Minimization Of Cost Function Research Articles

Global Social Cost Minimization With Possibly Nonconvex Objective Functions: An Extremum Seeking-Based Approach

A social cost minimization problem is addressed in this article. In the considered problem, a network of agents work collaboratively to minimize the social cost function, which is defined as the sum of the agents’ local objective functions. The engaged agents are supposed to be equipped with an undirected and connected communication graph. Different from most of the existing works, the social cost function in the considered problem is allowed to be nonconvex and possibly admits local extrema. To avoid local extrema and achieve the global minimization of the social cost function, an extremum-seeking-based approach is proposed by introducing a dynamic average consensus protocol to the sinusoidal-dither-signal-based extremum seeking scheme. The dynamic average consensus protocol is leveraged in the proposed extremum-seeker for information sharing and the sinusoidal probing signal is utilized for information extraction. For the avoidance of local extrema, the amplitude of the dither signal is designed to be adaptive. Through Lyapunov stability analysis, it is shown that the proposed method enables the decision variable to converge to a neighborhood of the global minimum point if the conditions on the network connectivity, the existence of unique global minimum and achievability of the global minimum are satisfied. The theoretical result is verified via simulating a numerical example.

Swarm-based chaotic gravitational search algorithm for solving mechanical engineering design problems

PurposeThe purpose of this paper is to investigate the performance of chaotic gravitational search algorithm (CGSA) in solving mechanical engineering design frameworks including welded beam design (WBD), compression spring design (CSD) and pressure vessel design (PVD).Design/methodology/approachIn this study, ten chaotic maps were combined with gravitational constant to increase the exploitation power of gravitational search algorithm (GSA). Also, CGSA has been used for maintaining the adaptive capability of gravitational constant. Furthermore, chaotic maps were used for overcoming premature convergence and stagnation in local minima problems of standard GSA.FindingsThe chaotic maps have shown efficient performance for WBD and PVD problems. Further, they have depicted competitive results for CSD framework. Moreover, the experimental results indicate that CGSA shows efficient performance in terms of convergence speed, cost function minimization, design variable optimization and successful constraint handling as compared to other participating algorithms.Research limitations/implicationsThe use of chaotic maps in standard GSA is a new beginning for research in GSA particularly convergence and time complexity analysis. Moreover, CGSA can be used for solving the infinite impulsive response (IIR) parameter tuning and economic load dispatch problems in electrical sciences.Originality/valueThe hybridization of chaotic maps and evolutionary algorithms for solving practical engineering problems is an emerging topic in metaheuristics. In the literature, it can be seen that researchers have used some chaotic maps such as a logistic map, Gauss map and a sinusoidal map more rigorously than other maps. However, this work uses ten different chaotic maps for engineering design optimization. In addition, non-parametric statistical test, namely, Wilcoxon rank-sum test, was carried out at 5% significance level to statistically validate the simulation results. Besides, 11 state-of-the-art metaheuristic algorithms were used for comparative analysis of the experimental results to further raise the authenticity of the experimental setup.

A few techniques to improve data-driven reduced-order simulations for unsteady flows

A key step to improve data-driven reduced-order simulations is to compute a transfer function that predicts the time evolution of the reduced-order modes accurately. We demonstrate a couple of useful techniques to achieve this objective: One is to pre-process time-series of reduced-order modes with a low-pass filter, e.g. a polynomial filter and B-spline, and the other is to compute a data-driven transfer function from multiple past time-steps, corresponding to a high-order temporal scheme. These techniques are exercised with POD modes generated from time-resolved planar PIV data. A fully separated flow past the NACA0012 airfoil at the angle of attack of 30∘ and Re=1000 is measured in a water tunnel, and non-periodic unsteady flow is analyzed in two dimensions. From the first 1000 frames, transfer functions are calculated based on minimization of different cost functions, which define least-squares errors in the predicted POD modes at the next time step; subsequently, their prediction capabilities are evaluated during the following 1000 frames based on the accuracy of the predicted POD modes at the next time steps. The multistep schemes can reduce the root-mean-square errors of the predicted mode coefficients by up to 10% without a low-pass filter. Combining a low-pass filter with the second-order temporal scheme can further reduce the errors by 7%, and introduction of an L1-norm constraint for the mode coefficients decreases it by extra 2%. In contrast, nonlinear transfer functions with more degrees of freedom deteriorate the prediction during time duration outside of the sampling period even relative to the linear prediction.

Image fusion via sparse regularization with non-convex penalties

The L1 norm regularized least squares method is often used for finding sparse approximate solutions and is widely used in signal restoration. Basis pursuit denoising (BPD) performs noise reduction in this way. However, the shortcoming of using L1 norm regularization is the underestimation of the true solution. Recently, a class of non-convex penalties have been proposed to improve this situation. This kind of penalty function is non-convex itself, but preserves the convexity property of the whole cost function. This approach has been confirmed to offer good performance in 1-D signal denoising. This paper demonstrates the aforementioned method to 2-D signals (images) and applies it to multisensor image fusion. The problem is posed as an inverse one and a corresponding cost function is judiciously designed to include two data attachment terms. The whole cost function is proved to be convex upon suitably choosing the non-convex penalty, so that the cost function minimization can be tackled by convex optimization approaches, which comprise simple computations. The performance of the proposed method is benchmarked against a number of state-of-the-art image fusion techniques and superior performance is demonstrated both visually and in terms of various assessment measures.

A Step-Up Multilevel Inverter Topology Using Novel Switched Capacitor Converters With Reduced Components

In this article, basic cell (BC) of a novel switched capacitor converter (SCC) has been proposed first. After that, the generalized structure of proposed SCC is developed. The developed SCC requires reduced number of switches, drivers, diodes, capacitors, and lower number of conducting switches in current flow paths and capacitor charging paths as compared to the other recently developed SCCs. A switched capacitor multilevel inverter (SCMLI) utilizing two numbers of generalized SCCs is developed next. Further, cascaded extension of proposed SCMLI is realized and analyzed for symmetric and asymmetric dc source configurations. A detail analysis of optimum selection of capacitance for switched capacitors of 13-level SCMLI is presented. An extensive comparison study shows that the proposed SCMLI requires lower number of components as compared to other SCMLIs. Further, the proposed structure has minimum cost function per level per boosting factor as compared to the other SCMLIs. Extensive experimental results considering fundamental switching frequency scheme are presented to validate the merits and effectiveness of the proposed structure.

Lyapunov-Krasovskii prescribed derivative and the Bellman functional for time-delay systems

The Dynamic Programming approach for optimal control problem establishes that the necessary condition for optimality is that the minimum cost function (the Bellman functional) must satisfy the Hamilton-Jacobi-Bellman equation. A sufficient condition is that if there exists a functional that satisfies the Hamilton-Jacobi-Bellman equation, then it is the minimum cost function. For linear time-delay systems, Krasovskii proposed the Bellman functional, and an optimal structure for the controller was reported. The Dynamic Programming combined with prescribed derivative functionals leads to an iterative procedure which allows finding suboptimal controls law at each step. There is numerical evidence that shows that these functionals are equivalent. However, their algebraic structure is different. The Bellman functional has only three terms and the iterative functional is composed by thirteen summands. The algebraic relation between both functionals is not easy to see. The present contribution gives a proof of this connection by using Fubini’s Theorem.

GA-Based Off-Line Parameter Estimation of the Induction Motor Model Including Magnetic Saturation and Iron Losses

This paper, starting from recent papers in the scientific literature dealing with Rotating Induction Motor (RIM) dynamic modelling, as a first step, improves its space-vector dynamic model, including both the magnetic saturation and iron losses; The main original aspects of the proposed model are the following: 1) the magnetic saturation of the iron core has been described on the basis of both current versus flux and flux versus current functions, 3) it includes the iron losses, separating them in hysteresis and eddy current ones, 4) it includes the effect of the load on the magnetic saturation. Afterwards, it proposes an off-line technique for the estimation of electrical parameters of this model, which is based on Genetic Algorithms (GA). The proposed method is based on input-output measurements and needs neither the machine design geometrical data nor a Finite Element Analysis (FEA) of the machine. It focuses on the application of an algorithm based on the minimization of a suitable cost function depending on the stator current error. The proposed electrical parameters estimation method has been initially tested in numerical simulation and further verified experimentally on a suitably developed test set-up.

Double-Talk Robust Multichannel Acoustic Echo Cancellation Using Least-Squares MIMO Adaptive Filtering: Transversal, Array, and Lattice Forms

In this paper, we address the problem of noise-robust multiple-input multiple-output (MIMO) adaptive filtering that is optimal in least-squares sense with application to multichannel acoustic echo cancellation. We formulate the problem as minimization of a multichannel least-squares cost function that incorporates near-end speech and noise statistics resulting in a novel noise-robust framework for MIMO adaptive filtering. Although the issue of numerical stability has been widely explored in the context of recursive least-squares (RLS) filtering, a rigorous mathematical treatment of the MIMO case in the context of numerically stable noise-robust multichannel echo cancellation remains absent. Guided by quantization-error modeling, we resolve the issue of numerical instability in our noise-robust scheme by utilizing transversal RLS filtering of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Type 2</i> . Thereafter, an explicit derivation of its inverse QR-decomposition (IQRD) counterpart based on Givens rotations is presented. We also derive computationally efficient lattice forms for our noise-robust RLS Type-2 and IQRD algorithms. It is highlighted that propagation of angle-normalized errors occurs <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">naturally</i> within the numerically stable least-squares lattice (LSL). Thus, our approach combines the four sought after attributes in a multichannel echo cancellation scheme, i.e., computational efficiency, numerical stability, fast convergence and tracking, and robustness against noise. We analyze our formulations using simulations in terms of convergence, re-convergence, robustness in the presence of double-talk, and numerical stability.

Distributed Frequency and Voltage Control for AC Microgrids based on Primal-Dual Gradient Dynamics

With the gradual transformation of power generation towards renewables, distributed energy resources are becoming more and more relevant for grid stabilization. In order to involve all participants in the joint solution of this challenging task, we propose a distributed, model-based and unifying controller for frequency and voltage regulation in AC microgrids, based on steady-state optimal control. It not only unifies frequency and voltage control, but also incorporates the classic hierarchy of primary, secondary and tertiary control layers with each closed-loop equilibrium being a minimizer of a user-defined cost function. By considering the individual voltage limits as additional constraints in the corresponding optimization problem, no superordinate specification of voltage setpoints is required. Since the dynamic model of the microgrid has a port-Hamiltonian structure, stability of the overall system can be assessed using shifted passivity properties. Furthermore, we demonstrate the effectiveness of the controller and its robustness against fluctuations in active and reactive power demand by means of numerical examples.

Novel Adaptive Filtering Algorithms Based on Higher-Order Statistics and Geometric Algebra

Adaptive filtering algorithms based on higher-order statistics are proposed for multi-dimensional signal processing in geometric algebra (GA) space. In this paper, the proposed adaptive filtering algorithms utilize the advantage of GA theory in multi-dimensional signal processing to represent a multi-dimensional signal as a GA multivector. In addition, the original least-mean fourth (LMF) and least-mean mixed-norm (LMMN) adaptive filtering algorithms are extended to GA space for multi-dimensional signal processing. Both the proposed GA-based least-mean fourth (GA-LMF) and GA-based least-mean mixed-norm (GA-LMMN) algorithms need to minimize cost functions based on higher-order statistics of the error signal in GA space. The simulation results show that the proposed GA-LMF algorithm performs better in terms of convergence rate and steady-state error under a much smaller step size. The proposed GA-LMMN algorithm makes up for the instability of GA-LMF as the step size increases, and its performance is more stable in mean absolute error and convergence rate.

Hybrid control of synchronization of fractional order nonlinear systems

AbstractUnder the existence of model uncertainties and external disturbance, finite‐time projective synchronization between two identical complex and two identical real fractional‐order (FO) chaotic systems are achieved by employing FO sliding mode control approach. In this paper, to ensure the occurrence of synchronization and asymptotic stability of the proposed methods, a sliding surface is designed and the Lyapunov direct method is used. By using integer and FO derivatives of a Lyapunov function, three different FO real and complex control laws are derived. A hybrid controller based on a switching law is designed. Its behavior is more efficient that if the individual controllers were designed based on the minimization of an appropriate cost function. Numerical simulations are implemented for verifying the effectiveness of the methods.

Variational Approach to Closure of Nonlinear Dynamical Systems: Autonomous Case

A general, variational approach to derive low-order reduced systems for nonlinear systems subject to an autonomous forcing, is introduced. The approach is based on the concept of optimal parameterizing manifold (PM) that substitutes the more classical notion of slow manifold or invariant manifold when breakdown of slaving occurs. An optimal PM provides the manifold that describes the average motion of the neglected scales as a function of the resolved scales and allows, in principle, for determining the best vector field of the reduced state space that describes e.g. the dynamics' slow motion. The underlying optimal parameterizations are approximated by dynamically-based formulas derived analytically from the original equations. These formulas are contingent upon the determination of only a few (scalar) parameters obtained from minimization of cost functionals, depending on training dataset collected from direct numerical simulation. In practice, a training period of length comparable to a characteristic recurrence or decorrelation time of the dynamics, is sufficient for the efficient derivation of optimized parameterizations. Applications to the closure of low-order models of Atmospheric Primitive Equations and Rayleigh-B\'enard convection are then discussed. The approach is finally illustrated --- in the context of the Kuramoto-Sivashinsky turbulence --- as providing efficient closures without slaving for a cutoff scale $k_c$ placed within the inertial range and the reduced state space is just spanned by the unstable modes, without inclusion of any stable modes whatsoever. The underlying optimal PMs obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by classical invariant manifold approximations in their standard forms when the latter are used at this cutoff wavelength.

Non-convex Optimization via Strongly Convex Majorization-minimization

AbstractIn this paper, we introduce a class of nonsmooth nonconvex optimization problems, and we propose to use a local iterative minimization-majorization (MM) algorithm to find an optimal solution for the optimization problem. The cost functions in our optimization problems are an extension of convex functions with MC separable penalty, which were previously introduced by Ivan Selesnick. These functions are not convex; therefore, convex optimization methods cannot be applied here to prove the existence of optimal minimum point for these functions. For our purpose, we use convex analysis tools to first construct a class of convex majorizers, which approximate the value of non-convex cost function locally, then use the MM algorithm to prove the existence of local minimum. The convergence of the algorithm is guaranteed when the iterative points $x^{(k)}$ are obtained in a ball centred at $x^{(k-1)}$ with small radius. We prove that the algorithm converges to a stationary point (local minimum) of cost function when the surregators are strongly convex.

Distributed Optimization for Disturbed Second-Order Multiagent Systems Based on Active Antidisturbance Control.

In this article, the distributed optimization problem is studied for second-order multiagent systems with both mismatched and matched disturbances. For this problem, a distributed active antidisturbance control framework is established, which consists of both disturbance estimation/compensation and distributed feedforward-feedback composite control design. In the first stage, some disturbance estimators are utilized to estimate various types of matched/mismatched disturbances for each agent. In the second stage, for each agent, based on the disturbance estimates, the information exchanges between the neighboring agents, and the gradient of the local cost function only accessible to itself, a kind of distributed composite controllers are proposed. Under these controllers, all the agents' outputs asymptotically reach consensus to the minimizer of the global cost function, which is the sum of all the local cost functions. The closed-loop system convergence is proven based on a new Lyapunov function, convex analysis, and an input-to-state stability criterion. Simulations demonstrate the effectiveness of the proposed control scheme.

Variable Action Period Predictive Flux Control Strategy for Permanent Magnet Synchronous Machines

This article introduces a method to achieve the variable action period (VAP) of voltage vector in the finite control set model predictive control (FCS-MPC). In the conventional FCS-MPC, the control behavior is limited since the action period of a voltage vector is normally fixed to the sampling period. This article aims to enhance the control performances by means of regulating the action period of selected voltage vectors, together with the selection of an optimal control action. A VAP-based predictive flux control strategy is thus proposed. The optimal voltage vector and its optimal action period are both obtained by means of a two-step cost function minimization. In this manner, the future states of the control objectives are taken into account in the construction of cost function. The action period for the selected voltage vectors is no longer fixed to the sampling period, which provides an additional control freedom to obtain improved reference tracking behaviors. Experimental validations and comparison studies demonstrate that enhanced steady-state performances as well as excellent dynamic behaviors can both be obtained with the proposed strategy adopted.

Low cogging torque design and simplified model-free predictive current control for radial-flux dual three-phase permanent magnet electric machines

This study presents two improved designs of eccentric-shaped permanent magnets and teeth-shaped stators in radial-flux dual three-phase permanent magnet electric machines to reduce cogging torque and torque ripple. The finite element analysis (ANSYS Electromagnetics) has been adopted in simulation, and real radial-flux dual three-phase permanent magnet electric machines have been fabricated in experiment to verify the study. Using the radial-flux dual three-phase permanent magnet electric machines in electric machine systems can improve the reliability and obtain higher output torque. In electrical drives and control, a simplified model-free predictive current control method has been proposed and implemented to drive the radial-flux dual three-phase permanent magnet electric machines, and the control law has been achieved by a TMS320F28377S microcontroller of Texas Instruments. The simplified model-free predictive current control method is senseless to parameter variations and back electromotive force of the permanent magnet electric machines, and only needs current sensors to detect six-phase current. The optimal one has been chosen by 14 various switching modes, which has the minimum cost function, and then the converter can be directly driven and controlled in the next sampling period. The features of the simplified model-free predictive current control method can reduce the algorithm calculation and avoid the defect of conventional model-based predictive current control scheme. A proportional-integral speed controller has also been designed to achieve the speed response of the fixed-speed tracking effect. Finally, the feasibility and effectiveness of the proposed simplified model-free predictive current control method for the dual three-phase permanent magnet electric machines can be verified in the experimental and quantitative results.

Robust Predictive Control of High-Density Cascaded DC Voltage Link Power Converters

Cascaded dc voltage link power converters typically employ a significant amount of energy storage at the intermediate link. The dc bus voltage is assumed to be stiff enough to provide decoupling between the source and load, allowing dynamic regulation functions to be implemented at the load and source terminals independently. In these cases, the size of the dc bus capacitance is dictated by requirements of steady-state ripple voltage, ripple current rating, ride-through considerations, and, more importantly, voltage overshoot during source and load transients. Design tradeoffs in realizing a stable system with acceptable transient performance often place a lower bound in the value of dc bus capacitance. In this paper, a robust predictive controller based on integrated and coupled control at the intermediate dc link, source, and load terminals is presented. The approach permits the use of a small amount of intermediate energy storage compared to conventional decoupled designs. The predictive control aspect of the proposed approach features cost function minimization and accurate duty ratio calculations based on converter model that accounts for small values of dc link capacitance. Although this allows stable operation of the system, it leads to persistent steady state error and transient overshoot voltages. The robust aspect of the proposed approach is based on the theory of variable structure systems. It introduces a capacitor charge restoration function that overcomes the drawbacks of the predictive control by squarely addressing uncertainty of model parameters and operating conditions. The resulting significant reduction in size of the capacitive reactive components opens the prospect of replacing electrolytic capacitors that have a limited life span with film capacitors with much longer life in emerging applications. The proposed approach is presented in a step-by-step manner using a dc-to-dc converter example and further extended to a dc-to-ac inverter case. Extensive computer simulations and experimental results from a laboratory-scale prototype of an example system are used to demonstrate the viability of the proposed controller.

Multi-item fuzzy economic production quantity model with multiple deliveries

Inventory control is one of the most critical issues in corporate management. Many mathematical models have been developed to optimize control strategies for the companies’ inventory. Economic production quantity (EPQ) is one of the classic models for inventory control, which is widely used. To deal with the uncertainty in the real world, we need to develop new and useful models for modeling systems of inventory management. In such cases, fuzzy models play a unique role in the field of inventory management. The main contribution of this study is to apply some well-known metaheuristic to solve an extended EPQ model based on fuzzy numbers considering multiple deliveries. This study aims to develop an EPQ model by considering demand as triangular fuzzy numbers and multiple deliveries (delivering in multiple packages) and by considering limitations in warehouse space as well as the total number of orders. Given these conditions, EPQ costs are calculated, and new modeling is presented. The obtained fuzzy model has been simplified by using the $$ \alpha{\text{-cut}} $$ and changing the variables, and finally, the most well-known metaheuristic algorithms, GA, PSO, GWO, and ICA, are applied in different problem sizes, and obtained results are analyzed in terms of minimizing cost function and CPU time. The result of this paper shows that GWO has superior performance in terms of various parameters.

Modelling and Design of Real-Time Energy Management Systems for Fuel Cell/Battery Electric Vehicles

Modelling and design of real-time energy management systems for optimising the operating costs of a fuel cell/battery electric vehicle are presented in this paper. The proposed energy management system consists of optimally sharing the propulsion power demand between the fuel cell and battery by enabling them to support each other for operating cost minimisation. The optimisation is achieved through real-time minimisation of a cost function, which accounts for fuel cell and battery degradation, hydrogen consumption and charge sustaining costs. A detailed analysis of each term of the overall cost function is performed and presented, which enables the development of a real-time, advanced energy management system for improving a previously presented simplified version using more accurate modelling and by considering cost function minimisation over a given time horizon. The performance of the proposed advanced energy management system are verified through numerical simulations over different driving cycles; particularly, simulations were performed in MATLAB-Simulink by considering a hysteresis-based energy management system and both simplified and advanced versions of the proposed energy management system for comparison.

Optimal power flow using multi-objective glowworm swarm optimization algorithm in a wind energy integrated power system

ABSTRACTThis paper solves a multi-objective optimal power flow (MO-OPF) problem in a wind-thermal power system. Here, the power output from the wind energy generator (WEG) is considered as the schedulable, therefore, the wind power penetration limits can be determined by the system operator. The stochastic behavior of wind power and wind speed is modeled using the Weibull probability density function (PDF). In this paper, three objective functions, i.e., total generation cost, transmission losses, and voltage stability enhancement index are selected. The total generation cost minimization function includes the cost of power produced by the thermal and WEGs, costs due to over-estimation and the under-estimation of available wind power. Here, the single objective optimization problems are solved using the Glowworm Swarm Optimization (GSO) algorithm, whereas the MO-OPF problems are solved using the multi-objective GSO (MO-GSO) algorithm. The proposed optimization problem is solved on a modified IEEE 30 and 300 bus test systems with wind farms located at different buses in the system. The simulation results obtained show the suitability and effectiveness of proposed MO-OPF approach.