Is multiple-objective model-predictive control “optimal”?

We consider multiple-objective model-predictive control (MPC) of a linear time-invariant (LTI) single-input single-output (SISO) system (for simplicity without input constraints and/or disturbances). The performance index is the sum of weighted convex functionals J = Σ_i=1ⁿw_iJ_i (with w_i ≥ 0). Although, by theory, the overall model-predictive control problem has a unique, globally optimal solution, this does not imply optimality of each sub-performance index Ji. To achieve “desirable” control performance, one has to find “decent” weighting factors wi; often done by “trial-and-error” which should be avoided, since weighting factor design might be not intuitive (even for LTI SISO systems without constraints; as we will show). The inherent difficulty lies in the mismatch between the “human performance index” (the optimality measure in the mind of the control engineer) and the implemented performance index J. In this paper, we illustrate these difficulties for a simple, linear third-order system and present some (old and new) approaches to ease weighting factor design. We do not give full answers but discuss first ideas which are admissible within the theoretical framework of standard MPC of LTI SISO systems.