On observers design for nonlinear time-delay systems

International audience

In this paper, we investigate the impulse effect in nonlinear singular system, since the existence of the nonlinear dynamics, we use the Differential Mean Value Theorem based method to transfer the nonlinear singular system into a linear parameter varying system. It should be noted that Differential Mean Value Theorem is an identical transmission, thus, for the nonlinear system, if we transfer the nonlinear system into a linear system via differential mean value theorem, the properties of the original system will not be changed. Based on this point, we try to analyze the transferred singular system by some existence method which applied to the linear singular system.

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time-delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H ∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, the practical stability and stabilization of a class of nonlinear neutral type time delay systems with multiple delays and bounded perturbations is discussed. Sufficient conditions based on Lyapunov-Krasovskii functionals are derived. They are formulated in terms of the feasibility of a set of Bilinear Matrix Inequalities (BMI’s). A practical exponential estimate of the system response is also obtained. This approach is shown to be useful in the solution of an engineering problem: the elimination of the stick-slip phenomenon in the drilling process.

The differential mean value theorem is one of the important basic theorems in differential calculus and a powerful tool to study the properties of the Han script. It reflects the relationship between functions and derivatives. Many proof questions encountered in the study of calculus need to use the relevant knowledge of the differential mean value theorem, and students are often at a loss. In this paper, several types of differential mean value theorems are discussed in depth, and it is found that the key point and difficulty in proving many propositions related to differential mean value theorems is to construct appropriate auxiliary functions. By constructing auxiliary functions, abstract and seemingly unrelated conclusions can be transformed into relatively straightforward views, which can easily find the path that meets the conditions of differential mean value theorems, and then solve the problem. Through these types of questions, this paper summarizes the common skills to find auxiliary functions, hoping to provide some help for students in learning, postgraduate entrance examination, etc.

Novel adaptive neural control design for nonlinear MIMO time-delay systems

The theoretical foundation of differential aspect in real analysis is the differential mean value theorem, which connects to both the local and total properties of the copula function. The link between the copula function's local and overall properties is differential calculus. The maximum value, minimum value, extreme value, monotonicity, and other difficulties may all be resolved with the help of these differential mean value theorems. Additionally, they aid in the computation of limits, the demonstration of inequality, and the identification of the curve's inflection point and concave-convex interval. They are also capable of mapping the function and locating the equation's root. To tackle these issues, one can apply a number of significant findings from the differential mean value theorem. This study gives the formulae for solving three distinct mean value theorems. The Cauchy, Lagrange, and Roller theorems are examples of these mean value theorems. Understanding the connections between the three mean value theorems is made easier by these studies, which also provide an explanation of the theory underlying the theorems and provide examples and visuals of their practical application.

Observers Synthesis Method for a Class of Nonlinear Discrete-Time Systems with Extension to Observer-Based Control

The leader-following consensus problem for a type of second-order nonlinear multi-agent systems (MASs) with input saturation, actuator faults, and sensor faults is examined in this study. An adaptive control strategy based on neural networks (NNs) is suggested to overcome the difficulties brought on by unknown nonlinear dynamics and real-world limitations. To handle the unknown nonlinear factors more precisely and flexibly than traditional approaches that depend on global Lipschitz requirements, we use the differential mean value theorem. This improvement enhances the system's capacity to deal with nonlinear behaviors that change quickly. Additionally, sensor faults that affect location and velocity readings and actuator faults that may decrease or deform the input signal are taken into consideration in the control design. The suggested design also clearly addresses input saturation, which may restrict the control authority. Complete state measurements are not required because a distributed adaptive NN-based controller is created only on relative location and velocity data. Using stability theory and appropriate Lyapunov function construction, we rigorously demonstrate that the suggested approach leads to leader-following consensus. The efficiency and resilience of the suggested control strategy against nonlinear uncertainty, actuator and sensor failures, and saturation effects in complicated multi-agent networks are confirmed by simulation results.

This paper has proposed a new state-dependent impulsive observer (SDIO) for nonlinear time-delay systems. This observer is based on extended pseudo-linearization, and its parameters are state-dependent. The SDIO is capable to estimate system states continuously by using system output that is just available at discrete impulse times. The stability of the proposed observer is proved by using time-varying Lyapunov function, and comparison system theory of impulsive differential equation systems. By new theorem, it is guaranteed that the estimation error asymptotically converges to zero under well-defined, and less-conservative sufficient conditions. Furthermore, the stability theorem gave an upper bound on the maximum allowable time interval between consequent impulses. The simulation results show effectiveness, and good performance of the proposed observer, for a wider classes of nonlinear time-delay systems.

This paper proposes an indirect adaptive fuzzy neural network (FNN) controller with state observer and supervisory controller for a class of uncertain nonlinear dynamic time delay systems. First, the approximate function of unknown time delay system is inferred by the adaptive time delay FNN system. Next, a state observer is designed to estimate the unknown system states and the indirect adaptive fuzzy controller is constructed. Finally, the closed loop controller is obtained by incorporating the supervisory controller with the indirect adaptive fuzzy controller. Therefore, if the system tends to unstable, i.e., error dynamics is larger than a prescribed constraint which is determined by designer, the supervisory controller will activate to force the state to be stable. The free parameters of the indirect adaptive FNN controller can be tuned online by observer-based output feedback control law and adaptive laws by means of Lyapunov stability criterion. The resulting simulation example shows that the performance of nonlinear time delay chaotic system is fully tracking the reference trajectory. Meanwhile, simulation results show that the adaptive control effort of the proposed control scheme is much less due to the assist of the supervisory controller.

This paper presents actuator fault detection of discrete-time nonlinear descriptor systems by means of nonlinear unknown input observers. The approach is based on the exact factorization of the estimation error in order to overcome the well-known problem of unmeasurable scheduling variables within the observation of convex models, thus avoiding the use of Lipschitz constants, differential mean value theorem, or robust techniques. As a result, the designing conditions are cast in terms of linear matrix inequalities and efficiently solved via commercially available software. Numerical as well as academic setups are provided to illustrate the advantages and performance of the proposal.

The dynamical degradation of digital chaotic systems (DCSs) often has serious negative influences on some digital chaos-based systems and then becomes one of the bottleneck problems stopping chaos from some applications. In this paper, we first restrict the Devaney’s chaos definition to finite state sets to describe the chaotic motion of digital systems. Then, we propose a novel control method for DCSs based on the differential mean value theorem and state feedback technology. Simulation results show the effectiveness, robustness, and superiority of the proposed method. Finally, we construct a new pseudorandom number generator (PRNG) and evaluate its randomness via NIST SP800-22 and TestU01 test suites. Statistical test results show that the proposed PRNG has high reliability of randomness, thus it can be used for cryptography and other potential applications.

A novel theorem for prescribed-time control of nonlinear uncertain time-delay systems

An Extension of Stability Criteria for Linear and Nonlinear Time Delay Systems