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A successive convexification approach for robust receding horizon control

Yana Lishkova, Mark Cannon

TL;DR

The paper develops a robust nonlinear model predictive control framework for systems with convex state and control constraints and dynamics expressible as a difference of convex functions, $f(x,u)=g(x,u)-h(x,u)$. A nominal strategy (cMPC) uses successive linearizations around a seed trajectory to transform the NLP into a sequence of convex programs, with tubes bounding linearization errors that ensure recursive feasibility and convergence to a local optimum in the disturbance-free setting. It then extends to additive disturbances by introducing RcMPC, which optimizes over seed tubes and linearization points to maintain feasibility under uncertainty, and proves a form of quadratic stability for the closed-loop system. The approach yields online convex subproblems, obviates the need for a priori error bounds, and supports early termination without compromising stability or feasibility, making it suitable for real-time robust receding-horizon control.

Abstract

A novel robust nonlinear model predictive control strategy is proposed for systems with nonlinear dynamics and convex state and control constraints. Using a sequential convex approximation approach and a difference of convex functions representation, the scheme constructs tubes that contain predicted model trajectories, accounting for approximation errors and disturbances, and guaranteeing constraint satisfaction. An optimal control problem is solved as a sequence of convex programs. We develop the scheme initially in the absence of external disturbances and show that the proposed nominal approach is non-conservative, with the solutions of successive convex programs converging to a locally optimal solution for the original optimal control problem. We extend the approach to the case of additive disturbances using a novel strategy for selecting linearization points. As a result we formulate a robust receding horizon strategy with guarantees of recursive feasibility closed-loop system stability.

A successive convexification approach for robust receding horizon control

TL;DR

The paper develops a robust nonlinear model predictive control framework for systems with convex state and control constraints and dynamics expressible as a difference of convex functions, . A nominal strategy (cMPC) uses successive linearizations around a seed trajectory to transform the NLP into a sequence of convex programs, with tubes bounding linearization errors that ensure recursive feasibility and convergence to a local optimum in the disturbance-free setting. It then extends to additive disturbances by introducing RcMPC, which optimizes over seed tubes and linearization points to maintain feasibility under uncertainty, and proves a form of quadratic stability for the closed-loop system. The approach yields online convex subproblems, obviates the need for a priori error bounds, and supports early termination without compromising stability or feasibility, making it suitable for real-time robust receding-horizon control.

Abstract

A novel robust nonlinear model predictive control strategy is proposed for systems with nonlinear dynamics and convex state and control constraints. Using a sequential convex approximation approach and a difference of convex functions representation, the scheme constructs tubes that contain predicted model trajectories, accounting for approximation errors and disturbances, and guaranteeing constraint satisfaction. An optimal control problem is solved as a sequence of convex programs. We develop the scheme initially in the absence of external disturbances and show that the proposed nominal approach is non-conservative, with the solutions of successive convex programs converging to a locally optimal solution for the original optimal control problem. We extend the approach to the case of additive disturbances using a novel strategy for selecting linearization points. As a result we formulate a robust receding horizon strategy with guarantees of recursive feasibility closed-loop system stability.
Paper Structure (11 sections, 9 theorems, 57 equations, 7 figures, 1 table)

This paper contains 11 sections, 9 theorems, 57 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Successive linearization MPC
  3. Proposed control law
  4. Feasibility and stability guarantees
  5. Robust MPC with additive disturbance
  6. Proposed robust control law
  7. Robust feasibility and stability guarantees
  8. Numerical examples
  9. Nominal cMPC
  10. Robust cMPC (RcMPC)
  11. CONCLUSIONS

Key Result

Theorem 1

If Assumptions assumpt:matrices and assumpt:two hold and the initial seed trajectory $(\mathbf{x}^0_0,\mathbf{u}^0_0)$ satisfies eqn:constr2-eqn:constr5, then the cMPC RHOCP is feasible at each iteration of Algorithm alg:cap, at all times $i \geq 0$.

Figures (7)

  • Figure 1: FPU system with two unit masses with coordinates $q_1,q_2$, and control forces $u_1,u_2$ applied directly to each mass.
  • Figure 2: Closed loop trajectories with cMPC (Algorithm \ref{['alg:cap']}). Left: Example 1. Right: Example 2.
  • Figure 3: Convergence of cMPC at successive iterations and time steps for Example 2. Upper plot: Optimal cost. Lower: $\|\mathbf{c}_i^\ast\|$.
  • Figure 4: Seed trajectories and tubes (${x}_{i|k}^0$ and $x_{i|k}^0 \oplus S_{i|k}^\ast$) for cMPC and Example 1 at iterations $n=1,2,3$ at time $i=0$.
  • Figure 5: Closed-loop state trajectories with RcMPC. Left: Example 1. Right: Example 2
  • ...and 2 more figures

Theorems & Definitions (20)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1: Recursive feasibility
  • Lemma 1
  • Theorem 2
  • Remark 4
  • Theorem 3
  • Remark 5
  • Theorem 4
  • ...and 10 more