Lyapunov Plotting: using 'mesh' and 'meshgrid' with matricies

John Williams answered . 2025-02-17 06:29:38

% Define P matrix (example values, modify as needed)
p1 = 1; p2 = 0.5; p3 = 0.5; p4 = 2;
P = [p1, p2; p3, p4];

% Define the range of e and ed values
e_range = -4:0.04:4; % Example range
ed_range = -4:0.04:4; % Example range

% Create a grid of e and ed values
[E, ED] = meshgrid(e_range, ed_range);

% Initialize V matrix
V = zeros(size(E));

% Compute V for each (e, ed) pair
for i = 1:size(E, 1)
    for j = 1:size(E, 2)
        x = [E(i, j); ED(i, j)];
        V(i, j) = x' * P * x; % Lyapunov function calculation
    end
end

% Plot the result
figure;
mesh(E, ED, V);
xlabel('e'); ylabel('ed'); zlabel('V(e, ed)');
title('Lyapunov Function V(e, ed)');
grid on;

Explanation:

1. Matrix Dimensions:

   • $E$ and $ED$ are $m\times n$ matrices representing the grid of $e$ and e?\dot{e}e? values.
   • $V$ is also an $m\times n$ matrix calculated for each grid point.

2. meshgrid Usage:

   • meshgrid generates two 2D matrices $E$ and $ED$ that define the grid of $e$ and e?\dot{e}e? values.

3. Double Loop:

   • The double for loop iterates over the grid points, computes $V$ for each (e,e?)(e, \dot{e})(e,e?), and stores it in the corresponding entry of $V$.

4. Plotting with mesh:

   • The mesh function plots V(e,e?)V(e, \dot{e})V(e,e?) as a 3D surface.

Optimization (Vectorized Calculation):

To avoid the double loop, you can vectorize the calculation of V:

% Vectorized computation of V
V = P(1,1)*E.^2 + (P(1,2) + P(2,1))*E.*ED + P(2,2)*ED.^2;

% Plot the result
figure;
mesh(E, ED, V);
xlabel('e'); ylabel('ed'); zlabel('V(e, ed)');
title('Lyapunov Function V(e, ed)');
grid on;

Output:

• A 3D plot showing the Lyapunov function $V$, where the x-axis is $e$, the y-axis is e?\dot{e}e?, and the z-axis is V(e,e?)V(e, \dot{e})V(e,e?).

Let me know if you need further assistance!

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