MATLAB Root Locus Stability Analysis 

The controller may be right at the boundary between stability and instability if there is a persistent oscillation.

You could do that, but it would lead to a potentially larger inverse response. The first two terms would be with (1-theta*s) in the numerator. The Taylor series approximation is more accurate with additional terms so it may improve if you used additional terms. There is also a higher-order Pade approximation that is likely more accurate. Here is additional information on linearization: apmonitor.com/pdc/index.php/Main/ModelLinearization

You can generate your own Root Locus plot with commands such as: s = tf('s'); G = 1/(s^3+2*s^2+s+1); % define transfer function K = linspace(-10,100,1000); % specify gain values [R,K] = rlocus(G,K); % compute root locus points (R is complex) plot(real(R),imag(R),'r.') % root locus plot Then you can add other elements to the plot. You could also try to add elements to the rlocus plot that MATLAB produces but I've had trouble doing that.

I'm not sure if there is a way to add to the default rlocus plot in MATLAB. Here is a customized version in Python: apmonitor.com/pdc/index.php/Main/StabilityAnalysis It may be easier here.

You shouldn't have a gain with an imaginary number. There is no gain with that root. Here is information on stability analysis: apmonitor.com/pdc/index.php/Main/StabilityAnalysis that is related to your question. I recommend that you create a root locus plot to determine the roots (solutions to the equation) that correspond to different gains.

@@apmThank you for the answer , Im really trying to find the gain at the imaginary axis, but in some books it is mentioned that we should substitute s=jw in ch. eq. to find the gain at the imaginary axis, or at least thats how are teacher's book says, and I cant find anything explaining how its done, I just end up with an odd complex No., not really knowing what Iam doing, So is this same as RH method explained in the link or am I just completely lost !?

@@sushibaitalmal9257 The book is likely talking about frequency analysis and maybe about Bode plots. A good resource is Seborg, Edgar, Mellichamp, and Doyle or RU-vid videos by Brian Douglas or Steven Brunton. Here is a session about learning resources: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-057Ev6cKLwE.html

redo your maths. C.E. -----> 1 + K.G(s).H(s) = 0 then K = magnitude of { (1) ÷ [G(s).H(s)] }. you should always have positive real number