PID Control
PID Control
Fundamental feedback control algorithm combining proportional, integral, and derivative actions to achieve desired system behavior.
Family: Control Algorithms Status: ๐ Planned
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Overview
PID Control (Proportional-Integral-Derivative Control) is a fundamental feedback control algorithm that combines three control actions to achieve desired system behavior. The algorithm continuously calculates an error value as the difference between a desired setpoint and a measured process variable, then applies a correction based on proportional, integral, and derivative terms.
This approach is widely used in industrial control systems, robotics, automotive applications, and many other domains where precise control is required. PID controllers are valued for their simplicity, effectiveness, and ability to handle a wide range of control problems with minimal tuning.
Mathematical Formulation¶
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Key Properties
Proportional Term
K_p e(t)
Provides immediate response proportional to current error
Integral Term
K_i โซโแต e(ฯ) dฯ
Eliminates steady-state error by accumulating past errors
Derivative Term
K_d (d/dt) e(t)
Improves transient response and reduces overshoot
Key Properties¶
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Linear Control
Simple linear combination of error terms
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Robust Performance
Works well across many different systems
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Easy Tuning
Three parameters provide intuitive control design
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Wide Applicability
Suitable for many control problems
Implementation Approaches¶
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Standard PID implementation with anti-windup and output limits
Complexity:
- Time: O(1)
- Space: O(1)
Advantages
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Simple and intuitive implementation
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Widely applicable to many systems
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Easy to tune with three parameters
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Includes anti-windup protection
Disadvantages
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May not work well for highly nonlinear systems
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Requires manual tuning for optimal performance
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Can be sensitive to measurement noise
PID with automatic gain adjustment based on system performance
Complexity:
- Time: O(1)
- Space: O(1)
Advantages
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Automatic gain adjustment
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Adapts to changing system conditions
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Reduces manual tuning effort
Disadvantages
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More complex than basic PID
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May require careful tuning of adaptation parameters
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 basic and adaptive variants:
src/algokit/control/pid_control.py -
Comprehensive test suite including performance tests:
tests/unit/control/test_pid_control.py
Complexity Analysis¶
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Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| Basic PID Controller | O(1) | O(1) | Constant time and space complexity per control step |
Performance Considerations
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PID controllers are computationally efficient
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Suitable for real-time applications
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Memory requirements are minimal
Use Cases & Applications¶
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Application Categories
Industrial Automation
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Process Control: Temperature, pressure, flow control in chemical plants
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Manufacturing: Position control in CNC machines and robotics
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Power Systems: Voltage and frequency regulation in power grids
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Quality Control: Maintaining product specifications in production lines
Robotics and Mechatronics
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Robot Control: Joint position and velocity control
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Autonomous Vehicles: Steering, throttle, and brake control
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Drones: Altitude, attitude, and position control
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Manipulation: Force and position control in robotic arms
Automotive Systems
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Engine Control: Fuel injection and ignition timing
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Cruise Control: Maintaining constant vehicle speed
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ABS Systems: Wheel slip control during braking
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Steering Control: Lane keeping and parking assistance
Consumer Electronics
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Camera Stabilization: Image stabilization in cameras and phones
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Audio Systems: Volume and equalizer control
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HVAC Systems: Temperature and humidity control
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Washing Machines: Water level and temperature control
Aerospace
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Flight Control: Aircraft attitude and altitude control
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Satellite Control: Attitude and orbit control
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Missile Guidance: Target tracking and navigation
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Spacecraft Control: Docking and station-keeping
Educational Value
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Control Theory: Introduction to feedback control concepts
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System Analysis: Understanding system response characteristics
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Parameter Tuning: Learning to adjust controller parameters
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Real-time Systems: Understanding continuous control implementation
References & Further Reading¶
:material-book: Core Textbooks
:material-library: PID 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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H-Infinity Control - Robust control design methodology that minimizes the worst-case performance of a system under bounded disturbances and uncertainties.
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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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Adaptive Control - Control strategy that automatically adjusts controller parameters based on real-time system identification and performance evaluation.