Adaptive tracking control of leader–follower systems with unknown dynamics and partial measurements

In this paper, a decentralized adaptive tracking control is developed for a second-order leader–follower system with unknown dynamics and relative position measurements. Linearly parameterized models are used to describe the unknown dynamics of a self-active leader and all followers. A new distributed system is obtained by using the relative position and velocity measurements as the state variables. By only using the relative position measurements, a dynamic output–feedback tracking control together with decentralized adaptive laws is designed for each follower. At the same time, the stability of the tracking error system and the parameter convergence are analyzed with the help of a common Lyapunov function method. Some simulation results are presented to validate the proposed adaptive tracking control.     

Communication free leader–follower formation control of unmanned aircraft systems

This paper proposes a formation control strategy for unmanned aircraft under which there is no need to information exchange among aircraft. Based on the measurement of relative information such as distances and orientations obtained from practical sensors, the formation is realized by employing the feedback linearization approach. By considering the leader maneuver to be unknown, a nonlinear estimator is designed and the stability of the combined controller–estimator is guaranteed. Simulation results confirm the effectiveness of the proposed formation flight control strategy.     

Manipulability of leader–follower networks with the rigid-link approximation

This paper introduces the notion of manipulability to mobile, multi-agent networks as a tool to analyze the instantaneous effectiveness of injecting control inputs at certain, so-called leader nodes in the network. Effectiveness is interpreted to characterize how the movements of the leader nodes translate into responses among the remaining follower nodes. This notion of effectiveness is a function of the interaction topologies, the agent configurations, and the particular choice of inputs used to influence the network. In fact, classic manipulability is an index used in robotics to analyze the singularity and efficiency of configurations of robot-arm manipulators. To define similar notions for leader–follower networks, we use a rigid-link approximation of the follower dynamics and, under this assumption, we prove that the instantaneous follower velocities can be uniquely determined from that of the leaders’, which allows us to define a meaningful and computable manipulability index for the leader–follower networks. This paper examines the property of the proposed index in simulation and with real mobile robots, and demonstrates how the index can be used to find effective interaction topologies.     

A distributed model predictive control scheme for leader–follower multi-agent systems

In this paper, we present a novel receding horizon control scheme for solving the formation problem of leader–follower configurations. The algorithm is based on set-theoretic ideas and is tuned for agents described by linear time-invariant (LTI) systems subject to input and state constraints. The novelty of the proposed framework relies on the capability to jointly use sequences of one-step controllable sets and polyhedral piecewise state-space partitions in order to online apply the ‘better’ control action in a distributed receding horizon fashion. Moreover, we prove that the design of both robust positively invariant sets and one-step-ahead controllable regions is achieved in a distributed sense. Simulations and numerical comparisons with respect to centralised and local-based strategies are finally performed on a group of mobile robots to demonstrate the effectiveness of the proposed control strategy.     

An adaptive control architecture for leader–follower multiagent systems with stochastic disturbances and sensor and actuator attacks

In this paper, we develop a novel distributed adaptive control architecture for addressing networked multiagent systems subject to stochastic exogenous disturbances with compromised sensor and actuators. Specifically, for a class of linear leader–follower multiagent systems, we develop a new structure of the neighbourhood synchronisation error for the control design protocol of each follower. The proposed control algorithm addresses time-varying multiplicative sensor attacks on the leader state measurements. In addition, the framework addresses time-varying multiplicative actuator attacks on the followers that do not have a communication link with the leader and additive actuator attacks on all follower agents in the network. The proposed adaptive controller guarantees uniform ultimate boundedness of the state tracking error for each agent in a mean-square sense.     

Distributed event-triggered cooperative attitude control of multiple rigid bodies with leader–follower architecture

In this note, the distributed event-triggered cooperative attitude control of multiple rigid bodies with leader–follower architecture is investigated, where both the cases of static and dynamic leaders are all considered. Two distributed triggering procedures are first introduced for the followers and leaders, and then the distributed cooperative controllers are designed under the proposed triggering schemes. Under the designed controllers with the event-triggered strategies, it is shown that the orientations of followers converge to the convex hull formed by the desired leaders’ orientations with zero angular velocities. Moreover, the communication pressure in network is reduced and the energy of each agent is saved. Simulation results show the effectiveness of the proposed method.     

On disturbance rejection proportional–integral–differential control with model-free adaptation

In this paper, an adaptive disturbance-rejection proportional–integral–differential (PID) control method is proposed for a class of nonlinear systems. First, PID-type criterion is introduced in a model-free adaptive control (MFAC) framework, which gives an optimal control interpretation for PID controller. Then, the design of adaptive disturbance rejection PID is proposed based on this new interpretation to realize controller gain auto-tuning. Due to the ingenious integration of active disturbance rejection and adaptive mechanism, the proposed adaptive disturbance rejection PID control scheme exhibits better control performance than MFAC case. Furthermore, the boundedness of controller gain, the convergence of tracking error and the bounded-input–bounded-output stability are proved for the proposed control system. Finally, the effectiveness of the proposed method is verified by numerical simulation.     

A leader–follower model for discrete competitive facility location

In this paper we investigate a leader–follower (Stackelberg equilibrium) competitive location model. The competitive model is based on the concept of cover. Each facility attracts consumers within a “sphere of influence” defined by a “radius of influence.” The leader and the follower, each has a budget to be spent on the expansion of their chains either by improving their existing facilities or constructing new ones. We find the best strategy for the leader assuming that the follower, knowing the action taken by the leader, will react by investing his budget to maximize his market share. The objective of the leader is to maximize his market share following the follower׳s reaction.     

Leader–follower fixed-time consensus of multi-agent systems with high-order integrator dynamics

The leader–follower fixed-time consensus of high-order multi-agent systems with external disturbances is investigated in this paper. A novel sliding manifold is designed to ensure that the tracking errors converge to zero in a fixed-time during the sliding motion. Then, a distributed control law is designed based on Lyapunov technique to drive the system states to the sliding manifold in finite-time independent of initial conditions. Finally, the efficiency of the proposed method is illustrated by numerical simulations.     

Leader–follower consensus of linear multi-agent systems with unknown external disturbances

This paper addresses the leader–follower consensus tracking problem for multi-agent systems with identical general linear dynamics and unknown external disturbances. First, a distributed extended state observer is proposed, where both the local states and disturbance of each agent are estimated simultaneously by using the relative output information between neighbors. Then a consensus algorithm is proposed for each agent based on the relative estimated states between neighbors and its own disturbance estimate. It is shown that, with the proposed observer-based consensus algorithm, the leader–follower consensus problem can be solved. Finally, we present a simulation example to demonstrate the effectiveness of the proposed algorithm.     

An output formation consensus control for leader–follower networked multi-agent systems under communication constraints and switching topologies

In this paper, the formation consensus problem for a class of leader–follower networked multi-agent systems under communication constraints and switching topologies is investigated. A networked predictive control scheme is proposed to achieve stability and output formation consensus with the switching topology, capable of compensating for data loss and time delays in the network. By equating the whole closed-loop networked multi-agent system with the proposed control scheme to the corresponding switched system, the sufficient and necessary condition of output formation consensus and stability for agents is given. Finally, using three-degree-of-freedom air-bearing spacecraft simulators as the control objects, the proposed scheme is demonstrated to be able to actively compensate for the communication constraints through numerical simulations, and it is also verified to have a good control performance by further realizing the formation task of the simulators through practical experiments.     

Leader–follower synchronisation for networked Lagrangian systems with uncertainties: a learning approach

This article addresses a leader–follower synchronisation problem of networked Lagrangian systems with uncertainties by an iterative learning control approach. The inherent properties of the systems are fully utilised in the controller design, and a directed acyclic graph is sufficient for communication among subsystems. The developed controller contains a proportional-plus-derivative (PD) term and two learning terms. The PD term drives the tracking error to zero, one learning term compensates for the model uncertainties, and the other one is used for disturbance rejection. It is shown that the synchronisation task can be achieved by the proposed controller, and all internal signals are either bounded or norm bounded. The theoretical results are supported by a numerical study.     

Leader–follower tracking control with guaranteed consensus performance for interconnected systems with linear dynamic uncertain coupling

This paper considers the leader–follower tracking control problem for linear interconnected systems with undirected topology and linear dynamic coupling. Interactions between the systems are treated as linear dynamic uncertainty and are described in terms of integral quadratic constraints (IQCs). A consensus-type tracking control protocol is proposed for each system based on its state relative to its neighbours. In addition, a selected set of subsystems is used to control their relative states with respect to the leader. Two methods are proposed for the design of this control protocol. One method uses a coordinate transformation to recast the protocol design problem as a decentralised robust control problem for an auxiliary interconnected large-scale system. Another method is direct; it does not employ coordinate transformation, rather it also allows for more general linear uncertain interactions. Using these methods, sufficient conditions are obtained which guarantee that the system tracks the leader. These conditions guarantee a suboptimal bound on the system consensus and tracking performance. The proposed methods are compared using a simulation example, and their effectiveness is discussed. Also, algorithms are proposed for computing suboptimal controllers.     

Decentralized linear–quadratic–Gaussian control of networked control systems with asymmetric information

The decentralized linear–quadratic–Gaussian (LQG) control problem for networked control systems (NCSs) with asymmetric information is investigated, where controller 1 shares its historical information with controller 2, and not vice versa. The asymmetry of the information structure leads to the coupling between controller 2 and estimator 1, and hence the classical separation principle fails. Through the assumption of linear control strategy, the coupling between controller 2 and estimator 1 (CCE) is decoupled, but the estimation gain is still coupled with the control gain. It is noted that the control gain conforms to the backward Riccati equation while estimation gain abides by the forward equation, which is computationally challenging. Applying the stochastic maximum principle, the solvability of the decentralized LQG control problem is reduced to that of corresponding forward and backward stochastic difference equations (FBSDEs). Further, necessary and sufficient conditions for the solvability of optimal control problem are presented by two Riccati equations, one of which is nonsymmetric. Moreover, a novel iterative forward method is proposed to calculate the coupled backward control gain and forward estimation gain.     

On smoothing in linear discrete systems with time delays†

An algorithm is presented for optimal linear fixed-point and fixed-lag smoothing in non-stationary linear discrete systems with multiple time delays. Relations derived previously in the problem of filtering are used directly to obtain the fixed-point and fixed-lag smoothing filters. The smoothing error covariance matrix is obtained via a recursive matrix equation. For the particular case of systems without delay terms, the algorithm defines new smoothing procedures.     

Variable Structure Feedback Construction Algorithm for controlling measurement-imploded leader–follower ground robots

This paper presents a systematic approach for decreasing the error growth of a formation system consisted of a leader and a number of followers which are open-loop in position measurements (position, linear velocity and relative distance to the leader). Using conventional algorithms to control such a system may lead to an unstable network with increasing consensus error. A 3-phase algorithm is proposed to address open-loop systems of the mentioned form. The algorithm is specifically beneficial to the case where linear velocity and position sensors get damaged. The main idea of the proposed algorithm is to use fundamental relations between feedbacks and construct some of them using the others which in case of car-like ground vehicles, angular sensors’ feedbacks are used to estimate linear feedbacks. The error and input mathematical relations are formulated using kinematic of car-like robots. The stability of the proposed method is proved using a quadratic CLF, and the numerical simulation results are presented to assess the effectiveness of the algorithm compared to a conventional open-loop form. The algorithm is also applied on a pair of ground car-like robots. The experimental results verify the conclusions based on simulation. Various simulation results show that the proposed algorithm has improved the divergence rate of follower’s path from leader’s, from 0.3 $\frac{m}{sec}$ for the unit of velocity’s uncertainty to about zero.     

Logic-based modeling of information transfer in cyber–physical multi-agent systems

In modeling multi-agent systems, the structure of their communication is typically one of the most important aspects, especially for systems that strive toward self-organization or collaborative adaptation. Traditionally, such structures have often been described using logic-based approaches as they provide a formal foundation for many verification methods. However, these formalisms are typically not well suited to reflect the stochastic nature of communication in the cyber–physical setting. In particular, their level of abstraction is either too high to provide sufficient accuracy or too low to be practicable in more complex models. Therefore, we propose an extension of the logic-based modeling language SALMA, which we have introduced recently, that provides adequate high-level constructs for communication and data propagation, explicitly taking into account stochastic delays and errors. In combination with SALMA’s tool support for simulation and statistical model checking, this creates a pragmatic approach for verification and validation of cyber–physical multi-agent systems.     

A dynamic holding strategy in public transit systems with real-time information

Holding strategies are among the most commonly used operation control strategies in transit systems. In this paper, a dynamic holding strategy is developed, which consists of two major steps: (1) judging whether an early bus should be held, and (2) optimizing the holding times of the held bus. A model based on support vector machine (SVM), which contains four input variables (time-of-day, segment, the latest speed on the next segment, and the bus speed on the current segment) for forecasting the early bus departure times from the next stop is also developed. Then, in order to determine the optimal holding times, a model aiming to minimize the user costs is constructed and a genetic algorithm is used to optimize the holding times. Finally, the dynamic holding strategy proposed in this study is illustrated with the microscopic simulation model Paramics and some conclusions are drawn.

Leader–follower synchronization of networked multi-agent systems via hybrid protocols

This paper studies the leader–follower synchronization problem of complex-valued networked multi-agent systems with time-delay. A new hybrid protocol including a continuous-time protocol which is based on the interaction topology of follower agents and a pinning delayed impulsive control protocol is proposed. By employing the Lyapunov functional method in complex domains and the mathematical analysis technique, several delay-dependent leader–follower synchronization criteria are established that take into account various sizes of delays. Particularly, our result shows that leader–follower synchronization of delayed complex-valued networked multi-agent systems can be achieved even if the proposed hybrid protocol is being subject to relatively large impulse delays. A numerical example is provided to illustrate the effectiveness of the theoretical results.     

Quantum discord in spin systems with dipole–dipole interaction

The behavior of total quantum correlations (discord) in dimers consisting of dipolar-coupled spins 1/2 are studied. We found that the discord $Q=0$ at absolute zero temperature. As the temperature $T$ increases, the quantum correlations in the system increase at first from zero to its maximum and then decrease to zero according to the asymptotic law $T^{-2}$ . It is also shown that in absence of external magnetic field $B$ , the classical correlations $C$ at $T\rightarrow 0$ are, vice versa, maximal. Our calculations predict that in crystalline gypsum $\hbox {CaSO}_{4}\cdot \hbox {2H}_{2}{\hbox {O}}$ the value of natural $(B=0)$ quantum discord between nuclear spins of hydrogen atoms is maximal at the temperature of 0.644  $\upmu $ K, and for 1,2-dichloroethane $\hbox {H}_{2}$ ClC– $\hbox {CH}_{2}{\hbox {Cl}}$ the discord achieves the largest value at $T=0.517~\upmu $ K. In both cases, the discord equals $Q\approx 0.083$  bit/dimer what is $8.3\,\%$ of its upper limit in two-qubit systems. We estimate also that for gypsum at room temperature $Q\sim 10^{-18}$  bit/dimer, and for 1,2-dichloroethane at $T=90$  K the discord is $Q\sim 10^{-17}$  bit per a dimer.