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Nonlinear Systems Analysis Lecture Note 16 Section 4.8 (Ultimate) Boundedness (Lyapunov Stability) Feng-Li Lian NTU-EE Sep05 – Jan06 NTUEE-NSA-Ch4.7-2 Feng-Li Lian © 2005 Outline Introduction (L9) Autonomous Systems (4.1 L9) Basic stability definitions Lyapunov’s stability theorems Variable gradient method Region of attraction Instability The Invariance Principle (4.2, L10) LaSalle’s theorem Linear Systems and Linearization (4.3, L11) Comparison Functions (4.4, L12) Non-autonomous Systems (4.5, L13) Linear Time-Varying Systems & Linearization (4.6, L14) Converse Theorems (4.7, L15) Boundedness & Ultimate Boundedness (4.8, L16) Input-to-State Stability (4.9, L17)
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Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

Dec 11, 2021

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Page 1: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

Nonlinear Systems Analysis

Lecture Note 16

Section 4.8

(Ultimate) Boundedness(Lyapunov Stability)

Feng-Li LianNTU-EE

Sep05 – Jan06

NTUEE-NSA-Ch4.7-2Feng-Li Lian © 2005Outline

Introduction (L9)

Autonomous Systems (4.1 L9)

• Basic stability definitions

• Lyapunov’s stability theorems

• Variable gradient method

• Region of attraction

• Instability

The Invariance Principle (4.2, L10)

• LaSalle’s theorem

Linear Systems and Linearization (4.3, L11)

Comparison Functions (4.4, L12)

Non-autonomous Systems (4.5, L13)

Linear Time-Varying Systems & Linearization (4.6, L14)

Converse Theorems (4.7, L15)

Boundedness & Ultimate Boundedness (4.8, L16)

Input-to-State Stability (4.9, L17)

Page 2: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-3Feng-Li Lian © 2005(Ultimate) Boundedness

NTUEE-NSA-Ch4.7-4Feng-Li Lian © 2005(Ultimate) Boundedness

Page 3: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-5Feng-Li Lian © 2005(Ultimate) Boundedness

NTUEE-NSA-Ch4.7-6Feng-Li Lian © 2005(Ultimate) Boundedness

Page 4: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-7Feng-Li Lian © 2005via Lyapunov Analysis

NTUEE-NSA-Ch4.7-8Feng-Li Lian © 2005via Lyapunov Analysis

Page 5: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-9Feng-Li Lian © 2005via Lyapunov Analysis

NTUEE-NSA-Ch4.7-10Feng-Li Lian © 2005via Lyapunov Analysis

Page 6: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-11Feng-Li Lian © 2005via Lyapunov Analysis

NTUEE-NSA-Ch4.7-12Feng-Li Lian © 2005Definitions of UB, GUB, UUB, GUUB

Page 7: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-13Feng-Li Lian © 2005Definitions of UB, GUB, UUB, GUUB

NTUEE-NSA-Ch4.7-14Feng-Li Lian © 2005Definitions of UB, GUB, UUB, GUUB

Page 8: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-15Feng-Li Lian © 2005Lyapunov Analysis - 1

NTUEE-NSA-Ch4.7-16Feng-Li Lian © 2005Lyapunov Analysis - 2

Page 9: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-17Feng-Li Lian © 2005Lyapunov Analysis - 3

NTUEE-NSA-Ch4.7-18Feng-Li Lian © 2005Lyapunov Analysis - 4

Page 10: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-19Feng-Li Lian © 2005Lyapunov Analysis - 5

NTUEE-NSA-Ch4.7-20Feng-Li Lian © 2005Lyapunov Analysis - 6

Page 11: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-21Feng-Li Lian © 2005Lyapunov Analysis - 7

NTUEE-NSA-Ch4.7-22Feng-Li Lian © 2005Lyapunov Analysis - 8

Page 12: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-23Feng-Li Lian © 2005Lyapunov Analysis - 9

NTUEE-NSA-Ch4.7-24Feng-Li Lian © 2005Lyapunov Analysis - 10

Page 13: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-25Feng-Li Lian © 2005Lyapunov Analysis - 11

NTUEE-NSA-Ch4.7-26Feng-Li Lian © 2005Theorem 4.18: G.U.U.B.

Page 14: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-27Feng-Li Lian © 2005Theorem 4.18: G.U.U.B.

NTUEE-NSA-Ch4.7-28Feng-Li Lian © 2005Theorem 4.18: G.U.U.B.

Page 15: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-29Feng-Li Lian © 2005Theorem 4.18: G.U.U.B.

NTUEE-NSA-Ch4.7-30Feng-Li Lian © 2005Theorem 4.18: G.U.U.B.

Page 16: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-31Feng-Li Lian © 2005Example 4.24

NTUEE-NSA-Ch4.7-32Feng-Li Lian © 2005Example 4.24

Page 17: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-33Feng-Li Lian © 2005Example 4.24

NTUEE-NSA-Ch4.7-34Feng-Li Lian © 2005Example 4.24

Page 18: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-35Feng-Li Lian © 2005Example 4.24

NTUEE-NSA-Ch4.7-36Feng-Li Lian © 2005Example 4.24

Page 19: Section 4.8 (Ultimate) Boundedness (Lyapunov Stability)

NTUEE-NSA-Ch4.7-37Feng-Li Lian © 2005Example 4.24