scholarly journals Reduced-Order Algorithm for Eigenvalue Assignment of Singularly Perturbed Linear Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Heonjong Yoo ◽  
Zoran Gajic ◽  
Kyeong-Hwan Lee
Keyword(s):  
Singular Perturbation ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Sylvester Equation ◽  
Singularly Perturbed Systems ◽  
Reduced Order ◽  
Recursive Solution ◽  
System Variables ◽  
Perturbed Linear Systems ◽  
Eigenvalue Assignment

In this paper, we present an algorithm for eigenvalue assignment of linear singularly perturbed systems in terms of reduced-order slow and fast subproblem matrices. No similar algorithm exists in the literature. First, we present an algorithm for the recursive solution of the singularly perturbed algebraic Sylvester equation used for eigenvalue assignment. Due to the presence of a small singular perturbation parameter that indicates separation of the system variables into slow and fast, the corresponding algebraic Sylvester equation is numerically ill-conditioned. The proposed method for the recursive reduced-order solution of the algebraic Sylvester equations removes ill-conditioning and iteratively obtains the solution in terms of four reduced-order numerically well-conditioned algebraic Sylvester equations corresponding to slow and fast variables. The convergence rate of the proposed algorithm is Oε, where ε is a small positive singular perturbation parameter.

scholarly journals Recursive Reduced-Order Algorithm for Singularly Perturbed Cross Grammian Algebraic Sylvester Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Intessar Al-Iedani ◽  
Zoran Gajic
Keyword(s):  
Recursive Algorithm ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Sylvester Equation ◽  
State Variables ◽  
Reduced Order ◽  
Aggregate Information ◽  
The Cross ◽  
Perturbed Linear Systems ◽  
Controllability And Observability

A new recursive algorithm is developed for solving the algebraic Sylvester equation that defines the cross Grammian of singularly perturbed linear systems. The cross Grammian matrix provides aggregate information about controllability and observability of a linear system. The solution is obtained in terms of reduced-order algebraic Sylvester equations that correspond to slow and fast subsystems of a singularly perturbed system. The rate of convergence of the proposed algorithm isOε, whereεis a small singular perturbation parameter that indicates separation of slow and fast state variables. Several real physical system examples are solved to demonstrate efficiency of the proposed algorithm.

scholarly journals Passive Control andε-Bound Estimation of Singularly Perturbed Systems with Nonlinear Nonlinearities

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Linna Zhou ◽  
Chunyu Yang
Keyword(s):  
Singular Perturbation ◽  
Passive Control ◽  
Controller Design ◽  
Design Method ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Storage Function ◽  
Rlc Circuit ◽  
Perturbed Systems

This paper considers the problems of passivity analysis and synthesis of singularly perturbed systems with nonlinear uncertainties. By a novel storage function depending on the singular perturbation parameterε, a new method is proposed to estimate theε-bound, such that the system is passive when the singular perturbation parameter is lower than theε-bound. Furthermore, a controller design method is proposed to achieve a predefinedε-bound. The proposed results are shown to be less conservative than the existing ones because the adopted storage function is more general. Finally, an RLC circuit is presented to illustrate the advantages and effectiveness of the proposed methods.

scholarly journals Additive fault tolerant control of nonlinear singularly perturbed systems against actuator fault

2017 ◽  
Vol 68 (1) ◽  
pp. 68-73 ◽  
Author(s):  
Adel Tellili ◽  
Aymen Elghoul ◽  
Mohamed Naceur Abdelkrim
Keyword(s):  
Singular Perturbation ◽  
Fault Tolerant ◽  
Fault Tolerant Control ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Actuator Faults ◽  
Singularly Perturbed Systems ◽  
Singular Perturbation Method ◽  
Numerical Application ◽  
Perturbed Systems

AbstractThis paper presents the design of an additive fault tolerant control for nonlinear time-invariant singularly perturbed systems against actuator faults based on Lyapunov redesign principle. The overall system is reduced into subsystems with fast and slow dynamic behavior using singular perturbation method. The time scale reduction is carried out when the singular perturbation parameter is set to zero, thus avoiding the numerical stiffness due to the interaction of two different dynamics. The fault tolerant controller is computed in two steps. First, a nominal composite controller is designed using the reduced subsystems. Secondly, an additive part is combined with the basic controller to overcome the fault effect. The derived ε - independent controller guarantees asymptotic stability despite the presence of actuator faults. The Lyapunov stability theory is used to prove the stability provided the singular perturbation parameter is sufficiently small. The theoretical results are simulated using a numerical application.

Stabilization of Discrete Singularly Perturbed Systems Under Composite Observer-Based Control

1999 ◽  
Vol 123 (1) ◽  
pp. 132-139 ◽  
Author(s):  
Feng-Hsiag Hsiao ◽  
Jiing-Dong Hwang ◽  
Shing-Tai Pan
Keyword(s):  
Singular Perturbation ◽  
Stability Criterion ◽  
Original System ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Stability Conditions ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Perturbed System ◽  
Perturbed Systems

New stability conditions for discrete singularly perturbed systems are presented in this study. The corresponding slow and fast subsystems of the original discrete singularly perturbed system are first derived. The observer-based controllers for the slow and the fast subsystems are then separately designed and a composite observer-based controller for the original system is subsequently synthesized from these observer-based controllers. Finally, a frequency domain ε-dependent stability criterion for the original discrete singularly perturbed system under the composite observer-based controller is proposed. If any one condition of this criterion is fulfilled, stability of the original system by establishing that of its corresponding slow and fast subsystems is thus investigated. An illustrative example is given to demonstrate that the upper bound of the singular perturbation parameter ε can be obtained by examining this criterion.

Discrete-Time Nonlinear Singularly Perturbed System Identification Using Coupled Multimodel Representation

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
Asma Ben Rajab ◽  
Nesrine Bahri ◽  
Majda Ltaief
Keyword(s):  
Discrete Time ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
System Model ◽  
Model Parameters ◽  
Singularly Perturbed Systems ◽  
Perturbed System ◽  
Marquardt Algorithm ◽  
Perturbed Systems ◽  
Full Knowledge

Abstract Many control and observability theories for singularly perturbed systems require the full knowledge of system model parameters exceptionally if the system is considered as black box. To overcome this problem and to obtain an accurate and faithful model, this paper describes a new identification method for discrete-time nonlinear singularly perturbed systems (NLSPS) using the coupled state multimodel representation. The Levenberg–Marquardt algorithm is used to identify not only the submodels parameters but also the perturbation parameter ε. Two cases are considered to identify these systems. The first one supposes that the perturbation parameter ε of the real system is known and thus only the submodels parameters are identified. The second case supposes that this perturbation parameter is unknown and has to be identified with the other submodels parameters. The simulation example demonstrates the effectiveness of the proposed identification.

scholarly journals Diagnosis of Discrete-Time Singularly Perturbed Systems Based on Slow Subsystem

2014 ◽  
Vol 8 (4) ◽  
pp. 175-180 ◽  
Author(s):  
Adel Tellili ◽  
Nouceyba Abdelkrim ◽  
Bahaa Jaouadi ◽  
Mohamed Naceur Abdelkrim
Keyword(s):  
Discrete Time ◽  
Diagnostic Algorithm ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Order System ◽  
Singularly Perturbed Systems ◽  
Perturbed Systems ◽  
Parity Space ◽  
The One ◽  
Slow Subsystem

Abstract This paper deals with the diagnosis of discrete-time singularly perturbed systems presenting two time scales property. Parity space method is considered to generate the fault detection residual. The focus is in two directions. First, we discuss the residual illconditioning caused by the singular perturbation parameter. Then, the use of the slow subsystem is considered to make the fault diagnosis easier. It is shown that the designed diagnostic algorithm based on reduced order model is close to the one synthesized using the full order system. The developed approach aims at reducing the computational load and the ill-conditioning for stiff residual generation problem. Two examples of application are used to demonstrate the efficiency of the proposed method.

scholarly journals On Output Feedback Multiobjective Control for Singularly Perturbed Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-28
Author(s):  
Mehdi Ghasem Moghadam ◽  
Mohammad Taghi Hamidi Beheshti
Keyword(s):  
Numerical Simulations ◽  
Output Feedback ◽  
Design Procedure ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Dynamic Output Feedback ◽  
Singularly Perturbed Systems ◽  
Dynamic Output ◽  
Perturbed Systems ◽  
And Performance

A new design procedure for a robust and control of continuous-time singularly perturbed systems via dynamic output feedback is presented. By formulating all objectives in terms of a common Lyapunov function, the controller will be designed through solving a set of inequalities. Therefore, a dynamic output feedback controller is developed such that and performance of the resulting closed-loop system is less than or equal to some prescribed value. Also, and performance for a given upperbound of singular perturbation parameter are guaranteed. It is shown that the -dependent controller is well defined for any and can be reduced to an -independent one so long as is sufficiently small. Finally, numerical simulations are provided to validate the proposed controller. Numerical simulations coincide with the theoretical analysis.

THREE-STEP TAYLOR GALERKIN METHOD FOR SINGULARLY PERTURBED GENERALIZED HODGKIN–HUXLEY EQUATION

2010 ◽  
Vol 01 (02) ◽  
pp. 257-276
Author(s):  
VIVEK SANGWAN ◽  
B. V. RATHISH KUMAR ◽  
S. V. S. S. N. V. G. K. MURTHY ◽  
MOHIT NIGAM
Keyword(s):  
Boundary Layer ◽  
Finite Element Method ◽  
Finite Element ◽  
Singular Perturbation ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Galerkin Finite Element Method ◽  
Third Order ◽  
Galerkin Finite Element ◽  
Huxley Equation

A numerical study is carried out for the singularly perturbed generalized Hodgkin–Huxley equation. The equation is nonlinear which mimics the ionic processes at a real nerve membrane. A small parameter called singular perturbation parameter is introduced in the highest order derivative term. Keeping other parameters fixed, as this singular perturbation parameter approaches to zero, a boundary layer occurs in the solution. Three-step Taylor Galerkin finite element method is employed on a piecewise uniform Shishkin mesh to solve the equation. To procure more accurate temporal differencing, the method employs forward-time Taylor series expansion including time derivatives of third order which are evaluated from the governing singularly perturbed generalized Hodgkin–Huxley equation. This yields a generalized time-discretized equation which is successively discretized in space by means of the standard Bubnov–Galerkin finite element method. The method is third-order accurate in time. The code based on the purposed scheme has been validated against the cases for which the exact solution is available. It is also observed that for the Singularly Perturbed Generalized Hodgkin–Huxley equation, the boundary layer in the solution manifests not only by varying the singular perturbation parameter but also by varying the other parameters appearing in the model.

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