As for the stability of the resulting adaptive control system,
we have the following theorem.
Theorem 2: Suppose that the augmented error system
(23) is ASPR with the PFC signal (24). Then all the
signals in the resulting control system with the control input
(30) are bounded. Further in the case where w(t)=0,
limt→∞ e(t) = 0 is attained by setting σi = 0 and σ = 0.
Proof: Defining the error between the ideal input a
(t) and
the actual input ua(t) by