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

CHAPTER 8 PERFORMANCE SPECIFICATIONS

The c components of c are the regulated variables that the commands c are

n

z

w

intended to control or regulate|the \output" in classical control terminology. The

a components of a are the actuator signals, which we remind the reader must be

n

z

included in in any sensible formulation of the controller design problem. The o

z

n

components of o are other critical signals such as sensor signals or state variables.

z

The vector signal etc contains all remaining regulated variables.

z

We conformally partition the closed-loop transfer matrix :

H

2

c 3 2 cc

cd

3

2

3

z

H

H

?

c

w

6

a 7 6 ac

ad

7

z

= H

H

?

6

7

6

7

4

d 5

w

:

4

o 5 4 oc

od

5

z

H

H

?

etc

etc

w

z

?

?

?

The symbol is used to denote a submatrix of that is not used to formulate

?

H

performance speci cations (some of these submatrices will be used in chapter 10).

In the next few sections we consider speci cations on each of the other submatrices

of . We remind the reader that a convex or a ne speci cation or functional on a

H

submatrix of corresponds to a convex or a ne speci cation or functional on the

H

entire matrix (see section 6.2.3).

H

8.1

Input/Output Specifications

In this section we consider speci cations on cc, the closed-loop transfer matrix

H

from the command or set-point inputs to the commanded variables, i.e. the variables

the commands are intended to control. In classical control terminology, cc consists

H

of the closed-loop I/O transfer functions. Of course, it is possible for a control

system to not have any command inputs or commanded variables ( c = 0), e.g. the

n

classical regulator.

The submatrix cc determines the response of the commanded variables c to

H

z

the command inputs c only c will in general also be a ected by the disturbances

w

z

( d) and the other exogenous input signals ( etc) (these e ects are considered in the

w

w

next section). Thus, the signal cc c is the noise-free response of the commanded

H

w

variables. In this section, we will assume for convenience that d = 0, etc = 0,

w

w

so that c = cc c. In other words, throughout this section c will denote the

z

H

w

z

noise-free response of the commanded variables.

8.1.1

Step Response Specifications

Speci cations are sometimes expressed in terms of the step response of cc, espe-

H

cially when there is only one command signal and one commanded variable ( c = 1).

n

The step response gives a good indication of the response of the controlled variable

to command inputs that are constant for long periods of time and occasionally

change quickly to a new value (sometimes called the set-point). We rst consider

the case c = 1. Let ( ) denote the step response of the transfer function cc.

n

s

t

H

8.1 INPUT/OUTPUT SPECIFICATIONS

173

Asymptotic Tracking

A common speci cation on cc is

H

lim

s

( ) = cc(0) = 1

s

t

H

!1

which means that for c constant (and as mentioned above, d = 0, etc = 0),

w

w

w

c( ) converges to c as

, or equivalently, the closed-loop transfer function

z

t

w

t

!

1

from the command to the commanded variable is one at = 0. We showed in

s

section 6.4.1 that the speci cation

asympt trk =

cc(0) = 1

H

fH

j

H

g

is ane, since the functional

( ) = cc(0)

H

H

is a ne.

A strengthened version of asymptotic tracking is asymptotic tracking of order

: cc(0) = 1, (j)

cc (0) = 0, 1

. This speci cation is commonly encountered

k

H

H

j

k

for = 1 and = 2, and referred to as \asymptotic tracking of ramps" or \zero

k

k

steady-state velocity error" (for = 1) and \zero steady-state acceleration error"

k

(for = 2). These higher order asymptotic tracking speci cations are also a ne.

k

Overshoot and Undershoot

We de ne two functionals of cc: the overshoot,

H

os( cc) = sup ( ) 1

H

t 0 s t ;

and the undershoot,

us( cc) = sup ( )

H

t 0 ;s t :

Figure 8.1 shows a typical step response and the values of these functionals.

These functionals are convex, so the speci cations

os =

os( cc)

H

fH

j

H

g

us =

us( cc)

H

fH

j

H

g

are convex: for example, if each of two step responses does not exceed 10% overshoot

( = 0 1) then neither does their average (see section 6.4.2 and gure 6.6).

:

174