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matlabmathdifferential-equations
lahidj
lahidj

Reputation: 127

Inverted pendelum matrix derivative approximation

Here I've written a dynamic function as:

function dAx = dynamic(t,x)
global u;

g = 9.8;
l = 0.5;
m = 0.5;
h = 2;

dx(1,1) = x(2);
dx(2,1) = g/l*sin(x(1))-h/(m*l^2)*x(2)+1/(m*l)*cos(x(1))*u(1,1);
dx(3,1) = g*lcos(x(3))-u(2,1);


A = [x(1)*x(2)+10*x(1);10*x(2)-5*x(1);x(3)] 

dx = 1e-3
dAx = [(((x(1)+dx)+(x(1)-dx))*((x(2)+dx)+(x(2)-dx)))/(2*dx)+(10*(x(1)+dx)+(x(1)-dx))/(2*dx);
       ((10*(x(2)+dx)+(x(2)-dx))-5*((x(1)+dx)+(x(1)-dx)))/(2*dx);
       ((x(3)+dx)+(x(3)-dx))/(2*dx)]; % dA/dx using central derivative method computation

Here there is a matrix A (3*1) and function out put is derivative of matrix A related to system states.

I've tried to use central difference method.Is my derivative matrix calculation correct?

Upvotes: 0

Views: 64

Answers (1)

user2971764
user2971764

Reputation: 37

@Lahidj dAx/dx is a matrix of the size of nA by nx. You compute this column by column. Take the first state (eg. x(1)). Increment and decrement it by a small value (dx), like 1e-3. Get the A vectors for increased and decreased values of alpha. Compute the difference between these two vectors and divide it by two times of dx. (A_plus - A_minus)/(2*dx). Repeat this for the rest of the states/columns, you would get dAx/dx.

Upvotes: 0

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