The Development of Neural Network Based System Identification ...

176 CHAPTER 6 NEURAL NETWORK BASED PREDICTIVE CONTROL SYSTEM
solution has satisfied the constraint condition Mx − γ < 0. In contrast, if the elements
in the Lagrange Multiplier is λ i > 0, the corresponding constraint is active.
The estimation of the constraints can be obtained using a dual method as suggested
in Wang and Young [2006]. Assuming there is a solution of ∆U that satisfy Mx − γ < 0,
the dual problem to the original QP problem can be stated as follows:
[ 1
max min
λ≥0 x 2 xT Ex + x T F + λ T (Mx − γ)]
(6.39)
The minimisation over decision variable x is assumed unconstrained and the optimal
solution is given as:
x = −E −1 F − E −1 M T λ (6.40)
By inserting Equation (6.40) into Equation (6.39), the dual problem can be written as:
[
max − 1
λ≥0 2 λT Lλ − λ T K − 1 ]
2 F T E −1 F
(6.41)
where the matrices L and K are given by:
L = ME −1 M T (6.42)
K = ME −1 F + γ (6.43)
The Equation (6.41) is also a QP problem and is equivalent to:
[
min λ T Lλ + λ T K + 1 ]
λ≥0
2 γT E −1 γ
(6.44)
The set of optimal Lagrange multipliers that minimise the dual function in Equation
(6.44) are denoted as λ ∗ . Using the value of λ ∗ , the decision variable x is obtained for
the MPC control using the following formulation:
x = −E −1 F − E −1 M T λ ∗ (6.45)
In order to obtain λ ∗ that approximates Equation (6.39), the Hildreth’s quadratic