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    25 October 2021, Volume 34 Issue 5
    Preface
    XIE Liang-Liang, ZHANG Ji-Feng
    2021, 34(5):  1597-1598. 
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    Year 2021 marks the 20th anniversary of this journal being retitled Journal of Systems Science and Complexity, under the suggestion of Professor Lei Guo, who became the Editorin- Chief in 2001 and led its development for three terms from 2001 to 2014. This year also sees the 60th birthday of Prof. Guo. Therefore, for a celebration of these happy occasions, we invited some of the most internationally renowned scholars in the field, who have connections with Prof. Guo in various ways, to contribute to this special issue. The contributions in this special issue cover many aspects of systems and control, including stochastic systems, system identification, optimal control, sampled-data control, control with delay, game theory, risk management, nonlinear systems, networked systems, etc. Besides recent progresses, the papers in this special issue also include reviews, reflections and perspectives. It is a celebration of Prof. Guo’s close to forty years’ contributions in the field by various authors. Prof. Guo has made fundamental contributions in adaptive control, system identification, adaptive signal processing, and stochastic systems. His current research interests include control of nonlinear uncertain systems, PID control theory, distributed filtering and estimation, capability of feedback, multi-agent systems, game-based control systems, and complex systems, among others. Besides his research contributions, Prof. Guo is also a leading figure in China’s advances in the field of systems and control. The 20 years’ development of this journal is just one example. We take this opportunity to wish Prof. Guo a very happy 60th birthday, and we look forward to many years of new contributions from him. XIE Liang-Liang, ZHANG Ji-Feng Beijing, 2021
    Revisiting Total Model Errors and Model Validation
    LJUNG Lennart
    2021, 34(5):  1598-1603.  DOI: 10.1007/s11424-021-1281-z
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    The paper contains a discussion of earlier work on Total Model Errors and Model Validation. It is maintained that the recent change of paradigm to kernel based system identification has also affected the basis for (and interest in) giving bounds for the total model error.
    Mathematical Models of Self-Appraisal in Social Networks
    ANDERSON Brian D. O. · YE Mengbin
    2021, 34(5):  1604-1633.  DOI: 10.1007/s11424-021-1193-y
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    In social networks where individuals discuss opinions on a sequence of topics, the selfconfidence an individual exercises in relation to one topic, as measured by the weighting given to their own opinion as against the opinion of all others, can vary in the light of the self-appraisal by the individual of their contribution to the previous topic. This observation gives rise to a type of model termed a DeGroot-Friedkin model. This paper reviews a number of results concerning this model. These include the asymptotic behavior of the self-confidence (as the number of topics goes to infinity), the possible emergence of an autocrat or small cohort of leaders, the effect of changes in the weighting given to opinions of others (in the light for example of their perceived expertise in relation to a particular topic under discussion), and the inclusion in the model of individual behavioral characteristics such as humility, arrogance, etc. Such behavioral characteristics create new opportunities for autocrats to emerge.
    Robust Designs Through Risk Sensitivity: An Overview
    BAS¸AR Tamer
    2021, 34(5):  1634-1665.  DOI: 10.1007/s11424-021-1242-6
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    This is an overview paper on the relationship between risk-averse designs based on exponential loss functions with or without an additional unknown (adversarial) term and some classes of stochastic games. In particular, the paper discusses the equivalences between risk-averse controller and filter designs and saddle-point solutions of some corresponding risk-neutral stochastic differential games with different information structures for the players. One of the by-products of these analyses is that risk-averse controllers and filters (or estimators) for control and signal-measurement models are robust, through stochastic dissipation inequalities, to unmodeled perturbations in controlled system dynamics as well as signal and the measurement processes. The paper also discusses equivalences between risk-sensitive stochastic zero-sum differential games and some corresponding risk-neutral three-player stochastic zero-sum differential games, as well as robustness issues in stochastic nonzero-sum differential games with finite and infinite populations of players, with the latter belonging to the domain of mean-field games.
    A Comprehensive Survey on STP Approach to Finite Games
    CHENG Daizhan · WU Yuhu · ZHAO Guodong · FU Shihua
    2021, 34(5):  1666-1680.  DOI: 10.1007/s11424-021-1232-8
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    Nowadays the semi-tensor product (STP) approach to finite games has become a promising new direction. This paper provides a comprehensive survey on this prosperous field. After a brief introduction for STP and finite (networked) games, a description for the principle and fundamental technique of STP approach to finite games is presented. Then several problems and recent results about theory and applications of finite games via STP are presented. A brief comment about the potential use of STP to artificial intelligence is also proposed.
    Revisiting the ODE Method for Recursive Algorithms: Fast Convergence Using Quasi Stochastic Approximation
    CHEN Shuhang · DEVRAJ Adithya · BERSTEIN Andrey · MEYN Sean
    2021, 34(5):  1681-1702.  DOI: 10.1007/s11424-021-1251-5
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    Several decades ago, Profs. Sean Meyn and Lei Guo were postdoctoral fellows at ANU, where they shared interest in recursive algorithms. It seems fitting to celebrate Lei Guo’s 60th birthday with a review of the ODE Method and its recent evolution, with focus on the following themes: • The method has been regarded as a technique for algorithm analysis. It is argued that this viewpoint is backwards: The original stochastic approximation method was surely motivated by an ODE, and tools for analysis came much later (based on establishing robustness of Euler approximations). The paper presents a brief survey of recent research in machine learning that shows the power of algorithm design in continuous time, following by careful approximation to obtain a practical recursive algorithm. • While these methods are usually presented in a stochastic setting, this is not a prerequisite. In fact, recent theory shows that rates of convergence can be dramatically accelerated by applying techniques inspired by quasi Monte-Carlo. • Subject to conditions, the optimal rate of convergence can be obtained by applying the averaging technique of Polyak and Ruppert. The conditions are not universal, but theory suggests alternatives to achieve acceleration. • The theory is illustrated with applications to gradient-free optimization, and policy gradient algorithms for reinforcement learning.
    Numerical Solutions for Optimal Control of Stochastic Kolmogorov Systems
    YIN George · WEN Zhexin · QIAN Hongjiang · NGUYEN Huy
    2021, 34(5):  1703-1722.  DOI: 10.1007/s11424-021-1170-5
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    This work is concerned with controlled stochastic Kolmogorov systems. Such systems have received much attention recently owing to the wide range of applications in biology and ecology. Starting with the basic premise that the underlying system has an optimal control, this paper is devoted to designing numerical methods for approximation. Different from the existing literature on numerical methods for stochastic controls, the Kolmogorov systems take values in the first quadrant. That is, each component of the state is nonnegative. The work is designing an appropriate discrete-time controlled Markov chain to be in line with (locally consistent) the controlled diffusion. The authors demonstrate that the Kushner and Dupuis Markov chain approximation method still works. Convergence of the numerical scheme is proved under suitable conditions.
    Safety-Critical Optimal Control for Autonomous Systems
    XIAO Wei · CASSANDRAS G. Christos · BELTA Calin
    2021, 34(5):  1723-1742.  DOI: 10.1007/s11424-021-1230-x
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    This paper presents an overview of the state of the art for safety-critical optimal control of autonomous systems. Optimal control methods are well studied, but become computationally infeasible for real-time applications when there are multiple hard safety constraints involved. To guarantee such safety constraints, it has been shown that optimizing quadratic costs while stabilizing affine control systems to desired (sets of) states subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). The CBF method is computationally efficient, and can easily guarantee the satisfaction of nonlinear constraints for nonlinear systems, but its wide applicability still faces several challenges. First, safety is hard to guarantee for systems with high relative degree, and the above mentioned QPs can easily be infeasible if tight or time-varying control bounds are involved. The resulting solution is also sub-optimal due to its myopic solving approach. Finally, this method works conditioned on the system dynamics being accurately identified. The authors discuss recent solutions to these issues and then present a framework that combines Optimal Control with CBFs, hence termed OCBF, to obtain near-optimal solutions while guaranteeing safety constraints even in the presence of noisy dynamics. An application of the OCBF approach is included for autonomous vehicles in traffic networks.
    Sampled-Data Semi-Global Robust Output Regulation for a Class of Nonlinear Systems
    LIU Wei · HUANG Jie
    2021, 34(5):  1743-1765.  DOI: 10.1007/s11424-021-1165-2
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    This paper investigates the semi-global robust output regulation problem for a class of uncertain nonlinear systems via a sampled-data output feedback control law. What makes the results interesting is that the nonlinearities of the proposed system do not have to satisfy linear growth condition and the uncertain parameters of our system are allowed to belong to some arbitrarily large prescribed compact subset. Two cases are considered. The first case is that the exogenous signal is constant. The second case is that the exogenous signal is time-varying and bounded. For the first case, the authors solve the problem exactly in the sense that the tracking error approaches zero asymptotically. For the second case, the authors solve the problem practically in the sense that the steady-state tracking error can be made arbitrarily small. Finally, an example is given to illustrate the effectiveness of our approach.
    Adaptive Optimization with Periodic Dither Signals
    XIE Siyu · WANG Le Yi
    2021, 34(5):  1766-1781.  DOI: 10.1007/s11424-021-1211-0
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    Optimization methods in cyber-physical systems do not involve parameter uncertainties in most existing literature. This paper considers adaptive optimization problems in which searching for optimal solutions and identifying unknown parameters must be performed simultaneously. Due to the dual roles of the input signals on achieving optimization and providing persistent excitation for identification, a fundamental conflict arises. In this paper, a method of adding a small deterministic periodic dither signal to the input is deployed to resolve this conflict and provide sufficient excitation for estimating the unknown parameters. The designing principle of the dither is discussed. Under dithered inputs, the authors show that simultaneous convergence of parameter estimation and optimization can be achieved. Convergence properties and convergence rates of parameter estimation and optimization variable updates are presented under the scenarios of uncertainty-free observations and systems with noisy observation and unmodeled components. The fundamental relationships and tradeoff among updating step sizes, dither magnitudes, parameter estimation errors, optimization accuracy, and convergence rates are further investigated.
    Prescribed-Time Control of Stochastic Nonlinear Systems with Reduced Control Effort
    LI Wuquan · KRSTIC Miroslav
    2021, 34(5):  1782-1800.  DOI: 10.1007/s11424-021-1217-7
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    A new prescribed-time state-feedback design is presented for stochastic nonlinear strictfeedback systems. Different from the existing stochastic prescribed-time design where scaling-free quartic Lyapunov functions or scaled quadratic Lyapunov functions are used, the design is based on new scaled quartic Lyapunov functions. The designed controller can ensure that the plant has an almost surely unique strong solution and the equilibrium at the origin of the plant is prescribed-time mean-square stable. After that, the authors redesign the controller to solve the prescribed-time inverse optimal mean-square stabilization problem. The merit of the design is that the order of the scaling function in the controller is reduced dramatically, which effectively reduces the control effort. Two simulation examples are given to illustrate the designs.
    Hidden Markov Models: Inverse Filtering, Belief Estimation and Privacy Protection
    LOURENC¸O Inˆes · MATTILA Robert · ROJAS Cristian R. · HU Xiaoming · WAHLBERG Bo
    2021, 34(5):  1801-1820.  DOI: 10.1007/s11424-021-1247-1
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    A hidden Markov model (HMM) comprises a state with Markovian dynamics that can only be observed via noisy sensors. This paper considers three problems connected to HMMs, namely, inverse filtering, belief estimation from actions, and privacy enforcement in such a context. First, the authors discuss how HMM parameters and sensor measurements can be reconstructed from posterior distributions of an HMM filter. Next, the authors consider a rational decision-maker that forms a private belief (posterior distribution) on the state of the world by filtering private information. The authors show how to estimate such posterior distributions from observed optimal actions taken by the agent. In the setting of adversarial systems, the authors finally show how the decision-maker can protect its private belief by confusing the adversary using slightly sub-optimal actions. Applications range from financial portfolio investments to life science decision systems.
    New Phase of Phase
    QIU Li · CHEN Wei · WANG Dan
    2021, 34(5):  1821-1839.  DOI: 10.1007/s11424-021-1249-z
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    Year 2021 is special. It sees the renaissance of concept of phase and the birth of a phase theory for matters much beyond complex numbers and single-input single-output (SISO) linear timeinvariant (LTI) systems while we celebrate the 60th birthday of Lei Guo, an exemplary research leader of our times. Here we give a short tutorial of the newly developed phase theory, as a birthday present.
    Identifiability and Solvability in Inverse Linear Quadratic Optimal Control Problems
    LI Yibei · WAHLBERG Bo · HU Xiaoming
    2021, 34(5):  1840-1857.  DOI: 10.1007/s11424-021-1245-3
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    In this paper, the inverse linear quadratic (LQ) problem over finite time-horizon is studied. Given the output observations of a dynamic process, the goal is to recover the corresponding LQ cost function. Firstly, by considering the inverse problem as an identification problem, its model structure is shown to be strictly globally identifiable under the assumption of system invertibility. Next, in the noiseless case a necessary and sufficient condition is proposed for the solvability of a positive semidefinite weighting matrix and its unique solution is obtained with two proposed algorithms under the condition of persistent excitation. Furthermore, a residual optimization problem is also formulated to solve a best-fit approximate cost function from sub-optimal observations. Finally, numerical simulations are used to demonstrate the effectiveness of the proposed methods.
    Stabilization of Continuous-Time Systems Against Stochastic Network Uncertainties: Fundamental Variance Bounds
    QI Tian · CHEN Jie
    2021, 34(5):  1858-1878.  DOI: 10.1007/s11424-021-1236-4
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    This paper studies the stabilizability and stabilization of continuous-time systems in the presence of stochastic multiplicative uncertainties. The authors consider multi-input, multi-output (MIMO) linear time-invariant systems subject to multiple static, structured stochastic uncertainties, and seek to derive fundamental conditions to ensure that a system can be stabilized under a mean-square criterion. In the stochastic control framework, this problem can be considered as one of optimal control under state- or input-dependent random noises, while in the networked control setting, a problem of networked feedback stabilization over lossy communication channels. The authors adopt a mean-square small gain analysis approach, and obtain necessary and sufficient conditions for a system to be meansquare stabilizable via output feedback. For single-input, single-output (SISO) systems, the condition provides an analytical bound, demonstrating explicitly how plant unstable poles, nonminimum phase zeros, and time delay may impose a limit on the uncertainty variance required for mean-square stabilization. For MIMO minimum phase systems with possible delays, the condition amounts to solving a generalized eigenvalue problem, readily solvable using linear matrix inequality optimization techniques.
    A Theory of the Risk for Empirical CVaR with Application to Portfolio Selection
    ARICI Giorgio · CAMPI Marco C. · CAR`E Algo · DALAI Marco · RAMPONI Federico A.
    2021, 34(5):  1879-1894.  DOI: 10.1007/s11424-021-1229-3
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    When decisions are based on empirical observations, a trade-off arises between flexibility of the decision and ability to generalize to new situations. In this paper, we focus on decisions that are obtained by the empirical minimization of the Conditional Value-at-Risk (CVaR) and argue that in CVaR the trade-off between flexibility and generalization can be understood on the ground of theoretical results under very general assumptions on the system that generates the observations. The results have implications on topics related to order and structure selection in various applications where the CVaR risk-measure is used. A study on a portfolio optimization problem with real data demonstrates our results.
    Optimal Control and Stabilization for Itˆo Systems with Input Delay
    WANG Hongxia · ZHANG Huanshui · XIE Lihua
    2021, 34(5):  1895-1926.  DOI: 10.1007/s11424-021-1226-6
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    The paper considers the linear quadratic regulation (LQR) and stabilization problems for Itˆo stochastic systems with two input channels of which one has input delay. The underlying problem actually falls into the field of asymmetric information control because of the nonidentical measurability induced by the input delay. In contrast with single-channel single-delay problems, the challenge of the problems under study lies in the interaction between the two channels which are measurable with respect to different filtrations. The key techniques conquering such difficulty are the stochastic maximum principle and the orthogonal decomposition and reorganization technique proposed in a companion paper. The authors provide a way to solve the delayed forward backward stochastic differential equation (D-FBSDE) arising from the maximum principle. The necessary and sufficient solvability condition and the optimal controller for the LQR problem are given in terms of a new Riccati differential equation established herein. Further, the necessary and sufficient stabilization condition in the mean square sense is provided and the optimal controller is given. The idea proposed in the paper can be extended to solve related control problems for stochastic systems with multiple input channels and multiple delays.
    Distributed Gradient Tracking Methods with Finite Data Rates
    MA Xiaoyu · YI Peng · CHEN Jie
    2021, 34(5):  1927-1952.  DOI: 10.1007/s11424-021-1231-9
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    This paper studies the distributed optimization problem over an undirected connected graph subject to digital communications with a finite data rate, where each agent holds a strongly convex and smooth cost function. The agents need to cooperatively minimize the average of all agents’ cost functions. Each agent builds an encoder/decoder pair that produces transmitted messages to its neighbors with a finite-level uniform quantizer, and recovers its neighbors’ states by a recursive decoder with received quantized signals. Combining the adaptive encoder/decoder scheme with the gradient tracking method, the authors propose a distributed quantized algorithm. The authors prove that the optimization can be achieved at a linear rate, even when agents communicate at 1-bit data rate. Numerical examples are also conducted to illustrate theoretical results.
    New Results in Stabilization of Uncertain Nonholonomic Systems: An Event-Triggered Control Approach
    LIU Tengfei · ZHANG Pengpeng · WANG Mengxi · JIANG Zhong-Ping
    2021, 34(5):  1953-1972.  DOI: 10.1007/s11424-021-1235-5
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    This paper reports latest developments in event-triggered and self-triggered control of uncertain nonholonomic systems in the perturbed chained form. In order to tackle the effects of drift uncertain nonlinearities, nonholonomic constraints and nonsmooth aperiodic sampling in eventbased control, a novel systematic design scheme is proposed by integrating set-valued maps with stateseparation and state-scaling techniques. The stability analysis of the closed-loop event-triggered control system is based on the cyclic-small-gain techniques that overcome the limitation of Lyapunov theory in the construction of Lyapunov functions for nonsmooth dynamical systems and enjoy inherent robustness properties due to the use of gain-based characterization of robust stability. More specifically, the closed-loop event-triggered control system is transformed into an interconnection of multiple input-tostate stable systems, to which the cyclic-small-gain theorem is applied for robust stability analysis. New self-triggered mechanisms are also developed as natural extensions of the event-triggered control result. The proposed event-based control design approach is new and original even when the system model is reduced to the ideal unperturbed chained form. Interestingly, the proposed methodology is also applicable to a broader class of nonholonomic systems subject to state and input-dependent uncertainties. The efficacy of the obtained event-triggered controllers is validated by a benchmark example of mobile robots subject to parametric uncertainties and a measurement noise such as bias in the orientation.
    Multi-Agent Control: A Graph-Theoretic Perspective
    CHEN Fei · REN Wei
    2021, 34(5):  1973-2002.  DOI: 10.1007/s11424-021-1218-6
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    Progress in development of multi-agent control is reviewed. Different approaches for multiagent control, estimation, and optimization are discussed in a systematic way with particular emphasis on the graph-theoretic perspective. Attention is paid to the design of multi-agent systems via Laplacian dynamics, as well as the role of the graph Laplacian spectrum, the challenges of unbalanced digraphs, and consensus-based estimation of graph statistics. Some emergent issues, e.g., distributed optimization, distributed average tracking, and distributed network games, are also reported, which have witnessed extensive development recently. There are over 200 references listed, mostly to recent contributions.
    Linear Quadratic Mean Field Games: Decentralized O(1/N)-Nash Equilibria
    HUANG Minyi · YANG Xuwei
    2021, 34(5):  2003-2035.  DOI: 10.1007/s11424-021-1266-y
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    This paper studies an asymptotic solvability problem for linear quadratic (LQ) mean field games with controlled diffusions and indefinite weights for the state and control in the costs. The authors employ a rescaling approach to derive a low dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The rescaling technique is further used for performance estimates, establishing an O(1/N)-Nash equilibrium for the obtained decentralized strategies.
    Biological Systems: Reliable Functions out of Randomness
    XIE Liang-Liang
    2021, 34(5):  2036-2047.  DOI: 10.1007/s11424-021-1271-1
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    What makes biological systems different from man-made systems? One distinction is explored in this paper: Biological systems achieve reliable functions through randomness, i.e., by both mitigating and exploiting the effects of randomness. The fundamental reason for biological systems to take such a random approach is the randomness of the microscopic world, which is dramatically different from the macroscopic world we are familiar with. To substantiate the idea, bacterial chemotaxis is used as an example.
    Distributed Recursive Projection Identification with Binary-Valued Observations
    WANG Ying · ZHAO Yanlong · ZHANG Ji-Feng
    2021, 34(5):  2048-2068.  DOI: 10.1007/s11424-021-1267-x
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    This paper investigates a distributed recursive projection identification problem with binaryvalued observations built on a sensor network, where each sensor in the sensor network measures partial information of the unknown parameter only, but each sensor is allowed to communicate with its neighbors. A distributed recursive projection algorithm is proposed based on a specific projection operator and a diffusion strategy. The authors establish the upper bound of the accumulated regrets of the adaptive predictor without any requirement of excitation conditions. Moreover, the convergence of the algorithm is given under the bounded cooperative excitation condition, which is more general than the previously imposed independence or persistent excitations on the system regressors and maybe the weakest one under binary observations. A numerical example is supplied to demonstrate the theoretical results and the cooperative effect of the sensors, which shows that the whole network can still fulfill the estimation task through exchanging information between sensors even if any individual sensor cannot.