Determination of the Area of Robust Stability of the System on the Basis of V. L. Kharitonov’s Theorem

2020 ◽  
Vol 21 (4) ◽  
pp. 208-212 ◽  
Author(s):  
A. P. Kutsyi ◽  
N. N. Kutsyi ◽  
T. V. Malanova
Keyword(s):  
Objective Function ◽  
Linear Systems ◽  
Robust Stability ◽  
Automatic System ◽  
Main Idea ◽  
Control Object ◽  
Single Model ◽  
The Stability ◽  
Robust Regulation

The parameters of the object of regulation during operation due to various reasons may vary. These changes can lead to a change in the performance indicators of the automatic system, as well as its stability. This article proposes an approach to determine the range of acceptable values of the parameters of the control object of an automatic system with a PID controller, in which the system will remain stable. Thus, the problem arises of analyzing an automatic control system given not only by a single model with clearly defined parameters, but by a family of models belonging to a given set — the task of robust regulation. The search for ranges in which the parameters of the regulated object can change is based on the solution of the nonlinear programming problem in this paper. The conclusion of the objective function and constraint system using the theorem of V. L. Kharitonova on the robust stability of linear systems. The main idea is that each parameter of the regulatory object can be changed by some value hi1 in the direction of decrease and by hi2 — in the direction of increase. Replacing the notation used in the theorem of V. L. Kharitonov, the lower and upper boundaries of the change of parameters by the sum and difference of the nominal values of the parameters and the corresponding hi1, hi2, we get a system of restrictions. Moreover, for the stability of Kharitonov polynomials, it is most convenient to use the Lienar-Shipar criterion. The larger the values of hi1, hi2, the wider the ranges of variation of the parameters, and the smaller the inverse of the sum of these values. Based on this statement, the objective function is formed. It should be noted that the condition for the considered automatic system on which the proposed approach is based is sufficient, but not necessary, since the coefficients of the polynomial are interdependent. An example with the help of which the proposed approach is demonstrated is considered. This approach can also be applied to other linear systems for which theconditions of V. L. Kharitonova. 

scholarly journals Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 148
Author(s):  
Vasile Dragan ◽  
Samir Aberkane
Keyword(s):  
Lower Bound ◽  
Linear Systems ◽  
Robust Stability ◽  
Stability Radius ◽  
Control Synthesis ◽  
Parametric Uncertainties ◽  
Jump Linear Systems ◽  
Markov Jump Linear Systems ◽  
Robust Stability Analysis ◽  
The Stability

This note is devoted to a robust stability analysis, as well as to the problem of the robust stabilization of a class of continuous-time Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations. The considered parametric uncertainties are of multiplicative white noise type with unknown intensity. In order to effectively address the multi-perturbations case, we use scaling techniques. These techniques allow us to obtain an estimation of the lower bound of the stability radius. A first characterization of a lower bound of the stability radius is obtained in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov differential equations. A second characterization is given in terms of the existence of positive solutions of adequately defined parameterized backward Lyapunov differential inequalities. This second result is then exploited in order to solve a robust control synthesis problem.

The practical use of variation principles in the determination of the stability of non linear systems

1961 ◽  
Vol 2 (2) ◽  
pp. 153-188 ◽  
Author(s):  
J. N. Lyness ◽  
J. M. Blatt
Keyword(s):  
Linear Systems ◽  
Initial Conditions ◽  
Mathieu Equation ◽  
Iteration Procedure ◽  
Variation Principle ◽  
Non Linear ◽  
Non Linear Systems ◽  
The Stability ◽  
Special Case

AbstractWe are interested in the motion of non linear systems. In this paper we use a variation principle and an iteration procedure in order to treat the stability of free oscillations against small disturbances of the initial conditions. It is found that approximations to the low lying stability lines can be obtained using the Rayleigh-Ritz variation principle and that these approximations can be consistently improved using an iteration procedure. These approximations are compared with the tabulated values in the special case of the Mathieu Equation. The results are in general a considerable improvement on those obtained using the usual Perturbation Theory, and have a much wider range of validity.

Measures of Robustness for Uncertain Time-Delay Linear Systems

1995 ◽  
Vol 117 (4) ◽  
pp. 633-635 ◽  
Author(s):  
Said Oucheriah
Keyword(s):  
Time Delay ◽  
Linear Systems ◽  
Robust Stability ◽  
Salient Feature ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Lyapunov Matrix ◽  
The Stability ◽  
Uncertain Time ◽  
Delay Dependent

Several delay-dependent criteria to test the stability of time-delay systems that were proposed require solving the Lyapunov matrix equation. This can be a troublesome task and often nontrivial. In this note, a delay-dependent sufficient condition that guarantees the robust stability of linear uncertain time-delay systems is presented. The stability test criterion derived in this paper is based on induced norms and matrix measures. The salient feature of the result obtained is its simplicity and ease in testing the robust stability of uncertain time-delay linear systems.

Determining the Norton’s Equivalent Model of Distribution System with Distributed Generation (DG) for Stability Analysis

2019 ◽  
Vol 12 (2) ◽  
pp. 190-198
Author(s):  
Sunny Katyara ◽  
Lukasz Staszewski ◽  
Faheem Akhtar Chachar
Keyword(s):  
Distributed Generation ◽  
Normal Condition ◽  
Distribution System ◽  
Distribution Networks ◽  
Equivalent Model ◽  
Stability Factor ◽  
Matlab Simulation ◽  
The Stability ◽  
Stability Issues

Background: Since the distribution networks are passive until Distributed Generation (DG) is not being installed into them, the stability issues occur in the distribution system after the integration of DG. Methods: In order to assure the simplicity during the calculations, many approximations have been proposed for finding the system’s parameters i.e. Voltage, active and reactive powers and load angle, more efficiently and accurately. This research presents an algorithm for finding the Norton’s equivalent model of distribution system with DG, considering from receiving end. Norton’s model of distribution system can be determined either from its complete configuration or through an algorithm using system’s voltage and current profiles. The algorithm involves the determination of derivative of apparent power against the current (dS/dIL) of the system. Results: This work also verifies the accuracy of proposed algorithm according to the relative variations in the phase angle of system’s impedance. This research also considers the varying states of distribution system due to switching in and out of DG and therefore Norton’s model needs to be updated accordingly. Conclusion: The efficacy of the proposed algorithm is verified through MATLAB simulation results under two scenarios, (i) normal condition and (ii) faulty condition. During normal condition, the stability factor near to 1 and change in dS/dIL was near to 0 while during fault condition, the stability factor was higher than 1 and the value of dS/dIL was away from 0.

Synthesis and solvatochromic studies of 2- amino-3-phenolazo1-(4-sulfophenyl)-3-methyi-5-pyrozolone and use it for the determination of trace amount of Nickel (II) in blood samples

2016 ◽  
Vol 5 (10) ◽  
pp. 4920
Author(s):  
Amar M. Ali ◽  
Hussain. J. Mohammed*
Keyword(s):  
Stability Constant ◽  
Trace Amount ◽  
Benzenesulfonic Acid ◽  
Determination Method ◽  
Blood Samples ◽  
Determination Of Nickel ◽  
Normal Human ◽  
The Stability ◽  
Normal Human Blood

A new, simple, sensitive and rapid spectrophotometric method is proposed for the determination of trace amount of Nickel (II). The method is based on the formation of a 1:2 complex with 4-(4-((2-hydroxy-6-nitrophenyl) diazenyl) -3-methyl-5-oxo-2, 5-dihydro-1H-pyrazol-1-yl) benzenesulfonic acid (2-ANASP) as a new reagent is developed. The complex has a maximum absorption at 516 nm and εmax of 1. 84 X 105 L. mol-1. cm-1. A linear correlation (0. 25 – 4. 0μg. ml-1) was found between absorbance at λmax and concentration. The accuracy and reproducibility of the determination method for various known amounts of Nickel (II) were tested. The results obtained are both precise (RSD was 1. 2 %) and accurate (relative error was 0. 787 %). The effect of diverse ions on the determination of Nickel (II) to investigate the selectivity of the method were also studied. The stability constant of the product was 0. 399 X 106 L. mol-1. The proposed method was successfully applied to the analysis of diabetes blood and normal human blood. 

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