Deep Polytopic Autoencoders for Nonlinear Feedback Design

Core Concepts

Polytopic autoencoders provide efficient nonlinear controller design through low-dimensional linear parameter-varying approximations.

Abstract

1. Introduction:

   • Designing controllers for nonlinear high-dimensional systems is challenging.
   • Existing methods require structural assumptions or are limited by system size.
   • State-dependent Riccati equation (SDRE) approximations have reduced computational complexity.

2. Polytopic Autoencoders:

   • Represent states in a polytope for advantages in reconstruction.
   • Provide low-dimensional parametrizations for controller design.
   • Combine learning approaches with local bases selection.

3. Series Expansions for SDRE Feedback Design:

   • Higher-order expansions improve nonlinear feedback design.
   • Truncated series expansions of the system dynamics.
   • Efficiently solve high-dimensional matrix equations for feedback design.

4. Numerical Experiments:

   • Evaluation of reconstruction errors for different models.
   • Comparison of SDRE feedback approximations with different orders.
   • Performance evaluation of feedback designs for stabilization.

5. Conclusion:

   • Nonlinear autoencoders with smooth clustering improve parametrization quality.
   • Higher-order expansions of LPV coefficients enhance feedback design.
   • Future work includes optimizing approximations for feedback control.

Stats

For example, for r = 10 and a first-order expansion, the solve of 11 matrix equations was required.
A second or third-order expansion would require 66 or 286 solves.
The property that Fe(0) = 0 and Fc(0) = 0 is achieved by using (convolutional) nonlinear layers without bias terms.

Quotes

"Polytopic autoencoders provide certain advantages over the reconstruction in a linear space."
"The nonlinear decoding has a Taylor expansion with vanishing second-order terms."

Key Insights Distilled From

Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback design

by Jan Heiland,... at arxiv.org 03-28-2024

https://arxiv.org/pdf/2403.18044.pdf

Deeper Inquiries

How can the use of polytopic autoencoders impact real-world control systems

Polytopic autoencoders can have a significant impact on real-world control systems by providing low-dimensional parametrizations of states within a polytope. This approach allows for more efficient and effective nonlinear controller design, especially for high-dimensional systems where traditional methods may be computationally intensive. By reconstructing states within a bounded set, polytopic autoencoders offer advantages in capturing system dynamics within observed regions, leading to tailored and bounded control strategies. The use of smooth clustering networks in polytopic autoencoders further enhances the reconstruction quality, improving the overall performance of control systems in real-world applications.

What are the potential drawbacks or limitations of relying on series expansions for feedback design

While series expansions are valuable for approximating nonlinear feedback designs, they come with certain drawbacks and limitations. One limitation is the computational complexity associated with solving high-dimensional matrix equations, especially for large-scale systems. The repeated solves required for higher-order expansions can be computationally expensive and time-consuming, impacting the real-time applicability of the feedback design. Additionally, the accuracy of the series expansions heavily relies on the choice of approximation schemes and parameter setups, which may introduce errors and uncertainties in the control system.

How might advancements in deep learning techniques influence the future development of nonlinear controller design

Advancements in deep learning techniques are poised to revolutionize the future development of nonlinear controller design. By leveraging deep neural networks, researchers can explore more complex and nonlinear relationships within control systems, leading to more accurate and adaptive control strategies. Deep learning models can handle high-dimensional data efficiently, enabling the extraction of intricate patterns and dynamics that traditional methods may overlook. The integration of deep learning with control theory opens up possibilities for data-driven control approaches, reinforcement learning-based controllers, and adaptive control systems that can continuously learn and optimize performance in real-time scenarios. These advancements hold the potential to enhance the robustness, adaptability, and intelligence of future nonlinear controller designs.