MATLAB Root Locus Stability Analysis

Stability analysis determines the range of controller gains that are expected to create a stable response. The stable response can either reach a new steady state with oscillations or no oscillations. A root locus plot is a quick way to show stability versus instability and any expected oscillations.

[@AJ-et3vf]

Thank you very much for this sir. Very helpful

[@fatimamasroor4904]

Cool.!really great explanation

[@amr.a-m8350]

Good video for Simulink the result on the scope when increasing from 10 to 300 sec shown on your video, this will give us sinewave with increasing or decreasing according to the gain of PID please why this occurred.

[@apm]

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

[@xursssulaxx]

can you use taylor approximation instead of padé if you wanted?

[@apm]

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

[@TobiTobeToby]

God bless you for this video!

[@ShahFahad-hj1ps]

very good tutorial

[@sushibaitalmal9257]

hello, I'm solving a problem (without using MATLAB) to find the gain at a specific point (s=1+3i) , but as I substitute the value of 's' in the characteristic eq. (S^3+7S^2+7S-15+K=0) , K comes up to be (90+45i) , but I never seen gain with a complex number, So do I take the magnitude or what ? ? ' _ '

[@apm]

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.

[@sushibaitalmal9257]

@@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 !?

[@apm]

@@sushibaitalmal9257 The book is likely talking about frequency analysis and maybe about Bode plots. A good resource is Seborg, Edgar, Mellichamp, and Doyle or KZbin videos by Brian Douglas or Steven Brunton. Here is a session about learning resources: kzbin.info/www/bejne/ZmaadqlsmLB_rac

[@KIMIRAIKKONE198N4]

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

[@TheLordsForce]

When designing compensators using the root locus, I usually will find min and max values of sigma, Beta and wd and then use them to find regions on the s plane where the closed loop poles need to fall within. Is there a way to plot additional lines and/or highlight regions of the root locus plot in Matlab?

[@apm]

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.

[@apm]

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.