trajectory-controlled-invariants

name: trajectory-controlled-invariants description: "Trajectory-based computation of controlled invariant sets for linear discrete-time systems and MPC. Use when computing maximal controlled invariant sets, designing MPC without terminal sets, or needing recursive feasibility guarantees. Keywords: controlled invariants, MPC, trajectory-based, convex feasible points, recursive feasibility, terminal sets."

Trajectory-Based Controlled Invariants for MPC

New approach to computing controlled invariant sets using trajectory-based characterization, enabling MPC without precomputed terminal sets.

Core Innovation

Convex Feasible Points (CFPs): New characterization of controlled invariance using finitely long state trajectories, not geometric set computation.

Key Concepts

Controlled Invariant Set

Set $S$ where: if $x \in S$, exists control $u$ such that next state $x' \in S$

Traditional approach: Backward fixed-point iteration (computationally expensive)

Convex Feasible Point

A point $x$ is CFP if exists trajectory $x_0, x_1, ..., x_n$ where:

• $x_0 = x$
• All $x_k$ satisfy constraints
• Trajectory length $n$ finite

Key insight: CFPs characterize controlled invariance without computing full set.

Algorithm

1. CFP Identification

Given: System Ax + Bu, constraints Cx ≤ d
Find: Trajectory from x satisfying constraints throughout

2. Maximal Controlled Invariant

Combine CFP notion with backward fixed-point algorithm:

• More efficient than pure geometric computation
• Produces maximal controlled invariant set

3. MPC Design

Two schemes with recursive feasibility guarantee:

• No terminal set required
• CFPs provide implicit terminal constraint

MPC Without Terminal Sets

Traditional MPC requires:

• Terminal set (precomputed)
• Terminal cost
• Complex offline computation

This approach:

• Uses CFPs as implicit terminal constraint
• Recursive feasibility guaranteed
• Less offline computation

Optimization Formulation

Search for CFPs as optimization problem:

$$\min_{u_0,...,u_{n-1}} |x_n - x_0|$$ $$\text{s.t. } x_k \in \mathcal{X}, u_k \in \mathcal{U}, x_{k+1} = Ax_k + Bu_k$$

Practical Benefits

1. Reduced offline computation: No terminal set computation
2. Guaranteed feasibility: Recursive feasibility from CFP structure
3. Flexible MPC: Can adjust horizon online
4. Scalable: Trajectory-based approach scales better than set computation

Design Procedure

1. Identify system dynamics $Ax + Bu$
2. Define constraint sets $\mathcal{X}, \mathcal{U}$
3. Solve optimization for CFPs
4. Design MPC using CFP structure
5. Verify recursive feasibility

Applications

• Constrained linear systems
• Autonomous vehicle control
• Process control
• Robotics
• Power systems

Comparison

Key Result

Controlled invariant sets computed through trajectory characterization:

• More efficient than geometric methods
• Enables MPC without terminal sets
• Practical for online implementation

References

• arXiv:2604.07225v1 - "A Trajectory-based Approach to the Computation of Controlled Invariants with application to MPC"
• Accepted at European Control Conference 2026
• Blanchini (1999) - Survey on invariant sets