Trasfer Fuctio Aalysis Free & Forced Resposes Trasfer Fuctio Syste Stability ME375 Trasfer Fuctios - 1
Free & Forced Resposes Ex: Let s look at a stable first order syste: y y Ku Take LT of the I/O odel ad reeber to keep tracks of the ICs: L y y L Ku ( ) K Rearrage ters s.t. the output Y(s) ) ters are o oe side ad the iput U(s) ( ) ad IC ters are o the other: Ys () Us () y(0) Factor out the output side of the equatio: ME375 Trasfer Fuctios - 2
Free & Forced Resposes Free Respose (u(t) = 0 & ozero ICs ) = 0 & ozero ICs) The respose of a syste to zero iput ad ozero iitial coditios. Ca be obtaied by Let u(t) ) = 0 ad use LT ad ILT to solve for the free respose. Forced Respose (zero( ICs & ozero u(t)) The respose of a syste to ozero iput ad zero iitial coditios. Ca be obtaied by Assue zero ICs ad use LT ad ILT to solve for the forced respose (replace differetiatio with s i the I/O ODE). ME375 Trasfer Fuctios - 3
I Class Exercise Fid the free ad forced resposes of the followig I/O odel: y 4 y y 5y 2u u u ME375 Trasfer Fuctios - 4
Trasfer Fuctio Give a geeral th order syste odel: ( ) ( 1) ( ) ( 1) a y a1y a1 y a0 y b u b1 u b1 u b0 u The forced respose (zero ICs) of the syste due to iput u(t) ) is: Takig the LT of the ODE: L y ( ) s Y( s) WHY? asys () a s Ys () asys () ays () bsus () b s Us () bsus () bus () 1 1 1 0 1 1 1 0 1 1 as a 1s as 1 a 0Ys () bs b 1s bs 1 b 0 Ds () Ns () 1 bs b 1s bs 1 b0 Ns () Ys () Us () Us () Gs () Us () 1 as a 1s as 1 a0 Ds () Gs () Trasfer Fuctio U() s ME375 Trasfer Fuctios - 5
Exaples (1) Recall the first order syste: (2) For the followig 2d order syste: 2 2 y y K u y 2 y y K u Fid the trasfer fuctio of the syste. Fid the trasfer fuctio of the syste. Takig LT of the ODE: Takig LT of the ODE: ME375 Trasfer Fuctios - 6
Trasfer Fuctio Give a geeral th order syste: a y a y a ya ybu b u bubu ( ) ( 1) ( ) ( 1) 1 1 0 1 1 0 The trasfer fuctio of the syste is: Gs () 1 bs b 1s bs 1 b0 1 as a 1 s as 1 a0 The trasfer fuctio ca be iterpreted as: u(t) Iput Differetial Equatio y(t) Output U(s) Iput G(s) Y(s) Output Tie Doai s -Doai ME375 Trasfer Fuctios - 7
Poles ad Zeros Give a trasfer fuctio (TF) of a syste: Gs () 1 bs b 1 s bs 1 b0 N s 1 as a 1s as 1 a0 Ds Poles The roots of the deoiator of the TF, i.e. the roots of the characteristic equatio. 1 D () s as a 1s as 1 a0 a ( s p )( s p ) ( s p ) 0 1 2 1 2 p, p,, p : poles of the TF () () Zeros The roots of the uerator of the TF. 1 Ns () bs b 1s bs 1 b0 b ( sz )( sz ) ( sz ) 0 1 2 1 2 z, z,, z : zeros of the TF Gs () bs b s bsb Ns ( ) as a s as a Ds 1 1 1 0 1 1 1 0 () ME375 Trasfer Fuctios - 8
Static Gai Static Gai ( G(0) ) Ex: For a secod order syste: 2 2 The value of the trasfer fuctio y2y y Ks u whe s = 0. If Fid the trasfer fuctio ad the static 1 b gai. s b1s bs 1 b0 Ns () Gs () 1 as a 1 s as 1 a 0 Ds () N(0) b0 KS G(0) D(0) a0 The static gai K S ca be iterpreted as the steady state value of the uit step respose. Ex: Fid the steady state value of the syste y 3 y 5y 7y u 2u u to a step iput of agitude 2. ME375 Trasfer Fuctios - 9
I Class Exercise Give a I/O odel of a 2d order syste: 5 y 2y 4y 6u (1) Fid the trasfer fuctio of the syste (2) Fid the poles ad zeros of the syste (3) What is the syste forced respose Y(s) ) to a uit step iput u(t) ) = 1? (4) As tie goes to ifiity, what is the steady state value of the uit step respose? ME375 Trasfer Fuctios - 10
A Closer Look at Free Respose Give a geeral th order syste odel: ( ) ( 1) ( ) ( 1) ay a y ayay bu b u bubu 1 1 0 1 1 0 The free respose (zero iput) of the syste due to ICs is: Takig the LT of the Hoogeeous ODE: ay a y ay ay ( ) ( 1) Ly Ly ( ) ( 1) 1 1 0 0 1 ( 1) 1 2 ( 2) a s Y( s) s y(0) y (0) a1 s Y( s) s y(0) y (0) a 1 sy s y a 0Y s ( ) (0) ( ) 0 1 as a1s a1s a 0YFree ( s) 1 2 ( a 1) s a1s a1 y(0) y (0) Y ( s) L y F ( s) A Polyoial of s That depeds o ICs F ( s ) 1 a s a s a s a Free 1 1 0 L y ME375 Trasfer Fuctios - 11
A Closer Look at Free Respose Ex: : Fid the free respose of the followig syste: 5 y 30y 50y 2u u Ex: : Perfor partial fractio expasio (PFE) of the above free respose whe: y( 0) 1 ad y( 0) 1 ME375 Trasfer Fuctios - 12
Free Respose ad Pole Positio The free respose of a syste ca be represeted by: Fs () Fs () Y Free () s 1 as a 1s a1sa0 a ( s p1 )( s p2 ) ( s p ) A1 A2 A s p1 s p2 s p Assue p1 p2 p i.e. distict poles R p1 t p2 t y () t Y () s Ae A e A e Free R S pi 0 pi is real pi 0 p 0 S T R S pi j T T 0 i 0 0 pt L 1 Free 1 2 Ig. Real ME375 Trasfer Fuctios - 13
Coplete Respose U(s) Iput Coplete Respose Gs () Ns () Ds () ( 1) ICs: y(0), y(0),, y (0) Y(s) Output Y() s Y () s Y () s G() s U() s Y () s Forced Free Free Us () Q: What part of the syste affects both the free ad forced respose? Q: If U(s) ) = 0 ad there are o-zero ICs, what will guaratee that y(t)? ME375 Trasfer Fuctios - 14
Stability Stability Cocept Describes the ability of a syste to stay at its equilibriu positio (for liear systes: all state variables = 0 or y(t) ) = 0) i the absece of ay iputs. A liear tie ivariat (LTI) syste is stable if ad oly if (iff) its free respose coverges to zero. Ex: Pedulu Ball o curved surface ME375 Trasfer Fuctios - 15
Stability of LTI Systes Stability Criterio for LTI Systes a y a y a ya ybu b u bubu ( ) ( 1) ( ) ( 1) 1 1 0 1 1 0 1 Ds as a 1s as 1 a0 Stable all poles of ( ) lie i the left-half coplex plae Coets o LTI Stability Characteristic Polyoial Stability of a LTI syste does ot deped o the iput. (why?) For 1st ad 2d order systes, stability is guarateed if all the coefficiets of the characteristic polyoial are positive. D() s as a : Stable a 0 i or a 0 i 1 0 2 Ds () as asa : Stable a0 i or a0 i 2 1 0 Effect of Poles ad Zeros o Stability Stability of a syste depeds oly o its poles. Zeros do ot affect syste stability. Zeros affect the trasiet respose of the syste. i i i i LHP ME375 Trasfer Fuctios - 16
I Class Exercises (1) Fid the trasfer fuctio of the followig I/O equatio: y2y 5y 2u u (2) Deterie the syste s stability. (3) () Plot the poles ad zeros of the syste o the coplex plae. (1) Fid the trasfer fuctio of the followig I/O equatio: y y 6y u 3u 4u (2) Deterie the syste s stability. (3) Plot the poles ad zeros of the syste o the coplex plae. ME375 Trasfer Fuctios - 17
Exaple (Iverted) Pedulu (1) Derive a atheatical odel for a pedulu. (2) Fid the equilibriu positios. (3) Discuss the stability of the equilibriu positios. ME375 Trasfer Fuctios - 18
Exaple (Iverted Pedulu) ME375 Trasfer Fuctios - 19