7.19

In control systems analysis, transfer functions are developed that mathematically relate the dynamics of a system's input to its output. A transfer function for robotic positioning system is given by

G(s) = C(s)/N(s) = s^3 +12.5 *s^2 + 50.5 *s +66/ s^4 + 19*s^3 +122*s^2 +296 * s +192

where G(s) = system gain, C(s) = system output, N(s) = system input, and s = Laplace transform complex frequency. Use a numerical technique to find the roots of the numerator and denominator and factor these into the form

G(s) = (s + a_1) (s + a_2) (s + a_3) / (s + b_1)(s + b_2)(s + b_3)(s + b_4)

where a_i and b_i = the roots of the numerator and denominator, respectively.

%roots(s^3 +12.5 *s^2 + 50.5 *s +66)

x = roots([1 12.5 50.5 66]);

%roots(s^4 + 19*s^3 +122*s^2 +296 * s +192)

y = roots([1 19 122 296 192]);

This results in 

EDU>> x


x =


   -5.5000
   -4.0000
   -3.0000


EDU>> y


y =


   -8.0000
   -6.0000
   -4.0000
   -1.0000


And so, 



G(s) =   (s + 5.5)  (s + 4) (s + 3)  / (s + 8)  (s + 6)  (s + 4)  (s + 1)