Rearrange state space system

2 min readAug 24, 2021

I’ve modelled a system in simulink and used operspec and linearise to extract a state space system. While the activity has worked well, I have a downstream process which requires the state space to be expressed in a different form.

Without modifying the simulink model, is there a matlab command or series of commands that is capable of rearranging a state space system such that some parameters which where previously ouputs become states. For context, my downstream process will also be dropping a number of states.

I’ve been using simulink, simulink control, system identification and the control system toolbox.

ANSWER

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Let’s assume that the state space model from Simulink is defined as:

xdot = A*x + B*u

[y1;y2] = [C1; C2] * x + [0;D2] u

where y1 are the outputs that you want to be state variables (you can always order the ouputs this way) and all matrices are of compatible dimensions. Note that D1 == 0.

Now we have at least two options.

The first option is to add additional states (w) with derivatives equal to the derivatives of the outputs

w = y1

wdot = y1dot = C1*xdot = C1*A*x + C1*B*u.

Now we can augment the state vector x witht the new states w. By definition, the new outputs that replace y1 are just w. Hence the new state space model is:

[xdot;wdot] = [A 0;C1*A 0] * [x;w] + [B;C1*B] * u

[ynew;y2] = [0 I;C2 0] * [x;w] + [0;D] * u

where the zero and identity matrices are of approropriate dimension. The result will be a non — minimal state space realization that preseves the original states, retains the input/output transfer function from u to y, and has outputs ynew that are state variables.

Here’s an example.

First, define some state space model

>> sys1 = minreal(ss(zpk([tf(1,[1 1]);tf(1,conv([1 1],[1 2]));tf(1,conv([1 1],[1 3]))])))sys1 =
 
  A = 
            x1       x2       x3
   x1   -2.067  -0.2775  -0.1388
   x2   0.2401   -1.212    0.894
   x3  0.03861   0.5588   -2.721
 
  B = 
           u1
   x1  0.9679
   x2  0.6052
   x3   1.303
 
  C = 
            x1       x2       x3
   y1   0.1629   0.6703   0.3352
   y2  -0.6095   0.4695   0.2348
   y3  0.06517   0.4681  -0.2659
 
  D = 
       u1
   y1   0
   y2   0
   y3   0
 
Continuous-time state-space model.

Let’s look at it

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