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Safe adaptive NMPC using ellipsoidal tubes

Johannes Buerger, Mark Cannon

TL;DR

The paper addresses safe learning-based NMPC for systems modeled as $x_{t+1}=f(x_t,u_t,\theta)+w_t$ with $\theta\in\Theta_0$ by introducing a successive-linearization framework that bounds prediction uncertainty with ellipsoidal tubes $\mathcal{E}(V,\beta^2)$. The online optimization is a convex SOCP that enforces tube-based constraint satisfaction and uses a backtracking line search to guarantee recursive feasibility; SME continuously refines the parameter set. A terminal-LDI based design with an SDP computes $V$, $K$, $\sigma$, and $\hat{\sigma}$ to ensure recursive feasibility and closed-loop stability, including ISS-type guarantees with a computable bound $\bar{\sigma}$. Numerical results show the approach scales favorably with problem size, delivering recursive feasibility and safety guarantees in a computationally efficient manner for moderate to large NMPC problems.

Abstract

A computationally efficient nonlinear Model Predictive Control (NMPC) algorithm is proposed for safe learning-based control with a system model represented by an incompletely known affine combination of basis functions and subject to additive set-bounded disturbances. The proposed algorithm employs successive linearization around predicted trajectories and accounts for the uncertain components of future states due to linearization, modelling errors and disturbances using ellipsoidal sets centered on the predicted nominal state trajectory. An ellipsoidal tube-based approach ensures satisfaction of constraints on control variables and model states. Feasibility is ensured using local bounds on linearization errors and a procedure based on a backtracking line search. We combine the approach with a set membership parameter estimation strategy in numerical simulations. We show that the ellipsoidal embedding of the predicted uncertainty scales favourably with the problem size. The resulting algorithm is recursively feasible and provides closed-loop stability and performance guarantees.

Safe adaptive NMPC using ellipsoidal tubes

TL;DR

The paper addresses safe learning-based NMPC for systems modeled as with by introducing a successive-linearization framework that bounds prediction uncertainty with ellipsoidal tubes . The online optimization is a convex SOCP that enforces tube-based constraint satisfaction and uses a backtracking line search to guarantee recursive feasibility; SME continuously refines the parameter set. A terminal-LDI based design with an SDP computes , , , and to ensure recursive feasibility and closed-loop stability, including ISS-type guarantees with a computable bound . Numerical results show the approach scales favorably with problem size, delivering recursive feasibility and safety guarantees in a computationally efficient manner for moderate to large NMPC problems.

Abstract

A computationally efficient nonlinear Model Predictive Control (NMPC) algorithm is proposed for safe learning-based control with a system model represented by an incompletely known affine combination of basis functions and subject to additive set-bounded disturbances. The proposed algorithm employs successive linearization around predicted trajectories and accounts for the uncertain components of future states due to linearization, modelling errors and disturbances using ellipsoidal sets centered on the predicted nominal state trajectory. An ellipsoidal tube-based approach ensures satisfaction of constraints on control variables and model states. Feasibility is ensured using local bounds on linearization errors and a procedure based on a backtracking line search. We combine the approach with a set membership parameter estimation strategy in numerical simulations. We show that the ellipsoidal embedding of the predicted uncertainty scales favourably with the problem size. The resulting algorithm is recursively feasible and provides closed-loop stability and performance guarantees.
Paper Structure (9 sections, 8 theorems, 45 equations, 1 table)

This paper contains 9 sections, 8 theorems, 45 equations, 1 table.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Linearization error bounds in prediction
  4. Tube membership conditions
  5. Ellipsoidal tube MPC subproblem
  6. Terminal constraints and terminal cost
  7. Recursive feasibility and stability
  8. Numerical results
  9. Conclusion

Key Result

Lemma 1

Condition (eq:tube_mem_cond_beta) holds for all $e\in\mathcal{E}(V,\beta_k^2)$ if for all $j\in\mathbb{N}_{\nu_{1}}$, $q\in\mathbb{N}_{\nu_\theta}$, with $\lambda_k$ defined by where $\Psi^{(r)} = (V^{-1} - w^{(r)} w^{(r)\, \top} \sigma^{-2})^{-1}$ and where $\sigma$ is a constant whose design is discussed in Section sec:termset.

Theorems & Definitions (20)

  • Definition 1
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 10 more