Numerical Solution for a system of Two Differential Equations

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.

Your help is appreciated.

equation 1

equation 2

[Sam Chak]

Hi @Suhaib Salah

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, .

Answer 1

[Paul]

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

[Suhaib Salah]

Dear Paul,

Thanks.

We got rid of the input, credit goes to you. But what does remain? How to go on with our code? We have 2 outputs which is hard for me.

Your help will be appreciated.

[Paul]

Basically, you need to do what @Sam Chak was suggesting.

You have two second order differential equations:

xddot = f(thetaddot, thetadot, theta, xddot, xdot, x, V)

thetaddot = g(thetaddot, thetadot, theta, xddot, xdot, x, V)

The first step is to manipulate the equations inot the form

M*[xddot;thetaddot) = h(thetadot, theta, xdot, x, V)

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

z4 = thetadot

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().