Analysis of Continuous Systems - 8

Question 1

Explain the differences in the stability analysis of linear and nonlinear systems.

Answer 1

Stability analysis of linearized systems only provides results near the linearization point. There are fundamental differences
in the stability analysis of linear and nonlinear systems:
Linear systems:
* There exists only no, one, or infinitely many steady states.
* Stability is a global property: The system is stable or not.
* Stability can be constructively proven (e.g., via eigenvalues), i.e., there exists a “recipe” for proving stability.
* Controllability and observability are also global properties.
Nonlinear systems:
* There exists no, or many steady states, or limit cycles.
* Stability is not a global property: Some steady states are stable, others are not.
* Stability cannot be constructively proven, we have to guess Lyapunov functions to show stability.
* Controllability and observability are not global properties.

Question 2

What are the fundamental differences in the stability analysis of
linear and nonlinear systems?

Answer 2

Linear systems
L1 There exists only no, one, or
infinitely many steady states.
L2 Stability is a global property:
The system is stable or not.
L3 Stability can be constructively
proven (e.g., via eigenvalues),
i.e., there exists a “recipe” for
proving stability.
L4 Controllability and observability
are also global properties.
Nonlinear systems
N1 There exists no, or many
steady states, or limit cycles.
N2 Stability is not a global
property: Some steady states
are stable, others are not.
N3 Stability cannot be
constructively proven, we have
to guess Lyapunov functions to
show stability.
N4 Controllability and observability
are not global properties.