08.Performance of control systems

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8.4

Disturbance rejection

337

338

8 Performance of control systems

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The idea here is very simple. We wish to allow a disturbance at any node which may

ˆ

to yˆ is

the transfer function from d1

enter the system at any other node. Just where the disturbance enters a system and where

yˆ(s)

RC (s)RP (s)

it is measured can drastically change the behaviour of the system. Typically, however, one

T1(s) =

=

,

ˆ

1 + R

(s)R

(s)

measures the disturbance at the output of the interconnected system. That is, typically

d1

(s)

C

P

in the above discussion j is taken to be m. Before getting to an illustration by example,

ˆ

to yˆ is

let us provide a result which gives some obvious consequences of the notion of disturbance

the transfer function from d2

type. The following result can be proved along the same lines as Proposition 8.11 and by

yˆ(s)

RP (s)

application of the Final Value Theorem.

T2(s) =

=

,

ˆ

1 + R

(s)R

(s)R

(s)

d2(s)

C1

C2

P

8.19 Proposition Let (S, G) be an IBIBO stable interconnected SISO linear system with input

ˆ

to yˆ is

in node 1 and output in node m. For j {2, . . . , m} the following statements are equivalent:

and the transfer function from d3

yˆ(s)

1

(i)

(S, G) is of j-disturbance type k with input at node i;

T3(s) =

=

.

ˆ

(s)

1 + R

C

(s)R

P

(s)

(ii)

limt→∞ y(t) exists and is nonzero for some j-disturbed output (d(t), y(t)) with input at

d3

node i, where d(t) is of type k;

Let’s attach some concrete transfer functions at this point. Our transfer functions are

(iii)

lims→0

1

Tji(s) exists and is nonzero.

pretty artificial, but serve to illustrate the point. We take

sk

The preceding three equivalent statements imply the following:

1

s

(iv)

lim

y(t) = 0 for every j-disturbed output (d(t), y(t)) with d(t) of type ` with ` < k.

RC (s) =

,

RP (s) =

.

t→∞

s

s + 2

Now let us consider a simple example.

Check

We compute the corresponding transfer functions to be

8.20 Example Let us consider the interconnected SISO linear system whose signal flow graph

s2

s2

s

s + 2

is shown in Figure 8.17. The system has three places where disturbances ought to naturally

TS,G(s) =

,

T1(s) =

,

T2(s) =

,

T3(s) =

.

s2 + s + 2

s2 + s + 2

s2 + s + 2

s2 + s + 2

r s

1

/•

RC (s)

/•

RP (s)

/•

1

/y s

A straightforward application of Proposition 8.11(iii) indicates that TS,G is of type 0. It is

ˆ(

)

f

ˆ(

)

also easy to see from Proposition 8.19(iii) that the system is of disturbance type 2 with

respect to the disturbance d1, of disturbance type 1 with respect to the disturbance d2, and

−1

of disturbance type 0 with respect to d3.

Figure 8.17

A system ready to be disturbed

As we see in this example, the ability of a system to reject disturbances may di er

from the ability of a system to represent a reference signal. Furthermore, for disturbance

be considered, as shown in Figure 8.18. Note that one should strictly add another node to

rejection, one typically wants the transfer function from the disturbance to the output to

have the property that the expected disturbance type gives zero steady-state error. Thus,

ˆ

(s)

ˆ

ˆ

(s)

if step disturbances are what is expected, it is su cient that the system be of disturbance

d1

d3(s)

d3

type 0.

1

1

1

Let us look at our falling mass example thinking of the gravitational force as a distur-

² RC (s)

²

RP

(s)

²

1

bance.

/

/

/

•

h

•

•

yˆ(s)

8.21 Example (Example 6.60 cont’d) The block diagram for the falling mass with the grav-

−1

Figure 8.18 Disturbances

itational force is shown in Figure 8.19. We think of the input −mg as a disturbance input

and so write d(t) = −mg1(t). This input is then a step input if we think of holding the mass

then letting it go. In fact, in our analysis, let’s agree to take y(0) = 0 and y˙(0) = 0 and see

the output node yˆ, but this does not change anything, so it can be safely omitted. This will

if the controller can return the mass to y = 0 after we let it fall under gravity. The transfer

function from the disturbance to the output is

generally be the case for an interconnected SISO linear system. In any case, the transfer

function for the signal flow graph is

yˆ(s)

ms2

Td(s) =

=

.

ˆ

ms2 + R

C

(s)

yˆ(s)

RC (s)RP (s)

d(s)

TS,G(s) =

=

,

rˆ(s)

1 + RC (s)(s)RP (s)

Let us look at the form of this transfer function for the various controllers we employed in

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8.5 The sensitivity function

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346

8 Performance of control systems

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8.6.1

Natural frequency-domain specifications There are a significant number of

formance are often the most useful:

performance specifications that are very easily presented in the frequency-domain. In this

section we list some of these. In a given design problem, one may wish to invoke only some

∞

1/2

of these constraints and not others. Also, the exact way in which the specifications are best

kek2 =

Z0

e

2

(t) dt

presented can vary in detail, depending on the particular application. However, it can be

kek∞ = sup{|e(t)| ≤ α for almost every t}.

hoped that what we say here is a helpful guide in getting started in a given problem.

α≥0

Before we begin, we remark that all of the specifications we give will be are what Helton

The first is of course the L2-norm of the error, and the second the L∞-norm of the error.

and Merrino [1998] call “disk constraints.” This means that the closed-loop transfer function

TL will be specified to satisfy a constraint of the form

The error measure

kek1 = Z0 ∞ |e(t)| dt

|M(iω) − TL(iω)| ≤ R(iω), ω [ω1, ω2]

is also used. For example, you will recall from Section 13.1 that the criterion used by Ziegler

R+ defined?

for functions M : i[ω1, ω2] → C M : i[ω1, ω2] → R+, and for ω1 < ω2 > 0.

Thus at each

and Nicols in their PID tuning was the minimisation of the L

1

-norm. In practice, one will

frequency ω in the interval [ω1, ω2], the value of TL at iω must lie in the disk of radius R(iω)

sometimes wish to consider alternative performance measures. For example, the following

with centre M(iω).

two, related to the L2 and L1-norms, will serve to penalise errors which occur for large times,

but deemphasise large initial errors:

Stability margin lower bound Recall from Proposition 7.15 that the point on the Nyquist

k

e

kt,2

=

∞

2

1/2

contour closest to −1 + i0 is a distance kSLk∞ away. Thus, to improve stability margins,

te (t) dt

one should provide a lower bound for this distance. This means specifying an upper bound

Z0

on the H

-norm of the sensitivity function. In terms of the transfer function this gives

kekt,1 = Z0

∞ t |e(t)| dt.

∞

|

1

−

TL(iω)

| ≤

ρsm,

ω > 0.

(8.3)

One can imagine choosing other weighting functions of time by which to multiply the error

It is possible that one may wish to enforce this constraint more or less at various frequencies.

to suit the particular nature of the problem with which one is dealing.

For example, if one knows that a system will be operating in a certain frequency range, then

One of the reasons for the usefulness of the L2 and L∞ norms is that they make it possible

stability margins in this frequency range will be more important that in others. Thus one

to formulate statements in terms of transfer functions. For example, we have the following

may relax (8.3) by introducing a weighting function W and asking that

result which was derived in the course of the proof of Theorem 5.22, if we recall that the

transfer function from the input to the error in Figure 8.28 is the sensitivity function.

|W (iω)(1 − TL(iω))| ≤ ρsm,