Fault coverage modeling in nonlinear dynamical systems

Fault tolerance has been an essential architectural attribute for achieving high reliability in many critical applications of digital systems. Automatic fault detection, location, isolation, recovery, and reconfiguration mechanisms play a crucial role in implementing fault tolerance because a not-covered fault may lead to a system or subsystem failure even when adequate redundancy exists. The probability of successfully recovering from a fault given that the fault has occurred is known as the coverage factor or coverage and this is used to account for the efficiency of fault-tolerant mechanisms. If the fault and error handling mechanisms cannot successfully cover all faults in the system, then the coverage factor becomes less than unity and the system is said to have imperfect coverage. The models that consider the effects of imperfect fault coverage are known as imperfect fault coverage models or simply imperfect coverage models, fault coverage models, or coverage models. For systems with imperfect fault coverage, an excessive level of redundancy may even reduce the system reliability. Therefore, an accurate analysis must account for not only the system structure but also the system fault and error handling behavior, which is often called coverage behavior. The appropriate coverage modeling approach depends on the type of fault tolerant techniques used. In this chapter, we present the status and trends of imperfect coverage models, and associated reliability analysis techniques. We also present the historical developments, modeling approaches, reliability algorithms, optimal design policies, and available software tools.

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

In this paper we investigate the problem of input-to-state stability (ISS) of nonlinear delayed impulsive systems. Both the continuous dynamics and the discrete dynamics of nonlinear impulsive systems are subjected to external input. By virtue of the Razumikhin technique in combination with Lyapunov functions, we obtain some Razumikhin-type theorems that warrant input-to-state stability of nonlinear impulsive systems with time-delays. The Razumikhin-type input-to-state stability theorems cover the cases where the delayed continuous dynamics are input-to-state stable or destabilizing and the discrete dynamics are input-to-state stable or destabilizing, such that nonlinear delayed impulsive systems are able to retain input-to-state stability under certain conditions. The applicability of the derived Razumikhin-type theorems is illustrated by numerical results.

Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach

Modeling of Microblogging Social Networks: Dynamical System vs. Random Dynamical System

An Ito–Taylor weak 3.0 method for stochastic dynamics of nonlinear systems

Linear quadratic regulator (LQR) and proportional-integral-derivative (PID) control methods, which are generally used for control of linear dynamical systems, are used in this paper to control the nonlinear dynamical system. LQR is one of the optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions. The nonlinear system states are fed to LQR which is designed using a linear state-space model. This is simple as well as robust. The inverted pendulum, a highly nonlinear unstable system, is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. In this paper, the modeling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using PID controller and LQR have been presented for both cases of without and with disturbance input. The Matlab-Simulink models have been developed for simulation and performance analysis of the control schemes. The simulation results justify the comparative advantage of LQR control method.

The problem of creating mathematical models of nonlinear dynamical systems does not have an unambiguous solution and requires the creation of a separate synthesis method for each such object. To develop a method for synthesizing mathematical models of an extensive class of nonlinear dynamical systems with polynomial nonlinearities. The work uses a method based on the solution of the Volterra integral equation in the ideology set forth in Van Trees H.L., according to which the structure of a nonlinear dynamical object present47s a series connection of the linear part, characterizing the inertial properties of the system, and the nonlinear element, given by static characteristic. The difference of the suggested version of the method from the classical one, proposed in the works of Van Trees H.L., is an expansion of their input and output signals into Fourier series and a representation of the inertial part of these systems by their Bode plots, connected into one structure with input and output signals and non-linearity by Volterra integral equation. The algorithm of the proposed method is disclosed by the example of solving the problem of identifying a nonlinear dynamical system which impulse response of the inertial part satisfies the separability requirement, the order of the polynomial nonlinearity is three, and the model of the input signal has the form of a sinusoid "raised" over the time axis on a priori given constant level. A computational experiment was carried out on the example of nonlinear dynamical systems with the third order of the nonlinear characteristic and the first and second orders of the model of the inertial part of these systems with the specified algorithms of their parametric identification. The suggested method allows to synthesis the mathematical model of a nonlinear dynamical system with the polynomial static characteristic to the case when the input signal has an arbitrary number of harmonics, and the model of the inertial part and the nonlinear polynomial function have an arbitrary order.

Generalized quadratic embeddings for nonlinear dynamics using deep learning

In this paper, we consider an observer design by using a formal linearization based on Fourier expansion for nonlinear dynamic and measurement systems. A non-linear dynamic system is given by a nonlinear ordinary differential equation, and a measurement sysetm is done by a nonlinear equation. Defining a linearization function which consists of the trigonometric functions considered up to the higher-order, a nonlinear dynamic system is transformed into an augmented linear one with respect to this linearization function by using Fourier expansion. Introducing an augmented measurement vector which consists of polynomials of measurement data, a measurement equation is transformed into an augmented linear one with respect to the linearization function in the same way. To these augmented linearized systems, a linear estimation theory is applied to design a new non-linear observer.

A strategy to nonlinear dynamic system identification based on Fuzzy Kalman Filter, is proposed. A mathematical formulation based on fuzzy Takagi-Sugeno structure is presented: the algorithm FCM estimates the fuzzy sets; from the data input and output of a nonlinear dynamic system, the ERA/DC algorithm based on clustering, estimated by FCM algorithm, is applied to obtain the matrices A, B, C, and D (state matrix, input influence matrix, output influence matrix, and direct transmission matrix, respectively) to each rule of the fuzzy model consequent. A Fuzzy Kalman Filter is applied to estimate states and output of a nonlinear dynamic system. Computational results show the efficiency of the proposed methodology, once that the TS Fuzzy Model Based on Fuzzy Kalman Filter follow the behavior relative to output and states of a nonlinear dynamic system, and compared it to other methodology it shows a better performance related to follow the behavior of states.

In this paper we present and compare different fault diagnosis algorithms using space space models for nonlinear dynamic systems. Most fault diagnosis and isolation algorithms for dynamic systems, which can be modeled using a set of state space equations, have relied on the system being linear and the noise and disturbances being Gaussian. In such cases, optimal filtering ideas based on Kalman filtering are utilized in estimation followed by a residual analysis, for which whiteness tests are typically carried out. Linearized approximations (e.g., Extended Kalman filters) have been used in the nonlinear dynamic systems case. However, linearization techniques, being approximate, tend to suffer from poor detection or high false alarm rates. In this paper, we use the sequential Monte Carlo filtering approach where the complete posterior distribution of the estimates are represented through samples or particles as opposed to the mean and covariance of an approximated Gaussian distribution. The particle filter is combined with the innovation-based fault detection techniques to develop a fault detection and isolation scheme. The advantage of particle filters is that they are capable of handling any functional nonlinearity and system or measurement noise of any distribution. An improvement on using a single Extended Kalman filter matched to a particular model is to use the the Interacting Multiple Model estimator, which consists of a number of EKFs running in parallel. Such a multiple model estimator can handle the abrupt changes in the system dynamics, which is essential for fault diagnosis. Here, we compare the fault detection performances of these algorithms on different nonlinear systems.

Guidance law development for aerospace vehicles can be significantly simplified by reducing the order of the vehicle dynamic model. Identifying timescale separation in the system dynamics is important in this regard. This paper introduces a systematic means of identifying the time-scale structure in a nonlinear dynamical system by using regional Lyapunov exponents. Regional Lyapunov exponents and associated vectors are used to analyze the linear variational equations associated with a nonlinear system, in order to draw conclusions about the time-scale structure in the nonlinear system. Examples are used to illustrate the the time-scale information provided by the regional Lyapunov exponents. The time scale structure in an example of closed-loop entry dynamics is investigated using the regional Lyapunov exponents. The exponents and their associated directions identify energy as the slow variable and radial distance and flight path angle as fast variables and thus provide the information needed for reduced order guidance law development.

In this paper a new protection scheme is introduced to detect and identify transformer winding faults. The new approach is based on artificial neural networks (ANNs) using radial basis functions (RBFs) and the principal component analysis (PCA). The nonlinear system's input and output data is manipulated without considering any model of the system. This approach is used to detect and identify internal short circuit faults of a three phase custom built transformer. The suggested technique is also able to distinguish between the fault and magnetizing inrush current. The test studies carried out shows that the proposed method leads to satisfactory results in terms of detecting and isolating parameter faults taking place in non-linear dynamical systems.

It is well-known that the controls of neural network require that all the inputs of their input layer are available. Under this circumstance, the architecture of the neural network is constrained. Hence, this kind of neurocontrol cannot apply to a wide class of nonlinear and unknown dynamical systems, or the control system that requires more sensors. Although the unknown system state and the unknown system dynamics can be achieved from an adaptive observer and a learning neural network, respectively, the simultaneous existence of the estimated state error and the modeling error makes the control system more likely to be unstable. In this paper, a novel neurocontroller based on the concept of sliding mode with estimated state is constructed to tackle a wide class of unknown and nonlinear dynamical systems. The focal stability of the overall system can be verified by the Lyapunov stability criteria. Finally, simulations are presented to verify the usefulness of the proposed method.