# How to find the compensating gain required for the 2nd order control system to have 5% overshoot

I have a unity feedback control system with $$G(s) = \frac{1.247}{s^2+9.76s+23.8}$$ in the forward path. Now Imagine, I added a static gain K in the forwards path to make the %overshoot 5%. How can I find the value of K? What if I added G'(s) instead of adding K where is $$G'(s) = \frac{K(s+0.1)}{s}$$What is the procedure to find it? Is there any way to do it using root locus plot?

• – jDAQ Apr 6 at 5:07
• I understood how to do it on paper just by algebra. But did not understand how to use root-locus plot to find gain. How is gain and damping factor represented on root-locus plot? – Michio Kaku Apr 6 at 5:40
• en.wikipedia.org/wiki/Root_locus "In addition to determining the stability of the system, the root locus can be used to design the damping ratio (ζ) and natural frequency (ωn) of a feedback system. Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arccosine whose center points coincide with the origin. By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency, ..." – jDAQ Apr 6 at 5:43
• Did you bother to read @jDAQ's linked answer? What other answer do you expect? – a concerned citizen Apr 6 at 6:25
• Introducing a PI controller, $\small G'(s)$, i.e. a zero, means that the CLTF cannot be expressed in standard 2nd order form (in fact, the CLTF is 3rd order). So the formulas for 2nd order characteristics such as % overshoot, settling time, time to first peak, etc... are inaccurate. With the forward path gain controller, $\small K$, only, express the CLTF in standard 2nd order form: $G(s)=\large \frac{K_{ss} \omega^2_n}{s^2+2\zeta\omega_ns+\omega^2_n}$, and look-up the value of $\zeta$ required for 5% overshoot. – Chu Apr 6 at 9:03