Core Concepts
Polytopic autoencoders provide efficient nonlinear controller design through low-dimensional linear parameter-varying approximations.
Abstract
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.
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.
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.
Numerical Experiments:
Evaluation of reconstruction errors for different models.
Comparison of SDRE feedback approximations with different orders.
Performance evaluation of feedback designs for stabilization.
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."