Design of Actuator Servo Controller - Hard Disk Drive: Mechatronics and Control - page 153

where x[i]=x(iT f ), z = e sT f , and according to sampled-data control theory,
the multirate input and output vectors u and y are defined as,

=
δ [u 1 [i], ···,u N [i]] T ,
= u(kT y ),u((k +1)T y ), ···,u((k + N −1)T y )] T ,

u[i]

(3.115)

∆ = y 1 [i], ···,y N [i]] T ,
= y(kT y ),y((k +1)T y ), ···,y((k + N −1)T y )] T ,

y[i]

(3.116)

and matrices A, B, C, D can be calculated by

⎡ ⎤

A s A N− s B s A N− s B s

··· B s

⎣ ⎦

C s D s 0 ··· 0

∙ ¸

A B
C D

C s A s C s B s D s ··· 0

=

, (3.117)

. . . .

···

C s A N−1 C s A N− s B s C s A N− s B s

··· D s

s

where P [z s ]=A s ,B s ,C s ,D s is the plant discretized by the zero order hold on
T y (= T u )andz s =e sT y . A s =e A c T N ,B s =

R

T f /N
0

e A c τ B c dτ.

Toverifytheabove,wehave

x[k +1] = A s x[k]+B s u[k], (3.118)

x[k +2] = A s x[k +1]+B s u[k +1],
= A s x[k]+A s B s u[k]+B s u[k +1], (3.119)

x[k +3] = A s x[k +2]+B s u[k +2],
= A s x[k]+A s B s u[k]+A s B s u[k +1]+B s u[k +2], (3.120)
.
x[k + N]=A s x[k]+A N− s B s u[k]+···
+A s B s u[k + N −2] + B s u[k + N −1], (3.121)

y[k]=C s x[k]+D s u[k], (3.122)

y[k +1] = C s x[k +1]+D s u[k +1],
= C s A s x[k]+C s B s u[k]+D s u[k +1], (3.123)

y[k +2] = C s x[k +2]+D s u[k +2],
= C s A s x[k]+C s A s B s u[k]+C s B s u[k +1]+D s u[k +2],
(3.124)

.

(3.125)

y[k + N −1] = C s A N−1

s x[k]+C s A N− s B s u[k]+
···+ C s B s u[k + N −2] + D s u[k + N −1].

(3.126)