Page 114 -

2.11 Linear Controller Design 95

(1)

is full rank, i. e. rankC=n

Next, We show that controllability is sufficient to stabilize the system
(2.11.1) by placing the eigenvalues of the A matrix

THEOREM 2.11–2: Let (A, b) be controllable. Then, there exists a constant
gain matrix K such that u=-Kx+v will place the eigenvalues of A-bk anywhere
in the s plane.

The state-feedback gain needed to place the eigenvalues may be found in
the single-input case using Ackermann’s formula (see for example [Antsaklis
and Michel 1997]). Assume the desired closed-loop eigenvalues are specified
as the roots of the equation

(2.11.3)

Then, the state-feedback controller is given by

(2.11.4)

where C is the controllability matrix of (2.11.1), and φ(Α) is the matrix
obtained by evaluating φ(s) at A, i.e.

(2.11.5)

EXAMPLE 2.11–1: Control Design for Double Integrator
Consider the double integrator system

Suppose that the desired closed-loop poles are the roots of the equation

Copyright © 2004 by Marcel Dekker, Inc.