Stability and Convergence of a Randomized Model Predictive Control Strategy

TL;DR

Stability and convergence estimates are derived for RBM-MPC of unconstrained linear systems and are validated in a numerical example that shows a clear computational advantage of RBM-MPC.

Abstract

RBM-MPC is a computationally efficient variant of Model Predictive Control (MPC) in which the Random Batch Method (RBM) is used to speed up the finite-horizon optimal control problems at each iteration. In this paper, stability and convergence estimates are derived for RBMMPC of unconstrained linear systems. The obtained estimates are validated in a numerical example that also shows a clear computational advantage of RBM-MPC.

Figures (3)

• Figure 1: The RBM-MPC control and state trajectory $u_{R-M}(\Omega_i,t)$ and $\bm{x}_{R-M}(\Omega_i, t)$ for 20 realizations of $\Omega_i$ compared to $u_M(t)$, $\bm{x}_M(t)$, $u^*_\infty(t)$, and $\bm{x}^*_\infty(t)$ for $n=100$, $h=1$, $\tau = 10$, and $T = 15$. The lines for $|\bm{x}_{R-M}(\Omega_i,t)|$ and $|\bm{x}_M(t)|$ in Figure \ref{['fig:results_time_state']} almost overlap.
• Figure 2: Differences between the RBM-MPC state trajectory $\bm{x}_{R-M}(\Omega_i, t)$, the MPC state trajectory $\bm{x}_M(t)$, and the infinite horizon state trajectory $\bm{x}^*_\infty(t)$ for $n =100$. The error bars indicate the $2\sigma$ confidence intervals estimated based on 20 realizations of $\Omega_i$.
• Figure 3: Differences between the RBM-MPC control $u_{R-M}(\Omega_i,t)$, the MPC control $u_M(t)$, and the infinite horizon control $u^*_\infty(t)$ for $n =100$. The error bars indicate the $2\sigma$ confidence intervals estimated based on 20 realizations of $\Omega_i$.

Theorems & Definitions (25)

• Remark 1
• Remark 2
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 3
• Lemma 3
• proof
• Lemma 4