Nash games for multiparameter singularly perturbed systems with uncertain small singular perturbation parameters

In this paper, the linear quadratic Nash games for infinite horizon multiparameter singularly perturbed systems with uncertain singular perturbation parameters are discussed. The main contribution is that a construction of high-order approximations to a strategy that guarantees a desired performance level on the basis of the successive approximation is proposed. It is newly shown that the proposed high-order approximate strategy improves the cost performance

Near-optimal kalman filters for multiparameter singularly perturbed linear systems

In this brief, we study the near-optimal Kalman filtering problem for multiparameter singularly perturbed system (MSPS). The attention is focused on the design of the near-optimal Kalman filters. It is shown that the resulting filters in fact remove ill-conditioning of the original full-order singularly perturbed Kalman filters. In addition the resulting filters can be used compared with the previously proposed result even if the fast state matrices are singular

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

Understanding of An Algorithm for Numerical Computation via Newton's Method for Basic Information Education

In this paper, the understanding of an algorithm for numerical computation via Newton's method is investigated. In order to demonstrate the efficiency of Newton's method, an example of how to use them is given. After introducing the mathematical tool related to matrix algebra, the new iterative technique that is based on Newton's method for solving a set of the algebraic Riccati equation is applied. It is shown that the proposed algorithm guarantees the local quadratic convergence. A numerical example to show the validity of the Newton's method is given for the practical control problem

An LMI approach to guaranteed cost control for uncertain delay systems

The guaranteed cost-control problem for uncertain linear systems which have delay in both state and control input is considered. Sufficient conditions for the existence of guaranteed cost controllers are given in terms of linear matrix inequality (LMI). It is shown that the state feedback controllers can be obtained by solving the LMI

Local uniqueness for Nash solutions of multiparameter singularly perturbed systems

In this brief, linear quadratic infinite-horizon Nash games for general multiparameter singularly perturbed systems are studied. The local uniqueness and the asymptotic structure of the solutions to the cross-coupled multiparameter algebraic Riccati equation (CMARE) are newly established. Utilizing the asymptotic structure of the solutions to the CMARE, the parameter-independent Nash strategy is established. A numerical example is given to demonstrate the efficiency and feasibility of the proposed analysis

A new design approach for solving linear quadratic Nash games of multiparameter singularly perturbed systems

In this paper, the linear quadratic Nash games for infinite horizon nonstandard multiparameter singularly perturbed systems (MSPS) without the nonsingularity assumption that is needed for the existing result are discussed. The new strategies are obtained by solving the generalized cross-coupled multiparameter algebraic Riccati equations (GCMARE). Firstly, the asymptotic expansions for the GCMARE are newly established. The main result in this paper is that the proposed algorithm which is based on the Newton's method for solving the GCMARE guarantees the quadratic convergence. In fact, the simulation results show that the proposed algorithm succeed in improving the convergence rate dramatically compared with the previous results. It is also shown that the resulting controller achieves O(∥μ∥2n) approximation of the optimal cost

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

Efficient numerical procedures for solving closed-loop Stackelberg strategies with small singular perturbation parameter

In this paper, the computation of the linear closed-loop Stackelberg strategies with small singular perturbation parameter that characterizes singularly perturbed systems (SPS) are studied. The attention is focused on a new numerical algorithm for solving a set of cross-coupled algebraic Lyapunov and Riccati equations (CALRE). It is proven that the new algorithm guarantees the local quadratic convergence. A numerical example is solved to show a reduction of the average CPU time compared with the existing algorithm