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
p_lAP 1 A2 A4(-9
P : , P-1BH = (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
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 tke t which appear in the solutions to linear differential equations. For example, given the
system x(t) = Fx(t), x(O) = xo with solution x(t) = 0(t,O)xo; the elements of the state transition matrix cD(t,O) are made up of modes of the form tK e X. Here x, which is generally complex valued, represents an eigenvalue of the matrix F.