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

Florian Messerer, Yunfan Gao, Jonathan Frey, Moritz Diehl

TL;DR

The paper tackles online optimization of internal feedback laws in tube-based robust and stochastic MPC by introducing Riccati-ZORO, a heuristic method that alternates between a nominal OCP with fixed backoffs and a tube-space optimization solved via a Riccati recursion. By focusing on ellipsoidal tubes under linear state feedback, the approach achieves substantial computational savings, reducing per-iteration complexity from $ obreakmath olinebreak ext{O}(n_x^6)$ to $ obreakmath olinebreak ext{O}(n_x^3)$, comparable to a nominal OCP. It offers two uncertainty-cost designs—trajectory-independent and constraint-adaptive weighting—to steer the tube optimization toward directions that most impact constraint backoffs, and provides both CasADi prototyping and acados high-performance implementations. The method enables heuristic online refinement of the feedback gains that govern the uncertainty tube, potentially improving constraint satisfaction and performance in robust and stochastic MPC settings. Overall, Riccati-ZORO advances real-time applicability of online tube-tuning by leveraging a Riccati-based decomposition and principled weighting strategies, with practical impact for safe and efficient MPC in uncertain environments.

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 in detail 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.

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

TL;DR

The paper tackles online optimization of internal feedback laws in tube-based robust and stochastic MPC by introducing Riccati-ZORO, a heuristic method that alternates between a nominal OCP with fixed backoffs and a tube-space optimization solved via a Riccati recursion. By focusing on ellipsoidal tubes under linear state feedback, the approach achieves substantial computational savings, reducing per-iteration complexity from to , comparable to a nominal OCP. It offers two uncertainty-cost designs—trajectory-independent and constraint-adaptive weighting—to steer the tube optimization toward directions that most impact constraint backoffs, and provides both CasADi prototyping and acados high-performance implementations. The method enables heuristic online refinement of the feedback gains that govern the uncertainty tube, potentially improving constraint satisfaction and performance in robust and stochastic MPC settings. Overall, Riccati-ZORO advances real-time applicability of online tube-tuning by leveraging a Riccati-based decomposition and principled weighting strategies, with practical impact for safe and efficient MPC in uncertain environments.

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 in detail 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 to , 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.

Paper Structure

This paper contains 14 sections, 1 theorem, 15 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem structure
  3. Related work
  4. Contribution and Outline
  5. Notation and preliminaries
  6. Linearization-based ellipsoidal tube OCP
  7. Linear quadratic regulator in tube space
  8. Riccati-ZORO
  9. Structure of the algorithm
  10. Nominal OCP with fixed backoffs
  11. Tube optimization via Riccati recursion
  12. Uncertainty cost design
  13. Trajectory-independent weighting
  14. Constraint-adaptive weighting

Key Result

lemma 1

Let $Q_N \succeq 0$, $C_k \succeq 0$, $Q_k \succeq 0$, $R_k \succ 0$, for $k=0, \dots, N-1$. Further, let $P_k \succ 0$ for all $P_k$ reachable with dynamics ocp:lqr_matrixform:dynamics, $k=0,\dots,N$. Then, the solution to ocp:lqr_matrixform is uniquely defined by the backward Riccati recursion for $k = N-1, \dots, 0$, followed, for $k=0, \dots, N-1$, by the forward Lyapunov recursion

Figures (1)

  • Figure 1: Schematic structure of Riccati-ZORO

Theorems & Definitions (3)

  • remark 1: Complexity of the standard OCP structure
  • remark 2: Stochastic interpretation
  • lemma 1: Messerer2021