Model predictive control of nonlinear hybrid systems with discrete inputs employing a hybrid fuzzy model

The paper deals with model predictive control (MPC) of nonlinear hybrid systems with discrete inputs based on reachability analysis. In order to implement a MPC algorithm, a model of the process that we are dealing with is needed. In the paper, a hybrid fuzzy modelling approach is proposed. The hybrid system hierarchy is explained and the Takagi–Sugeno fuzzy formulation for hybrid fuzzy modelling purposes is tackled. An efficient method of identification of the hybrid fuzzy model is also discussed.

An algorithm that is–due to its MPC nature–suitable for controlling a wide spectrum of systems (provided that they have discrete inputs only) is presented.

The benefits of the algorithm employing a hybrid fuzzy model are verified on a batch reactor example. The results suggest that by suitably determining the cost function, satisfactory control can be attained, even when dealing with complex hybrid–nonlinear–stiff systems such as the batch reactor.

Finally, a comparison between MPC employing a hybrid linear model and a hybrid fuzzy model is carried out. It has been established that the latter approach clearly outperforms the approach where a linear model is used.     

The Core of a Locally Extended Affine Lie Algebra

Locally extended affine Lie algebras are a general version of extended affine Lie algebras. In this article, we completely describe the structure of the core of a locally extended affine Lie algebra. We prove that the core of a locally extended affine Lie algebra is a direct limit of Lie tori.     

Intensity-based framework for surrender modeling in life insurance

In this paper, we propose an intensity-based framework for surrender modeling. We model the surrender decision under the assumption of stochastic intensity and use, for comparative purposes, the affine models of Vasicek and Cox–Ingersoll–Ross for deriving closed-form solutions of the policyholder’s probability of surrendering the policy. The introduction of a closed-form solution is an innovative aspect of the model we propose. We evaluate the impact of dynamic policyholders’ behavior modeling the dependence between interest rates and surrendering (affine dependence) with the assumption that mortality rates are independent of interest rates and surrendering. Finally, using experience-based decrement tables for both surrendering and mortality, we explain the calibration procedure for deriving our model’s parameters and report numerical results in terms of best estimate of liabilities for life insurance under Solvency II.     

A spectral method for numerical modeling of radial microwave heating in cylindrical samples with temperature dependent dielectric properties

In this paper, we develop a numerical model based on spectral methods for the simulation of heat transfer due to radial irradiation microwave applied to samples in cylindrical geometry. We solve the Maxwell’s equations and the resulting electric field distribution is incorporated as a source term in the heat transfer equation. The model includes the temperature dependence of the dielectric properties. The numerical model is validated with experimental temperature data from literature.     

Fuzzy modelling of a moving grate biomass furnace for simulation and control purposes

Takagi–Sugeno (TS) fuzzy models are developed for a moving grate biomass furnace for the purpose of simulating and predicting the main process output variables, which are heat output, oxygen concentration of flue gas, and temperature of flue gas. Numerous approaches to modelling biomass furnaces have been proposed in the literature. Usually their objective is to simulate the furnace as accurately as possible. Hence, very complex model architectures are utilized which are not suited for applications like model predictive control. TS fuzzy models are able to approximate the global non-linear behaviour of a moving grate biomass furnace by interpolating between local linear, time-invariant models. The fuzzy partitions of the individual TS fuzzy models are constructed by an axis-orthogonal, incremental partitioning scheme. Validation results with measured process data demonstrate the excellent performance of the developed fuzzy models.     

Modeling the reactive processes within a catalytic porous medium

A one-dimensional modelling approach to the reactive processes within a heated homogeneously premixed fuel–air mixture in its passage through a non-adiabatic catalytically reactive porous medium is described. The main focus of this contribution was comparison of the results obtained while using different modeling approaches that include mass diffusion to solid pores versus neglecting it; single step reaction versus detailed kinetic simulation; adiabatic versus non-adiabatic reactor operation; two different approaches accounting for radiation heat transfer. This model was tailored to our experimental results so as to obtain original kinetic data for corresponding global reactions for different types of catalysts and validate at the same time the predictive approaches.     

On calibrated representations and the Plancherel Theorem for affine Hecke algebras

This paper has two main purposes. Firstly, we generalise Ram’s combinatorial construction of calibrated representations of the affine Hecke algebra to the multi-parameter case (including the non-reduced BC _n case). We then derive the Plancherel formulae for all rank 1 and rank 2 affine Hecke algebras, using our calibrated representations to construct all representations involved.     

Splitting schemes for hyperbolic heat conduction equation

Rapid processes of heat transfer are not described by the standard heat conduction equation. To take into account a finite velocity of heat transfer, we use the hyperbolic model of heat conduction, which is connected with the relaxation of heat fluxes. In this case, the mathematical model is based on a hyperbolic equation of second order or a system of equations for the temperature and heat fluxes. In this paper we construct for the hyperbolic heat conduction equation the additive schemes of splitting with respect to directions. Unconditional stability of locally one-dimensional splitting schemes is established. New splitting schemes are proposed and studied for a system of equations written in terms of the temperature and heat fluxes.     

A generalization of Lelieuvre’s formula to equiaffine immersions of general codimension

A non-degenerate equiaffine immersion of codimension one into an equiaffine space is locally expressed in terms of its conormal map and its affine fundamental form. The expression is called the Lelieuvre’s formula. We recently defined the notions of an equiaffine immersion of general codimension and its transversal volume element map. In this paper, we locally express a non-degenerate equiaffine immersion of general codimension into an equiaffine space in terms of its transversal volume element map and its affine fundamental form.     

主动冷却点阵夹层防热结构温度响应计算模型北大核心CSCD

针对点阵夹层结构主动热防护问题,建立了夹层结构面板和芯体导热与冷却剂对流耦合的非稳态传热理论模型,利用有限体积法离散控制方程并在MATLAB中进行了迭代求解.模型首次考虑了面板与夹芯杆之间的收缩热阻,并利用分离变量法得到了收缩热阻的近似解析解.基于单胞模型和周期性边界条件,模拟得到了模型所需的表面对流传热系数h_(b)和h_(fin).最后,选取多单胞计算工况进行数值模拟和理论模型对比,并讨论了收缩热阻对模型预测精度的影响.结果表明:理论模型能够准确预测夹层结构及内部流体的温度变化,理论与仿真之间的最大误差不超过1%;随着外加热流密度不断增大,忽略收缩热阻使得计算结果造成的误差不断增大;与数值模拟相比,理论模型可显著地减少计算时间并节省计算资源,尤其适用于非均匀、非稳态复杂热载荷下点阵夹层结构的温度响应计算.     

Modelling on fuzzy control systems

A kind of modelling method for fuzzy control systems is first proposed here, which is called modelling method based on fuzzy inference (MMFI). It should be regarded as the third modelling method that is different from two well-known modelling methods, that is, the first modelling method, mechanism modelling method (MMM), and the second modelling method, system identification modelling method (SIMM). This method can, based on the interpolation mechanism on fuzzy logic system, transfer a group of fuzzy inference rules describing a practice system into a kind of nonlinear differential equation with variable coefficients, called HX equations, so that the mathematical model of the system can be obtained. This means that we solve the difficult problem of how to get a model represented as differential equations on a complicated or fuzzy control system.     

Direct Unions of Lie Tori (Realization of Locally Extended Affine Lie Algebras)

In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A₁, B, C, and BC.     

A simulation code for investigating 2D heating of material bodies by low frequency electric fields

The finite difference method has been used to simultaneously solve in two dimensions Maxwell's equations and the heat transfer equation in forms which are appropriate to modelling low frequency electrical heating of solid materials. The nonlinear coupling of these modelling equations, which is due to temperature dependent electrical conductivities, necessitates the use of an explicit-sequential solution method and the limiting of the timestep size to ensure stability. The finite difference equations were modified to account for sharp electrical conductivity differences between different media in the body being heated.The simulation code was tested by comparison of the simulator predictions with the measured results of a physical scale model experiment. The simulation code was able to accurately predict the resistance between the electrodes used for heating, the energy deposition and the temperature rise in the bulk of the physical model.     

Numerical investigation of transient heat transfer to hydromagnetic channel flow with radiative heat and convective cooling

This present study consists of a numerical investigation of transient heat transfer in channel flow of an electrically conducting variable viscosity Boussinesq fluid in the presence of a magnetic field and thermal radiation. The temperature dependent nature of viscosity is assumed to follow an exponentially model and the system exchanges heat with the ambient following Newton’s law of cooling. The governing nonlinear equations of momentum and energy transport are solved numerically using a semi-implicit finite difference method. Solutions are presented in graphical form and given in terms of fluid velocity, fluid temperature, skin friction and heat transfer rate for various parametric values. Our results reveal that combined effect of thermal radiation, magnetic field, viscosity variation and convective cooling have significant impact in controlling the rate of heat transfer in the boundary layer region.     

Heat transfer in a Walter’s liquid B fluid over an impermeable stretching sheet with non-uniform heat source/sink and elastic deformation

In this present article an analysis is carried out to study the boundary layer flow behavior and heat transfer characteristics in Walter’s liquid B fluid flow. The stretching sheet is assumed to be impermeable, the effects of viscous dissipation, non-uniform heat source/sink in the presence and in the absence of elastic deformation (which was escaped from attention of researchers while formulating the viscoelastic boundary layer flow problems)on heat transfer are addressed. The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. Analytical solutions are obtained for the resulting boundary value problems. The effects of viscous dissipation, Prandtl number, Eckert number and non-uniform heat source/sink on heat transfer (in the presence and in the absence of elastic deformation) are shown in several plots and discussed. Analytical expressions for the wall frictional drag coefficient, non-dimensional wall temperature gradient and non-dimensional wall temperature are obtained and are tabulated for various values of the governing parameters. The present study reveals that, the presence of work done by deformation in the energy equation yields an augment in the fluid’s temperature.     

Coupled flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model

This letter presents a research for coupled flow and heat transfer of an upper-convected Maxwell fluid above a stretching plate with velocity slip boundary. Unlike most classical works, the new heat flux model, which is recently proposed by Christov, is employed. Analytical solutions are obtained by using the homotopy analysis method (HAM). The effects of elasticity number, slip coefficient, the relaxation time of the heat flux and the Prandtl number on velocity and temperature fields are analyzed. A comparison of Fourier’s Law and the Cattaneo–Christov heat flux model is also presented.     

A note on the rigid core of affine surfaces

During recent decades, $$\mathbb {G}_a$$-actions, especially certain invariants of $$\mathbb {G}_a$$-actions, have been important tools in the study of affine varieties. The $$\mathbb {G}_a$$-actions are usually studied through locally nilpotent derivations in characteristic zero and exponential maps (see Definition 1.1) in arbitrary characteristic. The “Makar-Limanov invariant” of locally nilpotent derivations played a pivotal role in solving the linearization conjecture in the 1990s, while invariants of exponential maps were central to N. Gupta’s resolution of the Zariski cancellation problem in positive characteristic. In the study of locally nilpotent derivations on commutative algebras containing $$\mathbb {Q}$$, Freudenburg and Moser-Jauslin (Mich Math J 62:227–258, (2013), Theorem 6.1) have introduced a new invariant called “rigid core” and used it to formulate an alternative version of Mason’s theorem and to prove a well-known analogue of Fermat’s last theorem for rational functions (Freudenburg and Moser-Jauslin (2013), Corollary 6.1). In this note, we consider the concept of the rigid core in the framework of exponential maps on commutative algebras over an algebraically closed field k of arbitrary characteristic. We observe that for any factorial k-domain B with $${\text {tr.deg}}_k(B)=2$$, the concept of rigid core coincides with the Makar-Limanov invariant. We also show that over any affine two-dimensional normal k-domain B, its rigid core is a stable invariant.     

Quantifying vagueness in decision models

When modelling specific decision situations the decision maker often feels overstrained when he is asked for precise numerical quotations concerning the objectives or the constraints, whereas qualitative statements are easily given. In the recent past the theory of fuzzy sets has proven to be very useful for representing this type of information. Though it is quite advanced formally, the practical determination of its core elements, i.e. membership functions and operators, has only been explored to a very limited extent. This paper presents results of empirical research which focused on the problem of modelling vagueness, i.e. determining membership functions of fuzzy sets which are considered as quantitative representations of vague concepts such as ‘young man’, ‘long sticks’, ‘high profits’, etc.     

蔬菜B2C直销中基于批配送模式的拣货方案序贯求解方法

针对蔬果类商品B2C直销模式下拣货与配送环节拣货量大、订单个性化强、时间性强及批配送等特点,基于相似订单成组拣货这一现实需求,引入成组作业思想,建立最小化拣货成本和配送成本之和的成批成组拣货序列优化模型;针对该模型多阶段决策、多决策变量及NP-难等特点,以降低求解维度和减少求解时间为目标,基于逆序决策思想,提出序贯求解方法,并给出了客户成批聚类、批次内相似订单成组及成批成组拣货序列优化求解算法;通过应用实例验证本文模型和算法的可行性和有效性。研究结果表明,本文方法得到的方案比成组拣货与配送独立决策,以及批配送但非成组方法大大缩减了拣货时间,为蔬果类商品网上直销企业生成拣货作业计划提供理论指导。     

Fuzzy-hybrid modelling of an Ackerman steered electric vehicle

Physical system modelling with known parameters together with 2-D or high order look-up tables (obtained from experimental data), have been the preferred method for simulating electric vehicles. The non-linear phenomena which are present at the vehicle tyre patch and ground interface have resulted in a quantitative understanding of this phenomena. However, nowadays, there is a requirement for a deeper understanding of the vehicle sub-models which previously used look-up tables. In this paper the hybrid modelling methodology used for electric vehicle systems offers a two-stage advantage: firstly, the vehicle model retains a comprehensive analytical formulation and secondly, the ‘fuzzy’ element offers, in addition to the quantitative results, a qualitative understanding of specific vehicle sub-models. In the literature several hybrid topologies are reported, sequential, auxiliary, and embedded.In this paper, the hybrid model topology selected is auxiliary and within the same hybrid model, the first paradigm used is the vehicle dynamics together with the actuator/gearbox system. The second paradigm is the non-linear fuzzy tyre model for each wheel. In particular, conventional physical system dynamic modelling has been combined with the fuzzy logic type-II or type-III methodology. The resulting hybrid-fuzzy tyre models were estimated for a-priori number of rules from experimental data. The physical system modelling required the available vehicle parameters such as the overall mass, wheel radius and chassis dimensions. The suggested synergetic fusion of the two methods, (hybrid-fuzzy), allowed the vehicle planar trajectories to be obtained prior to the hardware development of the entire vehicle. The strength of this methodology is that it requires localised system experimental data rather than global system data. The disadvantage in obtaining global experimental data is the requirement for comprehensive testing of a vehicle prototype which is both time consuming process and requires extensive resources. In this paper the authors have proposed the use of existing experimental rigs which are available from the leading automotive manufacturers. Hence, for the ‘hybrid’ modelling, localised data sets were used. In particular, wheel-tyre experimental data were obtained from the University tyre rig experimental facilities. Tyre forces acting on the tyre patch are mainly responsible for the overall electric vehicle motion. In addition, tyre measurement rigs are a well known method for obtaining localised data thus allowing the effective simulation of more detailed mathematical models. These include, firstly, physical system modelling (conventional vehicle dynamics), secondly, fuzzy type II or III modelling (for the tyre characteristics), and thirdly, electric drive modelling within the context of electric vehicles. The proposed hybrid model synthesis has resulted in simulation results which are similar to piece-wise ‘look-up’ table solutions. In addition, the strength of the ‘hybrid’ synthesis is that the analyst has a set of rules which clearly show the reasoning behind the complex development of the vehicle tyre forces. This is due to the inherent transparency of the type II and type III methodologies. Finally, the authors discussed the reasons for selecting a type-III framework. The paper concludes with a plethora of simulation results.