z-transform of x*(k) and U*(z) denote the z-transform of u*(k) we have


                        X*(z)  -    1 z- 1


(5-24)


U*(z)         a _2z
        z- 2zcoswT + 1


The minimum polynomial having roots corresponding to the poles of both X*(z) and U*(z) determines the difference equation (5-17).        In our

example, this   polynomial  is obtained    by  multiplying together the denominator polynomials of X*(z) and U*(z). The result is the following


                 (z-1)(z2- 2zcoswT + 1) =


(5-25)


z3 - [l+2coswT]z2 + [l+2coswT]z - 1


Hence, the difference equation is


         s(k+3) - [1+2coswT]s(k+2) + [1+2coswT]s(k+1) - s(k) = 0   (5-26) s(j) = x*(j) or u*(j)


and the matrix Cd is


     1
     0
-(1+2coswT)


    0    1
    1
(1+2coswT)j


     We now consider selection of the feedback gains. Notice in Figure 5-4 that the feedback control law is


Cd      0


(5-27)