( ) ( ) ( ) = = = Lh t t h 0 ... 0 ( ) ( ) ( ) - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Outline Outline

• Deterministic Systems

Deterministic Systems

• Random Response of Linear Systems

Random Response of Linear Systems

• Stationary Response Analysis

Stationary Response Analysis

• Autocorrelation & Cross

Autocorrelation & Cross-

• Correlation

Correlation

• System Identification

System Identification

• Spectral Response

Spectral Response

• Examples

Examples

• G. Ahmadi

ME 529 - Stochastics

Deterministic Systems Deterministic Systems L L: deterministic linear differential operator : deterministic linear differential operator h(t h(t) ) : Impulse Response : Impulse Response

( ) ( )

t X t LY =

( )

... / ) ( = = = dt dY Y

( ) ( )

t t Lh δ = ( )

... = = h

( ) ( )

t L t h

t δ 1 −

=

• G. Ahmadi

ME 529 - Stochastics

.

( ) ( ) ( ) ( ) ( ) ( )

∫ ∫

+∞ ∞ − − +∞ ∞ − − −

− = − = = τ τ δ τ τ τ δ τ d t L X d t X L t X L t Y

t t t 1 1 1

( ) ( ) ( )

∫

+∞ ∞ −

− = τ τ τ d X t h t Y

( ) ( ) ( )

∫

− =

t

d X t h t Y τ τ τ

( ) ( ) ( )

∫

+∞ ∞ −

− = τ τ τ d t X h t Y

X(t X(t) = ) = h(t h(t) = 0 for t < 0 ) = 0 for t < 0 Response Response

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• G. Ahmadi

ME 529 - Stochastics

Systems Function Systems Function

.

( ) ( ) ( ) ( ) ( )

∫ ∫

+∞ ∞ − +∞ ∞ −

− = − = τ τ τ τ τ τ d t X h d X t h t Y

( ) ( ) ( )

∫

+∞ ∞ − −

= = dt e t h i H H

t iω

ω ω

h(t h(t) = 0 for t < 0 ) = 0 for t < 0

( ) ( ) ( ) ( ) ( )

∫ ∫

+∞ ∞ −

− = − = τ τ τ τ τ τ d t X h d X t h t Y

t

• G. Ahmadi

ME 529 - Stochastics

Second Order Statistics Second Order Statistics

.

( ) ( ) ( ) ( ) ( )

τ τ α α α τ τ h R d h R R

XX XX YX

* = − = ∫

+∞ ∞ −

( ) ( ) ( ) ( ) ( )

∫ ∫

+∞ ∞ − +∞ ∞ −

− − = + = α α α τ α α α τ τ d h R d h R R

YX YX YY

Response Mean Response Mean

( ) { } ( ) ( ) { } ( ) ( ) H d h d t X E h t Y E

X X

η τ τ η τ ε τ = = − =

∫ ∫

+∞ ∞ − +∞ ∞ −

• G. Ahmadi

ME 529 - Stochastics

.

( ) ( ) ( )

τ τ τ − = h R R

YX YY

*

( ) ( ) ( )

τ τ τ − = h R R

XX XY

*

( ) ( ) ( )

τ τ τ h R R

XY YY

* =

( ) ( ) ( ) ( )

τ τ τ τ − = h h R R

XX YY

* *

• G. Ahmadi

ME 529 - Stochastics

For White Input For White Input

( ) ( )

τ τ h RYX =

( ) ( ) { } ( ) ( ) ( )

τ τ τ

YX t

R dt t X t Y T t X t Y E ≈ + ≈ +

∫

1

( ) ( )

τ δ τ =

XX

R

Impulse Response Impulse Response

SLIDE 3

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• G. Ahmadi

ME 529 - Stochastics

.

( ) ( ) ( )

∫ ∫

+∞ ∞ − +∞ ∞ − −

= − = τ τ τ τ ω

ω ω

d e h d e h H

t i t i *

( ) ( )

∫

∞ + ∞ −

= ω ω π

ω

d H e t h

t i

2 1

System Function System Function Impulse Response Function Impulse Response Function

• G. Ahmadi

ME 529 - Stochastics

.

( ) ( ) ( )

ω ω ω H S S

XX YX

=

( ) ( ) ( )

ω ω ω

* YX YY

H S S = ( ) ( ) ( )

ω ω ω

* XX XY

H S S =

( ) ( ) ( )

ω ω ω H S S

XY YY

=

( ) ( ) ( )

2

ω ω ω H S S

XX YY

=

• G. Ahmadi

ME 529 - Stochastics

Power Spectrum Power Spectrum

.

Given Given System Function System Function ( )

, ...

1 1 1

t X Y a dt Y d a dt Y d a

n n n n n n

= + + +

− − −

( ) ( )

1 a ... i a H

n n

+ + = ω ω

( ) ( ) ( )

ω ω ω X H Y =

{ } ( ) { } { }

/ a X E X E H Y E = =

Expected Value Expected Value

( ) ( ) ( )

2

ω ω ω H S S

XX YY

=

• G. Ahmadi

ME 529 - Stochastics

Response Power Spectrum Response Power Spectrum Brownian Motion of a Particle Brownian Motion of a Particle

n V dt dV = + β { }

= n E

( )

α = w Snn

( ) ( ) ( )

ω ω ω

nn VV

S H S

2

=

( )

β ω ω + = i H 1

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• G. Ahmadi

ME 529 - Stochastics

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Response Autocorrelation Response Autocorrelation

( )

2 2 2

1 β ω ω + = H

( )

2 2

β ω α + = w SVV

( )

τ β

β α τ

−

= e RVV 2

{ }

β α 2

2 =

V E

{ }

= V E Response Power Spectrum Response Power Spectrum

• G. Ahmadi

ME 529 - Stochastics

Autocorrelation and power spectrum of particle velocity Autocorrelation and power spectrum of particle velocity

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics