Cost Function Research Articles

Mixed deterministic and stochastic disturbances in a discrete-time Nash game

This paper considers a discrete-time non-cooperative M-players linear affine quadratic game of pre-specified fixed duration, affected by stochastic noise and deterministic disturbances. The last one is seen as a pernicious fictitious player looking to maximise the expected cost functions of each player. Inspired in the discrete-time robust dynamic programming, sufficient conditions for the existence of a type of robust feedback NE are given in the solution of a set of discrete-time difference equations. A formal induction proof is provided for the closed form of the obtained robust set of strategies. Two illustrative simulation examples are included, one related to the problem of coordination of a two-echelon supply chain with uncertain seasonal demand. The goal of the agents is to reduce the expected cost of storage while satisfying a partially known demand. The second example is related to a game of government debt stabilisation. Comparing the simulation with the standard Nash strategy, the robust one achieves a better performance.

A Method of Dual-AGV-Ganged Path Planning Based on the Genetic Algorithm

The genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection, and it is known for its iterative optimization capabilities in both constrained and unconstrained environments. In this paper, a novel method for GA-based dual-automatic guided vehicle (AGV)-ganged path planning is proposed to address the problem of frequent steering collisions in dual-AGV-ganged autonomous navigation. This method successfully plans global paths that are safe, collision-free, and efficient for both leader and follower AGVs. Firstly, a new ganged turning cost function was introduced based on the safe turning radius of dual-AGV-ganged systems to effectively search for selectable safe paths. Then, a dual-AGV-ganged fitness function was designed that incorporates the pose information of starting and goal points to guide the GA toward iterative optimization for smooth, efficient, and safe movement of dual AGVs. Finally, to verify the feasibility and effectiveness of the proposed algorithm, simulation experiments were conducted, and its performance was compared with traditional genetic algorithms, Astra algorithms, and Dijkstra algorithms. The results show that the proposed algorithm effectively solves the problem of frequent steering collisions, significantly shortens the path length, and improves the smoothness and safety stability of the path. Moreover, the planned paths were validated in real environments, ensuring safe paths while making more efficient use of map resources. Compared to the Dijkstra algorithm, the path length was reduced by 30.1%, further confirming the effectiveness of the method. This provides crucial technical support for the safe autonomous navigation of dual-AGV-ganged systems.

Empirical optimal transport under estimated costs: Distributional limits and statistical applications

Optimal transport (OT) based data analysis is often faced with the issue that the underlying cost function is (partially) unknown. This is addressed in this paper with the derivation of distributional limits for the empirical OT value when the cost function and the measures are estimated from data. For statistical inference purposes, but also from the viewpoint of a stability analysis, understanding the fluctuation of such quantities is paramount. Our results find direct application in the problem of goodness-of-fit testing for group families, in machine learning applications where invariant transport costs arise, in the problem of estimating the distance between mixtures of distributions, and for the analysis of empirical sliced OT quantities.The established distributional limits assume either weak convergence of the cost process in uniform norm or that the cost is determined by an optimization problem of the OT value over a fixed parameter space. For the first setting we rely on careful lower and upper bounds for the OT value in terms of the measures and the cost in conjunction with a Skorokhod representation. The second setting is based on a functional delta method for the OT value process over the parameter space. The proof techniques might be of independent interest.

Myobolica: a stochastic approach to estimate physiological muscle control variability.

The inherent redundancy of the musculoskeletal systems is traditionally solved by optimizing a cost function. This approach may not be correct to model non-adult or pathological populations likely to adopt a "non-optimal" motor control strategy. Over the years, various methods have been developed to address this limitation, such as the stochastic approach. A well-known implementation of this approach, Metabolica, samples a wide number of plausible solutions instead of searching for a single one, leveraging Bayesian statistics and Markov Chain Monte Carlo algorithm, yet allowing muscles to abruptly change their activation levels. To overcome this and other limitations, we developed a new implementation of the stochastic approach (Myobolica), adding constraints and parameters to ensure the identification of physiological solutions. The aim of this study was to evaluate Myobolica, and quantify the differences in terms of width of the solution band (muscle control variability) compared to Metabolica. To this end, both muscle forces and knee joint force solutions bands estimated by the two approaches were compared to one another, and against (i) the solution identified by static optimization and (ii) experimentally measured knee joint forces. The use of Myobolica led to a marked narrowing of the solution band compared to Metabolica. Furthermore, the Myobolica solutions well correlated with the experimental data (R2 = 0.92, RMSE = 0.3 BW), but not as much with the optimal solution (R2 = 0.82, RMSE = 0.63 BW). Additional analyses are required to confirm the findings and further improve this implementation.

Characterizing barren plateaus in quantum ansätze with the adjoint representation

Variational quantum algorithms, a popular heuristic for near-term quantum computers, utilize parameterized quantum circuits which naturally express Lie groups. It has been postulated that many properties of variational quantum algorithms can be understood by studying their corresponding groups, chief among them the presence of vanishing gradients or barren plateaus, but a theoretical derivation has been lacking. Using tools from the representation theory of compact Lie groups, we formulate a theory of barren plateaus for parameterized quantum circuits whose observables lie in their dynamical Lie algebra, covering a large variety of commonly used ansätze such as the Hamiltonian Variational Ansatz, Quantum Alternating Operator Ansatz, and many equivariant quantum neural networks. Our theory provides, for the first time, the ability to compute the exact variance of the gradient of the cost function of the quantum compound ansatz, under mixing conditions that we prove are commonplace.

Demand Jump and Preemption under Irreversible Investment

Abstract This article shows that the possibility of preemption depends on the form of demand evolution and of the cost function. It characterizes the Markov perfect and open-loop equilibria of a two period game of capacity accumulation. When there is no demand evolution, there is no possibility of preemption under linear prices of investment. Preemption only appears when the demand shock between both periods is large enough.

Correlated Equilibria in Large Anonymous Bayesian Games

We consider multipopulation Bayesian games with a large number of players. Each player aims at minimizing a cost function that depends on this player’s own action, the distribution of players’ actions in all populations, and an unknown state parameter. We study the nonatomic limit versions of these games and introduce the concept of Bayes correlated Wardrop equilibrium, which extends the concept of Bayes correlated equilibrium to nonatomic games. We prove that Bayes correlated Wardrop equilibria are limits of action flows induced by Bayes correlated equilibria of the game with a large finite set of small players. For nonatomic games with complete information admitting a convex potential, we prove that the set of correlated and of coarse correlated Wardrop equilibria coincide with the set of probability distributions over Wardrop equilibria and that all equilibrium outcomes have the same costs. We get the following consequences. First, all flow distributions of (coarse) correlated equilibria in convex potential games with finitely many players converge to mixtures of Wardrop equilibria when the weight of each player tends to zero. Second, for any sequence of flows satisfying a no-regret property, its empirical distribution converges to the set of distributions over Wardrop equilibria, and the average cost converges to the unique Wardrop cost. Funding: This work was partially supported by European Cooperation in Science and Technology Action 16228 GAMENET. F. Koessler acknowledges the support of the Agence Nationale de la Recherche [Grant StratCom ANR-19-CE26-0010-01]. M. Scarsini acknowledges the support of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni project [Grant CUP_E53C22001930001], the Ministero dell’Università e della Ricerca Progetti di Rilevante Interesse Nazionale [Grant 2022EKNE5K], and the European Union-Next Generation EU, component M4C2, investment 1.1 (Ministero dell’Università e della Ricerca Progetti di Rilevante Interesse Nazionale Piano Nazionale di Ripresa e Resilienza) [Grant P2022XT8C8]. T. Tomala gratefully acknowledges the support of the HEC foundation and Agence Nationale de la Recherche/Investissements d’Avenir [Grant ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047].

Techno-economic analysis and optimization of a binary geothermal combined district heat and power plant for Puga Valley, India

This article investigates the energy, exergy, and economic performance of a Puga Valley-specific binary geothermal combined district heat and power plant, taking into account geotechnical data and related cost functions. The power plant uses an organic Rankine cycle (ORC) with R123 as the working fluid, utilizing waste heat for district heating to enhance efficiency. Variations in operating and design parameters such as geothermal fluid flow rate, temperature, and ORC temperature of the evaporator and condenser are found to have significant effects on the overall performance of the system and cost rate. A multi-objective grey wolf optimization technique based on artificial neural networks has been used to determine the ideal values of the above-mentioned parameters. The optimization study revealed a Pareto optimal curve with overall exergy efficiency and total cost rate as objective functions from which the optimal point was identified using the technique for order preference by similarity to the ideal solution (TOPSIS) method. While operating the proposed CHP plant at TOPSIS optimal point, the plant delivers a net electrical output of 1.08 MW alongside 4.19 MW of thermal energy, resulting in a total cost rate of 33.84 USD/h, with energy and exergy efficiencies peaking at 32.26 % and 36.8 %, respectively. The exergy analysis at the optimal point also revealed a notable reduction of 550 kW in the total exergy destruction rate compared with the base-case scenario. Furthermore, the design requirements of a 5 MW net power output CHP plant for effective recovery of the Puga Valley geothermal resource are discussed.

A Game-Theoretic Framework for Generic Second-Order Traffic Flow Models Using Mean Field Games and Adversarial Inverse Reinforcement Learning

A traffic system can be interpreted as a multiagent system, wherein vehicles choose the most efficient driving approaches guided by interconnected goals or strategies. This paper aims to develop a family of mean field games (MFG) for generic second-order traffic flow models (GSOM), in which cars control individual velocity to optimize their objective functions. GSOMs do not generally assume that cars optimize self-interested objectives, so such a game-theoretic reinterpretation offers insights into the agents’ underlying behaviors. In general, an MFG allows one to model individuals on a microscopic level as rational utility-optimizing agents while translating rich microscopic behaviors to macroscopic models. Building on the MFG framework, we devise a new class of second-order traffic flow MFGs (i.e., GSOM-MFG), which control cars’ acceleration to ensure smooth velocity change. A fixed-point algorithm with fictitious play technique is developed to solve GSOM-MFG numerically. In numerical examples, different traffic patterns are presented under different cost functions. For real-world validation, we further use an inverse reinforcement learning approach (IRL) to uncover the underlying cost function on the next-generation simulation (NGSIM) data set. We formulate the problem of inferring cost functions as a min-max game and use an apprenticeship learning algorithm to solve for cost function coefficients. The results show that our proposed GSOM-MFG is a generic framework that can accommodate various cost functions. The Aw Rascle and Zhang (ARZ) and Light-Whitham-Richards (LWR) fundamental diagrams in traffic flow models belong to our GSOM-MFG when costs are specified. History: This paper has been accepted for the Transportation Science Special Issue on ISTTT25 Conference. Funding: X. Di is supported by the National Science Foundation [CAREER Award CMMI-1943998]. E. Iacomini is partially supported by the Italian Research Center on High-Performance Computing, Big Data and Quantum Computing (ICSC) funded by MUR Missione 4-Next Generation EU (NGEU) [Spoke 1 “FutureHPC & BigData”]. C. Segala and M. Herty thank the Deutsche Forschungsgemeinschaft (DFG) for financial support [Grants 320021702/GRK2326, 333849990/IRTG-2379, B04, B05, and B06 of 442047500/SFB1481, HE5386/18-1,19-2,22-1,23-1,25-1, ERS SFDdM035; Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612; and Excellence Strategy of the Federal Government and the Länder]. Support through the EU DATAHYKING is also acknowledged. This work was also funded by the DFG [TRR 154, Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, Projects C03 and C05, Project No. 239904186]. Moreover, E. Iacomini and C. Segala are members of the Indam GNCS (Italian National Group of Scientific Calculus).

ε-Nash Equilibrium of Pursuer–Evader–Defender Missile Navigation Dynamic Games

This research is dedicated to developing a min–max robust control strategy for a dynamic game involving pursuers, evaders, and defenders in a multiple-missile scenario. The approach employs neural dynamic programming, utilizing multiple continuous differential neural networks (DNNs). The competitive controller devised addresses the robust optimization of a joint cost function that relies on the trajectories of the pursuer–evader–defender system, accommodating an uncertain mathematical model while adhering to control restrictions. The dynamic programming min–max formulation facilitates robust control by accounting for bounded modeling uncertainties and external disturbances for each game component. The value function of the Hamilton–Jacobi–Bellman (HJB) equation is approximated by a DNN, enabling the estimation of the closed-loop formulation for the joint dynamic game with state restrictions. The controller’s design is grounded in estimating the state trajectory under the worst possible uncertainties and perturbations, providing a robustness factor through the robust neural controller. The learning law class for the time-varying weights in the DNN is generated by studying the HJB partial differential equation for the missile motion for each player in the dynamic game. The controller incorporates the solution of the obtained learning laws and a time-varying Riccati equation, offering an online solution to the control implementation. A recurrent algorithm, based on the Kiefer–Wolfowitz method, adjusts the initial conditions for the weights to satisfy the final condition of the given cost function for the dynamic game. A numerical example is presented to validate the proposed robust control methodology, confirming the optimization solution based on the DNN approximation for Bellman’s value function.

On Uniqueness of an Optimal Solution to the Kantorovich Problem With Density Constraints

Abstract We study optimal transportation problems with constraints on densities of transport plans. We obtain a sharp condition for the uniqueness of an optimal solution to the Kantorovich problem with density constraints, namely that the Borel measurable cost function $h(x, y)$ satisfies the following non-degeneracy condition: $h(x, y)$ cannot be expressed as a sum of functions $u(x) + v(y)$ on a set of positive measure.

Time-Critical Unified Rendezvous Guidance for an Unmanned Autonomous Vehicle

Abstract This paper addresses the time-critical rendezvous problem for a pursuing autonomous unmanned vehicle, e.g., an unmanned aerial vehicle (UAV), guided using the concept of true-proportional navigation guidance, which is a variant of proportional-navigation guidance. In existing vehicle routing and flight time-constrained guidance techniques, specific rendezvous guidance commands are designed based on the specific motion of the target. In contrast to that, we propose a unified guidance command for a UAV that guarantees a time-critical rendezvous with a target that moves arbitrarily. We explore the purview of true-proportional navigation guidance and posit that a guidance law thus designed may be a potential candidate for designing time-critical rendezvous strategies against various target motions, even when the pursuer does not necessarily have a speed advantage over the target. We first derive a closed-form expression for the flight duration until rendezvous, over which we exercise control to make the pursuing vehicle rendezvous with the target at any feasible time prescribed a priori. Next, we ensure that the necessary flight time-based error variable converges to zero with an optimal convergence pattern with respect to a suitable cost function. We finally validate the efficacy of the proposed unified guidance command via numerical simulations.

Bayesian quadrature policy optimization for spacecraft proximity maneuvers and docking

Advancing autonomous spacecraft proximity maneuvers and docking (PMD) is crucial for enhancing the efficiency and safety of inter-satellite services. One primary challenge in PMD is the accurate a priori definition of the system model, often complicated by inherent uncertainties in the system modeling and observational data. To address this challenge, we propose a novel Lyapunov Bayesian actor-critic reinforcement learning algorithm that guarantees the stability of the control policy under uncertainty. The PMD task is formulated as a Markov decision process that involves the relative dynamic model, the docking cone, and the cost function. By applying Lyapunov theory, we reformulate temporal difference learning as a constrained Gaussian process regression, enabling the state-value function to act as a Lyapunov function. Additionally, the proposed Bayesian quadrature policy optimization method analytically computes policy gradients, effectively addressing stability constraints while accommodating informational uncertainties in the PMD task. Experimental validation on a spacecraft air-bearing testbed demonstrates the significant and promising performance of the proposed algorithm.

Towards more consistent volcano morphometry datasets: Assessing boundary delineation and DEM impact on geometric and drainage parameters

Composite volcanoes are dynamic landforms that require comprehensive morphological analysis to understand their formation, degradation and associated controlling processes. Establishing Digital Elevation Model (DEM) source, spatial resolution and edifice delineation method are the first essential steps to quantify volcano morphometry. The delineation and quantification of the morphology is a complex endeavor as a volcano’s edifice is the result of overlapping eruptive products, intrusions, and degradation by a range of erosional processes and thus needs to be assessed in different volcanic environments.In this study, sixteen volcanoes from four volcanic arcs are used to quantify and compare twelve different geometric and drainage parameters. We first perform an edifice delineation analysis where we compare the similarity of volcano boundaries drawn by seven volcano geomorphology experts. Afterwards, a second set of boundaries is drawn based on a slope threshold to guide the delineation between boundaries, which proves beneficial in enhancing consistency between expert-driven manual delineation. For the same volcanoes, we also extract automatic delineation algorithm-derived NETVOLC boundaries to complement the comparison. Afterwards, a comparative analysis of morphologic parameters is conducted for four free and globally available 30-m-resolution DEMs: ALOS (AW3D30), SRTM (SRTMGL1), ASTER (GDEM 003) and TanDEM-X. The impact of resolution is also assessed using the 12 m and 30 m grid TanDEM-X DEMs on the same parameters.Our results show that precise and consistent delineation of the volcanic edifice boundaries, and to a lesser degree the resolution of the DEM, holds greater significance than the specific DEM type used to extract morphometric parameters. The slope cost function of NETVOLC shows the lowest deviation from the expert-defined boundaries. The metrics most sensitive to the defined boundary are volumes and basin width, and the parameters that significantly differ between 12 m and 30 m TanDEM-X are irregularity index, eroded volume, slope and drainage density. Our analysis thus emphasizes the necessity of meticulous consideration when selecting the DEM, and more importantly, adopting a consistent approach to delineating edifice boundaries in comparative morphometric analyses of volcanoes.

Frequency-domain diffusion adaptation over networks with missing input data

Recently, a modified adapt-then-combine diffusion (mATC) strategy has been developed to handle distributed estimation problem with missing regressions (inputs). However, the mATC algorithm only considers the white input scenario and suffers from high complexity for long model filter lengths. To overcome these shortcomings, this paper proposes novel regularization-based frequency-domain diffusion algorithms for networks with missing input data. First, bias-eliminating cost function based on regularization is established by using the frequency-domain diagonal approximation. Then, with stochastic gradient descent, periodic update, and power normalization schemes, we design the regularization-based frequency-domain least mean square (R-FDLMS) algorithm as well as its normalized variant (R-FDNLMS). The latter converges faster than the former under colored inputs. The stability and steady-state behavior of the R-FDNLMS algorithm are also analyzed. Moreover, two effective power estimation methods are presented for both situations without and with the power ratio between the input signal and perturbation noise, along with a reset mechanism in the first case to enhance tracking performance. Finally, simulations are conducted to illustrate the superiority of the proposed algorithms and the validity of theoretical findings.

FOMCON Toolbox-Based Direct Approximation of Fractional Order Systems Using Gaze Cues Learning-Based Grey Wolf Optimizer

This study discusses a new method for the fractional-order system reduction. It offers an adaptable framework for approximating various fractional-order systems (FOSs), including commensurate and non-commensurate. The fractional-order modeling and control (FOMCON) toolbox in MATLAB and the gaze cues learning-based grey wolf optimizer (GGWO) technique form the basis of the recommended method. The fundamental advantage of the offered method is that it does not need intermediate steps, a mathematical substitution, or an operator-based approximation for the order reduction of a commensurate and non-commensurate FOS. The cost function is set up so that the sum of the integral squared differences in step responses and the root mean squared differences in Bode magnitude plots between the original FOS and the reduced models is as tiny as possible. Two case studies support the suggested method. The simulation results show that the reduced approximations constructed using the methodology under consideration have step and Bode responses more in line with the actual FOS. The effectiveness of the advocated strategy is further shown by contrasting several performance metrics with some of the contemporary approaches disseminated in academic journals.

Self-learning Markov prediction algorithm based aging-oriented gradient drop power control strategy for the transient modes of fuel cell hybrid electric vehicles

The power transients caused by switching from drive mode to brake mode in fuel cell hybrid electric vehicles (FCHEV) can result in significant degradation cost losses to the fuel cell. To address this issue, this study proposes a self-learning Markov prediction algorithm based aging-oriented gradient drop power control strategy. First, a real-time self-learning Markov predictor (SLMP) based on the traditional offline training Markov improvement is designed to predict the demand power and combined with the sequential quadratic programming (SQP) optimization algorithm to solve for the inner optimal demand power based on its global cost function minimization characteristic. On this basis, the fuel cell gradient drop power (FGDP) strategy is proposed to optimize the operating state of the vehicle powertrain under vehicle mode switching. This involves establishing a power gradient drop step based on considering the fuel cell hydrogen consumption cost and its lifetime degradation cost to further obtain the outer fuel cell demand power at the optimal step. And three execution modes are designed to trigger the FGDP strategy. Finally, by combining the above efforts, the SLMP-FGDP optimization control strategy is constructed. The numerical verification and hardware in loop experiments results show that the proposed improved SLMP can predict the vehicle demand power more accurately. Compared with the non-FGDP system, the SLMP-FGDP strategy can effectively near-eliminate the fuel cell power transient due to any braking scenario, thus effectively controlling the fuel cell lifetime degradation cost in a lower range and realizing a reduction of up to 52.21% of the fuel cell usage costs without significantly sacrificing the hydrogen fuel economy.

Optimal aggregation and disaggregation for coordinated operation of virtual power plant with distribution network operator

Virtual power plants (VPPs) offer an effective approach for managing distributed energy resources (DERs), including microturbines, distributed generators, demand response aggregators, and energy storage systems. This technology significantly enhances the economic efficiency and flexibility of distribution network systems. This study aims to facilitate a flexible power exchange between the distribution network system and the upper-level grid by aggregating the power flexibility of heterogeneous DERs via a VPP. Additionally, it introduces a methodology for optimal aggregation and disaggregation within a coordinated operation framework between the VPP and distribution network operator (DNO). On one hand, the VPP can determine its day-ahead feasible region and real-time flexible regulation power based on operational constraints of DERs and representative data provided by the DNO, circumventing the need for detailed network information. On the other hand, day-ahead and real-time correction procedures for the DER cost functions are proposed, effectively neutralizing the impact of network operational constraints on these cost functions. Consequently, precise cost functions for both the active power and the flexible regulation power aggregated by the VPP are derived. Employing this aggregated cost function enables the determination of a cost-minimized optimal scheduling solution in real-time by solving a fundamental economic dispatch problem, significantly alleviating computational demands. Finally, case studies demonstrate that the proposed method achieves an error in aggregated power of only 0.77%, compared to the precise computation method that requires comprehensive system information. The proposed optimal VPP disaggregation scheme exhibits power discrepancies of 0.63% and cost discrepancies of 0.91% relative to the precise method. Additionally, when implementing the most cost-effective demand response plan based on the proposed cost function, the average costs for aggregators are reduced by 19.7%.

Nonlinear RISE based integral reinforcement learning algorithms for perturbed Bilateral Teleoperators with variable time delay

In view of unknown Bilateral Teleoperators, the simultaneous satisfaction of not only synchronous control problem between two sides, but also optimal control performance under time-varying delays and external disturbance is a difficult task. Initially, in order to address this challenge, we proposed two control approaches including On-Policy and Off-Policy strategies after employing the sliding variable to reduce the order of dynamic model. Additionally, since the development of the unification between synchronous control problem and optimal control performance, it requires the consideration of discount factor addition to guarantee the bound of the infinite horizon cost function. In order to handle the dynamic uncertainties and the exponential cost function, by fully considering data collection technique in model-free On-Policy and Off-Policy strategies, Reinforcement Learning control schemes are proposed to guarantee the optimality effectiveness. Consequently, the Robust Integral of the Sign of the Error term is integrated into two proposed optimal control frames to improve the synchronous control performance and estimation capability of dynamic uncertainties. Moreover, the convergence of control policies and synchronous control problem of two proposed control frames are rigorously analyzed by Lyapunov stability theory. Finally, simulation results and the comparisons with the existing controller demonstrate the effectiveness of two proposed control frameworks.

Multi-agent flocking control with complex obstacles and adaptive distributed convex optimization

The optimization issue involving flocking control and obstacle avoidance behavior in continuous-time multi-agent systems is the main topic of this work. When an agent encounters an obstacle, a virtual agent is introduced to generate repulsive force, enabling the multi-agent system to navigate around obstacles smoothly. Under the local interaction between agents, a multi-agent flocking control and obstacle avoidance algorithm is proposed with utilizing adaptive distributed convex optimization. This algorithm can minimize the sum of local costs of the system. A symbolic function-based estimation method is introduced to effectively relax the constraints of the cost function. The Lyapunov approach is employed to show the stability of the multi-agent system. With the proposed algorithm, each agent can achieve flocking behavior for convex optimization in complex obstacle environments by interacting with their neighbors, virtual agents, and virtual leaders. Last but not least, simulation examples are given to further illustrate the effectiveness and efficiency of the suggested control algorithm.