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Extra info for Design of fuzzy PID controller for tracking control
R(x) _ (1-5)-t (E(x)) for each x E M . 2. It is unclear whether the hypotheses of the theorem imply that there exists a semigroup of *-morphisms a of N such that S is the generator of a and thus (1 - S)-' = ( dt e a, . In [Rot 1, Theorem 2] this is claimed to be true when M is abelian, but the proof has a gap: If H is the skew-symmetric operator implementing S on NQ , HxQ = S(x)S2 for x e D(S), it is clear that (1-H)-t exists and thus H generates a semigroup of contractions t z 0 -4 etH on NS2 .
N+l (n+l)! and hence 12 = 1. ) Z b and this gives the last inequality in 5. 5 => 4 is trivial. 4 => 4(p) for p = 0, 1, 2, ... 4(p) => 6(p+1) for p = 0, 1, 2, ... Again we may assume e = 1 is trivial. : Put R = (14)-1 and T = . 6(1+8)-1 = 1 - R . Then condition 4(p) implies that IITn RPII is uniformly bounded in n . Thus the series (I- 1)-1RP = j To n=O V RP converges provided I X I > 1 , where the left side of the above expression so far has only formal significance. But R = I - T , so formally (I- f)-1 RP = )-1 (I-a )P (1- + E (k) (1-X)P k %k(I- )k-1 k=1 which gives (I 1" )-1 1 ( Xn )P RP k=1 (p k l I2-21 k (I )k-1 X It is clear that algebraic manipulations will show that the expression to the right is the resol- vent (1 - I)-1 for I X I > 1 , hence the spectrum of T is contained in the unit circle.
But as U is an isometry, it follows that U commutes strongly with the components J and of the polar decomposition of JAIA. Hence T commutes with At for T E R , and as F(H) = TMS2 it follows that F commutes with At for t E R . By Takesaki's theorem the conditional expectation E exists and is unique, and E(x)fl = FxS2 for x E M. We next argue that R I N has the form (I - 5)-1 where S is a a-weakly densely defined, a-weakly closed *-derivation of N . Since Reynolds's identity is equivalent to the derivation property, it suffices to show that R(N) is dense in N and that R I N is 1 - 1 .