A machine-learning approach to synthesize virtual sensors for parameter-varying systems (2103.12324v1)

Abstract: This paper introduces a novel model-free approach to synthesize virtual sensors for the estimation of dynamical quantities that are unmeasurable at runtime but are available for design purposes on test benches. After collecting a dataset of measurements of such quantities, together with other variables that are also available during on-line operations, the virtual sensor is obtained using machine learning techniques by training a predictor whose inputs are the measured variables and the features extracted by a bank of linear observers fed with the same measures. The approach is applicable to infer the value of quantities such as physical states and other time-varying parameters that affect the dynamics of the system. The proposed virtual sensor architecture - whose structure can be related to the Multiple Model Adaptive Estimation framework - is conceived to keep computational and memory requirements as low as possible, so that it can be efficiently implemented in embedded hardware platforms. The effectiveness of the approach is shown in different numerical examples, involving the estimation of the scheduling parameter of a nonlinear parameter-varying system, the reconstruction of the mode of a switching linear system, and the estimation of the state of charge (SoC) of a lithium-ion battery.

• The paper presents a model-free ML framework that trains predictors using observer bank outputs to estimate unmeasurable parameters.
• It bridges MMAE techniques with data-driven methods by leveraging local LTI models and decision tree-based clustering for efficient sensor synthesis.
• The proposed architecture demonstrates robust performance in various benchmarks with low computational and memory requirements for embedded applications.

Virtual Sensor Synthesis via Machine Learning for Parameter-Varying Systems

This paper introduces a model-free approach to synthesize virtual sensors for estimating dynamical quantities that are unmeasurable at runtime but available during the design phase. The virtual sensor is synthesized using ML techniques by training a predictor. The inputs are measured variables and features extracted by a bank of linear observers fed with the same measurements. This architecture is designed to minimize computational and memory requirements for efficient implementation on embedded hardware.

MMAE and Data-Driven Frameworks

The paper bridges the gap between the MMAE framework and data-driven techniques for parameter-varying systems. The MMAE approach uses a bank of state estimators, each associated with a specific parameter value, along with a hypothesis testing algorithm to infer information about the dynamical system. The challenge lies in determining the set of parameter vectors and synthesizing the hypothesis tester without relying on a precise model of the process.

Virtual Sensor Architecture

The virtual sensor architecture comprises three main steps: learning a finite set of LTI models, designing a set of linear observers based on these models, and using ML to train a predictor that maps the estimates from the observers and raw input/output signals into an estimate of the parameter. This architecture addresses the challenge of estimating parameters in systems where direct measurements are unavailable during operation. The paper also introduces an off-line strategy for identifying local linear models based on interpreting decision tree regressors as a supervised clustering scheme.

Learning Local Models from Data

The paper addresses the challenge of learning local models from data when the system model is unknown. The approach involves training a functional approximator, $M_{LPV}$, to predict the correct parameter vector corresponding to a given parameter value. This is achieved by solving an optimization problem that minimizes the difference between predicted and actual outputs, subject to constraints on the model parameters.

$$\begin{array}{rrl} \underset{M_{LPV}}{\mathrm{min}} & \displaystyle \sum_{k=k_1}^{N}& L_{M_{LPV}}(\hat y_{k}, y_{k}) \ &\mathrm{subject~to} &\hat y_{k}=[0.75 y_{k-M},\dots,0.75 y_{k-1},u_{k-M},\dots,u_{k-1}~1]\gamma_{k}\ &&\gamma_{k}=M_{LPV}(\rho_k)\ &&k=k_1,\dots,N \end{array}$$

Feature Extraction and Prediction

After synthesizing the observers, a hypothesis testing scheme is constructed using a discriminative approach. The initial dataset is processed to generate information vectors, which are then used to train a predictor, $f_\theta$, that estimates the parameter vector. The predictor is designed to minimize a loss function that penalizes the distance between the measured and reconstructed parameter values. The paper explores two FE maps: one that uses the errors of each observer and one that compresses this information by summing the output prediction errors.

Experimental Results and Analysis

The paper presents numerical results on a synthetic benchmark system, a mode observer for switching linear systems, and a nonlinear state estimation problem. The performance of the proposed virtual sensor is evaluated in terms of FIT and NRMSE. The results demonstrate the effectiveness of the approach in various scenarios, including parameter estimation, mode discrimination, and state estimation.

Dependence on Hyper-parameters and Tuning

The paper discusses the hyper-parameters of the proposed virtual sensor, including the number of local models, the order of the local models, and the window size of the predictor. It also discusses the trade-offs between the complexity of the model and the accuracy of the estimation. Tuning these hyper-parameters is crucial for achieving optimal performance, and the paper provides practical guidance on how to do so.

Conclusion

The paper concludes by highlighting the potential of the proposed data-driven virtual sensor synthesis approach for reconstructing unmeasurable quantities in various systems. The key idea is to use past input and output data to synthesize a bank of linear observers, which serve as a basis for FE maps that simplify the learning process of the hypothesis testing algorithm. The architecture's low memory and CPU requirements make it suitable for embedded and fast-sampling applications.