This paper presents an overview of singular perturbations and time scales (SPaTS) in control theory and applications during the period 1984-2001 (the last such overviews were provided by [231, 371]). Due to the limitations on space, this is in way intended to be an exhaustive survey on the topic.

Singular perturbations and time scales in control theory and applications: An overview

This paper introduces a technique for simultaneous decomposition of a non-autonomous singularly perturbed system into three subsystems namely “slow”, “fast 1” and “fast 2” respectively. Design of a composite controller in terms of the individual subsystem controllers is derived and the decomposition of the optimal control problem of the original high order system into three smaller order optimal control problems, with separate quadratic performance indexes extracted from the quadratic performance index of the original system, is discussed.

A Three-Time-Scale Approach for Design of Linear State Regulator for Spatial Control of Advanced Heavy Water Reactor

This paper studies the reliable H ∞ control for linear time-invariant multiparameter singularly perturbed systems against sensor failures. By time-scale decomposition, the full-order system is decomposed into slow and fast subsystems. After designing a reliable H ∞ controller for the global system, three reduced reliable H ∞ sub-controllers based on the slow and fast problems are obtained through the manipulation of the algebraic Riccati equations. The resulting control systems are reliable in that they provide guaranteed asymptotic stability and H ∞ performance when all control components are operational as well as when sensor failures occur.

Reliable H ∞ control of multiple time scales singularly perturbed systems with sensor failure

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 O(/spl epsiv/) perturbations (where /spl epsiv/=/spl par//sub /spl epsiv/2//sup /spl epsiv/1//spl par/ and /spl epsiv//sub 1/, /spl epsiv//sub 2/ 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 O(/spl epsiv/) perturbed reduced-order symmetric algebraic Riccati equations that can be solved rather easily. The proposed decomposition scheme might facilitates new approaches to multimodeling control problems that are conceptually simpler and numerically more efficient than the ones previously used.

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

In this paper we show how to completely and exactly decompose the optimal Kalman filter of stochastic systems in multimodeling form in terms of one pure-slow and two pure-fast, reduced-order, independent, Kalman filters. The reduced-order Kalman filters are all driven by the system measurements. This leads to a parallel Kalman filtering scheme and removes ill-conditioning of the original full-order singularly perturbed Kalman filter. The results obtained are valid for steady state. In that direction, the corresponding algebraic filter Riccati equation is completely decoupled and solved in terms of one pure-slow and two pure fast, reduced-order, independent, algebraic Riccati equations. A nonsingular state transformation that exactly relates the state variables in the original and new coordinates (in which the required decomposition is achieved) is also established. The eighth order model of a passenger car under road disturbances is used to demonstrate efficiency of the proposed filtering technique.

S. G. Rajalakshmi

Papers

Diagonalization and stability of multi-time-scale singularly perturbed linear systems

Intelligent Polymeric Biomaterials Surface Driven by Plasma Processing