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

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.

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3194
Author(s):  
Natalia Bakhtadze ◽  
Igor Yadikin

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.


Author(s):  
Hossein Nejat Pishkenari ◽  
Mohammad Shahrokhi

In this paper an identification method which can estimate the unknown parameters of a general nonlinear system based on three techniques (gradient, least-squares and rapid identification) has been developed. The stability of the proposed schemes has been shown using the Lyapunov stability theorem. The properties of each identification technique have been discussed briefly. Open loop identification of the Lorenz chaotic system is presented to show the effectiveness of the proposed approach. To illustrate the efficiency of the identification method for control purposes, it has been applied for controlling the well-known Lorenz system. By exploiting the property of the system a novel singularity-free controller is proposed. The stability of controller has been shown by a Lyapunov function. The designed controller coupled with the proposed identification technique can stabilize the uncertain Lorenz system. The effectiveness of the approach has been shown through simulation.


2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Ahmad Banakar

The Lyapunov stability theorem is applied to guarantee the convergence and stability of the learning algorithm for several networks. Gradient descent learning algorithm and its developed algorithms are one of the most useful learning algorithms in developing the networks. To guarantee the stability and convergence of the learning process, the upper bound of the learning rates should be investigated. Here, the Lyapunov stability theorem was developed and applied to several networks in order to guaranty the stability of the learning algorithm.


2000 ◽  
Vol 45 (6) ◽  
pp. 1187-1191 ◽  
Author(s):  
Juing-Shian Chiou ◽  
Fan-Chu Kung ◽  
T.-H.S. Li

2014 ◽  
Vol 598 ◽  
pp. 714-717
Author(s):  
Sheng Qin Jiang ◽  
Guo Liang Cai ◽  
Shui Ming Cai

This paper studies synchronization of complex networks with multi-unknown parameters and disturbances via discrete control. Based on Lyapunov stability theorem and the stability analysis of impulse system, adaptive-impulse synchronization criteria for general multi-uncertain complex networks have been established. Finally, a numerical simulation is provided to support the theoretical results.


2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Shipei Huang ◽  
Zhengrong Xiang ◽  
Hamid Reza Karimi

This paper is concerned with the problem of robust stabilization for a class of uncertain two-dimensional (2D) discrete switched systems with state delays under asynchronous switching. The asynchronous switching here means that the switching instants of the controller experience delays with respect to those of the system. The parameter uncertainties are assumed to be norm-bounded. A state feedback controller is proposed to guarantee the exponential stability. The dwell time approach is utilized for the stability analysis and controller design. A numerical example is given to illustrate the effectiveness of the proposed method.


2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Guoliang Cai ◽  
Lingling Zhang ◽  
Lan Yao ◽  
Xiulei Fang

Modified function projective synchronization via adaptive impulsive controller between two different financial hyperchaotic systems is investigated, where the external uncertainties are considered. The updated laws of the unknown parameters are given and the sufficient conditions are deduced based on Lyapunov stability theorem and the stability analysis of impulsive system. Finally, the two financial hyperchaotic systems are taken for example and the numerical examples are worked through for illustrating the main results.


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