Robust MPC

Robust MPC

Model Predictive Control that handles model uncertainty and disturbances through robust optimization techniques to ensure constraint satisfaction and stability.

Family: Model Predictive Control Status: 📋 Planned

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Overview

Robust MPC extends the predictive control framework to handle systems with model uncertainty, parameter variations, and external disturbances. Unlike standard MPC, which assumes perfect knowledge of the system model, Robust MPC explicitly accounts for uncertainty in the system dynamics and ensures that constraints are satisfied and stability is maintained under all possible realizations of the uncertainty.

This approach is essential in applications where system models are uncertain or where the system operates in changing environments. Robust MPC provides theoretical guarantees for constraint satisfaction and stability, making it particularly valuable for safety-critical applications in aerospace, automotive, and process industries. The key challenge is to design controllers that are robust to uncertainty while maintaining good performance.

Mathematical Formulation

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Key Properties

Min-Max Optimization

min_{U} max_{θ,d} J(x, U, θ, d)

Optimizes worst-case performance over uncertainty set

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Tube-based Control

x(k) ∈ {x_nom(k)} ⊕ S

Ensures state remains within robust tube

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Robust Constraints

g(x,u) ≤ 0, ∀θ ∈ Θ, ∀d ∈ D

Constraints satisfied for all uncertainty realizations

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Key Properties

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• Uncertainty Handling

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  Explicitly accounts for model uncertainty and disturbances

• Robust Stability

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  Guarantees stability under all uncertainty realizations

• Constraint Satisfaction

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  Ensures constraints are met despite uncertainty

• Conservative Design

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  May be conservative to ensure robustness

• Computational Complexity

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  Higher computational cost due to robust optimization

Implementation Approaches

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Tube-based Robust MPC Controller

Robust MPC using tube-based approach with RPI sets

Complexity:

• Time: O(N⁴)
• Space: O(N³)

Advantages

• Handles model uncertainty and disturbances

• Guarantees constraint satisfaction under uncertainty

• Provides robust stability guarantees

• Suitable for safety-critical applications

Disadvantages

• Computationally intensive due to robust optimization

• May be conservative in performance

• Requires knowledge of uncertainty bounds

• Complex implementation and tuning

Complete Implementation

The full implementation with error handling, comprehensive testing, and additional variants is available in the source code:

• Main implementation with tube-based approach: src/algokit/mpc/robust_mpc.py

• Comprehensive test suite including robustness tests: tests/unit/mpc/test_robust_mpc.py

Complexity Analysis

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Time & Space Complexity Comparison

Approach	Time Complexity	Space Complexity	Notes
Tube-based Robust MPC	O(N⁴)	O(N³)	Complexity depends on prediction horizon N and uncertainty set size

Performance Considerations

• Robust optimization requires solving min-max problems

• Computational complexity scales with uncertainty set size

• Memory requirements scale with system dimensions and uncertainty

Use Cases & Applications

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Application Categories

Aerospace Systems

• Flight Control: Aircraft control with aerodynamic uncertainty

• Spacecraft Guidance: Orbital maneuvers with gravitational perturbations

• UAV Control: Autonomous flight with wind disturbances

• Satellite Control: Attitude control with environmental disturbances

Automotive Systems

• Vehicle Dynamics: Trajectory tracking with tire parameter uncertainty

• Engine Control: Combustion control with fuel quality variations

• Hybrid Vehicles: Energy management with battery degradation

• Autonomous Driving: Path planning with sensor uncertainty

Process Industries

• Chemical Plants: Process control with catalyst deactivation

• Oil Refineries: Distillation control with feed composition variations

• Power Plants: Load control with demand uncertainty

• Manufacturing: Quality control with material property variations

Robotics

• Manipulator Control: Joint control with payload uncertainty

• Mobile Robot Navigation: Path planning with map uncertainty

• Humanoid Robots: Balance control with ground contact uncertainty

• Industrial Robots: Precision control with tool wear

Energy Systems

• Smart Grids: Power flow control with renewable energy uncertainty

• Wind Turbines: Power control with wind speed variations

• Solar Systems: Power management with irradiance uncertainty

• Battery Management: Charging control with aging effects

Educational Value

• Control Theory: Robust control and uncertainty handling

• Optimization: Min-max optimization and robust optimization

• System Analysis: Uncertainty modeling and robust stability

• Real-time Systems: Implementation of robust control algorithms

References & Further Reading

:material-book: Core Textbooks

:material-book:

Model Predictive Control: Theory, Computation, and Design

2017 • Nob Hill Publishing • ISBN 978-0-9759377-0-9

:material-amazon: Buy on Amazon

:material-book:

Model Predictive Control: Classical, Robust and Stochastic

2017 • Springer • ISBN 978-3-319-42053-9

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:material-library: Robust MPC Theory

:material-book:

Robust model predictive control of constrained linear systems with bounded disturbances

2005 • Automatica • Volume 41, pages 219-224

:material-book:

Invariant approximations of the minimal robust positively invariant set

2005 • IEEE Transactions on Automatic Control • Volume 50, pages 406-410

:material-web: Online Resources

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Robust MPC

Wikipedia article on MPC

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Tube-based MPC

Control Engineering article on robust MPC

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Uncertainty in Control Systems

MATLAB Robust Control Toolbox

:material-code-tags: Implementation & Practice

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YALMIP

MATLAB toolbox for optimization modeling

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MOSEK

Optimization solver for robust optimization

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CVX

MATLAB software for disciplined convex programming

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:

• Distributed MPC - Model Predictive Control for large-scale systems using distributed optimization and coordination between multiple local controllers to achieve global objectives.

• Economic MPC - Model Predictive Control that optimizes economic objectives rather than tracking performance, focusing on profit maximization and cost minimization in process industries.

• Linear MPC - Model Predictive Control for linear time-invariant systems formulated as a Quadratic Programming problem with efficient real-time solution.

• Nonlinear MPC - Model Predictive Control for nonlinear systems using Sequential Quadratic Programming to handle complex dynamics and constraints.

• 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.

• Learning MPC - Model Predictive Control that learns system dynamics and improves performance through data-driven approaches, combining machine learning with predictive control.