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An iterative tangential interpolation algorithm for model reduction of MIMO systems

Jared Jonas, Bassam Bamieh

TL;DR

There is freedom in the interpolation point selection method, leading to multiple algorithms that have trade-offs between computational complexity and approximation performance, and it is proved that a weighted \(H_2\) norm of a representative error system is monotonically decreasing as interpolation points are added.

Abstract

We consider model reduction of large-scale multi-input, multi-output (MIMO) systems using tangential interpolation in the frequency domain. Our scheme is related to the recently-developed Adaptive Antoulas--Anderson (AAA) algorithm, which is an iterative algorithm that uses concepts from the Loewner framework. Our algorithm has two main features. The first is the use of freedom in interpolation weight matrices to optimize a proxy for an \(H_2\) system error. The second is the use of low-rank interpolation, where we iteratively add low-order interpolation data based on several criteria including minimizing maximum errors. We show there is freedom in the interpolation point selection method, leading to multiple algorithms that have trade-offs between computational complexity and approximation performance. We prove that a weighted \(H_2\) norm of a representative error system is monotonically decreasing as interpolation points are added. Finally, we provide computational results and some comparisons with prior work, demonstrating performance on par with standard model reduction methods.

An iterative tangential interpolation algorithm for model reduction of MIMO systems

TL;DR

There is freedom in the interpolation point selection method, leading to multiple algorithms that have trade-offs between computational complexity and approximation performance, and it is proved that a weighted norm of a representative error system is monotonically decreasing as interpolation points are added.

Abstract

We consider model reduction of large-scale multi-input, multi-output (MIMO) systems using tangential interpolation in the frequency domain. Our scheme is related to the recently-developed Adaptive Antoulas--Anderson (AAA) algorithm, which is an iterative algorithm that uses concepts from the Loewner framework. Our algorithm has two main features. The first is the use of freedom in interpolation weight matrices to optimize a proxy for an system error. The second is the use of low-rank interpolation, where we iteratively add low-order interpolation data based on several criteria including minimizing maximum errors. We show there is freedom in the interpolation point selection method, leading to multiple algorithms that have trade-offs between computational complexity and approximation performance. We prove that a weighted norm of a representative error system is monotonically decreasing as interpolation points are added. Finally, we provide computational results and some comparisons with prior work, demonstrating performance on par with standard model reduction methods.

Paper Structure

This paper contains 14 sections, 10 theorems, 90 equations, 4 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation and main results
  3. State space realizations
  4. Left-tangential interpolating systems
  5. Enforcing real coefficients
  6. The system $H$
  7. Weight Optimization
  8. Interpolation points selection
  9. Frequency selection strategies
  10. Random and discrete strategies
  11. Rank considerations
  12. Summary
  13. Computational results
  14. Discussion and conclusion

Key Result

Theorem III.1

Consider the assumptions in Assumptions 1 and the interpolant $R$ from equation eq:rtan_int, where each $W_i\in\mathbb{C}^{p\times r_i}$, $i=1,\ldots, \ell$ is a full-column-rank free parameter. Then, $R$ left-interpolates $G$ at each $j\omega_i$ for any valid weight $W_i\in\mathbb{C}^{p\times r_i}$

Figures (4)

  • Figure 1: A comparison of the performance of balanced reduction in blue, the generalized Loewner framework antoulas2017tutorial in red, the tangential-AAA (tAAA) algorithm Benner23 in yellow, and one of our proposed methods (max. error algorithm) in purple. The tAAA algorithm and the Loewner framework both were provided an input set of 1000 points logarithmically spaced between $\pm[10^{-1}, \, 10^2]j$. The graph shows the norm of the error system versus the reduced system's number of poles. The input model is the 3-input, 3-output, 270-state "ISS" model iss_model.
  • Figure 2: A comparison of the performance of the maximum error strategy (in blue) and the gridded frequency point selection (in varying colors) with varying $K$, the number of points in the grid.
  • Figure 3: A comparison of the performance of the maximum error strategy (in black) and random frequency point selection strategy (in varying colors) with varying $K$, the number of points tested per iteration, versus reduced system size. The solid lines represent the mean behavior of a number of random runs, with the shaded regions representing the middle 50% behavior.
  • Figure 4: A plot of the maximum singular value ($\sigma_1$) of the unreduced input model and each of our proposed algorithms reduced to 6 states on the upper sub-figure, along with the absolute error in the bottom sub-figure. In the top figure, the dotted black line indicates $\sigma_1$ of the input system, which is the aforementioned "ISS" model. In both plots, the maximum error approach is in red, the gridded approach is in purple, and the random approach is in blue. The diamonds indicate where the matching algorithm placed an interpolation point. The gridded approach uses a set of 20 frequencies logarithmically spaced between $[10^{-1}, \, 10^2]$. The random approach selects the best frequency of 20 random frequencies between $[10^{-1}, \, 10^2]$.

Theorems & Definitions (20)

  • Theorem III.1
  • Theorem III.2
  • proof
  • Remark III.1
  • Theorem III.3
  • Theorem IV.1
  • proof
  • Theorem IV.2
  • proof
  • Corollary IV.1
  • ...and 10 more