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UNIVERSITÁ DEGLI STUDI DI SALERNO FACOLTÀ DI INGEGNERIA Prof. Ing. Michele MICCIO Dip. Ingegneria Industriale (DIIn) Prodal Scarl (Fisciano) Transfer Function (TF) forms Rev.. 4.65 of September 4, 2014

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Transfer Functions (TFs) Rational Non-rational (e.g., trascendent)

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Non-factorized rational TFs non-factorized OR canonic Form the trailing coefficient is non-zero and equal to unity non-factorized OR canonic Form the trailing coefficient is non-zero and different from unity Examples

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Factorized rational TFs factorized rational Form with time constants factorized rational Form with zeroes and poles examples of type g=0

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Factorized rational TFs factorized Form with time constants OR Bode Form factorized Form with zeroes and poles where: K P = gain k ≠ K P = transfer constant g Z = “type” from: Bolzern, Scattolini e Schiavoni, "Fondamenti di controlli automatici", McGraw-Hill, 1998

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Factorized rational TFs of type g=0 factorized Form with time constants OR Bode Form factorized Form with zeroes and poles where: K P = gain k ≠ K P = transfer constant g Z = “type” from: Bolzern, Scattolini e Schiavoni, "Fondamenti di controlli automatici", McGraw-Hill, 1998

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The gain of factorized rational TFs from: Bolzern, Scattolini e Schiavoni, "Fondamenti di controlli automatici", McGraw-Hill, 1998 factorized Form with time constants gain definition and properties type g=0 generalized gain type g 0

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The gain of factorized rational TFs in MatLab® dcgain Computes low frequency (DC) gain of LTI system Syntax:K P = dcgain(sys) sys is the TF object in Matlab. The continuous-time DC gain is the transfer function value at the frequency corresponding to s=0. Remark: The DC gain is infinite for systems with integrators (type g 0). Transfer functions: s + 11 s + 1s – 1s + 1s - 1 ------- = -- ------- --------------------- 2 s + 22 s + 1 2 s + 22 s – 22 s – 2 dcgain 0.5-0.5-0.50.5

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Transfer Function forms in MatLab® G = tf(num,den) where num and den are row vectors listing the coefficients of the polynomials non-Factorized or Canonic Form ALTERNATIVE from Matlab 7.5 R2007b August 15, 2007 s=tf('s') G=tf(K * N(s) / D(s)) where N(s) and D(s) are polynomials typed according to Matlab algebraic rules Ex.:G=tf(3*(1/2*s+1)/(1/2*s^3+3/2*s^2+2*s+1)) Ex.:G=tf(3*[1/2 1], [1/2 3/2 2 1])

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Rational TF forms in MatLab® factorized form with zeroes, poles and transfer constant G = zpk(z,p,k) where z and p are the vectors of zeros and poles, and k is the transfer constant the transfer constant k is generally different from the static gain K P in Matlab K P =dcgain(G) G=zpk([-2],[-1+j -1-j -1],1) Zero/pole/gain: (s+2) -------------------- (s+1) (s^2 + 2s + 2) Example >> Kp=dcgain(G) Kp = 1.0000 Transfer constant: 1.0000 is the multiplying factor in the “zpk” TF

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Rational TF forms in MatLab® factorized form with zeroes, poles and transfer constant G = zpk(z,p,k) where z and p are the vectors of zeros and poles, and k is the transfer constant the transfer constant k is different from the static gain K P in Matlab K P =dcgain(G) G=zpk([-2],[-1+j -1-j -1],1) Example Transfer functions: s + 11 s + 1s – 1s + 1s - 1 ------- = -- ------- --------------------- 2 s + 22 s + 1 2 s + 22 s – 22 s – 2 transfer constants: 0.50.50.50.5

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Rational TF forms in MatLab® factorized form with zeroes, poles and transfer constant G = zpk(z,p,k) where z and p are the vectors of zeros and poles, and k is the transfer constant the transfer constant k is different from the static gain K P in Matlab K P =dcgain(G) Transfer functions: s + 11 s + 1s – 1s + 1s - 1 ------- = -- ------- --------------------- 2 s + 22 s + 1 2 s + 22 s – 22 s – 2 Transfer constants: 0.50.50.50.5

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Rational TF forms in MatLab® factorized form in form of time constants G = tf(num, conv(den1, den2) where num, den1 and den2 are row vectors listing the coefficients of the polynomials Example G=tf ( [1/2 1], conv([1/2 1 1], [1 1]) )

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Transfer Functions (TFs) Rational Non-rational (e.g., trascendent)

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Example: G=tf(3*(1/2*s+1)/(1/2*s^3+3/2*s^2+2*s+1)*exp(-20*s)) NON-rational Transfer Function in MatLab® non-factorized form G = tf(num, den, 'inputdelay',td) where num and den are row vectors listing the coefficients of the polynomials, td is the dead time Ex.:G=tf([1 8], [1 4 5], 'inputdelay',3) ALTERNATIVE from Matlab 7.5 R2007b August 15, 2007 s=tf('s') G=tf(K * N(s) / D(s) * exp(-t D *s))

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Transfer Functions vs. Frequency Response

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Frequency Response in MatLab® bode(Gp) % produces the pair of Bode plots of the transfer function Gp nyquist(Gp) % produces the polar plot of the transfer function Gp

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Frequency Response in MatLab® G = freqresp(Gp,w) % computes the frequency response Gp(jw) of the transfer function Gp at the frequencies specified by the vector w [mag,phase,w] = bode(Gp) % gives tabular representation of AR and phase shift as a function of frequency % e.g., w=100 --> phase = -266.5629; mag = 9.9930e-007 [GM,PM,Wco,Wgc] = MARGIN(Gp) % computes the gain margin GM, the phase margin PM, the crossover frequency Wco and gain crossover Wgc, for the SISO open-loop model Gp. % The gain margin GM is defined as 1/G where G is the gain at the -180° phase crossing. % The phase margin PM is in degrees. MARGIN(Gp) % plot the open-loop Bode plot with the gain and phase margins printed and marked with a vertical line.

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Frequency Response in MatLab® [Wn,Z,P] = damp(Gp) % returns vectors Wn, Z and P containing the natural (corner) frequencies, damping factors and poles, respectively, of the LTI model Gp Wn j = |p j |=1/ j for real p j Wn j =SQRT[Re(p j ) 2 + Im(p j ) 2 ] for complex p j Ex.:Transfer function: 1 --------------------------------- s^4 + 8 s^3 + 22.67 s^2 + 26.67 s >> [Wn,Z,P] = damp(Gp) Wn = 0 2.5820 4.0000 Z = -1.0000 0.7746 1.0000 P = 0 -2.0000 + 1.6330i -2.0000 - 1.6330i -4.0000

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Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise.

Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise.

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