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Computing and Optimizing the $H^2$-norm of Delay Differential Algebraic Systems

Evert Provoost, Wim Michiels

TL;DR

It is proved that a Lanczos tau method using a spline based on Legendre orthogonal polynomials preserves stability and guarantees convergence of the $H^2$-norm.

Abstract

We present a Lanczos tau method for the approximation and optimization of the $H^2$-norm of time-delay systems described by semi-explicit delay differential algebraic equations. The soundness of this approach is proven under the assumption of a finite strong $H^2$-norm. Furthermore, we prove convergence if the rational approximation of the exponential underlying the discretization is well-behaved and the discretization is stability preserving. Numerical results suggest that, for multiple delays, the method converges at cubic rate in the discretization degree for systems of retarded type and linearly for those of neutral type. In the single delay case, we note geometric convergence of the $H^2$-norm for systems of both retarded and neutral type when a symmetric basis is chosen. Explicit formulas are derived for the gradient of the approximation with respect to system parameters and delays. These allow us to compute the entire gradient using only about double the computational time of approximating the $H^2$-norm alone. We illustrate how these can be used to synthesize robust feedback controllers and stable approximate models. The article is concluded by a discussion of how the presented results extend and improve for approximations based on splines. We note acceleration of the convergence rate by about two orders for such a choice. Finally, we prove that a Lanczos tau method using a spline based on Legendre orthogonal polynomials preserves stability and guarantees convergence of the $H^2$-norm.

Computing and Optimizing the $H^2$-norm of Delay Differential Algebraic Systems

TL;DR

It is proved that a Lanczos tau method using a spline based on Legendre orthogonal polynomials preserves stability and guarantees convergence of the -norm.

Abstract

We present a Lanczos tau method for the approximation and optimization of the -norm of time-delay systems described by semi-explicit delay differential algebraic equations. The soundness of this approach is proven under the assumption of a finite strong -norm. Furthermore, we prove convergence if the rational approximation of the exponential underlying the discretization is well-behaved and the discretization is stability preserving. Numerical results suggest that, for multiple delays, the method converges at cubic rate in the discretization degree for systems of retarded type and linearly for those of neutral type. In the single delay case, we note geometric convergence of the -norm for systems of both retarded and neutral type when a symmetric basis is chosen. Explicit formulas are derived for the gradient of the approximation with respect to system parameters and delays. These allow us to compute the entire gradient using only about double the computational time of approximating the -norm alone. We illustrate how these can be used to synthesize robust feedback controllers and stable approximate models. The article is concluded by a discussion of how the presented results extend and improve for approximations based on splines. We note acceleration of the convergence rate by about two orders for such a choice. Finally, we prove that a Lanczos tau method using a spline based on Legendre orthogonal polynomials preserves stability and guarantees convergence of the -norm.
Paper Structure (14 sections, 15 theorems, 85 equations, 4 figures)

This paper contains 14 sections, 15 theorems, 85 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finiteness of the strong $H^2$-norm of a DDAE
  4. Computing the $H^2$-norm of a RDDE
  5. Computing the $H^2$-norm of a DDAE
  6. Lanczos tau methods for DDAEs
  7. Convergence properties
  8. $H^2$-optimal synthesis
  9. Computing the gradient
  10. Numerical examples
  11. Spline based Lanczos tau methods
  12. Conclusions
  13. Intermediate steps to compute the derivatives
  14. Full system matrices of the numerical examples

Key Result

Proposition 1.1

Let $\tilde{V}$ and $\tilde{U}$ be matrices with orthogonal columns such that $\mathop{\mathrm{span}}\nolimits \tilde{V} = \ker E$ and $\mathop{\mathrm{span}}\nolimits \tilde{U}^* = \ker E^*$, then $\tilde{V} A_0 \tilde{U}$ is non-singular if, and only if, the system eq:ddae has at most differentiat

Figures (4)

  • Figure 1: Convergence rate for different RDDE configurations of system \ref{['eq:conv-sys']}. The dashed curve corresponds to the single delay case, $\vec{\delta} = (1, 0; 0, 0)$, the solid curve to two delays, $\vec{\delta} = (1, 1; 0, 0)$.
  • Figure 2: Convergence rate for different NDDE configurations of system \ref{['eq:conv-sys']}. The dashed curve has a single delay, $\vec{\delta} = (0, 0; 1, 0)$, the solid curve two, $\vec{\delta} = (0, 0; 1, 1)$.
  • Figure 3: Convergence rate for different mixed configurations of system \ref{['eq:conv-sys']}. The dashed curve has a neutral term on the maximal delay, $\vec{\delta} = (1, 0; 0, 1)$, the solid curve on the interior delay, $\vec{\delta} = (0, 1; 1, 0)$.
  • Figure 4: The same experiment as on figure \ref{['fig:conv-mixed']} but using a spline discretization. The dashed curve has a neutral term on the maximal delay, the solid curve on the interior delay.

Theorems & Definitions (31)

  • Proposition 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Definition 2.3
  • Proposition 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 21 more