How do I write a delayed differential equation ?

Hi everyone,

I am currently trying to implement the following delayed differential equation:

y_dot(t)=K*yc(t-L)-(1/T)*y(t)

with yc the input of the associated transfer function and y the output.

My problem is how do I write this equation using a solver like ode ?

I saw the solver dde but it seems to be useful if my delay is in y but here it's in the input yc..

I also tried to rewrite like y_dot(t+L)=K*yc(t)-(1/T)*y(t+L) but I have the same issue writing it in ODE..

Thanks in advance

[Torsten]

Given t, can't you just evaluate yc(t-L) (e.g. by interpolation) ? Or is yc not explicitly given ?

Best wishes

Torsten.

[Vincent DUFOUR]

Actually, the yc is the command of a motor and the transfer function represent the dynamic of the motor (there is a dead time + a first order TF) so y is the real behaviour of the motor in respond to the order yc.

At each time t I have the order yc. So far I just solve the following equation:

y_dot(t)=K*yc(t)-(1/T)*y(t) and I give in input of the function the yc(t-L), it does work but I'd like to know if there is another way to do it or not ?

Because after that I want to correct it using PID etc..

Answer 1

[Shishir Reddy]

Hi Vincent,

To solve a delayed differential equation (DDE) in MATLAB where the delay is in the input signal yc(t−L) rather than in the state y(t−L) you can still use MATLAB's ‘dde23’ solver. However, the key point is that ‘dde23’ allows delays in any function used in the equation, not just the state variable.

Kindly refer the to the following steps to understand how this can be implemented in MATLAB using ‘dde23’.

1. Define the parameters and DDE as a function –

L = 1; % example delay value
yc = @(t) sin(t); % example input
K = 2;    % gain
T = 5;    % time constant
ddeFunc = @(t, y, Z) K * yc(t - L) - (1/T) * y;

2. Set history and time span.

y0 = 0; % initial condition
history = @(t) y0;
tspan = [0, 20];

3. Solve using ‘dde23’ and plot

sol = dde23(ddeFunc, L, history, tspan);
plot(sol.x, sol.y)
xlabel('Time t')
ylabel('Output y(t)')
title('Response of the DDE with delayed input').

For more information regarding the ‘dde23’ solver, kindly refer the following documentation - https://www.mathworks.com/help/matlab/ref/dde23.html

I hope this helps.