Nonlinear MPC
Nonlinear MPC
Model Predictive Control for nonlinear systems using Sequential Quadratic Programming to handle complex dynamics and constraints.
Family: Model Predictive Control Status: š Planned
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Overview
Nonlinear MPC extends the predictive control framework to handle systems with nonlinear dynamics, constraints, and objectives. Unlike Linear MPC, which can be formulated as a Quadratic Programming problem, Nonlinear MPC requires solving a Nonlinear Programming (NLP) problem at each time step.
This approach provides the ability to control complex nonlinear systems while maintaining the predictive and constraint-handling capabilities of MPC. Nonlinear MPC is essential in applications where system dynamics are inherently nonlinear, such as chemical processes, robotics, aerospace systems, and many other domains. While computationally more demanding than linear approaches, it offers superior performance for nonlinear systems and can handle complex constraints and objectives.
Mathematical Formulation¶
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Key Properties
Nonlinear Dynamics
x(k+1) = f(x(k), u(k))
Handles complex system behaviors
General Constraints
g(x,u) ā¤ 0, h_c(x,u) = 0
Supports nonlinear equality and inequality constraints
SQP Solution
min (1/2) ĪU^T H ĪU + g^T ĪU
Iteratively solves QP subproblems
Key Properties¶
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Nonlinear Dynamics
Handles complex system behaviors
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General Constraints
Supports nonlinear equality and inequality constraints
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Global Optimization
Can find globally optimal solutions
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Real-time Capability
Fast convergence with modern solvers
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Constraint Satisfaction
Ensures all constraints are met
Implementation Approaches¶
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Standard Nonlinear MPC implementation with SQP solver
Complexity:
- Time: O(Nā¶)
- Space: O(Nā“)
Advantages
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Handles nonlinear system dynamics
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Supports general nonlinear constraints
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Can achieve superior performance for nonlinear systems
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Flexible cost function formulation
Disadvantages
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Computationally intensive
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May not guarantee global optimality
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Requires careful tuning of optimization parameters
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Convergence not guaranteed in all cases
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 SQP solver:
src/algokit/mpc/nonlinear_mpc.py -
Comprehensive test suite including optimization tests:
tests/unit/mpc/test_nonlinear_mpc.py
Complexity Analysis¶
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Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| Basic Nonlinear MPC | O(Nā¶) | O(Nā“) | Complexity depends on prediction horizon N and system dimensions |
Performance Considerations
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Nonlinear optimization requires iterative 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
Chemical Process Control
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Reactor Control: Temperature and concentration control in chemical reactors
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Distillation Columns: Multi-component separation control
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Polymerization Processes: Molecular weight and conversion control
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Bioreactors: Cell growth and product formation control
Robotics and Mechatronics
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Manipulator Control: Joint trajectory tracking with dynamics
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Mobile Robot Navigation: Path planning with nonholonomic constraints
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Humanoid Robots: Balance and locomotion control
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Aerial Vehicles: Attitude and position control with aerodynamics
Aerospace Systems
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Aircraft Control: Flight dynamics and trajectory optimization
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Spacecraft Guidance: Orbital maneuvers with gravitational effects
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Missile Guidance: Target tracking with aerodynamic forces
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Satellite Control: Attitude control with environmental disturbances
Automotive Systems
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Vehicle Dynamics: Trajectory tracking with tire dynamics
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Engine Control: Combustion process optimization
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Hybrid Vehicles: Energy management with battery dynamics
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Autonomous Driving: Path planning with vehicle dynamics
Energy Systems
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Power Systems: Load frequency control with generator dynamics
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Renewable Energy: Wind turbine control with aerodynamic effects
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Battery Management: Charging control with electrochemical dynamics
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Smart Grids: Power flow optimization with network constraints
Educational Value
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Control Theory: Nonlinear system control and optimization
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Optimization: Nonlinear programming and SQP methods
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System Modeling: Nonlinear system representation and analysis
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Real-time Systems: Implementation of complex optimization algorithms
References & Further Reading¶
:material-book: Core Textbooks
:material-web: Nonlinear 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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Linear MPC - Model Predictive Control for linear time-invariant systems formulated as a Quadratic Programming problem with efficient real-time solution.
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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.