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)