(B.1) (F, G) is stabilizable
(B.2) rank F n + p ,
-H0
Conditions (B.1) 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.1) 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.1) and (B.2) are satisfied, the conditions given in Chapter Two for the nonlinear
formulation are also satisfied.
First consider assumptions (A.1) and (A.2) when applied to a linear system. The following theorem will relate these assumptions to conditions (B.1) 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.1) and (B.2) are both satisfied, there exists an input u*(t) and an initial state x*(O) = xo such that
tracking occurs. Furthermore, u*(t) and x° 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.1) and (A.2) are satisfied).