Finding range of values for gain for which dominant time constant of stable system is less than 0.2s

Question

I have a system with transfer function W(s) = K * 1/(s*(s+2)^2). Based on its root locus, I need to find range of values for gain K for which dominant time constant of stable system is less than 0.2s.

Answer 1

Your transfer function is $$W(s) = \frac{K}{s(s+2)^2},$$ which is a product of a first and second order system. If this is the TF whose timeconstant you want to modify, $K$ is not going to do anything because $K$ is a gain constant. If this is an open-loop transfer function which has unity, inverting feedback then the closed-loop transfer function is $$H_{CL}(s) = \frac{W(s)}{1+W(s)} = \frac{K}{s(s+2)^2+K},$$ and the timeconstants of the closed loop system are indeed functions of $K$. You now have a nontrivial 3rd-order chraracteristic equation $$C_{CL}(s) = s^3 + 4s^2+4s +K.$$ The root locus of this equation will give you the range of system parameters which are achievable with the feedback and compensation you have selected.

Now: there's no such thing as a "dominant time constant". Timeconstants for different parts of the response just give you an idea of the amount of time it takes for that part of the transient response to settle. You are probably thinking of dominant poles, which is a way of referring to poles which are close to the imaginary axis. If you write down the definition of a first/second order system and solve for the poles, you will see this implies the timeconstants for the dominant poles are typically the largest--e.g. the response time for dominant poles is the slowest, which means the transients for these parts of the response are the longest lasting. This is actually a justification for making dominant pole approximations of systems.

To solve your problem (at least how I've interpreted it), you need to plot the root locus and find the $K$ which produce roots of $C_{CL}$ with a timeconstant less than $0.2$s. MATLAB has built in code like SISOtool to do this, but personally I would simply write a script which, given a range of $K$, finds the roots of $C_{CL}(s)$ for each $K$ and then determines (1) the maximum timeconstant and (2) whether or not it satisfies the criteria. $K$ which pass this test form the solution set to the problem.