Nonlinear Control Sy.. - Free

6.7. LOOP TRANSFORMATIONS 175
Figure 6.9: The Feedback System S.
The same occurs with any other sufficient but not necessary stability condition, such as the
passivity theorem (to be discussed in Chapter 8). One way to obtain improved stability
conditions (i.e. less conservative) is to apply the theorems to a modified feedback loop that
satisfies the following two properties: (1) it guarantees stability of the original feedback
loop, and (2) it lessens the overall requirements over H1 and H2. In other words, it is
possible that a modified system satisfies the stability conditions imposed by the theorem in
use whereas the original system does not. The are two basic transformations of feedback
loops that will be used throughout the book, and will be referred to as transformations of
Types I and Type II.
Definition 6.15 (Type I Loop Transformation) Consider the feedback system S of Figure
6.9. Let H1, H2, K and (I + KH1)-1 be causal maps from Xe into X. and assume that K
is linear. A loop transformation of Type I is defined to be the modified system, denoted
SK, formed by the feedback interconnection of the subsystems Hi = Hl(I + KHl)-1 and
H2 = H2 - K, with inputs u'1 = ul - Ku2 and u'2 = u2i as shown in Figure 6.10. The
closed-loop relations of SK will be denoted EK and FK.
The following theorem shows that the system S is stable if and only if the system SK
is stable. In other words, for stability analysis, the system S can always be replaced by the
system SK.
Theorem 6.4 Consider the system S of Figure 6.9 and let SK be the modified system
obtained after a type I loop transformation and assume the K and (I+KH1)-1 : X -+ X.
Then (i) The system S is stable if and only if the system SK is stable.