Exact decomposition of the algebraic Riccati equation of deterministic multimodeling optimal control problems

Abstract

In this paper we show how to exactly decompose the algebraic Riccati equations of deterministic multimodeling in terms of one pure-slow and two pure-fast algebraic Riccati equations. The algebraic Riccati equations obtained are of reduced-order and nonsymmetric. However, their ( ) perturbations (where = and , are small positive singular perturbation parameters) are symmetric. The Newton method is perfectly suited for solving the nonsymmetric reduced-order pure-slow and pure-fast algebraic Riccati equations since excellent initial guesses are available from their ( ) perturbed reduced-order symmetric algebraic Riccati equations that can be solved rather easily. The proposed decomposition scheme might facilitates new approaches to mutimodeling control problems that are conceptually simpler and numerically more efficient than the ones previously used.

Feedback Control of Linear Multiparameter Singularly Perturbed Systems

“…If we know the values of the small parameters ε 1 , ε 2 , ε 12 and ε 21 , this optimal control problem could be solved (Coumarbatch and Gajić, 2000;Mukaidani et al 2002). However, it is impossible to obtain the optimal control when the small parameters are unknown.…”

“…The matrices A s , S s and Q s do not depend onP 11 andP 22 because their matrices can be computed by using T pq , p, q = 0, 1, 2 which are independent ofP 11 andP 22 (Coumarbatch and Gajić, 2000), that is,…”

“…In order to obtain the optimal solution to the multimodeling problems, we must solve the multiparameter algebraic Riccati equation (MARE), which are parameterized by the small positive same order parameters ε j , j = 1, 2, · · ·. Various reliable approaches for solving the MARE have been well documented in literatures (see e.g., Coumarbatch and Gajić, 2000;Mukaidani et al 2002). However, these results are limited to the case that the small parameters are assumed to be known.…”

“…It is well known that the solution of the linear quadratic control problem (1) and (3) is given by (Khalil and Kokotović, 1978;Coumarbatch and Gajić, 2000),…”

“…The deterministic and stochastic multimodeling stability, control, filtering and dynamic games have been investigated extensively by several researchers (see e.g., Kokotović, 1978, 1979;Coumarbatch and Gajić, 2000;Gajić, 1988;Wang et al, 1994). The multimodeling problems arise in large-scale dynamic systems.…”