A numerical analysis of the Nash strategy for weakly coupled large-scale systems

his note discusses the feedback Nash equilibrium of linear quadratic N-player Nash games for infinite-horizon large-scale interconnected systems. The asymptotic structure along with the uniqueness and positive semidefiniteness of the solutions of the cross-coupled algebraic Riccati equations (CAREs) is newly established via the Newton-Kantorovich theorem. The main contribution of this study is the proposal of a new algorithm for solving the CAREs. In order to improve the convergence rate of the algorithm, Newton's method is combined with a new decoupling algorithm; it is shown that the proposed algorithm attains quadratic convergence. Moreover, it is shown for the first time that solutions to the CAREs can be obtained by solving the independent algebraic Lyapunov equation (ALE) by using the reduced-order calculation

Order reduction approaches for the algebraic Riccati equation and the LQR problem

We explore order reduction techniques for solving the algebraic Riccati equation (ARE), and investigating the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a surrogate low dimensional model of the dynamical system, for instance by means of balanced truncation, and then solve the corresponding ARE. Alternatively, iterative methods can be used to directly solve the ARE and use its approximate solution to estimate quantities associated with the LQR. We propose a class of Petrov-Galerkin strategies that simultaneously reduce the dynamical system while approximately solving the ARE by projection. This methodology significantly generalizes a recently developed Galerkin method by using a pair of projection spaces, as it is often done in model order reduction of dynamical systems. Numerical experiments illustrate the advantages of the new class of methods over classical approaches when dealing with large matrices

The linear quadratic regulator problem for a class of controlled systems modeled by singularly perturbed Ito differential equations

This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of O(√ε) approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an O(ε) approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter ε is not precisely known. © 2012 Society for Industrial and Applied Mathematics.Vasile Dragan, Hiroaki Mukaidani and Peng Sh

Algorithms for Computing Nash Equilibria in Deterministic LQ Games

In this paper we review a number of algorithms to compute Nash equilibria in deterministic linear quadratic differential games.We will review the open-loop and feedback information case.In both cases we address both the finite and the infinite-planning horizon.Algebraic Riccati equations;linear quadratic differential games;Nash equilibria

Linear Quadratic Games: An Overview

In this paper we review some basic results on linear quadratic differential games.We consider both the cooperative and non-cooperative case.For the non-cooperative game we consider the open-loop and (linear) feedback information structure.Furthermore the effect of adding uncertainty is considered.The overview is based on [9].Readers interested in detailed proofs and additional results are referred to this book.linear-quadratic games;Nash equilibrium;affine systems;solvability conditions;Riccati equations