Page 846 - Mechatronics with Experiments

832 MECHATRONICS
A.1.6 Controller Design Functions: Transform Domain
and State-Space Methods

A transfer function is defined with numerator and denominator coefficients. A transfer
function description can be converted between three different ways of expressing it: (i)
numerator and denominator polynomials form, (ii) poles-zeros and gain form, (iii) partial
fraction expansion form (poles, residues, and DC gain).

num(s)
H(s) = ———————
den(s)

(s-z1)(s-z2)...(s-zn)
H(s) = K ——————————————————————
(s-p1)(s-p2)...(s-pn)

num(s) R(1) R(2) R(n)
- ——— = ———————— + ———————- + ... + ———————– + K(s)
den(s) s - P(1) s - P(2) s - P(n)

These functions are applied in s-domain as well as z-domain transfer functions.
numG = [1]; % Numerator coeficients
% of the transfer function.
denG = [ 1 2 4] ; % Denominator ......
% ...........................
Gs = tf(numG,denG) ; % Gs defines the transfer funtion.

[z,p,k] = tf2zp(numG, denG); % convert to zero-pole-gain form
% from the polynomial num(s)/den(s)
form
[numG, denG] = zp2tf(z,p,k); % do the reverse

[r,p,k] = residue(numG, denG) ; % convert from num(s)/den(s) form
% to Partial Fraction Expansion form

[num, den] = residue(r,p,k) ; % convert from partial fraction
expansion form
% to numerator and denominator
polynomial form

p = pole(Gs) ; % Given G(s), get poles,
[z, k] = zero(Gs) ; % .......... get zeros and gain
[num,den]=zp2tf(z,p,k) ; % obtain the num(s)/den(s) form
If we define complex variables “s” and “z” as follows for Laplace and Z-transforms, then
we can express the transfer functions in symbolic form,

s = TF(’s’) ; % specifies the transfer function H(s) = s
(Laplace variable).
z = TF(’z’,TS) ; % specifies H(z) = z with sample time TS.
You can then specify transfer functions directly as rational expressions in s or z, for example,

s = tf(’s’);
G = (s+1)/(sˆ2+3∗s+1) ;
z = tf(’z’);
H = (z+1)/(zˆ2+4∗z +2) ;