The impulse response of a state-space model is easily found by direct calculation using Eq. (E.1):
Note that we have assumed (zero initial state or zero initial conditions). The notation denotes a matrix having along the diagonal and zeros elsewhere.E.2
The impulse response of the state-space model can be summarized as
The impulse response terms for are known as the Markov parameters of the state-space model.
Note that each sample of the impulse response is a matrix.E.3 Therefore, it is not a possible output signal, except when . A better name might be ``impulse-matrix response''. In §E.4 below, we'll see that is the inverse z transform of the matrix transfer-function of the system.
Given an arbitrary input signal (and zero intial conditions ), the output signal is given by the convolution of the input signal with the impulse response: