MPC for tracking piecewise constant references for constrained linear systems

On real-time robust model predictive control

“…This problem formulation does, however, not provide recursive feasibility. In this paper, a realtime MPC method for tracking is developed based on the tracking approach in Limon et al (2008), and a new warm-start is proposed providing feasibility and stability in real-time for all time constraints.…”

“…In addition, no warm-start is available satisfying the state and input constraints after a reference change due to the fact that the terminal constraint depends on the current reference. In order to extend the real-time method to reference tracking, we propose a problem formulation based on the tracking approach presented in Limon et al (2008), which was included in a tube based robust MPC framework in Limon, Alvarado, Alamo, Fiacchini, and Camacho (2010). An artificial reference is introduced into the optimization problem, which may deviate from the real reference if the latter is not a feasible target from the current state.…”

“…In contrast to the regulation case discussed in Section 4, such a warm-start is not readily available for the tracking problem, which will be discussed in more detail in the following. We show that a warmstart strictly satisfying the Lyapunov constraint for the tracking MPC problem based on Limon et al (2008) exists at all iterations and how it can be generated by a modified warm-start procedure.…”

“…More details on the tracking formulation, the terminal set for tracking, and an illustrative example can be found in Limon et al (2008), for the combination with the robust formulation see Limon et al (2010) or also Zeilinger (2011). The role of the artificial reference can also be observed in the tracking example in Section 7.2.…”

“…While a feasible and stabilizing warm-start is well-known in the regulation case, a warm-start for tracking with these properties is not directly available. Employing the tracking approach presented in Limon, Alvarado, Alamo, and Camacho (2008), which provides recursive feasibility with respect to state and input constraints, we show that a warm-start strictly satisfying the stability constraint for the tracking MPC problem exists at all iterations and can be generated with negligible computational effort. A Lyapunov function is derived for the tracking approach, which has not been shown in the literature, but is crucial for proving input-to-state stability in the real-time case.…”