On the stability of receding horizon control with a general terminal cost

Abstract

Abstract-We study the stability and region of attraction properties of a family of receding horizon schemes for nonlinear systems. Using Dini's theorem on the uniform convergence of functions, we show that there is always a finite horizon for which the corresponding receding horizon scheme is stabilizing without the use of a terminal cost or terminal constraints. After showing that optimal infinite horizon trajectories possess a uniform convergence property, we show that exponential stability may also be obtained with a sufficient horizon when an upper bound on the infinite horizon cost is used as terminal cost. Combining these important cases together with a sandwiching argument, we are able to conclude that exponential stability is obtained for unconstrained receding horizon schemes with a general nonnegative terminal cost for sufficiently long horizons. Region of attraction estimates are also included in each of the results.

Stability and feasibility of state constrained MPC without stabilizing terminal constraints

“…It tells us that, for a sufficiently large horizon, the MPC algorithm provides a recursively feasible and asymptotically stable closed loop on every compact set in which the value function is finite, as long as we avoid 'small' areas of bad behaviour (close to O). Note that Theorem 6 is also applicable if state constraints are present and thus extends Jadbabaie and Hauser (2005). Compared to these references, the main additional ingredient is the quantitative information on the upper bound on V ∞ provided locally by Assumption 1 and globally by the requirement…”

“…This result is then extended to compact sets lying in the domain of V ∞ and avoiding suitable defined exceptional regions O. Overall, this part of the paper can be seen as a (discrete time) extension of Jadbabaie and Hauser (2005) to the state constrained case and with additional quantitative estimates for N .…”

“…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).…”

Combined Feedforward/Model Predictive Tracking Control Design for Nonlinear Diffusion-Convection-Reaction-Systems

“…Receding horizon formulations in model predictive control often use terminal set or equality constraints to achieve stability. In the case of a free end point formulation as it is the case in (12), stability can be shown, e. g., if the terminal cost function ||Δθ(t i,f )|| 2 P represents a (local) control Lyapunov function [1,11,8] or if the horizon length t f is sufficiently large [5]. For the error dynamics (12b), which is time-dependent due to the feedforward trajectories, the rigorous proof of stability [3] as well as the consistency of this finite-dimensional control with the original infinite-dimensional system is subject of current research.…”

Model Predictive Control for Nonlinear Sampled-Data Systems

“…Remark 4.3 An important problem is the choice of a good initial guess v [0,M −1] for the optimization, keeping in mind that we deal with a nonlinear optimization problem. Even though suboptimal solutions to this problem may be sufficient to ensure stability, see [6], here we also aim at good performance. Convergence to the global optimum, however, can only be expected when the initial solution is already close to it.…”