39
(B.l) (F, G) is stabilizable
(B.2) rank
V F
-H
G
0
n + p i=l,2,...,r
Conditions (B.l) and (B.2) are essential for a solution to the
linear servomechanism problem. Therefore, when the linear problem is
solved using the framework developed for the nonlinear problem,
conditions (B.l) and (B.2) should play important roles.
Solution to the Linear Problem via
the Nonlinear Formulation
We now proceed to show that when conditions (B.l) and (B.2) are
satisfied, the conditions given in Chapter Two for the nonlinear
formulation are also satisfied.
First consider assumptions (A.l) and (A.2) when applied to a linear
system. The following theorem will relate these assumptions to
conditions (B.l) and (B.2).
Theorem 3.2: Consider the linear system L and assume both the reference
r*(t) and the disturbance w*(t) satisfy the linear differential equation
(3-2). Then, if conditions (B.l) and (B.2) are both satisfied, there
exists an input u*(t) and an initial state x*(0) = x* such that
tracking occurs. Furthermore, u (t) and xQ can be chosen so that the
resulting state trajectory x*(t) and the input u*(t) satisfy the linear
differential equation (3-2) (i.e., assumptions (A.l) and (A.2) are
satisfied).