Simulink Constant Ramp Controller

Hi all,

In order to explain my problem, I've attached a Simulink model to this post. It consists of a classic PID control loop which calculates the response of the system to a signal step at the input. What I am looking for is a controller to replace the PID, which guarantees a constant slope over the entire step response, ideally without any remaining controll error. I've already tried different approaches, none of them seem to work reliably. Any ideas?:)

Cheers

Alex

[Nikhil]

Can you explain what you mean by ' a constant slope over the entire step response ' ?

[Alexander Reiter]

Does this help?

Answer 1

[Sam Chak]

Hi @Alexander Reiter

I didn't check the Simulink model. However, the desired response can be achieved with a nonlinear controller. See example below.

tspan  = linspace(0, 4, 10001);
x0     = 0;
opts   = odeset('RelTol', 1e-12, 'AbsTol', 1e-9);
[t, x] = ode45(@odefcn, tspan, x0, opts);

% Solution Plot
plot(t, x, 'linewidth', 1.5), hold on,
plot(t, heaviside(t - 1), 'r--'), grid on, 
xlabel('t'), ylabel('x(t)')
legend('output', 'input', 'location', 'east')

function xdot = odefcn(t, x)
    k    = 1;                       % parameter that determines the strength of u
    g    = 1000;                    % parameter that determines steepness of u
    ref  = heaviside(t - 1);        % reference input to be tracked
    
    % controller
    u    = - k*tanh(g*(x - ref));   % nonlinear controller
    
    % system
    xdot = u;       % equivalent to the plant transfer function, Gp(s) = 1/s
end

[Alexander Reiter]

Hi Sam,

thanks a lot, this works perfectly!

Answer 2

[Nikhil]

Yes, It helps. You can derive the transfer function for the controller since you know the transfer function for the system and the desired output from the equation:

where Y(s) is the output, which would be for a signal of constant slope m,

G(s) is C(s)*P(s), where P(s) is the system tranfer function, which in your case is as your system is an integrator,

H(s) is the feedback transfer function, which is 1,

U(s) is the input signal, which would be for the case of a step input.

The controller transfer function would then turn out to be

You can then use a Transfer Function block to implement the controller.

[Alexander Reiter]

Lovely! That works just perfectly! Thank you very much!

[Alexander Reiter]

Hi @Nikhil,

just one more question: While the transfer function produces a perfect ramp response, I won't stop after reaching the height of the step. Is there any chance to force the system to stop, as soon as the control deviation becomes zero? Check the attached file for the Simulink model.

Cheers

Alex

[Nikhil]

You can limit the output of the integrator block by enabling Limit output option.