Zero-Input Response of State Space Models

The response of a state-space model Eq.(1.8) to initial conditions, i.e., its initial state $\underline{x}(0)$ , is given by

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

and 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.

In our force-driven mass example, with the external force set to zero, we have, from Eq.(1.9) or Eq.(1.11),

$$\left[\begin{array}{c} x_n \\ [2pt] v_n \end{array}\right] = \left[\begin{array}{cc} 1 & T \\ [2pt] 0 & 1 \end{array}\right]\left[\begin{array}{c} x_{n-1} \\ [2pt] v_{n-1} \end{array}\right] = \left[\begin{array}{cc} 1 & T \\ [2pt] 0 & 1 \end{array}\right]^n\left[\begin{array}{c} x_0 \\ [2pt] v_0 \end{array}\right] = \left[\begin{array}{c} x_0+v_0 n T \\ [2pt] v_0 \end{array}\right].$$

Thus, any initial velocity $v_0$ remains unchanged, as physically expected. The initial position $x_0$ remains unchanged if the initial velocity is zero. A nonzero initial velocity results in a linearly growing position, as physically expected. This response to initial conditions can be added to any forced response by superposition. The forced response may be computed as the convolution of the input driving force $f_n$ with the impulse response Eq.(1.11).