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Part 1: SYSTEM RESPONSE (mainly revisions)

Summary of first order systems:

Specified by two parameters – Gain (A) and A

Time-constant (T).

Output reaches steady state after input, hence name ‘LAG’

1 sT+

A first order system has a gain of 10 and a time-constant of 1 msec,

What is its transfer function?

Part 1: SYSTEM RESPONSE - Example

Example: Describe the system, if the transfer function of a system is given as

204s +

Hence: A first order systemGain is 5Time-constant is 0.25 s

20 54 1 0.25s s=

+ +

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Part 1: SYSTEM RESPONSE

From Characteristic Equation to POLES

1 tTe−Loosely, the term ‘pole’ indicates that the

‘value’ of the transfer function is infiniteeT value of the transfer function is infinite.

The roots (or eigenvalues) of the CE locate the poles of the system.

We can represent the position of the pole(s) on the s-plane by means of a cross.

α+sk

Im (jω)

Our example (of the first order lag system) has one real pole, at s = - α

Re (σ)

Part 1: SYSTEM RESPONSE

Second Order Systems

In the previous examples ( )( )10110

++ ss( )( )

Characteristic Equation ( )( ) 0101 =++ ss

010112 =++ ss

Two real polesTwo real poles – two decaying exponential terms in transient response

More generally, roots of quadratic (of CE) are COMPLEX

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Part 1: SYSTEM RESPONSE

Second Order Systems - Example

525

2 ++ ssCharacteristic Equation

0522 =++ ss

A pair of conjugate poles

;211 js +−=

212 js −−=

212,1 js ±−=

- 1

Part 1: SYSTEM RESPONSE

Second Order Systems - Example

212,1 js ±−=

Amplitude decayingat e-t

- 1 Frequency of oscillation = 2 rad/s

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Part 1: SYSTEM RESPONSE

Second Order Systems – Standard Form

2

22

2

2 nn

n

ss ωζωω

++

ωn -- Natural Frequencyn y

ζ -- Damping ratio, or damping factor

ONLY for complex poles

Part 2: SYSTEM RESPONSE

Second Order Systems – Standard Form

02 22 =++ nnss ωζωPoles are roots of CE

122,1 −⋅±⋅−= ζωωζ nns

Four important casesζ = 0 undampedζ = 0 – 1, under-dampedζ = 1, critical dampingζ > 1, over-damped

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Part 2: SYSTEM RESPONSE

Pole locations

Pole far from the axisPole far from the axis Shorter time-constant Faster response.

Pole in Left Half-plane Stable System

Pole in Right Half-plane Unstable S stemUnstable System

Pole on axisMarginally Stable

Part 2: SYSTEM RESPONSE

Cascade systems and dominant pole

1 101

1+s 10+s

S stem C E

( )( )10110

++ ss

System C.E.(s+1)(s+10)=0

System poles at s = -1; and s = -10

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Revision: complex numbers or vectors

Im

bjba +=Y

Rea

bθjMe=Y

θ∠==→

MYY

22 baM +=

⎟⎠⎞

⎜⎝⎛= −

ab1tanθ

)cos(θMa =

)sin(θMb =

Revision: basic operations of complex numbers

)sin()cos( θθθ je j +=M=1: )()( j

)sin()cos( θθθ je j −=−

2)cos(

θθ

θjj ee −+

=

jee jj

2)sin(

θθ

θ−−

=

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Revision: basic operations of complex numbers

( ) ( ) ( ) ( )dbjcajdcjba +++=+++( ) ( ) ( ) ( )jjj

( ) ( ) ( ) ( )dbjcajdcjba −+−=+−+

( ) ( ) bdjbcjadacjdcjba −++=+×+

( ) ( )bcadjbdac ++−=

jdcjdc

jdcjba

jdcjba

−−

×++

=++ ( ) ( )

22 dcadbcjbdac

+−++

=

Revision: Basic operations of complex numbers

( ) 11

θjeMjba =+

( ) ( ) ( ) ( )2121

θθ jj eMeMjdcjba ⋅=+×+

( ) ( ) mjm

j eMeMM θθθ =⋅= + 2121

( ) 22

θjeMjdc =+

2

1

2

1θ

θ

j

j

eMeM

jdcjba

=++

( ) njn

j eMeMM θθθ =⎟⎟

⎠

⎞⎜⎜⎝

⎛= − 21

2

1

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Revision: Basic operations of complex numbers

( ) ( )tjttj eee βαβα −−+− ⋅= ( )eee

( ))sin()cos( tjte t ββα −⋅= −

( ) ( )tjttj eee βαβα +−−− ⋅= ( )eee

( ))sin()cos( tjte t ββα +⋅= −

tjtj BeAe )21()21( +−−− +