closed-loop convex if the set of closed-
De
nition
6.6:
D
loop transfer matrices that satisfy is convex.
D
One of the themes of this book is that many design specications are closed-loop
convex.
6.3.1
Open Versus Closed-Loop Formulation
We noted in section 3.1 that it is possible to formulate design speci cations in terms
of the (open-loop) controller transfer matrix , instead of the closed-loop transfer
K
matrix , as we have done. In such a formulation, a design speci cation is a
H
predicate on candidate controllers, and the feasibility problem is to nd a controller
that satis es a set of design speci cations. Such a formulation may seem more
K
natural than ours, since the speci cations refer directly to what we design|the
controller .
K
There is no logical di erence between these two formulations, since in sensible
problems there is a one-to-one correspondence between controllers and the closed-
loop transfer matrices that they achieve. The di erence appears when we consider
geometrical concepts such as convexity: a closed-loop convex speci cation will gen-
erally not correspond to a convex set of controllers. In chapters 13{16, we will see
that convexity of the speci cations is the key to computationally tractable solution
methods for the controller design problem. The same design problems, if expressed
in terms of the controller , have speci cations that are not convex, and hence do
K
not have this great computational advantage.
6.3.2
Norm-Bound Specifications
Many useful functionals are norms of an entry or submatrix of the transfer matrix
. A general form for a design speci cation is the norm-bound speci cation
H
xx
des
xx
(6.3)
kH
;
H
k
where
is some norm on transfer functions or transfer matrices (see chapter 5)
k
k
and xx is a submatrix or entry of (see section 6.2.3). We can interpret des
xx as
H
H
H
the desired transfer matrix, and the norm
used in (6.3) as our measure of the
k
k
deviation of xx from the desired transfer matrix des
xx . We often have des
xx = 0,
H
H
H
in which case (6.3) limits the size of xx.
H
We now show that the functional
( ) = xx
des
xx
H
kH
;
H
k
136