Dragutin Lj. Debeljkovic

On finite and practical stability of time delayed systems: Lyapunov-Krassovski approach, delay dependent criteria

This paper gives sufficient conditions for the practical and finite time stability of linear continuous time delay systems of the form X(t)=A0X(t)+A1X(t−τ). When we consider finite time stability, these new, delay independent conditions are derived using the approach based on Lyapunov-Krassovski functionals. In this case these functionals need not to have: a) properties of positivity in whole state space and b) negative derivatives along system trajectories. When we consider practical stability, before mentioned concept of stability, it is combined and supported by classical Lyapunov technique to guarantee attractivity properties of system behavior.

On non-Liapunov stability of descrete descriptor systems

In this paper, the singular (semi-state, descriptor) systems described by a mix of algebraic and difference equations are treated. Recent results on Liapunov stability are outlined and an analysis of finite-time stability presented.

Stability of singular time-delay systems in the sense of non-Lyapunov: Classical and modern approach

This paper provides sufficient conditions for both practical stability and finite-time stability of linear singular continuous time-delay systems which can be mathematically described as Ex(t)=Aox(t)+A1x(t-t). Considering a finite-time stability concept, new delay independent and delay dependent conditions have been derived using the approach based on the Lyapunov-like functions and their properties on the subspace of consistent initial conditions. These functions do not need to have the properties of positivity in the whole state space and negative derivatives along the system trajectories. When the practical stability has been analyzed the above mentioned approach was combined and supported by the classical Lyapunov technique to guarantee the attractivity property of the system behavior. Moreover an linear matrix inequality (LMI) approach has been applied in order to get less conservative conditions.

Necessary and Sufficient Conditions for Delay-Dependent Asymptotic Stability of Linear Discrete Time Delay Autonomous Systems

Abstract This paper offers new, necessary and sufficient conditions for delay-dependent asymptotic stability of systems of the form x ( k +1) = A 0 x ( k ) + A 1 x ( k – h ). The time-dependent criteria are derived by Lyapunovs direct method. Two matrix equations have been derived: matrix polynomial equation and discrete Lyapunov matrix equation. Also, modifications of the existing sufficient conditions of convergence of Traub and Bernoulli algorithms for computing the dominant solvent of the matrix polynomial equation are derived. Numerical computations are performed to illustrate the results obtained.

The application of different Lyapunov-like functionals and some aggregate norm approximations of the delayed states for finite-time stability analysis of linear discrete time-delay systems

Abstract In this paper, the problem of finite-time stability analysis for linear discrete time-delay systems is studied. By using the classical Lyapunov-like functional and Lyapunov-like functionals with power or exponential functions, some sufficient conditions for finite-time stability of such systems are proposed in the form of the linear matrix inequalities. The six aggregate norm approximations of the delayed states are introduced to establish the relations between the classical Lyapunov-like functional and its difference. To further reduce the conservatism of stability criteria, three inequalities with delayed states for the estimation of Lyapunov-like functional are proposed. A numerical example is included to illustrate the effectiveness and advantage of the proposed methods.

On non-Lyapunov stability of linear discrete time delay systems: LMIs approach

This paper gives new contributions to the area of non-Lyapunov (finite time stability, technical stability, practical stability, final stability) for the particular class of linear discrete time delay systems. The idea of attractive practical stability is introduced for the first time. Moreover, based on the matrix inequalities and Lyapunov-like functions, some new sufficient conditions under which the linear discrete time delay system is finite time stable are given. Finally, an example is employed to verify the efficiency of the proposed Theorems as well as to show that results derived upon LMIs are less restrictive than those based on a classical approach. To the best knowledge of authors, such results have not been reported yet.

A new approach to the stability of discrete descriptor time delay systems in the sense of non-lyapunov delay independent conditions

This paper gives sufficient conditions for the practical and finite time stability of linear singular continuous time delay systems of the form E x(k +1) = A0x(k ) + A1x(k - 1). When we consider finite time stability concept, these new, delay independent conditions are derived using approach based on Lyapunov - like functions and their properties on sub-space of consistent initial conditions. .

Asymptotic Stability Analysis of Linear Time- Delay Systems: Delay Dependent Approach

The problem of investigation of time delay systems has been exploited over many years. Time delay is very often encountered in various technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, long transmission lines, etc. The existence of pure time lag, regardless if it is present in the control or/and the state, may cause undesirable system transient response, or even instability. During the last three decades, the problem of stability analysis of time delay systems has received considerable attention and many papers dealing with this problem have appeared (Hale & Lunel, 1993). In the literature, various stability analysis techniques have been utilized to derive stability criteria for asymptotic stability of the time delay systems by many researchers (Yan, 2001; Su, 1994; Wu & Muzukami, 1995; Xu, 1994; Oucheriah, 1995; Kim, 2001). The developed stability criteria are classified often into two categories according to their dependence on the size of the delay: delay-dependent and delay-independent stability criteria (Hale, 1997; Li & de Souza, 1997; Xu et al., 2001). It has been shown that delaydependent stability conditions that take into account the size of delays, are generally less conservative than delay-independent ones which do not include any information on the size of delays. Further, the delay-dependent stability conditions can be classified into two classes: frequency-domain (which are suitable for systems with a small number of heterogeneous delays) and time-domain approaches (for systems with a many heterogeneous delays). In the first approach, we can include the two or several variable polynomials (Kamen 1982; Hertz et al. 1984; Hale et al. 1985) or the small gain theorem based approach (Chen & Latchman 1994). In the second approach, we have the comparison principle based techniques (Lakshmikantam & Leela 1969) for functional differential equations (Niculescu et al. 1995a; Goubet-Bartholomeus et al. 1997; Richard et al. 1997) and respectively the Lyapunov stability approach with the Krasovskii and Razumikhin based methods (Hale & Lunel 1993; Kolmanovskii & Nosov 1986). The stability problem is thus reduced to one of finding solutions to Lyapunov (Su 1994) or Riccati equations (Niculescu et al., 1994), solving linear matrix inequalities (LMIs) (Boyd et al. 1994; Li & de Souza, 1995; Niculescu et al., 1995b; Gu 1997) or analyzing eigenvalue distribution of appropriate finite-dimensional matrices (Su

Finite time stability of continuous time delay systems: Jensen's inequality-based approach

In this study, finite-time stability of the linear continuous time-delay systems was investigated. A novel formulation of the Lyapunov-like function was used to develop a new sufficient delay-dependent condition for finite-time stability. The proposed function does not need to be positive-definite in the whole state space, and it does not need to have negative derivatives along the system trajectories. The proposed method was compared with the previously developed and reported methodologies. It was concluded that the stability investigation using the novel condition for stability investigation was less complicated for numerical calculations. Furthermore, it gives comparable results in comparison with the ones obtained with other analyzed conditions, and it provides superior results for some systems.

Finite-time stability analysis of descriptor discrete time-delay systems using discrete convolution of delayed states

This paper provides sufficient conditions for the finite time stability of linear time invariant discrete descriptor time delay systems, mathematically described as Ex(k+1) = A0x(k) + A1x(t-h). A novel method was used to derive new delay dependent conditions. Stability of the system was analyzed using both the Lyapunov-like approach and the Jensens inequality, including convolution of delayed states. The established conditions were applied to analysis of the system stability. In this case, the aggregation functional does not have to be positive in the state space domain and does not need to have the negative derivatives along the system trajectories. The system stability conditions were applicable to investigation of the finite time stability using the novel conditions proposed in this paper. This mathematical formulation guaranteed that the states of the systems do not exceed the predefined boundaries over a finite time interval.