Discrete-Time Linear State Space Models

For linear, time-invariant systems, a discrete-time state-space model looks like a vector first-order finite-difference model:

$$\begin{eqnarray*} \underline{x}(n+1) & = & \mathbf{A}\, \underline{x}(n) + \mathbf{B}\, \underline{u}(n) \nonumber \\ \underline{y}(n) & = & \mathbf{C}\, \underline{x}(n) + \mathbf{D}\,\underline{u}(n) \end{eqnarray*}$$

where

• $\underline{x}(n)\in\mathbb{R}^N =$ state vector at time $n$
• $\underline{u}(n) = p\times 1$ vector of inputs
• $\underline{y}(n) = q\times 1$ output vector
• $\mathbf{A}= N\times N$ state transition matrix
• $\mathbf{B}= N\times p$ input coefficient matrix
• $\mathbf{C}= q\times N$ output coefficient matrix
• $\mathbf{D}= q\times p$ direct path coefficient matrix

The state-space representation is especially powerful for

• multi-input, multi-output (MIMO) linear systems
• time-varying linear systems
  (every matrix can have a time subscript $n$ )