Linear MPC
Linear MPC
Model Predictive Control for linear time-invariant systems formulated as a Quadratic Programming problem with efficient real-time solution.
Family: Model Predictive Control Status: 📋 Planned
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Overview
Linear MPC is a specialized form of Model Predictive Control that applies to linear time-invariant (LTI) systems. By leveraging the linearity of the system, Linear MPC can be formulated as a Quadratic Programming (QP) problem, which can be solved efficiently using specialized optimization algorithms.
This approach provides excellent performance for linear systems while maintaining the predictive and constraint-handling capabilities of MPC. Linear MPC is widely used in industrial process control, automotive applications, and aerospace systems where the system dynamics can be well-approximated by linear models. The QP formulation enables real-time implementation and provides theoretical guarantees for stability and performance.
Mathematical Formulation¶
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Key Properties
QP Formulation
min (1/2) U^T H U + f^T U
Convex optimization problem with efficient solvers
Linear Predictions
X_k = Φ x(k) + Γ U_k
Exact state predictions using matrix operations
Constraint Handling
G U_k ≤ w, A_eq U_k = b_eq
Linear constraints on inputs, outputs, and states
Key Properties¶
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QP Formulation
Convex optimization problem with efficient solvers
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Linear Predictions
Exact state predictions using matrix operations
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Real-time Capability
Fast solution using specialized QP algorithms
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Stability Guarantees
Theoretical stability under certain conditions
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Constraint Handling
Natural incorporation of linear constraints
Implementation Approaches¶
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Standard Linear MPC implementation with QP formulation
Complexity:
- Time: O(N³)
- Space: O(N²)
Advantages
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Efficient QP formulation with fast solvers
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Exact linear predictions
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Real-time implementation capability
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Theoretical stability guarantees
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Natural constraint handling
Disadvantages
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Limited to linear systems
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Requires accurate linear model
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Computational complexity scales with horizon
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 QP formulation:
src/algokit/mpc/linear_mpc.py -
Comprehensive test suite including QP solver tests:
tests/unit/mpc/test_linear_mpc.py
Complexity Analysis¶
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Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| Basic Linear MPC | O(N³) | O(N²) | Complexity depends on prediction horizon N and system dimensions |
Performance Considerations
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QP formulation enables efficient real-time solution
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Computational complexity scales with prediction horizon
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Memory requirements scale with system dimensions
Use Cases & Applications¶
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Application Categories
Industrial Process Control
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Chemical Plants: Temperature and pressure control in reactors
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Oil Refineries: Distillation column control
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Power Plants: Boiler and turbine control
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Manufacturing: Quality control in production lines
Automotive Systems
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Engine Control: Fuel injection and ignition timing
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Vehicle Dynamics: Trajectory tracking and stability
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Hybrid Vehicles: Energy management systems
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Cruise Control: Speed regulation and fuel efficiency
Aerospace Applications
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Flight Control: Aircraft attitude and altitude control
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Spacecraft Guidance: Orbital maneuvers and docking
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UAV Control: Autonomous flight and mission execution
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Satellite Control: Attitude and orbit maintenance
Robotics
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Manipulator Control: Joint trajectory tracking
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Mobile Robot Navigation: Path following and obstacle avoidance
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Industrial Robots: Pick and place operations
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Service Robots: Navigation and task execution
Energy Systems
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Smart Grids: Power flow optimization
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Renewable Energy: Wind and solar power management
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Battery Management: Charging and discharging control
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Microgrids: Distributed energy resource coordination
Educational Value
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Control Theory: Linear system control and optimization
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Optimization: Quadratic programming for control applications
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System Modeling: Linear time-invariant system representation
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Real-time Systems: Implementation of predictive control
References & Further Reading¶
:material-book: Core Textbooks
:material-library: Linear MPC Theory
: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 Model Predictive Control:
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Distributed MPC - Model Predictive Control for large-scale systems using distributed optimization and coordination between multiple local controllers to achieve global objectives.
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Economic MPC - Model Predictive Control that optimizes economic objectives rather than tracking performance, focusing on profit maximization and cost minimization in process industries.
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Robust MPC - Model Predictive Control that handles model uncertainty and disturbances through robust optimization techniques to ensure constraint satisfaction and stability.
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Nonlinear MPC - Model Predictive Control for nonlinear systems using Sequential Quadratic Programming to handle complex dynamics and constraints.
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Model Predictive Control - Advanced control strategy that uses system models to predict future behavior and optimize control actions over a finite horizon while handling constraints.
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Learning MPC - Model Predictive Control that learns system dynamics and improves performance through data-driven approaches, combining machine learning with predictive control.