The Development of Neural Network Based System Identification ...

6.8 MODEL PREDICTIVE CONTROL WITH CONSTRAINTS 175
with optimal control decision variable ∆U. The Active Set method requires an iterative
updating procedure to test the constraint conditions (active or inactive) before solving
for optimal decision variable. The main drawback of the Active Set method is that
it produces quite a high computation load if many constraints are imposed on the
optimisation problem [Wang, 2009d, Truong, 2007].
The Primal-Dual method such as Hildreth’s QP procedure can be used to overcome
the computation burden of the Active Set method. Generally, the Primal-Dual method
systematically identifies the inactive constraints and eliminated them directly in the
solution [Wang, 2009d]. This would lead to a much simpler programming implementation
for finding the optimal solution for the constrained control problem. To be consistent
with QP literature notation, the cost function in Equation (6.31) and the associate
constraints are reformulated as:
J = 1 2 xT Ex + x T F
subject to Mx ≤ γ (6.37)
where matrix E and F are compatible matrices and vectors in the quadratic programming
problem in Equation (6.31). Variable x denotes the decision variable or the control
decision variable ∆U. Matrix E is also known as the Hessian matrix with symmetric
mN c × mN c dimension and F is a column vector with mN c elements.
In order to minimise the objective function (6.31) subject to inequality constraints
Mx ≤ γ, the following Lagrange expression is considered:
J = 1 2 xT Ex + x T F + λ T (Mx − γ) (6.38)
where λ denotes the Lagrange Multiplier vector with a dimension equivalent to D
number of constraints. The Equation (6.38) is also known as the primal problem in
literature. The Lagrange Multiplier λ indicates whether a constraint is either active or
inactive. By definition, the elements in Lagrange Multiplier vector are non-negative
and if the element is λ i = 0 then the i th constraint is inactive, which indicates that a