Riccati-ZORO: An efficient algorithm for heuristic online optimization of internal feedback laws in robust and stochastic model predictive control

TL;DR

Riccati-ZORO is a novel algorithm that efficiently optimizes feedback laws in robust stochastic model predictive control by alternately solving nominal and tube-based optimal control problems, reducing computational complexity.

Contribution

It introduces a heuristic-based joint optimization approach for the nominal trajectory and uncertainty tube, significantly lowering computational complexity for ellipsoidal tubes.

Findings

Reduces complexity from O(n_x^6) to O(n_x^3) for linear feedback tubes.

Provides open-source implementations in CasADi and acados.

Demonstrates effectiveness through numerical experiments.

Abstract

We present Riccati-ZORO, an algorithm for tube-based optimal control problems (OCP). Tube OCPs predict a tube of trajectories in order to capture predictive uncertainty. The tube induces a constraint tightening via additional backoff terms. This backoff can significantly affect the performance, and thus implicitly defines a cost of uncertainty. Optimizing the feedback law used to predict the tube can significantly reduce the backoffs, but its online computation is challenging. Riccati-ZORO jointly optimizes the nominal trajectory and uncertainty tube based on a heuristic uncertainty cost design. The algorithm alternates between two subproblems: (i) a nominal OCP with fixed backoffs, (ii) an unconstrained tube OCP, which optimizes the feedback gains for a fixed nominal trajectory. For the tube optimization, we propose a cost function informed by the proximity of the nominal trajectory to constraints, prioritizing reduction of the corresponding backoffs. These ideas are developed for ellipsoidal tubes under linear state feedback. In this case, the decomposition into the two subproblems yields a substantial reduction of the computational complexity with respect to the state dimension from $\mathcal{O}(n_x^6)$ to $\mathcal{O}(n_x^3)$, i.e., the complexity of a nominal OCP. We investigate the algorithm in numerical experiments, and provide two open-source implementations: a prototyping version in CasADi and a high-performance implementation integrated into the acados OCP solver.