26
Stability of the Closed-Loop Transient System
In this section we investigate the stability of the system NCT.
First, the previously defined condition of controllability for the pair
(A,B) and the required observability of the pair (A, K2) are related to
the stability of NCT. Next, a method for checking local stability of
NCT using Liapunov's indirect method is presented.
Now consider controllability of the pair (A,B) which has already
been insured by the chosen structure for the internal model system.
Suppose, for the sake of example, that the pair (A,B) is not
controllable. This implies that the pair (A, BH) is not controllable.
Consequently, there exists a linear transformation matrix P such that
~A1
a2~
i ,
a4~
9
P ^BH =
_0
a3_
_0 _
(2-39)
Since the eigenvalues of A are in the closed right half-plane, the
eigenvalues of A3 are in the closed right half-plane. It is apparent
|f
that the modes associated with A3 are not affected by any control
law. Thus, we can conclude that when (A,B) is not controllable, the
system NCT can not be made asymptotically stable. This points out one
of the reasons behind the structure chosen for the internal model
system.
Now consider the situation which arises when the pair (A, K2) is
not observable. Since the eigenvalues of A are in the closed right
* Modes are components of the form t^ext which appear in the
solutions to linear differential equations. For example, given the
system x(t) = Fx(t), x(0) = xQ with solution x(t) = $(t,0)xo; the
elements of the state transition matrix $(t,0) are made up of modes of
the form tke Here X, which is generally complex valued, represents
an eigenvalue of the matrix F.