State-Space-Model from more dimensional Transfer-Function

Question

Im trying to extract the State-Space-Model for an cascaded Buck/Boost-Converter from an set of Transfer-functions \$ G(s)\$ (which I generated with Plecs/Matlab).

First of all I want to analyse the Buck-Modus (Sboost=0).

The State-Space-Model should look like this:

\$ \begin{align} \dot{\vec{\Delta\mathrm{x}}} = \begin{bmatrix} \dot{\Delta \mathrm{uC1}} \\[0.5em] \dot{\Delta\mathrm{iL}} \\[0.5em] \dot{\Delta\mathrm{uC2}}\\ \end{bmatrix} = \begin{bmatrix} \mathrm{A_{11}} & \mathrm{A_{12}} & \mathrm{A_{13}} \\[0.5em] \mathrm{A_{21}} & \mathrm{A_{22}} & \mathrm{A_{23}} \\[0.5em] \mathrm{A_{31}} & \mathrm{A_{32}} & \mathrm{A_{33}} \\ \end{bmatrix} \cdot \begin{bmatrix} \Delta\mathrm{uC1} \\[0.5em] \Delta\mathrm{iL} \\[0.5em] \Delta\mathrm{uC2} \\ \end{bmatrix} + \begin{bmatrix} \mathrm{B_{11}} & \mathrm{B_{12}} & \mathrm{B_{13}} \\[0.5em] \mathrm{B_{21}} & \mathrm{B_{22}} & \mathrm{B_{23}} \\[0.5em] \mathrm{B_{31}} & \mathrm{B_{32}} & \mathrm{B_{33}} \\ \end{bmatrix} \cdot \begin{bmatrix} \Delta\mathrm{Uin} \\[0.5em] \Delta\mathrm{Uac} \\[0.5em] \Delta\delta \end{bmatrix} \end{align}\$

As you can see, my Buck/Boost-Converter contains 3 State-Variables. I did an Impulse-Response Analysis in PLECS (Simulation) to obtain the Bode-Plot from the perturbation in an equilibrium Point.

After that I used \$\mathrm{tfest}()\$ in Matlab to generate \$G(s)\$ for the Bode-Plot.

I have following Transfer-Functions:

For \$\mathrm{uC1}\$: \$ \dfrac{\mathrm{uC1}}{\mathrm{Uin}}(s),\dfrac{\mathrm{uC1}}{\mathrm{Uac}}(s),\dfrac{\mathrm{uC1}}{\delta}(s) \$

For \$\mathrm{iL}\$: \$ \dfrac{\mathrm{iL}}{\mathrm{Uin}}(s),\dfrac{\mathrm{iL}}{\mathrm{Uac}}(s),\dfrac{\mathrm{iL}}{\delta}(s) \$

For \$\mathrm{uC2}\$: \$ \dfrac{\mathrm{uC2}}{\mathrm{Uin}}(s),\dfrac{\mathrm{uC2}}{\mathrm{Uac}}(s),\dfrac{\mathrm{uC1}}{\delta}(s) \$

and want to convert them into an State-Space-Model. How can I achieve that?

Answer 1

In Matlab, you could just apply ss() to your matrix of Transfer-functions and that should do it.

For example,

G = [tf(rand(1,1),rand(1,2)), tf(rand(1,1),rand(1,2)); tf(rand(1,1),rand(1,2)), tf(rand(1,1),rand(1,2))];

ss(G)

and you get

G

Transfer function 'G' from input 'u1' to output ...

           0.1182
 y1:  -----------------
      0.1398 s + 0.1585

           0.2366
 y2:  -----------------
      0.4479 s + 0.7206

Transfer function 'G' from input 'u2' to output ...

           0.8437
 y1:  ----------------
      0.446 s + 0.4288

           0.8192
 y2:  -----------------
      0.7782 s + 0.0429


ss(G)

ans.a =
               x1          x2          x3          x4
   x1   3.781e-17   5.228e-17  -6.661e-16     0.08869
   x2   6.074e-16   3.508e-16        1.09   9.021e-17
   x3           0          -1      -2.095   9.714e-17
   x4          -1  -5.551e-16           0      -1.664

ans.b =
             u1        u2
   x1  -0.01015   -0.5903
   x2   -0.5066    -1.337
   x3     0.527      1.18
   x4    0.1841    0.3669

ans.c =
          x1     x2     x3     x4
   y1      0      0  1.604      0
   y2      0      0      0  2.869

ans.d =
       u1  u2
   y1   0   0
   y2   0   0

Continuous-time model.