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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Control Systems

Stability and Routh Hurtwitz criterion


Stability introduction

• Requirements for design of a control system

– Transient response

– Stability

– Steady state errors

• Stability – most important parameter for design

• Total response

𝑐 𝑡 = 𝑐𝑓𝑜𝑟𝑐𝑒𝑑 𝑡 + 𝑐𝑛𝑎𝑡𝑢𝑟𝑎𝑙(𝑡)


System Stability Definition

Types of stability based on Natural response

definition:

1. A system is STABLE if the natural response

approaches zero as time approaches infinity

2. A system is UNSTABLE if the natural response

approaches infinity as time approaches infinity

3. A system is MARGINALLY STABLE if the natural

response neither decays nor grows but remains

constant or oscillates

BIBO Definition

1. A system is stable if every bounded input yields a

bounded output

2. A system is unstable if any bounded input yields an

unbounded output


How to define stability

H(s)

G(s)R(s) C(s)

+-

Stability with respect to G(s)? All poles in the left half plane

Stability with respect to 𝑮(𝒔)

𝟏+𝑮 𝒔 𝑯(𝒔)?

Poles of 1+G(s)H(s) in the left half.


System Stability Definition – Stable System

Time approaches

infinity the natural

response approaches

zero

Bounded input

yields bounded

output

Stable system

have poles

only in the left

hand plane


System Stability Definition – Unstable System

Time approaches

infinity the natural

response

approaches

infinity

Bounded input

yields an

unbounded

output

Unstable

system have

at least one

pole in the

right hand

plane And/or poles of multiplicity greater

than one on imaginary axis


System Stability Definition

Stable system –

closed loop

transfer function

poles only in the

left half plane

Unstable system –

closed loop transfer

function poles with at

least one pole in the

right half and/or poles of

multiplicity greater than

1 on the imaginary axis

𝑨𝒕𝒏𝒄𝒐𝒔(𝝎𝒕 + ∅)

Marginally stable –

closed loop transfer

function with only

imaginary axis poles

of multiplicity 1 and

poles in the left half

plane.

j

1

-1


Routh-Hurwitz Stability Criterion

Method to know how many closed-loop

system poles are in the left hand plane, how

many are in the right hand plane and how

many are on the imaginary axis

Step:

1. Generate Routh Table

2. Interpret Routh Table


Routh-Hurwitz Stability Criterion –Generate Routh Table

Given Routh Table


Routh-Hurwitz Stability Criterion –Generate Routh Table

Routh Table

The value in a

row can be

divided for

easy calculation


Routh-Hurwitz Stability Criterion –Generate Routh Table Example

Given


Routh-Hurwitz Stability Criterion –Generate Routh Table Example

10303110 23 ssS


Routh-Hurwitz Stability Criterion –Interpret Routh Table Example

The number of roots of the

polynomial that are in the

right-half plane is equal to

the number of sign

changes in the first column


Routh-Hurwitz Stability Criterion –Interpret Routh Table Example

Two sign changes = two right half plane poles, therefore unstable system


Routh-Hurwitz Stability Criterion –Example

How many roots are in the right-half plane and in the left-half plane?

62874693)( 234567 ssssssssP


Routh-Hurwitz Stability Criterion –Example

Determine the value of gain K to make the system stable


Routh-Hurwitz Criterion – Special Cases

Special cases:

1. Zero in the first column

2. Zero in the entire row


Routh-Hurwitz Criterion – Zero in the first column case

35632

10)(

2345

ssssssT

How many poles?

Five poles



How many sign

changes?

Two sign changes

Two poles are on the right half

planeThe system is

unstable



Alternative method Reverse the coefficients

35632

10)(

2345

ssssssT

35632 2345 sssss 123653 2345 sssss



123653 2345 sssss

How many sign changes?

Two sign changes

Same as previous result


Routh-Hurwitz Criterion – Zero in the entire row

5684267

10)(

2345

ssssssT

0 0 0

What to do?86)( 24 sssP ss

ds

sdP124

)( 3

4 12 0



How many sign changes?

No sign changes



What can we learn when the entire

row is zero?

An entire row of zero will appear in

the Routh Table when a purely even or a purely odd polynomial is

a factor of original polynomial

Even polynomial only has

roots symmetry about the origin

If we do not have row of

zeros, we don’t have roots on

imaginary axis



20384859392212 2345678 ssssssss

0 0 0 0

23)( 24 sssP ssds

sdP64

)( 3

4 6 0 0



20384859392212 2345678 ssssssss

Apply only

to even

polynomial

Apply to

original

polynomial



20384859392212 2345678 ssssssss

No sign

changes

No right

half plane

poles.

Because

symmetry,

no left-half

poles.

Two sign

changes

Two right

half poles


Routh-Hurwitz Criterion – Example





Ksss

KsT

7718)(

23

K < 1386, The system is stable

K > 1386, The system is unstable



K = 1386, the system is marginally stable

K = 1386

Engineering

Csl14 16 f15