You may recall from the study of differential equations governing continuous systems that a linear
Nth order differential equation can be decomposed in to a set of N first order differential
equations. The state-space method (in S&K called the state-space model) is the discrete systems,
difference equation equivalent.
Though this method is capable of dealing with much more complicated problems we will restrict our discussion (as in S&K Chap 7) to the solution of a single Nth order DE with no delays in the input :
For convenience we define a set of state variables (the state vector) :
So the 
represent previous output states which would be known. We want to advance one step and compute :
The state equations are a set of first-order DE's in the 
:
etc
Put this in matrix form for computational purposes :
Earliest at the top
Latest at bottom
Then reading off the bottom value 
:
This is normally calculated using the MATLAB function dlsim .
Example - use of dlsim
Find the response of a system described by the third-order DE :
for 
with inputs x(n)=u(n) and initial conditions y(-3)=-3 ,
y(-2)=-2 , y(-1)=-1
MATLAB - use dlsim with N=3.
