State Space Control
Housekeeping
close all; clear; clc; s = tf('s');
DEFINITIONS & DERIVATION:
Given a system provided by the state equation 
and the output equation 
the system can be represented in state-space by the equations
and
where:
A = [0 1; 1 0];
B = [0; 1];
C = [0 1];
D = [0];
E = [0];
F = [0];
sys = ss(A,B,C,D)
Handling the E matrix
Most methods for utilizing state-space systems do not lend themsevles well towards systems that have E and F matrices (usually derived from linearlized systems), but there are a few ways that these systems can be accomodated.
Appending it to the B matrix:
Consider the system 
. By creating an augmented BE matrix and including an artificial unity input, we can arrive at the system:
Augmenting the A matrix:
Consider the system 
By creating an artificial variable ϵ we can augment the system's A matrix in such a way that our artificial variable's derivative is zero. So long as our artificial variable has an initial condition of 
we can use it to add our constant terms into our A matrix.
As we can see, as long as ϵ has an initial condition of and a zero derivative such a system allows us to "fold" the E-matrix into our A-matrix without altering the dynamics of the system. It would also be prudent to increase the size of the accompanying C-matrix and/or D-matrix with an additional 0 term.
One thing to be wary of with this technique is that our row of 0 terms will mean that the augmented system will naturally not be fully controllable or observable due to there being no way to affect the value of ϵ. This may also cause issues with matrix-based techniques such as
Noise:
If using a system such as SIMULINK, the E-matrix can be included as noise in the system rather than integrating it directly into any of the other matrices.
