How do I plot the control signal "u" ?

Given a 2nd-order system:

x'' = u; subject to x(0)=1; x'(0)=0;

where "u" is a piecewise function given by

if |x| < 0.2, then u= - 0.5x' - x

if |x| > 0.2, then u= - 2.0x' - x

related matlab algorithm is

% calling ode45
t0 = 0 ;
tf = 20 ;
tspan = [t0 tf];
x0 = [1 0];

[t,x] = ode45(@(t,x) ode(t,x), tspan, x0);

figure(1)
plot(t, x(:,1), t, x(:,2))

% function file
function dx = ode(t,x)
dx = zeros(2,1);
dx(1) = x(2);
if abs(x(1)) < 0.2
    u = - 0.5*x(2) - x(1);
else
    u = - 2.0*x(2) - x(1);
end

dx(2) = u;

end

I want to plot “u“ the control signal, but I'm unable to plot this control signal "u". Anyone can help me please if you can. I already attach matlab code (Question.m) for better understanding.

Thank you.

Answer 1

[Sam Chak]

One workaround to plot the piecewise control signal u is to introduce an additional state into the system, where is the actuator. Technically, the piecewise control signal can be approximated by the actuator, which can be modeled as a 1st-order system.

where τ determines how fast you want the actuator to approximate the control signal u in seconds. So, if you want to catch up with u in 0.01 sec, then you would want to select . Here are the changes that you need to apply to the codes:

x0 = [1; 0; 0];
dx = zeros(3, 1);
dx(2) = x(3);
dx(3) = (-x(3) + u)/tau;