Markov Parameters

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

(G.2) |

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:

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