Maximal Controlled Invariant-MPC: Enhancing Feasibility and Reducing Conservatism through Terminal CBF Constraint in Safety-Critical Control

Technical Summary: Maximal Controlled Invariant-MPC

Problem Statement
Safety-critical control systems often rely on Control Barrier Functions (CBFs) to enforce constraints. However, constructing a minimally conservative CBF is computationally intractable for higher-order systems. Consequently, standard approaches often result in overly conservative safe sets ($X_{safe}$) that restrict the system to a subset of the true maximal controlled invariant safe set ($X^*_{safe}$). This conservatism reduces system optimality and feasibility, particularly in complex navigation tasks where the system must operate near the boundaries of known unsafe regions ($X_{avoid}$). Existing Model Predictive Control (MPC) formulations that integrate CBFs either rely on myopic geometric constraints, require slack variables and decay rate tuning, or impose multi-step quasi-CBF constraints, often failing to provide formal guarantees on reducing conservatism or expanding reachable sets with increasing prediction horizons.

Methodology
The authors propose a novel MPC formulation, termed MPC-MCI (Model Predictive Control with Maximal Controlled Invariant constraints), which decouples transient safety from terminal invariance to reduce conservatism while maintaining formal safety guarantees.

1. Formulation: The optimization problem minimizes a standard cost function over a prediction horizon $N$ subject to:

   • Transient Constraints: A signed distance function $d(x)$ is enforced to be non-negative ($d(x_{t+k}) \ge 0$) for all steps $k \in \{0, \dots, N-1\}$. This ensures the system never enters the known unsafe set $X_{avoid}$ during the prediction horizon.
   • Terminal Constraint: A single-step Control Barrier Function constraint is imposed only on the terminal state ($h(x_{t+N}) \ge 0$). This anchors the terminal state within the known safe set $X_{safe}$, which is a subset of the maximal controlled invariant set $X^*_{safe}$.

2. Theoretical Guarantees:

   • Recursive Feasibility: The authors provide a constructive proof showing that if the problem is feasible at $t=0$, it remains feasible for all $t > 0$. The proof relies on the fact that $X_{safe}$ is control invariant; thus, a valid control sequence exists to keep the system within $X_{safe}$ indefinitely once the terminal state is reached.
   • Safety: The closed-loop trajectory is proven to remain within $X^*_{safe}$ (the maximal controlled invariant subset of $X \setminus X_{avoid}$), ensuring safety even if the trajectory temporarily enters regions where the CBF value is negative ($h < 0$) but the signed distance remains positive ($d \ge 0$).
   • Reachability Enhancement: Through reachability analysis, the paper demonstrates that the one-step reachable set for MPC-MCI monotonically expands as the prediction horizon $N$ increases. In contrast, the reachable set for prior methods (specifically NMPC-DCBF) remains static regardless of horizon length.

3. Computational Efficiency: The constructive nature of the feasibility proof allows for warm-starting the nonlinear optimization. The optimizer uses the previously computed trajectory, appended with a recovery control policy, as the initial guess. This significantly reduces computational time without sacrificing feasibility guarantees.

Key Contributions

• Optimization Framework: Development of a recursively feasible MPC framework that guarantees safety by keeping the system within the maximal controlled invariant safe subset, addressing limitations in prior works [3, 4].
• Reduced Conservatism: The method requires only initial feasibility to guarantee future safety, allowing the system to operate in regions outside the conservative $X_{safe}$ (i.e., within $X^*_{safe} \setminus X_{safe}$) provided they are geometrically safe ($d \ge 0$).
• Simplified Constraints: Unlike approaches using two-step quasi-CBFs [7] or slack variables with decay rates [4], MPC-MCI imposes a single terminal CBF constraint and a distance constraint along the horizon, eliminating the need for parameter tuning (e.g., $\gamma$) or auxiliary constraint functions.
• Constructive Proofs: The paper provides constructive proofs for recursive feasibility that enable warm-starting, substantially speeding up computation.
• Reachability Analysis: Formal demonstration that the feasible and reachable sets expand with increasing prediction horizon, a property not guaranteed in previous NMPC-DCBF formulations.

Results
The proposed method was validated through simulations on a 2D nonholonomic unicycle system with a circular obstacle.

• Feasibility Enhancement: The number of infeasible initial states decreased by a factor of 1.74 to 2.73 compared to the NMPC-DCBF approach. Specifically, for a high-speed case, infeasible points dropped from 5,560 to 2,034.
• Horizon Dependence: As the prediction horizon $N$ increased (from 2 to 21), the fraction of feasible states for MPC-MCI increased monotonically (e.g., from 0.74 to 0.84 in Case 1), whereas the feasible fraction for NMPC-DCBF remained constant.
• Trajectory Tracking: The system successfully tracked a reference trajectory that lay entirely within the "unsafe" region of the CBF ($h < 0$) but remained geometrically safe ($d \ge 0$). The conservative NMPC-DCBF failed to track this trajectory, either deviating significantly or moving too slowly.

Significance
The paper claims that anchoring safety at the terminal state, rather than enforcing CBF constraints along the entire horizon, is a principled method to recover optimality lost to conservative CBF design. By leveraging the signed distance function for transient safety and the CBF only for terminal invariance, the MPC-MCI formulation achieves a provably safe controller that admits trajectories venturing into regions of "unknown safety" ($X^*_{safe} \setminus X_{safe}$). This approach eliminates the need for slack variables, decay-rate tuning, or complex multi-step constraints, offering a computationally efficient and less conservative alternative for safety-critical control in nonlinear systems.