We now decide to use a linear combination of the observable states contained in vector y to determine the input controlling voltage.

= 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 , 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.

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