Initial-Condition Response

Going back to the time domain, we have the linear discrete-time state-space model

$$\begin{eqnarray*} \underline{y}(n) & = & \mathbf{C}\, \underline{x}(n) + \mathbf{D}\,\underline{u}(n)\\ [10pt] \underline{x}(n+1) & = & \mathbf{A}\, \underline{x}(n) + \mathbf{B}\, \underline{u}(n) \end{eqnarray*}$$

and its ``impulse response''

$${\mathbf{h}}(n) \eqsp \left\{\begin{array}{ll} \mathbf{D}, & n=0 \\ [5pt] \mathbf{C}\mathbf{A}^{n-1}\mathbf{B}, & n>0 \\ \end{array} \right.$$

Given zero inputs and initial state $\underline{x}(0)\ne \underline{0}$ , we get

$$\underline{y}_x(n) \eqsp \mathbf{C}\mathbf{A}^n\underline{x}(0), \quad n=0,1,2,\ldots\,.$$

By superposition (for LTI systems), the complete response of a linear system is given by the sum of its forced response (such as the impulse response) and its initial-condition response