Observer-Based Soft Sensors Overview

• Observer-based soft sensors are algorithms that estimate unmeasured system states using partial and noisy measurements, integrating model-based and machine learning methods.
• They employ mathematical foundations such as Kalman filtering, geometric optimization, and neural observers to enhance accuracy and robustness.
• Applications span process control, robotics, and ecological monitoring, providing efficient, adaptive, and cost-effective state reconstruction.

Observer-based soft sensors are algorithms that estimate the internal states or key process variables of a dynamical system using noisy, partial, or indirect measurements, typically when direct sensing is infeasible or cost-prohibitive. These methodologies integrate sensor selection, model design, estimation algorithms, and practical constraints to achieve accurate, robust, and efficient state reconstruction—serving critical roles in process control, robotics, engineering, biology, and economics.

1. Mathematical Foundations and Observer Models

The mathematical basis for observer-based soft sensors lies in state estimation theory. The observer is an estimator of the unmeasured state vector $x$ of a dynamical system, using time-series measurements $y$ processed through a model or algorithm. The classical linear Kalman filter provides minimum mean-squared error (MMSE) estimation for systems with additive Gaussian noise:

$A^T K + K A - \gamma K c^T c K + Q = 0,$

where

• $A$ is the system dynamics matrix,
• $K$ is the steady-state error covariance,
• $c$ is the sensor measurement matrix,
• $\gamma$ models signal-to-noise ratio,
• $Q$ describes process noise covariance.

The Kalman filter’s performance is ultimately limited by the choice of $c$, motivating the need for optimal sensor design. More generally, observer-based soft sensors can be extended to nonlinear systems via neural network observers, sliding mode control, or physics-informed observers, which estimate states using both learned and mechanistic models (Farkane et al., 9 Jul 2025, Zheng et al., 2023).

2. Optimal Sensor and Actuator Design

Sensor selection and arrangement directly affect observability and estimation accuracy. The optimal sensor design problem is nonconvex and traditionally addressed via ad hoc or approximate methods. Geometric optimization over compact isospectral manifolds enables systematic sensor selection by minimizing the estimation error of the observer. The observation matrix $c$ (or Gram matrix $C = c^T c$) must commute with a positive definite operator $M = K R K$, where $K$ and $R$ solve the Riccati and Lyapunov equations, respectively:

$[C, M] = 0.$

The optimal sensor aligns with the largest eigenvectors of $M$, capturing the system’s most informative directions. The gradient flow equation,

$\dot{C} = \gamma [C, [C, M]],$

converges to the global optimum under stable dynamics and suitably small $\gamma$, yielding the minimal estimation error for a given signal-to-noise ratio. The duality principle extends these geometric methods to actuator placement for optimal control (Belabbas, 2015).

3. Calibration, Adaptation, and Update Strategies

Robust observer-based soft sensors require frequent calibration and adaptation to time-varying statistics and dynamic processes. Moving window calibration models—Partial Least Squares (PLS), Recursive PLS (RPLS), Random Forest (RF), and hybrids such as RF-PLS—operate over small sample windows to adapt rapidly to changing conditions, especially where historical data are scarce (Kneale et al., 2017). Update schemes can be continuous (per new sample) or delayed (per batch), balancing real-time responsiveness and predictive accuracy. The window size critically affects prediction error and stability, with small windows capturing fast covariance changes but potentially susceptible to noise.

Bi-objective identification methods can further improve performance—combining dynamic data and steady-state measurements in nonlinear models (MLPs, NARX polynomials) using weighted loss functions:

$J_{sd} = (1-\lambda) J_d + \lambda J_s,$

with $J_s$ computed via one-step-ahead static predictions, eliminating expensive fixed-point computations and improving extrapolation over wider operating ranges (Freitas et al., 2020).

4. Sensor Selection, Sparsity, and Dynamic Causality

Efficient soft sensor design not only estimates states but solves the sensor selection problem—identifying the minimal configuration yielding satisfactory observability. Regularized estimation techniques (e.g., LASSO/ℓ₁ norm penalties) enforce sensor sparsity, drive irrelevant parameter coefficients to zero, and merge similar operating condition models via inter-group penalties:

$$\min_{\theta_1,\dots,\theta_K} \sum_k \|Y_k - \Phi_k \theta_k\|_2^2 + \frac{1}{2}\lambda_1 \sum_{k,i} \|\theta_k - \theta_i\|_2 + \lambda_2 \sum_k \sum_j \|\theta_k^j\|_1.$$

This approach ensures computational efficiency and adaptivity in domains with sensor redundancy and many working conditions, as demonstrated in prototype vehicle applications (Wang et al., 6 May 2024).

Dynamic causality analysis advances sensor selection by quantifying the time-evolving impact ("causality score") of each input on state estimation. Liquid-time constant (LTC) neural networks, trained as continuous-time observers, facilitate perturbation-driven sensitivity analysis:

$$Score(I_j) = \frac{1}{T} \int_{t_0}^{t_0+T} \|\Delta x_j(t)\| dt,$$

where $\Delta x_j(t)$ measures the state trajectory change due to an input perturbation. Iterative pruning removes inputs with negligible causal effect, yielding minimal yet physically interpretable sensor sets that reflect system equations, as shown in mechanical, chemical, and ecological testbeds (Farlessyost et al., 14 Sep 2025).

5. Computational Methods and Modern AI Integration

Recent developments integrate ML, Reinforcement Learning (RL), and physics-informed neural networks (PINN) within observer frameworks. Transferable soft sensor architectures, such as Domain Adversarial Neural Network Regression (DANN-R), utilize adversarial training to extract domain-invariant latent features, minimizing distribution mismatch between training and deployment environments:

$$E(\theta_f, \theta_r, \theta_d) = \frac{1}{n_S} \sum L_r^i - \frac{\lambda}{n_S + n_T} \sum L_d^i,$$

where $L_r^i$ and $L_d^i$ denote regression and domain classification losses, respectively (Farahani et al., 2020). This enables robust adaptation to new plants and operational scenarios without retraining on target domain labels.

Coupled physics-informed neural networks with Akaike's information criterion (CPINN-AIC) facilitate PDE structure discovery for processes with spatiotemporal dependence. Training combines data-driven and physics-informed losses:

$MSE_N = MSE_{DN} + MSE_{PN},$

with model selection performed by minimizing

$AIC = 2p + n \ln(\hat{\sigma}^2),$

ensuring parsimonious modeling consistent with physical laws (Wang et al., 2023). This methodology effectively handles unmeasurable source terms and adapts the underlying process model to real data.

6. Robustness in Distributed and Adversarial Environments

Distributed observer architectures provide resilience to sensor faults and adversarial attacks. Sparse sensor attack models are addressed using consensus-based distributed ADMM algorithms, where each node collaboratively estimates both state and attack vectors:

• Primal update minimizes ℓ₁-regularized costs under consensus constraints.
• Auxiliary variable and dual updates enforce agreement and convergence.
• Adaptive penalty parameter schemes optimize primal-dual residual balances.

This ensures convergence and accurate estimation even when some sensors are compromised, with theoretical guarantees under 2$s$-sparse observability and empirical validation in networked scenarios (Prinse et al., 2022).

7. Domain-Specific Applications and Advanced Observer Designs

Observer-based soft sensors are implemented in diverse domains:

• Flexible structures: Modal observer design with Galerkin approximations, explicit Lyapunov stability, and finite-dimensional projections for reconstructing distributed states from partial outputs (Zuyev et al., 2021).
• Soft robotic manipulators: Cosserat-theoretic boundary observers estimate full continuum state from tip velocity alone, with boundary injection ensuring local input-to-state stability and discretization-invariant implementation (Zheng et al., 2023).
• Process control in chemical plants: Dynamic simulation combined with RL state observers and knowledge-driven simulators achieves robust online estimation, process optimization, and extrapolation to unrecorded conditions (Kubosawa et al., 2022).
• Nonlinear and uncertain systems: Neural observers with adaptive sliding mode correction provide robustness against noise, uncertainty, and model mismatches, with proven error convergence via Lyapunov analysis (Farkane et al., 9 Jul 2025).

These techniques yield precise state estimation with tailored guarantees and adaptation strategies that match the complexity and demands of modern industrial, robotic, ecological, and automotive environments.

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In sum, observer-based soft sensors synthesize algorithmic estimation, sensor/actuator optimization, machine learning, causal analysis, and robustness mechanisms. Their design and deployment draw upon geometric methods, adaptive windowing, regularized model selection, dynamic causal quantification, and hybrid AI-physical modeling, producing state-of-the-art solutions for real-time estimation challenges with rigorous theoretical and practical foundations across scientific and engineering domains.