Guideline for implementing PID and saturation

I have the following model where I have as a setpoint a step input with constant value 10 and stepme 0 and used initially a step input of value 600 in order to approximate an impulse; which are now the PID and saturation block guidelines in order to successfully track the setpoint (R=10)? How can I achieve an oscillating behaviour around the setpoint?

[Sam Chak]

Hi @Jo

It appears that the blocks in the 2nd image form a nonlinear function, ​, but this does not correspond to any of the three commonly expressed Hill equations. The exponent n is not provided. While the 3rd-order transfer function is linear, its output is connected to the Hill equation block, rendering the closed-loop system nonlinear.

It is unclear whether you intend for the process variable to be regulated at a constant setpoint of 10 or for the response of the process variable to oscillate around the setpoint like a sine wave.

t   = linspace(0, 20, 2001)';
sp  = 10;
y1  = 10*heaviside(t);
y2  = y1/27.*(27 - 2*exp(-10*t) + 15*exp(-4*t) - 40*exp(-t));
y3  = y2 + sin(2*pi/5*t);
plot(t, [y1, y2, y3]), grid on
xlabel('t')
ylabel('y(t)')
legend({'Setpoint', 'Flat response', 'Ocillating response'}, 'Location', 'east')

●

[Sam Chak]

Hi @Jo, any updates?

[Jo]

@Sam Chak, I was looking for the process to be regulated at a constant setpoint of 10; I managed to do some gain scheeduling in order to track that setpoint

[Sam Chak]

Hi @Jo, Thanks for your clarification. What is the value in this Constant block?

[Jo]

@SamChak It's 1.8499, below it's the expression of the Hill' equation

Answer 1

[Sam Chak]

Hi @Jo

This system is somewhat difficult to control. I have not explored how you successfully track the setpoint using a gain-scheduling control approach. If a relatively slow response is acceptable, it may be possible to design a series of gain schedules. Would you mind sharing how you achieved that?

In my control design in MATLAB, I assume that all system states are measurable for simplicity. I have not tested it in Simulink, but in theory, it should work as well if the system states are properly estimated using an observer.

%% settings
format long g
sympref("HeavisideAtOrigin", 1);

%% call ode45
[t, x]  = ode45(@ode, [0 30], [1; 1; 1]);
y       = zeros(1, numel(t));
for j = 1:numel(t)
    [~, y(j)] = ode(t(j), x(j,:).');
end

%% plot result
figure
plot(t, y)
grid on
xlabel({'$t$'},    'interpreter', 'latex')
ylabel({'$y(t)$'}, 'interpreter', 'latex')
title({'System''s Output'}, 'interpreter', 'latex')

%% steady-state value
y(end)

ans = 
    10

%% Lur'e System - Nonlinear feedback with Linear Time-invariant System
function [dx, y] = ode(t, x)
    % Parameters
    wn      = 0.0307;
    a1      = 40*wn^3;
    a2      = 54*wn^2;
    a3      = 15*wn;
    B       = a1;
    gamma   = 1.8499;
    
    % Nonlinear Output, y(t)
    y       = 100/(abs(real(x(1)))^gamma + 1);
    dy      = - (100*gamma*abs(real(x(1)))^(gamma - 1))/(abs(real(x(1)))^gamma + 1)^2;
    d2y     =   (100*gamma*abs(real(x(1)))^(gamma - 2)*(abs(real(x(1)))^gamma + gamma*(abs(real(x(1)))^gamma - 1) + 1))/(abs(real(x(1)))^gamma + 1)^3;
    dx3     = - a1*x(1) - a2*x(2) - a3*x(3);
    yinv    = abs(100*abs(real(x(1))) - 1)^(1/gamma);
    
    % Controller
    k       = 1/1.000014303460806;
    ref     = 10;       % setpoint
    yref    = k*ref;
    dyref   = 0;
    d2yref  = 0;
    d3yref  = 0;
    % u     = heaviside(t);
    u       = (yinv*(d3yref - 3*(dy*x(2) + y*x(3) - d2yref) - 3*(y*x(2) - dyref) - (y - yref) - d2y*x(2) - 2*dy*x(3)) - dx3)/(2*B);
    
    % Linear System
    dx      = zeros(3, 1);
    dx(1)   = x(2);
    dx(2)   = x(3);
    dx(3)   = B*u - a1*x(1) - a2*x(2) - a3*x(3);
end

Answer 2

[Sam Chak]

Hi @Jo

I revisited your Lur'e system and designed a combined PID controller with a lead compensator in Simulink to regulate the nonlinear process variable at the desired setpoint.

Sometimes, I find myself clouded by the overkill math I derived in MATLAB. The graphical model in Simulink allows me to "see" where the problem lies, enabling me to attack its weak points.