The Development of Neural Network Based System Identification ...

170 CHAPTER 6 NEURAL NETWORK BASED PREDICTIVE CONTROL SYSTEM
where,
⎡ ⎤ ⎡
⎤
CA
CB 0 0 · · · 0
CA 2
CAB CB 0 · · · 0
Γ =
CA 3
; Φ =
CA 2 B CAB CB · · · 0
⎢ . ⎥ ⎢ .
⎥
⎣ ⎦ ⎣
⎦
CA Np CA Np−1 B CA Np−2 B CA Np−3 B · · · CA Np−Nc B
(6.30)
Again, for the SISO case, the dimension of vector Ŷ is N p × 1 and the dimension of
vector ∆U is N c × 1. The dimension of matrix Γ is N p × n and matrix Φ is N p × N c .
Considering a MIMO case, the dimension for Ŷ and ∆U are vectors mN p × 1 and
qN c × 1. Subsequently, the dimension of matrix Γ and Φ for MIMO case is set to
qN p × n and qN p × mN c respectively.
6.7 MODEL PREDICTIVE CONTROL OPTIMISATION
The main objective of predictive control design is to bring the predicted output of the
system as close as possible to the reference signal within the prediction horizon, for a
given reference signal r(k) at sample time k. Here, the reference signal r(k) is assumed
to remain constant throughout the optimisation window. This could be achieved by
finding the optimal solution of control parameter vector ∆U such that the error cost
function between the reference signal r(k) and predicted output Ŷ is minimised.
In order to find the optimal solution of ∆U that minimises Equation (6.1), the cost
function is expressed as the following form after substituting Equation (6.29) into (6.1):
J = (R s − Γx(k)) T (R s − Γx(k)) − 2∆U T Φ T (R s − Γx(k))
+ ∆U T (Φ T Φ + ¯R)∆U (6.31)
Taking the first derivative of the cost function J:
∂J
∂∆U = −2ΦT (R s − Γx(k)) + 2(Φ T Φ + ¯R)∆U (6.32)