Distributed adaptive Nash equilibrium seeking for networked systems with unknown disturbance and actuator saturation

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.

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.

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.

This article investigates the Nash equilibrium (NE) seeking problems for multicoalition games with heterogeneous general linear dynamics over unbalanced digraphs. The coalition can be regarded as a virtual player but the true decision-makers are the actual players themselves which can only access to their own cost functions. To deal with the unbalanced digraphs, the left eigenvector is estimated and its prescribed-time convergence is illuminated with the time transfer approach. Then the NE seeking part and the output regulation part are designed to adapt the dynamics of the players. The initial values of the auxiliary vectors are selected to avoid utilizing the out-degree information. The steady state of the closed-loop system is analyzed and the prescribed-time convergence is proved based on the Lyapunov method. Finally, the simulation results of both the mobile sensor connectivity game and the electricity market game are presented to show the effectiveness of the proposed 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 paper investigates the Nash equilibrium (NE) seeking problem of the game in which the dynamics of all players are affected by external disturbances and the communication network among players may be influenced by persistent attacks. To solve this problem, a distributed robust switched algorithm is proposed. This distributed switched strategy consists of two modules: One is the internal-model based dynamics that can be used to compensate the effects caused by external disturbances; the other is gradient-based optimization module which can utilize the local information to estimate the players' decisions and dynamically seek the optimal solution. By utilizing average dwell-time and time-ratio constraints to characterize the network attacks, the proposed switched strategy is modeled into a hybrid dynamical system. Moreover, employing the hybrid systems approach, we prove that the decisions of all players can exponentially converge to the NE of the considered game. Finally, an example is provided to illustrate the theoretical results.

This brief studies the formation placement problems for multi-cluster system with convergent time requirements based on a time-varying Nash equilibrium (NE) seeking approach. In order to determine the placement of each agent that subject to formation constraints in each cluster within a prescribed setting time, a basic centralized time-varying NE seeking algorithm based on time transformation approach is designed. In addition, by applying a prescribed-time distributed observer, an extended distributed time-varying NE seeking algorithm is proposed. Compared with existing results, the convergence time can be appointed in advance according to specific mission requirements. Finally, the effectiveness of proposed algorithms is verified by numerical simulations.

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.

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.

In this paper, we study the generalized Nash equilibrium (GNE) seeking problem for an N-coalition game with inequality constraints. First, using full decision information, a finite-time average consensus-based approach is proposed. It is demonstrated that the algorithm converges to a GNE (specifically, a variational equilibrium) of the game. Then, using the finite-time consensus tracking technique, we propose a distributed algorithm that only requires neighboring action information for each agent. The solution by using partial decision information covers the distributed GNE seeking solutions of generalized non-cooperative games as a special case.

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 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.

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

An adaptive neural network (NN) event-triggered control scheme is proposed for nonlinear nonstrict-feedback multiagent systems (MASs) against input saturation, unknown disturbance, and sensor faults. Mean-value theorem and Nussbaum-type function are invoked to transform the structure of the input saturation and overcome the difficulty of unknown control directions, respectively. On the basis of the universal approximation property of NNs, a nonlinear disturbance observer is designed to estimate the unknown compounded disturbance composed of external disturbance and the residual term of input saturation. According to the measurement error defined by control signal, an event-triggered mechanism is developed to save network transmission resource and reduce the number of controller update. Then, an adaptive NN compensation control approach is proposed to tackle the problem of sensor faults via the dynamic surface control (DSC) technique. It is proved that all signals in the closed-loop system are semi-globally uniformly ultimately bounded. Finally, simulation results demonstrate the effectiveness of the presented control strategy.

In socio-technical multi-agent systems, deception exploits privileged information to induce false beliefs in “victims,” keeping them oblivious and leading to outcomes detrimental to them or advantageous to the deceiver. We consider model-free Nash-equilibrium-seeking for non-cooperative games with asymmetric information and introduce model-free deceptive algorithms with stability guarantees. In the simplest algorithm, the deceiver includes in his action policy the victim's exploration signal, with an amplitude tuned by an integrator of the regulation error between the deceiver's actual and desired payoff. The integral feedback drives the deceiver's payoff to the payoff's reference value, while the victim is led to adopt a suboptimal action, at which the pseudogradient of the deceiver's payoff is zero. The deceiver's and victim's actions turn out to constitute a “deceptive” Nash equilibrium of a different game, whose structure is managed — in real time — by the deceiver. We examine quadratic, aggregative, and more general games and provide conditions for a successful deception, mutual and benevolent deception, and immunity to deception (for a “non-generic” set of payoff functions). Stability results are established using techniques based on averaging and singular perturbations. Among the examples in the paper is a microeconomic duopoly in which the deceiver induces in the victim a belief that the buyers disfavor the deceiver more than they actually do, leading the victim to increase the price above the Nash price, and resulting in an increased profit for the deceiver and a decreased profit for the victim. A study of the deceiver's integral feedback for the desired profit reveals that, in duopolies with equal marginal costs, a deceiver that is greedy for very high profit can attain any such profit, and pursue this with arbitrarily high integral gain (impatiently), irrespective of the market preference for the victim.