Concept explainers
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 ℝm is the “input” to the system, y in ℝm is the “output.” and x in ℝn is the “state” vector. (Actually, the
19. Assumed 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 15.
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Linear Algebra and Its Applications (5th Edition)
- Let x=x(t) be a twice-differentiable function and consider the second order differential equation x+ax+bx=0(11) Show that the change of variables y = x' and z = x allows Equation (11) to be written as a system of two linear differential equations in y and z. Show that the characteristic equation of the system in part (a) is 2+a+b=0.arrow_forwardIn Exercises 1-12, find the solution of the differential equation that satisfies the given boundary condition(s). 10.arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning