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 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 New Designs of Reduced-Order Observer-Based Controllers for Singularly Perturbed Linear Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Heonjong Yoo ◽  
Zoran Gajic
Keyword(s):  
Perturbation Parameter ◽  
Fast Time ◽  
Feedback Controllers ◽  
Two Stage ◽  
Scale Separation ◽  
Reduced Order ◽  
Time Scale Separation ◽  
Reduced Order Observer ◽  
Ill Conditioning ◽  
Perturbed Linear Systems

The slow and fast reduced-order observers and reduced-order observer-based controllers are designed by using the two-stage feedback design technique for slow and fast subsystems. The new designs produce an arbitrary order of accuracy, while the previously known designs produce the accuracy of O(ϵ) only where ϵ is a small singular perturbation parameter. Several cases of reduced-order observer designs are considered depending on the measured state space variables: only all slow variables are measured, only all fast variables are measured, and some combinations of the slow and fast variables are measured. Since the two-stage methods have been used to overcome the numerical ill-conditioning problem for Cases (III)–(V), they have similar procedures. The numerical ill-conditioning problem is avoided so that independent feedback controllers can be applied to each subsystem. The design allows complete time-scale separation for both the reduced-order observer and controller through the complete and exact decomposition into slow and fast time scales. This method reduces both offline and online computations.

A Parameter Robust Finite Element Method for Fourth Order Singularly Perturbed Problems

2017 ◽  
Vol 17 (2) ◽  
pp. 337-349 ◽  
Author(s):  
Christos Xenophontos
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fourth Order ◽  
Hermite Polynomials ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
The Finite Element Method ◽  
Singularly Perturbed Problems ◽  
Exponentially Graded ◽  
Element Method

AbstractWe consider fourth order singularly perturbed problems in one-dimension and the approximation of their solution by the h version of the finite element method. In particular, we use piecewise Hermite polynomials of degree ${p\geq 3}$ defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error is measured in both the energy norm and a stronger, ‘balanced’ norm. Finally, we illustrate our theoretical findings through numerical computations, including a comparison with another scheme from the literature.

THE PARAMETER-ROBUST NUMERICAL METHOD BASED ON DEFECT-CORRECTION TECHNIQUE FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH LAYER BEHAVIOR

2010 ◽  
Vol 07 (04) ◽  
pp. 573-594 ◽  
Author(s):  
JUGAL MOHAPATRA ◽  
SRINIVASAN NATESAN
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Perturbation Parameter ◽  
Correction Method ◽  
Singularly Perturbed ◽  
Defect Correction ◽  
Central Difference ◽  
Correction Technique ◽  
Delay Differential

In this article, we consider a defect-correction method based on finite difference scheme for solving a singularly perturbed delay differential equation. We solve the equation using upwind finite difference scheme on piecewise-uniform Shishkin mesh, then apply the defect-correction technique that combines the stability of the upwind scheme and the higher-order central difference scheme. The method is shown to be convergent uniformly in the perturbation parameter and almost second-order convergence measured in the discrete maximum norm is obtained. Numerical results are presented, which are in agreement with the theoretical findings.

scholarly journals An adaptive mesh method for time dependent singularly perturbed differential-difference equations

2019 ◽  
Vol 8 (1) ◽  
pp. 328-339
Author(s):  
P. Pramod Chakravarthy ◽  
Kamalesh Kumar
Keyword(s):  
Prior Information ◽  
Grid Method ◽  
Adaptive Mesh ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Mesh Method

Abstract In this paper, a time dependent singularly perturbed differential-difference convection-diffusion equation is solved numerically by using an adaptive grid method. Similar boundary value problems arise in computational neuroscience in determination of the behaviour of a neuron to random synaptic inputs. The mesh is constructed adaptively by using the concept of entorpy function. In the proposed scheme, prior information of the width and position of the layers are not required. The method is independent of perturbation parameter ε and gives us an oscillation free solution, without any user introduced parameters. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme.

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 Reduced-Order DC Terminal Dynamic Model for Multi-Port AC-DC Power Electronic Transformer

Energies ◽  
2019 ◽  
Vol 12 (11) ◽  
pp. 2130 ◽  
Author(s):  
Zhe Wang ◽  
Yaohua Li ◽  
Zixin Li ◽  
Cong Zhao ◽  
Fanqiang Gao ◽  
...  
Keyword(s):  
Dominant Mode ◽  
The State ◽  
Reduced Order Model ◽  
Power Electronic ◽  
State Variables ◽  
Order Model ◽  
Energy Storage Devices ◽  
Reduced Order ◽  
Electronic Transformer ◽  
Power Electronic Transformer

As new electric power conversion equipment, a multi-port power electronic transformer (MP-PET), including a power electronic converter, high-frequency transformer, and multiple ac or dc interconnection interfaces, has a broad application in the hybrid distribution network. However, high integration and a large number of energy storage devices has led to very a high-order model of the system. To address this issue, a reduced-order small signal model of MP-PET is established in this paper. By taking the participation factors of the system mode to the state variables, the reduced-order model is derived based on the state variables, which are highly correlated with the dc voltage dominant mode. Compared with the full-order model, the proposed reduced-order model is accurate enough and simplified, and the validity of the simplified model is verified against simulations on a 10 kV/3 MVA MP-PET. The simulation results indicate that the proposed reduced-order model coincides well with the dynamic performance of the MP-PET.

scholarly journals Frequency Weighted Discrete-Time Controller Order Reduction Using Bilinear Transformation

2011 ◽  
Vol 62 (1) ◽  
pp. 44-48 ◽  
Author(s):  
Paknosh Karimaghaee ◽  
Navid Noroozi
Keyword(s):  
Discrete Time ◽  
Closed Loop ◽  
Linear Time ◽  
Order Reduction ◽  
Bilinear Transformation ◽  
Reduced Order ◽  
Linear Time Invariant ◽  
The Stability ◽  
Controllability And Observability ◽  
Time Invariant

Frequency Weighted Discrete-Time Controller Order Reduction Using Bilinear TransformationThis paper addresses a new method for order reduction of linear time invariant discrete-time controller. This method leads to a new algorithm for controller reduction when a discrete time controller is used to control a continuous time plant. In this algorithm, at first, a full order controller is designed ins-plane. Then, bilinear transformation is applied to map the closed loop system toz-plane. Next, new closed loop controllability and observability grammians are calculated inz-plane. Finally, balanced truncation idea is used to reduce the order of controller. The stability property of the reduced order controller is discussed. To verify the effectiveness of our method, a reduced controller is designed for four discs system.

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