Heat Partial Differential Equations Research Articles

Finite‐dimensional, output‐predictor‐based, adaptive observer for heat PDEs with sensor delay

SummaryWe are considering the problem of designing observers for heat partial differential equations (PDEs) that are subject to sensor delay and parameter uncertainty. In order to get finite‐dimensional observers, described by ordinary differential equations (ODE), we develop a design method based on the modal decomposition approach. The approach is extended so that both parameter uncertainty and sensor delay effects are compensated for. To cope more effectively with sensor delay, an output predictor is designed and the online provided output predictions are substituted to the future output values in the observer. To compensate for parameter uncertainty, we design a parameter estimator providing online parameter estimates, which are substituted to the unknown parameters in the observer. The parameter estimator design is made decoupled from the observer gain design by using an appropriate decoupling transformation. Using an analysis of the small‐gain‐theorem type, the whole (state and parameter) estimation error system is shown to be exponentially stable, under well‐defined conditions on the observer dimension, the sensor delay, and signal persistent excitation (PE).

Mechanically induced temperature oscillations in the coupled thermo-elasticity analysis of articulated dynamical systems

This article discusses a new approach for predicting and quantifying mechanically induced temperature oscillations in the coupled thermo-elasticity analysis of articulated mechanical systems (AMS). In this approach, the constrained equations of motion are solved simultaneously with discrete temperature equations obtained by converting heat partial-differential equation to a set of first-order ordinary differential equations. Dependence of the temperature gradients and their spatial derivatives on the position gradients, spinning motion, and curvatures is discussed. The approach captures dependence of the temperature-oscillation frequencies on the mechanical-displacement frequencies. The temperature field can be selected to ensure continuity of the temperature gradients at the nodal points. To generalize the AMS coupled thermo-elasticity formulation and capture the effect of the boundary and motion constraints (BMC) on the thermal expansion, the proposed method is based on integrating thermodynamics and Lagrange-D’Alembert principles. The absolute nodal coordinate formulation (ANCF) is used to describe continuum displacement and obtain accurate description of the reference-configuration geometry and change of this geometry due to deformations. A thermal-analysis large-displacement formulation is used to allow converting heat energy to kinetic energy, ensuring stress-free thermal expansion in case of unconstrained uniform thermal expansion. Cholesky heat coordinates are used to define an identity coefficient matrix for the efficient solution of the discretized heat equations. The approach presented is applicable to the two different forms of the heat equation used in the literature; one form is explicit function of the stresses while the other form does not depend explicitly on the stresses. Because of the need for using ANCF finite elements to achieve a higher degree of continuity in the coupled thermomechanical approach introduced in this article, the concept of the ANCF mesh topology is discussed.

Adaptive Observer Design for Heat PDEs with Discrete and Distributed Delays and Parameter Uncertainties

The problem of observer design is addressed for heat partial differential equations (PDEs) that are subject to parameter uncertainties and time delays. In addition to parameter uncertainty, the novelty also lies in the presence of time-delay in the domain. Both discrete and distributed delays are accounted for and dealt with using a unified integral representation. We design an adaptive observer that provides online estimates of the states and parameters. The observer is designed using the backstepping design method, involving a Volterra transformation, and a state-parameter decoupling transformation. The observer stability analysis is dealt with using a Lyapunov-Krasovskii functional. It is shown, under a persistent excitation (PE) assumption, that the resulting estimation error system is exponentially stable for small delays.

Generalized Lucas Tau Method for the Numerical Treatment of the One and Two-Dimensional Partial Differential Heat Equation

This paper is dedicated to proposing two numerical algorithms for solving the one- and two-dimensional heat partial differential equations ( PDEs ). In these algorithms, generalized Lucas polynomials ( GLPs ) involving two parameters are utilized as basis functions. The two proposed numerical schemes in one and two- dimensions are based on solving the corresponding integral equation to the heat equation, and after that employing, respectively, the tau and collocation methods to convert the heat equations subject to their underlying conditions into systems of linear algebraic equations that can be treated efficiently via suitable numerical procedures. In this article, the convergence analysis is examined for the proposed generalized Lucas expansion. Five illustrative problems are numerically solved via the two proposed numerical schemes to show the applicability and accuracy of the presented algorithms. Our obtained results compare favourably with the exact solutions.

Global Stabilization of Nonlinear Finite Dimensional System with Dynamic Controller Governed by 1 − d Heat Equation with Neumann Interconnection

This paper deals with the stabilization of a class of uncertain nonlinear ordinary differential equations (ODEs) with a dynamic controller governed by a linear 1−d heat partial differential equation (PDE). The control operates at one boundary of the domain of the heat controller, while at the other end of the boundary, a Neumann term is injected into the ODE plant. We achieve the desired global exponential stabilization goal by using a recent infinite-dimensional backstepping design for coupled PDE-ODE systems combined with a high-gain state feedback and domination approach. The stabilization result of the coupled system is established under two main restrictions: the first restriction concerns the particular classical form of our ODE, which contains, in addition to a controllable linear part, a second uncertain nonlinear part verifying a lower triangular linear growth condition. The second restriction concerns the length of the domain of the PDE which is restricted.

Performance output tracking for a one-dimensional heat equation with input delay

In this paper, we focus on the performance output tracking for a one-dimensional heat partial differential equation with unknown external disturbance and non-collocated configuration, in which the control input is subject to a time delay. By writing the time delay as a first-order hyperbolic equation, the control system can be modeled as a cascade construction in which the transport equation can be viewed as the actuator dynamics of the heat partial differential system. With the help of the actuator dynamics compensation, the difficulties caused by input delay are addressed. The full state feedback law is constructed to achieve the performance output tracking and an error based observer is designed successfully. The exponential stability of the closed-loop system and the well-posedness of the observer are obtained.

Two-dimensional mathematical modeling of an industrial ammonia synthesis reactor with CFD analysis

A three-bed ammonia reactor with the radial-axial flow and two heat exchangers in the rector's middle has been studied. According to the fluid regime type with two-dimensional axial-radial flow and intercooler, this reactor is different from the previous generation. Two-dimensional mathematical modelling is utilized to predict fluid flow, concentration, and temperature profiles along the reactor. The Navier-Stokes, mass, and heat partial differential equations are solved simultaneously via orthogonal collocation on the finite element. The results have been compared with industrial data, and good accordance has been observed. There are industrial data in the bed outlet, and results along the reactor cannot be validated. Therefore, CFD analysis of three beds is performed, and results are compared with results of two-dimensional mathematical modeling, and good agreement is observed.

Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations

One of the most challenging issues in applied mathematics is to develop and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is very hard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense that the number of computational operations needed by the algorithm to compute an approximation of accuracy $\epsilon > 0$ grows at most polynomially in both the reciprocal $1/\epsilon$ of the required accuracy and the dimension $d \in \mathbb{N}$ of the PDE. Recently, a new approximation method, the so-called full history recursive multilevel Picard (MLP) approximation method, has been introduced and, until today, this approximation scheme is the only approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of semilinear PDEs with general time horizons. It is a key contribution of this article to extend the MLP approximation method to systems of semilinear PDEs and to numerically test it on several example PDEs. More specifically, we apply the proposed MLP approximation method in the case of Allen-Cahn PDEs, Sine-Gordon-type PDEs, systems of coupled semilinear heat PDEs, and semilinear Black-Scholes PDEs in up to 1000 dimensions. The presented numerical simulation results suggest in the case of each of these example PDEs that the proposed MLP approximation method produces very accurate results in short runtimes and, in particular, the presented numerical simulation results indicate that the proposed MLP approximation scheme significantly outperforms certain deep learning based approximation methods for high-dimensional semilinear PDEs.

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

SummaryThis article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1−d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite‐dimensional backstepping design for state feedback stabilization design of coupled PDE‐ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right‐hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method.

Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme

We study the random heat partial differential equation on a bounded domain assuming that the diffusion coefficient and the boundary conditions are random variables, and the initial condition is a stochastic process. Under general conditions, this stochastic system possesses a unique solution stochastic process in the almost sure and mean square senses. To quantify the uncertainty for this solution process, the computation of the probability density function is a major goal. By using a random finite difference scheme, we approximate the stochastic solution at each point by a sequence of random variables, whose probability density functions are computable, i.e., we construct a sequence of approximating density functions. We include numerical experiments to illustrate the applicability of our method.

Heat equation parameter estimation based on the POD-Galerkin approach

The proper orthogonal decomposition (POD), a mathematical procedure initially developed to determine low-dimensional approximated descriptions of high-dimensional functions, has already proved its efficiency for model reduction of linear and non-linear systems. In this paper, the POD and the Galerkin projection methods are combined to estimate the thermal diffusivity parameter of a wall, the behavior of which is assumed to be modeled by the famous heat partial differential equation (PDE). According to this approach, empirical orthogonal functions are first extracted from the POD, then used by the Galerkin method to transform the initial PDE into a set of ordinary differential equations (ODEs). These ODEs explicitly contain the parameter to be identified which is estimated in this contribution by using the Levenberg-Marquardt algorithm. Simulation results are provided to illustrate the effectiveness of this multi-step approach.

Application of ANNs approach for wave-like and heat-like equations

AbstractArtificial neural networks are data processing systems which originate from human brain tissue studies. The remarkable abilities of these networks help us to derive desired results from complicated raw data. In this study, we intend to duplicate an efficient iterative method to the numerical solution of two famous partial differential equations, namely the wave-like and heat-like problems. It should be noted that many physical phenomena such as coupling currents in a flat multi-strand two-layer super conducting cable, non-homogeneous elastic waves in soils and earthquake stresses, are described by initial-boundary value wave and heat partial differential equations with variable coefficients. To the numerical solution of these equations, a combination of the power series method and artificial neural networks approach, is used to seek an appropriate bivariate polynomial solution of the mentioned initial-boundary value problem. Finally, several computer simulations confirmed the theoretical results and demonstrating applicability of the method.

Stability analysis of MHD outer velocity flow on a stretching cylinder

A study has been done on axisymmetric magnetohydrodynamic (MHD) flow of a viscous incompressible outer velocity fluid along a stretching cylinder. The momentum and heat partial differential equations are mold using similarity transformations into a set of nonlinear ordinary differential equations. These equations have been solved with the help of Runge-Kutta Fehlberg scheme numerically with shooting technique. It is found that the solution is not unique for stretching cylindrical surface. The dual solution exists for a certain range of fluid parameters on stretching surface. The stability analysis shown that the upper branch is a stable solution and the lower branch is a unstable solution. The critical point λc=-0.03211 of outer velocity parameter and the range of the unstable solution -0.03211=λc⩽λ⩽λr=0.1411 have been obtained for certain magnetic parameter. Also, the effects of MHD and outer velocity parameter with various other fluid parameters on heat transfer and flow characteristics have been analyzed and shown by figures and tables. The outcomes show that the outer velocity parameter has a substantial effect on MHD fluid flow in stretching cylindrical surface.

Stabilisation of unstable cascaded heat partial differential equation system subject to boundary disturbance

In this study, the authors consider boundary stabilisation for a cascade of unstable heat partial differential equation systems with interconnection at one end and general external disturbance at the control end. An unknown input type observer is first designed by sliding mode control method to estimate the state, and its Filippov type solution is determined. To design an output feedback, they need to know a state feedback which could be realised by the backstepping transformation. The transformation transforms the system into an equivalent target system where the governing equations are stable. To deal with the disturbance, they apply the active disturbance rejection control to estimate the disturbance by output. Based on observer and disturbance estimator, an output feedback control is then designed. The existence of solution to the closed-loop system is proved and the stability is concluded. Finally, some numerical simulations are presented for illustration.

Boundary Control for a Kind of Coupled PDE-ODE System

A coupled system of an ordinary differential equation (ODE) and a heat partial differential equation (PDE) with spatially varying coefficients is discussed. By using the PDE backstepping method, the state-feedback stabilizing controller is explicitly constructed with the assumptionsλ(x)∈P[x]nandλ(x)∈C[0, l]∞, respectively. The closed-loop system is proved to be exponentially stable by this controller. A simulation example is presented to illustrate the effectiveness of the proposed method.

Multiagent Deployment Over a Source

We consider the problem of deploying a group of autonomous vehicles (agents) in a formation that has higher density near the source of a measurable signal and lower density away from the source. The spatial distribution of the signal and the location of the source are unknown, but the signal is known to decay with the distance from the source. The vehicles do not have the capability of sensing their own positions, but they are capable of sensing the distance between them and their neighbors. We design a control algorithm based on a combination of two components. One component of the control law is inspired by the heat partial differential equation (PDE) and results in the agents deploying between two anchor agents. The other component of the control law is based on extremum seeking and achieves higher vehicle density around the source. By using averaging theory for PDEs, we prove that the vehicle density will be highest around the source. We also quantify the density function of the agents' deployment position. By discretizing the model with respect to the continuous agent index, we obtain decentralized control laws for discrete agents and illustrate the theoretical results with simulations.