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):

$$\begin{eqnarray*} \mathbf{h}(0) &=& C {\underline{x}}(0) + D\,\underline{\delta}(0) = D\\ [5pt] {\underline{x}}(1) &=& A\, {\underline{x}}(0) + B\,\underline{\delta}(0) = B\\ \mathbf{h}(1) &=& C B\\ [5pt] {\underline{x}}(2) &=& A\,{\underline{x}}(1) + B\,\underline{\delta}(1) = AB\\ \mathbf{h}(2) &=& C A B\\ [5pt] {\underline{x}}(3) &=& A\,{\underline{x}}(2) + B\,\underline{\delta}(2) = A^2B\\ \mathbf{h}(3) &=& C A^2 B\\ [5pt] &\vdots&\\ \mathbf{h}(n) &=& C A^{n-1} B, \quad n>0 \end{eqnarray*}$$

Note that we have assumed ${\underline{x}}(0)=0$ (zero initial state or zero initial conditions). The notation $\underline{\delta}(n)$ denotes a $q\times q$ matrix having $\delta (n)$ along the diagonal and zeros elsewhere.^G.2 Since the system input is a $q\times 1$ vector, we may regard $\underline{\delta}(n)$ as a sequence of $q$ 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

$$\fbox{$\displaystyle \mathbf{h}(n) = \left\{\begin{array}{ll} D, & n=0 \\ [5pt] CA^{n-1}B, & n>0 \\ \end{array} \right.$}$$ (G.2)

The impulse response terms $C A^n B$ for $n\geq 0$ are known as the Markov parameters of the state-space model.

Note that each ``sample'' of the impulse response $\mathbf{h}(n)$ is a $p\times q$ matrix.<sup>G.3</sup> Therefore, it is not a possible output signal, except when $q=1$ . A better name might be ``impulse-matrix response''. It can be viewed as a sequence of $q$ outputs, each $p\times 1$ . In §G.4 below, we'll see that $\mathbf{h}(n)$ is the inverse z transform of the matrix transfer-function of the system.

Given an arbitrary input signal $\underline{u}(n)$ (and zero intial conditions ${\underline{x}}(0)=0$ ), the output signal is given by the convolution of the input signal with the impulse response:

$$\underline{y}_u(n) = (\mathbf{h}\ast \underline{u})(n) = \left\{\begin{array}{ll} D\underline{u}(0), & n=0 \\ [5pt] \sum_{m=1}^nCA^{m-1}B\underline{u}(n-m), & n>0 \\ \end{array} \right. \protect$$ (G.3)