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    25 August 2021, Volume 34 Issue 4
    Exponential Sums over Finite Fields
    WAN Daqing
    2021, 34(4):  1225-1278.  DOI: 10.1007/s11424-021-0066-8
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    This is an expository paper on algebraic aspects of exponential sums over finite fields. This is a new direction. Various examples, results and open problems are presented along the way, with particular emphasis on Gauss periods, Kloosterman sums and one variable exponential sums. One main tool is the applications of various p-adic methods. For this reason, the author has also included a brief exposition of certain p-adic estimates of exponential sums. The material is based on the lectures given at the 2020 online number theory summer school held at Xiamen University. Notes were taken by Shaoshi Chen and Ruichen Xu.
    A Nonlinear Optimal Control Approach for Tracked#br# Mobile Robots#br#
    RIGATOS Gerasimos
    2021, 34(4):  1279-1300.  DOI: 10.1007/s11424-021-0036-1
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    The article proposes a nonlinear optimal (H-infinity) control approach for the model of a tracked robotic vehicle. The kinematic model of such a tracked vehicle takes into account slippage effects due to the contact of the tracks with the ground. To solve the related control problem, the dynamic model of the vehicle undergoes first approximate linearization around a temporary operating point which is updated at each iteration of the control algorithm. The linearization process relies on first-order Taylor series expansion and on the computation of the Jacobian matrices of the state-space model of the vehicle. For the approximately linearized description of the tracked vehicle a stabilizing H-infinity feedback controller is designed. To compute the controller’s feedback gains an algebraic Riccati equation is solved at each time-step of the control method. The stability properties of the control scheme are proven through Lyapunov analysis. It is also demonstrated that the control method retains the advantages of linear optimal control, that is fast and accurate tracking of reference setpoints under moderate variations of the control inputs.
    The Complex Dynamics of Hepatitis B Infected Individuals with Optimal Control
    DIN Anwarud · LI Yongjin · SHAH Murad Ali
    2021, 34(4):  1301-1323.  DOI: 10.1007/s11424-021-0053-0
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    This paper proposes various stages of the hepatitis B virus (HBV) besides its transmissibility and nonlinear incidence rate to develop an epidemic model. The authors plan the model, and then prove some basic results for the well-posedness in term of boundedness and positivity. Moreover, the authors find the threshold parameter R0, called the basic/effective reproductive number and carry out local sensitive analysis. Furthermore, the authors examine stability and hence condition for stability in terms of R0. By using sensitivity analysis, the authors formulate a control problem in order to eradicate HBV from the population and proved that the control problem actually exists. The complete characterization of the optimum system was achieved by using the 4th-order Runge-Kutta procedure.
    Distributed Nonsmooth Convex Optimization over Markovian Switching Random Networks with Two Step-Sizes
    YI Peng · LI Li
    2021, 34(4):  1324-1344.  DOI: 10.1007/s11424-020-0071-3
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    This paper investigates the distributed convex optimization problem over a multi-agent system with Markovian switching communication networks. The objective function is the sum of each agent’s local nonsmooth objective function, which cannot be known by other agents. The communication network is assumed to switch over a set of weight-balanced directed graphs with a Markovian property. The authors propose a consensus sub-gradient algorithm with two time-scale step-sizes to handle the Markovian switching topologies and the absence of global gradient information. With proper selection of step-sizes, the authors prove the almost sure convergence of all agents’ local estimates to the same optimal solution when the union graph of the Markovian network’ states is strongly connected and the Markovian chain is irreducible. The convergence rate analysis is also given for specific cases. Simulations are given to demonstrate the results.
    Robust Control for Discrete-Time T-S Fuzzy Singular Systems
    CHEN Jian · YU Jinpeng
    2021, 34(4):  1345-1363.  DOI: 10.1007/s11424-020-0059-z
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    This paper deals with the robust admissibility and state feedback stabilization problems for discrete-time T-S fuzzy singular systems with norm-bounded uncertainties. By introducing a new approximation technique, the initial membership functions are conveniently expressed in piecewiselinear functions with the consideration of the approximation errors. By utilizing the piecewise-linear membership functions, the fuzzy weighting-based Lyapunov function and the use of auxiliary matrices, the admissibility of the systems is determined by examining the conditions at some sample points. The conditions can be reduced into the normal parallel distributed compensation ones by choosing special values of some slack matrices. Furthermore, the authors design the robust state feedback controller to guarantee the closed-loop system to be admissible. Two examples are provided to illustrate the advantage and effectiveness of the proposed method.
    A Novel Networked Predictive Control Method for Systems with Random Communication Constraints
    PANG Zhonghua · BAI Chuandong · LIU Guoping · HAN Qinglong · ZHANG Xianming
    2021, 34(4):  1364-1378.  DOI: 10.1007/s11424-021-0160-y
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    This paper presents a novel observer-based predictive control method for networked systems where random network-induced delays, packet disorders and packet dropouts in both feedback and forward channels are considered. The proposed method has three significant features: i) A concept of destination-based lumped (DBL) delay is introduced to represent the combined effects of random communication constraints in each channel; ii) in view of different natures of the random DBL delays in the feedback and forward channels, different compensation schemes are designed; and iii) it is actual control inputs rather than predicted ones that are employed to generate future control signals based on the latest system state estimate available in the controller. For the resulting closed-loop system, a necessary and sufficient stability condition is derived, which is less conservative and also independent of random communication constraints in both channels. Simulation results are provided to demonstrate the effectiveness of the proposed method.
    Nonlinear Model Predictive Control-Based Guidance Algorithm for Quadrotor Trajectory Tracking with Obstacle Avoidance
    ZHAO Chunhui · WANG Dong · HU Jinwen · PAN Quan
    2021, 34(4):  1379-1400.  DOI: 10.1007/s11424-021-0316-9
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    This paper studies a novel trajectory tracking guidance law for a quadrotor unmanned aerial vehicle (UAV) with obstacle avoidance based on nonlinear model predictive control (NMPC) scheme. By augmenting a reference position trajectory to a reference dynamical system, the authors formulate the tracking problem as a standard NMPC design problem to generate constrained reference velocity commands for autopilots. However, concerning the closed-loop stability, it is difficult to find a local static state feedback to construct the terminal constraint in the design of NMPC-based guidance law. In order to circumvent this issue, the authors introduce a contraction constraint as a stability constraint, which borrows the ideas from the Lyapunov’s direct method and the backstepping technique. To achieve the obstacle avoidance extension, the authors impose a well-designed potential field function-based penalty term on the performance index. Considering the practical application, the heavy computational burden caused by solving the NMPC optimization problem online is alleviated by using the dynamical adjustment of the prediction horizon for the real-time control. Finally, extensive simulations and the real experiment are given to demonstrate the effectiveness of the proposed NMPC scheme.
    Q-Learning-Based Target Selection for Bearings-Only Autonomous Navigation
    XIONG Kai · WEI Chunling
    2021, 34(4):  1401-1425.  DOI: 10.1007/s11424-020-9265-y
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    This paper presents a Q-learning-based target selection algorithm for spacecraft autonomous navigation using bearing observations of known visible targets. For the considered navigation system, the position and velocity of the spacecraft are estimated using an extended Kalman filter (EKF) with the measurements of inter-satellite line-of-sight (LOS) vectors obtained via an onboard star camera. This paper focuses on the selection of the appropriate target at each observation period for the star camera adaptively, such that the performance of the EKF is enhanced. To derive an effective algorithm, a Q-function is designed to select a proper observation region, while a U-function is introduced to rank the targets in the selected region. Both the Q-function and the U-function are constructed based on the sequence of innovations obtained from the EKF. The efficiency of the Q-learning-based target selection algorithm is illustrated via numerical simulations, which show that the presented algorithm outperforms the traditional target selection strategy based on a Cramer-Rao bound (CRB) in the case that the prior knowledge about the target location is inaccurate.
    Distributed Optimal Control of Nonlinear Time-Delay System Subject to Delayed Measurements and Communication Disruptions
    SU Baili · DUAN Yuxing
    2021, 34(4):  1426-1437.  DOI: 10.1007/s11424-020-9302-x
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    This paper is focused on a distributed optimal control design for a class of nonlinear timedelay systems with delayed measurements and communication disruptions. The innovation lies in three aspects. The distributed optimal control method which includes an optimal controller and a bounded controller is designed based on Lyapunov function. The availability of data transmitted through the communication channel depends on a feasibility problem. And a sufficient condition to guarantee ultimate boundedness of the system is given based on appropriate assumptions. The significance of this paper is that this distributed optimal control method is applied to time-delay system. Finally, a simulation example is given to verify the effectiveness of the proposed method.
    Convergence of Distributed Gradient-Tracking-Based Optimization Algorithms with Random Graphs
    WANG Jiexiang · FU Keli · GU Yu · LI Tao
    2021, 34(4):  1438-1453.  DOI: 10.1007/s11424-021-9355-5
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    This paper studies distributed convex optimization over a multi-agent system, where each agent owns only a local cost function with convexity and Lipschitz continuous gradients. The goal of the agents is to cooperatively minimize a sum of the local cost functions. The underlying communication networks are modelled by a sequence of random and balanced digraphs, which are not required to be spatially or temporally independent and have any special distributions. The authors use a distributed gradient-tracking-based optimization algorithm to solve the optimization problem. In the algorithm, each agent makes an estimate of the optimal solution and an estimate of the average of all the local gradients. The values of the estimates are updated based on a combination of a consensus method and a gradient tracking method. The authors prove that the algorithm can achieve convergence to the optimal solution at a geometric rate if the conditional graphs are uniformly strongly connected, the global cost function is strongly convex and the step-sizes don’t exceed some upper bounds.
    Statistical Identification of Important Nodes in Biological Systems
    WANG Pei
    2021, 34(4):  1454-1470.  DOI: 10.1007/s11424-020-0013-0
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    Biological systems can be modeled and described by biological networks. Biological networks are typical complex networks with widely real-world applications. Many problems arising in biological systems can be boiled down to the identification of important nodes. For example, biomedical researchers frequently need to identify important genes that potentially leaded to disease phenotypes in animal and explore crucial genes that were responsible for stress responsiveness in plants. To facilitate the identification of important nodes in biological systems, one needs to know network structures or behavioral data of nodes (such as gene expression data). If network topology was known, various centrality measures can be developed to solve the problem; while if only behavioral data of nodes were given, some sophisticated statistical methods can be employed. This paper reviewed some of the recent works on statistical identification of important nodes in biological systems from three aspects, that is, 1) in general complex networks based on complex networks theory and epidemic dynamic models; 2) in biological networks based on network motifs; and 3) in plants based on RNA-seq data. The identification of important nodes in a complex system can be seen as a mapping from the system to the ranking score vector of nodes, such mapping is not necessarily with explicit form. The three aspects reflected three typical approaches on ranking nodes in biological systems and can be integrated into one general framework. This paper also proposed some challenges and future works on the related topics. The associated investigations have potential real-world applications in the control of biological systems, network medicine and new variety cultivation of crops.
    On the Properties of Cheng Projection
    FENG Jun-e · ZHANG Qingle · LI Yiliang
    2021, 34(4):  1471-1486.  DOI: 10.1007/s11424-021-9254-9
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    This paper mainly investigates some properties of Cheng projection, which was proposed recently by Prof. Cheng to reduce the dimension of vector. As a linear transformation from the original vector space to the target vector space, the matrix representation of Cheng projection is given. Then, the structure matrix of Cheng projection, called the Cheng projection matrix, is obtained. Algebraic properties of Cheng projection are derived via its projection matrix. Furthermore, the product and norm of Cheng projection matrices are concerned.
    A Guaranteed Cost Approach to Dynamic Output Feedback Control for Neutral-Type Markovian Jumping Stochastic Systems
    LI Yanbo · KAO Binghua · XIE Jing · KAO Yonggui
    2021, 34(4):  1487-1500.  DOI: 10.1007/s11424-020-9145-5
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    This paper is devoted to investigating the dynamic output feedback (DOF) control problem of Markovian jump neutral-type stochastic systems with a guaranteed cost function. Both of the state and measurement equations contain time delays. Mode-dependent DOF controllers are first designed such that the closed-loop system is asymptotically stable in mean-square and an adequate performance level of this system is guaranteed. Then, sufficient conditions for the solvability of this problem are derived in the form of linear matrix inequalities (LMIs). A numerical example is presented to reveal the effectiveness of our findings.
    Analysis on the Mutation and Time-Varying Characteristics of Coal Price System Evolution from the Perspective of Finance
    CHAI Jian · LEI Junchan · SHI Huiting · ZHANG Xuejun · LANG Meiling
    2021, 34(4):  1501-1521.  DOI: 10.1007/s11424-020-0061-5
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    Coal is essential to ensure China’s energy security. The sudden or gradual change of coal price reflects the degree of disequilibrium or expected disequilibrium of coal supply and demand, which will not be conducive to energy security. Therefore, it is important to analyze the change points of coal price and explore the reason of the price fluctuation. This paper analyses the coal price from January 2008 to June 2019 as the perspective of the financial market. Firstly, the PPM-DBSCAN model is used to identify the mutation point of coal price fluctuation. Secondly, path analysis is used to extract the core driving factors that affect coal price. Thirdly, the authors construct a time-varying and time-lag effect analysis model for structural changes of coal price based on the TVP-VAR model. The results show that there are 11 mutation points of coal price fluctuation. Financial market factors, coal supply and demand and alternative factors are the reasons of coal price mutation. The authors find that the imbalance of coal supply and demand in traditional view cannot fully explain the fluctuation of coal price. The impact of the financial market and non-thermal power generation have more influence on the coal price.
    The Role of White Precious Metals in the Allocation of General Asset
    SHEN Caisheng · ZHANG Tiancheng · CHEN Ying
    2021, 34(4):  1522-1537.  DOI: 10.1007/s11424-021-0044-1
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    White precious metals have often been regarded as industrial raw materials in the past. With the introduction of the Silver ETF in 2006 and Platinum as well as Palladium ETFs in 2010, their role in asset allocation has become more and more important. This paper selects five countries with rapid economic development (BRICS) as the research scope. The authors select silver, platinum and palladium to test whether these three kinds of white precious metals have the effect of hedging risk in general asset (stocks, bonds and foreign exchange). The authors find that, first of all, the three white precious metals have no stable effect on the BRICS stock market. Second, silver can better hedge the bond market of the BRICS, the authors suspect that this phenomenon is related to the fact that silver is a traditional financial asset and its trading volume is large. Third, white precious metals are good hedge assets for foreign exchange investment. Finally, the authors find that platinum can reduce the yield variance of all portfolios, indicating that platinum is an investment target that can effectively reduce the variance of portfolio returns within the BRICS.
    Inventory Games with Quantity Discount
    LI Wenzhong · XU Genjiu · SU Jun
    2021, 34(4):  1538-1554.  DOI: 10.1007/s11424-021-9235-z
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    Meca, et al. (2004) studied a class of inventory games which arise when a group of retailers who observe demand for a common good decide to cooperate and make joint orders. In this paper, the authors extend their model to the situation where the manufacturer provides the retailers with a price discount on purchases in excess of a certain order quantity. The authors define the corresponding inventory game with quantity discount, and show that the inventory game has a nonempty core. Then, the authors propose a core allocation rule, the demand-proportionality rule, and characterize it by introducing efficiency, the retailer splitting property and continuity.
    Functional Multiple-Outcome Model in Application to Multivariate Growth Curves of Infant Data
    YAN Xingyu · ZHOU Yingchun · PU Xiaolong · ZHAO Peng
    2021, 34(4):  1555-1577.  DOI: 10.1007/s11424-020-9319-1
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    Motivated by a medical study that attempts to analyze the relationship between growth curves and other variables and to measure the association among multiple growth curves, the authors develop a functional multiple-outcome model to decompose the total variation of multiple functional outcomes into variation explained by independent variables with time-varying coefficient functions, by latent factors and by noise. The latent factors are the hidden common factors that influence the multiple outcomes and are found through the combined functional principal component analysis approach. Through the coefficients of the latent factors one may further explore the association of the multiple outcomes. This method is applied to the multivariate growth data of infants in a real medical study in Shanghai and produces interpretable results. Convergence rates for the proposed estimates of the varying coefficient and covariance functions of the model are derived under mild conditions.
    Robust Estimation for Poisson Integer-Valued GARCH Models Using a New Hybrid Loss
    LI Qi · CHEN Huaping · ZHU Fukang
    2021, 34(4):  1578-1596.  DOI: 10.1007/s11424-020-9344-0
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    The Poisson integer-valued GARCH model is a popular tool in modeling time series of counts. The commonly used maximum likelihood estimator is strongly influenced by outliers, so there is a need to develop a robust M-estimator for this model. This paper has three aims. First, the authors propose a new loss function, which is a hybrid of the tri-weight loss for relatively small errors and the exponential squared loss for relatively large ones. Second, Mallows’ quasi-likelihood estimator (MQLE) is proposed as an M-estimator and its existence, uniqueness, consistency and asymptotic normality are established. In addition, a data-adaptive algorithm for computing MQLE is given based on a datadriven selection of tuning parameters in the loss function. Third, simulation studies and analysis of a real example are conducted to illustrate the performance of the new estimator, and a comparison with existing estimators is made.