system

System Characteristics

© Copyright 2008 Department of Chemical and Process Engineering University of Newcastle upon Tyne

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Part of a set of lecture notes on Process Dynamics by Ming T. Tham (2008)

INTRODUCTION

Transfer functions are ratios of polynomials in the Laplace operator 's'. Thus, the general

form of a transfer function )(sG that relates an output )(sY to an input )(sU can be written

as:

( )s

ns

n esAs

sBeasasasas

bsbsbsbsG

sUsY θ−θ−

−α−α

αα

−β−β

ββ ⋅=⋅

++++

++++==

)()()(

)()(

011

1

011

1 (1)

Note that it is assumed that the polynomials )(sA and )(sB do not have common factors. It is

more common, however, to find transfer functions expressed in more specific forms, so that

certain parameters of the system are highlighted. Typically, the )(sA and )(sB polynomials

are written as products of lower order terms, either as:

sn e

pspspsszszszs

sG θ−

α

β ⋅−−−

−−−=

)())(()())((

)(21

21

……

(2)

or

sn e

sasasassbsbsbK

sG θ−

α

β ⋅+++

+++=

)1()1)(1()1()1)(1(

)(21

21

……

(3)

Equations (2) and (3) are equivalent, each highlighting different characteristics of the system

being described by the transfer function. Which form of transfer representation we use

depends on the problem that we are trying to solve.

In the following sections, we will discuss the significance of the parameters that characterise

transfer functions. At the same time, you will be introduced to the terminology that is



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commonly used when discussing process dynamics and process control and automation

systems.

TRANSFER FUNCTION PARAMETERS

The parameters of a transfer function are the entities θand,, npz ii when using Eqn. (2), and

θand,,, nbaK ii when using Eqn. (3). If both equations describe the same system, then it is

obvious that the two sets of parameters will be related in some manner. The following

sections will describe the significance of these parameters.

Gain

Suppose a system is has the following ODE

τdy t

dty t Ku t( ) ( ) ( )+ = (4)

where the input and output, )(tu and )(ty respectively, are expressed as deviation variables.

At the steady state, when 0)(=

dttdy ,

)()( ∞=∞ Kuy (5)

Thus, at equilibrium, the final change in the output, )(∞y , is given by the final change in

input, )(∞u , multiplied by a constant, K . We are considering changes because Eqn. (4) is

expressed in terms of deviation variables, and the changes are deviations from the respective

initial values. The constant K is called the gain of the system, and is defined as:

inputin change finaloutputin change final

=K

At equilibrium, depending on the magnitude of K , the final change in the output, )(∞y , is

either an amplified or attenuated value of the final change in input, )(∞u . If K is greater



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than 1, then the input is amplified by the system, otherwise, attenuation occurs. The Laplace

transform of Eqn. (4) is:

τsY s Y s KU s( ) ( ) ( )+ = (6)

and the resulting the transfer function is:

sK

sUsYsG

τ+==

1)()()( (7)

The time domain solution to Eqn.(4), which is also the solution to Eqn. (6), depends on the

form of the input. As an example, assume that the input is a unit step change in )(tu , i.e.

1)( =tu

The Laplace Transform of )(tu is:

{ } { } ssULtuL /1)(1)( ===

Substituting into Eqn. (7), we have

Y s Ks s

Ks s

( )( )

( / )( / )

=+

⋅ = ⋅+1

1 11τ

ττ

(8)

From Laplace Transform tables, the inverse Laplace Transform of Eqn. (8) yields the time

domain solution of Eqn. (4) as:

y t K e t( ) /= − −1 τ or Y t K e Ytss( ) /= − +−1 τ (9)

Equation (9) shows that as time, t, tends to infinity, the exponential term will decay to zero

and hence )(ty will tend towards the gain of the process, K. This result can also be obtained

directly by applying the final value theorem to Eqn. (8), i.e.



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lim ( ) ( / )( / )s

sY s K ss s

K→

= ⋅+

=0

11

ττ

(10)

The following diagram shows the responses of several systems that have different gains to a

unit step change in input.

Figure 1. Step responses of several systems with different gains

Time Constant

Consider Eqn. (9) again. At time t = τ, the output has the value:

y t K e K( ) .= − =−1 0 6321 (11)

For a pure first-order system, the time constant, τ, is therefore the time taken for the process

to reach 63.2% of its final value. The system will reach this value in a shorter time, the

smaller the value of τ. Thus, the time constant governs the speed of response of a system, and

this is illustrated in Fig. 2, which shows the responses of 3 first order systems, each with a

different time constant, but with the same gains.



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Figure 2. Responses of first-order systems with different time constants

Time delay:

The time delay of the system is defined as the time taken, after the system has been

perturbed, before the system starts to react. It is a measure of the time inertia of the system.

The ODE of a first-order system, with gain K; time-constant τ; and time delay of magnitude θ

is given by:

)()()(θ−=+τ tKuty

dttdy (12)

The corresponding transfer function is:

sKe

sUsYsG

s

τ+==

θ−

1)()()(

That is, in the Laplace domain, a time delay of magnitude θ is expressed as se θ− . The

following figure shows the responses of 2 first-order systems; one without a time delay, and

one with a delay of 5 time units.



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Figure 3. Responses of first-order systems with and without a time delay

System order:

The order of the system, )(sG , is given by the order of the polynomial )(sA plus the integer

'n'. It is equivalent to the order of the differential equation that gives rise to the transfer

function after Laplace transformation.

Responses of higher order systems

A very common component of closed loop transfer functions is the following 2nd order

function:

Y sU s s s

n

n n

( )( )

=+ +

ωζω ω

2

2 22

ωn is called the natural frequency while ζ is called the damping factor. The step responses

of this system, for a fixed value of ωn = 1 and different values of damping factor, ζ. (0.5, 0.6,

0.7, 0.8, 0.9 and 1) are shown below. As expected, the smaller the damping factor, the more

oscillatory are the responses.



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DecreasingDamping

Responses of 2nd order systems with different damping factors

System type:

The system type is indicated by the integer 'n'. If 0=n , then )(sG is classified as a 'type-0'

system. If 1=n , )(sG is a 'type-1' system and so on. Type 1 systems and above are said to

have integrating properties, and have significant implications in process control systems.

Poles and Zeros:

Zeros are the roots of the numerator polynomial and are those values of 's' which sets the

transfer function to zero. Their locations on the s-plane determine the shape of the initial

response of the system.

Poles, on the other hand are the roots of the denominator polynomial and are those values of

's' which sets G(s) to infinity.



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Poles and zeros can either be real or complex. Complex poles or zeros always occur as a

complex conjugate pair. A system with complex poles will have an oscillatory response.

Characteristic Polynomial: The denominator of a transfer function is called the

characteristic polynomial and defines the characteristics of the system response. Setting it to

zero leads to the system characteristic equation and the poles of the transfer function are

solutions to this equation.

Stability

Consider the first order transfer function:

G s Ks

( ) =+1 τ

This has no zeros but has a pole that is determined by the solution of:

τ−=⇒=τ+ /101 ss (21)

From Eqn. (9), it has already been shown that the time-domain solution of )(sG to a unit step

change in input is:

[ ]τ−−= /1)( teKty

If τ is positive then the argument of the exponential term is negative and τ− /te defines a

decaying response. If τ is negative, the argument of the exponential term then becomes

positive and τ− /te will therefore yield an exponentially increasing response, i.e. )(ty will

increase without bounds. In other words, the system becomes unstable. System stability

therefore depends on the nature of the poles.

The larger the magnitude of the pole (the smaller the time constant), the faster is the time

response. In particular, the more negative the value of the pole, the faster the system reaches



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an equilibrium. Thus, the terminology 'fast' and 'slow' poles is sometimes used to simplify the

description of pole locations. When the value of the pole becomes more positive, the less

stable will be the time response. Systems with poles that have real parts greater than zero are

unstable.

Initial Response Characteristics

How a system responds initially is determined by its zeros, i.e. the roots of the numerator

polynomial of the transfer function. This can be illustrated by considering two systems with

the transfer functions:

G ss s11

1 2 1 3( )

( )( )=

+ + and G s s

s s21

1 2 1 3( )

( )( )=

++ +

Both systems, G1(s) and G2(s), have unit gains and the same poles, but G2(s) has a zero at

s = -1.

G1(s) and G2(s) can be decomposed to their first-order components by partial fraction

expansion to yield:

G ss s1

31 3

21 2

( )( ) ( )

=+

+−+

while )21(

1)31(

2)(2 sssG

+−

++

=

Note that the presence of the zero has altered the numerators of the first-order components.

These gain changes reveal that the faster component, i.e. that with the faster pole at s = -1/2,

becomes less suppressed. This effect is amplified since the influence of the slower

component, with a pole at s = -1/3, is also decreased. Thus, in this particular example, the

speed of response has been increased by the presence of a zero. The plots in Fig. 3 illustrate

the effects of different zeros on the response of the following transfer function:

G s ass s

( )( )( )

=+

+ +1

1 2 1 3



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with 'a' taking on values of 3, 2, 1, 0, -1 and -2 respectively.

Figure 3. Effects of zeros on system response

When the value of 'a' is negative, the time responses are initially in a different direction to

where they eventually settle. Such a response characteristics is called an inverse response

and occur when a zero has positive real parts. Systems that exhibit inverse response

properties are called non-minimum phase systems. Notice that as 'a' becomes more positive,

the responses become faster. In other words, the more negative the zero, the faster the initial

response. Notice also that when the value of 'a' is 2 or 3, first order responses result due to the

cancellation of the corresponding pole.

Systems that Oscillate

Systems oscillate because of restoring forces. These restoring forces always act in a direction

to restore the system to its equilibrium position. Typical examples from Physics are the



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pendulum and spring-damper systems. In a process environment, oscillating systems are also

common, and usually occur when feedback control is being applied. Here, the control

systems attempt to either drive a process variable of interest to a new desired value or to

maintain it at a fixed value, when the process is subject to external influences (disturbances).

The simplest system that can show oscillatory behaviour is a 2nd order system - first-order

systems do not oscillate. The general transfer function of a 2nd order system is:

22

2

2)(

nn

n

ssG

ω+ςω+ω

=

system

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