166 6. FILTERING AND SENSITIVITY MINIMIZATION
discrete analog of a strictly proper plant, is one whose transfer matrix is zero at z = 0. As a result,
in the formulation of the discretetime ﬁltering problem, the weighting matrices W
1
and W
2
can be
any rational H
∞
matrices and need not be strictly causal. Moreover, an optimal controller always
exists and the approximation procedure of Lemma 6.3.2 is not needed.
6.4 SENSITIVITY MINIMIZATION: SCALAR CASE
Because the sensitivity minimization problem is much more involved technically than the ﬁltering
problem, the discussion of the former is divided into three parts, namely the scalar case, the fat plant
case, and the general case. Accordingly, the problem studied in this section is the following: Suppose
p is a given scalar plant, and w is a given element of S. The objective is to ﬁnd a c ∈ S(p) that
minimizes the weighted sensitivity measure
5
J = sup
ω∈R
w(jω)
1 + p(j ω)c(j ω)
. (6.4.1)
As in the previous section, one can transform this problem into an afﬁne minimization problem.
Let (n, d) be a coprime factorization of p, and let x,y ∈ S be such that xn + yd = 1. Then the
problem becomes: Minimize
J(r) = sup
ω∈R
[(y − rn)dw](j ω)
=(y − rn)dw
∞
, (6.4.2)
by suitable choice of r ∈ S. Moreover, it can be assumed without loss of generality that the weighting
function w is outer. To see this, factor w in the form w = w
i
w
o
where w
i
and w
o
are inner and
outer, respectively. Then, since multiplication by the inner function w
i
is normpreserving, we have
J(r) =(y − rn)dw
o
∞
.
In order to take full advantage of the available results on H
p
spaces, a bilinear transformation
is now used to map the set S into H
∞
. Suppose f ∈ S, and deﬁne the function
˜
f by
˜
f(z)= f [(1 + z)/(1 − z)] . (6.4.3)
Since the bilinear transformation s = (1 + z)(1 − z) maps the open unit disc into the open right
halfplane, it follows that
˜
f is a rational element of H
∞
, i.e., f ∈ R
∞
. Moreover,
˜
f
∞
= sup
θ∈[0,2π ]

˜
f(e
jθ
)
= sup
ω∈R
f(jω)=f
S
, (6.4.4)
since the unit circle maps onto the imaginary axis plus the point at inﬁnity. Conversely, suppose
˜g ∈ H
∞
is rational, and deﬁne g by
g(s) =˜g[(s − 1)/(s + 1)] . (6.4.5)
5
Note that the two weights w
1
and w
2
are combined into one, since we are dealing with the scalar case.
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