Plot Root Locus for PD-Controller

Hi!

I'm looking to plot the Root Locus for my system.

With G =

G = tf([2],[1 2 1 0]);
G =
 
         2
  ---------------
  s^3 + 2 s^2 + s
 
Continuous-time transfer function.

I want to apply a D-regulator, i.e .

How do I use the rlocus plot to draw with respect to as my parameter and not a P-regulator which is standard?

Muchos gracias

Answer 1

[Jon]

I don't think you can do it directly with the rlocus command which is just for a variable overall open loop gain.

One way to do it is to make your own loop something like this (here I just put in some numerical values to give you the idea you will have to adapt to the details of your situation)

% proportional gain
kp = 3

% derivative filter constant
tauf = 20

% desired range of derivative gain
kd = linspace(0,30,100); % put your range in here

% define plant transfer function
G = tf([2],[1 2 1 0])

% loop to find poles (root locus) as a function of derivative gain
numPoints = length(kd) % number of points to evaluate on root locus

for k = 1:numPoints
    
    % form open loop transfer function for plant and controller
    Gol = (kp + kd(k)*s/(tauf*s + 1))*G;
    
    % find poles and zeros
    p = pole(Gol);
    z = zero(Gol);
    
    % plot the root locus points, need to add appropriate labels etc
    hold on
    plot(real(z),imag(z),'o')
    plot(real(p),imag(p),'*')
    
end

hold off

[Jon]

Sorry in the above I plot the open loop poles and zeros, you probably want the closed loop poles and zeros so

% proportional gain
kp = 3

% derivative filter constant
tauf = 20

% desired range of derivative gain
kd = linspace(0,30,100); % put your range in here

% define plant transfer function
G = tf([2],[1 2 1 0])

% loop to find poles (root locus) as a function of derivative gain
numPoints = length(kd) % number of points to evaluate on root locus

for k = 1:numPoints
    
    % form open loop transfer function for plant and controller
    Gol = (kp + kd(k)*s/(tauf*s + 1))*G;
    
    % form the closed loop transfer function
    Gcl = Gol/(1 + Gol);
    
    % find poles and zeros
    p = pole(Gcl);
    z = zero(Gcl);
    
    % plot the root locus points, need to add appropriate labels etc
    hold on
    plot(real(z),imag(z),'o')
    plot(real(p),imag(p),'*')
    
end

hold off

[Jon]

Also, in some cases you may be able to use the standard rlocus command rather than making your own loop if you are able to factor out the parameter of interest so it appears as an overall loop gain. I don't think that is possible though if you are interested in just the derivative gain, or derivative filter time constant.

Finally, in the above I assume you are implementing the derivative term using a realizable transfer function such as kd*s/(tauf*s + 1). For more theoretical work you could just use kd*s

[Oskar Mevik Päts]

Thanks for the answer.

I know how to do it the manual way, I was just wondering if there was any smart way to do it rlocus function.

[Jon]

From what I can determine from the documentation for rlocus can only be used when you can factor out the parameter of interest so that it can be applied as a multiplicative constant (overall loop gain). Otherwise you will have to make your own plot, for example as I show in my script above. It sounds like you already know how to do this though. If this answers your question please mark it as answered so others will know.