The Markov parameter sequence for a state-space model is a kind of matrix impulse response that easily found by direct calculation using Eq. (G.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.G.2 Since the system input is a vector, we may regard as a sequence of successive input vectors, each providing an impulse at one of the input components.
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.G.3 Therefore, it is not a possible output signal, except when . A better name might be ``impulse-matrix response''. It can be viewed as a sequence of outputs, each . In §G.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: