Explicit Feedback Synthesis for Nonlinear Robust Model Predictive Control with Guaranteed Uniform Approximation

Core Concepts

The authors present a novel grid-based technique called QuIFS (Quasi-Interpolation driven Feedback Synthesis) that provides strict guarantees of uniform approximation of the optimal feedback policy for nonlinear robust model predictive control problems.

Abstract

The key highlights and insights of the content are:

The QuIFS algorithm departs from conventional approximation techniques used in the MPC industry, such as multiparametric programming and kernel methods. It is driven by a particular type of grid-based quasi-interpolation scheme.

QuIFS provides one-shot uniform approximation guarantees of the optimal feedback policy, along with stability and recursive feasibility guarantees. These guarantees are robust and do not involve probabilistic (soft) bounds.

The algorithm applies to nonlinear systems and non-convex cost functions, and relies on coarse properties of the optimal feedback, such as Lipschitz continuity, rather than detailed local structural properties.

The complexity of the offline computations associated with QuIFS scales exponentially with the state dimension, unlike standard explicit MPC techniques where the complexity scales exponentially with the number of constraints.

QuIFS provides approximation error guarantees measured with respect to the uniform metric, which is necessary to ensure recursive feasibility. This is achieved without solving the associated Bellman equations/recursions.

The key technical tool used in QuIFS is a particular type of quasi-interpolation that conforms to neither parametric nor non-parametric function approximation techniques. It departs sharply from typical approximation theoretic tools that ensure asymptotic convergence.

For a prespecified uniform error margin, it is always possible to pick the quasi-interpolation parameters (discretization interval, shape parameter, and truncation parameter) such that the uniform error between the optimal feedback and the approximated one stays below the desired threshold.

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Key Insights Distilled From

Explicit feedback synthesis for nonlinear robust model predictive control driven by quasi-interpolation

by Siddhartha G... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2306.03027.pdf

Deeper Inquiries

How can the computational complexity of the QuIFS algorithm be further reduced, especially for high-dimensional systems, while maintaining the strong approximation guarantees

To reduce the computational complexity of the QuIFS algorithm for high-dimensional systems while maintaining strong approximation guarantees, several strategies can be employed:

Dimensionality Reduction Techniques: Utilize dimensionality reduction techniques such as Principal Component Analysis (PCA) or feature selection to reduce the number of dimensions in the system. By working with a lower-dimensional representation of the system, the computational burden can be significantly reduced.

Sparse Grids: Implement sparse grid techniques to selectively evaluate the system at specific grid points rather than a dense grid. This can help in reducing the number of evaluations required, especially in high-dimensional spaces.

Parallel Computing: Utilize parallel computing resources to distribute the computational load across multiple processors or nodes. This can help in speeding up the computations for high-dimensional systems.

Optimized Data Structures: Implement optimized data structures and algorithms tailored to the specific characteristics of the system. This can help in efficient storage and retrieval of data during the computation process.

Adaptive Grid Refinement: Implement adaptive grid refinement strategies to focus computational resources on regions of the state space that are more critical for the control problem. This can help in allocating computational resources more efficiently.

By incorporating these strategies, the computational complexity of the QuIFS algorithm can be effectively managed for high-dimensional systems while ensuring the desired approximation guarantees.

What are the potential limitations or drawbacks of the quasi-interpolation-based approach compared to other explicit MPC techniques, such as those based on neural networks or multiparametric programming

While the quasi-interpolation-based approach used in the QuIFS framework offers several advantages, such as strong approximation guarantees and recursive feasibility, it also has some potential limitations compared to other explicit MPC techniques like neural networks or multiparametric programming:

Limited Expressiveness: Quasi-interpolation techniques may have limitations in capturing complex nonlinear relationships compared to neural networks, which are known for their ability to approximate highly nonlinear functions.

Computational Efficiency: Neural network-based approaches can sometimes offer faster computation times, especially for large-scale systems, due to their parallel processing capabilities and optimized architectures.

Generalization: Neural networks have the ability to generalize well to unseen data, which may not be as straightforward with quasi-interpolation techniques that rely on specific grid points for interpolation.

Complexity Handling: Multiparametric programming techniques are known for their ability to handle complex constraints and system dynamics efficiently, which may be more challenging with quasi-interpolation methods.

Scalability: Neural networks can be easily scaled to handle larger and more complex systems, while the scalability of quasi-interpolation techniques may be limited in high-dimensional spaces.

While the QuIFS framework offers unique benefits, it's essential to consider these limitations when choosing between different explicit MPC techniques based on the specific requirements of the control problem.

Can the QuIFS framework be extended to handle more general classes of nonlinear systems and cost functions beyond the robust MPC setting considered in this work

The QuIFS framework can be extended to handle more general classes of nonlinear systems and cost functions beyond the robust MPC setting considered in the current work by incorporating the following enhancements:

Nonlinear System Dynamics: Extend the QuIFS algorithm to accommodate a wider range of nonlinear system dynamics by incorporating more sophisticated interpolation techniques that can capture complex nonlinear relationships.

Non-Convex Cost Functions: Modify the QuIFS framework to handle non-convex cost functions by adapting the interpolation scheme to approximate the cost function accurately within the given error bounds.

Constraint Handling: Enhance the QuIFS algorithm to efficiently handle a broader range of constraints, including non-convex and time-varying constraints, by incorporating advanced optimization techniques and constraint handling strategies.

Adaptive Approximation: Develop adaptive approximation strategies within the QuIFS framework to dynamically adjust the approximation accuracy based on the system's behavior and requirements, ensuring robust performance across different scenarios.

By incorporating these extensions and enhancements, the QuIFS framework can be tailored to address a wider range of nonlinear systems and cost functions, making it more versatile and applicable to diverse control problems.