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.
New approach to computing controlled invariant sets using trajectory-based characterization, enabling MPC without precomputed terminal sets.
Convex Feasible Points (CFPs): New characterization of controlled invariance using finitely long state trajectories, not geometric set computation.
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)
A point $x$ is CFP if exists trajectory $x_0, x_1, ..., x_n$ where:
Key insight: CFPs characterize controlled invariance without computing full set.
Given: System Ax + Bu, constraints Cx ≤ d
Find: Trajectory from x satisfying constraints throughout
Combine CFP notion with backward fixed-point algorithm:
Two schemes with recursive feasibility guarantee:
Traditional MPC requires:
This approach:
Search for CFPs as optimization problem:
$$\min_{u_0,...,u_{n-1}} |x_n - x_0|$$ $$\text{s.t. } x_k \in \mathcal{X}, u_k \in \mathcal{U}, x_{k+1} = Ax_k + Bu_k$$
| Aspect | Traditional | Trajectory-Based |
|---|---|---|
| Terminal set | Required | Not required |
| Offline computation | Heavy | Light |
| Feasibility guarantee | Via terminal set | Via CFPs |
| Flexibility | Limited | High |
Controlled invariant sets computed through trajectory characterization:
Use this skill for computing maximal controlled invariant sets for linear discrete-time systems. Apply the trajectory-based approach to enable MPC without explicit terminal set computation.
User: Help me with Trajectory Controlled Invariants Agent: [Activates trajectory-controlled-invariants skill and follows the instructions above]