By Daniel Axehill.

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**Extra resources for Applications of integer quadratic programming in control and communication**

**Example text**

From this analysis it follows that the decrease in the objective function value when a constraint is dropped is proportional to the multiplier associated with that constraint. 20b) are fulfilled because the initial point x0 is feasible and all subsequent αk are chosen such that primal feasibility is maintained. 20d) is fulfilled by the construction of the active set algorithm. In every iteration, x ˆk+1 fulfills the KKT condition in ˆ i , i ∈ Wk ∩ I set to zero. 20c) is fulfilled. In this case, all KKT conditions are fulfilled and hence a global optimal solution to the problem has been found.

Since the bandedness of the reduced Hessian matrix is reduced as the prediction horizon grows larger than one, the usefulness of a solver optimized for banded matrices decreases. A thorough description of QPSchur can be found in [7]. Another way of reducing the on-line computational effort is to precalculate the control law. Briefly, the procedure can be explained as given the state of the system, the controller gain is retrieved from a table. The control law has been computed off-line by solving a multi-parametric programming problem, where multi-parametric means that the problem depends on a vector of parameters [91].

It is hard to say exactly when MPC was invented, but probably the first patent was granted to Martin-Sanchez in 1976, [70]. An early academic publication containing the basic ideas was presented by Propoi 1963, [70]. There are also some methods similar to MPC, but with different names. One of the most well-known is Dynamic Matrix Control (DMC), [35]. The most commonly used variant of MPC is so-called linear MPC, where the dynamics is linear and a quadratic objective similar to the one used in Linear Quadratic (LQ) control is used.