State Space modelling from an ODE

How would I find the state variable description and transfer function of the 3rd order system of the following on MATLAB?

Answer 1

[Star Strider]

The easiest way is to let the Symbolic Math Toolbox do the heavy lifting:

syms y(t) u(t) t 
Dy = diff(y);
D2y = diff(y,2);
D3y = diff(y,3);
DEq = D3y + 6*D2y + 11*Dy + 6*y == 6*u;
[SS,Sbs] = odeToVectorField(DEq)

producing:

SS =
                               Y[2]
                               Y[3]
 6*u(t) - 6*Y[1] - 11*Y[2] - 6*Y[3]
 
Sbs =
   y
  Dy
 D2y

The ‘Sbs’ output simply tells you the substitutions the solver made, so for example ‘Y[1]=y’.

I’m sure you can take it from there.

[jokn buntue]

Can you please explain how to convert the results of odeToVectorField to the A,B,C,D matrices?

Thanks.

[Star Strider]

@jokn buntue — The ‘SS’ matrix is essentially a companion-form matrix, so eliminating the ‘6*u(t)’ term, it is the ‘A’ matrix. The ‘6*u(t)’ term becomes part of the ‘B’ matrix (vector here, since this is a SISO system).

That should get you started.

Answer 2

[djedoui Nassim]

Hey

You can follow this mathematical changement using your example,