The analysis shows that the constraint on control action is unnecessary. In Case 1, the control-free system (
) has two equilibrium points, with the stable one at the origin and the unstable one at 
. If the initial conditions are carefully selected such that 
and 
, then the trajectories are guaranteed to converge to the origin.
) has two equilibrium points, with the stable one at the origin and the unstable one at . If the initial conditions are carefully selected such that and , then the trajectories are guaranteed to converge to the origin.However, it is always desirable to find an easily implementable control action, denoted as u, such that the nonlinear system will converge to the equilibrium point from any possible starting non-equilibrium point. Since the control action is not provided, three types of control actions, 
, 
, and 
, in Case 2, are proposed, and the equilibrium at the origin is globally stable.
, , and , in Case 2, are proposed, and the equilibrium at the origin is globally stable.%% Finding equilibrium point
fun = @f; % function (refer to Nonlinear System)
x0 = [-2, 2]; % initial guess
z = fsolve(fun, x0)
[x1, x2] = meshgrid(-2:2);
delta = 1;
%% Case 1: Stream Plot of Control-free system
u = 0;
U = x1.*x2 - delta*x1;
V = - x1.*x2 - x2 + u;
figure(1)
l = streamslice(x1, x2, U, V); set(l, 'Color', '#c3829e');
xlabel({'$x_{1}$'}, 'interpreter', 'latex', 'fontsize', 12)
ylabel({'$x_{2}$'}, 'interpreter', 'latex', 'fontsize', 12)
title('Stream Plot of Control-free System')
axis tight
%% Case 2: Stream Plot of Control system
u1 = x1; % type-1 control action
u2 = x1 - x2; % type-2 control action
u3 = - (x1.^2 - x1.*x2); % type-3 control action
U = x1.*x2 - delta*x1;
V = - x1.*x2 - x2 + u3;
figure(2)
l = streamslice(x1, x2, U, V);
xlabel({'$x_{1}$'}, 'interpreter', 'latex', 'fontsize', 12)
ylabel({'$x_{2}$'}, 'interpreter', 'latex', 'fontsize', 12)
title('Stream Plot of Control System')
axis tight
%% Testing the system using Type-3 control action
tspan = [0 10];
x0 = [-2 2]; % divergent in the control-free system
[t, x] = ode45(@odefcn, tspan, x0);
figure(3)
plot(t, x, 'linewidth', 1.5), grid on
xlabel({'$t$'}, 'interpreter', 'latex', 'fontsize', 12)
ylabel({'$\mathbf{x}(t)$'}, 'interpreter', 'latex', 'fontsize', 12)
legend({'$x_{1}(t)$', '$x_{2}(t)$'}, 'interpreter', 'latex', 'fontsize', 14)
title({'Time responses under $u = - x_{1}^{2} + x_{1} x_{2}$'}, 'interpreter', 'latex', 'fontsize', 16)
%% Nonlinear System (for finding the equilibrium)
function y = f(x)
y = zeros(2, 1);
delta = 1;
y(1) = x(1)*x(2) - delta*x(1);
y(2) = - x(1)*x(2) - x(2);
end
%% Nonlinear ODEs
function dxdt = odefcn(t, x)
dxdt = zeros(2, 1);
delta = 1;
u = - (x(1)^2 - x(1)*x(2)); % Type-3
dxdt(1) = x(1)*x(2) - delta*x(1);
dxdt(2) = - x(1)*x(2) - x(2) + u;
end
