Exact slow–fast decomposition of the nonlinear singularly perturbed optimal control problem

We study the infinite horizon nonlinear quadratic optimal control problem for a singularly perturbed system, which is nonlinear in both, the slow and the fast variables. It is known that the optimal controller for such problem can be designed by finding a special invariant manifold of the corresponding Hamiltonian system. We obtain exact slow–fast decomposition of the Hamiltonian system and of the special invariant manifold into the slow and the fast ones. On the basis of this decomposition we construct high-order asymptotic approximations of the optimal state-feedback and optimal trajectory.     

Distributed Resource Allocation via Accelerated Saddle Point Dynamics

In this paper, accelerated saddle point dynamics is proposed for distributed resource allocation over a multi-agent network, which enables a hyper-exponential convergence rate. Specifically, an inertial fast-slow dynamical system with vanishing damping is introduced, based on which the distributed saddle point algorithm is designed. The dual variables are updated in two time scales, i.e., the fast manifold and the slow manifold. In the fast manifold, the consensus of the Lagrangian multipliers and the tracking of the constraints are pursued by the consensus protocol. In the slow manifold, the updating of the Lagrangian multipliers is accelerated by inertial terms. Hyper-exponential stability is defined to characterize a faster convergence of our proposed algorithm in comparison with conventional primal-dual algorithms for distributed resource allocation. The simulation of the application in the energy dispatch problem verifies the result, which demonstrates the fast convergence of the proposed saddle point dynamics.      

A method for solving systems of non-linear differential equations with moving singularities

We present a method for solving a class of initial valued, coupled, non-linear differential equations with ‘moving singularities’ subject to some subsidiary conditions. We show that these types of singularities can be adequately treated by establishing certain ‘moving’ jump conditions across them. We show how a first integral of the differential equations, if available, can also be used for checking the accuracy of the numerical solution.     

A customer-oriented organisational diagnostic model based on data mining of customer-complaint databases

The purpose of this paper is to develop a customer-oriented organisational diagnostic model, ‘PARA’ model, based on data mining of customer-complaint databases. The proposed ‘PARA’ model, which is designed to diagnose and correct service failures, takes its name from the initial letters of the four analytical stages of the model: (i) ‘primary diagnosis’; (ii) ‘advanced diagnosis’; (iii) ‘review’; and (iv) ‘action’. In the primary-diagnosis stage, the customer-complaint database is comprehensively analysed to identify themes and categories of complaints. In the advanced-diagnosis stage, a data-mining technique is employed to investigate the relationship between the categories of customer complaints and the deficiencies of the service system. In the review stage, the identified weaknesses of the service system are reviewed and awareness of these weaknesses is enhanced among the organisation’s employees. In the action stage, a strategy of action plans for improvement is developed. An empirical case study is conducted to demonstrate the practical efficacy of the ‘PARA’ model. The paper concludes by summarising the advantages of the proposed model and the implications for future research.     

Adaptive and nonadaptive Hermitian operator methods for combustion phenomena

Adaptive and nonadaptive, three-point, fourth-order accurate, compact or Hermitian operator methods are developed and used to study one-dimensional combustion phenomena. The nonadaptive Hermitian operator methods are based on the time linearization of the nonlinear partial differential equations, and employ an approximate factorization technique to reduce a three-dimensional reaction-diffusion operator to a sequence of three one-dimensional, linear, second-order differential operators in space. The three adaptive Hermitian operator techniques presented in this paper are based on the equidistribution of the arc length of the vector of dependent variables and use a subequidistribution principle to obtain smooth grids. The first adaptive technique uses quasilinearization and yields a block tridiagonal matrix for the values of the dependent variables at each iteration. The second technique employs partial quasilinearization and yields a system of uncoupled, linear algebraic equations for each dependent variable at each iteration. The third technique employs a predictor-corrector method to predict the grid point locations and a time linearization procedure to obtain the values of the dependent variables. It is shown that the efficiency and accuracy of adaptive Hermitian operator methods depend on the time step and number of grid points used in the calculations. It is also shown that adaptive methods which use a Crank-Nicolson scheme in time may yield oscillatory solutions, and that nonadaptive Hermitian operator methods require a much larger number of grid points than nonadaptive techniques if the solution of the governing equations is characterized by fast ignition phenomena and/or steep, fast moving flame fronts.     

On a model of three-dimensional bursting and its parallel implementation

A mathematical model for the simulation of three-dimensional bursting phenomena and its parallel implementation are presented. The model consists of four nonlinearly coupled partial differential equations that include fast and slow variables, and exhibits bursting in the absence of diffusion. The differential equations have been discretized by means of a second-order accurate in both space and time, linearly-implicit finite difference method in equally-spaced grids. The resulting system of linear algebraic equations at each time level has been solved by means of the Preconditioned Conjugate Gradient (PCG) method. Three different parallel implementations of the proposed mathematical model have been developed; two of these implementations, i.e., the MPI and the PETSc codes, are based on a message passing paradigm, while the third one, i.e., the OpenMP code, is based on a shared space address paradigm. These three implementations are evaluated on two current high performance parallel architectures, i.e., a dual-processor cluster and a Shared Distributed Memory (SDM) system. A novel representation of the results that emphasizes the most relevant factors that affect the performance of the paralled implementations, is proposed. The comparative analysis of the computational results shows that the MPI and the OpenMP implementations are about twice more efficient than the PETSc code on the SDM system. It is also shown that, for the conditions reported here, the nonlinear dynamics of the three-dimensional bursting phenomena exhibits three stages characterized by asynchronous, synchronous and then asynchronous oscillations, before a quiescent state is reached. It is also shown that the fast system reaches steady state in much less time than the slow variables.     

Parallel implementation of fast elliptic solver

A fast elliptic solver for separable elliptic equations on rectangular domains is considered. The method is referred to as FASV (fast algorithm for separation of variables) and is based on the odd-even block elimination technique in combination with the method for discrete separation of variables. The algorithm is connected with solving systems of algebraic equations with sparsity whose right-hand sides have only a few nonzero block components. The method is effective and stable by construction. Only a few of the block solution components are needed and hence these problems might be solved incompletely. Parallel implementation of the method proposed using the public domain PVM software is described in terms of decomposition of the original rectangular domain into a number of strips. Numerical results for a model problem on a cluster of a few IBM workstations are reported.     

Stereotype for control-burner relationship of four-burner ranges for Koreans

This study aims to investigate the population stereotype of control-burner arrangement of four-burner gas ranges for Koreans. A paper–pencil test and a real gas range experiment were adopted as methods of this research. In the paper–pencil test, 355 participants, including 116 females and 239 males, participated. The participants were shown a drawing of a four-burner gas range, in which each burner was labeled with meaningless symbols such as ‘☆’, ‘#’, ‘Δ’ and ‘□’. The participants were asked to indicate the corresponding burner to each control knob. Type III (a type of control-burner linkages, refer to Fig. 1) was most frequently chosen, followed by Type IV, Type V and Type II. In the real gas range experiment, 32 college students, consisting of 12 females and 20 males, volunteered to participate. Two independent variables of reaction time and error rate were used. The participants were requested to turn off the light on the burner by pushing the correct control as quickly as possible. The results revealed that although the linkage types were not significant on the reaction time and error rate (p > 0.20), both variables of Type III were the best. Hence, the most preferred arrangement of control-burner linkages for Koreans is Type III, irrespective of the three dependent variables such as proportion of responses, reaction time and error rate, which is different from that of Americans and Chinese.     

Decomposition of problems of optimal control and estimation for discrete systems with fast and slow variables

Consideration was given to the linear-quadratic problem of optimal control for the discrete linear system with fast and slow variables under incomplete information about system state. Decomposition of the discrete matrix Riccati equations was carried out. The proposed decomposition algorithm relies on a geometrical approach using the properties of the invariant manifolds of slow and fast motions of the nonlinear multirate discrete systems as basis. The splitting transformation was constructed in the form of asymptotic decomposition in the degrees of a small parameter.     

Exact slow-fast decomposition of a class of non-linear singularly perturbed optimal control problems via invariant manifolds

We study a Hamilton-Jacobi partial differential equation, arising in an optimal control problem for an affine non-linear singularly perturbed system. This equation is solvable iff there exists a special invariant manifold of the corresponding Hamiltonian system. We obtain exact slow-fast decomposition of the Hamiltonian system and of the special invariant manifold into slow and fast components. We get sufficient conditions for the solvability of the Hamiltonian-Jacobi equation in terms of the reduced-order slow submanifold, or, in the hyperbolic case, in terms of a reduced-order slow Riccati equation. On the basis of this decomposition we construct asymptotic expansions of the optimal state-feedback, optimal trajectory and optimal open-loop control in powers of a small parameter.     

H∞-norm and invariant manifolds of systems with state delays

The problem of finding bounds on the H_∞-norm of systems with a finite number of point delays and distributed delay is considered. Sufficient conditions for the system to possess an H_∞-norm which is less or equal to a prescribed bound are obtained in terms of Riccati partial differential equations (RPDE’s). We show that the existence of a solution to the RPDE’s is equivalent to the existence of a stable manifold of the associated Hamiltonian system. For small delays the existence of the stable manifold is equivalent to the existence of a stable manifold of the ordinary differential equations that govern the flow on the slow manifold of the Hamiltonian system. This leads to an algebraic, finite-dimensional, criterion for systems with small delays.     

Decomposition of a linear-quadratic optimal control problem with fast and slow variables

A linear-quadratic optimal control problem for a discrete different time-scale system is studied. The decomposition of the boundary value problem for the maximum principle is based on the geometric approach using the properties of invariant manifolds of slow and fast motions. This approach aids in constructing a transformation for reducing the initial problem to a boundary-value problem for slow variables and two initial-value problems for fast variables. The transformation is expressed as an asymptotic siries in powers of a small parameter.     

Area decomposition for electromechanical models of power systems

Coherency and time scale properties of power system models are shown to be related by the dichotomic solution of a matrix Riccati equation. A grouping algorithm is proposed which reduces the area decomposition problem to obtaining a basis for the slow subsystem and performing a Gaussian elimination. Since the slow coherency measure allows for a lack of coherency in fast parts of machine transients, the resulting area decomposition is independent of fault locations. The procedure is illustrated by a 16-machine example.     

Time-relevant stability of 2D systems

For many 2D systems, one of the independent variables plays a distinct role in the evolution of the trajectories; since often this special independent variable is time, we call such systems ‘time-relevant’. In this paper, we introduce a stability notion for time-relevant systems described by higher-order difference equations. We give algebraic tests in terms of the location of the zeros of the determinant of a polynomial matrix describing the system. We also give an LMI characterization of time-relevant stability involving only constant matrices.     

Learning and approximate inference in dynamic hierarchical models

A new variant of the dynamic hierarchical model (DHM) that describes a large number of parallel time series is presented. The separate series, which may be interdependent, are modeled through dynamic linear models (DLMs). This interdependence is included in the model through the definition of a ‘top-level’ or ‘average’ DLM. The model features explicit dependences between the latent states of the parallel DLMs and the states of the average model, and thus the many parallel time series are linked to each other. The combination of dependences within each time series and dependences between the different DLMs makes the computation time that is required for exact inference cubic in the number of parallel time series, however, which is unacceptable for practical tasks that involve large numbers of parallel time series. Therefore, two methods for fast, approximate inference are proposed: a variational approximation and a factorial approach. Under these approximations, inference can be performed in linear time, and it still features exact means. Learning is implemented through a maximum likelihood (ML) estimation of the model parameters. This estimation is realized through an expectation maximization (EM) algorithm with approximate inference in the E-step. Examples of learning and forecasting on two data sets show that the addition of direct dependences has a ‘smoothing’ effect on the evolution of the states of the individual time series, and leads to better prediction results. The use of approximate instead of exact inference is further shown not to lead to inferior results on either data set.     

Numerical simulation of the whispering gallery modes in prolate spheroids

In this paper, we discuss the progress in the numerical simulation of the so-called ‘whispering gallery’ modes (WGMs) occurring inside a prolate spheroidal cavity. These modes are mainly concentrated in a narrow domain along the equatorial line of a spheroid and they are famous because of their extremely high quality factor. The scalar Helmholtz equation provides a sufficient accuracy for WGM simulation and (in a contrary to its vector version) is separable in spheroidal coordinates. However, the numerical simulation of ‘whispering gallery’ phenomena is not straightforward. The separation of variables yields two spheroidal wave ordinary differential equations (ODEs), first only depending on the angular, second on the radial coordinate. Though separated, these equations remain coupled through the separation constant and the eigenfrequency, so that together with the boundary conditions they form a singular self-adjoint two-parameter Sturm–Liouville problem.     

An averaging approach to chattering

The singularly perturbed relay control systems (SPRCS) as mathematical models of chattering in the small neighborhood of the switching surface in sliding mode systems are examined. Sufficient conditions for existence and stability of fast periodic solutions to the SPRCS are found. It is shown that the slow motions in such SPRCS are approximately described by equations derived from equations for the slow variables of SPRCS by averaging along fast periodic motions. It is shown that In the general case, when the equations of a plant contain relay control nonlinearly, the averaged equations do not coincide with the equivalent control equations or with the Filippov's definition (1988) for the sliding motions in the reduced system; however, in the linear case, they coincide     

MINLIP for the identification of monotone Wiener systems

This paper studies the MINLIP estimator for the identification of Wiener systems consisting of a sequence of a linear FIR dynamical model, and a monotonically increasing (or decreasing) static function. Given T observations, this algorithm boils down to solving a convex quadratic program with O(T) variables and inequality constraints, implementing an inference technique which is based entirely on model complexity control, a technique which is dubbed as Einstein’s razor.² This technique is proven to yield, here, almost consistent estimates in case the given data is suitably rich (local Persistently Exciting of sufficient order), and if the samples obey a ‘true’ monotone Wiener system satisfying some regularity conditions. It is then indicated how to extend the method in order to cope with noisy data, and empirical evidence is given in order to support the claim of efficiency.     

Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems

A new efficient optimization method, called ‘Teaching–Learning-Based Optimization (TLBO)’, is proposed in this paper for the optimization of mechanical design problems. This method works on the effect of influence of a teacher on learners. Like other nature-inspired algorithms, TLBO is also a population-based method and uses a population of solutions to proceed to the global solution. The population is considered as a group of learners or a class of learners. The process of TLBO is divided into two parts: the first part consists of the ‘Teacher Phase’ and the second part consists of the ‘Learner Phase’. ‘Teacher Phase’ means learning from the teacher and ‘Learner Phase’ means learning by the interaction between learners. The basic philosophy of the TLBO method is explained in detail. To check the effectiveness of the method it is tested on five different constrained benchmark test functions with different characteristics, four different benchmark mechanical design problems and six mechanical design optimization problems which have real world applications. The effectiveness of the TLBO method is compared with the other population-based optimization algorithms based on the best solution, average solution, convergence rate and computational effort. Results show that TLBO is more effective and efficient than the other optimization methods for the mechanical design optimization problems considered. This novel optimization method can be easily extended to other engineering design optimization problems.     

Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.