91
z-transform of x (k) and U (z) denote the z-transform of u (k) we have
JL OtrtZ
u (z) = 2 (5-24)
z 2zcoswT + 1
The minimum polynomial having roots corresponding to the poles of both
jlf
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-l)(z2- 2zcoscdT + 1) =
z3 [l+2coso)T]z2 + [l+2coswT]z 1
(5-25)
Hence, the difference equation is
s(k+3) [l+2coswT]s(k+2) + [l+2cosuT]s(k+1) s(k) = 0 (5-26)
s(j) = x*(j) or u*(j)
and the matrix is
Ch =
0
0
1
1
0
0
1
-(l+2coswT) (1+2cosiT)
(5-27)
We now consider selection of the feedback gains. Notice in Figure
5-4 that the feedback control law is