Chapter 7, Problem 19P

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 a robotic positioning system is given by

$G\left(s\right)=\frac{C\left(s\right)}{N\left(s\right)}=\frac{s^3+9s^2+26s+24}{s^4+15s^2+77s^2+153s+90}$

where G(s) 5 system gain, C(s) 5 system output, N(s) 5 system input, and s 5 Laplace transform complex frequency. Use a numerical technique to find the roots of the numerator and denominator and factor these into the form

$G\left(s\right)=\frac{\left(s+a_1\right)\left(s+a_2\right)\left(s+a_3\right)}{\left(s+b_1\right)\left(s+b_2\right)\left(s+b_3\right)\left(s+b_4\right)}$

Where $a_i{\text{ and b}}_i=$ the roots of the numerator and denominator, respectively.