FDN and State Space Descriptions

When $M_1=M_2=M_3=1$ in Eq.(2.10), the FDN (Fig.2.28) reduces to a normal state-space model (§1.3.7),

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

$$\mathbf{y}_+(n) = \mathbf{x}(n) \protect$$ (3.11)

The matrix $\mathbf{A}={\bm \Gamma}\mathbf{Q}$ is the state transition matrix. The vector $\mathbf{x}(n) = [x_1(n), x_2(n), x_3(n)]^T$ holds the state variables that determine the state of the system at time $n$ . 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

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

to follow normal convention for 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 because tools for state-space analysis are well known and included in many software libraries such as with matlab.