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

Backstepping-based decentralized bounded-H∞ adaptive neural control for a class of large-scale stochastic nonlinear systems

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.

This paper is concerned with the probability-constrained filtering problem for a class of time-varying nonlinear stochastic systems with estimation error variance constraint. The stochastic nonlinearity considered is quite general that is capable of describing several well-studied stochastic nonlinear systems. The second-order statistics of the noise sequence are unknown but belong to certain known convex set. The purpose of this paper is to design a filter guaranteeing a minimized upper-bound on the estimation error variance. The existence condition for the desired filter is established, in terms of the feasibility of a set of difference Riccati-like equations, which can be solved forward in time. Then, under the probability constraints, a minimax estimation problem is proposed for determining the suboptimal filter structure that minimizes the worst-case performance on the estimation error variance with respect to the uncertain second-order statistics. Finally, a numerical example is presented to show the effectiveness and applicability of the proposed method.

Adaptive fault tolerant tracking control for a class of stochastic nonlinear systems with output constraint and actuator faults

In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well.

Stabilization of a class of nonlinear stochastic systems

Involves formulation of some problems arising in the analysis and control of a class of discrete-time nonlinear stochastic systems in terms of linear matrix inequalities. This allows one to utilize a wide variety of numerical techniques available for tackling both the feasibility problem and the actual numerical solution in an efficient manner.

This paper is concerned with the output feedback \(\mathcal {H}_\infty \) control problem for a class of stochastic nonlinear systems with time-varying state delays; the system dynamics is governed by the stochastic time-delay It\(\hat{o}\)-type differential equation with state and disturbance contaminated by white noises. The design of the output feedback \(\mathcal {H}_\infty \) control is based on the stochastic dissipative theory. By establishing the stochastic dissipation of the closed-loop system, the delay-dependent and delay-independent approaches are proposed for designing the output feedback \(\mathcal {H}_\infty \) controller. It is shown that the output feedback \(\mathcal {H}_\infty \) control problem for the stochastic nonlinear time-delay systems can be solved by two delay-involved Hamilton–Jacobi inequalities. A numerical example is provided to illustrate the effectiveness of the proposed methods.

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

A novel alleviating fuzzy control algorithm for a class of nonlinear stochastic systems in pure-feedback form

T–S fuzzy affine model based non-synchronized state estimation for nonlinear Itô stochastic systems

This paper is concerned with the problem of robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control of nonlinear stochastic systems with time-varying interval delay. Firstly, a novel kind of fuzzy model, stochastic fuzzy hyperbolic model (SFHM), is proposed to represent a class of nonlinear stochastic systems with time-varying interval delay. The SFHM is a overall nonlinear stochastic model, which is much fitter for the nonlinear multivariable plant whose expert knowledge is difficult to find. The identification of SFHM is more convenient than that of stochastic T-S fuzzy model. Specifically, it also can be seen as a neural network model and we can learn the model parameters by the learning method of neural network. Secondly, delay-range-dependent and delay-derivative-range-dependent criteria on stability and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance are developed for the SFHM by using new Lyapunov-Krasovskii functionals and improved free-weighting matrix technique for the SFHM case. Finally, simulation results show the validity of the proposed method.

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.

This paper investigates an improved fixed-time stability theory together with a state feedback controller for a class of nonlinear stochastic systems. First, a delicate transformation is performed, and next, a Gamma function is utilized to directly derive the value of the integral function, which ultimately yields a fixed-time stabilization theorem with a higher precision upper bound for the settling time. Unlike the existing estimation process of amplifying twice, we only performed one amplification, which weakens the effect of amplification. Then, a state feedback controller is constructed for stochastic systems by the method of adding a power integrator. Utilizing the proposed stochastic fixed-time stability theory, simulations show that the intended controller ensures that the trivial solution of the suggested system is fixed-time stable in probability. The results of the simulation demonstrate that the suggested control scheme is meaningful.