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Transfer Function Estimation in System Identification Toolbox via Vector Fitting
Ahmet Arda Ozdemir, Suat Gumussoy
TL;DR
Continuous-time transfer-function estimation from frequency-response data faces numerical issues in conventional SK implementations, while IV iterations can have worse conditioning. The paper combines orthonormal rational bases with a bilinear mapping, a minimal state-space realization, and coefficient constraints; it reports numerical improvements and demonstrates the algorithm in two examples.
Problem
Monomial-basis SK implementations can suffer numerical issues, and IV iterations have worse condition numbers than SK iterations.
Source: §1 · p. 1Method
The paper uses orthonormal rational basis functions with a bilinear mapping to the unit disk, a minimal state-space realization, and efficient coefficient constraints.
Source: §1 · pp. 1–2, §3.1 · p. 3, §3.2 · p. 4Results
Transforming the problem onto the unit disk frequently leads to important numerical improvements; measurement scaling reduced the worst observed condition number from 4.2·10^14 to 2.6·10^12.
Source: §3.3 · p. 4, §5 · p. 6Takeaways & Limitations
The method supports grey-box estimation and constraints including arbitrary relative degrees, integrators, and affine relationships between coefficients.
Source: §1 · p. 1For data with a small-magnitude valley, those measurements contribute little to the cost function; frequency-based weighting is a stated remedy.
Source: §4.1 · p. 6
Abstract
from arXiv · showhide
This paper considers black- and grey-box continuous-time transfer function estimation from frequency response measurements. The first contribution is a bilinear mapping of the original problem from the imaginary axis onto the unitdisk. This improves the numerics of the underlying Sanathanan-Koerner iterations and the more recent instrumental-variable iterations. Orthonormal rational basis functions on the unit disk are utilized. Each iteration step necessitates a minimal state-space realization with these basis functions. One such derivation is the second contribution. System identification with these basis functions yield zero-pole-gain models. The third contribution is an efficient method to express transfer function coefficient constraints in terms of the orthonormal rational basis functions. This allows for estimating transfer function models with arbitrary relative degrees (including improper models), along with other fixed and bounded parameter values. The algorithm is implemented in the tfest function in System Identification Toolbox (Release 2016b, for use with MATLAB) for frequency domain data. Two examples are presented to demonstrate the algorithm performance.
1. INTRODUCTION
The introduction motivates continuous-time transfer-function identification by numerical issues in conventional methods, then presents an approach combining orthonormal rational bases with domain mapping, coefficient constraints, and a minimal realization. The resulting algorithm is implemented in tfest for frequency-domain data.
1. INTRODUCTION
- Monomial-basis SK implementations can suffer numerical issues, while IV iterations have worse condition numbers than SK iterations.The paper addresses numerical issues in continuous-time transfer-function estimation from frequency-response measurements.Source: §1 · p. 1
- Orthonormal vector fitting uses orthonormal rational basis functions to further improve numerics beyond barycentric vector fitting.Vector fitting was introduced by Gustavsen et al. (1999); later work reported improved numerical properties for vector fitting (Drmaˇc et al. (2015a,b)) and introduced orthonormal rational bases (Deschrijver et al. (2007)).Source: §1 · p. 1
- The proposed coefficient-constraint method supports prior knowledge such as relative degree, integrators, and affine relationships between coefficients.It is designed for use within SK and IV iterations with any choice of basis functions.Source: §1 · p. 1
- The contributions include mapping the imaginary axis to the unit disk and deriving a minimal state-space realization for orthonormal rational basis functions there.The resulting algorithm is implemented in tfest for frequency-domain data in System Identification Toolbox (Release 2016b).Source: §1 · p. 2
2. PROBLEM STATEMENT AND CURRENT APPROACHES
The section formulates frequency-response transfer-function estimation and reviews how basis choices affect numerical conditioning. It contrasts SK iterations with IV iterations, whose converged fixed points have a stronger connection to the nonlinear least-squares problem.
2. PROBLEM STATEMENT AND CURRENT APPROACHES
- SK estimates a transfer function by solving a linear least-squares problem at each iteration with the previous denominator, while avoiding the trivial zero solution through a denominator constraint.The paper describes the estimation target as a MIMO system fitted to frequency-response measurements.Source: §2 · p. 2
- With noiseless data and estimated order at least the true order, SK fixed points coincide with nonlinear-problem stationary points, and one iteration can recover the true system absent numerical issues.With noise or data not captured by linear models, SK convergence is an open problem and a converged result need not be a local optimum; model-order estimation also remains open.Source: §2 · p. 2
2.1 Basis Functions and Vector Fitting
- VF improves monomial-basis numerics with a barycentric representation and interpolation points chosen as prior-iteration denominator zeros, avoiding the weight 1/d^(k−1)(s).These choices improve numerical stability, while repeated interpolation points can produce linearly dependent basis functions for closely packed or repeated poles.Source: §2.1 · pp. 2–3
- Orthonormal rational polynomials address close or repeated poles, except for multiple integrators; data-dependent orthogonal bases give condition number 1 but cost more to construct.The paper identifies the higher construction complexity of data-dependent bases as a future improvement direction for tfest.Source: §2.1 · p. 3
2.2 Instrumental Variable Iterations
- IV fixed points coincide with nonlinear least-squares stationary points under noise, nonlinearities, or incorrect model order; convergence is not guaranteed, but convergence yields a local optimum.SK fixed points do not generally have this property under those conditions.Source: §2.2 · p. 3
- tfest applies IV after SK, using the SK solution to initialize IV and seek a solution near a reasonable local optimum.The paper describes this sequence as a conservative choice for a broad user base and varied datasets.Source: §2.2 · p. 3
3. CONTRIBUTIONS
The section develops domain mapping, coefficient constraints, measurement scaling, and state-space realizations for orthonormal rational basis functions, then outlines the estimation sequence.
3.1 Domain Mapping
- 4.9·10^8 falls to 9.1·10^5 with transformed-domain barycentric bases, and column scaling reduces these condition numbers further to 1.2·10^3 and 8.1·10^1, respectively.The transformed-domain bases remain better conditioned with or without column scaling.Source: §3.1 · p. 4
3.2 Grey-Box Estimation: Parameter Constraints
- A numerical linear mapping relates estimated basis parameters to transfer-function coefficients, enabling fixed and bounded coefficient constraints in SK and constrained-cost treatment in IV.The constraints can encode known integrators, relative degree, and affine coefficient relationships.Source: §3.2 · p. 4
3.3 Measurement Scaling
- Measurement-magnitude scaling reduces the worst observed SK and IV condition number from 4.2·10^14 to 2.6·10^12, a modest gain on already well-scaled data.The method is described as straightforward, computationally cheap, and useful.Source: §3.3 · p. 4
3.4 State-Space Realization for Orthonormal Rational Basis Functions
- A minimal state-space realization of the unit-disk orthonormal rational basis supports extracting denominator zeros to set interpolation points for the next iteration.Conjugate-paired components are realized together to retain real coefficients.Source: §3.4 · p. 5
3.5 Algorithm Summary
- The algorithm scales measurements, performs an initial monomial fit followed by SK and IV iterations, selects the best solution, and converts it back to a transfer-function model.The final steps reverse the domain mapping and measurement scaling before converting the zero-pole-gain model.Source: §3.5 · p. 5
4. NUMERICAL EXPERIMENTS
The unconstrained 20th-order estimate captures the dynamics except for a narrow valley, while the constrained grey-box estimate captures system dynamics and honors the coefficient constraints. The experiments use noisy frequency-response measurements to demonstrate both settings.
4.1 Experiment Without Parameter Constraints
- In the experiment, parameters are randomly generated from a predetermined range, yielding a dynamically rich model with ten lightly damped modes spread over seven decades.The frequency response is sampled at 700 logarithmically spaced points in [10^-1, 10^6] rad/s. Multiplicative noise uses zero-mean Gaussian random values with variance 0.01, giving approximately 20 dB signal-to-noise ratio.Source: §4.1 · p. 5
- The 20th-order estimate captures the dynamics of interest, missing only the valley near 2.74·10^4 rad/s.The small-magnitude data around the valley contribute little to the cost function; frequency-based weighting is offered as a remedy.Source: §4.1 · pp. 5–6, Figure 2 · p. 6
4.2 Experiment with Parameter Constraints
- The constrained experiment fixes three leading numerator coefficients and the last denominator coefficient for a plant with an integrator and relative degree 3.The frequency response is sampled at 300 logarithmically spaced points from 1 to 10^4 rad/s with 20 dB signal-to-noise ratio.Source: §4.2 · p. 6
- The grey-box estimate captures system dynamics while enforcing the transfer-function coefficient constraints through poles and zeros in transformed domain q.The approach honors constraints specified in the s domain.Source: §4.2 · p. 6, Figure 3 · p. 6
5. CONCLUDING REMARKS
Mapping the problem to the unit disk and scaling measurement data improve the numerics of SK and IV iterations. The basis-function approach also supports grey-box estimation by efficiently relating basis functions to transfer-function coefficients.
5. CONCLUDING REMARKS
- Mapping the SK and IV iteration problem onto the unit disk often improves numerical behavior, and scaling measurement data is also helpful.Source: §5 · p. 6
- Vandermonde matrix properties provide an efficient way to relate barycentric or orthonormal rational basis functions to transfer-function coefficients for grey-box estimation.These bases yield zero-pole-gain models while retaining their numerical benefits for grey-box fits.Source: §5 · p. 6