Shorter horizons for model predictive control

Abstract

We provide a less conservative (compared with our previous results) lower bound on the horizon length for a stabilizing model predictive control algorithm where the terminal cost is not assumed to be a local control Lyapunov function. Our main additional assumption is that the value function is bounded by a linear function of a measure of the state uniformly in the horizon length.

Stability and feasibility of state constrained MPC without stabilizing terminal constraints

“…Tuna et al (2006) or Grüne (2012) where our local Assumption 1 was assumed on a (controlled) invariant subset of the viability kernel. We like to point out that no 'viability' conditions -such as forward invariance -nor regularity hypotheses on the dynamics f (·, ·) and the control constraint set U (·) are imposed on X in this section.…”

“…The proof is an adaptation of the arguments developed in Grüne (2012) to our setting. In particular, Variant II from Section 3.2 of this paper is used, whose idea was taken from Tuna et al (2006). Take any…”

“…In this paper we study stability and recursive feasibility of nonlinear MPC schemes without stabilizing terminal constraints or costs. For such schemes, it is known that stability for sufficiently large optimization horizons can be deduced from controllability assumptions or -alternatively and almost equivalently -bounds on the optimal value functions, see Jadbabaie and Hauser (2005); Grimm et al (2005); Tuna et al (2006); Grüne (2009) ;Grüne et al (2010); Grüne and Pannek (2011);Worthmann (2011).…”

Unconstrained nonlinear MPC: Performance estimates for sampled-data systems with zero order hold

“…In order to apply the existing results, two ways are possible. On the one hand, the system dynamics can be represented as a sampled-data system in order to treat them within the framework proposed in [15], [4]. On the other hand, the genuine continuous time approach as carried out in [13] yields significantly *This paper is a substantially revised version of [19] whose submission to the CDC was encouraged by SIAM.…”

“…[6]. Recently, techniques to determine the length of the prediction horizon such that stability is ensured were proposed, see [8] for linear and [15], [4] for nonlinear systems. In addition to stability guarantees, these methodologies yield performance estimates of the MPC closed loop compared to an infinite horizon optimal trajectory.…”