Ch. Rami Reddy

But for non linear controllers like fuzzy controllers is it possible to plot a root loci ??(even though matlab linear analysis was plotting poles and zeros)

You would have to linearize the system around an operating point. The poles and zero plot that you will get will only be for the linearized system around that operating point.

Hi Joseph Jose,

From your question, I guess you want to perform a stability analysis on the fuzzy control system.

In linear control design, the Root Locus is a graphical method to show all possible closed-loop pole locations and the root loci when the loop gain parameter is varied. Since a fuzzy control system is inherently nonlinear, it would be pretty difficult to plot the Root Locus.

But you mentioned that the system (plant) has poles in the right-half plane. So I guess the plant is linear and I think you maybe able to plot the locations of the closed-loop poles. Consider a double integrator system m·x" = u, where it can be stabilized with a typical PD controller, u = − Kd·x' − Kp·x.

If the mass m = 1 and Kp is fixed at 1, then the closed-loop transfer function is given by 1/(s2 + Kd·s + 1). If the gain Kd is varied by a fuzzy logic system that has a range of 1 ≤ Kd ≤ 3, then the locations of the closed-loop poles can be computed and plotted (see figure above).

Note that this not a Root Locus plot because each pair of poles will have its own root loci. However, the locations of the closed-loop poles can tell us the behavior of the fuzzy control system.

• If 1 ≤ Kd < 2, then the closed-loop system is under-damped.
• If Kd = 2, then the closed-loop system is critically-damped.
• If 2 < Kd ≤ 3, then the closed-loop system is over-damped.

thank you Yew-Chung Chak for the long descriptive answer

the system in highly non linear but i linearized it around some operating points for root locus analysis (using simulink linear analysis trim option) some poles were at right half of the plane at a certain operating point (say t=5sec and input=50% step)

Now, If i want to compare the stability of the fuzzy controller with the model with no control model,(at the above mentioned operating point) i will have to check for the stability of the system with all possible values of Kp and Kd (its a PD controller) from (Kp(min) to Kp(max) and Kd(min) to Kd(max)) and compare its pole location with the one with no control. isn't it ?

Hi Joseph Jose,

It is unnecessary to compare with the unstable nonlinear plant model if you want to prove the stability of the fuzzy control system. Since the plant model is nonlinear, it would be extremely tedious to linearize it at all operating points.

For example, if there are 21 points on the operating range -1 ≤ x ≤ 1 and 21 points on -1 ≤ x' ≤ 1, then you need 441 linearized models. On top of that, you need to compute the locations of the closed-loop poles for all possible values of Kp and Kd in each linearized model.

A relatively simpler approach is to use the Lyapunov method to show that the stability holds for the designed fuzzy PD controller. Pick a small value of gamma (γ > 0+) and plot the right hand side of the last inequality (I'll call this the Lyapunov surface). Then, use 'gensurf' to generate the control surface of the ufuzzy and compare it with the Lyapunov surface.

If the inequality holds, then the asymptotic stability holds. This graphical method is explained in the book, "Fuzzy Control" by Kevin M. Passino and Stephan Yurkovich.

Excellent answers