Solution: In the study of engineering control of physical

Chapter 2, Problem 19E

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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 ?m is the “input” to the system, y in ?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.)Assume A – sIn 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