Nonlinear Control Sy.. - Free

240 CHAPTER 9. DISSIPATIVITY
Figure 9.3: Feedback interconnection.
Having laid out the ground rules, we can now state the main theorem of this section.
Theorem 9.5 Consider the feedback interconnection (Figure 9.3) of the systems Y'1 and
7'2, and assume that both systems are dissipative with respect to the supply rate
wt(ut, yi) = yT Q;yti + 2yT Slut + u'Rut.
(9.26)
Then the feedback interconnection of 01 and 02 is Lyapunov stable (asymptotically stable)
if the matrix Q defined by
Ql + aR2 -S1 + aS2
-ST + aS2 R1 + aQ2 ]
is negative semidefinite (negative definite) for some a > 0.
Proof of Theorem 9.5: Consider the Lyapunov function candidate
O(x1, X2) = O(x1) + aO(x2)
a > 0
(9.27)
where O(x1) and O(x2) are the storage function of the systems a/il and z/b2i respectively. Thus,
O(x1i X2) is positive definite by construction. The derivative of along the trajectories
of the composite state [x1, x2]T is given by
_
w1(u1,yl)+aw2(u2,y2)
(y1 Qlyl + 2yTSlu1 + u1 R1ul) + a(yz Q2y2 + 2y2 S2u2 + u2 R2u2) (9.28)