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.

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.

A Simplified Two-Stage Design of Linear Discrete-Time Feedback Controllers With Applications to Systems With Slow and Fast Modes

2014 ◽  
Author(s):  
Verica Radisavljevic-Gajic
Keyword(s):  
Discrete Time ◽  
Design Procedure ◽  
Linear Feedback ◽  
Fast Time ◽  
Singularly Perturbed Systems ◽  
Feedback Controllers ◽  
Two Stage ◽  
Stage Design ◽  
Two Stage Design ◽  
Time Systems

In this paper we have shown how to simplify an algorithm for the two stage design of linear feedback controllers by reducing computational requirements. The algorithm is further simplified for linear discrete-time systems with slow and fast modes (multi-time scale systems or singularly perturbed systems) providing independent and accurate designs in slow and fast time scales. The simplified design procedure and its very high accuracy are demonstrated on the eigenvalue assignment problem of a steam power system.

Implicit Quasi-Steady-State Approximation and Application to a Power Plant Evaporator

2006 ◽  
Vol 129 (1) ◽  
pp. 66-71 ◽  
Author(s):  
Eduard Eitelberg ◽  
Edward Boje
Keyword(s):  
Steady State ◽  
Power Plant ◽  
Nonlinear Models ◽  
Low Frequency ◽  
Perturbation Parameter ◽  
Perturbation Approach ◽  
Quasi Steady State ◽  
Reduced Order ◽  
First Order ◽  
Time Scale Separation

Construction of reduced order models using the conventional quasi-steady-state (QSS) or singular perturbation approach may not yield good low frequency approximations, especially if there is not a distinct time scale separation into slow and fast subsystems. An implicit QSS technique is proposed for general nonlinear models. The resulting reduced order model is accurate to first order in the perturbation parameter and its linearization is accurate to first order in frequency. An example is included showing the application of the proposed method to model reduction on a power plant evaporator.

scholarly journals On Singular Perturbations of Flexible and Variable-Speed Wind Turbines

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
R. Oulad Ben Zarouala ◽  
C. Vivas ◽  
J. Á. Acosta ◽  
L. El Bakkali
Keyword(s):  
Singular Perturbations ◽  
Time Scale ◽  
Reduced Model ◽  
Physical Parameters ◽  
Scale Separation ◽  
Reduced Order ◽  
Physical Mechanisms ◽  
Time Scale Separation ◽  
Successful Approach ◽  
Mechanical Dynamics

A model for the mechanical dynamics of a wind turbine is developed, which is the composition of three physical mechanisms: flexion, torsion, and rotational dynamics. A first contribution is the identification of the essential physical parameters that provide a time-scale separation of these three mechanisms. Under the assumption of singular perturbations the time-scale separation allows to work with a reduced model of order one. This reduction has been essential for the control of this system allowing to control designers to take into account only the reduced-order model. A second contribution consists in employing a measurement of the fore-aft nacelle acceleration with the reduced model, together with a Kalman filter to estimate the flexible DOFs of the system (tower and average blade deflection). The successful approach is tested on high-order nonlinear aeroelastic simulator (FAST).

A Simplified Two-Stage Design of Linear Discrete-Time Feedback Controllers

2014 ◽  
Vol 137 (1) ◽  
Author(s):  
Verica Radisavljevic-Gajic
Keyword(s):  
Discrete Time ◽  
Design Procedure ◽  
Linear Feedback ◽  
Fast Time ◽  
Singularly Perturbed Systems ◽  
Feedback Controllers ◽  
Two Stage ◽  
Stage Design ◽  
Two Stage Design ◽  
Time Systems

In this paper, we have shown how to simplify an algorithm for the two-stage design of linear feedback controllers by reducing computational requirements. The algorithm is further simplified for linear discrete-time systems with slow and fast modes (multitime scale systems or singularly perturbed systems), providing independent and accurate designs in slow and fast time scales. The simplified design procedure and its very high accuracy are demonstrated on the eigenvalue assignment problem of a steam power system.

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