MPC as a Copilot: A Predictive Filter Framework with Safety and Stability Guarantees

Yunda Yan; Chenxi Tao; Jinya Su; Cunjia Liu; Shihua Li

eess.SYMar 29, 2026

MPC as a Copilot: A Predictive Filter Framework with Safety and Stability Guarantees

Authors: Yunda Yan, Chenxi Tao, Jinya Su, Cunjia Liu, Shihua Li

Organizations: Department of Computer Science, University College London, London WC1E 6BT, UK · School of Automation, Southeast University, Nanjing 210096, China · Department of Aeronautical and Automotive Engineering, Loughborough University, Leicestershire LE11 3TU, UK

Abstract

Ensuring both safety and stability remains a fundamental challenge in learning-based control, where goal-oriented policies often neglect system constraints and closed-loop state convergence. To address this limitation, this paper introduces the Predictive Safety--Stability Filter (PS2F), a unified predictive filter framework that guarantees constraint satisfaction and asymptotic stability within a single architecture. The PS2F framework comprises two cascaded optimal control problems: a nominal model predictive control (MPC) layer that serves solely as a copilot, implicitly defining a Lyapunov function and generating safety- and stability-certified predicted trajectories, and a secondary filtering layer that adjusts external command to remain within a provably safe and stable region. This cascaded structure enables PS2F to inherit the theoretical guarantees of nominal MPC while accommodating goal-oriented external commands. Rigorous analysis establishes recursive feasibility and asymptotic stability of the closed-loop system without introducing additional conservatism beyond that associated with the nominal MPC. Furthermore, a time-varying parameterisation allows PS2F to transition smoothly between safety-prioritised and stability-oriented operation modes, providing a principled mechanism for balancing exploration and exploitation. The effectiveness of the proposed framework is demonstrated through comparative numerical experiments.

Figures & tables

Figure 1: Conceptual illustration of the proposed PS 2 F framework. The external controller generates a goal-oriented command

u_{\mathrm{ext}}

, which may prioritise efficiency or high-level task performance but can potentially violate constraints or compromise stability. The PS 2 F framework acts as a predictive filter that projects

u_{\mathrm{ext}}

onto the safety–stability set

\mathbb{S}(x)

(defined in ( 16 ) and implicitly constructed via the two OCPs), ensuring that the applied control input remains both safe and stable.

Figure 2: Schematic of the proposed PS 2 F framework. The nominal MPC serves as a copilot , generating safety- and stability-certified trajectories. The filtering OCP $\mathbb{P}_{f,M}(\cdot)$ then refines the external command within a safe and stable region before applying the final control action to the system plant. The dashed arrow above the external controller indicates that the external command may be independent of the system state, i.e., operating in open-loop mode.

u(k)\leftarrow u^{*}(0;x(k))\quad\text{(cf.\ \eqref{equ_finalcontrol}).}

Algorithm 1 PS 2 F

u(k)\leftarrow u^{*}(0;x(k))\quad\text{(cf.\ \eqref{equ_finalcontrol}).}

Algorithm 2 PS 2 F with Mode Scheduling

Figure 3: Three-dimensional illustration of the PS 2 F operation over time. The S 2 -set $\mathbb{S}(x(k))$ (grey surface) evolves with the system state, while the external command $u_{\mathrm{ext}}(k)$ (red circle) and the filtered control $u(k)$ (blue square) are visualised along the time axis.

Param.	Effect on $\mathbb{S}(x)$	Side Effect
$a$	Enlarges $\mathbb{S}(x)$	Typically slower convergence
$M$	Enlarges $\mathbb{S}(x)$	Higher computational cost

Table 1: Effect of increasing each PS 2 F design parameter.

Figure 5: Effect of parameter

a

\mathbb{S}(x)

, with

N=M=5

Q=10I_{2}

, and

R=I_{2}

Figure 6: Effect of prediction horizon

M

\mathbb{S}(x)

, with

N=5

a=0.95

Q=10I_{2}

, and

R=I_{2}

Figure 7: Effect of MPC weighting

Q=\rho I_{2}

\mathbb{S}(x)

, with

N=M=5

a=0.95

, and

R=I_{2}

Figure 8: Effect of prediction horizon

N

\mathbb{S}(x)

, with

Q=10I_{2}

R=I_{2}

a=0.95

, and

M=N

Panel of Figure 9 — Figure 9: Closed-loop evolution of the robot system under the baseline controller and the proposed PS 2 F method. (a) States; (b) Control inputs; (c) Value function.

Figure 10: Comparison of robot trajectories under the baseline controller and the proposed PS 2 F method when both are scheduled to return at

k=30

. A yellow robot marker denotes that the state violates the safety constraints. (a) Baseline. (b) Proposed.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 11: Schematic illustration of the recursive feasibility analysis.

Safety-Critical Control

Safety-critical control designs controllers that keep physical systems within hard safety constraints despite uncertainty, disturbances, and limited actuation.

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Autonomous systems are increasingly deployed in real-world environments, where they must achieve high performance while maintaining safety under state and input constraints. Although Model Predictive Control (MPC) provides a principled framework for constrained optimal control, guaranteeing safety beyond its finite planning horizon remains a fundamental challenge. In this work, we augment MPC with a safety value function-based terminal constraint that enforces membership in a control-invariant safe set at the end of each planning horizon. This formulation enables real-time synthesis of trajectories that are both high-performing and provably safe. We show that, under an exact safety value function and a feasible initialization, the proposed MPC scheme is recursively feasible, thereby ensuring persistent safety. In contrast to traditional terminal set constructions that rely on local linearizations or conservative approximations, our approach incorporates a reachability-based safety value function for terminal constraints, yielding less conservative and more expressive safety guarantees. We validate the proposed framework through simulation and hardware experiments on a Flexiv Rizon 10s manipulator. Results demonstrate improved constraint satisfaction and robustness compared to standard state-constrained MPC and reactive safety filtering, while maintaining competitive task performance. The full implementation and experiments are available on the project website.

Abstract

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