Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards a MATLAB Toolbox to compute backstepping kernels using the power series method

Xin Lin, Rafael Vazquez, Miroslav Krstic

TL;DR

A technique is introduced that mitigates this behaviour by computing the expansion at different points, denoted as localized power series, which effectively navigates around singularities, and can also accelerates convergence by using more local approximations.

Abstract

In this paper, we extend our previous work on the power series method for computing backstepping kernels. Our first contribution is the development of initial steps towards a MATLAB toolbox dedicated to backstepping kernel computation. This toolbox would exploit MATLAB's linear algebra and sparse matrix manipulation features for enhanced efficiency; our initial findings show considerable improvements in computational speed with respect to the use of symbolical software without loss of precision at high orders. Additionally, we tackle limitations observed in our earlier work, such as slow convergence (due to oscillatory behaviors) and non-converging series (due to loss of analiticity at some singular points). To overcome these challenges, we introduce a technique that mitigates this behaviour by computing the expansion at different points, denoted as localized power series. This approach effectively navigates around singularities, and can also accelerates convergence by using more local approximations. Basic examples are provided to demonstrate these enhancements. Although this research is still ongoing, the significant potential and simplicity of the method already establish the power series approach as a viable and versatile solution for solving backstepping kernel equations, benefiting both novel and experienced practitioners in the field. We anticipate that these developments will be particularly beneficial in training the recently introduced neural operators that approximate backstepping kernels and gains.

Towards a MATLAB Toolbox to compute backstepping kernels using the power series method

TL;DR

A technique is introduced that mitigates this behaviour by computing the expansion at different points, denoted as localized power series, which effectively navigates around singularities, and can also accelerates convergence by using more local approximations.

Abstract

In this paper, we extend our previous work on the power series method for computing backstepping kernels. Our first contribution is the development of initial steps towards a MATLAB toolbox dedicated to backstepping kernel computation. This toolbox would exploit MATLAB's linear algebra and sparse matrix manipulation features for enhanced efficiency; our initial findings show considerable improvements in computational speed with respect to the use of symbolical software without loss of precision at high orders. Additionally, we tackle limitations observed in our earlier work, such as slow convergence (due to oscillatory behaviors) and non-converging series (due to loss of analiticity at some singular points). To overcome these challenges, we introduce a technique that mitigates this behaviour by computing the expansion at different points, denoted as localized power series. This approach effectively navigates around singularities, and can also accelerates convergence by using more local approximations. Basic examples are provided to demonstrate these enhancements. Although this research is still ongoing, the significant potential and simplicity of the method already establish the power series approach as a viable and versatile solution for solving backstepping kernel equations, benefiting both novel and experienced practitioners in the field. We anticipate that these developments will be particularly beneficial in training the recently introduced neural operators that approximate backstepping kernels and gains.
Paper Structure (12 sections, 1 theorem, 49 equations, 6 figures, 2 tables)

This paper contains 12 sections, 1 theorem, 49 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Towards a MATLAB toolbox
  3. A review of the basic results of Power-Series
  4. A MATLAB double series vector-matrix framework: definitions and operators
  5. Basic examples revisited
  6. Example 1: Parabolic equation with space-varying reaction
  7. Example 2: Parabolic equation with space-varying diffusion
  8. Example 3: $2\times2$ 1-D linear hyperbolic system with space-varying coefficients
  9. Example 4: Motion planning kernels for $(0+2)\times(0+2)$ 1-D linear hyperbolic system with space-varying coupling
  10. Changing the point of expansion to accelerate convergence and avoid singularities
  11. Example 5: Parabolic equation with space-varying reaction not verifying Theorem 1 in Power-Series
  12. Concluding remarks

Key Result

Theorem 1

If there exists $\delta>0$ such that $\lambda(\xi)$ is analytic on the disc $\mathcal{D}_{L+\delta}(\xi_0)$ centered at $\xi_0$, and such disc covers the interval $[0,1]$, then there exists a $x_0$ and power series solution in the form of eqn-series_patch0 which converges and defines an analytic fun

Figures (6)

  • Figure 1: Convergent case with $\lambda(x)=3+x^2\sin(3x)$ for Example 1.
  • Figure 2: Convergent space-varying diffusion case, $\lambda(x)=3+x\sin(6x)$, $\epsilon(x)=2+x^2$ (Example 2).
  • Figure 3: Hyperbolic 2x2 kernel gains $K_{vu}(L,\xi)$ (left) and $K_{vv}(L,\xi)$ (right) for different orders, showing convergence (Example 3).
  • Figure 4: Solutions of gain kernels for Example 4. Note the discontinuous kernel $L_{12}(L,\xi)$ and the piecewise differentiable kernel $L_{11}(L,\xi)$.
  • Figure 5: Divergent case with $\lambda(x)=\sqrt{0.5+x^2}$ (Example 1).
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 1: Localized power series