# Simulating PID controller in Matlab

I am trying to explore PID control with a simulation. My problem is that the derivative term does not behave as expected in my simulation: No matter how large its coefficient it seems unable to prevent overshoots.

I tried to tune the system with the Ziegler-Nichols method and using the parameters the should produce "no overshoot" according to the Wikipedia entry of the Ziegler-Nichols method.

My "process" is simply the 5-sample exponential moving average of the PID controller output. Controller output is limited to [-1,1]

Can anybody see what the problem with this code is:

PVset = 0.5;            % set point for process variable

steps = 100;          % time periods/steps to simulate

PV = 0;               % process variable (time series)
error = 0;            % error in PV (time series)
der = 0;              % derivative of error (time series)

Ku = 1;
Tu = 2;

Kp = Ku*.2;
Ki = 2*Kp/Tu;
Kd = Kp*Tu/3;

for k=2:steps
% calculate error
error(end+1) = PVset-PV(end);
% P term
delta = Kp*error(end);
% I term
delta = Ki*sum(error) + delta;
% D term
% instantaneous derivative time series
der = error(2:end)-error(1:end-1);
% average derivative over x periods
per = 4;
if k >= per+1
avgDer = mean(der(end-(per-1):end));
delta = Kd*avgDer+delta;
end
% limit PID controller output to -1 -> 1
if abs(delta) > 1
delta = sign(delta);
end
% simulate process behaviour
p = 5;
PV(k+1) = (p-1)/p*PV(k-1) + 1/p*delta;
end

figure
hold on
plot((0:steps), PV)
plot((1:steps), error, 'r')


Tu=8