1. Eigenvalues, Zeros and Poles
(1)
Definition: Roots of the characteristic polynomial of a system are called eigenvalues of the system.
characteristic
polynomial
eigenvalues
nxn
nx1

2. Transfer function of the system (1):
In general is a matrix is then the transfer function from the input to the output
After cancelling the common terms in nominator and denominator, one gets
Definition:
Poles of (or the system (1)) are the roots of
Zeros of (or the system (1)) are the roots of
Result: Poles of a system are a subset of system eigenvalues.

3. Generalization of the notion ...........
Stability
Zero input stability:
(2)
Definition: (Equilibrium) A constant solution of the system (2) is called an equilibrium of the system
How to find ?
Solve !
What is for linear systems?
If A is invertible then is zero. Otherwise, there are infinitely many equilibria. Solve
Definition (Norm): Let V be a vector space (a space with appropriate addition and multiplication by scalars operations). Norm of a vector is defined as a positive valued function that satisfies.
Generalization of the notion

4. Definition: (Lyapunov stability)
Let be an equilibrium of the system
The equilibrium is called Lyapunov stable if for every arbitrary small
, there exists a such that
Definition: (Asymptotic stability)
Let be a Lyapunov stable equilibrium of the system
Then is asymptotically stable if there exists a such that

5. The zero equilibrium of the system is Lyapunov stable if and only if
Theorem:
The zero equilibrium of the system is Lyapunov stable if and only if
Proof:
Solution:
Norm property
Let be Lyapunov stable but not bounded.
Then there exist a which is not bounded. Choose i.
Then,
This result is contraditory. Therefore,

6. Theorem:
The zero equilibrium of the system is Lyapunov stable if and only if
Proof: Similar to the previous proof.
For the system , let the eigenvalues of the system be Then,
Theorem: 1)
is Lyapunov stable
and eigenvalues with have multiplicity one. 2)
is asymptotically stable

7. Definition: (BIBO sability) The system is bounded input bounded output (BIBO) stable if and only if output of the system is bounded for all bounded inputs.
Theorem:
is BIBO stable All poles of have negative real part.
Theorem:
is asymptotically stable is BIBO stable.

8. Example:
Find the equilibria and analyze the stability for the following system!

9. Example:
Find the transfer function for the following system! Is the system
Lyapunov stable?
asymptotically stable?
BIBO stable?