Observer design and practical stability of nonlinear systems under unknown time‐delay

Abstract

In the present paper, we study observer design and we establish some sufficient conditions for practical exponential stability for a class of time-delay nonlinear systems written in triangular form. In case of delay, the exponential convergence of the observer was confirmed. Based on the Lyapunov-Krasovskii functionals, the practical stability of the proposed observer is achieved. Finally, a physical model and simulation findings show the feasibility of the suggested strategy.Mathematics Subject Classification. 93C10, 93D15, 93D20.Time-delay systems are one of the basic mathematical models of real phenomena such as nuclear reactors, chemical engineering systems, biological systems [16], and population dynamics models [18]. The analysis of systems without delays is generally simple as compared to nonlinear systems under time delays. However, there are a number of problems relating to nonlinear observer for time delay system. In particular, a problem of theoretical and practical importance is the design of observer-based for time-delay systems. In literature, a separation principle, for nonlinear free-delay systems, using high gain observers is provided in [1] and [2].Much attention has been paid to solve such a problem and many observer designs for time-delay nonlinear systems approaches have been used. In [26], it is shown that observer design for a class of nonlinear time delay systems is solved by using linear matrix inequality. In [11], some sufficient conditions for practical uniform stability of a class of uncertain time-varying systems with a bounded time-varying state delay were provided using the Lyapunov stability theory.[4] and [9] show that state and output feedback controllers of time-delay systems, written in a triangular linear growth condition are reached under delay independent conditions and under delay dependent conditions respectively.The observer design problem for nonlinear systems satisfying a Lipschitz continuity condition has been a topic of numerous papers, such as for nonlinear free-delay systems [1,2,27,25], for nonlinear systems with unknown, time-varying [19,14,12]. A reduced-order observer design method is presented in [27] for a class of Lipschitz nonlinear continuous-time systems without time delays which extend the results in [25].However, in practice, dynamics, measurement, noises or disturbances often prevent the error signals from tending to the origin. Thus, the origin is not a point of equilibrium of the system. An additive term on the right-hand side of the nonlinear system is used to present the uncertainties systems. For this reason, the property is referred to as practical stability which is more suitable for nonlinear free-delay systems ( see [5,7]) and for nonlinear systems with time-delay ( see [10,14,19,24]). Under unknown, bounded timedelay, an observer design for a class of nonlinear system is presented in [19]. [10] concluded that a class of nonlinear time delay systems is conformed due to some assumptions and the time varying delay bounded the practical exponentia...

A Separation Principle for the Stabilisation of a Class of Fractional Order Time Delay Nonlinear Systems

“…The high-gain observer design framework established in [11] for free delay systems can be properly extended to this class of time-delay fractional differential systems. For the same class of systems (3.4) with q = 1, a separation principle and observer-based stabilisation were studied in [2] and [8,12] respectively.…”

Lyapunov matrix‐based method to global robust practical exponential r‐stability for a class of delayed continuous‐time nonlinear systems: Theory and applications

“…The analysis of practical stability for nonlinear systems has become one of the important research topics in the field of control theory and its applications. Recent years, this subject has achieved a number of excellent research results, see References 20‐24 for delay free systems, and References 25‐31 for time‐delay systems. Stamova 27 used the vector Lyapunov function method and the differential inequalities of piecewise continuous functions to study the practical stability of solutions of impulsive nonlinear functional differential equations.…”

“…Stamova 27 used the vector Lyapunov function method and the differential inequalities of piecewise continuous functions to study the practical stability of solutions of impulsive nonlinear functional differential equations. Echi 29 studied the observer design problem for a class of time‐delay nonlinear systems written in triangular form, and sufficient conditions for the practical stability of the error system are given by applying Lyapunov–Krasovskii functional (LKF) method. By applying the LKF method, Villafuerte et al 30 studied the practical stability of neutral time‐delay systems.…”

Observer based control for strong practical stabilization of a class of uncertain time delay systems

“…It proposes a robust exponential stability criterion which is delay dependent. In [10] sufficient conditions are provided to prove the practical stability for a class of nonlinear delay systems satisfying some relaxed triangular-type condition. According to the Lyapunov-Krasovskii functional, the problem of global exponential stability of a class of nonlinear time-delay systems written in triangular form that satisfies a linear growth condition is achieved by [5].…”

“…The initial conditions for the system are x(0) = [−10, −20] T , and the initial conditions the observer have been given byx(0) =[10,10] T . We have the Lipschitz constant defined in(10) equal to Now, select K = [−4 − 9] and L = [−5 − 5] T , A K and A L are Hurwitz. Using Matlab, the solutions of the Lyapunov equations (5) and (17) are given by P = 0.1200 0.1000 0.1000 1.100 S = 1.1944 −0.5000 −0.5000 0.2778 .…”