Reshaping of the probability density function of nonlinear stochastic systems against abrupt changes

In this work, a novel shape control approach of the probability density function (PDF) for nonlinear stochastic systems is presented. First, we provide the formula for the PDF shape controller without devising the control law of the controller. Then, based on the exact analytical solution of the Fokker-Planck-Kolmogorov (FPK) equation, the product function of the polynomial and the exponential polynomial is regarded as the stationary PDF of the state response. To validate the performance of the proposed control approach, we compared it with the exponential polynomial method and the multi-Gaussian closure method by implementing comparative simulation experiments. The results show that the novel PDF shape control approach is effective and feasible. Using an equal number of parameters, our method can achieve a similar or better control effect as the exponential polynomial method. By comparison with the multi-Gaussian closure method, our method has clear advantages in PDF shape control performance. For all cases, the integral of squared error and the errors of first four moments of our proposed method were very small, indicating superior performance and promising good overall control effects of our method. The approach presented in this study provides an alternative for PDF shape control in nonlinear stochastic systems.

An approximation theory of optimal control for nonlinear stochastic dynamic systems has been established. Based on the generalized Hamilton-Jacobi-Bellman equation of the cost function of nonlinear stochastic systems, general iterative procedures for approximating the optimal control are developed by successively improving the performance of a feedback control law until a satisfactory suboptimal solution is achieved. A successive design scheme using upper and lower bounds of the exact cost function has been developed for the infinite-time stochastic regulator problem. The determination of the upper and lower bounds requires the solution of a partial differential inequality instead of equality. Therefore it provides a degree of flexibility in the design method over the exact design method. Stability of the infinite-time sub-optimal control problem was established under not very restrictive conditions, and stable sequences of controllers can be generated. Several examples are used to illustrate the applicati on of the proposed approximation theory to stochastic control. It has been shown that in the case of linear quadratic Gaussian problems, the approximation theory leads to the exact solution of optimal control.

Approximate entropy （ApEn） is well known to be an effective abrupt change detection method in dynamic structure. Based on this, we compare the performances between the moving detrended fluctuation analysis （MDFA） and ApEn in detecting abrupt dynamic change. The results show that the MDFA results almost do not depend on length of subseries, and while the ApEn could identify dynamic structure to some extent but still depends on the length of subseries. At the same time, there exists a huge drift for the ApEn results which means the actual time-instants of abrupt change dont coincide with the detected one. Therefore, compared with ApEn, MDFA is more suitable to be used to detect abrupt dynamic change, and the advantage of MDFA is obvious.

In this paper, a new method for the control of the shape of the conditional output probability density function (pdf) for general nonlinear dynamic stochastic systems is presented using two-step neural networks (NNs). Following the square-root B-spline NN approximation to the measured output pdf, the problem is transferred into the tracking of dynamic weights. Different from the previous related works, time-delay dynamic NNs with undetermined parameters are employed to identify the nonlinear relationships between the control input and the weighting vectors. In order to achieve the required control objective and satisfy the state constraints due to the property of output pdfs, a constrained PI tracking controller is designed by solving a class of linear matrix inequalities and algebraic equations. With the proposed tracking controller and adaptive projection algorithms, both identification and tracking errors can be made to converge to zero, and the state constraints can also be simultaneously guaranteed. Finally, two simulated examples are given, which effectively demonstrate the use of the proposed control algorithm.

The effect of yeast supplementation on equine fecal microbial population dynamics following abrupt dietary changes

Construction of Lyapunov functions for nonlinear planar systems by linear programming

This article presents a fuzzy finite-time command filtering output feedback control method for a class of nonlinear systems. A fast convergent output feedback control algorithm based on backstepping finite-time command filtering is developed. Fuzzy logic system is used to estimate uncertain functions in nonlinear systems. A fuzzy state observer is designed to measure the unknown state. The developed finite-time command filtering feedback control method overcomes well the computational complexity problem due to the calculation of the derivatives of virtual control signals. A compensation mechanism is also introduced to compensate for the error caused by the filter. The proposed method ensures not only that all signals in the closed-loop system are finite-time bound, but also that the tracking error converges to a small neighborhood around the origin. The effectiveness of the proposed method is demonstrated in the simulation results.

Non-linear output frequency response functions of MDOF systems with multiple non-linear components

In this paper, the structure of several convex cones that arise in the study of Lyapunov functions are investigated. In particular, the cones associated with quadratc Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases some technical results are presented on the structure of individual cones and it is shown how these insights can lead to new results on the problem of common Lyapunov function existence. 

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.

Stability analysis for a class of stochastic delay nonlinear systems driven by G-Brownian motion

Focusing on the high-precision motion control of time-delay nonlinear systems actuated by smart-material-based actuators and inspired by the flourishing development of fractional-order control theory, this article proposes an adaptive fractional-order command filter backstepping approximate inverse control scheme with the following key features. First, to the best of the authors’ knowledge, by combining the limited coverage lemma with the neural network, the unknown time-delay functions in the fractional-order nonlinear systems are precisely online estimated for the first time. This leads to the control problem of fractional-order nonlinear systems with time delays being overcome. Second, a new fractional-order filter is constructed to compute fractional differentiation of the designed virtual control laws, and simultaneously, the compensation signal is designed to eliminate the negative impact of filtering errors on the controller. Third, the approximate inverse model of asymmetric hysteresis for fractional-order asymmtric hysteretic nonlinear systems is derived, and the asymmetric hysteresis nonlinearities are precisely mitigated. Fourth, the independently developed giant magnetostrictive actuator-based motion platform is constructed, and a new fractional-order model of this platform is developed. Then, the motion control experiments are conducted to show the effectiveness of the proposed control scheme.

Adaptive output-feedback control design with prescribed performance for switched nonlinear systems

In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochastic differential equations (SDE), therefore the Fokker-Planck-Kolmogorov (FPK) equation is expressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of δ-correlated excitations, the existence of multiplicative excitations, and the dimension of SDE or FPK equation. Examples are given and numerical results are provided for comparing with known exact solution to show the effectiveness of the method.

Fault detection and diagnosis based on particle filters combined with interactive multiple-model estimation in dynamic process systems