Nondifferentiable system

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This problem is taken from the PROPT manual, example 72.

$J = x_3(t_f)$ $\dot{x_1}(t) = x_2$ $\dot{x_2}(t) = -x_1-x_2+u+d$ $\dot{x_3}(t) = 5 x_1^2 + 2.5 x_2^2 + 0.5 u^2$ $d = 100 \cdot [U(t-0.5)-U(t-0.6)]$

Where U is the unit step function so that $U(t-\alpha) = 0$ when $t < \alpha$ and $U(t-\alpha) = 0$ when $t > \alpha$ .

$x_1(0)=0$ $x_2(0)=0$ $x_3(0)=0$

Solving the problem

This problem introduces how to solve nondifferentiable systems. The problem can be solved by using phases where the model is altered when the unit steps get triggered. The problem can also be solved by using the casadi function if_else and thus modeling the unit step as well. By doing this only one phase is needed.

Main file

%% Nondifferentiable system
% Author: Dennis Edblom
sys = YopSystem(...
    'states', 3, ...
    'controls', 1, ...
    'model', @nonDiffModel);
% Symbolic variables
t = sys.t;
x = sys.x;
u = sys.u;

% Phase description
ocp = YopOcp();
ocp.min({ t_f( x(3) ) });
ocp.st(...
     'systems', sys, ...
     ... % Initial conditions
    { 0 '==' t_0(x(1)) }, ...
    { 0 '==' t_0(x(2)) }, ...
    { 0 '==' t_0(x(3)) }, ...
    ... % Final conditions
    { 2 '==' t_f( t )  } ...
    );

w0 = YopInitialGuess(...
    'signals', [t; x; u], ...
    'signalValues', [0; 0; 0; 0; -5] ...
    );

sol = ocp.solve(...
    'controlIntervals', 100, ...
    'collocationPoints', 'radau', ...
    'polynomialDegree', 3, ...
    'initialGuess', w0);

%% Plot the results
figure(1)
subplot(2,1,1)
sol.plot(t,x);
legend('x1','x2','x3');
title('Nondiff System state variable');

subplot(2,1,2)
sol.plot(t,u);
legend('u');
title('Nondiff System control');

%% Model
function dx = nonDiffModel(t, x, u)
% Nondifferentiable part
d = 100*(if_else(t > 0.5,1,0)-if_else(t > 0.6,1,0));

dx1 = x(2);
dx2 = -x(1)-x(2)+u+d;
dx3 = 5*x(1)^2+2.5*x(2)^2+0.5*u^2;
dx = [dx1;dx2;dx3];
end

Note: The model uses the CasADi function if_else to model $d = 100 \cdot [U(t-0.5)-U(t-0.6)]$ .