Internal Stability

Chapter-6

EE-826 Linear Control System

College of Electrical & Mechanical Engineering

National University of Sciences & Technology (NUST)

Internal Stability

� Concerned with the boundedness properties and

asymptotic behaviour of the unforced system

� Stability behaviour of the system irrespective of the initial

time and initial states

3

Uniform Stability

Internal Stability

Example

Solution

4

Internal Stability

Uniform Exponential Stability

5

Uniform Exponential Stability

Internal Stability

6

Internal Stability

Linear time invariant systems

Marginal Stability

� Eigenvalue(s) with zero real part

7

� Eigenvalue(s) with zero real part

� Stable for all initial conditions

� Stable for all inputs except those with resonant frequency

Internal Stability

Lyapunov Transformation

Equivalent Condition

8

Equivalent Condition

Internal Stability

Lyapunov Transformation: Application

( ) ( )A t T A t+ =

9

Lyapunov Stability Criteria

Chapter-7

EE-826 Linear Control System

College of Electrical & Mechanical Engineering

National University of Sciences & Technology (NUST)

Lyapunov Stability Criteria

Motivation

11

Lyapunov Stability Criteria

Proof

12

Proof

Lyapunov Stability Criteria

Example 7.3

13

Lyapunov Stability Criteria

Uniform Exponential Stability

14

Proof

Uniform Exponential Stability

Proof

15

Lyapunov Stability Criteria

Example

16

Lyapunov Stability Criteria

17

Lyapunov Stability Criteria

18