The response of a state-space model Eq.(1.8) to *initial
conditions*, *i.e.*, its *initial state*
, is given by

and the

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

Thus, any initial velocity remains unchanged, as physically expected. The initial position 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 with the impulse response Eq.(1.11).

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