Distributed least-distance formation control for Euler–Lagrange systems: A game-based approach

This paper investigates the adaptive Nash equilibrium (NE) seeking problems for networked systems, in which the players are troubled by actuator saturation and unknown disturbance. Based on the inverse function of hyperbolic tangent function, an augmented plant is constructed and the dynamic surface control technique is utilised to deal with nonlinear characteristics brought by the actuator saturation. By introducing adaptive gains to gradient, consensus estimation and disturbance compensation parts, a fully distributed NE seeking strategy is proposed. According to the Lyapunov stability analysis, the convergence error of the adaptive NE seeking under actuator saturation and disturbance is uniformly bounded. Finally, a simulation is provided to justify the effectiveness of the proposed method.

This paper presents a study on Nash equilibrium seeking in noncooperative games within multi-robot systems, a topic of increasing importance in diverse sectors including civil, security, and military. Unlike conventional approaches where players can directly observe the actions of others, our method assumes limited visibility, where players can only communicate through an undirected and connected communication graph. We introduce a novel distributed control approach, integrating gradient play with a consensus protocol. This method facilitates effective Nash equilibrium seeking by leveraging information shared among neighboring robots in heterogeneous linear dynamic systems. The proposed solution employs a high-level distributed Nash equilibrium-seeking algorithm, serving as an optimal reference generator for each robot to track the Nash equilibrium, and an advanced output regulation technique, aiming to regulate the output (e.g., position) of the robots with respect to the obtained references. Theoretical analysis confirms the convergence of our algorithm through Lyapunov stability analysis. The effectiveness and practical applicability of our approach are validated through numerical simulations and empirical testing with physical robots, highlighting its efficacy and utility in real-world scenarios.

Distributed Nash Equilibrium (NE) seeking for networked games has been widely investigated in recent years. In particular, convergence speed and communication cost are two of the critical concerns for the design of the seeking algorithms. To achieve fast convergence while saving communication resources, this paper investigates the appointed-time distributed NE seeking problem for networked games under a discrete-time communication scenario. By utilizing the idea of motion planning, a new continuous-time distributed NE seeking algorithm, which ensures convergence to NE at the prescribed time, is proposed based on sampled-data information exchange. Compared with the existing related works, the established algorithm has advantages on both the communication cost and convergence speed (i.e., appointed-time convergence). Finally, the effectiveness of the proposed method is verified via numerical simulations.

SummaryIn this paper, a bipartite consensus problem is considered for a high‐order multiagent system with cooperative‐competitive interactions and unknown time‐varying disturbances. A signed graph is used to describe the interaction network associated with the multiagent system. The unknown disturbances are expressed by linearly parameterized models, and distributed adaptive laws are designed to estimate the unknown parameters in the models. For the case that there is no exogenous reference system, a fully distributed adaptive control law is proposed to ensure that all the agents reach a bipartite consensus. For the other case that there exists an exogenous reference system, another fully distributed adaptive control law is also developed to ensure that all the agents achieve bipartite consensus on the state of the exogenous system. The stability of the closed‐loop multiagent systems with the 2 proposed adaptive control laws are analyzed under an assumption that the interaction network is structurally balanced. Moreover, the convergence of the parameter estimation errors is guaranteed with a persistent excitation condition. Finally, simulation examples are provided to demonstrate the effectiveness of the proposed adaptive bipartite consensus control laws for the concerned multiagent system.

In this article, the prescribed-time distributed Nash equilibrium (NE) seeking problem for multicoalition games with heterogeneous general linear players is investigated. In such games, there exist multiple coalitions that consist of heterogeneous players. The players have to seek the NE of the entire coalition while being limited to access only their individual cost functions. The proposed algorithm involves a two-layer structure composed of the NE seeking part and the tracking control part. To address the challenge of global information dependency, the leader-following-based estimates are designed. The NE seeking part is designed based on the distributed estimates and concurrently performed over the weight-balanced digraphs. Subsequently, by introducing the time-varying parametric Lyapunov equation, the strategies of the players are driven to the NE solution within a manually prescribed settling time, and the prescribed-time convergence is proved. Finally, the proposed algorithm is executed on the heterogeneous coalitions made up of quadrotor unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) in a group formation scenario. Both the simulation and the experiment results are presented to show the effectiveness of the algorithm.

In this paper, we consider distributed Nash equilibrium (NE) seeking in potential games over a multi-agent network, where each agent can not observe the actions of all its rivals. Based on the best response dynamics, we design a distributed NE seeking algorithm by incorporating the non-smooth finite-time average tracking dynamics, where each agent only needs to know its own action and exchange information with its neighbours through a communication graph. We give a sufficient condition for the Lipschitz continuity of the best response mapping for potential games, and then prove the convergence of the proposed algorithm based on the Lyapunov theory. Numerical simulations are given to verify the result and illustrate the effectiveness of the algorithm.

This article investigates the distributed Nash equilibrium seeking problem of quadratic time‐varying games with Euler–Lagrange (EL) players, where external disturbances and parametric uncertainties are involved. A gradient‐based hierarchical algorithm consisting of a game layer and a control layer is proposed. Specifically, in the game layer, EL players communicate with neighbors through a graph to reach the consensus on potential aggregate values, which will be employed to calculate the gradient of each player's objective function, and then, a gradient‐based sliding mode controller is developed to track time‐varying gradient in the control layer. Thus, the convergence results are hierarchically obtained through the Lyapunov stability method. In addition, the hierarchical control strategy is extended to address the constrained problems through the utilization of a smooth penalty function. By appropriately choosing control parameters, the Nash equilibrium seeking errors can be arbitrarily small. The relation between the optimal solutions of the original problem and the dual one is further discussed. Finally, the proposed methods are numerically verified.

This article studies a distributed Nash equilibrium (NE) seeking problem for multiple agents of aggregative games. A novel distributed algorithm is presented to assure both fast convergence and nonovershoot performance by the combination of an accelerated method and a gradient-based NE seeking algorithm. To be specific, an accelerated method (i.e., the heavy-ball method or Nesterov’s accelerated method) is introduced in the designed algorithm for the fast convergence of agents’ strategies to the NE. Moreover, a suitable switching mechanism is proposed to improve the transient performance by switching the distributed algorithm based on the accelerated method to the gradient-based algorithm. As a result, the presented distributed algorithm is modeled by a hybrid dynamical system (HDS). The semi-globally practical convergence is established by analyzing the stability of a parameterized HDS. An example of distributed energy resources is taken to verify the presented algorithm.

This paper considers a simple Nash-equilibrium-seeking (NES) scheme acting on a family of plants that can be represented as non-cooperative games. Specifically, the family of plants under investigation have a number of inputs, each of which can be associated with a measured cost. The NES scheme consists of a number of decentralised extremum-seeking (ES) agents which, without requiring knowledge of the underlying plant dynamics, control each input in order to minimise its associated steady-state cost. A non-local stability result for the NES scheme is provided which allows two agents to use the same dither signal if the effect of each agent on the other's steady-state cost function is sufficiently weak. It is then demonstrated in simulation that, by reducing the number of distinct dither signals, the proposed scheme is able to more quickly seek the Nash equilibrium.

A payoff-based learning approach for Nash equilibrium seeking in continuous potential games

This paper presents a sliding mode control for the motion control of tethered space robot. The input saturation of tension and thrusters is well taken into account, which is dealt with by a modified adaptive law to aid the expression of ideal control input in Lyapunov-based stability analysis. The adaptive law is designed to be naturally bounded such that the boundary of adaptive law is calculable and fixed to make the ideal input bounded, which satisfies the condition of the partial expression of the ideal input. For reducing the effect from disturbance and uncertainties, the relay-type control law combined with the adaptive law is used to drive the system states onto the sliding surface although the saturation occurs. Numerical simulation illustrate the results of deployment process to verify the effectiveness of the stability analysis.

In this article, a noncooperative game in which the engaged players are of double integrator-type dynamics, is investigated. The dynamics of the players are considered to be subject to unknown time-varying disturbances and unmodeled terms. Besides, the communication topology among the players suffers from attacks and time-varying communication delays. To find a Nash equilibrium for such games, a distributed robust Nash equilibrium seeking algorithm is proposed. By utilizing average dwell-time and time-ratio constraints to tackle the attacks, the closed-loop system is modeled as a hybrid system with memory. To analyze the stability of the distributed switched algorithm, a Lyapunov functional is constructed, by which we show the uniform global asymptotic stability of the equilibrium. Finally, an example is given to illustrate our results.

In this paper, we deal with the problem of finding a Nash equilibrium for a generalized convex game. Each player is associated with a convex cost function and multiple shared constraints. Supposing that each player can exchange information with its neighbors via a connected undirected graph, the objective of this paper is to design a Nash equilibrium seeking law such that each agent minimizes its objective function in a distributed way. Consensus and singular perturbation theories are used to prove the stability of the system. A numerical example is given to show the effectiveness of the proposed algorithms.

This paper proposes a novel adaptive sliding mode control (ASMC) for a class of polynomial systems comprising uncertain terms and input nonlinearities. In this approach, a new polynomial sliding surface is proposed and designed based on the sum‐of‐squares (SOS) decomposition. In the proposed method, an adaptive control law is derived such that the finite‐time reachability of the state trajectories in the presence of input nonlinearity and uncertainties is guaranteed. To do this, it is assumed that the uncertain terms are bounded and the input nonlinearities belong to sectors with positive slope parameters. However, the bound of the uncertain terms is unknown and adaptation law is proposed to effectively estimate the uncertainty bounds. Furthermore, based on a novel polynomial Lyapunov function, the finite‐time convergence of the sliding surface to a pre‐chosen small neighborhood of the origin is guaranteed. To eliminate the time derivatives of the polynomial terms in the stability analysis conditions, the SOS variables of the Lyapunov matrix are optimally selected. In order to show the merits and the robust performance of the proposed controller, chaotic Chen system is provided. Numerical simulation results demonstrate chattering reduction in the proposed approach and the high accuracy in estimating the unknown parameters.

A planar 4R-foldable quadrotor, i.e., foldable quadrotor with four independent revolute joints for wings rotation has the edge over conventional quadrotor in confined space navigation due to its morphing capability and multimodal behaviors. However, the morphing induced inertial and center of gravity variations cause attitude destabilization, which impedes the trajectory tracking efficiency and reduces the flight stability in confined environments. In this paper, we propose an adaptive flight stabilization framework for a planar 4R-foldable morphing quadrotor that comprises the adaptive control and geometric tracking control. The adaptive control laws are developed to estimate the morphing induced inertial and center of gravity variations with respect to the desired attitude dynamics. These adaptive laws are then used to develop the feedback component of geometric tracking control which improves the trajectory tracking efficiency. Together with the controller design, the boundness of the attitude tracking errors is proved. Moreover, the simulations were conducted to validate the effectiveness of the proposed control framework in confined environments. Initial results show that the adaptive stabilization framework is effective in attitude stabilization and enhancing the trajectory tracking performance of a planar 4R-foldable morphing quadrotor.