Stability conditions of discrete system

[jumpboy]

Homework Statement

Consider a discrete-time system, driven by:

x[k+1] = Ax[k]

for non-zero inital conditions x[0]

a) write the closed-form solution of x[k]. If the system is asymptotically stable how should x(k) behave?
b) what is condition for asymptotic stability?

Homework Equations

although this is a general theoretical question, for the remainder of the problem that I was able to work out A is a real 3x3 matrix.
x_e is an equilibrium point of the system
delta is region that initial conditions place the system at
epsilon is barrier of the stability region

The Attempt at a Solution

a)
Closed-form solution = I have no clue what this even means.

asymptotic stability: x[k] should converge to an X_e for t>= 0 as t approaches infinity. I believe that I need to specify a relationship between the initial conditions and the system A but I am unsure how to relate them outside:

||x[0] - x_e|| <= delta => ||x[k] - x_e|| <= epsilon

b)
A system of this form will be asymptotic stable if the eigenvalues of A have a modulus less than 1.

however my textbook (Feedback Systems, Astrom & Murray) states that the eigenvalues must lie in the open left half plane and not necessarily have modulus of 1.
thanks in advance for any help

 

[lanedance]

so you can write
x[k+1] = Ax[k] = A^k.x[0]

now assuming A has a complete basis of eigenvectors, consider expanding x[0] in terms of the eigenvectors...

as for what the textbook says, make you sure you are comparing apples with apples... see below
http://en.wikipedia.org/wiki/Lyapunov_stability#Stability_for_linear_state_space_models

there is a difference if it is written in terms of derivatives, or discrete timesteps