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Scalable computation of input-normal/output-diagonal balanced realization for control-affine polynomial systems

Nicholas A. Corbin, Arijit Sarkar, Jacquelien M. A. Scherpen, Boris Kramer

Abstract

We present a scalable tensor-based approach to computing input-normal/output-diagonal nonlinear balancing transformations for control-affine systems with polynomial nonlinearities. This transformation is necessary to determine the states that can be truncated when forming a reduced-order model. Given a polynomial representation for the controllability and observability energy functions, we derive the explicit equations to compute a polynomial transformation to induce input-normal/output-diagonal structure in the energy functions in the transformed coordinates. The transformation is computed degree-by-degree, similar to previous Taylor-series approaches in the literature. However, unlike previous works, we provide a detailed analysis of the transformation equations in Kronecker product form to enable a scalable implementation. We derive the explicit algebraic structure for the equations, present rigorous analyses for the solvability and algorithmic complexity of those equations, and provide general purpose open-source software implementations for the proposed algorithms to stimulate broader use of nonlinear balanced truncation model. We demonstrate that with our efficient implementation, computing the nonlinear transformation is approximately as expensive as computing the energy functions.

Scalable computation of input-normal/output-diagonal balanced realization for control-affine polynomial systems

Abstract

We present a scalable tensor-based approach to computing input-normal/output-diagonal nonlinear balancing transformations for control-affine systems with polynomial nonlinearities. This transformation is necessary to determine the states that can be truncated when forming a reduced-order model. Given a polynomial representation for the controllability and observability energy functions, we derive the explicit equations to compute a polynomial transformation to induce input-normal/output-diagonal structure in the energy functions in the transformed coordinates. The transformation is computed degree-by-degree, similar to previous Taylor-series approaches in the literature. However, unlike previous works, we provide a detailed analysis of the transformation equations in Kronecker product form to enable a scalable implementation. We derive the explicit algebraic structure for the equations, present rigorous analyses for the solvability and algorithmic complexity of those equations, and provide general purpose open-source software implementations for the proposed algorithms to stimulate broader use of nonlinear balanced truncation model. We demonstrate that with our efficient implementation, computing the nonlinear transformation is approximately as expensive as computing the energy functions.

Paper Structure

This paper contains 14 sections, 6 theorems, 41 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonlinear balancing theory
  4. Kronecker product definitions and notation
  5. Main result: input-normal/output-diagonal transformation
  6. Analysis of the solvability and scalability of the proposed approach
  7. Existence of solutions
  8. FLOP count
  9. Numerical results
  10. Illustrative two-dimensional example
  11. $N$ coupled Duffing oscillators
  12. Diagonalization Performance
  13. Scaling
  14. Conclusion

Key Result

Theorem 1

Fujimoto2010 Suppose the Jacobian linearization of the nonlinear system eq:FOM-NL is minimal, asymptotically stable, and has distinct Hankel singular values. Then there is a neighborhood $\mathcal{W}$ of the origin and a smooth coordinate transformation $\mathbf{x} = \Phi(\mathbf{z})$ on $\mathcal{W

Figures (6)

  • Figure 1: Visualization of non-diagonal and diagonal tensors in three dimensions.
  • Figure 2: System of $N$ coupled nonlinear mass-spring-dampers.
  • Figure 3: (Case $n=6$) 3D and 2D projection of $\mathbf{w}_4$ (original coordinates) and $\tilde{\mathbf{w}}_4$ (transformed coordinates), which can be regarded as a symmetric 4D tensor. As a result of repeated Hankel singular values, $\tilde{\mathbf{w}}_4$ still has off-diagonal entries.
  • Figure 4: (Case $n=8$) 3D and 2D projection of $\mathbf{w}_4$ (original coordinates) and $\tilde{\mathbf{w}}_4$ (transformed coordinates), which can be regarded as a symmetric 4D tensor.
  • Figure 5: 2D projection of $\mathbf{w}_4$ (original coordinates) and $\tilde{\mathbf{w}}_4$ (transformed coordinates), which can be regarded as a symmetric 4D tensor ($n=50$).
  • ...and 1 more figures

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Definition 1: Diagonal monomial entries
  • Definition 2: Off-diagonal monomial entries
  • Definition 3: Diagonal polynomial
  • Definition 4: Minimal monomial representation
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • ...and 9 more