Page 184 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Output Feedback Control of Pure-Feedback Systems 181

Consequently, the pure-feedback system (11.1) can be redescribed as

⎧
= z i+1 , i = 1,...,n − 1
⎪ ˙ z i
⎨
λ n
˙ z n = a n (¯x n ) + b n (¯x n ,u )u (11.12)
y = z 1 .
⎪
⎩
To facilitate the controller design, the function b n (¯x n ,u ) in (11.12)
λ n
is assumed to be positive and bounded satisfying 0 < b 1 < b n (¯x n ,u )< b 2,
λ n
where b 1 and b 2 are positive constants. It should be noted that this condition
has been widely used in the literature [1–3] as a necessary condition to
guarantee the controllability of (11.1).
Substituting (11.3)into(11.12)yields
⎧
=z i+1 , i = 1,...,n − 1
⎪ ˙ z i
⎨
λ n
λ n
˙ z n =a n (¯x n ) + b n (¯x n ,u )d(t)v + b n (¯x n ,u )ρ(t) (11.13)
⎪
⎩
y = z 1
Moreover, we can rewrite system (11.13) in the form of
⎧
= z i+1 , i = 1,...,n − 1
⎪ ˙ z i
⎪
⎪
λ n
⎨ = a(¯x n ) + b(¯x n ,u )v
˙ z n
(11.14)
= F(¯x n ,u ,v) + b 0v
λ n
⎪
⎪
⎪
⎩
y = z 1
where a(¯x n ) = a n (¯x n ) + b n (¯x n ,u )ρ(t), b(¯x n ,u ) = b n (¯x n ,u )d(t),and F(¯x n ,
λ n
λ n
λ n
λ n
λ n
λ n
u ,v) = a n (¯x n ) + (b(¯x n ,u ) −b 0 )v,and b 0 is the estimation of b(¯x n ,u ) and
can be obtained based on the prior modeling knowledge. Considering the
positive boundedness of b n (¯x n ,u ), we know that b(¯x n ,u ) is also positive
λ n
λ n
¯
and bounded with the lower boundary b = b 1 and upper boundary b =
b 2 (d l1 + d r1 ).
As shownin(11.14), the original system (11.14)isnowrewrittenas
a canonical form, which is more suitable for control design. However, in
the proposed coordinate transformation, the system states z i ,i = 2,··· ,n
are not available though the original system state x i ,i = 1,··· ,n may be
λ n
measurable. Moreover, the lumped dynamics F(¯x n ,u ,v) in (11.14)are
unknown. Hence, they should be specifically addressed in the following
control design. It is noted that ESO can be used to cope with these two
issues simultaneously. Hence, the following subsection will introduce the
design of an ESO.