Time-optimal Problem Research Articles

One Time-Optimal Problem for a Set-Valued Linear Control System

We consider a time-optimal problem for a set-valued linear control system in the case where a section of the solution of the system coincides with a target set. For this problem, we establish both the solvability conditions and the optimal time and optimal controls. The results are illustrated by model examples.

A quantum class topper optimization algorithm to solve combined emission economic dispatch problem

A combined emission and economic dispatch problem is referred as a complex multi-objective problem related to power system. In this article, an hybridized version of class topper optimization known as quantum class topper optimization is proposed to solve this problem. In quantum class topper optimization, quantum mechanism is used to enhance the searching ability of the proposed algorithm. The exploration, exploitation and convergence behavior of quantum class topper optimization is validated using benchmark functions. Later, four test cases on combined emission and economic dispatch are used to test the effectiveness of quantum class topper optimization. These four tests prove that the proposed optimization algorithm is effective to solve this real time optimization problem.

A statistical learning approach to determine optimal sizing & investment timing of commercial-scale distributed energy resources

The problem of optimal sizing and investment timing for a portfolio of gas-fired and photovoltaic (PV) generation is presented and solved using an optimization- classification technique. It is assumed that the demand, the prices of natural gas, and PV technology are stochastic processes. Our methodology takes advantage of decomposing the problem into two sub-problems. First, the optimal sizing problem is solved using dynamic programming and most likely solutions are identified as clusters. Then, the Investment time problem is formulated as a Real Option for each cluster to determine optimal timing. Although the two-step optimization approach can successfully close the loop between operational dynamics and investment decisions, we are also interested in discovering patterns in a multidimensional space of input parameters that make a certain combination of assets optimal among dozens of discrete choices. On that note, by applying Recursive Partitioning algorithm, decision trees are developed to estimate the structure of solutions rendered from optimization models by a rule-based system. Despite the high level of accuracy, the initial model is biased in favor of highly frequent clusters, and discards the optimal clusters resulted from extreme market behaviors. Value at Risk (VaR) is employed as a risk measure to demonstrate the enhancement risk performance. Finally, in order to investigate the robustness of the results, we conduct extensive sensitivity analysis over different parameter settings. The proposed model can be thought of as a statistically optimal summarizer of optimization models that enables decision makers to have an insight into optimal investment strategies according to characteristics of the building and the long-term energy market outlook.

MobileVisFixer: Tailoring Web Visualizations for Mobile Phones Leveraging an Explainable Reinforcement Learning Framework.

We contribute MobileVisFixer, a new method to make visualizations more mobile-friendly. Although mobile devices have become the primary means of accessing information on the web, many existing visualizations are not optimized for small screens and can lead to a frustrating user experience. Currently, practitioners and researchers have to engage in a tedious and time-consuming process to ensure that their designs scale to screens of different sizes, and existing toolkits and libraries provide little support in diagnosing and repairing issues. To address this challenge, MobileVisFixer automates a mobile-friendly visualization re-design process with a novel reinforcement learning framework. To inform the design of MobileVisFixer, we first collected and analyzed SVG-based visualizations on the web, and identified five common mobile-friendly issues. MobileVisFixer addresses four of these issues on single-view Cartesian visualizations with linear or discrete scales by a Markov Decision Process model that is both generalizable across various visualizations and fully explainable. MobileVisFixer deconstructs charts into declarative formats, and uses a greedy heuristic based on Policy Gradient methods to find solutions to this difficult, multi-criteria optimization problem in reasonable time. In addition, MobileVisFixer can be easily extended with the incorporation of optimization algorithms for data visualizations. Quantitative evaluation on two real-world datasets demonstrates the effectiveness and generalizability of our method.

Optimal dividend and risk control policies in the presence of a fixed transaction cost

In this paper, we consider a large insurance company whose cumulative cash flow process is described by a drifted Brownian motion. The preference of the insurer is to maximize his/her firm’s value, which corresponds to the expected present value of the dividend payments up to the ruin time. In the business process, the insurer has the option to draw up a dividend payment policy and to purchase proportional reinsurance at a certain point in time. In view of some typical practical expenses (e.g., consultant commission), it is assumed that a fixed transaction cost occurs at the beginning of the reinsurance commitment which, once made, is irreversible. This leads to a mixed stochastic control problem of optimal stopping time and singular control. For this mixed problem, we derive closed-form solutions for the optimal time to purchase reinsurance, the optimal retained risk proportion, the optimal dividend barrier, and the value function. The optimal solution shows that reinsurance is valueless to the insurer when the fixed cost is larger than a threshold, and comes into play when the fixed cost is less than this threshold. We also perform some numerical calculations to assess the impacts of fixed costs on the value function and the optimal policies.

Improved social spider algorithm for large scale optimization

Heuristic algorithms can give optimal solutions for low, middle, and large scale optimization problems in an acceptable time. The social spider algorithm (SSA) is one of the recent meta-heuristic algorithms that imitate the behaviors of the spider to perform global optimization. The original study of this algorithm was proposed to solve low scale continuous problems, and it is not be solved to middle and large scale continuous problems. In this paper, we have improved the SSA and have solved middle and large scale continuous problems, too. By adding two new techniques to the original SSA, the performance of the original SSA has been improved and it is named as an improved SSA (ISSA). In this paper, various unimodal and multimodal standard benchmark functions for low, middle, and large-scale optimization are studied for displaying the performance of ISSA. ISSA’s performance is also compared with the well-known and new evolutionary methods in the literature. Test results show that ISSA displays good performance and can be used as an alternative method for large scale optimization.

Coadjoint Orbits and Time-Optimal Problems for Step-$$2$$ Free Nilpotent Lie Groups

The coadjoint orbits and the Casimir functions are described for free nilpotent Lie groups of step $$2$$ . The symplectic foliation consists of affine subspaces in the Lie coalgebra. Left-invariant time-optimal problems are considered on Carnot groups of step $$2$$ for which the set of admissible velocities is a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. The first integrals of the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle are described. For two-dimensional coadjoint orbits, the constancy and periodicity properties of the solutions of this subsystem, as well as the phase flow, are described.

Casimir Functions of Free Nilpotent Lie Groups of Steps 3 and 4

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3, 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group.

Age of Information in a Cellular Internet of UAVs: Sensing and Communication Trade-Off Design

In this paper, we consider the cellular Internet of unmanned aerial vehicles (UAVs), where UAVs sense data with onboard sensors for multiple sensing tasks and transmit the data to the base station (BS). To quantify the “freshness” of the data at the BS, we bring in the concept of the age of information (AoI). The AoI is determined by the time for UAV sensing and that for UAV transmission, which gives rise to a trade-off within a given period. To minimize the AoI, we formulate a joint sensing time, transmission time, UAV trajectory, and task scheduling optimization problem. This NP-hard problem can be decoupled into two subproblems. We first propose an iterative algorithm to optimize the sensing time, transmission time, and UAV velocity for completing a specific task. Afterwards, we design the order in which the UAV performs data updates for multiple sensing tasks. The convergence and complexity of the proposed algorithm, together with the trade-off between UAV sensing and UAV transmission, are analyzed. Simulation results show that the AoI with the proposed algorithm is about 15% lower than that of the greedy algorithm, and over 40% lower than that of the random algorithm.

An analytical optimization approach to the joint trajectory and signal optimization problem for connected automated vehicles

Traffic conflict points (e.g., intersections, work-zones) cause travel delay, stop-and-go traffic, and excessive energy consumption. Efforts have been taken to improve traffic conflict point performance via trajectory control of connected automated vehicles (CAV) as the CAV technology emerges. One major challenge to these efforts is the complexity in optimization of CAV trajectories, particularly with joint signal timing optimization. This challenge poses barriers to real-time application requirements, scaling them up to address network level problems and drawing analytical insights into problem structures. To overcome this challenge, this paper aims to seek for an efficient and analytical solution to a joint vehicle trajectory and signal timing optimization problem. This problem simultaneously optimizes CAV trajectories and signal timing to minimize travel delay and energy consumption at a conflicting point with two traffic approaches. This study modifies the original complex formulation in two ways. First, the vehicle trajectory shape is simplified into a piece-wise quadratic function with no more than five segments. Second, instead of using the highly non-linear instantaneous fuel consumption function, a simplified macroscopic measure is proposed to approximate fuel consumption as an analytical quadratic function of signal red interval. These simplifications provide elegant theoretical properties that enable solving an analytical exact solution to this complex problem with parsimonious analytical insights. Numerical examples reveal that the proposed model can significantly reduce travel delay and fuel consumption. Moreover, it is demonstrated that the presented algorithm is highly efficient and appropriate for real-world traffic applications.

Sense-Store-Send: Trajectory Optimization for a Buffer-Aided Internet of UAVs

In this letter, we study a buffer-aided Internet of unmanned aerial vehicles (UAVs) in which a UAV performs data sensing, stores the data, and sends it to the base station (BS) in cellular networks. To minimize the overall completion time for all the sensing tasks, we formulate a joint trajectory, sensing location and sensing time optimization problem. To solve this NP-hard problem efficiently, we propose an iterative trajectory, sensing location and sensing time optimization (ITLTO) algorithm, and discuss the trade-off between sensing time and flying time. Simulation results show that the proposed algorithm can effectively reduce the completion time for the sensing tasks.

A Bisection Algorithm for Time-Optimal Trajectory Planning Along Fully Specified Paths

The time-optimal trajectory planning problem involves minimizing the time required to follow a path defined in space, subject to kinematic and dynamic constraints. Here, we introduce a novel technique, called the bisection algorithm (BA), which is fully implemented in C++ and extends dynamic programming approaches to the problem. These approaches, which rely on dividing the global problem into a series of simpler subproblems, become increasingly advantageous compared to direct transcription methods as the number of problem constraints increases. In contrast to nearly all other dynamic programming approaches, BA does not rely on finding a maximum-velocity curve or explicitly finding acceleration switching points during the trajectory planning process. Additionally, only one forward and one backward integration are used, during which all constraints are imposed. This approach is made feasible through careful control of the numerical integration process and the use of a bisection algorithm to resolve constraint violations during integration. BA is shown to be significantly simpler, faster, and more robust than recently proposed algorithms: a direct comparison is made for a series of paths to be followed by a serial manipulator, subject to kinematic constraints. The wide applicability of BA is then established by solving the time-optimal problem for a parallel manipulator following a complex path, subject to both kinematic and dynamic constraints.

Minimum Time—Minimum Jerk Optimal Traffic Management for AGVs

A combined minimum time - minimum jerk traffic management system for the vehicle coordination in an automated warehouse is presented. The algorithm is organised in two steps: in the first, a simple minimum time optimisation problem is solved, in the second step, this time-optimal solution is refined into a smooth minimum jerk plan for the autonomous forklifts in order to avoid impulsive forces that may unbalance the vehicle. For the first step, we propose a novel approach based on Linear Programming, which guarantees convergence to the optimal solution starting from a feasible point, and a low computational overhead, which makes it suitable for real-time applications. The output of this step is a piecewise constant velocity profile for all the moving robots that ensures collision avoidance. The second step takes such speed profile and generates its smoothed version, which minimises the jerk while respecting the same levels of safety of the solution generated by the first step. We discuss the different solutions with simulation and experimental data.

Coadjoint Orbits of Three-Step Free Nilpotent Lie Groups and Time-Optimal Control Problem

We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories with initial covectors lying in two-dimensional coadjoint orbits is studied. Under some broad conditions on the set of admissible velocities (in particular, in the sub-Riemannian case) the corresponding extremal controls are periodic, constant, or asymptotically constant.

Resource Allocation for Wireless Cooperative IoT Network With Energy Harvesting

In this paper, resource allocation is studied for a fully sustainable cooperative IoT network, in which a relay powered by renewable energy forwards data to a destination while charging multiple IoT nodes by radio-frequency (RF) signals. An optimal joint time and power allocation problem is formulated to maximize the long-term sum-throughput of IoT nodes, taking into consideration the bounded transmit power of IoT nodes, the stochastic characteristic of energy harvesting (EH) process, and dynamic wireless channel conditions. To solve the formulated problem, we analyze the time allocation for cooperative communications with EH, considering both data and energy dependency of the two hop transmissions in three cases with different network settings. Based on the analysis, we derive the closed-form solutions of optimal time and power allocation in a network with symmetric links. Then, we extend our solution to a general network with asymmetric links. By employing Lyapunov optimization, an online stochastic resource allocation algorithm is proposed to obtain the maximum network throughput. It has been shown that the proposed algorithm can achieve close-to-optimal network throughput while maintaining the stability of the system. Finally, extensive simulations validate the analysis and demonstrate the effectiveness of the proposed algorithm.

Extremals of a Left-Invariant Sub-Finsler Metric on the Engel Group

Using the Pontryagin maximum principle for the time-optimal problem in coordinates of the first kind, we find the extremals of an arbitrary left-invariant sub-Finsler metric on the Engel group defined by a distribution of rank 2.

Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables

The paper deals with the problem of optimal control with a Boltz-type quality index over a finite time interval for a linear steady-state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A.B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady-state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time-optimal problems.

Integrated Robust Control of the Global Toroidal Rotation and Total Plasma Energy in Tokamaks

Integrated-control solutions will play a significant role in future tokamaks, in which a variety of coupled control problems will need to be solved simultaneously by means of a limited number of actuators. In this article, the problem of simultaneously regulating the global toroidal rotation, Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ</sub> , and total plasma energy, W, is tackled. These two 0-D variables, Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ</sub> and W, depend on the ion toroidal rotation and electron temperature profiles, respectively. Both Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ</sub> and W also depend on the electron density and safety factor profiles. The actuation methods considered in this article are co-current and counter-current neutral beam injection. A nonlinear, robust controller that makes use of Lyapunov redesign techniques is synthesized based on 0-D, control-oriented models of the Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ</sub> and W dynamics. In addition, an actuator management scheme is designed to handle variations in the control priorities and availability of the neutral beam injectors. The actuator manager solves an optimization problem in real time in order to find the most appropriate course of action when unexpected changes occur. The integrated control architecture is tested for a DIII-D scenario by means of the 1-D code Control-Oriented Transport Simulator (COTSIM), which predicts the time evolution of the electron temperature, electron density, ion toroidal rotation, and safety factor profiles.

Periodic Time-Optimal Controls on Two-Step Free-Nilpotent Lie Groups

For two-step free nilpotent Lie algebras, we describe symplectic foliations and Casimir functions. A left-invariant time-optimal problem is considered in which the set of admissible controls is given by a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. We describe integrals for the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle. The properties of solutions to this system for low ranks of the Poisson bivector are described.

Optimal time task in saccadic eye movement

This article is devoted to fast goal-directed human eye movements. Such movements are described in few ways. One of them is the time-optimal problem model. It this work, the saccade – fast single conformable ballistic eye movement, is under consideration. Saccadic eye trajectory is described by time-optimal task solution. Differential equations system are based on pendulum model. It defines control moment applied from extraocular muscles to eye globe and movement of it. Main feature of the system is all values in suggested system have physical meaning and were found in published experimental investigations. Consideration of the moment applied from extraocular muscles to eye glob as control is main difference of the system from other models for eye movement in submitted works. In this case, the open-loop system has two real roots and one zero. The particular case of the ratio of real roots of 1:3 is analyzed in the article. As a result of modelling, trajectory characteristics closed to the saccades observed in experimental studies is obtained. On the basis of the constructed model, the parameters of the synthesized saccades were compared with the parameters obtained on the experimental sample. The obtained models can be used both for problems of physiology and medicine, and for constructing virtual reality environments. For example, when developing 3D interfaces, the use of a mathematical model makes it possible to optimize the elements arrangement using game theory.