create a state space solution for a double inverted pendulum in matlab

To create a state space solution for a double inverted pendulum in MATLAB using Control System Toolbox, you can follow these steps:

1. Define the system parameters: mass of the pendulums, length of the rods, etc.
2. Write the equations of motion for the system using Lagrangian mechanics.
3. Linearize the equations of motion around the upright position using Taylor series expansion.
4. Convert the linearized equations into state-space form using matrices.
5. Construct the state-space model in MATLAB using ss command.
6. Compute the system poles and check the stability of the system.

Here is the sample code:

main.m
% Define system parameters
m1 = 1; % mass of upper pendulum
m2 = 1; % mass of lower pendulum
l1 = 1; % length of upper rod
l2 = 1; % length of lower rod
g = 9.81; % acceleration due to gravity

% Write the equations of motion
% ...

% Linearize the equations of motion
% ...

% Convert the linearized equations into state-space form
A = [0 0 1 0; 0 0 0 1; 0 (m1*g)/((m1+m2)*l1) 0 0; 0 (m1+m2)*g/((m1+m2)*l1) 0 0];
B = [0;0;1/(m1*l1^2);1/((m1+m2)*l1^2)];
C = [1 0 0 0; 0 1 0 0];
D = [0;0];

% Construct the state-space model
sys = ss(A,B,C,D);

% Compute the system poles
poles = eig(A);
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This will give you the state-space model of the double inverted pendulum system in MATLAB.