Discussion of Feedback
We now decide to use a linear combination of the observable states contained in vector y to
determine the input controlling voltage.
Our original state space equation was
x(k+1) = Gx(k) + Hv(k)
y = Cx(k)
Now observe that our input v(k) is actually
, the input controlling voltage to the servo motor.
We also have decided to set
= KY, and we know y = Cx(k)
Thus we now have:
x(k+1) = Gx(k) + HkCx(k)
x(k+1) = (G + HkC)x(k)
Note that in this case HK is the outer product of two vectors producing a square matrix.
If we assume an initial x vector x(0), we then have
As k increases, the components of vector x(k) will tend toward zero only if the
eigenvalues of the matrix (G + HKC) all have magnitude less than one. It is now
our job to chose the four values,
such that the matrix (G + HKC) is stable.