temporal-logic-control

name

temporal-logic-control

description

Temporal logic control synthesis for nonlinear stochastic systems using finite-state abstractions (IMDP). Safety-critical system control with formal guarantees. Use when designing controllers for autonomous systems, safety-critical applications, or systems requiring formal verification. Keywords: temporal logic, control synthesis, nonlinear systems, stochastic systems, safety-critical, IMDP, formal verification.

Temporal Logic Control

Control synthesis for nonlinear stochastic systems with temporal logic specifications using finite-state abstractions.

Problem Statement

Autonomous systems in safety-critical environments require:

• Formal guarantees on control policy correctness
• Complex temporal logic specifications (reachability, safety, liveness)
• Handling of stochastic disturbances and nonlinear dynamics
• Provably correct policies despite uncertainty

Solution Approach

Finite-state abstraction-based control synthesis:

1. Continuous → Discrete: Abstract nonlinear stochastic system to IMDP
2. Policy Synthesis: Compute policy on IMDP satisfying temporal logic
3. Refinement: Refine abstraction for accuracy
4. Implementation: Map discrete policy to continuous controller

Core Methodology

Step 1: System Modeling

# Nonlinear discrete-time stochastic system:
x_{k+1} = f(x_k, u_k) + w_k

where:
- x_k: state (continuous)
- u_k: control input
- w_k: stochastic disturbance
- f: nonlinear dynamics

Step 2: Finite-State Abstraction

Construct Interval MDP (IMDP):
1. Partition state space into regions
2. Compute transition probability intervals
3. Account for nonlinearity and stochasticity
4. Bound abstraction error

Step 3: Temporal Logic Specification

Common specifications:
- Safety: □(unsafe → avoid)
- Reachability: ◇(target)
- Reach-avoid: ◇(target) ∧ □(unsafe → avoid)
- Recurrence: □◇(goal)
- Response: □(request → ◇(response))

Step 4: Policy Synthesis

# IMDP policy synthesis:
policy = synthesize(IMDP, specification)

# Returns policy satisfying specification
# with probability >= threshold

Step 5: Controller Implementation

Map discrete policy to continuous:
1. Identify current state region
2. Apply discrete policy action
3. Refine to continuous control input
4. Handle boundary cases

Key Techniques

Approximate Stochastic Simulation

# Quantify abstraction accuracy:
simulation_relation(original_system, abstraction)
→ accuracy_bound

IMDP Construction

# Interval MDP:
States: {S1, S2, ..., Sn}
Transitions: P(s'|s,a) ∈ [p_low, p_high]
Actions: {a1, a2, ..., am}

Online Performance Optimization

Online refinement:
1. Monitor system performance
2. Detect specification violations
3. Refine abstraction locally
4. Update policy online

Workflow Example

Scenario: Autonomous drone navigation in uncertain environment.

1. Model: Drone dynamics + wind disturbance
2. Specification: Reach target while avoiding obstacles
3. Abstraction: IMDP with state regions
4. Synthesis: Compute safe policy
5. Implementation: Discrete actions → continuous thrust
6. Online: Refine if wind changes

Best Practices

1. Bound abstraction error: Critical for formal guarantees
2. Iterative refinement: Start coarse, refine as needed
3. Online adaptation: Handle changing conditions
4. Conservative synthesis: Account for worst-case transitions
5. Verification: Validate policy on original system

Temporal Logic Operators

Operator	Meaning	Example
□ (always)	Always true	□(safe)
◇ (eventually)	Eventually true	◇(goal)
U (until)	P until Q	safe U goal
→ (implies)	P implies Q	request → response

Applications

• Autonomous vehicle control
• Robotics navigation
• Power grid management
• Medical device control
• Aerospace systems
• Industrial automation

Safety-Critical Considerations

Formal guarantees:
- Probability of satisfaction >= threshold
- Conservative abstraction bounds
- Worst-case scenario handling
- Fail-safe mechanisms

Tools Reference

• SySCoRe: Toolset for formal control synthesis
• Stochastic Abstraction: IMDP construction
• Policy Synthesis: IMDP solver
• Verification: Model checking

Related Work

• Abstraction-based control: Finite MDP/IMDP methods
• Stochastic MPC: Receding horizon control
• Safe RL: Learning with safety constraints
• Formal methods: Model checking, verification

Source Paper

Temporal Logic Control of Nonlinear Stochastic Systems with Online Performance Optimization

• arxiv ID: 2604.01372
• Authors: Riccardi, Badings, Laurenti, Abate, De Schutter
• Published: April 2026

Related Skills

• kg-research-workflow: Import papers to knowledge graph
• arxiv-search: Search for control systems papers
• skill-creator: Create skills from research

Notes

• Abstraction accuracy critical for guarantees
• IMDP handles uncertainty intervals
• Online optimization enables adaptation
• Formal verification essential for safety-critical
• Nonlinearity requires careful abstraction