Since W > 0 and Q > 0 this is a convex quadratic program in z parameterized by p. Convexity guarantees that a global solution can be found. The optimal solution z*(p) is a continuous piecewise linear function z*(p) defined on any subset,

of the parameter space. Moreover, an exact representation of this piecewise linear function can be computed off-line using multi-parametric QP algorithms (T0ndel and Johansen, 2003b) or the Matlab Multi-Parametric Toolbox (MPT) by Kvasnica, Grieder and Baotic (2004). Consequently, it is not necessary to solve the QP (1.36) in real time for the current value of T and the parameters fmin, fmax and ยก3 , if they are allowed to vary.

In fact it suffices to evaluate the known piecewise linear function z*(p) as a function of the given parameter vector p which can be done efficient with a small amount of computations. For details on the implementation aspects of the mp-QP algorithm; see Johansen et al. (2003) and references therein. An on-line control allocation algorithm is presented in T0ndel et al. (2003a).

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