FDN and State Space Descriptions

When $M_1=M_2=M_3=1$, the FDN in Fig.1.22 reduces to a standard state-space description [455, Appendix E] for a three-input, three-output, order 3 system:

$$\mathbf{x}(n+1) = \mathbf{A}\mathbf{x}(n) + \mathbf{u}_+(n)$$

$$\mathbf{y}_+(n) = \mathbf{x}(n) \protect$$ (2.11)

The matrix $\mathbf{A}={\bm \Gamma}\mathbf{Q}$ is called the state transition matrix. The vector $\mathbf{x}(n) = [x_1(n), x_2(n), x_3(n)]^T$ holds the state variables which completely determine the state of the system at time $n$.^2.7 The order of a state-space system is equal to the number of state variables, i.e., the dimensionality of $\mathbf{x}(n)$. The input and output signals have been trivially redefined as

$$\begin{eqnarray*} \mathbf{u}_+(n) &\isdef & \mathbf{u}(n+1)\\ \mathbf{y}_+(n) &\isdef & \mathbf{y}(n+1) \end{eqnarray*}$$

in order to obtain standard state-space form.

Thus, an FDN can be viewed as a generalized state-space model for a class of $N$th-order linear systems--``generalized'' in the sense that unit delays are replaced by arbitrary delays. This correspondence is valuable for analysis since tools for state-space analysis are well known (and included in software such as Matlab, among others).

An introduction to state-space models is given in [455, available online]. A more detailed treatment appears in [221].