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Online input design for discrimination of linear models using concave minimization

Jacques Noom, Oleg Soloviev, Carlas Smith, Michel Verhaegen

TL;DR

The paper tackles online input design for discriminating among linear models under Gaussian disturbances by recasting CLAFD as a stochastic control problem. It introduces a concave upper bound on the predicted misdiagnosis probability via the Bhattacharyya coefficient, and proves concavity in a subdomain relevant to hard discrimination. The method leverages Disciplined Convex-Concave Programming (DCCP) to handle convex constraints (polytopic and energy-based) and proposes a quadratic Taylor approximation to speed computation. Simulation studies show closed-loop approaches outperform open-loop strategies and a prior Bhattacharyya-distance-based method, with applicability to feedback-controlled systems. The work enables real-time, accurate model discrimination with reduced computational burden and flexible constraint handling.

Abstract

Stochastic Closed-Loop Active Fault Diagnosis (CLAFD) aims to select the input sequentially in order to improve the discrimination of different models by minimizing the predicted error probability. As computation of these error probabilities encompasses the evaluation of multidimensional probability integrals, relaxation methods are of interest. This manuscript presents a new method that allows to make an improved trade-off between three factors -- namely maximized accuracy of diagnosis, minimized number of consecutive measurements to achieve that accuracy, and minimized computational effort per time step -- with respect to the state-of-the-art. It relies on minimizing an upper bound on the error probability, which is in the case of linear models with Gaussian noise proven to be concave in the most challenging discrimination conditions. A simulation study is conducted both for open-loop and feedback controlled candidate models. The results demonstrate the favorable trade-off using the new contributions in this manuscript.

Online input design for discrimination of linear models using concave minimization

TL;DR

The paper tackles online input design for discriminating among linear models under Gaussian disturbances by recasting CLAFD as a stochastic control problem. It introduces a concave upper bound on the predicted misdiagnosis probability via the Bhattacharyya coefficient, and proves concavity in a subdomain relevant to hard discrimination. The method leverages Disciplined Convex-Concave Programming (DCCP) to handle convex constraints (polytopic and energy-based) and proposes a quadratic Taylor approximation to speed computation. Simulation studies show closed-loop approaches outperform open-loop strategies and a prior Bhattacharyya-distance-based method, with applicability to feedback-controlled systems. The work enables real-time, accurate model discrimination with reduced computational burden and flexible constraint handling.

Abstract

Stochastic Closed-Loop Active Fault Diagnosis (CLAFD) aims to select the input sequentially in order to improve the discrimination of different models by minimizing the predicted error probability. As computation of these error probabilities encompasses the evaluation of multidimensional probability integrals, relaxation methods are of interest. This manuscript presents a new method that allows to make an improved trade-off between three factors -- namely maximized accuracy of diagnosis, minimized number of consecutive measurements to achieve that accuracy, and minimized computational effort per time step -- with respect to the state-of-the-art. It relies on minimizing an upper bound on the error probability, which is in the case of linear models with Gaussian noise proven to be concave in the most challenging discrimination conditions. A simulation study is conducted both for open-loop and feedback controlled candidate models. The results demonstrate the favorable trade-off using the new contributions in this manuscript.
Paper Structure (24 sections, 1 theorem, 50 equations, 6 figures)

This paper contains 24 sections, 1 theorem, 50 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notation
  4. Bayesian formulation
  5. Infinite horizon stochastic control problem
  6. Methodology
  7. Model predictive control
  8. Bound on predicted error probability
  9. Domain of concavity of the Bhattacharyya coefficient
  10. Online check for concavity of \ref{['optprob']}
  11. For polytopic constraints
  12. For energy constraints
  13. Quadratic Taylor approximation of \ref{['optprob']}
  14. Summary of approaches
  15. Simulation Experiment
  16. ...and 9 more sections

Key Result

Lemma 3.1

Expression firstb is concave in the domain where is satisfied.

Figures (6)

  • Figure 1: The compliance of \ref{['conc']} against measurement noise variance $\mathtt{R}$ and model differences $\mathtt{\left\lVert\Gamma^{[01]}\right\rVert_F}$, generated with initial conditions, horizon $N$ and model $\mathtt{M^{[0]}}$ as in Sect. \ref{['clamduncontrolled']}, $\mathtt{C^{[1]}} = \text{constant} \cdot \mathtt{C^{[0]}}$, and with inputs $\mathbf{u}_k = \mathtt{[-1,\,-1,\,-1,\,-1,\,1,\,1,\,-1,\,-1,\,0,\,0]^\top}$. Large noise contribution and small model differences are favorable for satisfying \ref{['conc']}.
  • Figure 2: Number of measurements before decision in experiment with polytopic constraints for the four closed-loop methods, compared to open-loop. The medians are from left to right $\mathtt{\{119,\, 400,\, 78,\, 78,\, 88\}}$ and the distributions of all methods differ significantly with MWW approximated $p$-value $\mathtt{< 0.001}$, except the distribution pair $(\text{QTA},\text{BC})$. The average computational time per measurement for the closed-loop methods was $\mathtt{\{10.8,\, 12.4,\, 26.1,\, 89.1\}}$ milliseconds for BD, QTA, BC and $\Sigma$BC, respectively.
  • Figure 3: Number of measurements before decision in experiment with quadratic constraints for the four closed-loop methods, compared to open-loop. The medians are from left to right $\mathtt{\{122,\, 59,\, 54.5,\, 55,\, 55\}}$. Although the distributions of QTA, BC and $\Sigma$BC do not differ significantly from each other, they do differ significantly from OL and BD with MWW approximated $p$-value $\mathtt{< 0.001}$. The average computational time per measurement for the closed-loop methods was $\mathtt{\{223,\, 175,\, 310,\, 928\}}$ milliseconds for BD, QTA, BC and $\Sigma$BC, respectively.
  • Figure 4: Applied inputs $u_k = [u_{k,1}, u_{k,2}]^\top$ for one realization of the approaches summarized in Sect. \ref{['summary']} in quadratic constraint set, with corresponding probability of true model $\mathtt{M^{[3]}}$. The black vertical bars indicate time instances at which the input pattern changes significantly in main frequency. The colored circles indicate time instances when the final decision is made. The BD approach did not present the change in main frequency and failed to decide after 200 time steps.
  • Figure 5: Online input design for discrimination of models for a feedback (FB) controlled system with observer, feedforward (FF) and feedback gains $K^{[0]}$, $G^{[0]}$ and $F^{[0]}$, respectively. The online input design method BC can also be replaced with one of the other methods proposed in this manuscript $\Sigma$BC or QTA.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 3.1: Domain of concavity of a multivariate Gaussian
  • proof