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 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Response Response Deterministic Systems Deterministic Systems L : deterministic linear differential operator : deterministic linear differential operator L ( ) ( ) +∞ ( ) ( ) +∞ ( ) ( ) ∫ ∫ − − − = = τ δ − τ τ = τ δ − τ τ 1 1 1 Y t L X t L X t d X L t d ( ) ( ) t t t − ∞ − ∞ = ( ) = = = LY t X t Y 0 dY ( 0 ) / dt ... 0 X(t) = X(t ) = h(t h(t) = 0 for t < 0 ) = 0 for t < 0 h(t) ) : Impulse Response : Impulse Response h(t ( ) +∞ ( ) ( ) ∫ = − τ τ τ Y t h t X d ( ) ( ) ( ) − ∞ = δ = = Lh t t h 0 ... 0 ( ) ( ) ( ) ( ) +∞ ( ) ( ) ∫ t ∫ = − τ τ τ = τ − τ τ . Y t h X t d Y t h t X d ( ) ( ) − ∞ − = t δ 0 1 h t L t ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 1
( ) ( ) +∞ ( ) Response Mean Response Mean ∫ Systems Function Systems Function ω = ω = − i ω t H H i h t e dt − ∞ { ( ) } +∞ ( ) { ( ) } +∞ ( ) ( ) ∫ ∫ = τ − ε τ = η τ τ = η E Y t h E X t d h d H 0 X X − ∞ − ∞ ( ) +∞ ( ) ( ) +∞ ( ) ( ) ∫ ∫ = − τ τ τ = τ − τ τ Y t h t X d h X t d − ∞ − ∞ Second Order Statistics Second Order Statistics = ∫ ( ) +∞ ( ) ( ) ( ) ( ) τ τ − α α α = τ τ h(t) = 0 for t < 0 h(t ) = 0 for t < 0 R R h d R * h YX XX XX − ∞ ( ) ( ) ( ) +∞ ( ) ( ) . . ∫ t ∫ = − τ τ τ = τ − τ τ ( ) +∞ ( ) ( ) +∞ ( ) ( ) Y t h t X d h X t d ∫ ∫ τ = τ + α α α = τ − α − α α R R h d R h d − ∞ 0 YY YX YX − ∞ − ∞ ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ( ) ( ) ( ) τ = τ τ R R * h For White Input For White Input YY XY ( ) ( ) ( ) ( ) τ = δ τ τ = τ ( ) ( ) ( ) R R YX h τ = τ − τ XX R R h * XY XX ( ) ( ) ( ) Impulse Response Impulse Response τ = τ − τ R R h * YY YX { ( ) ( ) } 1 ( ) ( ) ( ) ∫ t + τ ≈ + τ ≈ τ E Y t X t Y t X t dt R ( ) ( ) ( ) ( ) YX . T 0 τ = τ τ − τ R R * h * h YY XX ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 2
( ) ( ) ( ) ω = ω ω S S H System Function System Function YX XX ( ) ( ) ( ) ω = ω ω +∞ +∞ ( ) ( ) ( ) * ∫ ∫ S S H ω = − τ − ω τ = τ ω τ * i t i t H h e d h e d YY YX − ∞ − ∞ ( ) ( ) ( ) ( ) ( ) ( ) ω = ω ω ω = ω ω * S S H S S H YY XY Impulse Response Function Impulse Response Function XY XX ( ) ( ) ( ) ( ) 1 + ∞ ( ) ∫ ω = ω ω 2 ω = ω ω i t h t e H d S S H . . π − ∞ YY XX 2 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi − Brownian Motion of a Particle n n 1 Brownian Motion of a Particle d Y d Y ( ) Given Given + + + = a a a Y X t ... , − − n n 1 0 n n 1 dt dt { } dV ( ) = + β = = α E n 0 S nn w V n ( ) 1 ω = System Function System Function H dt ( ) ω + + n a i ... a n 0 Response Power Spectrum Response Power Spectrum ( ) ( ) ( ) ω = ω ω Y H X ( ) ( ) ( ) ( ) 1 ω = ω 2 ω { } ( ) { } { } ω = i = = S H S Expected Value Expected Value H E Y H 0 E X E X / a ω + β 0 VV nn . ( ) ( ) ( ) ω = ω ω 2 S S H Power Spectrum Power Spectrum YY XX ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 3
Response Power Spectrum Response Power Spectrum α ( ) ( ) 1 = ω 2 = S VV w H ω + β ω + β 2 2 2 2 Response Autocorrelation Response Autocorrelation α { } α ( ) 2 = − β τ τ = R VV e E V β β . 2 2 Autocorrelation and power spectrum of particle velocity { } Autocorrelation and power spectrum of particle velocity = E V 0 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 4
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