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MESB374System Modeling and Analysis 

System Stability and Steady StateResponse

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• Stability Concept

Describes the ability of a system to stay at its equilibrium position in the

absence of any inputs. 

Stability

Ex: Pendulum

where the derivatives of all states are zeros

invertedpendulum

simplependulum

hill plateau valley

 –  A linear time invariant (LTI) system is stable if and only if (iff) itsfree response converges to zero for all ICs.

Ball on curved surface

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Examples (stable and unstable 1st ordersystems)Q: free response of a 1 st order system. 

05 (0) y y u t y y

1

5( )0

t 

 y t  y e

Q: free response of a 1 st order system. 

1

5 1G s TF:

Pole: 0.2 p

t 

y 

t 

y 

05 (0) y y u t y y

1

5( )0

t 

 y t  y e

1

5 1G s TF:

Pole: 0.2 p

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• Stability Criterion for LTI Systems

Stability of LTI Systems

( ) ( 1) ( ) ( 1)

1 1 0 1 1 0

1

1 1 0

Characteristic Polynomial

Stable All poles lie in the left-half complex plane (LHP)

All roots of ( ) 0 l

n n m m

n m m

n n

n

 y a y a y a y b u b u b u b u

 D s s a s a s a

ie in the LHP

Complex (s-plane)

Re

Im

Marginallystable/``unstable’’ 

RelativeStability(gain/phase margin)

AbsolutelyStable

Unstable

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• Comments on LTI Stability

 –  Stability of an LTI system does not depend on the input (why?)

 –  For 1st and 2nd order systems, stability is guaranteed if all the

coefficients of the characteristic polynomial are positive (of same

sign).

 –  Effect of Poles and Zeros on Stability

• Stability of a system depends on its poles only.• Zeros do not affect system stability. 

• Zeros affect the specific dynamic response of the system.

Stability of LTI Systems

0 02

1 0 1 2

( ) : Stable 0( ) : Stable 0 and 0

 D s s a a D s s a s a a a

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• Passive systems are usually stable

 –  Any initial energy in the system is usually dissipated in real-worldsystems (poles in LHP);

 –  If there is no dissipation mechanisms, then there will be poles on

the imaginary axis

 –  If any coefficients of the denominator polynomial of the TF are

zero, there will be poles with zero RP

System Stability (some empirical guidelines)

• Active systems can be unstable

 –  Any initial energy in the system can be amplified by internal source

of energy (feedback)

 –  If all the coefficients of the denominator polynomial are NOT thesame sign, system is unstable

 –  Even if all the coefficients of the denominator polynomial are the

same sign, instability can occur (Routh’s stability criterion for 

continuous-time system)

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In Class Exercises(1) Obtain TF of the following system:

(2) Plot the poles and zeros of the systemon the complex plane.

(3) Determine the system’s stability. 

L

2 5 y y y u u  y y y u u u 6 3 4

22 5s Y s sY s Y s sU s U s

2

1

2 5

Y s sG s

U s s s

Poles: 

Zero: 

22 5 0s s 1,2

2 4 201 2

2 p j

1 0s 1 z

Real

Img.

11 2 p j

2 1 2 p j

1 z

Stable 

(1) Obtain TF of the following system:

(2) Plot the poles and zeros of the systemon the complex plane.

(3) Determine the system’s stability. 

TF: 

2

3 2

3 4

6

Y s s sG s

U s s s s

Poles: 

Zeros: 

3 26 0s s s 1 2,3

1 230,

2

 j p p

2 3 4 0s s 1,2

3 7

2

  j z

Real

Img.

3

1 23

2 2 p j Marginally

Stable 

2

1 23

2 2 p j

1 0 p

1

3 7

2 2

 z j

2

3 7

2 2 z j

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Example

Inverted Pendulum(1) Derive a mathematical model for

a pendulum.

(2) Find the equilibrium positions.

(3) Discuss the stability of theequilibrium positions.

 

 B 

mg

l

EOM:  sino I mgl B  

is very small 

Equilibrium

position: 0

0

0

  

 

Assumption:   

Linearized EOM: 

0o

o

K 

 I mgl B

 I B mgl

 

 

Characteristic

equation: 2

0o

s I sB K  

Poles: 2

1,2

4

2

o

o

 B B KI  p

 I 

Real

Img.

Unstable 

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Transient and Steady State ResponseEx:

5u t t 

5 10 y y u

to a ramp input:

Let’s find the total response of a stable first order system: 

with I.C.: 0 2 y

- total response

20

Transfer FunctionFree Response

Forced Response

10 5 12

5 5 y

U s

Y ss s s

- PFE 31 2

2

2

5 5

aa aY s

s s s s

22 forced

0( ) 10

sa s Y s

0

2 12

1 forced 22 1 00

1 50 50( ) 2

2 1 ! 5 5s

ss

d d a s Y s

ds ds s s

3 forced5

5 ( ) 2s

a s Y s

5 5

3

Transient responseSteady state response Transient responsefree responsefrom Forced response from Forced response

2 10 2t t  y t t a e e

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Transient and Steady State Response

to a input u(t) can be decomposed into two parts

In general, the total response of a STABLE LTI system

Transient Response Steady State Response

T SS y t y t y t 

( ) ( 1) ( ) ( 1)

1 1 0 1 1 0

n n m m

n m m y a y a y a y b u b u b u b u

where

• Transient Response

 –  contains the free response of the system plus a portion of forced response

 –  will decay to zero at a rate that is determined by the characteristic roots (poles)

of the system

• Steady State Response

 –  will take the same (similar) form as the forcing input

 –  Specifically, for a sinusoidal input, the steady response will be a sinusoidal

signal with the same frequency as the input but with different magnitude and

phase.

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Transient and Steady State ResponseEx:

5sin 3u t t 

4 3 6 y y y u

to a sinusoidal input:

Let’s find the total response of a stable second order system: 

with I.C.: 0 0, 0 2 y y

- total response

2 2 2 2

Forced Response Free Response

6 5 3 2 4 2

4 3 3 4 3

sY s

s s s s s

- PFE

31 2 4 1 2

3 1 3 3 3 1

aa a a b bY s

s s s j s j s s

2

9

2a 1

5

2a

3

11

2a j

3 3

3 1 1 2 2

Steady state response Transient response

1 3

2Re

7 155sin 3 tan 2

2 2

  jt t t 

t t 

 y t a e a b e a b e

t e e

4

11

2a j

2 3b 1 1b

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Steady State Response

 f f t sF st s

( ) lim ( ) lim ( ) 0

4 12 4 3 y y y u u

2

4 3 5

4 12

sY s

s s s

• Final Value Theorem (FVT)

Given a signal’s LT F(s), if all of the poles of sF(s) lie in the LHP, then f(t) converges to a constant value as given in the following form

 Ex.

(1). If a constant input u=5 is applied to the sysetm at time t=0, determine

whether the output y(t) will converge to a constant value?

(2). If the output converges, what will be its steady state value?

We did not consider the effects of IC since •it is a stable system •we are only interested in steady state response 

A linear system is described by the following equation:

0

5( ) lim ( ) lim ( )

4t s y y t sY s

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Steady State ResponseGiven a general n-th order stable system

( ) ( 1) ( ) ( 1)

1 1 0 1 1 0

n n m m

n m m y a y a y a y b u b u b u b u

1

1 1 0

1

1 1 0

m m

m m

n n

n

b s b s b s bG s

s a s a s a

1

1 1 0

( )( )

Free n n

n

F sY s

s a s a s a

Free Response 

Transfer Function 

Steady State Value of Free Response (FVT) 

10 01 1 0

00

( )lim ( ) lim

0 (0)lim 0

SS Free n ns sn

s

sF s y sY ss a s a s a

F 

a

In SS value of a stable LTI system, there is NO contribution from ICs.