Transfer Function Analysis
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1 Trasfer Fuctio Aalysis Free & Forced Resposes Trasfer Fuctio Syste Stability ME375 Trasfer Fuctios - Free & Forced Resposes Ex: Let s s look at a stable first order syste: τ y + y = Ku Take LT of the I/O odel ad reeber to keep tracks of the ICs: L[ τ y + y] = L[ Ku] τ ( ) + = 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() ( ) ( ) ( ) Factor out the output side of the equatio: Y ( s) = U ( s) + y() ME375 Trasfer Fuctios - 2
2 Free & Forced Resposes Free Respose (u(t)( ) = & ozero ICs) The respose of a syste to zero iput ad ozero iitial coditios. Ca be obtaied by Let u(t) ) = 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 2
3 Trasfer Fuctio Give a geeral th order syste odel: ( ) ( ) ( ) ( ) a y + a y + + ay + a y = bu + b u + + bu + bu 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 ( ) () as + a s + + as+ a Ys = bs + b s + + bs+ b Us () Ds ( ) Ns ( ) bs + b s + + bs + b Ns () Ys () = Us () = Us () = Gs () Us () as + a s + + as + a Ds () Gs () Trasfer Fuctio ME375 Trasfer Fuctios - 5 Exaples () Recall the first order syste: τ y + y = K u Fid the trasfer fuctio of the syste. Takig LT of the ODE: (2) For the followig 2d order syste: y + 2 ζ ω y + ω y = Kω u Fid the trasfer fuctio of the syste. Takig LT of the ODE: 2 2 ME375 Trasfer Fuctios - 6 3
4 Trasfer Fuctio Give a geeral th order syste: ay + a y + + ay + ay= bu + b u + + bu + bu ( ) ( ) ( ) ( ) The trasfer fuctio of the syste is: bs + b s + + bs+ b Gs () as + a s + + as+ a 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: bs + b s + + bs + b N() s Gs () = as + a s + + as + a Ds () Poles Zeros The roots of the deoiator of the The roots of the uerator of the TF. TF, i.e. the roots of the characteristic equatio. Ns () = bs + b s + + bs + b Ds () = as + a s + + as + a = b( s z)( s z2) ( s z) = = a( s p)( s p2) ( s p) = z, z2,, z : zeros of the TF p, p,, p : poles of the TF 2 bs + b s + + bs+ b Ns ( ) Gs () = = as + a s + + as+ a Ds () ME375 Trasfer Fuctios - 8 4
5 Static Gai Static Gai ( G() ) The value of the trasfer fuctio whe s =. If bs + b s + + bs + b N() s Gs () = as + a s + + as + a Ds () N() b KS = G() = = D() a The static gai K S ca be iterpreted as the steady state value of the uit step respose. Ex: For a secod order syste: 2 2 y+ 2ζ ωy + ω y = Ksω u Fid the trasfer fuctio ad the static gai. 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 + 2 y + 4 y = 6 u () 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) ) =? (4) As tie goes to ifiity, what is the steady state value of the uit step respose? ME375 Trasfer Fuctios - 5
6 A Closer Look at Free Respose Give a geeral th order syste odel: ( ) ( ) ( ) ( ) ay + a y + + ay + ay = bu + b u + + bu + bu The free respose (zero iput) of the syste due to ICs is: ( ) ( ) Takig the LT of the Hoogeeous ODE: ay + a y + + ay + ay= ( ) ( ) L y L y ( ) 2 ( 2) a s Y( s) s y() y () + a s Y( s) s y() y () [ ] + + a sy( s) y() + a Y( s) = L y L y as + a s + + as+ a YFree ( s) 2 ( ) = ( as + a s + + a) y() + + y () Y ( s) = F( s) F( s) A Polyoial of s That depeds o ICs Free as + a s + + as+ a ME375 Trasfer Fuctios - A Closer Look at Free Respose Ex: : Fid the free respose of the followig syste: 5 y + 3y + 5y = 2 u + u Ex: : Perfor partial fractio expasio (PFE) of the above free respose whe: y ( ) = ad y( ) = ME375 Trasfer Fuctios - 2 6
7 Free Respose ad Pole Positio The free respose of a syste ca be represeted by: Fs () Fs () YFree() s = = as + a s + + as + a a( s p)( s p2) ( s p) A A2 A = s p s p2 s p Assue p p p i.e. distict poles R S T 2 p t p2 t y ( t) = Y ( s) = A e + A e + + A e p p i i is real = σ + jβ Free R S T i R S T pi < pi = p > σ < σ = σ > p t L Free 2 Ig. Real ME375 Trasfer Fuctios - 3 Coplete Respose U(s) N() s Y(s) Gs () Iput D() s Output ( ) ICs: y(), y (),, y () Coplete Respose Y ( s) = Y ( s) + Y ( s) = G ( s) U ( s) + Y ( s) Forced Free Free = U( s) + Q: What part of the syste affects both the free ad forced respose? Q: If U(s) U ) = ad there are o-zero ICs, what will guaratee that y(t) y? ME375 Trasfer Fuctios - 4 7
8 Stability Stability Cocept Describes the ability of a syste to stay at its equilibriu positio (for liear systes: all state variables = or y(t) ) = ) 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 - 5 Stability of LTI Systes Stability Criterio for LTI Systes ( ) ( ) ( ) ( ) ay + a y + + ay + ay = bu + b u + + bu + bu Stable all poles of Ds ( ) = as + a s + + as + a lie i the left-half coplex plae Coets o LTI Stability Stability of a LTI syste does ot deped o the iput. (why?) For st ad 2d order systes, stability is guarateed if all the e coefficiets of the characteristic polyoial are positive. D( s) = as + a : Stable ai > i or ai < i 2 Ds ( ) = as + as+ a : Stable a> i or a< i 2 Characteristic Polyoial 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 LHP ME375 Trasfer Fuctios - 6 8
9 I Class Exercises () Fid the trasfer fuctio of the followig I/O equatio: y 2 y 5y = 2u + u (2) Deterie the syste s s stability. (3) Plot the poles ad zeros of the syste o the coplex plae. () Fid the trasfer fuctio of the followig I/O equatio: y + y + 6y = u 3u + 4u (2) Deterie the syste s s stability. (3) Plot the poles ad zeros of the syste o the coplex plae. ME375 Trasfer Fuctios - 7 Exaple (Iverted) Pedulu () Derive a atheatical odel for a pedulu. (2) Fid the equilibriu positios. (3) Discuss the stability of the equilibriu positios. ME375 Trasfer Fuctios - 8 9
10 Exaple (Iverted Pedulu) ME375 Trasfer Fuctios - 9
School of Mechanical Engineering Purdue University. ME375 Transfer Functions - 1
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
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