Exercise: Properties of linear discrete time systems
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Consider the following discrete time system
where
In this problem, we will explore some of the properties of this discrete time system as a function of the parameters, the initial conditions, and the inputs.
Assume that the off diagonal element a12 = 0 and that there is no input, u = 0. Write a closed form expression for the output of the system from a nonzero initial condition z[0] = (z1[0],z2[0]) and give conditions on a11 and a22 under which the output gets smaller as k gets larger.
Now assume that
and write a closed form expression
for the response of the system from a nonzero initial conditions. Given a
condition on the elements of A under which the output gets smaller as k
gets larger.
Write a MATLAB program to plot the output of the system in response to a unit step input, u[k] = 1,
. Plot the response of your system
with z[0] = 0 and A given by
