scholarly journals Analysis and Prediction of Electric Power System’s Stability Based on Virtual State Estimators

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3194
Author(s):  
Natalia Bakhtadze ◽  
Igor Yadikin
Keyword(s):  
Linear Part ◽  
Bilinear System ◽  
Lyapunov Equation ◽  
Bilinear Systems ◽  
Spectral Techniques ◽  
State Estimators ◽  
Generalized Matrix ◽  
The Stability ◽  
Bounded Input ◽  
Associative Search

The stability of bilinear systems is investigated using spectral techniques such as selective modal analysis. Predictive models of bilinear systems based on inductive knowledge extracted by big data mining techniques are applied with associative search of statistical patterns. A method and an algorithm for the elementwise solution of the generalized matrix Lyapunov equation are developed for discrete bilinear systems. The method is based on calculating the sequence of values of a fixed element of the solution matrix, which depends on the product of the eigenvalues of the dynamics matrix of the linear part and the elements of the nonlinearity matrixes. A sufficient condition for the convergence of all sequences is obtained, which is also a BIBO (bounded input bounded output) systems stability condition for the bilinear system.

scholarly journals Robust Stabilization and Observer-Based Stabilization for a Class of Singularly Perturbed Bilinear Systems

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2380
Author(s):  
Ding-Horng Chen ◽  
Chun-Tang Chao ◽  
Juing-Shian Chiou
Keyword(s):  
State Feedback ◽  
Robust Stabilization ◽  
Lyapunov Equation ◽  
Bilinear Systems ◽  
Stability Theorem ◽  
Singularly Perturbed ◽  
Lyapunov Stability Theorem ◽  
Input Gain ◽  
The Stability ◽  
Bounded Input

An infinite-bound stabilization of a system modeled as singularly perturbed bilinear systems is examined. First, we present a Lyapunov equation approach for the stabilization of singularly perturbed bilinear systems for all ε∈(0, ∞). The method is based on the Lyapunov stability theorem. The state feedback constant gain can be determined from the admissible region of the convex polygon. Secondly, we extend this technique to study the observer and observer-based controller of singularly perturbed bilinear systems for all ε∈(0, ∞). Concerning this problem, there are two different methods to design the observer and observer-based controller: one is that the estimator gain can be calculated with known bounded input, the other is that the input gain can be calculated with known observer gain. The main advantage of this approach is that we can preserve the characteristic of the composite controller, i.e., the whole dimensional process can be separated into two subsystems. Moreover, the presented stabilization design ensures the stability for all ε∈(0, ∞). A numeral example is given to compare the new ε-bound with that of previous literature.

OBSERVER-BASED FUZZY CONTROL DESIGN FOR DISCRETE-TIME T-S FUZZY BILINEAR SYSTEMS

2013 ◽  
Vol 21 (03) ◽  
pp. 435-454 ◽  
Author(s):  
JIANGRONG LI ◽  
JUNMIN LI ◽  
ZHILE XIA
Keyword(s):  
Fuzzy Control ◽  
Discrete Time ◽  
Control Design ◽  
Bilinear System ◽  
Single Step ◽  
Bilinear Systems ◽  
Linear Matrix ◽  
Fuzzy Observer ◽  
Nonlinear Minimization ◽  
The Stability

This paper is concerned with the problem of observer-based fuzzy control design for discrete-time T-S fuzzy bilinear systems. Based on the piecewise quadratic Lyapunov function (PQLF), the piecewise fuzzy observer-based controllers are designed for T-S fuzzy bilinear systems. It is shown that the stability for discrete T-S fuzzy bilinear system can be established if there exists a PQLF can be constructed and the fuzzy observer-based controller can be obtained by solving a set of nonlinear minimization problem involving linear matrix inequalities(LMIs) constraints. An iterative algorithm making use of sequential linear programming matrix method (SLPMM) to derive a single-step LMI condition for fuzzy observer-based control design. Finally, an illustrative example is provided to demonstrate the effectiveness of the results proposed in this paper.

scholarly journals Characterization of Cocycle Attractors for Nonautonomous Reaction–Diffusion Equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1650135 ◽  
Author(s):  
C. A. Cardoso ◽  
J. A. Langa ◽  
R. Obaya
Keyword(s):  
Chaotic Dynamics ◽  
Linear Part ◽  
Reaction Diffusion ◽  
Positive Cone ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Minimal Set ◽  
Full Characterization ◽  
The Stability

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

Stabilizing Feedback Controls for Nonlinear Hamiltonian Systems and Nonconservative Bilinear Systems in Elasticity

1982 ◽  
Vol 104 (1) ◽  
pp. 27-32 ◽  
Author(s):  
S. N. Singh
Keyword(s):  
Asymptotic Stability ◽  
Hamiltonian Systems ◽  
Attitude Control ◽  
Sufficient Conditions ◽  
Bilinear Systems ◽  
Feedback Controls ◽  
Control Laws ◽  
Nonlinear Maps ◽  
The Stability ◽  
Necessary And Sufficient

Using the invariance principle of LaSalle [1], sufficient conditions for the existence of linear and nonlinear control laws for local and global asymptotic stability of nonlinear Hamiltonian systems are derived. An instability theorem is also presented which identifies the control laws from the given class which cannot achieve asymptotic stability. Some of the stability results are based on certain results for the univalence of nonlinear maps. A similar approach for the stabilization of bilinear systems which include nonconservative systems in elasticity is used and a necessary and sufficient condition for stabilization is obtained. An application to attitude control of a gyrostat Satellite is presented.

Finite-time synchronization of delayed fractional-order quaternion-valued memristor-based neural networks

2020 ◽  
pp. 2150032
Author(s):  
Dawei Ding ◽  
Ziruo You ◽  
Yongbing Hu ◽  
Zongli Yang ◽  
Lianghui Ding
Keyword(s):  
Neural Networks ◽  
Fractional Order ◽  
Finite Time ◽  
Time Synchronization ◽  
Linear Part ◽  
State Feedback Control ◽  
Razumikhin Technique ◽  
Quaternion Multiplication ◽  
Control Schemes ◽  
The Stability

This paper mainly concerns with the finite-time synchronization of delayed fractional-order quaternion-valued memristor-based neural networks (FQVMNNs). First, the FQVMNNs are studied by separating the system into four real-valued parts owing to the noncommutativity of quaternion multiplication. Then, two state feedback control schemes, which include linear part and discontinuous part, are designed to guarantee that the synchronization of the studied networks can be achieved in finite time. Meanwhile, in terms of the stability theorem of delayed fractional-order systems, Razumikhin technique and comparison principle, some novel criteria are derived to confirm the synchronization of the studied models. Furthermore, two methods are used to obtain the estimation bounds of settling time. Finally, the feasiblity of the synchronization methods in quaternion domain is validated by the numerical examples.

A generalized finite-time analytical approach for the synchronization of chaotic and hyperchaotic systems

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Muhammad Haris ◽  
Muhammad Shafiq ◽  
Adyda Ibrahim ◽  
Masnita Misiran
Keyword(s):  
Finite Time ◽  
Chaotic Systems ◽  
Closed Loop ◽  
Time Synchronization ◽  
Linear Part ◽  
Content Type ◽  
Nonlinear Part ◽  
Analytical Stability ◽  
Chaotic Signals ◽  
The Stability

PurposeThe purpose of this paper is to develop some interesting results in the field of chaotic synchronization with a new finite-time controller to reduce the time of convergence.Design/methodology/approachThis article proposes a finite-time controller for the synchronization of hyper(chaotic) systems in a given time. The chaotic systems are perturbed by the model uncertainties and external disturbances. The designed controller achieves finite-time synchronization convergence to the steady-state error without oscillation and elimination of the nonlinear terms from the closed-loop system. The finite-time synchronization convergence reduces the hacking duration and recovers the embedded message in chaotic signals within a given preassigned limited time. The free oscillation convergence keeps the energy consumption low and alleviates failure chances of the actuator. The proposed finite-time controller is a combination of linear and nonlinear parts. The linear part keeps the stability of the closed-loop, the nonlinear part increases the rate of convergence to the origin. A generalized form of analytical stability proof is derived for the synchronization of chaotic and hyper-chaotic systems. The simulation results provide the validation of the accomplish synchronization for the Lu chaotic and hyper-chaotic systems.FindingsThe designed controller not only reduces the time of convergence without oscillation of the trajectories which can run the system for a given time domain.Originality/valueThis work is originally written by the author.

Analysis of Stability for Generalized Discrete Large-Scale Systems

2013 ◽  
Vol 467 ◽  
pp. 627-632
Author(s):  
Chen Fang ◽  
Jiang Hong Shi ◽  
Shuang Yu ◽  
Jian Fang Mao
Keyword(s):  
Lyapunov Function ◽  
Large Scale ◽  
Lyapunov Equation ◽  
Large Scale Systems ◽  
Stability And Instability ◽  
Analysis Of Stability ◽  
The Stability

Based on the condition that all independent subsystems of the generalized large-scale systems are regular and causal, this thesis studied both stability and instability of discrete linear generalized large-scale systems through Lyapunov equation and Lyapunov function, and proposed the criterion theorem for the stability or instability of discrete linear generalized large-scale systems.

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