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A 2-D spatial construction method is proposed for the online modeling of distributed parameter systems (DPSs), such as battery thermal process. The proposed method can combine the advantages of spectral method and Karhunen–Loève decomposition (KLD) method. First, the continuous spatial basis functions are designed by the 2-D spatial construction to keep the information between sensing locations. With the 2-D space-time separation and recursive learning, the derived model can preserve the couplings between spatial dimensions and update over time. The radial basis function network is utilized to identify the low-dimensional temporal dynamics. After the space-time synthesis, the constructed spatiotemporal model can provide continuous modeling of the DPS with satisfactory performance. Convergence analysis has been carried out, which proves that the proposed method can guarantee bounded errors. Finally, simulations and experiments on a pouch-type lithium-ion battery with unknown partial differential equations prove the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A spatial-construction-based fault diagnosis method is proposed to detect and locate the abnormality for unknown distributed parameter systems (DPSs). To accurately locate the abnormality, the continuous spatial basis functions (SBFs) are derived by the proposed spatial construction method from empirical data. Theoretical analysis proves that the B-spline curve is a proper solution to the spatial construction problem. Two new statistics are constructed based on the derived continuous SBFs and the improved independent component analysis algorithm. The abnormality can be timely detected according to the reference signals derived by the central limit theorem and hypothesis testing. With the continuous SBFs, the probability distribution of statistic contribution can be constructed to reveal the actual position of the abnormality. The proposed method can timely detect and locate the abnormality under fewer sensors without the knowledge of PDE and boundary conditions. The internal short circuit experiment on a lithium-ion battery demonstrates the effectiveness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

Modeling high-spatial dimensional (high-D) distributed parameter systems (DPSs) is very difficult because of the spatially distributed characteristic and complex spatiotemporal coupling. In this article, a new framework based on high-order singular vector decomposition (HOSVD) is proposed to model a high-D DPS. A modified HOSVD is designed for separation of time and multiple spatial variables. It can well preserve the spatially distributed nature of the high-D DPS. It also considers the interaction across different spatial modes, which is an important part of spatiotemporal coupling in the high-D DPS. Experiments of a two-spatial dimensional thermal curing process are used to verify the effectiveness of the proposed method. An average prediction error near 1% of the curing temperature can be achieved. [ABSTRACT FROM AUTHOR]

For complex distributed parameter systems (DPSs) with strong nonlinearities and time-varying dynamics, the conventional spatiotemporal modeling methods become ill-suited since the elementary assumption that the process data follow a unimodal Gaussian distribution usually becomes invalid. In this paper, a multimode method is proposed for modeling of such systems. First, the original operating space is partitioned along the time dimension into several subspaces via modified dissimilarity analysis. Each subspace represents the local spatiotemporal characteristics of the original system. Second, the Karhunen–Loève decomposition (KLD)-based spatiotemporal modeling approach is applied to approximate the local dynamics of each subspace. Finally, an ensemble model is obtained using the soft weighting sum of the local ones, where the corresponding weights are calculated by principal component regression. By properly decomposing the original space into several local parts, the ensemble model is capable of handling the strong nonlinearities and time-varying dynamics of the system. The validity and efficiency of the proposed method are verified on two representative applications: 1) a one-dimensional parabolic catalytic rod and 2) a two-dimensional curing thermal process. The experimental results show that the proposed method provides a superior performance regarding modeling accuracy compared to several baselines. [ABSTRACT FROM AUTHOR]

The construction of spatial basis functions (BFs) is critical to the time/space separation of the distributed parameter system (DPS). The spatial BFs constructed by traditional Karhunen–Loéve may not work satisfactorily for two spatial-dimensional (2-D) DPS, because of a distorted mapping of the original sensor array in the row-wise vectorization process. In this article, a novel time/space separation based method is proposed to construct dimension embedded BFs (DE-BFs) for modeling 2-D DPS. The DE-BFs are first formulated according to the spatial sensor array structure, and sequentially optimized with alternating least squares by minimizing the reconstruction error. The mapping relationship between the DE-BFs and the spatial sensor array is well preserved. In addition, the coupling across temporal and different spatial dimensions is sufficiently captured. A satisfactory model accuracy can be achieved by the DE-BFs, even with limited training data. Experiments of a 2-D curing thermal process are used to verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A reinforcement learning-based method is proposed for optimal sensor placement in the spatial domain for modeling distributed parameter systems (DPSs). First, a low-dimensional subspace, derived by Karhunen–Loève decomposition, is identified to capture the dominant dynamic features of the DPS. Second, a spatial objective function is proposed for the sensor placement. This function is defined in the obtained low-dimensional subspace by exploiting the time-space separation property of distributed processes, and in turn aims at minimizing the modeling error over the entire time and space domain. Third, the sensor placement configuration is mathematically formulated as a Markov decision process (MDP) with specified elements. Finally, the sensor locations are optimized through learning the optimal policies of the MDP according to the spatial objective function. The experimental results of a simulated catalytic rod and a real snap curing oven system are provided to demonstrate the feasibility and efficiency of the proposed method in solving the combinatorial optimization problems, such as optimal sensor placement. [ABSTRACT FROM AUTHOR]

The thermal effect has a significant impact on the performance of a lithium-ion battery. Thus, modeling the thermal process, which always involves unknown boundary heat exchange, is significant to battery management. Two critical issues should be addressed for the thermal process modeling: 1) the nominal model, which is constructed offline, can be updated efficiently to compensate for any online disturbances; and 2) the influence of previous and recent spatiotemporal dynamics may be varying and should be handled properly. Bearing these in mind, in this paper, a spatial correlation-based incremental learning technique is designed for spatiotemporal modeling. First, the incremental learning technique is developed to update the dominant spatial basis functions of the nominal model, which is constructed by a time/space separation-based method. Then, a forgetting factor is incorporated into the incremental learning technique to handle time-varying dynamics. Additionally, the popular approximator, that is, the radial basis function neural network, is utilized to identify the low-dimensional temporal model. Simulations and experiments on a pouch type battery with boundary heat exchange have demonstrated the accuracy and efficiency of the proposed modeling method. [ABSTRACT FROM AUTHOR]

An incremental spatiotemporal learning scheme is proposed for online modeling of distributed parameter systems (DPSs). A novel incremental learning method is developed to recursively update the spatial basis functions and the corresponding temporal model based on the Karhunen–Loève decomposition for time-space separation. The time-space synthesis continually evolves by adding new increment data with more updated information and revising the existing parameters of the dynamic system. In this way, the spatiotemporal structure is inherited and updated efficiently as output data increases over time. The adaptive nature of this evolving structure makes it promising for online modeling of DPSs under streaming data environment. The proposed incremental modeling scheme is evaluated on the classical benchmark of a catalytic rod problem. The simulation results demonstrate the viability and efficiency of the proposed method for online modeling of DPSs. [ABSTRACT FROM AUTHOR]

Distributed parameter systems widely exist in many industrial thermal processes. Estimation of their temperature distribution in the entire operating area is not easy as the dynamics are time/space coupled, and there are only a few sensors available for measurement. In this paper, an effective spatiotemporal model is proposed for prediction of the temperature distribution. After the dominant spatial basis functions are obtained by the Karhunen–Loève method under the time/space separation, a kernel-based random vector functional-link network is developed for learning unknown temporal dynamics. After time/space synthesis, the spatiotemporal model can be constructed to effectively estimate the temperature distribution in high learning speed. The generalization performance of this model is discussed using Rademacher complexity. Simulations on two typical industrial thermal processes show that the proposed method has superior model performance than neural networks and least square support vector machine. [ABSTRACT FROM AUTHOR]

Time/space separation based spatiotemporal modeling methods have been proven to be effective and efficient for modeling a class of distributed parameter systems (DPSs). However, these conventional methods may not work satisfactorily for DPSs with time-dependent boundary conditions. A sliding window based dynamic spatiotemporal modeling method is proposed for this kind of DPSs. First, the sliding window is appropriately designed to capture the most recent spatiotemporal data. Then, the conventional Karhunen–Loève method can be used to construct the analytical model. Besides, a more general sliding window method can be achieved by using a forgetting factor to adjust different influence of the current and previous data. This analytical model can be utilized for online performance prediction. Simulation experiments on a benchmark and a battery with unknown boundary cooling have demonstrated the superior performance of the proposed method on the DPSs with time-dependent boundary conditions. [ABSTRACT FROM AUTHOR]

The temperature distribution of a lithium-ion battery (LIB) belongs to a nonlinearly distributed parameter system (DPS), which is infinite dimensional in the space direction. Modeling of such systems often leads to the following challenges: time/space-coupled dynamics; time-varying dynamics; and spatial nonlinearity. For the first two challenges, Karhunen–Loève (KL) decomposition based data-driven spatiotemporal models have been widely researched and successfully applied to industrial thermal processes. However, this method is a global linear model reduction method that ignores the nonlinear properties of measurements. This will prohibit the model performance of a strong nonlinear DPS, which has strong spatial nonlinearity refers to the aforementioned third challenge. To address this problem, a local-properties-embedding-based modeling method is developed for the nonlinear DPS. First, the finite-dimensional nonlinear spatial basis functions that can be the representative of the nonlinear feature of the original space are learned by the local-properties-embedding-based method. Then, the low-dimensional representative can be derived using the time/space separation and the unknown temporal coefficients can be identified through the traditional neural learning algorithm. Finally, the spatiotemporal temperature distribution can be reconstructed by the time/space synthesis. Since the nonlinear structure feature of the spatiotemporal datasets has been considered, the proposed modeling method can be more effective than the traditional KL-based method for modeling the nonlinear DPS. Numerical simulations on the LIB showed that the proposed methods are more robust than the KL-based method in both the time direction and the space direction. [ABSTRACT FROM AUTHOR]

This paper employs a Takagi–Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial $L^\infty$ norm $\Vert \cdot \Vert _\infty$ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov–Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincaré–Wirtinger inequality in one-dimensional spatial domain, as well as a vector-valued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of $\Vert \cdot \Vert _\infty$ , if sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally, the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples. [ABSTRACT FROM AUTHOR]

The real-time monitoring of temperature distribution in lithium-ion batteries (LIBs) is crucial for their safety and optimal operation in electrical vehicles. An accurate and effective thermal model is needed for online temperature monitoring since limited sensors are available in vehicle application. In this paper, a data-based spatiotemporal modeling method is researched for online estimation of temperature distribution of LIBs. First, Isometric Mapping (ISOMAP) method is used for time/space separation and model reduction. Then, the low-dimensional representation can be obtained in terms of ISOMAP based mapping functions. The unknown temporal dynamics in the low-dimensional space can be approximated using neural network model with parameters trained using extreme learning machine (ELM) algorithm. Finally, the spatiotemporal model of the thermal process can be reconstructed by integrating the neural network model and the mapping functions. The generalization bound of the proposed spatiotemporal model can be analyzed using Rademacher complexity. Simulation results showed that the proposed modeling method can model the LIB thermal process very well. [ABSTRACT FROM PUBLISHER]

This paper presents a deep learning-based model reduction method for distributed parameter systems (DPSs). The proposed method includes three phases. In phase I, numerical or experimental data of the spatiotemporal distribution is reduced into low-dimensional representations using the deep auto-encoder (DAE). In phase II, the low-dimensional representations are used to establish the reduced-order model. In phase III, the reduced model is then used to reconstruct the high-dimensional DPS. Experimental studies are conducted to validate the proposed method. The proposed method is compared with the classical proper orthogonal decomposition method and demonstrates better modeling accuracy and efficiency in the experiments. [ABSTRACT FROM PUBLISHER]

Effective thermal management is crucial to the optimal operation and health management of lithium-ion batteries. The online estimation of the temperature distribution in vehicle battery systems is not easy, as there are only a few sensors available on site. Furthermore, the thermal behaviors of batteries are difficult to predict, as their dynamics are strongly time-varying. In this paper, a hybrid model is developed for spatiotemporal estimation of temperature distribution in lithium-ion batteries. A simple but effective nominal model is first developed for real-time thermal management using a time/space separation method. Subsequently, a data-based neural model is proposed to compensate the model-plant mismatch caused by the spatial nonlinearity and other model uncertainties. The developed algorithm is simple and can be readily integrated into existing battery management systems. Simulation studies demonstrate the effectiveness of the proposed method. [ABSTRACT FROM PUBLISHER]