% i
clc;
clear;
syms x z;
tspan = [0,10];      % Visualise solution from 0 to 5 seconds
x0 = [1,1];             % Initial condition of 1

L = 3;
kappa = 2;
rho = L + kappa/sqrt(2);
sigma(x,z) = x^2 + 4*x - 0.5*z;
v(x,z) = -rho*sign(sigma(x,z));
u(x,z) = v(x,z) - 2*x^3 - 10*x^2 - 12*x + x*z + 2*z;
u_func = matlabFunction(u(x,z));
[t,x] = ode45(@(t,x) f_1(x,t,u_func),tspan,x0);    % Solve for x in the un-perturbed system

x_surface = linspace(0,2,10);
z_surface = 2*x_surface^2 + 8*x_surface;

% Visualise solutions
figure('Name','i_1','NumberTitle','off','Position', [100, 100, 700, 400]);
subplot(1,2,1);
plot(x(:,1),x(:,2),'Linewidth',2);
title('x(t), z(t)');
xlabel('x(t)');
ylabel('z(t)');

subplot(1,2,2);
plot(x_surface,z_surface,'Linewidth',2);
title('Sliding surface sigma = 0');
xlabel('x(t)');
ylabel('z(t)');
%

figure('Name','i_2','NumberTitle','off','Position', [100, 100, 700, 400]);
subplot(1,3,1);
plot(t,x(:,1),'Linewidth',2);
title('t, x(t)');
xlabel('Time (s)');
ylabel('x(t)');
xlim(tspan);

subplot(1,3,2);
plot(t,x(:,2),'Linewidth',2);
title('t, x(t)');
xlabel('Time (s)');
ylabel('x(t)');
xlim(tspan);

subplot(1,3,3);
plot(t,sigma(x(:,1),x(:,2)),'Linewidth',2);
title('t, x(t)');
xlabel('Time (s)');
ylabel('x(t)');
xlim(tspan);

%%
function dX = f_1(x,t,u_func)
u = u_func(x(1),x(2));
dx = x(1)^2 + 3*x(1) - 0.5*x(2);
dz = -2*u + cos(t) + 2*sin(3*x(1)*x(2));
dX = [dx;dz];
end