Linearly discounted economic MPC without terminal conditions for periodic optimal operation

TL;DR

This work introduces a linearly discounted economic MPC without terminal conditions to achieve optimal performance when the system operates around a periodic orbit. By weighting the stage costs linearly across the horizon, the method mitigates end-of-horizon effects and does not require offline knowledge of the optimal period. Under standard dissipativity and controllability assumptions, the authors prove that the resulting scheme attains optimal asymptotic average performance up to a vanishing error and guarantees practical asymptotic stability of the optimal periodic orbit, with rigorous support from weak turnpike analysis and a rotated Lyapunov framework. Numerical experiments on a harmonic oscillator and an economic growth model illustrate improved transient behavior and near-optimal long-run performance compared to undiscounted MPC, while highlighting slower convergence speed in some growth scenarios. Overall, LDE-MPC offers a terminal-condition-free, robust approach to achieving near-optimal, stable periodic operation in economic control problems without requiring extensive offline design.

Abstract

In this work, we study economic model predictive control (MPC) in situations where the optimal operating behavior is periodic. In such a setting, the performance of a standard economic MPC scheme without terminal conditions can generally be far from optimal even with arbitrarily long prediction horizons. Whereas there are modified economic MPC schemes that guarantee optimal performance, all of them are based on prior knowledge of the optimal period length or of the optimal periodic orbit itself. In contrast to these approaches, we propose to achieve optimality by multiplying the stage cost by a linear discount factor. This modification is not only easy to implement but also independent of any system- or cost-specific properties, making the scheme robust against online changes therein. Under standard dissipativity and controllability assumptions, we can prove that the resulting linearly discounted economic MPC without terminal conditions achieves optimal asymptotic average performance up to an error that vanishes with growing prediction horizons. Moreover, we can guarantee practical asymptotic stability of the optimal periodic orbit under the additional technical assumption that dissipativity holds with a continuous storage function. We complement these qualitative guarantees with a quantitative analysis of the transient and asymptotic average performance of the linearly discounted MPC scheme in a numerical simulation study.

Figures (8)

• Figure 1: Illustration of the states $x$ (nodes) and feasible transitions (edges) with corresponding input $u$ and cost $\ell$ in Example \ref{['exmp:motivation_disc']}. This diagram is taken from Mueller2016.
• Figure 2: Comparison of the cost of strategies $u^1$ and $u^2$ for different $N$ depicted for both, the linearly discounted and the undiscounted cost functionals. A negative value indicates that strategy $u^1$ results in a better cost than strategy $u^2$.
• Figure 3: The cost $\ell^p(\Pi^p)$ in Example \ref{['exmp:oscillator']} of the optimal periodic orbit $\Pi^p$ of length $p$. We observe that the optimal period length is $p^\star = 6$ as $\ell^p(\Pi^p)$ is minimal for all $p\in 6\mathbb{N}$.
• Figure 4: Closed-loop trajectories $x_{\mu_N^1}(\cdot, x_0)$ in Example \ref{['exmp:oscillator']} of the undiscounted MPC scheme for different initial conditions $x_0\in \{x_0^1, x_0^2, x_0^3\}$.
• Figure 5: Asymptotic average performance in Example \ref{['exmp:oscillator']} of the undiscounted MPC $\mu_N^1$, the linearly discounted MPC $\mu_N^\beta$, and the $p^\star$-step MPC $\nu_N$.

Theorems & Definitions (24)

• Definition 2: Optimal periodic orbit
• Example 6: Motivating example
• Definition 7: Weak turnpike property
• Theorem 8: Weak turnpike property
• Corollary 9: Recursive feasibility
• Theorem 10: Asymptotic average performance
• Definition 11: Practical asymptotic stability
• Remark 12
• Definition 13: Practical Lyapunov function
• Theorem 14: Lyapunov function $\Rightarrow$ stability