Calculate K for best settling time?

I'm practising on compensators and the root locus. A very common problem in my textbook is to find the value of a variable for the minimum settling time of the system.

I have no trouble drawing the root locus for the variable but I can't decide on the right value. I only found a formula for second order systems but that doesn't cover most of the cases.

I also tried inverse transforming to find a result in the time domain but I usually have an exponentially decaying component multiplied by a sinusoid and it gets more complicated.

So , what's a general technique to calculate settling time ?

• Root locus is a way of plotting the poles as gain (usually) is varied and, for a 2nd order system it's easy to find the best pole positions for a given target response. Are you applying it to higher order systems or, ultimately are you just trying to find the damping ratio that gives you the overshoot you want? I'm unclear what you are trying to do. – Andy aka Jul 18 '17 at 12:56
• Settling time is defined as "the time required for the response curve to reach and stay within a range of certain percentage (usually 5% or 2%) of the final value". For second order systems it's approximately equal to 4/ζω. My question is how do I find the values of the parameter that will give me the settling time I'm aiming for for any order system. – John Katsantas Jul 18 '17 at 14:50
• For system other than a standard 1st or 2nd order the response equation must be solved. Even if a 2nd order system has a finite zero, the $\frac{4}{\zeta \omega _n}$ ROT doesn't work. – Chu Jul 18 '17 at 15:10