Here is a proportional control system.

The root locus for this system is shown below.

• Determine the transfer function, G(s), in the system above.  If there are any constants which can not be identified numerically, indicate those constants with symbols.  Use usual symbols, i.e. K for gain, p for a pole, z for a zero, etc.  Express the transfer function in terms of a ratio of factors.
• Draw arrows on every portion of the root locus to  indicate the direction of increasing gain.  If there are points missing, fill in the missing portion of the locus.
• Four closed loop poles are shown for the same gain.  What is the root locus gain that produces the poles that are shown?  The root locus gain is the constant in the transfer function when the transfer function is expressed as a ratio of polynomials with leading coefficients of 1.
• What is the damping ratio for the four poles shown?
• Will this system become unstable.  If so, why?  If not, why not?

• This system has four poles at -1, -3, -7 and -8, so the denominator polynomial is s⁴ + 19s³ + 110s² + 269s + 168.
• This system has two zeros at -4 + j2 and - 4 -j2, so the numerator polynomial is s² + 8s + 20.
• There is a possibility of a non-unity constant, Kp, so the transfer function is:
• K(s² + 8s + 20)/(s⁴ + 19s³ + 110s² + 269s + 168) (or you can leave it factored!)
• The root locus gain is 35.
• The damping ratio for the poles closer to the origin is 0.776.
• The damping ratio for the poles farther from the origin is 0.702.

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