Chapter-6
Internal Stability
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
Uniform Stability
3
Uniform Stability
4
Internal Stability
Example
Solution
5
Internal Stability
6
Uniform Exponential Stability
7
Internal Stability
8
Internal Stability
9
Internal Stability
Linear time invariant systems
Marginal Stability



Eigenvalue(s) with zero real part
Stable for all initial conditions
Stable for all inputs except those with resonant frequency
10
Internal Stability
Uniform Asymptotic Stability
11
Internal Stability
Lyapunov Transformation
Equivalent Condition
12
Internal Stability
Lyapunov Transformation: Application
A(t  T )  A(t )
14
Chapter-7
Lyapunov Stability Criteria
EE-826 Linear Control System
College of Electrical & Mechanical Engineering
National University of Sciences & Technology (NUST)
17
Lyapunov Stability Criteria
Motivation
18
Quadratic Lyapunov Function
19
Lyapunov Stability Criteria
Proof
20
Lyapunov Stability Criteria
Example 7.3
21
Lyapunov Stability Criteria
Uniform Exponential Stability
Proof
22
Uniform Exponential Stability
Proof
23
Lyapunov Stability Criteria
Example
24
Lyapunov Stability Criteria
25
Lyapunov Stability Criteria
26