Singularly Perturbed Systems Research Articles

Application of the concordance method of asymptotic expansions to solving boundary-value problems

The concordance method of asymptotic expansions applied for constructing uniform asymptotic expansions of singularly-perturbed partial differential equations and systems is presented.

Second-order sliding mode controllers for nonlinear singular perturbation systems

Nonlinear systems are decomposed into slow and fast response subsystems using the singular perturbation theory and second-order sliding mode controllers are designed for each of the subsystems. The combined control action forms a suboptimal controller that can eliminate chattering in the continuous control output. Simulated examples are included to demonstrate the effectiveness of the proposed algorithm.

Dynamical Behavior and Singularities of a Single-machine Infinite-bus Power System

This paper uses the geometric singular perturbation theory to investigate dynamical behaviors and singularities in a fundamental power system presented in a single-machine infinite-bus formulation. The power system can be approximated by two simplified systems S and F, which correspond respectively to slow and fast subsystems. The singularities, including Hopf bifurcation (HB), saddle-node bifurcation (SNB) and singularity induced bifurcation (SIB), are characterized. We show that SNB occurs at P Tc = 3.4382, SIB at P T0 = 2.8653 and HB at P Th = 2.802 for the singular perturbation system. It means that the power system will collapse near SIB which precedes SNB and that the power system will oscillate near HB which precedes SIB. In other words, the power system will lose its stability by means of oscillation near the HB which precedes SIB and SNB as P T is increasing to a critical value. The boundary of the stability region of the system can be described approximately by a combination of boundaries of the stability regions of the fast subsystem and slow subsystem.

Sensor Fault Diagnosis of Singularly Perturbed Systems Applied to F-8 Flight Dynamic

The paper outlines sensor fault-detection and isolation of singularly perturbed systems (SPS) using the slow subsystem. In deterministic case, the fast subsystem is considered as a modeling error by the approximation of the SPS with the slow subsystem. A second approach was to design a set of observer-based residuals using the slow subsystem. In stochastic case, the slow subsystem is used to approximate the initial system and to the sensor fault-diagnosis. The goal of the paper is to generate slow subsystem based residuals to fault-detection of SPS. We applied the results on a F-8 aircraft-dynamic to validate the proposed approaches.

Numerical computation of cross‐coupled algebraic Riccati equations related to H2/H∞ control problem for singularly perturbed systems

AbstractIn this paper, we present a numerical algorithm to the cross‐coupled algebraic Riccati equations(CARE) related to H2/H∞ control problems for singularly perturbed systems (SPS) by means of Newton's method. The resulting algorithm can be widely used to solve Nash game problems and robust control problems because the CARE is solvable even if the quadratic term has an indefinite sign. We prove that the resulting iterative algorithm has the property of the quadratic convergence. Using the solution of the CARE, we construct the high‐order approximate H2/H∞ controller. Copyright © 2004 John Wiley & Sons, Ltd.

The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system

This paper is concerned with the numerical solution for singular perturbation system of two coupled ordinary differential equations with first and second orders and with initial and boundary conditions, respectively. Finite difference scheme on a special non-uniform mesh, whose solution converges pointwise independently of the singular perturbation parameter is constructed and analyzed. Numerical results supporting the theory are presented.

Nonstandard singular perturbation systems and higher index differential-algebraic systems

In order to obtain trajectory approximation results for a given singular perturbation system (SPS), two systems are derived from it: the slow and the fast one. Tikhonov's theorem gives sufficient conditions on them to ensure a good approximation for a standard SPS, i.e., its corresponding slow system is a differential-algebraic system (DAS) of index 1. In this paper it is shown that a nonstandard SPS with the parameter set to zero can be seen as a DAS of higher index. This connection allows us to obtain a Tikhonov's theorem when this DAS is of index 2.

Multivariable circle criteria for multiparameter singularly perturbed systems

A direct approach to the Lur'e problem for singularly perturbed systems (SPS) is proposed. In contrast to previous results, the feedback connection between the linear and nonlinear parts of SPS is allowed to depend essentially on both the slow and the fast variables. The Lur'e problem for multiparameter SPS is studied by the same framework.

Time-suboptimal control design of singularly perturbed systems by reduced order feedback design

In the paper a new method for feedback control of singularly perturbed systems (SPS) is developed; it is based on the separation of slow and fast motions. The main problems concerned with SPS are the difficulties with high-order systems, singularly perturbed switching hypersurfaces and limit cycles. This obstacle can be alleviated by means of decoupling the slow and fast variables. The control goal is the reduction of dimensionality. The composite control design is elaborated. The first step is the time-optimal design for the slow subsystem. The second step consists in a Lyapunov-type control for the fast system. The advantage is an important reduction of computational amount and a simplified implementation of the control. The disadvantage is the suboptimality but the error is in order of the small parameter ε. Simulation results have shown the effectiveness of the composite algorithm, compared with the reduced order control.

On a state-space approach in robust control for singularly perturbed systems

A direct state-space approach for an estimate of the graph metric of singularly perturbed systems (SPS) is proposed. Based on a new version of the Tikhonov Theorem on uniform convergence of the slow variables in a stable SPS over the L unit ball of inputs in, an estimate of the first order to the small parameter for a 2 graph metric between SPS and its 'state decoupled form' is obtained for both time varying and time invariant cases. Moreover, a similar estimate of a graph metric between SPS and its slow system when the fast system satisfies some additional H conditions is given. Furthermore, the robust state feedback control for SPS is also considered in an analogous manner. An advantage there is that infinitely many composite feedback controls are obtained directly, bypassing the traditional arguments used in SPS theory such as approximation of solutions of Riccati equations and the Implicit function theorem.

Robust Dynamic Output Feedback Sliding-Mode Control of Singular Perturbation Systems

In this paper, the design of the robust dynamic output feedback sliding-mode controller (SMC) for singular perturbation systems is investigated. All the gain matrices of the dynamic sliding-mode controller can be determined via the static output feedback control for the alternative model of the reduced sliding dynamics. The results obtained differ from those of Refs. (6), (7) and (13) with respect to dynamic output feedback and also from those of Ref. (31) with respect to sliding-mode control. An active suspension system for a quarter-car is exploited to illustrate the derived schemes.

A unified theory of full-order and low-order observers based on singular system theory

The standard construction of the minimal-order state observer for a linear time-invariant plant is shown to be a degenerate form of the ordinary full-order observer. The naive state estimation technique of successively differentiating the plant output is also shown to be a limiting case of the same full-order structure. Various results concerning the set of all degenerate forms of the full-order observer are presented. In particular, conditions are established under which shifting observer eigenvalues far to the left in the complex plane implies proximity to a successively differentiating system. In addition, it is shown that such a scheme can never yield satisfactory estimation. Singular perturbation and singular system theory are used throughout the analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robustness of dynamic output feedback control for singular perturbation systems

The problem of the robustness of dynamic output feedback control for singular perturbation systems is investigated. The solution to this problem is reduced to the simultaneous design of static output feedback controllers for the fast subsystem and the so-called auxiliary system of the slow subsystem. Some conditions are proposed to ensure the robustness of the actual closed-loop system. The exact upper bound of the parasitic parameter for the controlled system is also determined. Finally, an actual model which failed in dynamic output feedback control in [4] is reexamined successfully here.

Stability margins of singular perturbation systems via state-feedback control

The gain and phase margins of singular perturbation systems are analysed under unmodelled high-frequency dynamics control, composite control, and the original full-order linear quadratic (LQ) control. The analysis is on the basis that there is a good relation between the minimum singular value of return difference transfer matrix and the stability margins. We begin with the examination of stability margins of subsystems and then show that state-feedback control design of subsystems could preserve gain and phase margins for the original full-order singularly perturbed system if the singular perturbation parameter epsiv; is sufficiently small. The effectiveness of e on stability margins is formulated and determined. It is found that the effectiveness can be evaluated by a simple method. Two examples are exploited to illustrate the analytic results.

Analysis of stability problems via matrix Lyapunov functions

The stability of nonlinear systems is analyzed by the direct Lyapunov′s method in terms of Lyapunov matrix functions. The given paper surveys the main theorems on stability, asymptotic stability and nonstability. They are applied to systems of nonlinear equations, singularly‐perturbed systems and hybrid systems. The results are demonstrated by an example of a two‐component system.

Almost Periodic Solutions of Singularly Perturbed Systems of Differential Equations with Impulse Effect

The present paper considers the problem for existence, uniqueness and asymptotic representation of periodic solutions of some singularly-perturbed systems of differential equations with impulses. The study is carried out with the help of a modification of the boundary functions method, adequately worked out to match the specific issues of the considered problem.

Iterative time-scale separation of synchronous machines

A scheme for iterative separation of time-scales in linear singularly-perturbed systems is developed and applied to synchronous machines. Both the current and flux-linkage dynamic models are considered and it is shown that they are linearly related for small disturbances. Simulation studies on a typical system have indicated the usefulness of time-scale separation schemes in analysing the dynamic behaviour of synchronous machines.

Periodic solutions of singularly-perturbed systems of differential equations with impulse effect

The present paper considers the problem for existence, uniqueness and asymptotic representation of periodic solutions of some singularly-perturbed systems of differential equations with impulses. The study is carried out with the help of a modification of the boundary functions method, adequately worked out to match the specific issues of the considered problem.

On approximate methods of analysing certain singularly-perturbed systems

A certainclass of sigularly-perturbed systems which have a variety of m-dimensional stationary positions is considered. When a small parameter disappears, the system also has an m-dimensional manifold of stationary positions and, therefore, the corresponding characteristic equation has m zero roots. The conditions under which the solution of a stability problem reduces to the same problem for a degenerate system are defined. As an application in practice gyroscopic stabilizing systems (the critical case corresponds to such systems) with elastic elements of high stiffness are discussed. The conditions under which the solution of the problem of the stability of steady motion follows from the solution of this problem for an ideal system (with absolutely rigid elements) are obtained. The problem of the closeness of the corresponding solutions of the complete and a simplified system of differential equations over an infinite time interval is discussed.