UNIVERSIT

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UNIVERSITÁ DEGLI STUDI DI SALERNODipartimento di
Ingegneria Industriale
Transfer Function (TF) forms

• Prof. Ing. Michele MICCIO
• Dip. Ingegneria Industriale (DIIn)
• Prodal Scarl (Fisciano)

Rev.. 6.2 of June 7, 2016
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Transfer Functions (TFs)

• Rational
• Non-rational (e.g., trascendent)

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Non-factorized rational TFs
Examples
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
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Factorized rational TFs
examples of type g0
factorized rational Form ? with time constants
factorized rational Form ? with zeroes and poles
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Factorized rational TFs
factorized Form ? with time constants OR Bode Form
factorized Form ? with zeroes and poles
where KP gain k ? KP 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 g0
factorized Form ? with time constants OR Bode Form
factorized Form ? with zeroes and poles
where KP gain k ? KP 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
factorized Form ? with time constants
type g?0
type g0
generalized gain
gain definition and properties
? from Bolzern, Scattolini e Schiavoni,
"Fondamenti di controlli automatici",
McGraw-Hill, 1998
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Transfer Function forms in MatLab
non-Factorized or Canonic Form
G tf(num,den) where num and den are row vectors
listing the coefficients of the polynomials
Ex. Gtf(31/2 1, 1/2 3/2 2 1)
ALTERNATIVE from Matlab 7.5 R2007b August 15,
2007 stf('s') Gtf(K N(s) / D(s)) where N(s)
and D(s) are polynomials typed according to
Matlab algebraic rules Ex. Gtf(3(1/2s1)/(1/2
s33/2s22s1))
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The gain of factorized rational TFs in MatLab
dcgain Computes low frequency (DC) gain of LTI
system Syntax KP dcgain(sys) sys is the TF
object in Matlab. Remarks The continuous-time
DC gain is the transfer function value at the
frequency corresponding to s0 ??The DC gain is
zero for systems with zeroes at origin (type
glt0) ??The DC gain is infinite for systems with
integrators (type ggt0) Examples of Transfer
Functions s 1 1 s 1 s 1 s 1 s -
1 ------- -- ------- ------- ------- ----
--- 2 s 2 2 s 1 2 s 2 2 s 2 2 s
2 dcgain 0.5 -0.5 -0.5 0.5
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Poles and Zeroes in MatLab

• p pole(Gp)
• computes the poles pj of the LTI model Gp(s)
• z zero(Gp)
• returns the transmission zeros of the LTI model
  Gp(s).
• p and z are column vectors

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Poles and Zeroes in MatLab
pzmap(sys) pzmap(sys) computes the poles and
zeros of the LTI model SYS and plots them in the
complex plane The poles are plotted as x
and the zeros are plotted as o
from prof. Pribeiro
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Rational TF forms in MatLab
Example
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
Gzpk(-2,-1j -1-j -1,1) Zero/pole/gain
(s2) -------------------- (s1) (s2 2s
2)
??the transfer constant k is generally different
from the static gain KP in Matlab KPdcgain(G)
gtgt Kpdcgain(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
Example
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
Gzpk(-1,-1,1/2)
??the transfer constant k is different from the
static gain KP in Matlab KPdcgain(G)
Transfer functions s 1 1 s 1 s 1 s
1 s - 1 ------- -- -------
------- ------- ------- 2 s 2 2 s 1 2
s 2 2 s 2 2 s 2 transfer
constants 0.5 0.5 0.5 0.5
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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
Gtf(1/2 1, conv(1/2 1 1, 1 1))
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Transfer Functions in MatLab
from prof. Pribeiro
sys minreal(sys1) cancels pole-zero pairs
in transfer functions. the output sys has
minimal order and the same response
characteristics as the original model sys1.
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Transfer Functions in MatLab
Gparallel(G1,G2) or GG1G2
from prof. Pribeiro
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Feedback generation in MatLab
Tfeedback(GcG, 1, sign) with unity
feedback
from prof. Pribeiro
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Feedback generation in MatLab
Tfeedback(GcG, H, sign) with H(s) as the
feedback block
from prof. Pribeiro
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Transfer Functions (TFs)

• Rational
• Non-rational (e.g., trascendent)

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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
Example 1
Gtf(1 8, 1 4 5, 'inputdelay',3)
ALTERNATIVE from Matlab 7.5 R2007b August 15,
2007 stf('s') Gtf(K N(s) / D(s) exp(-tDs))
Example 2 G3(1/2s1)/(1/2s33/2s22s
1)exp(-20s)
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Transfer Functions (TFs)

• Use of Transfer Functions inFrequency Response
  methods

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Angle of a complex number in MatLab
gtgt phiangle(p) computes the argument/s of the
complex number/s in "p gtgt modulusabs(p)
returns an array such that each element is the
absolute value of the corresponding element of
"p". if "p" is complex, abs(p) returns the
complex modulus (magnitude)
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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 mag,phase,w bode(Gp) gives tabular
representation of AR and phase shift as a
function of frequency e.g., w100 --gt phase
-266.5629 mag 9.9930e-007 G
freqresp(Gp,w)  computes the frequency response
Gp(jw) of the transfer function Gp at the
frequencies specified by the vector w
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Frequency Response in MatLab

• 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.
• systems)

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Frequency Response in MatLab

• S ALLMARGIN(Gp)
• provides detailed information about the
• gain, phase, and delay margins and the
  corresponding
• crossover frequencies of the SISO open-loop
  model SYS.
• The output S is a structure with the
  following fields
• GMFrequency all -180 deg crossover
  frequencies (in rad/sec)
• GainMargin corresponding gain margins
  (g.m. 1/G where G is the gain at crossover)
• PMFrequency all 0 dB crossover
  frequencies (in rad/sec)
• PhaseMargin corresponding phase margins
  (in degrees)
• DelayMargin, DMFrequency delay margins
  (in seconds for continuous-time systems, and
  multiples of the sample time for discrete-time
  systems) and corresponding critical frequencies
• Stable 1 if nominal closed loop is
  stable, 0 if unstable, and NaN if stability
  cannot be assessed (as in the case of most FRD
  systems)

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Frequency Response in MatLab
Wn,zeta,P damp(Gp) returns vectors Wn,
zeta and P containing the natural (corner)
frequencies, damping factors and poles,
respectively, of the LTI model Gp ? Wnj
pj1/?j for real pj Wnj SQRTRe(pj)2
Im(pj)2 for complex pj Ex. Transfer function
1 --------------------------------- s
4 8 s3 22.67 s2 26.67 s gtgt Wn,zeta,P
damp(Gp) Wn 0 2.5820 2.5820
4.0000 zeta -1.0000 0.7746 0.7746
1.0000 P 0 -2.0000 1.6330i
-2.0000 - 1.6330i -4.0000
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Transfer Functions (TFs)

• Use of Transfer Functions forTime-domain
  Responses

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The unit step response
step(SYS) plots the unit step response of the
LTI model SYS (created with either tf, zpk, or
ss)
from prof. Pribeiro
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The impulse response
impulse(SYS) plots the impulse response of the
LTI model SYS (created with either tf, zpk, or
ss)
from prof. Pribeiro