In this paper, we consider a Nash equilibrium seeking problem for a class of high-order multi-agent systems with unknown dynamics. Different from existing results for single integrators, we aim to steer the outputs of this class of uncertain high-ord
er agents to the Nash equilibrium of some noncooperative game in a distributed manner. To overcome the difficulties brought by the high-order structure, unknown nonlinearities, and the regulation requirement, we first introduce a virtual player for each agent and solve an auxiliary noncooperative game for them. Then, we develop a distributed adaptive protocol by embedding this auxiliary game dynamics into some proper tracking controller for the original agent to resolve this problem. We also discuss the parameter convergence problem under certain persistence of excitation condition. The efficacy of our algorithms is verified by numerical examples.
This paper studies the distributed optimization problem where the objective functions might be nondifferentiable and subject to heterogeneous set constraints. Unlike existing subgradient methods, we focus on the case when the exact subgradients of th
e local objective functions can not be accessed by the agents. To solve this problem, we propose a projected primal-dual dynamics using only the objective functions approximate subgradients. We first prove that the formulated optimization problem can only be solved with an approximate error depending upon the accuracy of the available subgradients. Then, we show the exact solvability of this optimization problem if the accumulated approximation error is not too large. After that, we also give a novel componentwise normalized variant to improve the transient behavior of the convergent sequence. The effectiveness of our algorithms is verified by a numerical example.
In this paper, an optimal output consensus problem is studied for discrete-time linear multiagent systems subject to external disturbances. Each agent is assigned with a local cost function which is known only to itself. Distributed protocols are to
be designed to guarantee an output consensus for these high-order agents and meanwhile minimize the aggregate cost as the sum of these local costs. To overcome the difficulties brought by high-order dynamics and external disturbances, we develop an embedded design and constructively present a distributed rule to solve this problem. The proposed control includes three terms: an optimal signal generator under a directed information graph, an observer-based compensator to reject these disturbances, and a reference tracking controller for these linear agents. It is shown to solve the formulated problem with some mild assumptions. A numerical example is also provided to illustrate the effectiveness of our proposed distributed control laws.
This paper investigates an optimal consensus problem for a group of uncertain linear multi-agent systems. All agents are allowed to possess parametric uncertainties that range over an arbitrarily large compact set. The goal is to collectively minimiz
e a sum of local costs in a distributed fashion and finally achieve an output consensus on this optimal point using only output information of agents. By adding an optimal signal generator to generate the global optimal point, we convert this problem to several decentralized robust tracking problems. Output feedback integral control is constructively given to achieve an optimal consensus under a mild graph connectivity condition. The efficacy of this control is verified by a numerical example.
In this paper, we investigate a constrained optimal coordination problem for a class of heterogeneous nonlinear multi-agent systems described by high-order dynamics subject to both unknown nonlinearities and external disturbances. Each agent has a pr
ivate objective function and a constraint about its output. A neural network-based distributed controller is developed for each agent such that all agent outputs can reach the constrained minimal point of the aggregate objective function with bounded residual errors. Two examples are finally given to demonstrate the effectiveness of the algorithm.
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 distribute
d 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.
This paper studies an optimal consensus problem for a group of heterogeneous high-order agents with unknown control directions. Compared with existing consensus results, the consensus point is further required to an optimal solution to some distribut
ed optimization problem. To solve this problem, we first augment each agent with an optimal signal generator to reproduce the global optimal point of the given distributed optimization problem, and then complete the global optimal consensus design by developing some adaptive tracking controllers for these augmented agents. Moreover, we present an extension when only real-time gradients are available. The trajectories of all agents in both cases are shown to be well-defined and achieve the expected consensus on the optimal point. Two numerical examples are given to verify the efficacy of our algorithms.
In this technical note, we investigate an optimal output consensus problem for heterogeneous uncertain nonlinear multi-agent systems. The considered agents are described by high-order nonlinear dynamics subject to both static and dynamic uncertaintie
s. A two-step design, comprising sequential constructions of optimal signal generator and distributed partial stabilization feedback controller, is developed to overcome the difficulties brought by nonlinearities, uncertainties, and optimal requirements. Our study can not only assure an output consensus, but also achieve an optimal agreement characterized by a distributed optimization problem.
Nonlinear nanostructured surfaces provide a paradigm shift in nonlinear optics with new ways to control and manipulate frequency conversion processes at the nanoscale, also offering novel opportunities for applications in photonics, chemistry, materi
al science, and biosensing. Here, we develop a general approach to employ sharp resonances in metasurfaces originated from the physics of bound states in the continuum for both engineering and enhancing the nonlinear response. We study experimentally the third-harmonic generation from metasurfaces composed of symmetry-broken silicon meta-atoms and reveal that the harmonic generation intensity depends critically on the asymmetry parameter. We employ the concept of the critical coupling of light to the metasurface resonances to uncover the effect of radiative and nonradiative losses on the nonlinear conversion efficiency.
Metasurface, a kind of two-dimensional structured medium, represents a novel platform to manipulate the propagation of light at subwavelength scale. In linear optical regime, many interesting topics such as planar metalens, metasurface optical hologr
aphy and so on have been widely investigated. Recently, metasurfaces go into nonlinear optical regime. While it is recognized that the local symmetry of the meta-atoms plays vital roles, its relationship with global symmetry of the nonlinear metasurfaces remains elusive. According to the Penrose tiling and the newly proposed hexagonal quasicrystalline tiling, here we designed and fabricated the nonlinear optical quasicrystal metasurfaces based on the geometric phase controlled plasmonic meta-atoms with local rotational symmetry. The second harmonic waves will be determined by both the tiling schemes of quasicrystal metasurfaces and the local symmetry of meta-atoms they consist of. The proposed concept opens new routes for designing nonlinear metasurface crystals with desired optical functionalities.
