Quadratic Cost Research Articles

Operator theoretic approach to optimal control problems characterized by the Caputo fractional differential equations

In this paper, our aim is to solve an optimal control problem by using an operator theoretic approach in the sense of Caputo fractional derivative. We first reduce the fractional dynamical system into an equivalent Hammerstein operator equation and then we provide a sufficient condition for the operators to ensure the existence of an optimal pair for the fractional dynamical system. By analyzing minimum conditions for the fractional optimal control problem (FOCP), the existence results of an optimal pair are obtained for both convex and non-convex cases. We derive an optimality system with a quadratic cost functional by using the Gateaux derivative. Finally, we relate our optimality system with the Hamiltonian system of minimum principle, and some examples are included to demonstrate the effectiveness of theoretical results. The primary motivation for this research is to investigate a new approach to solving the fractional optimal control problem employing functional analysis and operator theory techniques.

A novel fault diagnosis scheme for a ballast system

The Back Propagation (BP) neural network model is one of the neural network models suitable for model recognition and classification. However, a single BP network model cannot fully tap the feature information hiding in the data and is vulnerable to the uncertainty associated with system parameter changes and also suffers from the effects of long diagnosis times, inaccurate and undesired diagnosis results in applying a large amount of high-dimensional feature data. In addition, the standard BP algorithm has limitations in pattern recognition and classification. For example, it adopts the quadratic cost function to train the neural network, which can lead to low convergence accuracy, low convergence rate, and so on. Compared with the quadratic cost function, the cross-entropy cost function can effectively improve the convergence accuracy and the convergence rate of the neural network. The Dempster–Shafer (D–S) evidence theory is a powerful tool for handling incomplete, imprecise, and uncertain information and merging it when it derives from multiple BP networks. In order to overcome the drawbacks of a single BP network, it is necessary to divide the high-dimensional feature data into several different categories of low-dimensional feature data and to input the same dimensional feature data into the same BP network model, as well as to design the information fusion algorithms based on D–S evidence theory and multiple improved BP networks with the cross-entropy cost function. For this reason, this paper proposes a novel information fusion diagnosis approach based on D–S evidence theory and multiple improved BP networks for fault diagnosis and classification of multiple devices in a ballast system subjected to extreme environmental conditions. The proposed diagnosis approach utilizes multiple neural networks to carry out local fault diagnosis, adopts the D–S evidence theory to fuse the results of local diagnosis and derive the decision results, and introduces an improved BP algorithm based on the cross-entropy cost function into the individual neural networks to enhance the convergence rate and diagnosis accuracy of the individual neural networks. This method can overcome the deficiencies of individual BP networks in multiple BP networks in their poor diagnostic ability to handle incomplete, imprecise, and uncertain sample information. According to the design idea of the proposed diagnosis approach, a new fusion modeling for fault diagnosis of multiple devices is constructed for a ballast system. The validity of the proposed diagnosis method and model is illustrated through both simulation studies and experimental tests.

Incremental Entity Blocking over Heterogeneous Streaming Data

Web systems have become a valuable source of semi-structured and streaming data. In this sense, Entity Resolution (ER) has become a key solution for integrating multiple data sources or identifying similarities between data items, namely entities. To avoid the quadratic costs of the ER task and improve efficiency, blocking techniques are usually applied. Beyond the traditional challenges faced by ER and, consequently, by the blocking techniques, there are also challenges related to streaming data, incremental processing, and noisy data. To address them, we propose a schema-agnostic blocking technique capable of handling noisy and streaming data incrementally through a distributed computational infrastructure. To the best of our knowledge, there is a lack of blocking techniques that address these challenges simultaneously. This work proposes two strategies (attribute selection and top-n neighborhood entities) to minimize resource consumption and improve blocking efficiency. Moreover, this work presents a noise-tolerant algorithm, which minimizes the impact of noisy data (e.g., typos and misspellings) on blocking effectiveness. In our experimental evaluation, we use real-world pairs of data sources, including a case study that involves data from Twitter and Google News. The proposed technique achieves better results regarding effectiveness and efficiency compared to the state-of-the-art technique (metablocking). More precisely, the application of the two strategies over the proposed technique alone improves efficiency by 56%, on average.

Operator theoretic approach in fractional‐order delay optimal control problems

This paper investigates the fractional optimal control problem with a single delay in the state by using the operator theoretic approach. In this approach, we first reduce the delay fractional dynamical system into an equivalent operator equation, and then, by providing sufficient conditions to the operators, the existence of an optimal pair is proved for the abstract system. The optimality system for the quadratic cost functional is derived by using the Frechet derivative. Then we relate the operator theoretic approach optimality system to a Hamiltonian system of Pontryagin's minimum principle. The primary goal of this article is to demonstrate the existence of an optimal pair for the fractional‐order delay dynamical system by using a functional minimization theorem in functional analysis. Likewise, the optimality systems for the fractional‐order delay dynamical system are derived by using the functional (either Gateaux or Frechet) derivatives. Finally, we provide two numerical examples that support our theoretical findings.

Inverse linear-quadratic discrete-time finite-horizon optimal control for indistinguishable homogeneous agents: A convex optimization approach

The inverse linear-quadratic optimal control problem is a system identification problem whose aim is to recover the quadratic cost function and hence the closed-loop system matrices based on observations of optimal trajectories. In this paper, the discrete-time, finite-horizon case is considered, where the agents are also assumed to be homogeneous and indistinguishable. The latter means that the agents all have the same dynamics and objective functions and the observations are in terms of “snap shots” of all agents at different time instants, but what is not known is “which agent moved where” for consecutive observations. This absence of linked optimal trajectories makes the problem challenging. We first show that this problem is globally identifiable. Then, for the case of noiseless observations, we show that the true cost matrix, and hence the closed-loop system matrices, can be recovered as the unique global optimal solution to a convex optimization problem. Next, for the case of noisy observations, we formulate an estimator as the unique global optimal solution to a modified convex optimization problem. Moreover, the statistical consistency of this estimator is shown. Finally, the performance of the proposed method is demonstrated by a number of numerical examples.

A Controllability Method for Maxwell's Equations

We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a nonintrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap postprocessing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence toward the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.

Instabilities in multi-asset and multi-agent market impact games

We consider the general problem of a set of agents trading a portfolio of assets in the presence of transient price impact and additional quadratic transaction costs and we study, with analytical and numerical methods, the resulting Nash equilibria. Extending significantly the framework of Schied and Zhang (2019) and Luo and Schied (2020), who considered the single asset case, we prove the existence and uniqueness of the corresponding Nash equilibria for the related mean-variance optimization problem. We then focus our attention on the conditions on the model parameters making the trading profile of the agents at equilibrium, and as a consequence the price trajectory, wildly oscillating and the market unstable. While Schied and Zhang (2019) and Luo and Schied (2020) highlighted the importance of the value of transaction cost in determining the transition between a stable and an unstable phase, we show that also the scaling of market impact with the number of agents J and the number of assets M determines the asymptotic stability (in J and M) of markets.

Economics of scale and optimal size of hospitals (case study of shahid Rahnemoun hospital)

Economies of scale are one of the important behavioral indicators of production units. In the presence of economies of scale, the scale curve is descending, which indicates a decrease in the average cost of production with an increase in production. Theoretically, the optimal size of a production unit is the volume of production at which the average cost is at its lowest. The issue of the need to pay attention to economies of scale and optimal allocation of resources on the one hand and the role and position of hospitals in providing medical services on the other hand and the high share of treatment costs in public spending in Iran, compared to the world average, The motivation was to evaluate the efficiency conditions in production and determine the optimal size of the hospital. In this study, in order to determine the optimal size of Shahid Rahnemoun Hospital in Yazd according to the number of beds, the cost quadratic function of the hospital translog was estimated based on production factors using the quadratic cost function method. The study period is 1390-1398, which uses Eviews econometric software.

Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm

We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and H\older continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general conditions including quadratic costs and unbounded marginals. The results for the value extend to regularization by an arbitrary divergence. As an application, we show convergence of Sinkhorn's algorithm in Wasserstein sense, including for quadratic cost. Two techniques are presented: The first compares an optimal coupling with its so-called shadow, a coupling induced on other marginals by an explicit construction. The second transforms one set of marginals by a change of coordinates and thus reduces the comparison of differing marginals to the comparison of differing cost functions under the same marginals.

Stochastic MPC With Dynamic Feedback Gain Selection and Discounted Probabilistic Constraints

This paper considers linear discrete-time systems with additive disturbances, and designs a Model Predictive Control (MPC) law incorporating a dynamic feedback gain to minimise a quadratic cost function subject to a single chance constraint. The feedback gain is selected online and we provide two selection methods based on minimising upper bounds on predicted costs. The chance constraint is defined as a discounted sum of violation probabilities on an infinite horizon. By penalising violation probabilities close to the initial time and assigning violation probabilities in the far future with vanishingly small weights, this form of constraints allows for an MPC law with guarantees of recursive feasibility without a boundedness assumption on the disturbance. A computationally convenient MPC optimisation problem is formulated using Chebyshev's inequality and we introduce an online constraint-tightening technique to ensure recursive feasibility. The closed loop system is guaranteed to satisfy the chance constraint and a quadratic stability condition. With dynamic feedback gain selection, the closed loop cost is reduced and conservativeness of Chebyshev's inequality is mitigated. Also, a larger feasible set of initial conditions can be obtained. Numerical simulations are given to show these results.

Batch-to-Batch Adaptive Iterative Learning Control-Explicit Model Predictive Control Two-Tier Framework for the Control of Batch Transesterification Process.

To harness energy security and reduce carbon emissions, humankind is trying to switch toward renewable energy resources. To this extent, fatty acid methyl esters, also known as biodiesel, are popularly used as a green fuel. Fatty acid methyl esters can be produced by a batch transesterification reaction between vegetable oil and alcohol. Being a batch process, fatty acid methyl esters production is beset with issues such as uncertainties and unsteady state behavior, and therefore, adequate process control measures are necessitated. In this study, we have proposed a novel two-tier framework for the control of the fatty acid methyl esters production process. The proposed approach combines the constrained batch-to-batch iterative learning control technique and explicit model predictive control to obtain the desired concentration of the fatty acid methyl esters. In particular, the batch-to-batch iterative learning control technique is used to generate reactor temperature set-points, which is further utilized to obtain an optimal coolant flow rate by solving a quadratic objective cost function, with the help of explicit model predictive control. Our simulation results indicate that the fatty acid methyl esters concentration trajectory converges to the desired batch trajectory within four batches for uncertainty in activation energy and six batches for uncertainty in both inlet concentration of triglyceride and in activation energy even in the presence of process disturbances. The proposed approach was compared to the heuristic-based approach and constraint iterative learning control approach to showcase its efficacy.

Model predictive control and protection of MMC-based MTDC power systems

Meshed offshore grids (MOGs) present a viable option for a reliable bulk power transmission topology. The station-level control of MOGs requires faster dynamics along with multiple objective functions, which is realized by the model predictive control (MPC). This paper provides control, and protection design for the Modular Multilevel Converter (MMC) based multi-terminal DC (MTDC) power system using MPC. MPC is defined using a quadratic cost function, and a dqz rotating frame voltage inputs are represented using Laguerre orthonormal functions. MPC has been applied for the control of both grid forming and grid following converters in a four-terminal MTDC setup, implemented for real-time Electromagnetic Transient (EMT) simulation. By applying numerous time-domain simulations, the advantages of the MPC when dealing with AC and DC side disturbances are investigated. The investigation highlights the MPC’s inherent feature of fast response and high damping during- and post-disturbance, which is compared to the traditional PI controller performance. The analysis provides a comprehensive insight into the transient behavior of the MTDC during disturbances.

Lyapunov-Based Nonlinear Model Predictive Control for Attitude Trajectory Tracking of Unmanned Aerial Vehicles

This paper presents a new Lyapunov-based nonlinear model predictive controller (LNMPC) for the attitude control problem of unmanned aerial vehicles (UAVs), which is essential for their functioning operation. The controller is designed based on a quadratic cost function integrating UAV dynamics and system constraints. An additional contraction constraint is then introduced to ensure closed-loop system stability. That constraint is fulfilled via a Lyapunov function derived from a sliding mode controller (SMC). The feasibility and stability of the LNMPC are finally proved. Simulation and comparison results show that the proposed controller guarantees the system stability and outperforms other state-of-the-art nonlinear controllers such as the backstepping controller (BSC) and SMC. In addition, the proposed controller can be integrated into an existing UAV model in the Gazebo simulator to perform software-in-the-loop tests. The results show that the LNMPC is better than the built-in PID controller of the UAV, which confirms the validity and applicability of our proposed approach.

A hybrid electricity pricing mechanism for joint system optimization and social acceptance within energy communities

This article presents a framework for local electricity pricing mechanisms for households designed for social acceptance. The optimization goal of the mechanisms is to flatten the neighborhood electricity profile. The motivation and need for such mechanisms result from the expectation that the energy transition leads to high peaks in the distribution grid, both in electricity consumption and renewable generation, posing a significant challenge to these grids.Following the literature, quadratic cost functions have the potential to achieve the envisioned system optimization. However, their drawback is that consumers find the resulting pricing mechanisms too complicated and are generally not willing to participate in systems offering such prices. In contrast, the simpler pricing mechanisms currently used in practice are socially accepted. However, these mechanisms lack sufficient incentive to reduce electricity peaks in the distribution grids.Our approach is to combine these concepts in a hybrid pricing mechanism for local energy communities, using a piecewise linear cost function, that approximates a quadratic function. The resulting pricing mechanism is evaluated in a field test, and, based on the feedback of participating consumers and other criteria defined in literature, we can conclude that the proposed mechanism is socially accepted.In the implementation of this hybrid pricing mechanism, its performance can be improved such that it obtains results comparable to those of quadratic cost functions. Based on these findings, a detailed numerical evaluation, and the results from the field test, we conclude that the presented pricing mechanism has the potential for being used in practice.

Navigation of a quadratic potential with ellipsoidal obstacles

Given a convex quadratic potential of which its minimum is the agent’s goal and a Euclidean space populated with ellipsoidal obstacles, one can construct a Rimon–Koditschek (RK) artificial potential to navigate. Its negative gradient attracts the agent toward the goal and repels the agent away from the boundary of the obstacles. This is a popular approach to navigation problems since it can be implemented with local spatial information that is acquired during operation time. However, navigation is only successful in situations where the obstacles are not too eccentric (flat). This paper proposes a modification to gradient dynamics that allows successful navigation of an environment with a quadratic cost and ellipsoidal obstacles regardless of their eccentricity. This is accomplished by altering gradient dynamics with a Hessian correction that is intended to imitate worlds with spherical obstacles in which RK potentials are known to work. The resulting dynamics simplify by the quadratic form of the obstacles. Convergence to the goal and obstacle avoidance is established from almost every initial position (up to a set of measure one) in the free space, with mild conditions on the location of the target. Results are corroborated empirically with numerical simulations.

High-precision computational guidance in terminal phase with impact angle, lead angle and lateral acceleration constraints

This paper proposes a high-precision three-dimensional nonlinear optimal computational guidance law in the terminal phase of an interceptor that ensures near-zero miss-distance as well as the desired impact angle. Additionally, it achieves these ambitious objectives while ensuring that the lead angle and lateral acceleration constraints are not violated throughout its trajectory. This ensures (i) the target does not escape the field of view of its seeker at any point in time (a state constraint) and (ii) it does not demand unreasonable lateral acceleration that cannot be generated (a control constraint). The guidance problem is formulated and solved using newly proposed Path-constrained Model Predictive Static Programming (PC-MPSP) framework. All constraints, both equality and inequality, are equivalently represented as linear constraints in terms of the errors in the control history, thereby reducing the complexity and dimensionality of the problem significantly. Coupled with a quadratic cost function in control, the problem is then reduced to a standard quadratic optimization problem with linear constraints, which is then solved using the computationally efficient interior-point method. Results clearly demonstrate the advantage of the proposed guidance scheme over the conventional Biased PN as well as the recently proposed GENeralized EXplicit (GENEX) guidance techniques. Numerical simulations with variation in initial conditions and Monte–Carlo simulations with parametric uncertainty demonstrate the robustness of the proposed guidance scheme.

Linear approximations to improve lower bounds of a physician scheduling problem in emergency rooms

The physician assignment process consists of coverage of shifts and duties allocated to physicians in a planning period, taking into account work regulations, individual preferences, and organizational rules, which mostly conflict with each other. In this work, we propose a reformulated mixed-integer programming model based on the literature to tackle fairness in physician scheduling in Emergency Rooms (ERs). In particular, we propose two mixed-integer quadratic programming formulations that consider quadratic costs and two models with linear costs. Our approaches provide balanced schedules concerning target hours and weekends in terms of fairness. Our models also provide a high degree of demand coverage, providing decision-makers a significant advantage.

The original hotelling model with linear and quadratic firms logistics costs

We introduce a modified Hotelling model setup that incorporates the original model and considers logistics’ related costs incurred by the firms. Both the linear and quadratic logistics related costs cases are studied. We find that the relative magnitudes of firms’ logistics costs to customers’ transportation and/or search costs have significant effects on firms’ decisions in terms of location choices. In the linear case, we show, under a large set of cost structures, that substantial differentiation exists and is stable. In the quadratic case, maximum differentiation is the only equilibrium for all cost structures in the model but firms’ profits are higher than in the original case. We also consider the case where customers transportation costs are quadratic while firms logistics costs are linear.

Free time evolution of the quantum wave function and optimal transportation

It is shown that, in the absence of nodes and under regularity assumptions, a solution in a finite interval of time of the free Schrodinger equation solves a minimization problem, which is a stochastic generalization of the classical optimal transportation problem with quadratic cost.

Distributed Time-Varying Quadratic Optimal Resource Allocation Subject to Nonidentical Time-Varying Hessians With Application to Multiquadrotor Hose Transportation

This article considers the distributed time-varying optimal resource allocation problem with time-varying quadratic cost functions and a time-varying coupled equality constraint for multiagent systems. The objective is to design a distributed algorithm for agents with single-integrator dynamics to cooperatively satisfy the coupled equality constraint and minimize the sum of all local cost functions. Here, both the coupled equality constraint and cost functions depend explicitly on time. The cost functions are in quadratic form and may have nonidentical time-varying Hessians. To solve the problem in a distributed manner, an estimator based on the distributed average tracking method is first developed for each agent to estimate certain global information. By leveraging the estimated global information and an adaptive gain scheme, a distributed continuous-time algorithm is proposed, which ensures the agents to find and track the time-varying optimal trajectories with vanishing errors. We illustrate the applicability of the proposed method in the optimal hose transportation problem using multiple quadrotors.