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Balanced Truncation of Descriptor Systems with a Quadratic Output

Jennifer Przybilla, Igor Pontes Duff, Pawan Goyal, Peter Benner

Abstract

This work discusses model reduction for differential-algebraic systems with quadratic output equations. Under mild conditions, these systems can be transformed into a Weierstraß canonical form and, thus, be decoupled into differential equations and algebraic equations. The corresponding decoupled states are referred to as proper and improper states. Due to the quadratic function of the state as an output, the proper and improper states are coupled in the output equation, which imposes a challenge from a model reduction viewpoint. Keeping the coupling in mind, our goal in this work is to find important subspaces of the proper and improper states and to reduce the system accordingly. To that end, we first propose the system's matrices, the so-called Gramians, to characterize the system's dominant subspaces. We pay particular attention to the computation of the observability Gramians that take into account the nonlinear coupling between the proper and the improper states. We furthermore show that the proposed Gramians are related to certain kernel functions, which are used to identify important subspaces. This allows us to propose a reduction algorithm to obtain reduced-order systems by removing the subspaces that are difficult to reach, as well as, difficult to observe. Moreover, we quantify the error between the full-order and reduced-order models and demonstrate the proposed methodology using three numerical experiments.

Balanced Truncation of Descriptor Systems with a Quadratic Output

Abstract

This work discusses model reduction for differential-algebraic systems with quadratic output equations. Under mild conditions, these systems can be transformed into a Weierstraß canonical form and, thus, be decoupled into differential equations and algebraic equations. The corresponding decoupled states are referred to as proper and improper states. Due to the quadratic function of the state as an output, the proper and improper states are coupled in the output equation, which imposes a challenge from a model reduction viewpoint. Keeping the coupling in mind, our goal in this work is to find important subspaces of the proper and improper states and to reduce the system accordingly. To that end, we first propose the system's matrices, the so-called Gramians, to characterize the system's dominant subspaces. We pay particular attention to the computation of the observability Gramians that take into account the nonlinear coupling between the proper and the improper states. We furthermore show that the proposed Gramians are related to certain kernel functions, which are used to identify important subspaces. This allows us to propose a reduction algorithm to obtain reduced-order systems by removing the subspaces that are difficult to reach, as well as, difficult to observe. Moreover, we quantify the error between the full-order and reduced-order models and demonstrate the proposed methodology using three numerical experiments.
Paper Structure (19 sections, 11 theorems, 120 equations, 5 figures, 1 algorithm)

This paper contains 19 sections, 11 theorems, 120 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weierstraß-canonical form
  4. BT for DAEs with linear output
  5. BT for linear dynamical systems with quadratic outputs
  6. Gramians for DAE systems with quadratic output
  7. Proper observability Gramian
  8. Improper observability Gramians
  9. Kernel functions and balanced truncation
  10. Energy norms of kernel functions
  11. Balanced truncation
  12. Error Estimation
  13. The proper-proper output error
  14. The improper-proper output error
  15. Extension to the multiple output case
  16. ...and 4 more sections

Key Result

Theorem 3.1

Consider the asymptotically stable DAE system with a quadratic output equation from eq:DAE_q and the corresponding proper controllability Gramian $\boldsymbol{\mathcal{P}}_{\mathrm{p}}$ as defined in eq:Gramian_contr. The proper observability Gramian $\boldsymbol{\mathcal{Q}}_{\mathrm{p}\mathrm{p}}$ where the projection matrices $\mathbf{P}_{\mathrm{l}}$ and $\mathbf{P}_{\mathrm{r}}$ are defined a

Figures (5)

  • Figure 1: An illustrative example: Output responses of the full-order and reduced-order models and the corresponding error.
  • Figure 2: Index--2 Stokes example: the decay of Hankel singular values.
  • Figure 3: Index--2 Stokes example: the outputs of the full-order model and the obtained reduced-order model of order $15$ for a test input $\mathbf{u}(t) = \mathrm{sin}(t)^3e^{-t/2}$. The plot also shows the error between the outputs and our error estimate.
  • Figure 4: Index--3 Mechanical system example: the decay of Hankel singular values.
  • Figure 5: Index--3 Mechanical system example: the outputs of the full-order model and the obtained reduced-order model of order $21$ for the input $\mathbf{u}(t) = \mathrm{sin}(2t)^2e^{-t/2}$. The plot also shows the error between the outputs and our error estimate.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Theorem 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.3
  • ...and 19 more