| 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
- Reduced offline computation: No terminal set computation
- Guaranteed feasibility: Recursive feasibility from CFP structure
- Flexible MPC: Can adjust horizon online
- Scalable: Trajectory-based approach scales better than set computation
Design Procedure
- Identify system dynamics $Ax + Bu$
- Define constraint sets $\mathcal{X}, \mathcal{U}$
- Solve optimization for CFPs
- Design MPC using CFP structure
- Verify recursive feasibility
Applications
- Constrained linear systems
- Autonomous vehicle control
- Process control
- Robotics
- Power systems
Comparison
| Aspect | Traditional | Trajectory-Based |
|---|
| Terminal set | Required | Not required |
| Offline computation | Heavy | Light |
| Feasibility guarantee | Via terminal set | Via CFPs |
| Flexibility | Limited | High |
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
