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Register a new userFixed-Time Nash Equilibrium Seeking in Non-Cooperative Games
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We introduce a novel class of Nash equilibrium seeking dynamics for non-cooperative games with a finite number of players, where the convergence to the Nash equilibrium is bounded by a KL function with a settling time that can be upper bounded by a positive constant that is independent of the initial conditions of the players, and which can be prescribed a priori by the system designer. The dynamics are model-free, in the sense that the mathematical forms of the cost functions of the players are unknown. Instead, in order to update its own action, each player needs to have access only to real-time evaluations of its own cost, as well as to auxiliary states of neighboring players characterized by a communication graph. Stability and convergence properties are established for both potential games and strongly monotone games. Numerical examples are presented to illustrate our theoretical results.
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In this paper, we aim to develop distributed continuous-time algorithms under directed graphs to seek the Nash equilibrium of a noncooperative game. Motivated by the existing consensus-based designs in Gadjov and Pavel (2019), we present a distributed algorithm with a proportional gain for weight-balanced directed graphs. By further embedding a distributed estimator of the left eigenvector associated with zero eigenvalue of the graph Laplacian, we extend it to the case with arbitrary strongly connected directed graphs having possible unbalanced weights. In both cases, the Nash equilibrium is proven to be exactly reached with an exponential convergence rate.
We introduce a new class of extremum seeking controllers able to achieve fixed time convergence to the solution of optimization problems defined by static and dynamical systems. Unlike existing approaches in the literature, the convergence time of the proposed algorithms does not depend on the initial conditions and it can be prescribed a priori by tuning the parameters of the controller. Specifically, our first contribution is a novel gradient-based extremum seeking algorithm for cost functions that satisfy the Polyak-Lojasiewicz (PL) inequality with some coefficient kappa > 0, and for which the extremum seeking controller guarantees a fixed upper bound on the convergence time that is independent of the initial conditions but dependent on the coefficient kappa. Second, in order to remove the dependence on kappa, we introduce a novel Newton-based extremum seeking algorithm that guarantees a fully assignable fixed upper bound on the convergence time, thus paralleling existing asymptotic results in Newton-based extremum seeking where the rate of convergence is fully assignable. Finally, we study the problem of optimizing dynamical systems, where the cost function corresponds to the steady-state input-to-output map of a stable but unknown dynamical system. In this case, after a time scale transformation is performed, the proposed extremum seeking controllers achieve the same fixed upper bound on the convergence time as in the static case. Our results exploit recent gradient flow structures proposed by Garg and Panagou in [3], and are established by using averaging theory and singular perturbation theory for dynamical systems that are not necessarily Lipschitz continuous. We confirm the validity of our results via numerical simulations that illustrate the key advantages of the extremum seeking controllers presented in this paper.
With the proliferation of distributed generators and energy storage systems, traditional passive consumers in power systems have been gradually evolving into the so-called prosumers, i.e., proactive consumers, which can both produce and consume power. To encourage energy exchange among prosumers, energy sharing is increasingly adopted, which is usually formulated as a generalized Nash game (GNG). In this paper, a distributed approach is proposed to seek the Generalized Nash equilibrium (GNE) of the energy sharing game. To this end, we convert the GNG into an equivalent optimization problem. A Krasnoselski{v{i}}-Mann iteration type algorithm is thereby devised to solve the problem and consequently find the GNE in a distributed manner. The convergence of the proposed algorithm is proved rigorously based on the nonexpansive operator theory. The performance of the algorithm is validated by experiments with three prosumers, and the scalability is tested by simulations using 123 prosumers.
In this paper, we present a novel Newton-based extremum seeking controller for the solution of multivariable model-free optimization problems in static maps. Unlike existing asymptotic and fixed-time results in the literature, we present a scheme that achieves (practical) fixed time convergence to a neighborhood of the optimal point, with a convergence time that is independent of the initial conditions and the Hessian of the cost function, and therefore can be arbitrarily assigned a priori by the designer via an appropriate choice of parameters in the algorithm. The extremum seeking dynamics exploit a class of fixed time convergence properties recently established in the literature for a family of Newton flows, as well as averaging results for perturbed dynamical systems that are not necessarily Lipschitz continuous. The proposed extremum seeking algorithm is model-free and does not require any explicit knowledge of the gradient and Hessian of the cost function. Instead, real-time optimization with fixed-time convergence is achieved by using real time measurements of the cost, which is perturbed by a suitable class of periodic excitation signals generated by a dynamic oscillator. Numerical examples illustrate the performance of the algorithm.
In this paper we propose a new operator splitting algorithm for distributed Nash equilibrium seeking under stochastic uncertainty, featuring relaxation and inertial effects. Our work is inspired by recent deterministic operator splitting methods, designed for solving structured monotone inclusion problems. The algorithm is derived from a forward-backward-forward scheme for solving structured monotone inclusion problems featuring a Lipschitz continuous and monotone game operator. To the best of our knowledge, this is the first distributed (generalized) Nash equilibrium seeking algorithm featuring acceleration techniques in stochastic Nash games without assuming cocoercivity. Numerical examples illustrate the effect of inertia and relaxation on the performance of our proposed algorithm.
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