SS.03 / The Matrix Transfer Function +

A multiple-input multiple-output system is written in terms of the $r$ component input vector $\mathrm{U}(s)$

(1) $\begin{equation*} \mathrm{X}(s) = [s \mathrm{I} - \mathrm{A}]^{-1} \mathrm{BU}(s) \end{equation*}$

generating a set of $n$ simultaneous linear equations, where the matrix $\mathrm{B}$ is $n \times r$ . The $\mathrm{m}$ component system output vector $\mathrm{Y}(s)$ may be found by substituting this solution for $\mathrm{X}(s)$ into the output equation as in Eq. 2:

(2) $\begin{equation*} \begin{split} & \mathrm{Y}(s) = \mathrm{C}[s \mathrm{I} - \mathrm{A}]^{-1}\mathrm{B}\{\mathrm{U}(s)\} + \mathrm{D}\{\mathrm{U}(s)\} \\ & \mathrm{Y}(s) = [\mathrm{C}[s \mathrm{I} - \mathrm{A}]^{-1}\mathrm{B} + \mathrm{D}]\{\mathrm{U}(s)\} \end{split} \end{equation*}$

and expanding the inverse in terms of the determinant and the adjoint matrix

(3) $\begin{equation*} \begin{split} & \mathrm{Y}(s) = \frac{\mathrm{C} \; \mathrm{adj}(s\mathrm{I} - \mathrm{A}) \mathrm{B} + \mathrm{det} [s \mathrm{I} - \mathrm{A}] \mathrm{D}}{\mathrm{det}[s \mathrm{I} - \mathrm{A}] \mathrm{D}} \mathrm{U}(s) \\ & \mathrm{Y}(s) = \mathrm{H}(s)\mathrm{U}(s) \end{split} \end{equation*}$

where $\mathrm{H}(s)$ is defined to be the matrix transfer function relating the output vector $\mathrm{Y}(s)$ to the input vector $\mathrm{U}(s)$ :

(4) $\begin{equation*} \mathrm{H}(s) = \frac{(\mathrm{C} \; \mathrm{adj}(s \mathrm{I} - \mathrm{A})\mathrm{B} + \mathrm{det}[s \mathrm{I} - \mathrm{A}]\mathrm{D})}{\mathrm{det}[s \mathrm{I} - \mathrm{A}]} \end{equation*}$

For a system with $r$ inputs $U_{1}(s), \dots ,U_{r}(s)$ and $\mathrm{m}$ outputs $Y_{1}(s), \dots, Y_{m}(s), \mathrm{H}(s)$ is a $m \times r$ matrix whose elements are individual scalar transfer functions relating a given component of the output $\mathrm{Y}(s)$ to a component of the input $\mathrm{U}(s)$ . Expansion of Eq. 3 generates a set of equations:

(5) $\begin{equation*} \begin{bmatrix} Y_{1}(s) \\ Y_{2}(s) \\ \vdots \\ Y_{m}(s) \end{bmatrix} = \begin{bmatrix} H_{11}(s) & H_{12}(s) & \dots & H_{1r}(s) \\ H_{21}(s) & H_{22}(s) & \dots & H_{2r}(s) \\ \vdots & \vdots & \ddots & \vdots \\ H_{m1}(s) & H_{m2}(s) & \dots & H_{mr}(s) \end{bmatrix} \begin{bmatrix} U_{1}(s) \\ U_{2}(s) \\ \vdots \\ U_{r}(s)\end{bmatrix} \end{equation*}$

where the $i$ th component of the output vector $\mathrm{Y}(s)$ is:

(6) $\begin{equation*} Y_{i}(s) = H_{i1}(s)U_{1}(s) + H_{i2}(s)U_{2}(s) + \dots + H_{ir}(s)U_{s}(s). \end{equation*}$

The elemental transfer function $H_{ij}(s)$ is the scalar transfer function between the $i$ thoutput component and the $j$ th input component. Equation 5 shows that all of the $H_{ij}(s)$ transfer functions in $\mathrm{H}(s)$ have the same denominator factor $\mathrm{det} [s \mathrm{I} - \mathrm{A}]$ , giving the important result that all input-output differential equations for a system have the same characteristic polynomial, or alternatively have the same coefficients on the left-hand side.
If the system has a single-input and a single-output, $\mathrm{H}(s)$ is a scalar, and the procedure generates the input/output transfer operator directly.