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Let's say we have a system $S$ which we model by a nth order ODE and its Lapace transform a transfer function $H(s)$.

A state space model $(A,B,C,D)$ is another way of modeling this and we have the canonical forms among many other ways to model it.

The question is weather there are restrictions in choosing state variables? According to the definitons of them it donst look like it has have to be related to the in/out-put equations very strongly.

Are we simply satisfied if the transfer functions agree and that the output $y$ is a sum of the states?

In the somewhat similar situation for ODE's when you go from ODE to first order system the systems are equivalent but it donst seem that the restiction is the same here in chooing state variables.

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If we consider a controllable and observable state space model

$$\dot{\boldsymbol{x}}(t)=\boldsymbol{Ax}(t)+\boldsymbol{Bu}(t)$$ $$\boldsymbol{y}=\boldsymbol{Cx}(t)+\boldsymbol{Du}(t)$$

then it can be expressed in the frequency domain by a transfer function matrix $\boldsymbol{G}(s)$.

The representation in the state space formulation is not unique because you can always use a regular similarity transformation $\boldsymbol{z}=\boldsymbol{Tx}$ on the states such that the corresponding state space representation has changed while $\boldsymbol{G}(s)$ and the actual dynamics do not change. The dynamics are just described in a different coordinate system.

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  • $\begingroup$ So you are saying that any minimal state space representation is unique up to a similiarty tranformation? Well there are alot of other state models right? All which capture different aspects of the internal behaviour which the transfer function "misses". If you have cascade and a cancelation for instance. $\endgroup$ – user21312 Mar 1 '18 at 11:15
  • $\begingroup$ @user21312 But cancellation would imply that the transfer function is not minimal. $\endgroup$ – Kwin van der Veen Mar 1 '18 at 12:04
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    $\begingroup$ @user21312 For transfer functions, they form an equivalence class. just like 4/8 is 1/2 there is no point in keeping the pole zero cancellations in transfer functions. If this is happening then either there is an engineering problem or a modeling problem. Similar argument can be made for state models $\endgroup$ – percusse Mar 4 '18 at 21:24
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    $\begingroup$ @user21312 If there are additional inputs between the blocks you can also model them via transfer functions, but the idea is that transfer functions are input/output models and there are no "hidden" modes. State models also model the internal dynamics but the minimal representations coincide with the transfer functions. $\endgroup$ – percusse Mar 5 '18 at 9:29
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    $\begingroup$ @user21312 Not really. They are by definition spurious dynamics that can be removed and the overall behavior will not be changed. That's why the name "minimal" is used; the minimal number of states that will represent the total dynamics. There won't be richer dynamics just spurious ones. Otherwise they won't be cancelled anyways. $\endgroup$ – percusse Mar 5 '18 at 12:39
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The state space representation is not unique. This is clear as the degrees of freedom for a state space is much higher than the transfer function.

Some of the most famous representations are:

  • Controllable Canonical Form [1]
  • Observable Canonical Form [2]
  • Diagonal Canonical Form [3]
  • Jordan Canonical Form [4]

However, it is important to represent the system in a minimal form. You can add many meaningless states which do not influence the output. Isn't it? They do not change the result but they waste time and memory of simulations or the controller.

For better numerical results, balancing is popular. See Gramian balancing of state space [5].

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