Robust Dichotomy Analysis and Synthesis With Application to an Extended Chua's Circuit

This paper focuses on uncertain pendulum-like systems subject to norm constraint structured uncertainties, and addresses the robust gradient-like behaviour analysis and synthesis problems for such systems. By applying the structured singular value of an operator to structured uncertainty, the results on robust analysis for the gradient-like behaviour of uncertain pendulum-like systems are obtained. Based on the analysis results, using KYP lemma, a dynamic output feedback controller for the closed loop system guaranteeing gradient-like behaviour is designed and the controller parameters are explicitly expressed by a set of feasible solutions of corresponding linear matrix inequalities. Finally two numerical examples are presented to demonstrate the applicability and validity of the proposed approach.

The onset of jump resonance in the driven Chua's circuit is deeply described in this paper, further confirming the role of Chua's circuit in nonlinear circuits and systems theory. The study of the conditions under which this peculiar behavior occurs is based on the possibility of obtaining a Lur'e representation of the driven Chua's circuit and applying the describing function approach. Design considerations to select the Chua's circuit parameters in order to shape the jump resonance features are given, providing the nonlinear core for highly selective frequency drift sensors.

The sufficient conditions for chaos synchronization of two nonidentical systems by replacing variables control have not been proposed until now. In this paper, synchronization of two chaotic Lur'e systems with parameter mismatch by replacing variables control is studied. First of all, we present a master-slave Lur'e systems synchronization scheme with both parameter mismatch and replacing variables control, and derive a responsive error system for the scheme. A new definition of synchronization with finite L 2-gain is then introduced. Based on the definition, the sufficient synchronization criteria which are in the form of linear matrix inequality (LMI) are proved using a quadratic Lyapunov function. By means of MKY lemma the frequency domain criteria are further derived from the obtained LMIs. These frequency domain criteria are illustrated on the master-slave Chua's circuits with parameter mismatch so that the ranges of the parameters of Chua's circuit are analytically solved in the sense of the synchronization with finite L 2-gain by replacing singe-variable control. The illustrative examples verify that within the ranges of the parameters it is possible to synchronize the master-slave Chua's circuits up to a small synchronization error bound, even the qualitative behaviors of the slave circuit are different from that of the master one, such as the trajectory of the master circuit is chaotic and that of the slave divergent. The relation between the synchronization error bound and parameter mismatch is shown.

An improved model-free adaptive control for nonlinear systems: An LMI approach

International audience

A technique for analyzing Chua's nonlinear circuit is presented. The method involves: 1. Determination of possible behaviors by the harmonic balance method and estimation of their stability; 2. Rough estimation of possible distortions for determined regimes. An exact method for estimating distortions is applied to a system with a cubical nonlinearity.

Built a map relating single atoms in the Periodic Table and different nonlinear Chua's circuits. We show that is possible to connect energy levels in atoms, (as calculated by the simple quantization Bohr's rules), with certain special values of the circuit's resistors. These values are the ones at which the system undergoes a bifurcation, characterizing the period-doubling sequence leading to the chaotic state. We found a quite universal relationship relating atoms to electronic nonlinear circuits that allows one to calculate spectroscopy levels starting from the bifurcation analysis or, vice-versa, bifurcation values from the knowledge of atomic energy levels.

This paper introduces an $H_\infty$ fuzzy control synthesis method for a nonlinear large-scale system with a reduced number of linear matrix inequalities (LMIs). It is well known that a nonlinear large-scale system can be transformed to a Takagi–Sugeno (T–S) fuzzy system by using “sector nonlinearity” or “local approximation in fuzzy partition spaces” methods. Next, in order to achieve the fuzzy control design for this T–S fuzzy system, we solve the stabilization conditions that are represented by the LMIs. However, if the number of LMIs is large, the control design process may become very complicated. In this study, based on the Lyapunov method and S -procedure, several theorems are proposed for the synthesis of parallel distributed compensation (PDC)-type fuzzy control such that the nonlinear large-scale system achieves $H_\infty$ control performance, and the number of LMIs to be solved is reduced explicitly. As a result, the control design process will become much easier. Furthermore, if the modeling error between the nonlinear system and T–S fuzzy system exists, the robust $H_\infty$ control performance and the number reduction of LMIs are also achieved by the proposed theorem. Several examples are presented in this paper to show the number reduction effect of LMIs and the effectiveness of the proposed controller synthesis.

In this paper, a new method for robust Lagrange stability analysis of a class of nonlinear discrete-time systems is proposed. Both the linear part structured and unstructured uncertainties are considered in a unified way. Sufficient, conditions for robust Lagrange stability are established in terms of linear matrix inequalities (LMIs). With this LMI approach, the largest allowable magnitude of the admissible perturbation was given explicitly by solving a, generalized eigenvalue minimization problem which is essentially a convex optimization problem and numerically efficient, illustrative example confirmed the efficiency and accuracy of the proposed approach

A new circuitry design based on Chua's circuit for generating n-scroll attractors (n = 1, 2, 3, …) is proposed. In this design, the nonlinear resistor in Chua's circuit is constructed via a systematical procedure using basic building blocks. With the proposed construction scheme, the slopes and break points of the v–i characteristic of the circuit can be tuned independently, and chaotic attractors with an even or an odd number of scrolls can be easily generated. Distinct attractors with n-scrolls (n = 5, 6, 7, 8, 9, 10) obtained with this simple experimental set-up are demonstrated.

This brief addresses the problem of designing a robust tracking controller for a nonlinear Chua's circuit involving plant uncertainties and external disturbances. Given a bounded desired trajectory, a robust tracking controller, which is based on modeling-error compensation is developed such that all the states and signals of the closed-loop system are bounded and the tracking error can be made as small as desired. In contrast to the previous proposed controllers for tracking control of Chua's circuit, the controller developed here is linear and equivalent to a traditional PI compensator.

This paper introduces two novel fractional-order chaotic systems with cubic nonlinear resistor and investigates its adaptive sliding mode synchronization. Firstly the novel two equilibrium chaotic system with cubic nonlinear resistor (NCCNR) is derived and its dynamic properties are investigated. The fractional-order cubic nonlinear resistor system (FONCCNR) is then derived from the integer-order model and its stability and fractional-order bifurcation are discussed. Next a novel no-equilibrium chaotic cubic nonlinear resistor system (NECNR) is derived from NCCNR system. Dynamic properties of NECNR system are investigated. The fractional-order no equilibrium cubic nonlinear resistor system (FONECNR) is derived from NECNR. Stability and fractional-order bifurcation are investigated for the FONECNR system. The non-identical adaptive sliding mode synchronization of FONCCNR and FONECNR systems are achieved. Finally the proposed systems, adaptive control laws, sliding surfaces and adaptive controllers are implemented in FPGA.

A systematic approach to construct two chaotically synchronized systems was developed. An object matrix was found to satisfy a Riccati inequality to design the synchronized chaotic Lipschitz state-observer through using the LMI (linear matrix inequality) toolbox. The proposed method, combining chaotic masking and chaotic modulation, was then applied to secure communication. The complex encrypted information signal was injected into the chaos system and simultaneously transmitted to the receiver after chaotic masking. The information signal was recovered through a complex decrypter in the synchronized receiver. Furthermore, the well-known Chua's circuit was considered as illustration examples to demonstrate the effectiveness of the proposed approach.

This paper investigates anticipating synchronization of a general form of chaotic lur'e master-slave systems. Through the analysis of the set-valued nonlinearities, linear matrix inequalities (LMI)-based output feedback is firstly proposed to achieve globally asymptotic prediction of the given system's future states. Then, based on the ideology of active control, sliding mode controllers are adopted to ensure accurate anticipation of Lur'e systems, which are provided with high robustness against possible disturbances. The control techniques are both verified through Lyapunov stability analysis, and illustrated by numerical simulation of Chua's circuit. Simulation results show that active sliding mode control scheme has better performance.

The analyze of the stationary regime of non-linear circuits excited with periodical signals is difficult, on one hand because into the general solutions of the differential non-linear equation system the components of the stationary regime can not be separated, and on the other hand because of the different structure of the response signal and of the excitation signal. Characteristically for the non-linear element is the fact that between the input parameter and the output parameter do not exist a proportionality relation. A particular case is the non-linear resistance. This circuit element is defined by the fundamental terms of the trigonometric series of voltage and current. The characteristics of non-linear resistance are approximated by a polynomial and the paper presents a method for determining the polynomial's coefficients to compute the non-linear and the static resistance, corresponding to a periodical current excitation applied to the non-linear resistor, by the technique of multiple synchronous detection. In the second part of the paper is presented an instrument for measurement in virtual technique the non-linear and static resistance, implemented in LabVIEW graphical programming software. The solution allows a quick and convenient measurement of these important circuit components, together with an appropriate graphical representation. The performances of this measurement technique are: easiness of design, flexibility of interconnection of signal processing software blocks, clear presentation of the results in the front panel of the instrument and, last but not least, accuracy of the measurement data software processing.