H-Infinity Control
H-Infinity Control
Robust control design methodology that minimizes the worst-case performance of a system under bounded disturbances and uncertainties.
Family: Control Algorithms Status: 📋 Planned
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Overview
H-Infinity Control is a robust control design methodology that minimizes the worst-case performance of a system under bounded disturbances and uncertainties. The controller is designed to minimize the H-infinity norm of the closed-loop transfer function from disturbances to performance outputs, ensuring robust performance across a wide range of operating conditions.
This approach is particularly valuable in aerospace applications, automotive systems, power electronics, and other domains where system performance must be guaranteed despite parameter variations, external disturbances, and model uncertainties. H-infinity controllers provide excellent robustness properties and are widely used in safety-critical applications.
Mathematical Formulation¶
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Key Properties
Robust Performance
||T_zw(s)||_∞ < γ
Guarantees performance under bounded disturbances
Frequency Domain Design
sup_{ω} σ_max(T_zw(jω))
Direct design in frequency domain
Optimal Robustness
min_K ||T_zw(s)||_∞
Minimizes worst-case performance
Key Properties¶
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Robust Performance
Guarantees performance under bounded disturbances
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Frequency Domain Design
Direct design in frequency domain
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Optimal Robustness
Minimizes worst-case performance
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Multi-Objective Design
Handles multiple performance criteria
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Stability Guarantees
Ensures closed-loop stability
Implementation Approaches¶
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H-infinity controller design using state-space methods
Complexity:
- Time: O(n³)
- Space: O(n²)
Advantages
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Robust performance guarantees
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Handles multiple performance objectives
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Frequency domain design approach
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Theoretical stability guarantees
Disadvantages
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Computationally intensive design
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May be conservative
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Requires good system model
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Complex implementation
H-infinity design focusing on sensitivity and complementary sensitivity
Complexity:
- Time: O(n³)
- Space: O(n²)
Advantages
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Direct control over sensitivity functions
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Good disturbance rejection
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Robust stability
Disadvantages
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Requires careful weight selection
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May be conservative
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Complex weight tuning
Complete Implementation
The full implementation with error handling, comprehensive testing, and additional variants is available in the source code:
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Main implementation with state-space and mixed sensitivity variants:
src/algokit/control/h_infinity_control.py -
Comprehensive test suite including robustness tests:
tests/unit/control/test_h_infinity_control.py
Complexity Analysis¶
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Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| State-Space H-Infinity | O(n³) | O(n²) | Time complexity dominated by Riccati equation solution |
Performance Considerations
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Design phase is computationally intensive
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Runtime control computation is efficient
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Memory requirements grow with system size
Use Cases & Applications¶
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Application Categories
Aerospace Systems
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Flight Control: Robust control for aircraft with uncertain dynamics
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Satellite Control: Attitude control under varying conditions
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Missile Guidance: Robust tracking despite target maneuvers
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Spacecraft Docking: Precise control under uncertain contact
Automotive Systems
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Active Suspension: Robust control for varying road conditions
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Engine Control: Robust performance under engine wear
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Steering Control: Robust lane keeping despite wind disturbances
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Brake Control: Robust ABS under varying road conditions
Power Electronics
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Motor Drives: Robust control under load variations
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Power Converters: Robust voltage regulation
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Grid Control: Robust power system stabilization
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Renewable Energy: Robust control under weather variations
Robotics and Automation
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Robot Control: Robust manipulation under payload changes
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Manufacturing: Robust control despite tool wear
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Autonomous Vehicles: Robust navigation under environmental changes
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Precision Machining: Robust control for high accuracy
Process Control
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Chemical Plants: Robust control under catalyst degradation
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Power Plants: Robust control under load variations
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Water Treatment: Robust control under varying water quality
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Food Processing: Robust control under ingredient variations
Educational Value
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Robust Control Theory: Understanding worst-case design approaches
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Frequency Domain Analysis: Learning H-infinity norm and singular values
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Riccati Equations: Understanding optimal control solutions
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Multi-Objective Optimization: Balancing competing performance criteria
References & Further Reading¶
:material-book: Core Textbooks
:material-library: H-Infinity Control
:material-web: Online Resources
:material-code-tags: Implementation & Practice
Interactive Learning
Try implementing the different approaches yourself! This progression will give you deep insight into the algorithm's principles and applications.
Pro Tip: Start with the simplest implementation and gradually work your way up to more complex variants.
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Navigation¶
Related Algorithms in Control Algorithms:
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Sliding Mode Control - Robust control strategy that forces the system state to reach and remain on a predefined sliding surface, regardless of parameter uncertainties and external disturbances.
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Robust Control - Comprehensive control design methodology that ensures system stability and performance despite model uncertainties, parameter variations, and external disturbances.
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PID Control - Fundamental feedback control algorithm combining proportional, integral, and derivative actions to achieve desired system behavior.
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Adaptive Control - Control strategy that automatically adjusts controller parameters based on real-time system identification and performance evaluation.