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NONLINEAR CONTROL SYSTEMSWEEK6 (2/3, 3/3)

10.18.2017

TENG-HU CHENG

OUTLINES

• Some Lemmas and Theorems

• UUB (Uniformly ultimately bounded)

• Barbalat’s Lemma (similar to Lasalle’s Theorem)

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LEMMAS (FROM KHALIL’S BOOK)

• Lemma 4.3:

• Lemma 4.5:

The eq. pt. 0 of , is U.A.S. iff there exist a class

function

and a positive constant c, independent of , such that

, , ∀ 0, ∀

is radially unbounded if and are

THEOREM 4.8 (FROM KHALIL’S BOOK)

• Let 0 be an eq. pt. for , and D ⊂ be a domain

containing 0. Let 0,∞ → be a continuously

differentiable function such that

,

, 0

∀ 0 and ∀ ∈ D, where and are continuous

positive definite functions on D. Then, 0 is uniformly stable.

P.D. and Decrescent

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THEOREM 4.9 (FROM KHALIL’S BOOK)

• Let 0 be an eq. pt. for , and D ⊂ be a domain

containing 0. Let 0,∞ → be a continuously

differentiable function such that

,

,

∀ 0 and ∀ ∈ D, where , and are continuous

positive definite functions on D. Then, 0 is uniformly

asymptotically stable.

EXPLANATION OF THEOREM 4.9

,

,

• Value of , depends on and , but we only care about .

• Given a fix , , is upper and lower bounded from the 1st ineq.

• Taking the time derivative, , can be a function of time due to the

substitution of nonautonomous dynamics

• Given a fix , if , 0 which implies , → 0 as → ∞, and x → 0

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PROOF OF THEOREM 4.9

1. ,

2. , ,

3. , , is a class function.

4. Base on 2. and 3., , ≜ , · ≜ ∘ · .

5. Let y(t) satisfy the autonomous 1st ODE , , 0.

6. Using Lemma 4.3, Lemma 4.4, 4, and 5 yields , , ,, ∀ , ∈ 0, ,where , isaclass functiondefinedon

0, 0,∞ .

7. , , , , ,

where is a class function.

UUB (UNIFORMLY ULTIMATELY BOUNDED)

• δ sin

Solution: δ sin d

δ 1

δ,

is uniformly bounded

Independent of time

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UUB (UNIFORMLY ULTIMATELY BOUNDED)

What if the dynamics is highly nonlinear compared to the one in

the previous slide (e. g. , δ sin c ) and have no

analytical solutions?

UUB – FROM LYAPUNOV PERSPECTIVE

• Consider a dynamical system δ sin . Let , then

δ sin

δ sin δ

δ as → ∞

Case 1: I.C. δ, 0, then 0 until δ

Case 2: I.C. δ, 0

Case 3: I.C. δ, 0, then 0 until δ

Uniformly Ultimately Bounded

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UUB-EXAMPLE

UUB

• Uniformly Bounded:

If exist a positive constant , independent of 0 and ∀ ∈ 0, , ∃ s.t.

0 independent of , then

⇒

• Uniformly Ultimately Bounded:

If exist a positive constant , independent of 0 and ∀ ∈ 0, , ∃ s.t.

0 independent of , and ∃ ≜ , 0 independent of , then

⇒ , ∀ +

Can be reduced by increasing control gainInitial condition

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PRELIMINARY OF BARBALAT’S LEMMA

COMMON MISCONCEPTIONS

• → 0 ⇏ converge or has a limit

EX1: ⟹ sin log t

EX2: sin log cos log ⟹ sin log unbounded

Does not have a limit

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COMMON MISCONCEPTIONS (CONT’D)

→ 0 ⇏ → 0

EX1: sin → 0 ⟹ sin 2 sin

Does not have a limit

BARBALAT’S LEMMA

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BARBALAT’S LEMMA

• Note that Lasalle’s Theorem is only designed for autonomous system

• For nonautonomous system, one should apply Barbalat’s Lemma

BARBALAT’S LEMMA

• Integral form:

If is uniformly continuous (U.C.) and if →

exists and is finite then → 0

• Non-Integral form:

If is U.C. and → exists and is finite, then

→ 0

∈ (bounded)

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BARBALAT’S LEMMA - NOTES

Definitions:

1. ≜

2. ≜ . (Note: implies )

3. if ∈ ⇒ ∈ ,

→ ??

Note: , ,where ≜

BARBALAT’S LEMMA - EXAMPLE

Ex:

Consideraclosed‐loopsystemwithdynamicsdescribedby

, ,where ≜ .

TheLyapunov functionisdesignedas , , and

substitutingthedynamicsinto , yields

,

1. is P.D., is N.S.D.

2. According to the dynamics and 1.

is bounded, , ∈

N.S.D.

∈U.C.

3. According to , ∈

∈ and is U.C. implies → 0