The accessibility property of the class of driftless single input nonlinear time-delay systems is characterized. This result is obtained within a newly introduced geometric approach. In this context all those possible autonomous elements, which can depend on the variables with time-delay, are also characterized when the system is not accessible.
This paper addresses the problem of the estimation of unknown time-varying delays of discrete-time systems. The problem is reformulated as a mode detection for hybrid systems with a switching law depending on the unknown delay. Some existing data-based residual methods used for the estimation of the mode of switching linear discrete-time systems are briefly reviewed and presented in a unified way. Furthermore, it is shown that the usual existence conditions of some detectors can be relaxed. Next, results concerning the so-called discernibility property are particularized to the hybrid formulation of the estimation problem. Finally, it is shown that the issue is especially interesting in the context of chaotic secure communications both for information recovery and cryptanalysis purposes.
In this paper, we analyze the stability properties of a class of systems governed by linear difference equations. Lyapunov stability and asymptotic stability are analyzed using two different approaches, namely Lyapunov-Krasovskii techniques and spectral techniques. In the case of commensurable delays, we carry out an analytical form of the solution, allowing us to determine necessary and sufficient conditions for stability. A link with discrete-time systems is made, and some comments on time-varying delays are proposed.***Copyright©2013 IFAC
This paper addresses some properties of simple characteristic roots of quasipolynomials including commensurate delays. Although such a problem seems easy and was largely treated in the literature, to the best of the authors' knowledge, the invariance and the ultimate stability properties have not been fully investigated. In this paper, we propose a new frequency-sweeping framework, which simultaneously considers the Taylor series of the critical imaginary roots as well as the Puiseux series of singular points of the frequency-sweeping curves. Through analyzing the algebraic properties of the corresponding frequency-sweeping curves, we are able to completely characterize the invariance property of the simple imaginary roots with respect to the corresponding critical delay values. As a consequence of the invariance property, the ultimate stability property can be easily derived. Finally, as a byproduct of the approach proposed in the paper, the complete stability for time-delay systems with only simple imaginary roots can be systematically derived. Some illustrative examples complete the presentation.
In the huge literature dedicated to stability of time-delay systems, the most popular approach remains the use of Lyapunov-Krasovskii functionals. This framework allows to study a large class of time-delay systems including constant or time-varying delays. Since several years, the main challenge is to propose new functionals and techniques for deriving less and less conservative stability conditions. Nevertheless, all these approaches usually adopt the same procedure which is based on the well-known Jensen's inequality which generally induces some conservatism difficult to overcome. This paper analyses firstly the conservatism induced by the Jensen's inequality and secondly proposes a wide class of new parametrized inequalities. All these are based on an extensive use of Wirtinger inequality which has been recently introduced in Liu and Fridman [2012] and Seuret and Gouaisbaut [2012b] for stability analysis.
This paper is an analysis of 1D hyperbolic partial differential equation with moving interface as a delay system. The model derives from the mass balance of an extrusion process that describes the strong coupling between the mass transport equation and an ordinary differential equation which represents the interface motion. Solving the transport equation by the method of characteristics, we obtain an state-dependent-input-delay control problem. The stabilization of the whole system around an equilibrium is done by using a state predictor.
This paper deals with the H∞ optimal controller design for a magnetic suspension system model derived in Knospe and Zhu [2011], with added input/output delay. The plant is a fractional order system with time delay i.e., the transfer function of the plant involves infinite dimensional terms including a rational function of √s and e–hs, where h > 0 represents the delay. The H∞ optimal controller is designed by using the recent formulation given in özbay [2012] for the mixed sensitivity minimization problem for unstable infinite dimensional plants with low order weights. The effect of time delay on the achievable performance level is illustrated.
In this paper some new robust stability conditions for the exponential stability of some classes of integral delay systems with exponential kernels are derived by using the Lyapunov-Krasovskii functional approach. Copyright ©***2013 IFAC
As shown by its title, the aim of this paper is twofold. Firstly, the equation of Nicholson describing population dynamics is interesting in itself since it is a straightforward example of time delay equation where the delay appears naturally in the models as a consequence of the (here biological) phenomenon. On the other hand the nonlinearity occurring in this equation is a sector restricted one and incorporating the stability problems in an absolute stability one makes sense. The main result of the paper is the positive answer to the associated delay independent Aizerman conjecture.
In this paper, the finite constructive method for linear differential-difference systems stability and instability analysis is proposed, and its convergence is proved. This method is based on obtaining of the quadratic lower bound for the quadratic Lyapunov – Krasovskii functionals on some special set of functions. The method proposed is applied to the stability domain construction in a parameter space, and to the critical delay values numerical determination.
In this paper, we show that the controller synthesis of delayed systems can be formulated and solved in a convex manner through the use of a duality transformation, a structured class of operators, and the Sum-of-Squares (SOS) methodology. The contributions of this paper are as follows. We show that a dual stability condition can be formulated in terms of Lyapunov operators which are positive, self-adjoint and preserve the structure of the state-space. Second, we provide a class of such operators which can be parameterized using Sum-of-Squares. Next, we show how any operator in this class can be inverted using simple operations on the SOS variables which can be performed in Matlab. Next we use SOS and semidefinite programming to formulate a dual stability test for time-delay systems. Next, we use the dual stability results to formulate a convex test for stabilizability and show how SOS can be used to solve this test and recover the controller. Finally, we give a numerical example. The results of this paper are significant in that they open the way for dynamic output H∞ optimal control of infinite-dimensional systems by giving the first truly convex, numerically realizable full-state feedback controller synthesis criterion.
It is shown that the necessary stability conditions of one delay n dimensional linear systems obtained in a previous contribution via Lyapunov functionals of complete type, are also sufficient in the scalar case. The obtained criteria depends on a positivity condition in terms of the Lyapunov function, in analogy with the delay free case.
The paper considers the convergence of the Hill method for a single loop linear periodic system with delay. The relation between the Hill method and investigation methods by applying the theory of Fredholm integral equations of the second kind is indicated and used for accuracy estimations.
This paper analyzes simple toy systems, consisting of a difference equation in continuous time, but where the delay depends on the current state, and gives some preliminary results for the higher dimensional case. These difference equations in continuous time are conceptually simpler than the corresponding differential systems. The central tenet is that a state space should encode the minimal sufficient information that is needed to solve the Cauchy problem associated with the system. The state space is rigorously constructed for some typical state dependent difference equations in continuous time. Dynamics are represented by the associated infinitesimal generator, which is subsequently derived. In passing, we discuss a class of scalar and vector iterated functional differential equations, and present some new characterizations of their solution. Finally, we present some preliminary results for a scalar differential delay system with state dependent delay. This contribution is purely theoretical.
This paper focuses on the concept of delay-independent stability for dynamical systems described by continuous-time linear delay-difference equations and the corresponding stability notion in discrete-time domain. The problem will be formulated with respect to delay-parameter space. Our intention is to summarize delay-independent stability condition and to provide, in a compact formulation, an appropriate numerical method for its computation, at least for two dimensional delay case. Obtained results are applied in stability analysis of the discrete-time delay-difference equations. Such a strong stability condition appears to be necessary for the existence of specific invariant regions in the state-space. Some illustrative examples complete the paper.
In the framework of the goal adaptation concept, and with the aim to provide additional desirable properties of the closed-loop system, a new technique for performance forming in model reference adaptive control of uncertain linear state delayed plants is proposed. We develop a unified adaptive control scheme for the state feedback and output feedback control cases. The control law leads to a tractable design formulation, where the performance is adapted on-line to satisfy new requirements in addition to the usual stability properties. The desired system performance is illustrated by simulation results.
Recently the problem of estimating the initial state of some linear infinite-dimensional systems from measurements on a finite interval was solved by using the sequence of forward and backward observers [14]. In the present paper, we introduce a direct Lyapunov approach to the problem and extend the results to the class of semilinear systems governed by 1-d wave equations with boundary measurements from a finite interval. We first design forward observers and derive Linear Matrix Inequalities (LMIs) for the exponential stability of the estimation errors. Further we find LMIs for an upper bound T* on the minimal time, that guarantees the convergence of the sequence of forward and backward observers on [0, T*] for the initial state recovering. For observation times bigger than T*, these LMIs give upper bounds on the convergence rate of the iterative algorithm in the norm defined by the Lyapunov functions. The efficiency of the results are illustrated by a numerical example.
This paper deals with the input-to-state stabilization, with respect to a disturbance acting on the control input, of stabilizable systems described by nonlinear coupled delay differential and difference equations. These equations describe, for instance, lossless propagation phenomena in electrical and hydraulic engineering, and include, as special cases, neutral functional differential equations in Hale's form and retarded functional differential equations. A recent Lyapunov-Krasovskii characterization of the global asymptotic stability, in the Lp norm, of these systems is exploited. Such a characterization is obtained by means of one only functional for the overall system, though both differential and difference equations are involved in the system model. In the spirit of Sontag's feedback control redesign method, it is shown that the disturbance can be attenuated, in the sense of input-to-state stability in the Lp norm, by adding to the control law a term obtained by the Lyapunov-Krasovskii functional for the global asymptotic stability, in the Lp norm, of the disturbance-free closed-loop system. An example is studied in order to show the effectiveness of the proposed methodology.
This paper focuses on the construction of a Smith predictor for network-based haptic systems. Roughly speaking, the idea is to use a predictor just on the haptic side in order to compensate the viscosity effect and to provide an accurate feeling in the case of contacts. A new approach is presented by using the available information on the distance from the virtual reality simulator and introducing it in the predictor in order to maintain the similarity between the “real” and the “predicted model”. In order to validate the approach, experimental results are presented for constant and random varying delays (normal distributed and gamma with gap distributed), for a simple virtual environment and for a virtual box.
