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

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

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

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

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

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

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

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 = s + 22 s s + 22 s – 22 s – 2 dcgain

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])

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 = Transfer constant: is the multiplying factor in the “zpk” TF

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 = s + 22 s s + 22 s – 22 s – 2 transfer constants:

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 = s + 22 s s + 22 s – 22 s – 2 Transfer constants:

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]) )

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

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))

Transfer Functions vs. Frequency Response

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

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 = ; mag = e-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.

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: s^4 + 8 s^ s^ s >> [Wn,Z,P] = damp(Gp) Wn = Z = P = i i