Stability Criteria For Linear Systems Research Articles

Improved stability criteria for linear systems with time-varying delay

This study is concerned with the stability analysis of systems with time-varying delay in a given interval. A new type of augmented Lyapunov functional which contains some triple-integral terms is proposed. By introducing free-weighting matrices, a new delay-range-dependent stability criterion is derived in terms of linear matrix inequality. The rate-range of the delay is considered, so the stability criterion is also delay-rate-range dependent. Numerical examples are given to illustrate the effectiveness of the proposed method.

Improved delay-range-dependent stability criteria for linear systems with time-varying delays

This paper is concerned with the stability analysis of linear systems with time-varying delays in a given range. A new type of augmented Lyapunov functional is proposed which contains some triple-integral terms. In the proposed Lyapunov functional, the information on the lower bound of the delay is fully exploited. Some new stability criteria are derived in terms of linear matrix inequalities without introducing any free-weighting matrices. Numerical examples are given to illustrate the effectiveness of the proposed method.

Novel robust stability criteria for uncertain systems with time-varying delay

This paper is concerned with the delay-dependent stability and robust stability criteria for linear systems with time-varying delay and norm-bounded uncertainties. Through constructing a general form of Lyapunov–Krasovskii functional, and using integral inequalities, some slack matrices and newly established convex combination condition in the calculation, the delay-dependent stability criteria are derived in terms of linear matrix inequalities. Numerical examples are given to illustrate the improvement on the conservatism of the delay bound over some reported results in the literature.

New stability criteria for linear systems with interval time-varying delay

This paper investigates robust stability of uncertain linear systems with interval time-varying delay. The time-varying delay is assumed to belong to an interval and is a fast time-varying function. The uncertainty under consideration includes polytopic-type uncertainty and linear fractional norm-bounded uncertainty. A new Lyapunov–Krasovskii functional, which makes use of the information of both the lower and upper bounds of the interval time-varying delay, is proposed to drive some new delay-dependent stability criteria. In order to obtain much less conservative results, a tighter bounding for some term is estimated. Moreover, no redundant matrix variable is introduced. Finally, three numerical examples are given to show the effectiveness of the proposed stability criteria.

Modified linear–nonlinear decomposition method for chaotic synchronization

This Letter proposes a general linear–nonlinear decomposition method for chaotic synchronization. The method expands the concept of chaotic synchronization based on the stability criterion of linear systems, proposed earlier. Retaining the basic idea of the standard linear–nonlinear decomposition method—the nonlinear part of a given chaotic system is used as a synchronization signal, hence the error system is always linear and allows precise stability analysis—the proposed method allows to design not only one, but a large number of coupling variants, thus offering the researcher the possibility to choose the best possible coupling variant for a given chaotic synchronization problem.

Projective synchronization of the fractional order unified system

This paper calculates the largest Lyapunov exponent and correction dimension of the fractional order unified system when changing the system parameter, and studies the dynamical behavior of this system. Based on the stability criterion of linear systems, a new approach for constructing projective synchronization of fractional order unified system is proposed. Numerical simulations of fractional order Chen system, fractional order Lü system and fractional order Lorenz-like system are provided to demonstrate the effectiveness of the proposed scheme.

SYNCHRONIZATION AND ANTI-SYNCHRONIZATION OF CHAOTIC SYSTEMS BASED ON LINEAR SEPARATION AND APPLICATIONS IN SECURITY COMMUNICATION

This paper analyzes a coupled dynamic system. Based on the stability criterion of linear systems, a new approach for constructing chaotic synchronization and anti-synchronization is proposed. The complete synchronizations and anti-synchronizations of the coupled dynamic system are achieved via the linear separation method. Finally, the method is applied in security communication. Numerical simulations are provided for illustration and verification of the proposed method.

Chaos synchronization of the energy resource system

This paper presents the chaos synchronization problem for new dynamical system (that is, energy resource demand–supply system), where the controller is designed using two different control methods. Firstly, based on stability criterion of linear system, chaotic synchronization is achieved with the help of the active theory, and accordingly, the simulation results are given for verifying the feasibility of the method. Secondly, based on Lyapunov stability theory, on the assumption that all the parameters of the system are unknown, adaptive control approach is proposed to make the states of two chaotic systems asymptotic synchronization. In the end, numerical simulations are used to show the effectiveness of the proposed control method.

Projective synchronization of fractional order chaotic system based on linear separation

Abstract This Letter analyses the dynamical behavior of fractional order unified system, based on the stability criterion of linear systems, a new approach for constructing projective synchronization of fractional order unified system is proposed. Numerical simulations of fractional order Chen system, fractional order Lu system and fractional order Lorenz-like system are achieved via the linear separation method.

Projective synchronization of autonomous chaotic system based on linear separation

This paper analyses the projective synchronization in autonomous chaotic system. Base on the stability criterion of linear systems, a new approach for constructing chaotically projective synchronization is proposed. The projective synchronizations of Lorenz system, Rssler system and hyperchaotic chen's system are achieved via the linear separation method. Numerical simulations are provided for illustration and verification of the proposed method.

Delay-dependent stability criteria for linear systems with multiple time delays

The problem of the delay-dependent stability of linear systems with multiple time delays is discussed. A new method is first presented for a system with two time delays, in which free weighting matrices are used to express the relationships among the terms of the Leibniz-Newton formula. Next, this method is used to show the equivalence between a system with two identical time delays and a system with a single time delay. Then, a numerical example verifies that the criterion given is effective and is a significant improvement over the existing ones. Finally, the basic idea is extended to a system with multiple time delays.

PROJECTIVE SYNCHRONIZATION OF UNIDENTICAL CHAOTIC SYSTEMS BASED ON STABILITY CRITERION

A new method of projective synchronization of unidentical chaotic systems is proposed in this letter. This method is based on the stability criterion of linear systems. The response of two unidentical chaotic systems can synchronize up to any desired scaling factor by a suitable separation of the systems. The new method of projective synchronization is suitable not only for the three-dimensional coupled partially linear systems, but also for higher dimensional even hyperchaotic systems. The simplicity and effectiveness of the proposed method are illustrated by the Lorenz system, the four-dimensional partially linear system, the four-dimensional hyperchaotic Rösser system and Chua's circuit system as four numerical examples.

Chaotic attitude motion and its control of spacecraft in elliptic orbit and geomagnetic field

The chaotic attitude motion and its control of a magnetic spacecraft with internal damping in an elliptic orbit and geomagnetic field is discussed. Based on the dynamical equations, the Melnikov method is used to predict the existence of chaos in the sense of Smale horseshoe. The chaotic behavior is numerically identified by means of Poincaré map and Lyapunov exponents. The chaotic attitude motion of spacecraft can be controlled by a method based on the stability criterion of linear system. The method can stabilize the chaos onto any desired periodic orbit by a small time-continuous perturbation feedback. The linearization of the system around the stabilized orbit is not required. The desired periodic solution can be automatically detected in the control process. The numerical simulation demonstrates the stabilization of chaotic attitude motion to period-1 or period-2 motion and shows the effectiveness and flexibility of the proposed control method.

Stability criteria for linear systems with multiple time-varying delays

New delay-independent and delay-dependent stability criteria for linear systems with multiple time-varying delays are established by using the time-domain method. The results are derived based on a new-type stability theorem for general retarded dynamical systems and new analysis techniques developed in the author's previous work. Unlike some results in the literature, all of the established results do not depend on the derivative of time-varying delays. Therefore, they are suitable for the case with very fast time-varying delays. In addition, some remarks are also given to explain the obtained results and to point out the limitations of the previous results in the literature.

Stability criteria for linear systems with uncertain delays

New delay-independent and delay-dependent stability criteria for linear systems with multiple uncertain delays are established by using both the time-domain and the frequency-domain methods. The results are derived based on the established new preliminary lemmas and by using new-type stability theorems for general retarded dynamical systems and new analysis techniques in the time-domain and the frequency-domain. All the established stability criteria depend only on the eigenvalues related to the coefficient matrices of the systems and do not involve any free tuning parameters. In addition, some remarks are given to explain in detail the obtained results and to point out the limitations of the existing results in the literature.

Chaotic synchronization based on stability criterion of linear systems

A new approach for constructing chaotically synchronizing systems is proposed in this Letter. This method is based upon the stability criterion of linear systems. The complete synchronization and anti-synchronization for chaotic systems are discussed. The Lorenz system, the Duffing oscillator and the Rössler system are treated as examples. It is shown that the proposed method is simpler than the conventional synchronization approach, and is effective for autonomous or non-autonomous systems. Since the Lyapunov exponents are not required to calculate, this method is convenient particularly for the synchronization of hyperchaotic systems with more two positive Lyapunov exponents.

Delay-dependent/delay-independent stability of linear systems with multiple time-varying delays

Delay-dependent/delay-independent uniform asymptotic stability and uniform stability criteria for linear systems with multiple time-varying delays are established respectively in this note. The results are derived based on a new-type stability theorem for retarded dynamical systems and a new analysis technique for estimating the derivative of a Lyapunov function along the solution of a system at certain specific instants. Four remarks together with an illustrative example are given to compare the obtained results with and to show their superiority to those in the literature.

Stability Criteria for Linear Systems with Partly Uncertain Periodic Coefficients

Stability criteria of partly uncertain linear systems with variable coefficients play an important part in various fields of robust control. In this paper we consider linear systems with periodic coefficients that may vary within their known lower and upper bounds depending upon time as well. Stability analysis of systems with variable coefficients is often based on the Lyapunov functions method leading in many cases to rigid sufficient stability conditions. A new approach to stability analysis of such systems is described in this paper. Using this method we obtain an upper bound for the real parts of the characteristic exponents of the system solutions yielding sufficient stability criteria and find the cases in which these bounds are reached. This explicitly determines the worst possible periodic coefficients yielding the largest value of the maximal real part of the characteristic exponents. Certain results are extended to the case of nonperiodic time-dependent bounded coefficients.

Further Results on the Stability of Linear Systems with Multiple Delays

This paper has two purposes. One is to establish new stability criteria for linear systems with multiple time-varying/constant delays, and the other is to point out the limitations of the author's previous results. The stability criteria, including delay-dependent and delay-independent ones, are derived by using both time-domain and frequency-domain techniques. All of the established stability criteria are given in the form of linear matrix inequality problems, including only one tuning parameter matrix P>0∈Rn×n.

A GENERALIZED MARKOV STABILITY CRITERION FOR LINEAR SYSTEMS

This paper presents a generalized Markov stability criterion for linear systems by determining the Cauchy index in arbitrary segments. Based on the proposed result, several new algebraic criteria on Schur stability and strict aperiodicity for linear systems are obtained.