Model Predictive Control System Design and Implementation Using ...

306 9 Classical MPC Systems in State-space Formulation
where the polynomial F (z) andH(z) are defined as
G m (z) = H(z)
F (z) z−d , (9.20)
F (z) =z n + f 1 z n−1 + ...+ f n
H(z) =h 1 z n−1 + h 2 z n−2 + ...+ h n .
As the number of time delays is counted as part of the model order in the
discrete-time system, the minimal realization of a state space model has the
number of state variables equal to n+d. However, in a non-minimal realization
by taking the state variables as
x m (k) T = [ y(k) y(k − 1) ... y(k − n +1)u(k − 1) ... u(k − n − d +1) ] ,
the number of state variables is equal to 2n + d − 1. 2 With this choice of state
variables, we obtain the particular state-space realization as
x m (k +1)=A m x m (k)+B m u(k)
y(k) =C m x m (k). (9.21)
The matrices A m , B m , C m are defined as the block matrices
[ ] [ ]
A1 A
A m = 2
B1
; B
A 3 A m = ; C
4 B m = [ ]
C 1 C 2 ,
2
where the matrix A 1 has dimension n × n, A 4 has the dimension (n + d − 1) ×
(n + d − 1), A 2 has the dimension n × (n + d − 1), and A 3 is a zero matrix
and has the dimension (n + d − 1) × n. More specifically, for d ≠0,
⎡
⎤
−f 1 −f 2 ...−f n−1 −f n
⎡
⎤
1 0 ... 0 0
0 ... h 1 ... h n
A 1 =
0 1 ... 0 0
0 0 ... 0 0
; A 2 = ⎢
⎥
⎢ .
⎣
.
. .. . .. . .. . ..
⎥ ⎣
⎦ . . . . . ⎦
0 0 ... 0 0
0 0 ... 1 0
⎡
⎤
0 0 ... 0 0
1 0 ... 0 0
A 4 =
0 1 ... 0 0
⎢
⎣
. . ,
. . . .. . .. . ..
⎥
⎦
0 0 ... 1 0
B 1 has the dimension n × 1andB 2 has the dimension (n + d − 1) × 1with
the following forms:
2 Note that in the case that n = 1, the number of state variables in the NMSS is
equal to the number of states in the minimal realization.