Poles and ZerosC

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• The dynamic behavior of a transfer function model can be characterized by the numerical value of its poles and zeros.

• Two equivalent general representation of a TF:

1 20

1 2

0

mi

im mi

nn ni

ii

b sb s z s z s z

G sa s p s p s p

a s

where {zi} are the “zeros” and {pi} are the “poles”.

We will assume that there are no “pole-zero” cancellations. That is, that no pole has the same numerical value as a zero.

Note that, for system to be physically realizable, n>m.

Example: 4 poles (denominator is 4th order polynomial) & 0 zero (numerator is a const)

1 2

2 21 2 2

2

1 2 3 41 2 2

2 2

0 1 1 22 2

( )( 1)( 2 1)

110; ; ,

If , then

1 1sin cos

tt

KG s

s s s s

s s s s j

Y s G s U s

u t MS t

y t A At Be e C t C t

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Effects of Poles on System Response

1 2

2 1

1(1) decays slower than

(2) RHP pole unstable system

(3) complex conjugate poles oscillation

(4) origin integrating element

tt

e e

Example of Integrating Element

0qqdt

dhA i

0

0

( ) ( ) ( )

if ( ) 0

( ) 1then

( )

i

i

AsH s Q s Q s

Q s

H s

Q s As

pure integrator (ramp) for step change in qi

Cause of Zeros – Input Dynamics

1

1

1 0

1

[Example 1]

1

1

1[Example 2]

1

1

a

a

t

a

a

a

y y K u u

K sG s

s

y y K u d u

K sG s

s s

Some Facts about Zeros

• Zeros do not affects the number and locations of the poles, unless there is an exact cancellation of a pole by a zero.

• The zeros exert a profound effect on the coefficients of the response modes.

Example of 2nd-Order Overdamped System with One (1) Zero

1 2

1 2

1 2

1 2

1 2 2 1

1

1

1

1 1

where

1

and

Case (a): overshoot

Case (b): 0 similar to 1st-order step response

Case (c): 0 inverse

a

t t

a a

a

a

a

K s MY s G s U s

s s s

y t KM e e

y KM

response

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Step Response of 2nd-Order Overdamped System without Zeros

1 2

1 2

1 2

1 2

1 2

/ /1 1 2

1 2

11 1

1

=2

1t t

Mu t MS t U s

sK

G ss s

e ey t G s U s KM

L

Further Analysis of Inverse Response

1 2

1 2

1 2

0 1 2

1 2

1

1 1

where 0, 0, 0

1

1 1

By initial value theorem

1lim

1 1

0

a

a

a

a

st

a

K s MY s G s U s

s s s

KM

KM sdysY s

dt s s

KM sdy dys s

dt dt s s

KM

L

L

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Common Properties of Overshoot and Inverse Responses

1 2

1 2

Overshoot or inverse response can be expected

whenever there are two physical effects that act

on the same output in opposite ways and with

different time scales, i.e.

(1) sgn sgn

(2)

K K

Another Example

1

1

1

1

1

1

1

1

1

1 1

0 and

Case (a): 0 decr

(jump

easing

Case (b): 0 increasing

Case (c): 0 increasing

)

a

t

a

a

a

a

a

K s MY s G s U s

s s

y t KM e

y KM y KM

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Time DelaysTime delays occur due to:

1. Fluid flow in a pipe

2. Transport of solid material (e.g., conveyor belt)

3. Chemical analysis

- Sampling line delay

- Time required to do the analysis (e.g., on-line gas chromatograph)

Mathematical description:

A time delay, , between an input u and an output y results in the following expression:

θ

0 for θ

θ for θst Y s

y t eu t t U s

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Implication of Time Delay

The presence of time delay in a process means that we cannot factor the transfer function in terms of simple poles and zeros!

Polynomial Approximation of Time Delays

2 2 3 3

2 2

2 2

Talor series expansion:

12! 3!

1/1 Pade approximation:

1 / 2

1 / 22/2 Pade approximation:

1 / 2 /12

1 / 2 /12

s

s

s

s se s

se

s

s se

s s

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Approximation of nth-Order Systems

11

0

A n-th order system with n equal time constants:

1

/, 1

!

,

lim

n n

int n

ni

sn

n

KG s

sn

ntMy n t G s KM e

s i

y t KMS t

G s e

L

Approximation of Higher-Order Transfer Functions

0θ0(A) 1 θ (zero)se s

In this section, we present a general approach for approximating high-order transfer function models with lower-order models that have similar dynamic and steady-state characteristics.

Previously we showed that the transfer function for a time delay can be expressed as a Taylor series expansion. For small values of s,C

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An alternative first-order approximation is

0

0

θθ

0

1 1(B) (pole)

1 θs

se

se

Skogestad’s “Half Rule”

1. Largest neglected time constant

• One half of its value is added to the existing time delay (if any) .

• The other half is added to the smallest retained time constant.

2. Time constants that are smaller than those in item 1.

• Use (B)

3. RHP zeros.

• Use (A)

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Example 6.4

Consider a transfer function:

0.1 1

5 1 3 1 0.5 1

K sG s

s s s

Derive an approximate first-order-plus-time-delay (FOPDT) model,

θ

τ 1

sKeG s

s

using two methods:

(a) The Taylor series expansions (A) and (B).

(b) Skogestad’s half rule

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Compare the normalized responses of G(s) and the approximate models for a unit step input.

Solution

(a) The dominant time constant (5) is retained. Applying the approximations in (A) and (B) gives:

0.10.1 1 ss e

and

3 0.51 1

3 1 0.5 1s se e

s s

Substitution into G(s) gives the Taylor series approximation,

0.1 3 0.5 3.6

5 1 5 1

s s s s

TSKe e e Ke

G ss s

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(b) To use Skogestad’s method, we note that the largest neglected time constant in G(s) has a value of three.

θ 1.5 0.1 0.5 2.1 Ch

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• According to “half rule” (Rule 1), half of this value is added to the next largest time constant to generate a new time constant

• Rule 1: The other half provides a new time delay of 0.5(3) =

1.5. • The approximation of the RHP zero in Rule 3 provides an

additional time delay of 0.1. • Approximating the smallest time constant of 0.5 in G(s) by

Rule 2 produces an additional time delay of 0.5. • Thus the total time delay is, • Therefore

τ 5 0.5(3) 6.5.

2.1

6.5 1

s

SkKe

G ss

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Example

1

21 2

1

12

(FOPDT)

(SO

1 3 1 0.2 1 0.05 1

(a) 1

(b) PDT) 1 1

s

s

s

K s eG s

s s s s

KeG s

s

KeG s

s s

Part (a)

3.75

1

31 0.2 0.05 1 3.75

23

12 13.52

13.5 1

sKeG s

s

Part (b)

1

2

2.15

2

0.21 0.05 1 2.15

212

0.23 3.1

2

12 1 3.1 1

sKeG s

s s