Linear Controller Design: Limits of Performance by Stephen Boyd and Craig Barratt - HTML preview

CHAPTER 13 ELEMENTS OF CONVEX ANALYSIS

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