Numerical Solution for a system of Two Differential Equations

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Suhaib Salah on 6 Nov 2021
Edited: Paul on 8 Nov 2021
Dear,
I have a robotic system which has 2 outputs: 1- the position of a system. (X) 2- the tilt angel of the robot. (theta)
Following are the 2 equations, one for each output. The input is the voltage V. However, any other symbol is a constant.
We want to use ODE solvers. But we couldn't find an ODE example containing an input. Attached are the photos of the equation. equation 1 equation 2
Sam Chak on 6 Nov 2021
Generally, you need to design a feedback control law for the input voltage, V. But first, the ODEs have to rearranged so that the double-dot terms do not appear on the right-hand side of the equation. For example, the following system can be decoupled to become  which can be expressed in the state space model  where is the input force.
For the general nonlinear system, .

Paul on 6 Nov 2021
Edited: Paul on 6 Nov 2021
Because V(t) is an input presumably it can be expressed as a function of time, in which case the simplest appraoch is to just compute it inside odefun(). For example, suppose the differential equation to be solved is
dx/dt = -x(t) + V(t)
where V(t) = sin(t). Then the odefun is
function dxdt = odefun(t,x)
dxdt = -x + sin(t)
end
Paul on 8 Nov 2021
Basically, you need to do what @Sam Chak was suggesting.
You have two second order differential equations:
The first step is to manipulate the equations inot the form
where M is 2x2 and h is 2x1. The equations appear to be linear in the second derivatives, so it should be doable. The Symbolic Math Toolbox can help if don't want to do it by hand.
Next, define a state vector:
z1 = x
z2 = theta
z3 = xdot
With these definitons, odefun would look something like this:
function dzdt = odefun(t,z)
V = % compute V(t) here
dzdt = zeros(4,1);
dzdt(1) = z(3);
dzdt(2) = z(4);
dzdt(3:4) = M\h(z4,z2,z3,z1,V); % assumes M is invertible
end
The state vector, z, is returned from the ode solver, like ode45() or ode23().