Algorithm of the resolving of a boundary-value problem for a nonlinear controlled system and its numerical modeling

An algorithm to construct a differentiable control function guaranteeing the transfer nonlinear stationary systems of ordinary differential equations from the initial state to a given final state of the phase space such that restrictions for the control are satisfied is proposed. The proposed algorithm is convenient for numerical implementation and is applicable to a broad class of systems. A sufficient condition of the existence of a desired transfer function is constructed. A certain practical problem is considered and simulated numerically by means of the presented method.     

Optimal control of a rotary crane

This paper is concerned with the optimal control of a rotary crane, which makes two kinds of motion (rotation and hoisting) at the same time. The optimal control which transfers a load to a desired place as fast as possible and minimizes the swing of the load during the transfer, as well as the swing at the end of transfer, is calculated on the basis of a dynamic model. A new computational technique is employed for computing the optimal control, and several numerical results are presented.The authors wish to thank Professor D. G. Hull and the reviewers for their valuable comments and suggestions.     

A method for solving a boundary problem for a nonlinear controlled system

An algorithm is proposed for constructing a control function that transfers a wide class of nonlinear systems of ordinary differential equations from an initial state to an arbitrarily small neighborhood of a given terminal state. The algorithm is convenient for numerical implementation. Taking into account the restrictions on the control and phase coordinates, a constructive criterion is obtained for choosing terminal states for which this transfer is possible. The problem of an interorbital flight is considered and modeled numerically.     

Method for solving a boundary problem for a nonlinear control system with incomplete information

An easy-to-implement numerical method is proposed for synthesizing a feedback control function that transfers a wide class of nonlinear stationary systems from an initial state to a given terminal state with allowance for measured data. A constructive criterion is obtained for choosing terminal states for which this transfer is guaranteed. The problem of an interorbital flight is considered and numerically simulated.     

Numerical modelling of unsteady convective–diffusive heat transfer with a control volume hybrid method

Presented in this paper is a numerical methodology for the solution of the parabolic governing partial differential equation that describes unsteady advection–diffusion heat transfer. The formulation presented here is shown to be free from the numerical oscillation commonly associated with advection–diffusion heat transfer regardless of the value of the Peclet number. The formulation involves the absorption of the advection term in the unsteady heat equation into the capacitance term. This process is achieved with the use of a control volume methodology applied to each nodal element on a finite-volume mesh. This is shown to ensure that spurious energy losses and gains are avoided and provides for consistency between temperature and energy change. This approach provides unconditional stability and it is shown that good accuracy is achievable with relatively large time-steps.     

Synthesis of discrete stabilization for a nonlinear stationary control system under incomplete information

The paper presents an algorithm for synthesizing discrete control functions in the case of incomplete information and discrete measurer. Under these control functions, solutions of a wide class of nonlinear stationary systems are transferred from the given initial state into an arbitrary neighborhood of the origin with due account taken of the constraints imposed on the control. Constructive criteria for the selection of the initial states and discretization step are derived, which ensure implementation of the suggested algorithm. The efficiency of the method is illustrated with numerical simulation of the problem of interorbital transfer.     

Iterative learning control for a class of nonlinear systems with random packet losses

This paper considers the problem of iterative learning control (ILC) for a class of nonlinear systems with random packet dropouts. It is assumed that an ILC scheme is implemented via a networked control system (NCS), and that during the packet transfer between the remote nonlinear plant and the ILC controller packet dropout occurs. A new formulation is employed to model the packet dropout case, where the random dropout rate is transformed into a stochastic parameter in the system’s representation. Through rigorous analysis, it is shown that under some given conditions, the iterative learning control can guarantee the convergence of the tracking error although some packets are missing. The analysis is also supported by a numerical example.     

Analysis of dual-phase-lag heat conduction in cylindrical system with a hybrid method

The dual-phase-lag heat transfer model is applied to investigate the transient heat conduction in an infinitely long solid cylinder for an exponentially decaying pulse boundary heat flux and for a short-pulse boundary heat flux. A hybrid application of the Laplace transform method and the control volume scheme is used to obtain the numerical solutions. Comparison between the numerical results and the analytic solution for an exponentially decaying heat flux pulse evidences the accuracy of the present numerical results. Results further show that the present numerical scheme can overcome the mathematical difficulties to analyze such problems. Effects of the thermal lag ratio τ_q/τ_T, the shift time τ_q–τ_T, the function form of heating pulse, and geometry of medium on the behavior of heat transfer are investigated.     

Non-Newtonian phase-change heat transfer of nano-enhanced octadecane with mesoporous silica particles in a tilted enclosure using a deformed mesh technique

In the present study, the melting phase-change heat transfer of nano-enhanced phase-change octadecane by using mesoporous silica particles is investigated in an inclined cavity, theoretically. The presence of mesoporous silica particles induces non-Newtonian effects in the molten octadecane. A phase-change interface-tracking approach, deformed mesh technique, is employed to track the phase-change interface and heat transfer in the cavity. The Arbitrary Lagrangian-Eulerian (ALE) moving mesh technique along with the finite element method is adopted to solve the governing equations for conservation of mass, momentum, and energy during the phase-change process. A re-meshing technique and an automatic time step control approach are employed to control the quality of the deformed mesh and the computed numerical solution. The effect of various mass fractions of nanoparticles and various inclination angles of the enclosure on the heat transfer and phase-change behavior of nano-enhanced octadecane are addressed. The outcome reveals that using the mesoporous silica particles diminish the heat transfer in the enclosure. Although the presence of nanoparticles improved the conductive heat transfer, a reduction in the phase-change heat transfer performance of the enclosure can be observed, which is due to the increase of the viscosity (consistency parameter) of the liquid and suppression of natural convective flows. Moreover, the presence of nanoparticles reduces the latent heat capacity of octadecane as they do not contribute to the phase-change energy storage. Dispersing 5% mass fraction of nanoparticles in octadecane can reduce the heat transfer up to 50% and increase the consistency parameter by three folds. The angle of inclination of the cavity also plays an important role in the heat transfer characteristics. Tilting the cavity by -75° leads to an 80% reduction in the heat transfer.     

Parametric optimization of a four-link close-chain manipulator with active and passive actuators

We investigate the problem of optimization of motion laws and design parameters of a four-link manipulator with a closed-chain kinematic structure. The manipulator performs cyclic transfer operations in a horizontal plane under the action of active and passive (springs and dampers) actuators. As a minimization criterion, we take a quadratic (with respect to control moments of forces) functional. An algorithm is proposed for constructing a suboptimal solution of the formulated problem based on parametrization of the generalized coordinates of the manipulator with a family of given functions and on the use of numerical procedures of mathematical programming.     

The role of data transfer on the selection of a single vs. multiple mesh architecture for tightly coupled multiphysics applications

Data transfer from one mesh to another may be necessary in a number of situations including spatial adaptation, remeshing, arbitrary Lagrangian-Eulerian (ALE), and multiphysics simulation. Data transfer has the potential to introduce error into a simulation; the magnitude and impact of which depends on the application, transfer scenario, and the algorithm used to perform the data transfer. During a transient simulation, data transfer may occur many times, with the potential of error accumulation at each transfer. This paper examines data transfer error and its impact on a set of simple multiphysics problems. Data transfer error is examined using analytical functions to compare schemes based on interpolation, area-weighted averaging, and L² minimization. An example error analysis is performed to illustrate data transfer error and behavior for a simple problem. Data transfer error is also investigated for a one-dimensional time-dependent system of partial differential equations. This study concludes that data transfer error can be significant in coupled multiphysics systems. These numerical experiments suggest that error is a function of data transfer scheme, and characteristics of the field data and mesh. If there are significant differences in the meshes in a multiple mesh simulation, this study suggests that data transfer may lead to error and instability if care is not taken. Further, this work motivates that data transfer error should be included in the estimation of numerical error when data transfer is employed in a simulation.     

An optimal control problem for a parabolic equation in the class of smooth controls

We consider an optimal control problem for a parabolic equation with a differential constraint on the boundary. We study this problem in the class of smooth controls satisfying certain pointwise constraints. Such problems describe mass transfer processes in column-type apparatuses, taking into account the longitudinal mixing. Control functions in these problems represent flows of raw materials or finished products. For the problem under consideration we obtain a necessary optimality condition, propose a method for improving admissible controls, and carry out the numerical experiment.     

Verified reduction of a model for a continuous casting process

The casting of metals is known to involve the complex interaction of turbulent momentum and heat transfer in the presence of solidification, and it is believed that computational fluid dynamical (CFD) techniques are required to model it correctly. Here, using asymptotic methods, we demonstrate that the key quantities obtained in an earlier CFD model for a particular continuous casting process – ostensibly for a pure metal, but equally for an alloy of eutectic composition – can be recovered using a much simpler model that takes into account just the heat transfer, requiring the numerical solution of a two-phase Stefan problem. Combining this with a more recent asymptotic thermomechanical model for the same continuous casting process, we postulate that it should be possible, with the additional help of algebraic manipulation, to reduce a model that takes into account turbulent momentum and heat transfer in the melt and the thermomechanics in the solid shell to one formulated in terms of only heat transfer, without adversely affecting model predictions.     

SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

Optimal boundary control of glass cooling processes

In this paper, an optimal control problem for glass cooling processes is studied. We model glass cooling using the SP1 approximations to the radiative heat transfer equations. The control variable is the temperature at the boundary of the domain. This results in a boundary control problem for a parabolic/elliptic system which is treated by a constrained optimization approach. We consider several cost functionals of tracking‐type and formally derive the first‐order optimality system. Several numerical methods based on the adjoint variables are investigated. We present results of numerical simulations illustrating the feasibility and performance of the different approaches. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical Method for Damping String Vibrations with Unknown Initial State in the Class of Weak Generalized Solutions

The problem of positional boundary control is considered for the spatially one-dimensional wave equation. The objective of control is to transfer the system from an unknown initial state into the state of rest in finite time. A specific feature of the statement of the problem is the weakening of the requirements on the regularity of generalized solutions, observations, and control. The smoothing procedure is used to transfer the problem into the class of strong generalized solutions, where the method previously developed by the authors can be used. The procedure is mathematically justified, and the corresponding results of numerical experiments are given.     

Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux

In this paper, a powerful analytical method, called homotopy analysis method (HAM) is used to obtain the analytical solution for a nonlinear ordinary deferential equation that often appear in boundary layers problems arising in heat and mass transfer which these kinds of the equations contain infinity boundary condition. The boundary layer approximations of fluid flow and heat transfer of vertical full cone embedded in porous media give us the similarity solution for full cone subjected to surface heat flux boundary conditions. Nonlinear ODE which is obtained by similarity solution has been solved through homotopy analysis method (HAM). The main objective is to propose alternative methods of solution, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The obtained analytical solution in comparison with the numerical ones represents a remarkable accuracy. The results also indicate that HAM can provide us with a convenient way to control and adjust the convergence region.     

On a Double Porosity Model of Fractured-Porous Reservoirs Based on a Hybrid Flow Function

In this paper, a model of double porosity for a fractured porous medium using a combination of classical and gradient functions of mass transfer between the cracks and porous blocks in a weakly compressible single-phase fluid flow is considered. As compared to the wellknown models, the model with such a mass transfer function allows one to take into account the anisotropic properties of filtration in a more general form. The results of numerical tests for two- and three-dimensional model problems are presented. The computational algorithm is based on a finite element approximation with respect to space and a completely implicit approximation with respect to time.     

Numerical study of a mathematical model of a catalytic fuel processor with cocurrent oxidation and conversion flows

The results are presented of the numerical study of a mathematical model in the form of a nonlinear boundary value problem describing the stationary regimes in a catalytic fuel processor. We study a two-dimensional model for the endoblock, with the longitudinal heat and mass transfer by the gas and the transversal heat conductivity along the catalyst in the two-temperature approximation. For the exochannel, a model is considered with the longitudinal heat and mass transfer by the gas flow and the longitudinal heat transfer along the catalytic wall. These two blocks are related to each other through the equality of the temperature and heat flux on the boundary. The results obtained are in good agreement with experimental data.     

On the homotopy analysis method for solving a particle transport equation

The purpose of this paper consists in the finding of the solution for a stationary neutron transport equation that is accompanied by the homogeneous boundary conditions, using the techniques of homotopy analysis method (HAM) and a numerical integration formula. Also, algorithm presented can be used for solving the integral–differential equations in which the unknown function depends on two variables, such as a radiative transfer equation. Results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.