How to get PD parameters

Hi

I've a question regarding PD controller in order to enhance a liquid level control system in a tank. It is required to ensure the steady state error in the liquid level in the tank when it undergoes a step change is always less than ±0.5% (0.005). The settling time and overshoot of the system response should also be less than 1.5 seconds and 5% respectively.

what would be the appropriate parameters of PD. I tried a lot to find by changing the parameters of PD but unfortunately no improvement.

below is the code:

s = tf('s');

The Transfer functions of the components

Gp = 2 / (0.2 * s + 1); %pump

Gv = (2.2 * 10^-3) * 2; %valve

Gt = 24.2 / (3.3 * s + 1); %tank

Gl = 1; % level sensor

The Open loop transfer function

oltf = Gp * Gv * Gt * Gl;

%PD controller parameters

kP = 6; % Start with an initial guess for kP

TD = 0.2; % Start with an initial guess for TD

% PD controller transfer function

PD = kP * (1 + TD * s);

% Closed loop transfer function with PD

CLTF_PD = (PD * oltf) / (1 + PD * oltf);

Answer 1

[Paul]

Hi Hasan,

Let's take a look at the code.

s = tf('s');

Gp = 2 / (0.2 * s + 1); %pump
Gv = (2.2 * 10^-3) * 2; %valve
Gt = 24.2 / (3.3 * s + 1); %tank
Gl = 1; % level sensor

oltf = Gp * Gv * Gt * Gl;
%PD controller parameters
kP = 6; % Start with an initial guess for kP
TD = 0.2; % Start with an initial guess for TD
% PD controller transfer function
PD = kP * (1 + TD * s);

Assuming that the goal is to control the true level and that the level sensor is in the feedback path, the closed-loop transfer function is:

H(s) = PD(s)*Gp(s)*Gv(s)*Gt(s)/( 1 + PD(s)*Gp(s)*Gv(s)*Gt(s)*Gl(s) )

So this line should really be:

% Closed loop transfer function with PD
CLTF_PD = (PD * Gp * Gv * Gt) / (1 + PD * Gp * Gv * Gt * Gl);

Of course, with Gl(s) = 1 it doesn't really matter, but maybe you're going to change Gl(s) later. Also, transfer function algebra in Matlab is discouraged. Instead, better to get in the habit of using interconnection commands, such as feedback

CLTF_PD = feedback(PD * Gp * Gv * Gt, Gl);

Step response

figure
stepplot(CLTF_PD)

We see that the steady state error is quite a bit larger (~0.45) than the requirement (0.005), so let's start with that.

What is the expression for the final value of the output , in terms of H(s), in response to a step input ? Or alternatively, what is the expression for the final value of the error, e(t) = r(t) - y(t), in terms of H(s), when r(t) is a step input ?