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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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