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.


           vsubi = KY

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 vsubi , the input controlling voltage to the servo motor. We also have decided to set vsubi = 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, p33_7 such that the matrix (G + HKC) is stable.