On Computation of Approximate Solutions to Large-Scale Backstepping Kernel Equations via Continuum Approximation

TL;DR

This paper introduces two novel methods for approximating backstepping kernels in large-scale PDE control, enabling efficient computation without complexity growth with system size, and validates their effectiveness through numerical examples.

Contribution

It presents explicit power series solutions and closed-form solutions for continuum backstepping kernels, facilitating scalable control design for large PDE systems.

Findings

Power series convergence established for continuum kernels

Computational complexity reduced via series truncation

Numerical validation shows effective stabilization using approximate kernels

Abstract

We provide two methods for computation of continuum backstepping kernels that arise in control of continua (ensembles) of linear hyperbolic PDEs and which can approximate backstepping kernels arising in control of a large-scale, PDE system counterpart (with computational complexity that does not grow with the number of state components of the large-scale system). In the first method, we provide explicit formulae for the solution to the continuum kernels PDEs, employing a (triple) power series representation of the continuum kernel and establishing its convergence properties. In this case, we also provide means for reducing computational complexity by properly truncating the power series (in the powers of the ensemble variable). In the second method, we identify a class of systems for which the solution to the continuum (and hence, also an approximate solution to the respective large-scale) kernel equations can be constructed in closed form. We also present numerical examples to illustrate computational efficiency/accuracy of the approaches, as well as to validate the stabilization properties of the approximate control kernels, constructed based on the continuum.