System Of Parabolic Partial Differential Equations Research Articles

Sensor Fault Detection and Diagnosis of Linear Parabolic PDE Systems With Unknown Inputs

This paper investigates the sensor bias fault detection and diagnosis problem for linear parabolic partial differential equation (PDE) systems under the existence of unknown input signals. A variation of Wirtinger's inequality is used to design a Luenberger-type PDE observer and radial basis function (RBF) neural networks are applied to approximate the unknown inputs, guaranteeing the boundedness of the state estimation error. Taking the measurement error as the fault indication signal, a residual evaluation logic is developed to achieve fault detection. An adaptive diagnostic law is constructed to estimate the values of sensor bias faults once they are detected. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -modification RBF neural networks are designed to compensate for the unknown inputs in the presence of sensor faults, ensuring that both state and sensor bias faults estimation errors converge to some adjustable sets. The effectiveness and applicability of the developed fault diagnosis strategy are demonstrated by simulation results on a heat diffusion process.

Adaptive Neural Consensus Observer Networks Design for a Class of Semilinear Parabolic PDE Systems.

This article concerns the investigation on the consensus problem for the joint state-uncertainty estimation of a class of parabolic partial differential equation (PDE) systems with parametric and nonparametric uncertainties. We propose a two-layer network consisting of informed and uninformed boundary observers where novel adaptation laws are developed for the identification of uncertainties. Particularly, all observer agents in the network transmit their information with each other across the entire network. The proposed adaptation laws include a penalty term of the mismatch between the parameter estimates generated by the other observer agents. Moreover, for the nonparametric uncertainties, radial basis function (RBF) neural networks are employed for the universal approximation of unknown nonlinear functions. Given the persistently exciting condition, it is shown that the proposed network of adaptive observers can achieve exponential joint state-uncertainty estimation in the presence of parametric uncertainties and ultimate bounded estimation in the presence of nonparametric uncertainties based on the Lyapunov stability theory. The effects of the proposed consensus method are demonstrated through a typical reaction-diffusion system example, which implies convincing numerical findings.

Bayesian filter for failure times identification of moving heat sources in 2D geometry

This study investigates the application of the Bayesian filter method for identifying failure times in a 2D parabolic partial differential equation system. The identification of failure times in thermal systems, which are subject to partial differential equations, presents significant difficulties, especially due to their ill-posed nature, which makes them highly sensitive to measurement errors. A Bayesian inference framework was previously developed in a related study, aiming to solve inverse heat conduction problems by utilising temperature measurements from sensors to estimate failure times or potential restarts of fixed heat sources. This paper focuses on the case of mobile sources, where a set of fixed sensors is considered and the trajectories of the heating sources are known and follow a constant velocity. The main objective is to accurately identify the failing heat sources and determine the exact failure time, as well as the possibility of resuming normal operation. A Monte Carlo simulation is performed to assess the impact of sensor measurements.

Adaptive boundary observer network design for the consensus on the estimation of a class of parabolic partial differential equation systems

AbstractThis work develops a network of adaptive boundary observers and studies the agreement between state and parameter estimates for a single target parabolic partial differential equation (PDE) system in the presence of structured and unstructured uncertainties. It is assumed that the unknown parameters take the form of either a structured uncertainty with unknown constant parameters or an unstructured uncertainty that can be neutralized by a radial basis function neural networks with unknown weights. The proposed adaptive observers consisting of agents in the network follow the structure of adaptive identifiers for the considered target PDE systems with the insertion of a penalty term in both the state and parameter estimates. Different from earlier efforts, the proposed adaptive laws include a penalty term of the mismatch between the parameter and state estimates generated by the other adjacent agents, which helps to accelerate the estimation of uncertainties. Additionally, the effects of these modifications on the agreement amongst the state and parameter estimates are investigated. Theoretical proofs are provided to show that the proposed approach guarantees the exponential convergence of estimation errors in the case of structured uncertainties and the ultimate boundedness of estimation errors in the case of unstructured uncertainties. Finally, numerical simulations are carried out to verify the effectiveness of the design methods.

Dynamic event–triggered security control for stochastic partial differential equation systems via a unified pointwise measurement/control strategy

This article focuses on the security control for parabolic partial differential equation systems driven by fractional Brownian motion based on a unified pointwise measurement/control strategy. Initially, by employing the Takagi–Sugeno fuzzy rule, nonlinear stochastic systems are transformed into a series of linear subsystems. Then, based on the improved dynamic event-triggered mechanism and pointwise measurements method, a security controller is presented to resist deception attacks. In addition, the Lyapunov direct method and some novel inequalities are used to ensure the considered system is mean-square exponentially stable, and the gain of the point controller is obtained. Finally, an application example is provided to verify the effectiveness of the proposed method.

Observer-Based Boundary Fuzzy Control Design of Nonlinear Parabolic PDE Systems Using Mobile Sensors

This paper studies the boundary fuzzy control problem for nonlinear parabolic partial differential equation (PDE) systems under spatially noncollocated mobile sensors. In a real setup, sensors and actuators can never be placed at the same location, and the noncollocated setting may be beneficial in some application scenarios. The control design is very difficult due to the noncollocated mobile observation, which can be solved by an observer-based technique. At first, a Takagi-Sugeno fuzzy PDE model is devoted to accurately representing the nonlinear parabolic PDE system. Next, we present a state estimation scheme including fuzzy Luenberger-type PDE state observer plus mobile sensor guidance. Then, an observer-based boundary fuzzy controller is posed to render the resulting closedloop system exponentially stable, and the exponential decay rate is increased by the designed mobile sensor guidance laws. At last, two examples verify the proposed method.

Error-triggered spatial model predictive control for oven temperature

An error-triggered spatial model predictive control (MPC) strategy is present for the parabolic partial differential equation (PDE) system, utilizing input/output data. The initial model is first constructed, in which the spatial basis functions (SBFs) are obtained through Karhunen-Loève expansion, and the temporal models are trained by the neural network. Due to the time-varying nature of the PDE system and the poor control performance during model updates, an error-triggered mechanism is developed to update the SBFs. Besides, a temporal sliding window is designed to update the temporal models, which can strike a balance between capturing the latest dynamics and maintaining temporal model accuracy. Based on the aforementioned model, a spatial MPC incorporating spatial distribution information is proposed. Theoretical analysis affirms the stability of the proposed controller. To demonstrate the superiority of this method, experiments during the thermal process in an oven are carried out. Compared with the methods with fixed update step size, the proposed method has the smoother actuator response and better control performance.

Lyapunov-based stabilization mobile control design of linear parabolic PDE systems

This article considers the exponential mobile stabilization control of linear parabolic partial differential equation (PDE) systems. Firstly, a stabilization scheme based on static output feedback (SOF) control and mobile actuator/sensor guidance is proposed. Next, the operator theory is used to discuss the well-posedness of closed-loop PDE systems. Thus, the SOF control and mobile guidance design approach is simultaneously proposed to guarantee that the closed-loop PDE system is exponentially stable. In final, the effectiveness of the design method is verified by numerical simulation.

Galerkin-Bernstein Approximations of the System of Time Dependent Nonlinear Parabolic PDEs

The purpose of the research is to find the numerical solutions to the system of time dependent nonlinear parabolic partial differential equations (PDEs) utilizing the Modified Galerkin Weighted Residual Method (MGWRM) with the help of modified Bernstein polynomials. An approximate solution of the system has been assumed in accordance with the modified Bernstein polynomials. Thereafter, the modified Galerkin method has been applied to the system of nonlinear parabolic PDEs and has transformed the model into a time dependent ordinary differential equations system. Then the system has been converted into the recurrence equations by employing backward difference approximation. However, the iterative calculation is performed by using the Picard Iterative method. A few renowned problems are then solved to test the applicability and efficiency of our proposed scheme. The numerical solutions at different time levels are then displayed numerically in tabular form and graphically by figures. The comparative study is presented along with L2 norm, and L&amp;infin; norm.

Fault-Tolerant Stochastic Sampled-Data Fuzzy Control for Nonlinear Delayed Parabolic PDE Systems

For nonlinear delayed parabolic partial differential equation (PDE) systems, this paper addresses fault-tolerant stochastic sampled-data (SD) fuzzy control under spatially point measurements (SPMs). Initially, a T–S fuzzy PDE model is given to accurately describe the nonlinear delayed parabolic PDE system. Secondly, in consideration of possible actuator failure, a fault-tolerant SD fuzzy controller with stochastic sampling under SPMs is designed for nonlinear delayed parabolic PDE system, where two sampling periods are considered whose occurrence probabilities are given constants and satisfy the Bernoulli distribution. Then, by constructing a novel time-dependent Lyapunov functional, sufficient conditions that guarantee the mean square exponential stability of closed-loop delayed PDE system are obtained based on linear matrix inequalities (LMIs). Lastly, three examples are given to illustrate the designed approach.

Matrix-oriented FEM formulation for reaction-diffusion PDEs on a large class of 2D domains

For the spatial discretization of elliptic and parabolic partial differential equations (PDEs), we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order k∈N. On a quite general class of 2D domains, namely separable domains, and even on special surfaces, the discrete problem is then reformulated as a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. By considering the IMEX Euler method for the PDE time discretization, we obtain a sequence of these equations. To solve each multiterm Sylvester equation, we apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method with a matrix-oriented preconditioner that captures the spectral properties of the Sylvester operator. Solving the Poisson problem and the heat equation on some separable domains by MO-FEM-PCG, we show a gain in computational time and memory occupation wrt the classical vector PCG with same preconditioning or wrt a LU based direct method. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns on some separable domains and cylindrical surfaces for a morphochemical reaction-diffusion PDE system for battery modelling.

Dynamic Fuzzy Boundary Output Feedback Control for Nonlinear Delayed Parabolic Partial Differential Equation Systems Under Noncollocated Boundary Measurement

For nonlinear space-varying parabolic partial differential equation systems (PPDESs) with random time-varying delay, this paper introduces a dynamic fuzzy boundary output feedback (DFBOF) control under non-collocated boundary measurement (NCBM). Initially, the nonlinear delayed PPDESs are represented by Takagi-Sugeno (T–S) fuzzy models and random time-varying delay is considered by taking the influence of uncertain factors, which belongs to two intervals in a probabilistic way. Since the system state is not fully available and NCBM makes the boundary control design very difficult, a fuzzy observer under NCBM is presented to surmount the design difficulty. Subsequently, an observer-based fuzzy boundary controller is proposed and spatial linear matrix inequality (SLMI) based sufficient conditions to ensure mean square exponential stability are obtained for closed-loop delayed PPDESs by utilizing the Lyapunov direct method and Wirtinger inequality. Then, to solve the SLMIs, the feasibility conditions of DFBOF controller design for nonlinear delayed PPDES are expressed in LMIs. Lastly, two examples are offered to demonstrate the validity of the presented dynamic fuzzy boundary control approach.

Adaptive Synchronization for Networked Parabolic PDE Systems With Uncertain Nonlinear Actuator Dynamics

This article focuses on the synchronization control of networked uncertain parabolic partial differential equations (PDEs) with uncertain nonlinear actuator dynamics. Compared to existing networked PDE systems, control input occurs in ordinary differential equation (ODE) subsystems rather than in PDE ones. Compared to existing results, where the exact system parameters must be known for the entire system, this paper further considers parabolic PDE-ODE systems with unknown parameters affecting the interior of the PDE domain. Due to the unknown parameters and uncertain nonlinear actuator dynamics, the existing distributed algorithms and stability analysis tools cannot be utilized to solve the synchronization problem of cascaded parabolic systems. To address this difficulty, this study designs a novel passive identifier to estimate the states and unknown parameters. Subsequently, based on the passive identifier and Lyapunov function method, a synchronization controller is presented for cascaded parabolic PDE systems to ensure that the synchronization control and the boundedness of all the closed-loop signals are achieved. Lastly, the effectiveness of the obtained results is illustrated using simulation.

Model Predictive Control of Parabolic PDE Systems under Chance Constraints

Model predictive control (MPC) heavily relies on the accuracy of the system model. Nevertheless, process models naturally contain random parameters. To derive a reliable solution, it is necessary to design a stochastic MPC. This work studies the chance constrained MPC of systems described by parabolic partial differential equations (PDEs) with random parameters. Inequality constraints on time- and space-dependent state variables are defined in terms of chance constraints. Using a discretization scheme, the resulting high-dimensional chance constrained optimization problem is solved by our recently developed inner–outer approximation which renders the problem computationally amenable. The proposed MPC scheme automatically generates probability tubes significantly simplifying the derivation of feasible solutions. We demonstrate the viability and versatility of the approach through a case study of tumor hyperthermia cancer treatment control, where the randomness arises from the thermal conductivity coefficient characterizing heat flux in human tissue.

Observer-Based Fuzzy Quantized Control for a Stochastic Third-Order Parabolic PDE System

The work addresses an observer-based fuzzy quantized control for stochastic third-order parabolic partial differential equations (PDEs) using discrete point measurements. For the first time, we contribute in introducing three types of quantizer—logarithmic quantizer, uniform quantizer, and hysteresis quantizer into the controller designs for the stochastic PDE system. The main advantage of the quantized control law lies in reducing the energy that the system spends. The stability analysis is nontrivial due to the complex stochastic PDE model. Different stability results are presented for each case. Sufficient LMI-based conditions are investigated to ensure the internal mean-square exponential stability and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance of the perturbed actuated system. Consistent simulation results that support the proposed theoretical statements are provided.

Hybrid‐driven‐based fuzzy secure filtering for nonlinear parabolic partial differential equation systems with cyber attacks

SummaryThe problem of hybrid‐driven fuzzy filtering for nonlinear semi‐linear parabolic partial differential equation systems with dual cyber attacks is investigated. First, a Takagi‐Sugeno (T‐S) fuzzy model is employed to reconstruct the original nonlinear systems. Second, a hybrid‐driven mechanism is applied for filter design to balance the systems' performance and the limited network resource consumption under dual cyber attacks composed of deception and denial of service attacks. Based on the Lyapunov direct method, sufficient conditions to guarantee the stability of the augmented system are obtained, and the parameters of the designed filter are earned with explicit form. Finally, a simulation example with robust analysis and comparative analysis is provided to illustrate the effectiveness of the proposed method.

Comparison of Two Aspects of a PDE Model for Biological Network Formation

We compare the solutions of two systems of partial differential equations (PDEs), seen as two different interpretations of the same model which describes the formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the network, and we compute the solution of an elliptic–parabolic PDE system for the conductivity vector m, the conductivity tensor C and the pressure p. We use finite differences schemes in a uniform Cartesian grid in a spatially two-dimensional setting to solve the two systems, where the parabolic equation is solved using a semi-implicit scheme in time. Since the conductivity vector and tensor also appear in the Poisson equation for the pressure p, the elliptic equation depends implicitly on time. For this reason, we compute the solution of three linear systems in the case of the conductivity vector m∈R2 and four linear systems in the case of the symmetric conductivity tensor C∈R2×2 at each time step. To accelerate the simulations, we make use of the Alternating Direction Implicit (ADI) method. The role of the parameters is important for obtaining detailed solutions. We provide numerous tests with various values of the parameters involved to determine the differences in the solutions of the two systems.

Adaptive Event-Triggered Control of Networked Fuzzy PDE Systems Under Hybrid Cyber Attacks

This study focuses on the security control problem of networked fuzzy partial differential equation (PDE) systems under hybrid cyber attacks. First, the nonlinear parabolic PDE system is exactly represented by a Takagi–Sugeno fuzzy PDE model. Second, with taking the insecure transmission network into consideration, a more practical stochastic hybrid cyber attacks model for PDE-based networked control systems is introduced. This model contains both denial-of-service attacks and deception attacks in a unified framework. Then, based on spatially pointwise nonuniform sampled-data measurements, an adaptive event-triggered control scheme with an adjustable threshold is newly designed to counter the hybrid cyber attacks’ impact while saving communication resources. Moreover, by constructing a novel Lyapunov–Krasovskii functional and employing some inequality techniques, stability conditions with less conservatism are derived. Finally, the developed method is applied to the Fisher equation to demonstrate its viability and superiority.

A reduced model for compressible viscous heat-conducting multicomponent flows

In the present paper we propose a reduced temperature non-equilibrium model for simulating multicomponent flows with inter-phase heat transfer, diffusion processes (including the viscosity and the heat conduction) and external energy sources. We derive three equivalent formulations for the proposed model. The first formulation consists of balance equations for partial densities, the mixture momentum, the mixture total energy, and phase volume fractions. The second formulation is symmetric and obtained by replacing the equations for the mixture total energy and volume fractions in the first formulation with balance equations for the phase total energy. Replacing one of the phase total energy equation of the second formulation with the mixture total energy equation gives the third formulation. All the three formulations assume velocity and pressure equilibrium across the material interface. These equivalent forms provide different physical perspectives and numerical conveniences. Temperature equilibration and continuity across the material interfaces are achieved with the instantaneous thermal relaxation. Temperature equilibrium is maintained during the heat conduction process. The proposed models are proved to respect the thermodynamical laws. For numerical solution, the model is split into a hyperbolic partial differential equation (PDE) system and parabolic PDE systems. The former is solved with the high-order Godunov finite volume method that ensures the pressure–velocity–temperature (PVT) equilibrium conduction. The parabolic PDEs are solved with both the implicit and the explicit locally iterative method (LIM) based on Chebyshev parameters. Numerical results are presented for several multicomponent flow problems with diffusion processes. Furthermore, we apply the proposed model to simulate the target ablation problem that is of significance to inertial confinement fusion. Comparisons with one-temperature models in literature demonstrate the ability to maintain the PVT property and superior convergence performance of the proposed model in solving multicomponent problems with diffusions.

Fault Isolation of a Class of Uncertain Nonlinear Parabolic PDE Systems

This paper proposes a novel fault isolation (FI) scheme for distributed parameter systems modeled by a class of parabolic partial differential equations (PDEs) with nonlinear uncertain dynamics. A key feature of the proposed FI scheme is its capability of dealing with the effects of system uncertainties for accurate FI. Specifically, an approximate ordinary differential equation (ODE) system is first derived to capture the dominant dynamics of the original PDE system. An adaptive dynamics identification approach using radial basis function neural network is then proposed based on this ODE system, to achieve locally-accurate identification of the uncertain system dynamics under faulty modes. A bank of FI estimators with associated adaptive thresholds are finally designed for real-time FI decision making. Rigorous analysis on the fault isolatability is provided. Simulation study on a representative transport-reaction process is conducted to demonstrate the effectiveness of the proposed approach.