Hiroaki Mukaidani

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

This 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, Newtons 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

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.

Brief paper: Soft-constrained stochastic Nash games for weakly coupled large-scale systems

In this paper, we discuss infinite-horizon soft-constrained stochastic Nash games involving state-dependent noise in weakly coupled large-scale systems. First, we formulate linear quadratic differential games in which robustness is attained against model uncertainty. It is noteworthy that this is the first time conditions for the existence of robust equilibria have been derived based on the solutions of sets of cross-coupled stochastic algebraic Riccati equations (CSAREs). After establishing an asymptotic structure with positive definiteness for CSAREs solutions, we derive a recursive algorithm by means of Newtons method so that it can be used to obtain solutions for CSAREs. As another important feature, we propose a high-order approximate Nash strategy based on iterative solutions. Finally, we provide a numerical example to verify the efficiency of the proposed algorithms.

New iterative algorithm for algebraic Riccati equation related to H/sub /spl infin// control problem of singularly perturbed systems

We present the solution to the algebraic Riccati equation (ARE) with indefinite sign quadratic term related to the H/sub /spl infin// control problem for singularly perturbed systems by means of a Kleinman type algorithm. The resulting algorithm is very efficient from the numerical point of view because the ARE is solvable even if the quadratic term has an indefinite sign. Moreover, the resulting iterative algorithm is quadratically convergent. We also present an algorithm for solving the generalized algebraic Lyapunov equation on the basis of the fixed point algorithm.

Pareto Optimal Strategy for Stochastic Weakly Coupled Large Scale Systems With State Dependent System Noise

This note is concerned with the decentralized infinite horizon stochastic Pareto-optimal static output feedback strategy for a class of weakly coupled systems with state-dependent noise. First, Pareto-optimal control problems are formulated using a static output feedback strategy. The necessary conditions are given by the cross-coupled stochastic algebraic Riccati-type equations (CSAREs) for proving the existence of the static output feedback strategy that minimizes the quadratic cost function. After determining the asymptotic structure of the solutions of the CSAREs, a new sequential numerical algorithm and Newtons method for solving the CSAREs are described. The resulting numerical solution is used to develop the Pareto-optimal strategy. Finally, the efficiency of the proposed algorithm is demonstrated by solving a numerical example for a practical megawatt-frequency control problem.

Brief paper: Robust guaranteed cost control for uncertain stochastic systems with multiple decision makers

The guaranteed cost control (GCC) problem for uncertain stochastic systems with N decision makers is investigated. It is noteworthy that the necessary conditions, which are determined from Karush-Kuhn-Tucker (KKT) conditions, for the existence of a guaranteed cost controller have been derived on the basis of the solutions of cross-coupled stochastic algebraic Riccati equations (CSAREs). It is shown that if CSAREs have an optimal solution, then the closed-loop system is exponentially mean square stable (EMSS) and has a cost bound. In order to simplify computations and attain a global optimum, the linear matrix inequality (LMI) technique is also considered. Finally, a numerical example for a practical megawatt-frequency control problem shows that the proposed methods can help in attaining an adequate cost bound. Furthermore, the features of these methods are characterized.

Recursive approach of H control problems for singularly perturbed systems under perfect- and imperfect-state measurements

In this paper, we study the H control problem for singularly perturbed systems under both perfect- and imperfect-state measurements by using the recursive approach of Gacjic et al. We construct a controller that guarantees a disturbance attenuation level larger than a boundary value of the reduced-order slow and fast subsystems when the singular perturbation parameter epsilon1 approaches zero. In order to obtain the controller, we must solve the generalized algebraic Riccati equations. The main results in this paper are to propose a new recursive algorithm to solve the generalized algebraic Riccati equations and to find sufficient conditions for the convergence of the proposed algorithm. Using the recursive algorithm, we show that the solution of the generalized algebraic Riccati equation converges to a positive semidefinite stabilizing solution with the rate of convergence of O(epsilon1k) under sufficient conditions. Furthermore, in the case of perfect-state measurements, we also show that the controller...

New method for composite optimal control of singularly perturbed systems

Abstract In this paper, a new method, based on a generalized algebraic Riccati equation arising in descriptor systems, is presented to solve the composite optimal control problem of singularly perturbed systems. Contrary to the existing method, the slow subsystem is viewed as a special kind of descriptor system. A new composite optimal controller is obtained which is valid for both standard and non-standard singularly perturbed systems. It is shown that the composite optimal control can be obtained simply by revising the solution of the slow regulator problem. It is proven that the composite optimal control can achieve a performance which is O(ee2) close to the optimal performance. Although this result is well-known for the standard singularly perturbed systems, it is new in the non-standard case. The equivalence between the new composite optimal controller and the existing one is also established for the standard singularly perturbed systems.

THE GUARANTEED COST CONTROL FOR UNCERTAIN LARGE–SCALE INTERCONNECTED SYSTEMS

Abstract The guaranteed cost control problem of the decentralized robust control for a class of large–scale interconnected systems with norm–bounded time–varying parameter uncertainties is considered. Based on the LMI design approach, a class of decentralized local state feedback controllers is proposed, and some sufficient conditions for the existence of guaranteed cost controllers are derived by making use of the LMI.

Numerical computation of sign-indefinite linear quadratic differential games for weakly coupled large-scale systems

In this paper, N-player linear quadratic differential games that are sign-indefinite for infinite horizon weakly coupled large-scale systems are discussed. After establishing the asymptotic structure and local uniqueness of the solution for cross-coupled sign-indefinite algebraic Riccati equations (CSARE), a new algorithm for solving CSARE is provided. It is shown that the proposed algorithm attains linear convergence. Moreover, in order to reduce the computational workspace, the recursive algorithm is combined. Finally, a high-order approximation strategy based on the proposed iterative solutions is described. As a result, it was recently proved that the numerical strategy achieves a high-order approximation of the equilibrium value. As another important feature, when the small parameters are unknown, a parameter-independent strategy is developed.