trajectory-based-controlled-invariants

name	trajectory-based-controlled-invariants
description	Compute controlled invariant sets for linear discrete-time systems using trajectory-based approach with convex feasible points. Use for MPC design, control invariant set computation, feasibility guarantees, and discrete-time system analysis. Keywords: controlled invariant, MPC, trajectory-based, convex feasible points, discrete-time systems, recursive feasibility.

Trajectory-Based Controlled Invariants

Description

A trajectory-based approach to computing controlled invariant sets for linear discrete-time systems. Introduces convex feasible points concept for invariant set characterization and provides MPC schemes with recursive feasibility guarantees without precomputed terminal sets.

Activation Keywords

• controlled invariant
• controlled invariant set
• trajectory-based invariants
• convex feasible points
• MPC recursive feasibility
• discrete-time system invariance
• 控制不变集
• MPC 可行性
• 轨迹方法不变集

Tools Used

• exec: Run Python scripts for system analysis
• read: Load system models, reference papers
• write: Save invariant set computations

Core Concepts

1. Convex Feisible Points

Definition: A finitely long state trajectory that provides new characterization of controlled invariance.

Key Properties:

• Provides necessary condition for control invariance
• Can be verified with finite trajectory length
• Combines with backward fixed-point algorithm

2. Controlled Invariant Sets

Sets where system state can remain indefinitely under proper control.

Traditional Methods:

• Backward reachability
• Fixed-point iteration

Trajectory-Based Innovation:

• Uses finite trajectories instead of infinite analysis
• Reduces computational complexity
• Enables practical MPC design

3. MPC without Terminal Sets

Two proposed MPC schemes:

1. Direct feasibility guarantee via trajectory constraints
2. Recursive feasibility through convex feasible points

Advantages:

• No need for precomputed terminal sets
• Online feasibility verification
• Reduced offline computation burden

Instructions for Agents

Step 1: System Modeling

Given discrete-time linear system:

x(k+1) = Ax(k) + Bu(k)

Identify:

• State matrix A
• Input matrix B
• State constraints X
• Input constraints U

Step 2: Trajectory Analysis

1. Generate candidate trajectories
2. Verify convex feasible point conditions
3. Check trajectory constraints satisfaction

Verification criterion:

• Trajectory must remain in X for all steps
• Control inputs must satisfy U constraints
• Must provide convex hull feasible for continuation

Step 3: Invariant Set Computation

Use backward fixed-point algorithm:

def compute_maximal_invariant(A, B, X, U, iterations=100):
    """Compute maximal controlled invariant set."""
    current_set = X
    for i in range(iterations):
        # Backward reachable set
        predecessor = backward_reach(A, B, current_set, U)
        # Intersection with constraints
        current_set = predecessor ∩ X
        if current_set.is_empty():
            break
    return current_set

Step 4: MPC Design

Construct MPC scheme:

def mpc_trajectory_based(x_current, horizon, A, B, X, U, convex_feasible_point):
    """MPC with trajectory-based feasibility guarantee."""
    # Optimization problem
    # Variables: x(0:N), u(0:N-1)
    # Constraints:
    #   - Dynamics: x(k+1) = Ax(k) + Bu(k)
    #   - State: x(k) ∈ X
    #   - Input: u(k) ∈ U
    #   - Feasibility: convex_feasible_point condition
    # Objective: minimize cost function
    
    return optimal_sequence

Step 5: Feasibility Verification

Check recursive feasibility:

• Verify convex feasible point at each iteration
• Ensure existence of feasible continuation
• Validate constraint satisfaction

Use Cases

Use Case 1: Safety-Critical Control

Scenario: Robot navigation with obstacle avoidance

Application:

• Define safety constraints (avoid obstacles)
• Compute controlled invariant safe set
• Design MPC guaranteeing perpetual safety
• No precomputation of terminal safe set needed

Use Case 2: Resource Management

Scenario: Tank level control with limits

Application:

• State constraints: tank level bounds
• Input constraints: pump capacity
• Compute invariant operating region
• MPC maintains feasibility indefinitely

Use Case 3: Power System Control

Scenario: Generator scheduling with stability constraints

Application:

• State: power output, frequency
• Input: control signals
• Invariant set ensures stability
• Trajectory-based simplifies computation

Mathematical Formulation

Convex Feisible Point Definition

A trajectory {x(0), x(1), ..., x(N)} is a convex feasible point if:

1. State constraints: x(k) ∈ X for all k
2. Input constraints: u(k) ∈ U satisfies x(k+1) = Ax(k) + Bu(k)
3. Convexity: There exists convex combination enabling continuation

Optimization Problem

minimize   Σ J(x(k), u(k))
subject to x(k+1) = Ax(k) + Bu(k), k = 0, ..., N-1
           x(k) ∈ X, k = 0, ..., N
           u(k) ∈ U, k = 0, ..., N-1
           convex_feasible_point(x(N)) = true

Error Handling

Infeasible Initial State

Problem: Current state outside invariant set

Solution:

• Check feasibility before MPC execution
• If infeasible, use recovery maneuver
• Plan trajectory to invariant set boundary

Invariant Set Computation Failure

Problem: Algorithm doesn't converge

Solution:

• Increase iteration limit
• Check constraint compatibility
• Verify system controllability
• Adjust numerical precision

Numerical Issues

Problem: Convex hull computation unstable

Solution:

• Use robust convex hull algorithms
• Increase trajectory length
• Apply regularization techniques

Examples

Example 1: Double Integrator

System:

A = [[1, 1], [0, 1]]
B = [[0], [1]]
X = {x: |x₁| ≤ 10, |x₂| ≤ 2}
U = {u: |u| ≤ 1}

Process:

1. Generate trajectories with N=10
2. Identify convex feasible points
3. Compute maximal invariant set
4. Design MPC with horizon 5

Example 2: Room Temperature Control

System:

A = 0.95  # Decay rate
B = 1.0   # Heating effect
X = [18°C, 25°C]
U = [0kW, 5kW]

Application:

• Compute temperature invariant interval
• MPC maintains comfortable temperature
• No terminal set precomputation

Resources

• Paper: arXiv:2604.07225 - "A Trajectory-based Approach to Controlled Invariants"
• Code: scripts/compute_invariant_set.py
• References: references/discrete_time_control.md

Related Skills

• kuramoto-control-theory: Synchronization control
• robust-regret-control: Robust MPC design
• ml-complexity-management: Complexity analysis for control

Notes

• Applicable to linear discrete-time systems only
• Numerical stability depends on trajectory length
• Convexity assumption required
• MPC horizon affects computational load