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Nonlocal Stability of a Nash Equilibrium Seeking Scheme with Dither Reuse
"... Abstract — This paper considers a simple Nashequilibriumseeking (NES) scheme acting on a family of plants that can be represented as noncooperative 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 ..."
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Abstract — This paper considers a simple Nashequilibriumseeking (NES) scheme acting on a family of plants that can be represented as noncooperative 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 extremumseeking (ES) agents which, without requiring knowledge of the underlying plant dynamics, control each input in order to minimise its associated steadystate cost. A nonlocal 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 steadystate 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. I.
University of Toronto Systems Control Group
"... We introduce a general analytic framework for decentralized extremumseeking control schemes involving N dynamic agents. The agents cooperate in achieving a coordination objective which is abstracted as a collective optimization problem. Each agent has access only to the measurements of its individu ..."
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We introduce a general analytic framework for decentralized extremumseeking control schemes involving N dynamic agents. The agents cooperate in achieving a coordination objective which is abstracted as a collective optimization problem. Each agent has access only to the measurements of its individual objective function values, and may communicate with neighboring agents. We provide a set of sufficient conditions under which such schemes exhibit semiglobal practical asymptotic stability with respect to an equilibrium point which is parameterized by the collective optimizer. I.
Dither Reuse in Nash Equilibrium Seeking
"... Abstract—The Nash equilibrium seeking (NES) scheme consists of a number of decentralised extremumseeking (ES) “agents”, each controlling an input such that an associated (selfish) cost is regulated to its steadystate Nash equilibrium. A nonlocal stability result for the NES scheme is provided wh ..."
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Abstract—The Nash equilibrium seeking (NES) scheme consists of a number of decentralised extremumseeking (ES) “agents”, each controlling an input such that an associated (selfish) cost is regulated to its steadystate Nash equilibrium. A nonlocal 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 steadystate cost function is sufficiently weak. Its application to plants with quadratic cost functions is presented as an example. It is then demonstrated in simulation that, by reducing the number of distinct dither signals, the proposed scheme still has acceptable convergence properties, while the design effort is reduced. Index Terms—Adaptive control, extremumseeking, Nash equilibrium. I.
eo dNewtonbased extremum seeking Averaging Singular perturbation Stability
"... multivariable extremum seeking, the proposed algorithm removes the dependence of the convergence rate on the unknown Hessian matrix and makes the convergence rate, of both the parameter estimates and of the estimates of theHessian inverse, userassignable. In particular, the new algorithm allows all ..."
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multivariable extremum seeking, the proposed algorithm removes the dependence of the convergence rate on the unknown Hessian matrix and makes the convergence rate, of both the parameter estimates and of the estimates of theHessian inverse, userassignable. In particular, the new algorithm allows all the parameters to converge with the same speed, yielding straight trajectories to the extremum even with maps that have highly elongated level sets, in contrast to curved ‘‘steepest descent’ ’ trajectories of the gradient algorithm. Simulation results show the advantage of the proposed approach over gradientbased extremum seeking, by assigning equal, desired convergence rates to all the parameters using Newton’s approach.
Lie Bracket Approximation of Extremum Seeking Systems
"... Extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map. It often requires advanced systemtheoretic tools to understand the qualitative behavior of extremum seeking systems. In this paper, a novel interpretation of extremum ..."
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Extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map. It often requires advanced systemtheoretic tools to understand the qualitative behavior of extremum seeking systems. In this paper, a novel interpretation of extremum seeking is introduced. We show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain Lie brackets of the vector fields of the extremum seeking system. It turns out that the Lie bracket system directly reveals the optimizing behavior of the extremum seeking system. Specifically, we establish a theoretical foundation and prove that uniform asymptotic stability of the Lie bracket system implies practical uniform asymptotic stability of the corresponding extremum seeking system. We use the established results in order to prove local and semiglobal practical uniform asymptotic stability of the extrema of a certain map for multiagent extremum seeking systems. 1.