Parabolic Parabolic Partial Differential Equation Systems 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.

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.

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.

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.

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.

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.

Machine learning-based model predictive control of diffusion-reaction processes

In this work, we develop a machine-learning-based predictive control design for nonlinear parabolic partial differential equation (PDE) systems using process state measurement time-series data. First, the Karhunen-Loève expansion is used to derive dominant spatial empirical eigenfunctions of the nonlinear parabolic PDE system from the data. Then, these empirical eigenfunctions are used as basis functions within a Galerkin's model reduction framework to derive the temporal evolution of a small number of temporal modes capturing the dominant dynamics of the PDE system. Subsequently, feedforward neural networks (FNN) are used to approximate the reduced-order dominant dynamics of the parabolic PDE system from the data within a desired operating region. Lyapunov-based model predictive control (MPC) scheme using FNN models is developed to stabilize the nonlinear parabolic PDE system. Finally, a diffusion-reaction process example is used to demonstrate the effectiveness of the proposed machine-learning-based predictive control method.

Control Design for Parabolic PDE Systems via T–S Fuzzy Model

In this article, we investigate the parabolic partial differential equations (PDEs) systems with Neumann boundary conditions via the Takagi–Sugeno (T–S) fuzzy model. On the basis of the obtained T–S fuzzy PDE model, a novel fuzzy state controller which is associated with the boundary state of position and the mean value coefficient matrix derived through the mean value theorem of integral is designed to analyze the asymptotic stability of the parabolic PDE system. Without sampling the nonlinear parameter of the system, new stability conditions are deduced in the form of linear matrix inequalities (LMIs). Moreover, compared with the novel fuzzy state controller, more conservative conditions based on another fuzzy state controller are also provided. Finally, we explore the state-feedback controller into the Fisher equation as an application. Simulation results show that the proposed method is effective.

Fault Detection of a Class of Nonlinear Uncertain Parabolic PDE Systems

This letter investigates the fault detection (FD) problem of a class of uncertain distributed parameter systems modeled by nonlinear parabolic partial differential equations (PDEs). A novel FD scheme is proposed with a neural network-based adaptive dynamics learning approach. Specifically, based on the Galerkin method, a finite dimensional ordinary differential equation (ODE) system is first derived to capture the dominant dynamics of the PDE system. An adaptive dynamics learning approach using radial basis function neural networks (RBF NN) is then developed to achieve locally accurate identification of the associated uncertain system dynamics. The learned knowledge can be obtained and stored in a bank of constant RBF NN models, which can be used to further construct a bank of FD estimators. A threshold is finally designed for real-time FD decision making. One important feature of the proposed FD scheme is that the spatio-temporal uncertain dominant dynamics of the parabolic PDE system can be locally accurately identified, such that occurring faults can be effectively distinguished from system uncertainties for accurate FD. Extensive simulation studies are conducted to verify the effectiveness and advantages of the proposed FD scheme.

Guaranteed cost design for controlling semilinear parabolic PDE systems with mobile collocated actuators and sensors

This paper investigates the guaranteed cost design problem for controlling a class of semilinear parabolic partial differential equation (PDE) systems using mobile collocated actuators and sensors. Initially, a mode indicator function is employed to indicate the different modes for all actuator/sensor pairs according to whether each actuator/sensor pair is static or mobile. Subsequently, a mode-dependent switching control scheme is proposed and the well-posedness of the closed-loop PDE system is also analysed. Then, based on Lyapunov direct method, an integrated design of switching controllers and mobile actuator/sensor guidance laws is developed in the form of linear matrix inequalities, such that the closed-loop PDE system is exponentially stable while providing an upper bound for the prescribed quadratic cost function. Moreover, a suboptimal guaranteed cost design problem is also addressed to make the cost bound as small as possible. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method.

Fuzzy Control Design of Nonlinear Time-Delay Parabolic PDE Systems Under Mobile Collocated Actuators and Sensors.

Via mobile sensing measurements, this study applies the Takagi-Sugeno (T-S) fuzzy model to deal with the mobile fuzzy control design problem for nonlinear time-delay parabolic partial differential equation (PDE) systems. Initially, we use a T-S fuzzy model to accurately represent the nonlinear time-delay parabolic PDE system. Subsequently, under the assumption that the actuators and sensors are collocated while the spatial domain is divided by several subdomains, a control scheme containing the fuzzy controllers and the guidance of mobile actuator/sensor pairs is proposed based on the obtained T-S fuzzy model, where the projection modification guidance to be designed can guarantee that each mobile actuator/sensor pair moves within the prescribed area. Then, using the Lyapunov direct method and integral inequalities, a membership-function-dependent design of fuzzy controllers plus mobile actuator/sensor guidance laws is developed to render the resulting closed-loop time-delay system exponentially stable. Moreover, the exponential decay rate can also be increased by the proposed mobile guidance laws. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method and the application of mobile actuator/sensor pairs contributes to accelerating the convergence speed of the closed-loop state.

Static output feedback stabilization for a linear parabolic PDE system with time-varying delay via mobile collocated actuator/sensor pairs

This paper addresses a static output feedback (SOF) stabilization problem for a linear parabolic partial differential equation (PDE) system with time-varying delay via mobile collocated actuator/sensor pairs. Initially, an SOF stabilization scheme via mobile actuator/sensor pairs is proposed for the delayed PDE system, where the spatial domain is divided into multiple subdomains according to the number of actuator/sensor pairs and the projection modification algorithm is employed to ensure each collocated actuator/sensor pair only can move within the respective subdomain. Subsequently, the well-posedness of the closed-loop delayed PDE system is analyzed using the operator semigroup theory. Then, by constructing an appropriate Lyapunov–Krasovskii functional candidate, a delay-dependent control-plus-guidance design is developed in the form of bilinear matrix inequalities (BMIs), such that the resulting closed-loop system is exponentially convergent and the mobile actuator/sensor guidance can improve the transient response of closed-loop state. Moreover, an iterative algorithm based on linear matrix inequalities is provided to solve the BMIs. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed method.

Design of Fuzzy State Observer and Mobile Sensor Guidance for Semilinear Parabolic PDE Systems.

This article investigates the issue of the fuzzy observer design for the semilinear parabolic partial differential equation (PDE) systems with mobile sensing measurements. Initially, we employ a Takagi-Sugeno (T-S) fuzzy PDE model to represent the semilinear parabolic PDE system accurately in a local region. Afterward, via the T-S fuzzy model and under the hypothesis that the spatial domain is divided by several subdomains in the light of the number of sensors, a state observation scheme which contains a fuzzy observer and the mobile sensor guidance is proposed. Then, by means of the Lyapunov direct method and integral inequalities, a design method of the fuzzy observer and mobile sensor guidance is provided to render the resulting state estimation error system exponentially stable, while the designed mobile sensor guidance can increase the exponential decay rate. Finally, numerical simulations are presented to show that the proposed fuzzy observer design approach is effective and the employment of mobile sensors contributes to improving the response speed of the state estimation error in comparison with the static ones.

Observer Design for Lipschitz Nonlinear Parabolic PDE Systems With Unknown Input

In this article, a novel method to design the observer for a class of uncertain Lipschitz nonlinear parabolic partial differential equations (PDE) systems is investigated. First, the observer and the dynamic errors with undetermined parameters for the parabolic PDE systems subject to appropriate boundary conditions are presented. The conditions of the designed observer are involved. Then the analysis of asymptotic stability and $\mathcal {H}_\infty $ performance conditions for the observer design of uncertain nonlinear parabolic PDE systems are studied and presented in terms of matrix inequalities based on the Lyapunov stability theory. Finally, the effectiveness of the proposed method is validated by a numerical parabolic PDE system.

Switching state observer design for semilinear parabolic PDE systems with mobile sensors

This paper studies the problem of state observation for a class of semilinear parabolic partial differential equation (PDE) systems using mobile sensors, where the spatial domain is decomposed into multiple subdomains according to the number of sensors and each sensor is able to move within the respective spatial subdomain. Initially, the well-posedness of the original parabolic PDE system is discussed. Subsequently, a mode indicator function is used to indicate the different modes for all sensors and a mode-dependent observation scheme is presented to reduce the conservatism of observer design. Then, by employing Lyapunov direct technique and Wirtinger’s inequality, an integrated design of switching state observers and mobile sensor guidance laws is developed in terms of linear matrix inequalities, such that the resulting state estimation error system is exponentially stable. Moreover, the mobile sensor guidance can enhance the transient performance of the error system. Finally, numerical simulations are given to illustrate the effectiveness of the proposed design method.

Sampled-data fuzzy control with space-varying gains for nonlinear time-delay parabolic PDE systems

This paper introduces a sampled-data fuzzy control (SDFC) with space-varying gains for nonlinear time-delay parabolic partial differential equation (PDE) systems. First, a Takagi-Sugeno (T-S) fuzzy time-delay parabolic PDE model is employed to represent the nonlinear time-delay PDE system. Second, with the aid of the T-S fuzzy time-delay PDE model, a SDFC design with space-varying gains is developed in the formulation of space-dependent linear matrix inequalities (LMIs) by constructing an appropriate Lyapunov functional, which can stabilize exponentially the time-delay PDE system. These stabilization conditions can be applied to either slowing-varying time delay or fast-varying one. Finally, simulation results on two examples are presented to demonstrate the effectiveness of the proposed method.

Fuzzy Stabilization Design for Semilinear Parabolic PDE Systems With Mobile Actuators and Sensors

This paper deals with a fuzzy stabilization design problem for a class of semilinear parabolic partial differential equation (PDE) systems using mobile actuators and sensors. Initially, a Takagi–Sugeno (T–S) fuzzy PDE model is employed to accurately represent the semilinear parabolic PDE system. Subsequently, based on the T–S fuzzy model, a stabilization scheme containing the fuzzy controllers and the guidance of mobile actuator/sensor pairs is proposed, where the spatial domain is decomposed into multiple subdomains according to the number of actuator/sensor pairs and each actuator/sensor pair is capable of moving within the respective subdomain. Then, by a Lyapunov direct technique, an integrated design of fuzzy controllers plus mobile actuator/sensor guidance laws is developed in the form of bilinear matrix inequalities (BMIs), such that the resulting closed-loop system is exponentially stable and the mobile actuator/sensor guidance can enhance the transient performance of the closed-loop system. Furthermore, an iterative algorithm based on linear matrix inequalities is proposed to solve the BMIs. Finally, two examples are given to illustrate the effectiveness of the proposed method.