make a contraction controller for a cart pole system in matlab

To make a contraction controller for a cart-pole system in MATLAB, we will use state feedback with pole placement. This involves designing a control law that uses state variables (position and velocity of the cart and pole) to determine the control input (force applied on the cart) to bring the system to a desired stable state.

Here's how you can do it in MATLAB:

First, define the system dynamics using the cart-pole equations of motion. Let x = [theta, dtheta, x, dx] be the state vector, where theta is the angle of the pole from vertical, dtheta is its derivative, x is the position of the cart and dx is its derivative. Let u be the force applied on the cart. Then the state-space equations are:

main.m
xdot = [dtheta; a*dtheta + b*u; dx; c*dtheta + d*u]
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where a, b, c, and d are constants that depend on the system parameters.

Next, choose the desired poles of the system. These are the eigenvalues of the desired closed-loop system, which determine its stability and response characteristics. In this case, we want the poles to have negative real parts to ensure stability.

Then, compute the feedback gains using the pole placement method. MATLAB has a built-in function 'place' that can be used for this purpose.

main.m
p = [-1 -2 -3 -4]; % desired poles
K = place(A, B, p); % compute feedback gains
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Finally, implement the control law using state feedback:

main.m
u = -K*x; % control input
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This will cause the system to converge to the desired stable state.

Note: The choice of poles and the design of the state feedback controller depend on the specific requirements of the system and the desired performance.