The Development of Neural Network Based System Identification ...

6.7 MODEL PREDICTIVE CONTROL OPTIMISATION 171
the necessary condition to obtain the minimisation of J is by setting:
∂J
∂∆U = 0
which leads to the following solution for the incremental control signal within one
optimisation window:
∆U = (Φ T Φ + ¯R) −1 Φ T (R s − Γx(k)) (6.33)
where the matrix Φ T Φ has dimension of mN c ×mN c , matrix Φ T Γ has dimension mN c ×n
and matrix Φ T ¯Rs has dimension mN c × q. Matrix ¯R denotes a diagonal matrix with
dimension equal to Φ T Φ. The matrix ¯R takes the form of ¯R = r w I mNc×mNc (r w ≥ 0)
with r w used as control penalty factor to achieve the desired closed loop performance.
The reference signal is given by r(k) = [r 1 (k) r 2 (k) r 3 (k) · · · r q (k)] T , which indicates
the q reference signals to the multi output system.
Since the MPC control only uses the first m elements in ∆U, the final form of
incremental optimal control is given as follows:
∆U = [I m O m · · · O m ](Φ T Φ +
} {{ }
¯R) −1 Φ T (R s − Γx(k))
N c
= K y r(k) − K mpc x(k) (6.34)
where I m and O m are the identity and zero matrix with dimension m × m respectively.