System Of Parabolic Partial Differential Equations Research Articles

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.

A Computation Approach to Chance Constrained Optimization of Boundary-Value Parabolic Partial Differential Equation Systems

Abstract This work studies chance constrained optimization of boundary-value parabolic partial differential equations (CCPDE) with random data, where the PDE model is treated as equality constraint and chance constraints are imposed on inequality constraints involving state variables. Since such a CCPDE problem is generally non-smooth, non-convex and difficult to solve directly, we use our recently proposed smoothing approximation method to solve the problem. As a result, the probability function of the chance constraints is approximated in two different ways by a family of differentiable functions. This leads to two smooth parametric optimization problems IAτ and OAτ, where the feasible sets of IAτ are always subsets (inner approximation) and the feasible sets of OAτ always supersets (outer approximation). The feasible sets of IAτ (resp. OAτ ) converge asymptotically to the feasible set of the CCPDE. Moreover, any limit point of a sequence of optimal solutions of IAτ (resp. OAτ ) is a stationary point of CCPDE. The viability of the approximation approach is numerically demonstrated by optimal thermal cancer treatment as a case study.

A space-time adaptive finite element method with exponential time integrator for the phase field model of pitting corrosion

In this paper we propose a space-time adaptive finite element method for the phase field model of pitting corrosion, which is a parabolic partial differential equation system consisting of a phase variable and a concentration variable. A major challenge in solving this phase field model is that the problem is very stiff, which makes the time step size extremely small for standard temporal discretizations. Another difficulty is that a high spatial resolution is required to capture the steep gradients within the diffused interface, which results in very large number of degrees of freedom for uniform meshes. To overcome the stiffness of this model, we combine the Rosenbrock–Euler exponential integrator with Crank–Nicolson scheme for the temporal discretization. Moreover, by exploiting the fact that the speed of the corroding interface decreases with time, we derive an adaptive time stepping formula. For the spatial approximation, we propose a simple and efficient strategy to generate adaptive meshes that reduces the computational cost significantly. Thus, the proposed method utilizes local adaptivity and mesh refinement for efficient simulation of the corrosive dissolution over long times in heterogeneous media with complex microstructures. We also present an extensive set of numerical experiments in both two and three dimensional spaces to demonstrate efficiency and robustness of the proposed method.

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 Guaranteed Cost for Nonlinear Parabolic PDE Systems via Static Output Feedback

This article introduces a sampled-data (SD) static output feedback fuzzy control (FC) with guaranteed cost for nonlinear parabolic partial differential equation (PDE) systems. First, a Takagi–Sugeno (T–S) fuzzy parabolic PDE model is employed to represent the nonlinear PDE system. Second, with the aid of the T–S fuzzy PDE model, a SD FC design with guaranteed cost under spatially averaged measurements is developed in the formulation of linear matrix inequalities by utilizing a time-dependent Lyapunov functional and inequality techniques, which can stabilize exponentially the PDE system while providing an optimized upper bound on the cost function. The membership functions of the proposed controller are determined by the measurement output and independent of the fuzzy PDE plant model. Finally, simulation results are presented to control the diffusion equation and the FitzHugh–Nagumo equation for demonstrating the effectiveness of the proposed method.

Event-triggered reliable [formula omitted] fuzzy filtering for nonlinear parabolic PDE systems with Markovian jumping sensor faults

This paper is devoted to even-triggered reliable H∞ filter design for a class of nonlinear partial differential equation (PDE) systems with Markovian jumping sensor faults. Initially, a Takagi-Sugeno (T-S) fuzzy model is adopted to reconstruct the nonlinear systems. Then, the time delays and signal quantization that often occur in network transmission are taken into account, and an integral-type event-triggered scheme is developed to improve the transmission channel utilization. Furthermore, non-parallel-distributed-compensation technique is introduced to increase the flexibility of filter design and the filter’s parameters can be obtained by solving several linear matrix inequalities. Finally, two numerical simulations and an application study to Catalytic Rod are provided to demonstrate the effectiveness and practicability of the proposed methodology.

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.

Control design for non‐linear distributed parameter systems with mobile actuators and sensors

This study deals with the H ∞ control design problem for a class of non-linear distributed parameter systems described by parabolic partial differential equations (PDEs) via mobile collocated actuators and sensors. Initially, the spatial domain is decomposed into multiple subdomains according to the number of actuator/sensor pairs and the projection modification algorithm is employed to guarantee each actuator/sensor pair is only capable of moving within the respective subdomain. Subsequently, the well-posedness of the closed-loop PDE system is analysed by means of the operator semigroup theory. Then, a control-plus-guidance design method for the non-linear PDE system is developed in the form of bilinear matrix inequalities, such that the resulting closed-loop system is exponentially stable while satisfying a prescribed H ∞ performance of disturbance attenuation, and the mobile actuator/sensor guidance can enhance the transient performance of the closed-loop system. Finally, a numerical example and a practical application example are respectively given to show the effectiveness of the proposed design method.

Static Collocated Piecewise Fuzzy Control Design of Quasi-Linear Parabolic PDE Systems Subject to Periodic Boundary Conditions

This paper presents a Lyapunov and partial differential equation (PDE)-based methodology to solve static collocated piecewise fuzzy control design of quasi-linear parabolic PDE systems subject to periodic boundary conditions. Two types of piecewise control, i.e., globally piecewise control and locally piecewise control are considered, respectively. A Takagi–Sugeno (T–S) fuzzy PDE model that is constructed via local sector nonlinearity method is first employed to accurately describe spatiotemporal dynamics of quasi-linear PDEs. Based on the T–S fuzzy PDE model, a static collocated piecewise fuzzy feedback controller is constructed to guarantee the locally exponential stability of the resulting closed-loop system. Sufficient conditions for the existence of such fuzzy controller are developed by applying vector-valued Poincare–Wirtinger inequality and its variants and a linear matrix inequality (LMI) relaxation technique. These sufficient conditions are presented in terms of standard LMIs. Finally, the performance of the suggested fuzzy controller is illustrated by numerical simulation results of a nonlinear PDE system described by quasi-linear FitzHugh–Nagumo equation with periodic boundary conditions.

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.

Extended BACOLI

BACOLI is a Fortran software package for solving one-dimensional parabolic partial differential equations (PDEs) with separated boundary conditions by B-spline adaptive collocation methods. A distinguishing feature of BACOLI is its ability to estimate and control error and correspondingly adapt meshes in both space and time. Many models of scientific interest, however, can be formulated as multiscale parabolic PDE systems, that is, models that couple a system of parabolic PDEs describing dynamics on a global scale with a system of ordinary differential equations describing dynamics on a local scale. This article describes the Fortran software eBACOLI, the extension of BACOLI to solve such multiscale models. The performance of the extended software is demonstrated to be statistically equivalent to the original for purely parabolic PDE systems. Results from eBACOLI are given for various multiscale models from the extended problem class considered.

Sampled-data fuzzy control for a class of nonlinear parabolic distributed parameter systems under spatially point measurements

In this paper, the exponential stabilization problem is addressed for a class of nonlinear parabolic partial differential equation (PDE) systems via sampled-data fuzzy control approach. Initially, the nonlinear PDE system is accurately represented by the Takagi–Sugeno (T–S) fuzzy PDE model. Then, based on the T–S fuzzy PDE model, a novel time-dependent Lyapunov functional is used to design a sampled-data fuzzy controller under spatially point measurements such that the closed-loop fuzzy PDE system is exponentially stable with a given decay rate. The stabilization conditions are presented in terms of a set of linear matrix inequalities (LMIs). Finally, simulation results on the control of the diffusion equation and the FitzHugh–Nagumo (FHN) equation to illustrate the effectiveness of the proposed design method.

Improved H∞ sampled‐data control for semilinear parabolic PDE systems

SummaryIn this paper, an H∞ sampled‐data control problem is addressed for semilinear parabolic partial differential equation (PDE) systems. By using a time‐dependent Lyapunov functional and vector Poincare's inequality, a sampled‐data controller under spatially averaged measurements is developed to stabilize exponentially the PDE system with an H∞ control performance. The stabilization condition is presented in terms of a set of linear matrix inequalities. Finally, simulation results on the control of the diffusion equation and the FitzHugh‐Nagumo equation are given to illustrate the effectiveness of the proposed design method.

Adding Integral Action for Open-Loop Exponentially Stable Semigroups and Application to Boundary Control of PDE Systems

The paper deals with output feedback stabilization of exponentially stable systems by an integral controller. We propose appropriate Lyapunov functionals to prove exponential stability of the closed-loop system. An example of parabolic PDE (partial differential equation) systems and an example of hyperbolic systems are worked out to show how exponentially stabilizing integral controllers are designed. The proof is based on a novel Lyapunov functional construction which employs the forwarding techniques.

GPU-Based Model Predictive Control of Nonlinear Parabolic Partial Differential Equations System and Its Application in Continuous Casting

Temperature control of steel billets plays an important role in the quality of steel billets. This paper was motivated by the necessity for setting a value of spray cooling water flow with regard to the accuracy, fast applicability to real-time optimization, and capability of non-steady operation scenarios, especially for the change of casting speed. Therefore, this paper is focused on the GPU-based model predictive control (MPC) for temperature control of steel billets. The system dynamics in MPC is a heat transfer model characterized by the nonlinear parabolic partial differential equations (PDEs). This paper presented two algorithms. First, an adaptive trust-region Levenberg-Marquardt method (ATR-LM) based on the measured surface temperature is presented to estimate the unknown parameters in the heat transfer model. The corresponding experimental results indicate that the presented method can reduce the iteration time. Second, for the purpose of satisfying the real-time requirement, the stream parallel sparse Jacobian method (SP-SJ) is presented to solve the dynamic optimization problem associated with the PDEs. The corresponding experimental results show that the presented method exhibits satisfactory computational performance and achieves satisfactory control performance.

Boundary stabilization of a class of reaction–advection–diffusion systems via a gradient-based optimization approach

In this paper, the boundary stabilization problem of a class of unstable reaction–advection–diffusion (RAD) systems described by a scalar parabolic partial differential equation (PDE) is considered. Different the previous research, we present a new gradient-based optimization framework for designing the optimal feedback kernel for stabilizing the unstable PDE system. Our new method does not require solving non-standard Riccati-type or Klein–Gorden-type PDEs. Instead, the feedback kernel is parameterized as a second-order polynomial whose coefficients are decision variables to be tuned via gradient-based dynamic optimization, where the gradients of the system cost functional (which penalizes both kernel and output magnitude) with respect to the decision parameters are computed by solving a so-called “costate” PDE in standard form. Special constraints are imposed on the kernel coefficients to ensure that the optimized kernel yields closed-loop stability. Finally, three numerical examples are illustrated to verify the effectiveness of the proposed approach.

Spatially Piecewise Fuzzy Control Design for Sampled-Data Exponential Stabilization of Semilinear Parabolic PDE Systems

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 Poincare–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.

Dynamic Boundary Fuzzy Control Design of Semilinear Parabolic PDE Systems With Spatially Noncollocated Discrete Observation.

The problem of dynamic boundary fuzzy control design is investigated in this paper for nonlinear parabolic partial differential equation (PDE) systems with spatially noncollocated discrete observation. Two cases of noncollocated discrete observation in space (i.e., pointwise observation in space and local piecewise uniform observation in space) are considered, respectively. The spatially noncollocated discrete observation makes the control design very difficult. Such design difficulty can be surmounted by an observer-based feedback control technique. It is assumed that the semilinear PDEs are accurately represented by a Takagi-Sugeno fuzzy PDE model. On the basis of the obtained fuzzy PDE model, a fuzzy Luenberger-type PDE state observer with above two cases of noncollocated discrete observation in space is first proposed for exponential estimation of the PDE system state. An observer-based dynamic fuzzy controller is then constructed such that the resulting closed-loop system is exponentially stable. Sufficient conditions on existence of such fuzzy controller are developed by Lyapunov technique with variations of vector-valued Poincaré-Wirtinger inequality, and presented in terms of linear matrix inequalities. Finally, extensive numerical simulation results of two examples are provided to support the proposed design method.

Modeling Flow in Porous Media With Double Porosity/Permeability: Mathematical Model, Properties, and Analytical Solutions

Geomaterials such as vuggy carbonates are known to exhibit multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores with different hydromechanical properties. Moreover, these pore-networks are connected through fissures and conduits. Although some models are available in the literature to describe flows in such media, they lack a strong theoretical basis. This paper aims to fill this gap in knowledge by providing the theoretical foundation for the flow of incompressible single-phase fluids in rigid porous media that exhibit double porosity/permeability. We first obtain a mathematical model by combining the theory of interacting continua and the maximization of rate of dissipation (MRD) hypothesis, and thereby provide a firm thermodynamic underpinning. The governing equations of the model are a system of elliptic partial differential equations (PDEs) under a steady-state response and a system of parabolic PDEs under a transient response. We then present, along with mathematical proofs, several important mathematical properties that the solutions to the model satisfy. We also present several canonical problems with analytical solutions which are used to gain insights into the velocity and pressure profiles, and the mass transfer across the two pore-networks. In particular, we highlight how the solutions under the double porosity/permeability differ from the corresponding ones under Darcy equations.

Observer-Based <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$</tex-math> </inline-formula> Sampled-Data Fuzzy Control for a Class of Nonlinear Parabolic PDE Systems

In this paper, an observer-based $H_{\infty }$ sampled-data fuzzy control problem is addressed for a class of nonlinear parabolic partial differential equation (PDE) systems. With the aid of the modal decomposition technique, a nonlinear ordinary differential equation (ODE) model is initially derived to describe the dominant (slow) dynamics of the PDE system. Subsequently, the resulting nonlinear ODE model is accurately represented by the Takagi–Sugeno (T–S) fuzzy model. Then, based on the T–S fuzzy model, a finite-dimensional observer-based sampled-data fuzzy control design with $H_\infty$ performance is developed for the PDE system via employing a novel time-dependent functional. The outcome of the observer-based $H_{\infty }$ sampled-data fuzzy control problem can be formulated as a bilinear matrix inequality optimization problem. Moreover, an iterative optimization algorithm based on the linear matrix inequalities is given to obtain a suboptimal $H_\infty$ sampled-data fuzzy controller. Finally, simulation results on the Fisher equation and a temperature cooling fin of high-speed aerospace vehicle illustrate that the proposed design method is effective.