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Estimating Closed-Loop Multiple-Input Single-Output Dynamic Systems using Gaussian Processes

Erik-Anant Stedjan Narayan, Sigurd Hofsmo Jakobsen, Kjetil Obstfelder Uhlen

arXiv:2610.07919v1eess.SYcs.SY

TL;DR

Individual transfer functions are difficult to identify in closed-loop MISO systems when feedback and correlated inputs obscure their contributions. This paper develops a complex-GPR estimator with analytical identifiability conditions and closed-loop bias handling; simulations show the identifiability requirement predicts module fit and reveal a regressor-construction trade-off.

  • Problem

    Recovering individual modules in closed-loop MISO systems is challenging because feedback and correlated inputs can obscure each input's contribution.

    Source: §1 · pp. 1, 3

  • Method

    The paper extends complex-GPR transfer-function estimation to MISO systems, deriving module-identifiability conditions and adapting closed-loop bias-handling techniques.

    Source: §1 · p. 2, §5 · p. 28

  • Results

    Monte Carlo simulations show that the theoretical identifiability requirement strongly predicts system-module fit.

    Source: §5 · p. 29

  • Takeaways & Limitations

    Closed-loop regressor construction involves a trade-off, so the paper concludes it should be selected on a per-channel basis rather than used as default preprocessing.

    Source: §5 · p. 29

    The developed binary pass/fail identifiability requirement remains to be refined into a continuous metric alongside robust regressor selection and handling.

    Source: §5 · p. 29

Abstract

from arXiv · show

This paper extends complex Gaussian process regression-based transfer function (TF) estimation from the single-input single-output setting to closed-loop multiple-input single-output (MISO) systems. Accurate estimation of individual TFs in closed-loop MISO systems is challenging as feedback and correlation between measured signals can obscure the contributions of each input. We formulate a Gaussian process-based estimator for MISO systems, provide analytical conditions under which each system module can be separately identified, and show that the estimator converges to the true TFs in the zero noise limit. Moreover, we demonstrate how conventional closed-loop bias handling techniques relying on external excitation can be employed to construct appropriate regressors. Monte Carlo simulations support the analytical identifiability results and highlight the trade-off between closed-loop bias and error introduced by constructing exogenous regressors partly from endogenous signals. The presented work offers an expressive, nonparametric alternative to existing estimation approaches for MISO systems, while providing a probabilistic characterization of the estimation uncertainty.

1. Introduction

The paper extends Gaussian-process transfer-function estimation to closed-loop MISO systems, where correlated inputs and feedback complicate recovery of individual modules. It develops identifiability conditions and closed-loop bias handling alongside the estimator.

1. Introduction

  • The work extends open-loop SISO Gaussian-process transfer-function estimation to closed-loop MISO systems with noisy outputs and possibly correlated inputs.The motivating application is power-system management, though the approach addresses system identification more broadly.Source: §1 · pp. 1–3
  • Separate module recovery requires sufficiently informative excitation, but conventional persistence-of-excitation conditions do not directly transfer to complex-GPR frequency-domain estimation.The paper frames the issue as whether input-weighted GP-supported latent spaces uniquely decompose the observations.Source: §1 · p. 3
  • Existing GP-based transfer-function estimators address SISO systems but lack a suitable solution for closed-loop estimation when output disturbances can bias estimates.The paper identifies bridging closed-loop bias-handling methods with GPR-based estimation as an open need.Source: §1 · p. 3
  • The estimator conditions GP models of the transfer functions and transient on measurements of the single output and multiple inputs.The frequency-domain estimator models the transfer functions and transient as independent GPs and estimates them simultaneously.Source: §1 · pp. 2–3
  • The work derives direct-sum or rank conditions for exact module identifiability, establishes zero-noise convergence, and adapts conventional closed-loop bias handling to construct regressors.Monte Carlo simulations examine the developments in open- and closed-loop power-system models.Source: §1 · p. 3

2. Methods

The methods section introduces the central theoretical developments, while its notation subsection defines element-wise vectorization for scalar functions and Cartesian-product evaluation for two-argument functions.

2. Methods

  • The methods section formulates a complex-GPR MISO estimator, derives exact-identifiability requirements, and addresses closed-loop bias.Source: §2 · p. 4

2.1. Notation

  • For a scalar function ζ, its vectorized extension maps the function element-wise over the entries of a vector.Source: §2.1 · p. 4
  • For a two-argument scalar function, vectorized evaluation maps all pairs of entries from the two input vectors into an n × m matrix.Source: §2.1 · p. 4

2.2. Problem Formulation

The problem formulation moves the MISO input-output relation into the frequency domain and accounts for finite-recording transients. It treats measured inputs as fixed regressors and output as random, conditional on those inputs.

2.2. Problem Formulation

  • The time-domain MISO output is the sum of each module's input-response convolution and additive output noise.Source: §2.2 · p. 4
  • The DFT representation evaluates continuous-time transfer functions at frequency bins and includes per-channel transients arising from finite-window effects.The mismatch between the DFT's implicit periodic extension and measured trajectories gives rise to the transient terms.Source: §2.2 · pp. 4–5
  • For estimation, module and noise transients can be combined into one smooth equivalent transient represented as an artificial input channel with a unit-valued regressor.Source: §2.2 · pp. 5–6
  • The probabilistic model conditions on observed input spectra treated as fixed regressors, while the output is random through its dependence on unknown modules and stochastic noise.Source: §2.2 · p. 6
  • The conditional observation model assumes input conditioning does not change the output-noise model, including its zero mean and covariance.Source: §2.2 · p. 6

2.3. MISO Estimator Based on Complex Gaussian Process Regression

The MISO-GPTF estimator uses complex Gaussian-process priors and Gaussian conditioning to infer module transfer functions from output measurements. Its posterior is expressed in real coordinates, with independent priors across modules.

2.3. MISO Estimator Based on Complex Gaussian Process Regression

  • The MISO-GPTF estimator places complex-GP priors on unknown transfer functions and derives their posterior by conditioning on the observed output spectrum.The required covariances include module priors, output spectrum, and output-to-module cross-covariance.Source: §2.3 · pp. 6–7

2.3.2. Real-Expanded Representation

  • The complex-valued inference is represented in real coordinates so standard multivariate Gaussian conditioning formulas can be applied.Source: §2.3 · p. 7, §2.3.2 · pp. 8–9

2.3.1. Covariance Matrices

  • The covariance construction assumes mutually independent transfer-function GPs and output noise conditional on the regressors.Independent module priors encode within-module frequency relationships but do not share information between different modules.Source: §2.3.1 · p. 7

2.3.2. Real-Expanded Representation

  • Conditioning yields the MISO-GPTF posterior GP, whose pointwise maximum a posteriori estimate equals its posterior mean.The posterior also retains covariance information, including cross-covariances that may arise from conditioning on shared output measurements.Source: §2.3.2 · pp. 9–10
  • With one module, MISO-GPTF reduces to existing complex-GPR SISO estimators, while covariance-function selection and hyperparameter optimization remain outside this paper's scope.Source: §2.3.2 · p. 10

2.4. Identifiability of Individual Modules

The section derives conditions for separating MISO modules at the structural and finite-grid Gaussian-process levels, then establishes when the posterior mean recovers the grid-evaluated transfer functions as noise vanishes.

2.4.1. Structural Module Identifiability

  • Module contributions are uniquely identifiable when their output spaces form a direct sum; identifying the modules themselves also requires each input-to-contribution map to be injective.Together, these conditions make the combined observation operator injective when the inputs are known.Source: §2.4.1 · pp. 11–12

2.4.2. Identifiability in a MISO GPR Setting

  • On a finite frequency grid, GP-supported contributions require a direct-sum condition, while exact identification of module vectors additionally requires condition (57).The theorem combines separability and injectivity requirements for module identifiability on the prior-supported subspaces.Source: §2.4.2 · p. 13
  • Effective GP-supported identifiability uniquely separates retained dominant prior directions, but does not establish exact identifiability of the full GP model.The effective representation discards directions below the selected eigenvalue tolerance.Source: §2.4.2 · pp. 13–14

2.4.3. Zero-Noise Consistency of the Posterior Mean

  • Under module identifiability and fixed positive-definite noise shaping, the posterior mean converges in L2 and probability to the grid-evaluated module vectors as noise variance tends to zero.If the forward model is rank-deficient, the estimator recovers only observable components; effective-prior recovery does not imply recovery of discarded low-variance directions.Source: §2.4.3 · pp. 16–17

2.5. Closed-Loop Identifiability

Closed-loop feedback can make measured regressors statistically predictable from output disturbance, risking biased module estimates. The section describes using external excitation and least-squares projection to construct regressors for MISO-GPTF.

2.5. Closed-Loop Identifiability

  • In closed loop, disturbance predictable from measured regressors may be attributed to module dynamics, potentially biasing estimates.The open-loop conditional likelihood is no longer valid in this setting.Source: §2.5 · p. 17
  • A least-squares projection method uses external excitation to construct regressors that are predominantly excitation-generated rather than disturbance-generated.When external excitation is independent of disturbance, a sufficiently flexible model approximately retains excitation-induced input components while rejecting disturbance-induced components.Source: §2.5 · pp. 17–19
  • Projected or structural inputs can be used in the subsequent MISO-GPTF procedure, but projected-regressor construction trades closed-loop bias against approximation and estimation error.The estimator's derivation assumes independence between regressors and disturbances.Source: §2.5 · p. 19

3. Results

The two-bus case study tests MISO-GPTF estimation in open- and closed-loop configurations, using Monte Carlo simulations to assess identifiability and projected-regressor bias handling. Open-loop fits depend strongly on the exact identifiability condition, while projection effects in closed loop vary by module and SNR.

3.1. Dynamic Power System Model

  • The test case models frequency-active power dynamics in a two-bus system, comparing open loop with directly applied U1 and U2 against closed loop with external R1 and R2.In closed loop, U1 and U2 become endogenous signals; G1 and G2 are the target transfer functions.Source: §3.1 · p. 21, Figure 1 · p. 22

3.2. Simulation Setup

  • The simulations vary input similarity in open loop and SNR in closed loop, comparing direct endogenous regressors with inputs projected onto external excitation signals.Each open- and closed-loop condition uses 100 Monte Carlo runs.Source: §3.2 · pp. 22–23

3.3. Open-Loop Identifiability Results

  • When the exact identifiability requirement is met, both modules fit close to ideally; failed cases show substantially worse fits, with module 1 especially weak.The maximum failed module 2 fit is 0.94, below the minimum fit among passing cases.Source: §3.3 · p. 23, Figure 2 · p. 24

3.4. Closed-Loop Projection Results

  • Projection helps module 1 most at low SNR, but its benefit decreases as SNR rises; its median effect on module 2 is consistently negative.At higher SNR, FIR approximation error may become more influential than closed-loop bias.Source: §3.4 · p. 24, Figure 5 · p. 27
  • Closed-loop fits depend strongly on SNR, with module 2 generally fitting better than module 1; the gap narrows as SNR increases and nearly disappears at 20 dB.Figure 5 compares module fits from direct and projected inputs and shows the change after projection across SNR levels.Source: §3.4 · p. 24, Figure 5 · p. 27

4. Discussion

The discussion identifies limits to the exact identifiability criterion, closed-loop regressor construction, and independent module modeling.

4. Discussion

  • The exact rank requirement may be too strict: acceptable fits may still occur when it fails, potentially only over limited frequency ranges.The discussion proposes stable identifiability and singular-value-based conditions as directions for more nuanced assessment.Source: §4 · p. 28
  • Module separability and regressor choice are distinct: full rank may still yield poor closed-loop estimates with endogenous contaminated regressors, while exogenous regressors cannot recover inseparable modules.Projection trades closed-loop bias against approximation and estimation errors, and its effects differ by channel.Source: §4 · p. 28
  • The analysis treats module estimates as independent, leaving potential gains from modeling known or suspected inter-module relationships unexplored.The discussion notes that coregionalization could let the estimator borrow information between modules.Source: §4 · p. 28

5. Conclusions

The paper extends Gaussian-process transfer-function estimation to closed-loop MISO systems, deriving conditions for module identifiability and examining closed-loop regressor construction. Monte Carlo results link identifiability to module fit and show that projection’s effects vary by channel and SNR.

5. Conclusions

  • The measured input spectra and covariance-function choice determine the conditions under which separate modules are exactly identifiable.The paper also reports convergence of its estimator to the true modules in the noise-free limit.Source: §5 · p. 28
  • Monte Carlo simulations show that the theoretical identifiability requirement strongly predicts system-module fit.The estimator provides continuous frequency-domain transfer functions with quantified uncertainty bounds.Source: §5 · pp. 28–29
  • Projecting regressors most clearly improved module 1 at low SNR, with a diminishing advantage as SNR rose, but slightly worsened module 2 fit consistently.The authors identify a trade-off between bias from endogenous inputs and error from constructing projected exogenous regressors.Source: §5 · p. 29
  • Regressor construction should be chosen per channel rather than applied as a default preprocessing step.Further work is proposed to replace the binary pass/fail requirement with a continuous metric and to refine robust regressor selection and handling.Source: §5 · p. 29

Appendix A. Proof of Theorem 1

The proof characterizes unique recovery of the transfer-function tuple through two requirements: uniquely separable output contributions and injective module maps. It establishes that these conditions are equivalent to injectivity of the overall observation map.

Appendix A. Proof of Theorem 1

  • Unique recovery of the transfer-function tuple requires both uniquely separable contributions and injective module maps.If a module map is not injective, distinct transfer functions can produce the same output contribution.Source: Appendix A · pp. 29–30
  • A direct sum of module contribution spaces makes each contribution unique in any decomposition of the observed output.The proof shows that two decompositions must agree contribution by contribution.Source: Appendix A · p. 29
  • Injectivity of the overall map implies each module map is injective and the contribution spaces form a direct sum.The argument tests a single nonzero module at a time, then uses a zero-sum collection of contributions to establish directness.Source: Appendix A · p. 30
  • Conversely, if the contribution spaces form a direct sum and every module map is injective, the overall map is injective.A tuple mapped to zero must have zero contribution in every space, and injectivity then forces every transfer function to be zero.Source: Appendix A · p. 30

Appendix B. Proof of Proposition 2

The proof uses orthogonal projections to separate the relevant subspaces and simplify inversion. As the noise variance tends to zero, the resulting estimates converge to the true quantities, including the MISO transfer-function estimates.

Appendix B. Proof of Proposition 2

  • Mutually orthogonal projection matrices that sum to the identity decouple the vector space into independent subspaces, allowing the inverse to be found by inverting each projection weight.This gives the expression identified as equation (78).Source: Appendix B · p. 31
  • When L has full column rank, Vs is square and orthogonal, so VsVs^T equals the identity matrix.Source: Appendix B · p. 31
  • As the noise variance tends to zero, the estimate of W converges in L2, and convergence in probability follows.The proof then uses this convergence to establish convergence of the MISO-GPTF estimate; the transfer-function estimates converge in L2 and hence in probability.Source: Appendix B · p. 32
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