In the study of engineering control of physical systems, a standard set of differential...

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:

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 this does not affect the algebraic calculations in Exercises 19 and 20.)

Suppose the transfer function W(s) in Exercise 19 is invertible for some s. It can be shown that the inverse transfer function W(s)^–1 which transforms outputs into inputs, is the Schur complement of A – BC – sI_n for the matrix below. Find this Schur complement. See Exercise 16.

Exercise 19:

Assume 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 16.

Exercise 16:

Let If A₁₁ is invertible, then the matrix is called the Schur complement of A₁₁. Likewise, if A₂₂ is invertible, the matrix is called the Schur complement of A₂₂. Suppose A₁₁ is invertible. Find X and Y such that