Question

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

G(s) = C(s)/N(s) = s³ + 9s² + 26s + 24 / s⁴ + 15s³ + 77s² + 153s + 90

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

G(s) = (s + a1)(s + a2)(s + a3)/
(s + b1)(s + b2)(s + b3)(s + b4)

where ai and bi = the roots of the numerator and denominator, respectively.

Answer 1

To find the roots of the numerator and denominator of a transfer function, we can use factoring. The question is regarding the factorization of a transfer function's numerator and denominator into their root polynomials in control systems engineering.

Step-by-step explanation:

To find the roots of the numerator and denominator of the given transfer function, we can use a numerical technique like factoring. The transfer function given is G(s) = (s³ + 9s² + 26s + 24) / (s⁴ + 15s³ + 77s² + 153s + 90). We can factor the numerator and denominator to get the transfer function in the desired form: G(s) = (s + a1)(s + a2)(s + a3) / (s + b1)(s + b2)(s + b3)(s + b4).

The question is regarding the factorization of a transfer function's numerator and denominator into their root polynomials in control systems engineering. Numerical techniques and software tools such as MATLAB or Python can be used to find the roots, which will then be used to express the function in a factorized form.

The question involves finding the roots of the numerator and denominator of a transfer function in control systems analysis. The transfer function given is G(s) = s³ + 9s² + 26s + 24 / s´ + 15s³ + 77s² + 153s + 90. To find the roots numerically, software tools such as MATLAB or Python libraries like NumPy may be used, as these calculations can be complex and time-consuming by hand.

Once the roots are determined, they are denoted as ¡ and ¢ for the numerator and as £, ¤, ¥, and ¦ for the denominator. The resulting factorized transfer function will thus be in the form of G(s) = (s + a1)(s + a2)(s + a3) / (s + b1)(s + b2)(s + b3)(s + b4), with each ai and bi representing the roots of the numerator and denominator, respectively.

Practical applications of such an analysis may involve designing controllers that can ensure the robotic positioning system behaves as intended, providing particular attention to stability and response times.