stability of transfer functions in the Laplace domain

stability of transfer functions in the Laplace domain

The following describes SISO (single input single output) system descriptions.
More

complex systems

require a more sophisticated

analysis

.

For a general

transfer function

,

H

⁢

(

s

)

, in Laplace domain, we have

H

⁢

(

s

)

=

∏

i

=

0

Z

(

s

-

z

i

)

∏

i

=

0

P

(

s

-

p

i

)

(1)

The

conclusions

below can all be derived and understood by expansion of

H

⁢

(

s

)

in
terms of

partial fractions

, and then doing a inverse Laplace
transform fraction by fraction.

z

i

denotes the zeros and

p

i

denotes the poles of the

linear time invariant system

(LTI). Stability of the system

H

⁢

(

s

)

is
characterized by the location of the poles in the

complex

s-plane.
There are many definitions
of stability in the

control system

literature, the most common one
used (for transfer functions) is the bounded-input-bounded-output
stability (BIBO), which states that for a BIBO stable system, for any
bounded input, or finite amplitude input, the output of the system
will also be bounded.

For example, a typical

second order

system such as the
mass-spring-dashpot system has the following transfer function,

H

⁢

(

p

)

=

ω

2

p

2

+

2

⁢

ζ

⁢

ω

⁢

p

+

ω

2

(2)

with a pair of complex poles at

p

=

-

ζ

⁢

ω

±

ω

⁢

ζ

2

-

1

. In common control system literature,

ζ

is
usually denoted as the

damping ratio

and

ω

is denoted as
the

natural frequency

of the system. In the case of the
mass-spring-dashpot system, we can tell that the oscillation of the
mass attached to a spring should be a

function

of the weight of the
mass and the stiffness of the spring, hence in the literature we have

ω

=

K

M

, where

K

is the spring

constant

and

M

is the weight of the mass. From the same logic, the amount time
for the mass to stop oscillating should be a function of the spring
stiffness and the

characteristics

of the dashpot, so we have

ζ

=

D

⁢

ω

2

⁢

K

, where

D

is the dashpot constant.

To determine is the system

H

⁢

(

p

)

BIBO stable, the simplest

solution

is to scrutinize the time domain solution of

2

via
inverse Laplace transform, and such discussion can easily be found in
common control system related text books and online lecture notes.
However, such approach tells nothing about the

physical nature

of the dynamic system, so here I will try to establish relationship
between the location of the poles and the stability of the system.
Here I first state that the system in

2

,
specifically the mass-spring-dashpot system, is stable,
since it is physically impossible for the system to produce an output

increasing

in amplitude forever as time goes to

infinity

.

First let’s consider an ideal mass-spring system. If we push the mass
once (impulse response), the mass will oscillate
forever with the same amplitude and frequency, since there are no
dashpot to dampen the motion. In this case,

D

=

0

→

ζ

=

0

, which the pair of complex poles of the system will be located on
the

imaginary axis

of the complex s-plane, and the stronger the spring
(

K

is large) the further away the poles from the origin. So the

imaginary part

of the poles

i

⁢

m

⁢

g

⁢

(

p

i

)

dominates the oscillatory nature of the
system. Now let’s assume that we have a relatively weak spring compare
to the dashpot that we are going to add to the system, so the

real
part

of the poles

r

⁢

e

⁢

a

⁢

l

⁢

(

p

i

)

will be dominating. If we push the mass
the same way as before, now we can expect the mass to oscillate for
sometime then come to a rest, depending on the

strength

of the
dashpot. The stronger the dashpot, the further away the poles from
the origin along the

real axis

on the s-plane. We have almost covered
the whole left-hand side of the s-plane, and so far we have the following
observations.

•

The imaginary part of complex poles correspond to oscillatory energy storage
mechanisms. Note if the original

differential equation

has real

coefficients

,
the complex poles are always in

complex conjugate

pairs . As a result
these can always be represented by sine and cos terms in the time domain.

•

The real parts of poles govern the decay rate of responses to
stimulations; with negative values corresponding to decaying terms, and positive
parts to growth terms (which are not stable).

•

The ratio between the real and imaginary parts of the poles govern the shape of the
response. In particular ratios less than one hide the visible part of the oscillatory terms in the
time domain.

•

Poles that have neither real or imaginary parts, i.e. at the origin,
correspond to successive integrations in the time domain. One integration for
each pole. Since these have an

unbounded

output for a step input, systems like these are
not BIBO; but are common.

For a system consisted of poles with imaginary parts only,

p

i

=

±

j

⁢

ω

, it is usually referred to as a

marginally stable
system. Notice that the real part of the poles for system

2

are always negative, since both

ζ

and

ω

are always positive. If a stable system has negative poles
(real or complex),

unstable

system must has positives poles, in the
right-hand side of the s-plane. This can easily be visualize if we
place a amplifier (with

-

ζ

) instead of a dashpot in the system.

Title

stability of transfer functions in the Laplace domain

Canonical name

StabilityOfTransferFunctionsInTheLaplaceDomain

Date of creation

2013-03-22 14:22:41

Last modified on

2013-03-22 14:22:41

Owner

rrogers (21140)

Last modified by

rrogers (21140)

Numerical id

11

Author

rrogers (21140)

Entry type

Definition

Classification

msc 93A10

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