Page 319 - The Mechatronics Handbook

0066_frame_C14.fm Page 33 Wednesday, January 9, 2002 1:51 PM

and a reference-output map can be found. Our goal is to find the bounded controller such that the tracking
error e(⋅):[t 0 ,∞)→E with E 0 ⊆ E evolves in the specified closed set

(
1
S e δ() = { e ∈ : e 0 ∈ E 0 , x ∈ X X 0 ,U, R, D, Z, P), t ∈ [t 0 , ∞)
(
(
(
∀
(
(
∀
et() ≤ ρ e t, e 0 ) + ρ r r ) + ρ d d ) + ρ y y ) + δ, δ ≥ 0, e ∈ E E 0 , R, D, Y), t ∈ [t 0 , ∞)}
Here, ρ e (⋅) is the KL-function; ρ r (⋅), ρ d (⋅) and ρ y (⋅) are the K-functions.
A positive-invariant domain of stability is found for the closed-loop system with x 0 ∈X 0 , e 0 ∈Ε 0 , u ∈U,
r ∈R, d ∈D, z ∈Z and p ∈P. In particular,
+
(
(
(
S s = { x ∈ , e ∈ : xt() ≤ ρ x t, x 0 ) + ρ r r ) + ρ d d ) δ,
1
4
∀ x ∈ XX 0 ,U, R, D, Z, P), t ∈ [t 0 , ∞), et() ≤ ρ e t, e 0 ) + ρ r r )
(
∀
(
(
(
(
(
∀
∀
+ ρ d d ) + ρ y y ) + δ, e ∈ EE 0 , R, D, Y ), t ∈ [t 0 , ∞)},
where ρ x (⋅) is the KL-function.
To study the robustness, tracking, and disturbance rejection, we consider a state-error set
(
XE X 0 , E 0 ,U, R, D, Z, P) = ( { x, e) ∈ X × E : x 0 ∈ X 0 , e 0 ∈ E 0 , u ∈ U,
r ∈ R, d ∈ D, z ∈ Z, p ∈ P, t ∈ [t 0 , ∞)}
The robust tracking, stability, and disturbance rejection are guaranteed if XE ⊆ S s . The admissible set
S s is found by using the Lyapunov stability theory [9], and

S s =  x ∈ ,e ∈ : x 0 ∈ X 0 , e 0 ∈ E 0 , u ∈ U, r ∈ R, d ∈ D, z ∈ Z, p ∈ P
4
1
 dV t, x, e)
(
(
r 1 x + r 2 e ≤ V t, x, e) ≤ r 3 x + r 4 e , ------------------------- ≤ – r 5 x – r 6 e ,
dt
∀
(
∀
(
∀ x ∈ XX 0 ,U, R, P, Z, P), e ∈ EE 0 , R, D, Y ), t ∈ [t 0 , ∞) 


where ρ 1 (⋅), ρ 2 (⋅), ρ 3 (⋅) and ρ 4 (⋅) are the K ∞ -functions; and ρ 5 (⋅) and ρ 6 (⋅) are the K-functions.
κ
If in XE there exists a C Lyapunov function V(t, x, e) such that for all x 0 ∈ X 0 , e 0 ∈ E 0 , u ∈ U,
r ∈ R, d ∈ D, z ∈ Z, and p ∈ P on [t 0 , ∞) sufficient condition for stability (s1)
r 1 x + r 2 e ≤ V t, x, e) ≤ r 3 x + r 4 e
(
and inequality
dV t, x, e)
(
------------------------- ≤ – r 5 x – r 6 e
dt
which is the sufficient condition for stability s2, hold, then
1. solution x(⋅):[t 0 ,∞)→X for closed-loop system is robustly bounded and stable,
2. convergence of the error vector e(⋅):[t 0 , ∞)→E to S e is ensured in XE,
3. XE is convex and compact, and XE ⊆ S s .
That is, if criteria (s1) and (s2) are guaranteed, we have XE ⊆ S s .

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