Chapter 2.4, Problem 19E

In the study of engineering control of physical systems, a standard set of differential equations is transformed by Laplace transforms into the following system of linear equations;

$\left[\begin{array}{cc}A-s{I}_n& B\\ C& I_m\end{array}\right]\left[\begin{array}{c}\begin{array}{c}x\\ u\end{array}\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}0\\ \text{y}\end{array}\end{array}\right]$    (8)

where A is n × n. B is n × m, C is m × n, and s is a variable. The vector u in ℝ^m is the “input” to the system, y in ℝ^m is the “output.” and x in ℝⁿ is the “state” vector. (Actually, the vectors x, u, and y are functions of s, but we suppress this fact because it does not affect the algebraic calculations in Exercises 19 and 20.)

19. Assumed A– sI_n is invertible and view (8) as a system of two matrix equations. Solve the top equation for x and substitute into the bottom equation. The result is an equation of the form W(s)u = y, where W(s) is a matrix that depends on s. W(s) is called the transfer function of the system because it transforms the input u into the output y. Find W(s) and describe how it is related to the partitioned system matrix on the left side of (8). See Exercise 15.