This article focuses on the constrained optimization problem for second-order multiagent systems that experience heterogeneous communication delays. Specifically, the involved agents work together to find the optimal solution of a global payoff function, which is summed by multiple strongly convex local payoff functions, with each function being exclusively owned by an individual agent. However, the feasible solutions must satisfy a coupled equality constraint, formulated by individual parameters assigned to each agent. Initially, a basic anti-delay distributed protocol is developed, which leverages a scattering transformation to enhance the generation of received information. Using the Lyapunov framework, we demonstrate that the agents coordinated by this anti-delay distributed protocol can effectively reach a consensus on the expected optimal solution, despite the presence of communication delays. In addition, we present two results that extend the basic anti-delay distributed protocol. First, we consider the scenario of lacking velocity and develop a velocity-free anti-delay distributed protocol to achieve the concerned constrained optimization objective. Next, we augment the system order and develop an anti-delay distributed optimization protocol for integrator chain multiagent systems. Finally, we confirm the anti-delay performance of the developed distributed protocols through simulations.

The cooperative tracking control issue of quadrotor unmanned aerial vehicles (UAVs) is investigated, where a cluster of UAVs is required to maintain a preassigned pattern while tracking a reference trajectory. Since only a part of UAVs can access the reference trajectory, an adaptive distributed estimator is developed such that each UAV could obtain the accurate estimate exponentially. Based on the hierarchical development, a parallel optimal learning control strategy is proposed by introducing virtual artificial systems that generate the practical control commands. In particular, a saturated force command is exploited for the position loop tracking to the estimated trajectory and a torque command is utilized to ensure the command attitude tracking, respectively. Moreover, a data-based learning law is designed for the critic weight under the finite excitation (FE), which is made available for the inadmissible initial control. It is shown that the overall closed-loop system is uniformly ultimately bounded and the tracking errors eventually converge to small sets around zero. Simulation and experiment results further verify the proposed control strategy.Note to Practitioners—This paper is inspired by the cooperative formation control issue of quadrotor UAVs. In practical missions like surveillance and reconnaissance, a UAV comes into or leaves out of the cluster may lead to re-tune the control parameters for all the involved UAVs. To save the onboard resource, the communication topology among the UAVs cannot be tedious. The proposed parallel learning strategy is implemented as follows. First, we design an adaptive distributed estimator for each UAV that could obtain the reference information accurately. Then, we propose a data-based parallel optimal learning strategy for each UAV that tracks the estimated reference trajectory. Since the proposed strategy does not require any global information regarding the communication topology, the parameters can be remained when the UAV cluster needs to extend or reduce, which makes it more practical. Finally, the proposed strategy is validated by both simulation and experiment results.

This article studies the generalized Nash equilibrium (GNE) seeking problem of second-integrator multiplayer systems. In particular, each player is endowed with an individual payoff function with respect to collective decision variables, and simultaneously, a coupling inequality constraint and a set constraint are imposed to each player. The players communicate with their local neighbors over a directed topology. To begin with, a distributed-observer-based seeking strategy is synthesized by leveraging a proper composite variable. It is first demonstrated using nonsmooth analysis that the established distributed observer enables each player to accurately estimate the decision variables of others in terms of a strongly connected topology condition. Upon this basis, all the decision variables are then shown to converge to the expected GNE asymptotically borrowing from convex theory. In addition, three extension results are also given under the built GNE seeking framework. First, under the postulation that the velocity information is unavailable, a velocity-free distributed GNE seeking strategy is synthesized for second-integrator systems by implementing a proper auxiliary dynamics. Second, we consider nonlinear Euler-Lagrange systems with unknown inertia parameters and synthesize an improved distributed GNE seeking strategy resorting to an adaptation technique. Third, we focus on integrator chain systems and synthesize a modified distributed GNE seeking strategy using a new composite variable based on a proper coordinate transformation. For three extension cases, we all show in detail the achievement of the GNE seeking objective. Finally, a practical example is simulated to confirm the built GNE seeking results.

The optimization problem of second-order discrete-time multiagent systems with set constraints is studied in this article. In particular, the involved agents cooperatively search an optimal solution of a global objective function summed by multiple local ones within the intersection of multiple constrained sets. We also consider that each pair of local objective function and constrained set is exclusively accessible to the respective agent, and each agent just interacts with its local neighbors. By borrowing from the consensus idea, a projection-based distributed optimization algorithm resorting to an auxiliary dynamics is first proposed without interacting the gradient information of local objective functions. Next, by considering the local objective functions being strongly convex, selection criteria of step size and algorithm parameter are built such that the unique solution to the concerned optimization problem is obtained. Moreover, by fixing a unit step size, it is also shown that the optimization result can be relaxed to the case with just convex local objective functions given a properly chosen algorithm parameter. Finally, practical and numerical examples are taken to verify the proposed optimization results.

This article investigates the noncooperative game of multiple clusters composed by double-integrator agents subject to set constraints. In particular, by decomposing each cluster cost function into multiple individual ones and allocating them to respective agents, the generalized Nash equilibrium (GNE) associated with the upper cluster layer is sought by the lower agent layer in terms of a hierarchical structure. A particular agent in each cluster is appointed as the messenger that is specialized in interacting with adjacent messengers from other clusters. It is first shown that the concerned seeking objective is achieved provided that the agents from the same cluster reach a consensus on their corresponding component of the GNE. Motivated by this consensus idea, a distributed seeking strategy through the intracluster interaction is first developed, which is applicable to the case where each agent has access to the information of the messengers from other clusters. Based on this result, we next consider a relaxed case where the messenger information is absent for each agent and the inter-cluster information interaction is only allowed via their messengers. Another distributed seeking strategy resorting to a distributed observer is developed via both the inter- and intra-cluster interactions. It is shown that both the developed distributed seeking strategies are capable of seeking out the concerned GNE. As extensions to the double-integrator model, we further consider two evolved models, i.e., Euler–Lagrange model and multi-integrator model. Two modified distributed seeking strategies are developed for achieving the concerned NE seeking objective. Finally, the effectiveness of the developed distributed seeking strategies is validated by an application.

The time-varying economic dispatch over a network is considered where both the cost function and the equality constraint are time-varying. A distributed algorithm is first designed combing the prediction-correction framework and the consensus + innovations approach. In the prediction steps, the sensitivity analysis method is introduced to calculate the shift between the estimated and the exact optimization solution of the correction steps. Afterwards, the overall optimal errors in two steps are calculated by the sub-optimal analysis method. The convergence analysis shows that the dispatch errors are bounded and the convergence process is Q-linear. A dynamic prediction-correction algorithm is then presented to improve the accuracy at every time instant. Finally, numerical examples using the IEEE 118-bus system are presented to illustrate the validity of the algorithms.

This article investigates the targeted bipartite consensus problem of opinion dynamics in cooperative-antagonistic networks. Each agent in the network is assigned with a convergence set to represent a credibility interval, in which its opinion is trustworthy. The network topology is characterized by a signed switching digraph. The objective is to achieve the bipartite consensus targeted within these credibility intervals. A gradient term is introduced in the opinion dynamics besides the consensus term. Under the assumption that the underlying graph is uniformly jointly strongly connected and structurally balanced, it is first shown that the considered opinion dynamics reaches the targeted bipartite consensus within the intersections of the convergence sets associated with two antagonistic groups. Next, by relaxing the connectivity condition to uniform joint quasistrong connectivity, the targeted bipartite consensus result is also proven with an additional convergence set assumption. Numerical examples are provided to validate the proposed theoretical results.

This article focuses on the solution to the coordinated formation problem of heterogeneous vertical takeoff and landing (VTOL) unmanned aerial vehicles (UAVs) in the presence of parametric uncertainties. In particular, their inertial parameters are distinct and unavailable. For the sake of the accomplishment of the coordinated formation objective of multiple underactuated VTOL UAVs through local information exchange, an adaptive distributed control algorithm is developed under a cascaded structure. Specifically, by introducing an immersion and invariance (I&I) adaption strategy for the exponential mass estimation, a distributed command force is first synthesized in the position loop. Next, an applied torque with adaption is synthesized for the attitude tracking to a command attitude. This command attitude, as well as the applied thrust, is extracted from the synthesized command force without singularity. It is shown in terms of the Lyapunov theory that driven by the proposed adaptive distributed control algorithm, the concerned coordinated formation control of multiple VTOL UAVs is achieved asymptotically. Finally, an illustrative example is simulated to validate the effectiveness of the proposed control algorithm.

In this article, a class of distributed quadratic games is considered over an undirected graph. The issue on the communication topology restriction is introduced and the players’ dynamics are with nonlinear dynamics and unknown time-varying perturbation. A distributed Nash equilibrium seeking algorithm is proposed based on a high-gain observer method, and the convergence is analyzed by the Lyapunov stability theory. It is shown that each player estimates the rival players’ states, and the errors between the players’ states and the Nash equilibrium are ultimately bounded by a small bound. Moreover, the presented algorithm is free of chattering phenomena because it is designed using the hyperbolic tangent function instead of the signum function to dominant the perturbation. The effectiveness of the proposed algorithm is validated via a simulation of the oligopoly game in which five firms produce the same products in a duopoly market structure.

A distributed control algorithm is proposed in this paper to achieve the leader-follower cooperative attitude tracking of multiple rigid bodies without using absolute and relative angular velocity information. The nonlinear manifold SO(3) is applied to represent the attitude of each rigid body. By considering the case of limited information exchange, i.e., not all the rigid bodies have access to the leader's information, a novel continuous nonlinear distributed estimator satisfying the rotation matrix dynamics is first introduced for each rigid body to estimate the leader's information asymptotically. Then, two auxiliary systems driven by proper inputs are designed such that the issues due to the angular velocity unavailability are effectively solved. Next, a feasible distributed protocol is designed by implementing the outputs from the distributed estimator and the auxiliary systems. It is demonstrated in terms of Lyapunov theory that, the developed distributed control algorithm ensures the asymptotically stable cooperative attitude tracking of multiple rigid bodies under the weak tree connectivity assumption. Finally, illustrative examples are provided to validate and highlight the proposed results.

This paper develops a novel immersion and invariance (I&I)-based adaptive controller for nonlinear underactuated quadrotor systems subject to uncertain inertial parameters. The objective is to drive the quadrotor to accurately track a reference trajectory. The controller synthesis is based on a hierarchical framework. First, the solution to the trajectory tracking problem is transformed to stabilizing its corresponding error system asymptotically. Second, to avoid singularity in the command attitude extraction, a saturated backstepping control strategy with smooth hyperbolic tangent functions is developed to synthesize a command force in the position loop, where an I&I adaptive methodology is introduced to update the mass estimation. The I&I adaption guarantees the asymptotic convergence of the mass estimation to its actual value. Third, in the attitude loop, the projection algorithm is used to update the inertial tensor estimations, which enables the estimations to be maintained within a prescribed range. Then, a rotation matrix-based applied torque with an initial condition constraint is synthesized. It is proven that, with appropriate control parameters, the proposed controller guarantees the asymptotic stability of the error system and then the trajectory tracking of the quadrotor is achieved. Finally, numerical simulations are given to verify and highlight the performance of the proposed controller.