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For the sake of simplicity the following notation $a_k := a[k]$ is assumed for time sequences.

A completely general discrete-time (DT) non-linear(NL) time-invariant (TI) dynamical system can be described with a system of vector difference equations, where $x[k]$ is the system state, $u[k]$ is the system input and $y[k]$ is the system output: $$\begin{cases} x[k+1] = f(x[k],u[k])\\ y[k] = g(x[k], u[k]) \end{cases} \quad\forall k \in \mathbb{Z} \ge0$$ $f(\cdot)$ and $g(\cdot)$ are general nonlinear functions.

The first equation is very close to a general autonomous non linear differential equation with $\dot x(t)=f(x(t),u(t))$ where $u(t)$ is the source term.

Now, the same dynamical DT NL TI system could be described also through an input output relationship, involving more among their "samples" (current and past): $$y[k] = h(y[k-1],y[k-2],...,y[k-n],u[k],u[k-1],...,u[k-m]), \quad\forall k \in \mathbb{Z} \ge 0 \quad$$ where again, $h$ is a general non linear function, and $n,m$ are integer positive values.

How can one prove that this input-output relationship form is equivalent to the former state-space representation, at least when $x[0]=0$ (without any care about a possible relationship between $f,g$ and $h$)?

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Well, any input-output representation obviously admits a state-sapce form. for your equation in $y[k]$ you can easily construct one as follows. Create a "shift" system (an integrator chain) as $$ \begin{aligned} x_1[k+1] &= x_2[k],\\ x_2[k+1] &= x_3[k],\\ &\vdots\\ x_n[k+1] &= y[k] \end{aligned} $$ In this way indeed you have $x_n[k] = y[k-1]$, $x_{n-1}[k]=y[k-2]$, ..., $x_1[k] = y[k-n]$. You can do the same by defining another shift system to carry the information in $u[k]$, namely $$ \begin{aligned} z_1[k+1] &= z_2[k],\\ z_2[k+1] &= z_3[k],\\ &\vdots\\ z_m[k+1] &= u[k] \end{aligned} $$ Thus, with $\xi := col(x_1,...,n_n,z_1,...,z_m)$ the IO representation is equivalent to $$ \begin{aligned} \xi[k+1] &= A\xi[k],\\ y[k] &= h(\xi_1[k],\dots,\xi_n[k],u[k],\xi_1[k],\dots,\xi_m[k]) \end{aligned} $$ The converse is instead not trivial. Pick the LTI system $$ \begin{aligned} x[k+1] &= \begin{bmatrix}2 & -1\\0 & 0.1\end{bmatrix}x[k] + \begin{bmatrix}1\\1\end{bmatrix}u[k],\qquad (1)\\ y[k] &= x_2[k] \end{aligned} $$ Then the output is the same as $$ y[k] = 0.1 y[k-1] + u[k],\qquad (2) $$ however, (2) and (1) are not equivalent. In general, given a IO representation you can always find a state-space representation, however for general state-space forms a IO representation might be not enough to describe the whole state information. You should find a diffeomorphism among the two representations to assess equivalence

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  • $\begingroup$ Thank you for your answer, your reasoning is perfectly clear. Of course, the IO representation misses the information about the state, and it says nothing about the initial state. But, assuming that I don't care about the state, and that $x[0]=0$, can I prove that, given a certain state-space representation with $x\in\mathbb{R}^n$ (so the system has order $n$) there exists always at least one IO representation where the output depends on input and output past samples up to $n$ backward? (I mean, using, also using a path similiar to the one that allows to go from IO to a state space) $\endgroup$ – Vexx23 Oct 28 '16 at 14:22
  • $\begingroup$ In general I think you need some observability conditions. In such a way that the state space model is diffeomorphic to the "shift" form above $\endgroup$ – LJSilver Oct 28 '16 at 17:23
  • $\begingroup$ Sorry I am not so familiar with the concept of diffeomorphism: is it referred to the possibility to invert the function relating the derivative of the state to the current state and input? Of course in this case we have not derivatives but differences, so maybe the the concept is simply extended to the function involved in the difference equation. Anyway, provided that this function has 'good' properties, how can I find a way to obtain the IO relationship from the state space model? and, above all, why the order n as number of states is reflected into the higher backward shift? $\endgroup$ – Vexx23 Oct 28 '16 at 20:43
  • $\begingroup$ I mean, when the state is not in the trivial form where each $i$-th component of the state in the instant $k$ is simply the $(i-1)$-th component in the next time instant $k+1$, how one can derive that $n$ as dimension of the state implies that that a 'possible' (if this can be done, under the property of observability) IO relationship links the output with its past samples and input past samples up to $n$ backward shift? $\endgroup$ – Vexx23 Oct 28 '16 at 20:46
  • $\begingroup$ The fact is that if a IO form is the same as the shifting state space representation, then asking for a general state space representation to admit a well defined IO description is the same as askibg that there exists an invertible change of coordinates (i.e. a diffeomorphism) trabsforming the system to a shift form. Try to have a look to basic nonlinear system books, like "nonlinear control systems" by A.Isidori or "nonlinear system" by khalil $\endgroup$ – LJSilver Oct 29 '16 at 17:13

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