Nonlinear Cost Function Research Articles

Estimation fusion of nonlinear cost functions with application to multisensory Kalman filtering

This paper focuses on four fusion algorithms for the estimation of nonlinear cost function (NCF) in a multisensory environment. In multisensory filtering and control problems, NCF represents a nonlinear multivariate functional of state variables, which can indicate useful information of the target systems for automatic control. To estimate the NCF using multisensory information, we propose one centralized and three decentralized estimation fusion algorithms. For multivariate polynomial NCFs, we propose a simple closed-form computation procedure. For general NCFs, the most popular procedure for the evaluation of their estimates is based on the unscented transformation. The effectiveness and estimation accuracy of the proposed fusion algorithms are demonstrated with theoretical and numerical examples.

An interval type-2 fuzzy logic system-based method for prediction interval construction

This paper introduces a new non-parametric method for uncertainty quantification through construction of prediction intervals (PIs). The method takes the left and right end points of the type-reduced set of an interval type-2 fuzzy logic system (IT2FLS) model as the lower and upper bounds of a PI. No assumption is made in regard to the data distribution, behaviour, and patterns when developing intervals. A training method is proposed to link the confidence level (CL) concept of PIs to the intervals generated by IT2FLS models. The new PI-based training algorithm not only ensures that PIs constructed using IT2FLS models satisfy the CL requirements, but also reduces widths of PIs and generates practically informative PIs. Proper adjustment of parameters of IT2FLSs is performed through the minimization of a PI-based objective function. A metaheuristic method is applied for minimization of the non-linear non-differentiable cost function. Performance of the proposed method is examined for seven synthetic and real world benchmark case studies with homogenous and heterogeneous noise. The demonstrated results indicate that the proposed method is capable of generating high quality PIs. Comparative studies also show that the performance of the proposed method is equal to or better than traditional neural network-based methods for construction of PIs in more than 90% of cases. The superiority is more evident for the case of data with a heterogeneous noise.

MIP-Based Stochastic Security-Constrained Daily Hydrothermal Generation Scheduling

This paper presents the application of a mixed-integer programming (MIP) approach for solving stochastic security-constrained daily hydrothermal generation scheduling (SCDHGS). Power system uncertainties including generating units and branch contingencies and load uncertainty are explicitly considered in the stochastic programming of SCDHGS. The roulette wheel mechanism and lattice Monte Carlo simulation (LMCS) are first employed for random scenario generation wherein the stochastic SCDHGS procedure is converted into its respective deterministic equivalents (scenarios). Then, the generating units are scheduled through MIP over the set of deterministic scenarios for the purpose of minimizing the cost of supplying energy and ancillary services over the optimization horizon (24 h) while satisfying all the operating and network security constraints. To a more realistic modeling of the DHGS problem, in the proposed MIP formulation, the nonlinear valve loading effect, cost, and emission function are modeled in linear form, and prohibited operating zones (POZs) of thermal units are considered. Furthermore, a dynamic ramp rate of thermal units is used, and for the hydro plants, the multiperformance curve with spillage and time delay between reservoirs is considered. To assess the efficiency and powerful performance of the aforementioned method, a typical case study based on the standard IEEE-118 bus system is investigated, and the results are compared to each other in different test systems.

Robust method for camera self-calibration by an unkown planar scene

In this paper we present a self-calibration method for a CCD camera with varying intrinsic parameters based on an unknown planar scene. The advantage of our method is reducing the number of images (two images) needed to estimate the parameters of the camera used. Moreover, self-calibration equations are related to the number of points matched (very numerous and easy to detect) rather than to the number of images, since the use of a large number of images requires high computation time. On the other hand, we base on the points matched, which are numerous, when estimating the projection matrices and homographies between the images. The latter are used with the images of the absolute conic to formulate a system of non-linear equations (self-calibration equations depend on the number of matched pairs). Finally, the intrinsic parameters of the camera can be obtained by minimizing a non-linear cost function in a two-step procedure: initialization and optimization. Experiment results show the robustness of our algorithms in terms of stability and convergence.

Joint optimisation of price, warranty and recovery planning in remanufacturing of used products under linear and non-linear demand, return and cost functions

We investigate joint optimisation of remanufacturing, pricing and warranty decision-making for end-of-life products. A novel mathematical–statistical model is proposed where decisions involve pricing of returned used products (cores), degree of their remanufacturing, selling price and the warranty period for the final remanufactured products. The virtual age reliability improvement approach is chosen to model the upgrading of the cores to higher quality levels. We consider price- and warranty-dependent demand, price- and age-dependent return, and age-dependent remanufacturing cost in the model development. Both linear and non-linear forms of these functions are investigated. First, under some restrictive conditions of upgrade level and age distribution of received cores, special cases of the problem, which can be solved using a recently developed non-linear optimisation solver, are presented. We also implement a particle swarm optimisation algorithm for the solution of the original problem when all the restrictive assumptions are dropped. Finally, numerical experiments and sensitivity analysis are presented to address different aspects of the model and the solution approaches.

On Volterra functional operator games on a given set

The paper is devoted to finding sufficient conditions for an ?-equilibrium in the sense of piecewise-program strategies in antagonistic games associated with nonlinear controlled functional operator equations and cost functionals of a rather general form. The concept of piecewise-program strategies is defined based on the concept of Volterra set chain for operators involved in equations controlled by opponent players. Reduction of distributed control systems to equations of this type is illustrated by examples.

An Efficient Non-Linear Cost Compression Algorithm for Multi Level Cell Memory

This paper defines a non-linear cost compression problem, proposes an efficient algorithm, and applies it to a real application of multi level cell memory to minimize energy consumption and latency. The non-linear cost compression problem extends the traditional cost compression problem to allow a non-linear cost function of symbol frequencies, while it is a weighted linear combination of symbol frequencies in the cost compression problem. In order to solve the non-linear cost compression problem efficiently, we propose an encoding symbol frequency based approach. We first compute frequencies of encoding symbols to minimize a cost function. To achieve the computed frequencies of a cost-compressed message, we deploy existing size-decompression algorithms. The proposed algorithm is optimal and as fast as the existing size compression algorithms. Our experimental results show that it reduces the energy consumption and latency by 70 percent for a text file in multi level cell memory. Furthermore, it increases the lifetime of endurance limited memory.

Determining the optimum production quantity in three-echelon production system with stochastic demand

This paper considers a three-echelon integrated production inventory model consisting of a central warehouse and a manufacturer including two independent departments as processing and assembly stages. To assemble the finished product, materials are carried to the assembly stage through two different flow channels. One group requires preprocessing in the processing stage, and the other one directly arrives from an outside supplier. Storing inventories either in a warehouse (as finished products) or in the stages (as work-in-process) and shipping them between the stages incur inventory costs that must be balanced to achieve minimum joint total cost. The central warehouse faces stochastic demand, which is assumed to be a generally distributed demand. It is controlled by continuous review (R,Q) policy. Additionally, warehouse ordering cost can be reduced through further investment. To analyze, we formulate a nonlinear cost function to aggregate all the costs. After, a simulated annealing algorithm has been suggested to solve the problem. Numerical experiments are presented to evaluate the performance and effectiveness of the proposed algorithm. Also, we use the branch and bound technique and a nonlinear optimization technique—generalized reduced gradient—for solving this problem. The experimental results show a fine performance of the proposed algorithm in comparison with two other methods.

The Optimal Design of HTS Devices

In the design of high-temperature superconductor (HTS)-based electromagnetic devices, some of the major challenges include ac loss reduction, minimization of heat leakage, and reduction of the amount of HTS material used to decrease costs. This paper considers a computer model of HTS-based leads involving a multiphysics scenario that considers the electromagnetic and thermal behavior of the system. This paper attempts to address the requirements mentioned and provides an optimum solution by applying an approach based on multiobjective optimization. The differential evolution technique, a stochastic method, widely known for the ease of use and its ability to handle adequately non-differentiable, non-linear, and multimodal cost functions was used. The proposed framework provides a technique to optimize effectively HTS leads, which not only deals with the non-linear aspect of HTS materials, but also includes a multiphysics environment.

Multisensor Estimation Fusion of Nonlinear Cost Functions in Mixed Continuous-Discrete Stochastic Systems

We propose centralized and distributed fusion algorithms for estimation of nonlinear cost function (NCF) in multisensory mixed continuous-discrete stochastic systems. The NCF represents a nonlinear multivariate functional of state variables. For polynomial NCFs, we propose a closed-form estimation procedure based on recursive formulas for high-order moments for a multivariate normal distribution. In general case, the unscented transformation is used for calculation of nonlinear estimates of a cost functions. To fuse local state estimates, the mixed differential difference equations for error cross-covariance between local estimates are derived. The subsequent application of the proposed fusion estimators for a multisensory environment demonstrates their effectiveness.

Dynamics Analysis of Game and Chaotic Control in the Chinese Fixed Broadband Telecom Market

This paper considers a dynamic duopoly Cournot model based on nonlinear cost functions. The model with heterogeneous players and the spillover effect is applied to the Chinese fixed broadband telecom market. We have studied its dynamic game process. The existence and stability of the Nash equilibrium of the system have been discussed. Simulations are used to show the complex dynamical behaviors of the system. The results illustrate that altering the relevant parameters of system can affect the stability of the Nash equilibrium point and cause chaos to occur. With the use of the delay feedback control method, the chaotic behavior of the model has been stabilized at the Nash equilibrium point. The analysis and results will be of great importance for the Chinese fixed broadband telecom market.

Optimistic Planning for the Near-Optimal Control of General Nonlinear Systems with Continuous Transition Distributions

Optimistic planning is an optimal control approach from artificial intelligence, which can be applied in receding horizon. It works for very general nonlinear dynamics and cost functions, and its analysis establishes a tight relationship between computation invested and near-optimality. However, there is no optimistic planning algorithm that searches for closed-loop solutions in stochastic problems with continuous transition distributions. Such transitions are essential in control, where they arise e.g. due to continuous disturbances. Existing algorithms only search for open-loop input sequences, which are suboptimal. We therefore propose a closed-loop algorithm that discretizes the continuous transition distribution into sigma points, and call it sigma-optimistic planning. Assuming the error introduced by sigma-point discretization is bounded, we analyze the solution returned, showing that it is near-optimal. The algorithm is evaluated in simulation experiments, where it performs better than a state-of-the-art open-loop planning technique; a certainty-equivalence approach also works well.

The Chaos Dynamic of Multiproduct Cournot Duopoly Game with Managerial Delegation

Although oligopoly theory is generally concerned with the single-product firm, what is true in the real word is that most of the firms offer multiproducts rather than single products in order to obtain cost-saving advantages, cater for the diversity of consumer tastes, and provide a barrier to entry. We develop a dynamical multiproduct Cournot duopoly model in discrete time, where each firm has an owner who delegates the output decision to a manager. The principle of decision-making is bounded rational. And each firm has a nonlinear total cost function due to the multiproduct framework. The Cournot Nash equilibrium and the local stability are investigated. The tangential bifurcation and intermittent chaos are reported by numerical simulations. The results show that high output adjustment speed can lead to output fluctuations which are characterized by phases of low volatility with small output changes and phases of high volatility with large output changes. The intermittent route to chaos of Flip bifurcation and another intermittent route of Flip bifurcation which contains Hopf bifurcation can exist in the system. The study can improve our understanding of intermittent chaos frequently observed in oligopoly economy.

A New modified particle swarm Optimization (PSO) Technique for Non-Convex Economic Dispatch

This paper gives a new modified technique for evaluating non-convex economic dispatch (NCED) problem having non-linear, discontinuous cost function. This new particle swarm optimization technique incorporates time varying acceleration coefficients (NPSOTVAC). The highly non-linear cost function due to ramp rate limit, prohibited operating zones and valve point loading effect is considered in the proposed method. The effectiveness of proposed NPSOTVAC algorithm is proved through test on 6 and 15 unit test systems.

A Direct Search Method to solve Economic Dispatch Problem with Valve-Point Effect

Scarcity of energy resources, increasing power generation cost and ever-growing demand for electric energy necessitates optimal economic dispatch in today’s power systems. The main objective of economic dispatch is to reduce the total power generation cost, while satisfying various equality and inequality constraints. Traditionally in economic dispatch problems, the cost function for generating units has been approximated as a quadratic function which doesn’t provide accurate results. Moreover, to obtain accurate fuel cost, valve-point effect in thermal power plant has to be taken into account. The inclusion of valve-point effect makes the modeling of the fuel cost functions of generating units more practical. In this paper a new evolutionary algorithm called Pattern Search Technique, has been employed to solve economic dispatch problem with the valve-point effect. Using this technique the non-linear cost function is solved for three unit system and the results are compared with the traditional PSO and GA method. These results prove that Pattern Search method is capable of getting higher quality solution including mathematical simplicity, fast convergence, and robustness to solve hard optimization problems.

Speed control of optimal designed PMBLDC motor using improved fuzzy particle swarm optimization

Permanent brushless dc motors have been used in many areas. Considering to their vast advantages, researchers have studied extensively for speed control and reducing the torque ripple of this motors. But a little study was done for both speed control and optimum design of them. This paper presents for the optimal design of a PMBLDC motor with goal of reducing volume and building cost. In addition the speed control aim is considered using a multi-objective nonlinear cost function which is solved by fuzzy particle swarm optimization. First characteristics of motor are expressed as functions of motor geometries. Then cost function which combines the step response characteristic of motor speed, building cost and its volume is constructed and minimized. To reach this goal in this application the new improved fuzzy particle swam optimization is used for the first time. The results of simulations show that this method has good ability and efficiency in reaching global best point in compare of GA and PSO methods.

Unit commitment problem: A new formulation and solution method

In this paper, we present a new formulation and solution method for the well known unit commitment problem (UCP) for scheduling the thermal generators in a day-ahead electricity market. Compared to the traditional approach, our approach has several advantages such as: (a) reducing the combinatorial complexity (i.e., the size of the binary state space) significantly, (b) eliminating the need for linearizing the constraints associated with the minimum ON time and minimum OFF time for any thermal generator, (c) eliminating the need for defining new binary decision variables to represent the startup and shutdown decisions for any thermal generator in each hour and (d) eliminating the need to linearize the non-linear cost functions associated with any thermal generator (e.g., time dependent exponential startup cost function). According to our formulation, the UCP can be stated as finding a feasible path of ON–OFF states for each generator (i.e., a sequence of unit commitment states that satisfy the corresponding minimum ON time and minimum OFF time constraints over the scheduling horizon) such that the total generation cost is minimized while meeting the demand and reserve requirement in each hour for the next day. We show how a near optimal solution for the UCP can be constructed using our solution method which is based on the Lagrangian relaxation (LR) method. Although only a near optimal solution is found, we show that our solution is comparable to that obtained when the UCP is modeled as a mixed integer linear program (MILP).

Parallel Covariance Intersection Fusion Optimal Kalman Filter

For multisensor network systems with unknown cross-covariances, a novel multi-level parallel covariance intersection (PCI) fusion Kalman filter is presented in this paper, which is realized by the multi-level parallel two-sensor covariance intersection (CI) fusers, so it only requires to solve the optimization problems of several one-dimensional nonlinear cost functions in parallel with loss computation burden. It can significantly reduce the computation time and increase data processing rate when the number of sensors is very large. It is proved that the PCI fuser is consistent, and its accuracy is higher than that of each local filter and is lower than that of the optimal Kalman fuser weighted by matrices. The geometric interpretation of accuracy relations based on the covariance ellipses is given. A simulation example for tracking systems verifies the accuracy relations.

A Hamiltonian approach to compute an energy efficient trajectory for a servomotor system

This paper considers a nonlinear constrained optimal control problem (NCOCP) originated from energy optimal trajectory planning of servomotor systems. Solving the exact optimal solution is challenging because of the nonlinear and switching cost function, and various constraints. This paper proposes a method to manage the switching cost function to establish a set of necessary conditions of an NCOCP. Specifically, a concept “sub-trajectory” is introduced to match multiple Hamiltonian due to switches in the cost function. Necessary conditions on the optimal trajectory are established as a union of conditions for all sub-trajectories and Weierstrass–Erdmann corner conditions between sub-trajectories. The set of feasible structures of optimal trajectories is further identified and represented by various state transition diagrams for the servomotor application. A decomposition-based shooting method is proposed to compute an optimal trajectory by solving multi-point boundary value problems. Simulations and experiments validate the effectiveness of the methodology and the energy saving benefit.

Camera self-calibration with varying intrinsic parameters by an unknown three-dimensional scene

This work proposes a method of camera self-calibration having varying intrinsic parameters from a sequence of images of an unknown 3D object. The projection of two points of the 3D scene in the image planes is used with fundamental matrices to determine the projection matrices. The present approach is based on the formulation of a nonlinear cost function from the determination of a relationship between two points of the scene and their projections in the image planes. The resolution of this function enables us to estimate the intrinsic parameters of different cameras. The strong point of the present approach is clearly seen in the minimization of the three constraints of a self-calibration system (a pair of images, 3D scene, any camera): The use of a single pair of images provides fewer equations, which minimizes the execution time of the program, the use of a 3D scene reduces the planarity constraints, and the use of any camera eliminates the constraints of cameras having constant parameters. The experiment results on synthetic and real data are presented to demonstrate the performance of the present approach in terms of accuracy, simplicity, stability, and convergence.