Control barrier functions basic example in 1d

Hi,

I'm trying to program a basic example of a control barrier functions, for a system of 1D. The idea is that this system should not go for the negatives values .

The dynamic of my system is defined by ,

The function h that is the constinst restriction is given by , such that

The enforce condition is given by,

The question is that I don't know how to program this example, I tried with the assumption that

But my code is not working.

This is my code

%% CASE: x \in R, x can only take positive values
%time interval
t_int = 0:0.001:20;
%initial condition
x_0 = 10;
b = 0.00000001;
%differential eqation
dxdt = @(t, x) -1 - x;
[t, x] = ode45(dxdt, t_int, x_0);
% Graficar la solución
figure;
plot(t, x);
xlabel('Time');
ylabel('x(t)');
grid on;

Can someone help me?

[Torsten]

I'm completely unable to follow what you try to do.

[William Rose]

@Hortencia Alejandra Ramírez Vázquez,

I agree with @Torsten. I am unable to understand the dynamics of hte system you wish to simulate. You define b in your code but uyou do not use b in the differential equation, and you do not refer to b in your explanation.

The differential equation you use in your code

dx/dt=-x-1

is a decaying exponential which decays to x=-1, with time constant=1.

%% CASE: x \in R, x can only take positive values
%time interval
tSpan = [0,20];
%initial condition
x_0 = 10;
b = 0.00000001;  % not used
%differential eqation
dxdt = @(t, x) -1 - x;
[t, x] = ode45(dxdt, tSpan, x_0);
% Graficar la solución
figure;
plot(t, x);
xlabel('Time');
ylabel('x(t)');
grid on;

Answer 1

[Altaïr]

Hey @Hortencia Alejandra Ramírez Vázquez,

From the equations mentioned in the query, it seems the condition for the system is incorrectly stated as , and should instead be , where is a class-K function. Additionally, the equation , suggests that the class-K function is chosen to be .

To ensure the condition on the control input, u can be chosen as , where ρ is a positive constant. However, the below line in the shared code suggests that the constant is chosen as which is contradictory.

dxdt = @(t, x) -1 - x;

To resolve the issue, replace ‘-1’ in the above line with some positive constant. Here’s a plot showing the system response for different values of ρ.

Kindly refer the following link to read more about Barrier Certificate Enforcement:

https://www.mathworks.com/help/slcontrol/ug/barrier-certificate-enforcement-for-control-design.html