Approximation Properties of Receding Horizon Optimal Control

Abstract

In this survey, receding horizon control is presented as a method for obtaining approximately optimal solutions to infinite horizon optimal control problems by iteratively solving a sequence of finite horizon optimal control problems. We investigate conditions under which we can obtain mathematically rigorous approximation results for this approach. A key ingredient of our analysis is the so-called turnpike property of optimal control problems.

Turnpike Properties and Strict Dissipativity for Discrete Time Linear Quadratic Optimal Control Problems

“…Such a tour consists of three phases: driving to the highway (i.e., approaching the equilibrium), driving on the highway (i.e., staying near the equilibrium) and leaving the highway (i.e., moving away from the equilibrium). Turnpike phenomena have attracted the attention of researchers because of the structural insights they allow on the structure of the optimal solutions, particularly in mathematical economy, see, e.g., [12], but also as a method for synthesizing long term optimal trajectories [1,13,16] and in recent years for analyzing model predictive control schemes [9], [11,Chapter 8]. In this paper we investigate this property for optimal control problems with linear dynamics and a cost function consisting of quadratic, linear and constant terms.…”

Feedback, dynamics, and optimal control in climate economics

“…Conceptually, fixing the horizon a priori rules out discount rates below a certain threshold since numerically significant behavior occurs on long time scales but is not rendered insignificant by the discounting. increases 7 [29]. Extending the approximation results of [29] to include time-varying systems and discounted optimal control problems is the subject of ongoing work and some specific indications are provided at the end of this section and in Section 6.…”

“…increases 7 [29]. Extending the approximation results of [29] to include time-varying systems and discounted optimal control problems is the subject of ongoing work and some specific indications are provided at the end of this section and in Section 6.…”

On the Relation Between Turnpike Properties for Finite and Infinite Horizon Optimal Control Problems

“…Figure 3.1 shows finite horizon optimal trajectories on different horizons N which exhibit the turnpike property. For the details of the optimal control problems behind these figures we refer to [7]. Definition 3.2: (infinite horizon turnpike property) The optimal control problem (2.1) has the infinite horizon robust turnpike property at an equilibrium x e ∈ X, if for each δ > 0, each ε > 0 and each bounded set X b ⊂ X there is a constant C ∞ δ,ε,X b ∈ N such that [7] all trajectories (…”

“…This interest stems from the fact that it was realized that this property considerably simplifies the computation of (approximately) optimal trajectories in all areas of optimal control, either directly by constructive synthesis techniques as in [1] or indirectly via a receding horizon approach as in economic model predictive control [6,7]. 1 Moreover, the turnpike property can also be rigorously established in control systems governed by partial differential equations [11], significantly enlarging the classes of systems for which these methods are applicable.…”