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

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 (11)

• Figure 1: Conceptual illustration of the proposed PS2F 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 PS2F framework acts as a predictive filter that projects $u_{\mathrm{ext}}$ onto the safety--stability set $\mathbb{S}(x)$ (defined in \ref{['eq:admissible_controls_sim']} and implicitly constructed via the two OCPs), ensuring that the applied control input remains both safe and stable.
• Figure 2: Schematic of the proposed PS2F 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.
• Figure 3: Three-dimensional illustration of the PS2F operation over time. The S2-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.
• Figure 4: Evolution of the PS2F filtering process along the $u_1$--$u_2$ axes. Each panel shows the S2-set $\mathbb{S}(x(k))$ (grey surface), the external command $u_{\mathrm{ext}}(k)$ (red circle), and the filtered control $u(k)$ (blue square). The orange arrow depicts the corrective action generated by PS2F, which corresponds to the perpendicular projection of $u_{\mathrm{ext}}(k)$ onto the admissible set. This projection direction is orthogonal to the local tangent of the boundary $\partial\mathbb{S}(x(k))$ (purple dash-dotted line), ensuring that the filtered input remains within the safety--stability domain while staying as close as possible to the external command.
• Figure 5: Effect of parameter $a$ on $\mathbb{S}(x)$, with $N=M=5$, $Q = 10I_2$, and $R = I_2$.

Theorems & Definitions (14)

• Proposition 1: rawlings2020model
• Remark 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Remark 2
• Remark 3
• Proposition 7