Transition Matrix Properties

Chapter-4

EE-826 Linear Control System

College of Electrical & Mechanical Engineering

National University of Sciences & Technology (NUST)

State Transition Matrix

Unforced Linear System

State Equation Solution

Peano Backer Series

3

Peano Backer Series

State Transition Matrix for Time Invariant Systems

kth term

4

Property 4.1 State transition matrix for time invariant systems

where with guaranteed convergence

Proof

Bounds on individual terms of matrix series

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Convergence of scalar series

Example: Harmonic Oscillator

0 1

1 0A

= −

3A A= −

2A I= −Solution

6

3A A= −4A I=

Transition Matrix2 2 3 3

.......2! 3!

At A t A te I At= + + + +

Example: Harmonic Oscillator2 2 4 4

3 3 5 5

...........2! 4!

............3! 5!

At A t A te I

A t A tAt

= + + +

+ + +

2 4 3 5t t t t

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2 4 3 5

1 ........... ............2! 4! 3! 5!

At t t t te I A t

= − + + + − + +

.cos .sinI t A t= +

cos sin

sin cos

t t

t t

= −

Property 4.2

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Proof

( )

( ) ( , )

t

A d

o ox t t x e xτ

σ στ

∫= Φ =

Proposed Solution

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2 3

( ) ( )

( ) ( ) ....2! 3!

t t

t

o

A d A d

x t I A d xτ τ

τ

σ σ σ σσ σ

= + + + +

∫ ∫∫

Differentiating

( ) . ( ) ( ). ( )

( ) 0 ( ) ......2!

t t

o

A d A t A t A d

x t A t xτ τ

σ σ σ σ

+

= + + +

∫ ∫ɺ

Condition of Property 4.2

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2. ( ). ( )

( ) ( ) ............2!

t

o

A t A d

x t A t xτ

σ σ

= + +

∫ɺ

Condition of Property 4.2

Implies ( ) ( ) ( )x t A t x t=ɺ

Example 4.3( ) ( )

( )0 0

a t a tA t

=

1 1( ) ( ) ( )

0 0A t a t a t M

= =

Solution

Property 4.2 holds

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( ) ( )

( )( )

2 3

( , )

2! 3!

tt

a d MA dMt e e e

M MI M

ττ

σ σσ σλτ

λ λλ

∫∫

Φ = = =

= + + +

2 3.....M M M= =

Property 4.2 holds

since

Example 4.3 (contd)2 3

2 3

( , ) ...2! 3!

1 ...2! 3!

t I M

I M M

I M e Mλ

λ λτ λ

λ λλ

Φ = + + + +

= − + + + + +

= − +

12

I M e Mλ= − +

Special case

General Properties

Property 4.4

Matrix Differential Equation

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Property 4.5

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Linear state equation

Example 4.6

Solution

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( )( ,0)

0 1

t

A

e f tt

Φ =

* ( )( ,0)

* *A

f tdt

dt

Φ =

ɺ

Example 4.62

2

sin 1 cos( ,0) 0 ( ) .....2

0 00 0

sin 1 cos( ) .....2

0 00 0

A

tt td t

t A tdt

tt t t

A t I I

− Φ = + + + +

− = − + + + +

16

( ) ( ,0)AA t I t

= − +Φ

* ( ) cos( ,0)

* *A

f t tdt

dt

+ Φ =

( ) ( ) cosf t f t t= +ɺ

Implies

Example 4.6

sin cos( )

2

te t tf t

+ −=

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State Transition Matrix for

Property 4.7 Composition Property

Property 4.9 Determinant of state transition matrix

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Property 4.10 Inverse of state transition matrix

Example 4.11

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Example 4.11

Alternate Solution using Property 4.5

0 0( , ) ( , )TTAA t t t t−Φ = Φ

implies (0, ) ( ,0)T

T

A At t−Φ = Φ

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State Variable Changes

State transformation

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State Variable Changes

Property 4.13 Transition matrix for transformed states

Example 4.14

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Example 4.14

Proposed state transformation

State Variable Changes

Example 4.14 (contd)

Integrating both sides

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Discrete-time State Equations

Chapter-20

EE-826 Linear Control System

College of Electrical & Mechanical Engineering

National University of Sciences & Technology (NUST)

State Equation Solution

( 1) ( ) ( ) ( ) ( )x k A k x k B k u k+ = +

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State Equation Solution

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Transition Matrix Properties

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Transition Matrix Properties

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Transition Matrix Properties

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