where s is the unit step response of the transfer function H. We will determine a
subgradient at H0. The unit step response of H0 will be denoted s0.
We will use the rule that involves a maximum of a family of convex functionals.
For each t 0, we de ne a functional step t as follows: step t(H) = s(t). The
functional step t evaluates the step response of its argument at the time t it is a
linear functional, since we can express it as
step t
Z
(H) = 1 1 ej!t
2
j! H(j!)d!:
;1
Note that we can express the overshoot functional as the maximum of the
a ne functionals step t 1:
;
(H) = sup step t(H) 1:
t 0
;
Now we apply our last rule. Let t0 0 denote any time such that the overshoot
is achieved, that is, (H0) = s0(t0) 1. There may be several instants at which
;
the overshoot is achieved t0 can be any of them. (We ignore the pathological case
where the overshoot is not achieved, but only approached as a limit, although it is
possible to determine a subgradient in this case as well.)
Using our last rule, we nd that any subgradient of the functional step t0 1
;
is a subgradient of at H0. But the functional step t0 1 is a ne its derivative
;
is just step t0. Hence we have determined that the linear functional step t0 is a
subgradient of at H0.
Let us verify the basic subgradient inequality (13.3). It is
(H)
(H0) + step t0(H H0):
;
Using linearity of step t0 and the fact that (H0) = s0(t0) 1 = step t0(H0) 1,
;
;
the subgradient inequality is
(H) s(t0) 1:
;
Of course, this is obvious: it states that for any transfer function, the overshoot is
at least as large as the value of the unit step response at the particular time t0,
minus 1.
A subgradient of other functionals involving the maximum of a time domain
quantity, e.g., maximum envelope violation (see section 8.1.1), can be computed in
a similar way.
13.4.3
Quasigradient for Settling Time
Suppose that is the settling-time functional, de ned in section 8.1.1:
(H) = inf T 0:95 s(t) 1:05 for t T
f
j
g
13.4 COMPUTING SUBGRADIENTS
303
We now determine a quasigradient for at the transfer function H0. Let T0 =
(H0), the settling time of H0. s0(T0) is either 0.95 or 1.05. Suppose rst that
s0(T0) = 1:05. We now observe that any transfer function with unit step response
at time T0 greater than or equal to 1.05, must have a settling time greater than or
equal to T0, in other words,
(H) T0 whenever s(T0) 1:05:
Using the step response evaluation functionals introduced above, we can express
this observation as
(H)
(H0) whenever step T0(H H0) 0:
;
But this shows that the nonzero linear functional step T0 is a quasigradient for
at H0.
In general we have the quasigradient qg for at H0, where
qg =
step T0 if s0(T0) = 1:05
step T0 if s0(T0) = 0:95
;
and T0 = (H0).
13.4.4
Maximum Magnitude of a Transfer Function
We rst consider the case of SISO H. Suppose that
(H) = H = sup H(j!)
k
k
j
j
1
!2R
provided H is stable (see section (5.2.6)). (We leave to the reader the modi cation
necessary if is a weighted
norm.) We will determine a subgradient of at
H
1
the stable transfer function H0 = 0.
6
For each !
, consider the functional that evaluates the magnitude of its
2
R
argument (a transfer function) at the frequency j!:
mag !(H) = H(j!) :
j
j
These functionals are convex, and we can express the maximum magnitude norm
as
(H) = sup mag !(H):
!2R
Thus we can use our maximum tool to nd a subgradient.
Suppose that !0
is any frequency such that H0(j!0) = (H0). (We
2
R
j
j
ignore the pathological case where the supremum is only approached as a limit. In
this case it is still possible to determine a subgradient.) Then any subgradient of
304