Stable adaptive control for nonlinear multivariable systems with a triangular control structure

1221 x(T) is nonzero. Based on this remark, it was erroneously claimed 1 that W n has no usual extension. From Proposition 1 one obtains the following result. Proposition 2: The norm of ET x in W n is kE T xk W = kxk 2 2;T + k _ xk 2 2;T + kx(T)k 2 1=2 (1) where k1k2;T denotes the seminorm in L n 2;e defined by k1k2;T = kP T 1k 2. Set k1k W;T := kE T 1 k W (and notice the difference with the definition of k1k2;T). The family (k1kW;T) T 0 is a nondecreasing family of seminorms; it defines over W n a topology T which is strictly finer than the one defined by the norm k1kW [because if a sequence (xp) p0 of W n is such that kxpkW tends to zero, then kxpkW;T tends to zero for every T 0, but not conversely]. In accordance with [2], we take the following definition. Definition 1: The extended resolution space W n e is the completion of (W n ; T).