6. Stability and Suboptimality Without Stabilizing Terminal Conditions

In this chapter, we present a comprehensive stability and suboptimality analysis for NMPC schemes without stabilizing terminal conditions. After defining the setting and presenting motivating examples, we introduce a boundedness condition on the optimal value function and an asymptotic controllability assumption. Moreover, we give a detailed derivation of stability and performance estimates based on these assumptions and the relaxed dynamic programming framework introduced before. We show that our stability criterion is tight for the class of systems satisfying the controllability assumption and give conditions under which the level of suboptimality and a bound on the optimization horizon length needed for stability can be explicitly computed from the parameters in the controllability condition. As a spinoff we recover the well-known result that—under suitable conditions—stability of the NMPC closed loop can be expected if the optimization horizon is sufficiently large. We further deduce qualitative properties of the stage cost which lead to stability with small optimization horizons and illustrate by means of two examples how these criteria can be used even if the parameters in the controllability assumption cannot be evaluated precisely. Finally, we give weaker conditions under which semiglobal and semiglobal practical stability of the NMPC closed loop can be ensured.