State Space Filters

An important representation for discrete-time linear systems is the state-space formulation

$$\underline{y}(n) = C {\underline{x}}(n) + D\underline{u}(n) \protect$$

$${\underline{x}}(n+1) = A {\underline{x}}(n) + B \underline{u}(n)$$ (G.1)

where ${\underline{x}}(n)$ is the length $N$ state vector at discrete time $n$ , $\underline{u}(n)$ is a $q\times 1$ vector of inputs, and $\underline{y}(n)$ the $p\times 1$ output vector. $A$ is the $N\times N$ state transition matrix,^G.1and it determines the dynamics of the system (its poles or resonant modes).

The state-space representation is especially powerful for multi-input, multi-output (MIMO) linear systems, and also for time-varying linear systems (in which case any or all of the matrices in Eq.(G.1) may have time subscripts $n$ ) [37]. State-space models are also used extensively in the field of control systems [28].

An example of a Single-Input, Single-Ouput (SISO) state-space model appears in §F.6.