Limit cycles in relay systems with fractional order plants

2019 ◽  
Vol 41 (15) ◽  
pp. 4424-4435
Author(s):  
Ali Yüce ◽  
Nusret Tan ◽  
Derek P Atherton
Keyword(s):  
Time Delay ◽  
Transfer Function ◽  
Limit Cycle ◽  
Fractional Order ◽  
Limit Cycles ◽  
Transfer Functions ◽  
Feedback Systems ◽  
Relay Feedback ◽  
Cycle Frequency ◽  
Plant Transfer

In this paper, limit cycle frequency, pulse width and stability analysis are examined using different methods for relay feedback nonlinear control systems with integer or fractional order plant transfer functions. The describing function (DF), A loci, a time domain method formulated in state space notation and Matlab/Simulink simulations are used for the analysis. Comparisons of the results of using these methods are given in several examples. In addition, the work has been extended to fractional order systems with time delay. Programs have been developed in the Matlab environment for all the theoretical methods. In particular, Matlab programs have been written to obtain a graphical solution for the A loci method, which can precisely calculate the limit cycle frequency. The developed solution methods are shown in various examples. The major contribution is to look at finding limit cycles for relay feedback systems having plants with a fractional order transfer function (FOTF). However, en route to this goal new assessments of limit cycle stability have been done for a rational plant transfer function plus a time delay.

scholarly journals Fitting of Generic Process Models by an Asymmetric Short Relay Feedback Experiment—The n-Shifting Method

2021 ◽  
Vol 11 (4) ◽  
pp. 1651
Author(s):  
José Sánchez Moreno ◽  
Sebastián Dormido Bencomo ◽  
José Manuel Díaz Martínez
Keyword(s):  
Time Delay ◽  
Transfer Function ◽  
Dynamic Systems ◽  
Transfer Functions ◽  
Process Models ◽  
Transfer Function Model ◽  
Minimum Phase ◽  
Relay Feedback ◽  
Function Model ◽  
Generic Process

This paper presents the generalization of the shifting method for relay feedback identification of dynamic systems of any order. The original shifting method enables the fitting of a maximum of five parameters of a transfer function model from the information obtained from a short relay test and without prior knowledge of the process to identify. The generalization, known as n-shifting, allows the estimation of the parameters of transfer functions of any order by applying one short relay test to the process to identify. Without loss of generality, the n-shifting approach is applied to fit an n-order plus time delay (n-OPTD) model but the approach can be also developed to identify models with other structures (non-minimum phase, unstable, integrators). Some examples of estimations are presented.

Stability of Limit Cycles for Time-Delay Relay Feedback Systems

2001 ◽  
Vol 34 (6) ◽  
pp. 395-400
Author(s):  
Chong Lin ◽  
Qing-Guo Wang ◽  
Tong Heng Lee ◽  
James Lam
Keyword(s):  
Time Delay ◽  
Limit Cycles ◽  
Feedback Systems ◽  
Relay Feedback

Complementary Sensitivity Design of PID Controllers for Arbitrary-Order Transfer Functions With Time Delay

2008 ◽  
Author(s):  
Tooran Emami ◽  
John M. Watkins
Keyword(s):  
Time Delay ◽  
Transfer Functions ◽  
Linear Time ◽  
Order System ◽  
Pid Controllers ◽  
Single Input Single Output ◽  
Linear Time Invariant ◽  
Single Output ◽  
Plant Transfer ◽  
Time Invariant

A graphical technique for finding all proportional integral derivative (PID) controllers that stabilize a given single-input-single-output (SISO) linear time-invariant (LTI) system of any order system with time delay has been solved. In this paper a method is introduced that finds all PID controllers that also satisfy an H∞ complementary sensitivity constraint. This problem can be solved by finding all PID controllers that simultaneously stabilize the closed-loop characteristic polynomial and satisfy constraints defined by a set of related complex polynomials. A key advantage of this procedure is the fact that it does not require the plant transfer function, only its frequency response.

On the approximate inverse Laplace transform of the transfer function with a single fractional order

2020 ◽  
pp. 014233122097766
Author(s):  
Ali Yüce ◽  
Nusret Tan
Keyword(s):  
Transfer Function ◽  
Laplace Transform ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Transfer Functions ◽  
Inverse Laplace Transform ◽  
Fractional Order Systems ◽  
Approximate Inverse ◽  
Classical Mathematics ◽  
Curve Fitting Method

The history of fractional calculus dates back to 1600s and it is almost as old as classical mathematics. Although many studies have been published on fractional-order control systems in recent years, there is still a lack of analytical solutions. The focus of this study is to obtain analytical solutions for fractional order transfer functions with a single fractional element and unity coefficient. Approximate inverse Laplace transformation, that is, time response of the basic transfer function, is obtained analytically for the fractional order transfer functions with single-fractional-element by curve fitting method. Obtained analytical equations are tabulated for some fractional orders of [Formula: see text]. Moreover, a single function depending on fractional order alpha has been introduced for the first time using a table for [Formula: see text]. By using this table, approximate inverse Laplace transform function is obtained in terms of any fractional order of [Formula: see text] for [Formula: see text]. Obtained analytic equations offer accurate results in computing inverse Laplace transforms. The accuracy of the method is supported by numerical examples in this study. Also, the study sets the basis for the higher fractional-order systems that can be decomposed into a single (simpler) fractional order systems.

scholarly journals On the Essential Instabilities Caused by Fractional-Order Transfer Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
Farshad Merrikh-Bayat ◽  
Masoud Karimi-Ghartemani
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Characteristic Equation ◽  
Stability Condition ◽  
Transfer Functions ◽  
Branch Points ◽  
The Stability ◽  
Unstable Branch

The exact stability condition for certain class of fractional-order (multivalued) transfer functions is presented. Unlike the conventional case that the stability is directly studied by investigating the poles of the transfer function, in the systems under consideration, the branch points must also come into account as another kind of singularities. It is shown that a multivalued transfer function can behave unstably because of the numerator term while it has no unstable poles. So, in this case, not only the characteristic equation but the numerator term is of significant importance. In this manner, a family of unstable fractional-order transfer functions is introduced which exhibit essential instabilities, that is, those which cannot be removed by feedback. Two illustrative examples are presented; the transfer function of which has no unstable poles but the instability occurred because of the unstable branch points of the numerator term. The effect of unstable branch points is studied and simulations are presented.

Flame Transfer Function for Swirled LPP Combustion From Experiments and CFD

2004 ◽  
Author(s):  
Carol A. Armitage ◽  
Alex J. Riley ◽  
R. Stewart Cant ◽  
Ann P. Dowling ◽  
Simon R. Stow
Keyword(s):  
Time Delay ◽  
Transfer Function ◽  
Gas Turbines ◽  
Transfer Functions ◽  
Low Frequency ◽  
Analytical Models ◽  
Delay Spread ◽  
Good Representation ◽  
Atmospheric Conditions ◽  
Spread Model

Combustion oscillations that arise in gas turbines can lead to plant damage. One method used to predict these oscillations is to analyse the acoustics using a simple linear model. This model requires a transfer function to describe the response of the heat release to flow perturbations. A transfer function has been obtained for a swirled premixed combustion system using experiments under atmospheric conditions and CFD. These results have been compared with analytical models. The experimental and computational transfer functions both indicate a low frequency zero. A time-delay spread model gives a good representation of the computational transfer function. The experimental transfer function is described well by a model that combines a time-delay spread with a constant gain.

Describing Function Analysis of Limit Cycles in a Multiple Flame Combustor

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Frédéric Boudy ◽  
Daniel Durox ◽  
Thierry Schuller ◽  
Grunde Jomaas ◽  
Sébastien Candel
Keyword(s):  
Transfer Function ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Function Analysis ◽  
Combustion Instabilities ◽  
Describing Function ◽  
Variable Size ◽  
Theoretical Predictions ◽  
Flame Describing Function ◽  
Oscillation Frequencies

A recently developed nonlinear flame describing function (FDF) is used to analyze combustion instabilities in a system where the feeding manifold has a variable size and where the flame is confined by quartz tubes of variable length. Self-sustained combustion oscillations are observed when the geometry is changed. The regimes of oscillation are characterized at the limit cycle and also during the onset of oscillations. The theoretical predictions of the oscillation frequencies and levels are obtained using the FDF. This generalizes the concept of flame transfer function by including dependence on the frequency and level of oscillation. Predictions are compared with experimental results for two different lengths of the confinement tube. These results are, in turn, used to predict most of the experimentally observed phenomena and in particular, the correct oscillation levels and frequencies at limit cycles.

scholarly journals Approximation method for a fractional order transfer function with zero and pole

2014 ◽  
Vol 24 (4) ◽  
pp. 447-463 ◽  
Author(s):  
Krzysztof Oprzędkiewicz
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Approximation Method ◽  
Transfer Functions ◽  
Control Algorithms ◽  
Model Based ◽  
Interval Approach ◽  
Special Control

Abstract The paper presents an approximation method for elementary fractional order transfer function containing both pole and zero. This class of transfer functions can be applied for example to build model - based special control algorithms. The proposed method bases on Charef approximation. The problem of cancelation pole by zero with useful conditions was considered, the accuracy discussion with the use of interval approach was done also. Results were depicted by examples.

Describing Function Analysis of Limit Cycles in a Multiple Flame Combustor

2010 ◽  
Author(s):  
Fre´de´ric Boudy ◽  
Daniel Durox ◽  
Thierry Schuller ◽  
Grunde Jomaas ◽  
Se´bastien Candel
Keyword(s):  
Transfer Function ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Function Analysis ◽  
Combustion Instabilities ◽  
Describing Function ◽  
Variable Size ◽  
Theoretical Predictions ◽  
Flame Describing Function ◽  
Oscillation Frequencies

A recently developed nonlinear Flame Describing Function (FDF) is used to analyze combustion instabilities in a system where the feeding manifold has a variable size and where the flame is confined by quartz tubes of variable length. Self-sustained combustion oscillations are observed when the geometry is changed. Regimes of oscillation are characterized at the limit cycle and also during the onset of oscillations. Theoretical predictions of the oscillation frequencies and levels are obtained using the FDF. This generalizes the concept of flame transfer function by including a dependence on the frequency and on the level of oscillation. Predictions are compared with experimental results for two different lengths of the confinement tube. These results are in turn used to predict most of the experimentally observed phenomena and in particular the correct oscillation levels and frequencies at limit cycles.

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