Order Nonlinear Systems Research Articles

Unified Krylov–Bogoliubov–Mitropolskii method for solving nth order non-linear systems with slowly varying coefficients

The unified Krylov–Bogoliubov-Mitropolskii (KBM) method is extended for obtaining the transient response of an n-th order ( n⩾2) non-linear system with slowly varying coefficients. The method is a generalization of KBM method and covers all the three cases when the eigenvalues of the unperturbed equation are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the eigenvalues in the general result that the solution corresponding to each of the three cases can be obtained. The method is illustrated by examples.

Convergent semidiscretization of a nonlinear fourth order parabolic system

A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.

A numerical technique to predict periodic and quasi-periodic response of nonlinear dynamic systems

A frequency domain based algorithm using Fourier approximation and Galerkin error minimization has been used to obtain the periodic orbits of large order nonlinear dynamic systems. The stability of these periodic response is determined through a bifurcation analysis using Floquet theory. This technique is applicable to dynamic systems having both analytic and nonanalytic nonlinearities. This technique is compared with numerical time integration and is found to be much faster in predicting the steady state periodic response.

Upper and lower solutions for periodic problems

Upper and lower solutions method, topological degree and Miranda fixed point theorem are applied to obtain the existence of at least one periodic solution for second order and first order nonlinear differential systems.

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows with postmilking teat disinfection

A mathematical model for the transmission of two interacting classes of mastitis causing bacterial pathogens in a herd of dairy cows is presented and applied to a specific data set. The data were derived from a field trial of a specific measure used in the control of these pathogens, where half the individuals were subjected to the control and in the others the treatment was discontinued. The resultant mathematical model (eight non-linear simultaneous ordinary differential equations) therefore incorporates heterogeneity in the host as well as the infectious agent and consequently the effects of control are intrinsic in the model structure. A structural identifiability analysis of the model is presented demonstrating that the scope of the novel method used allows application to high order non-linear systems. The results of a simultaneous estimation of six unknown system parameters are presented. Previous work has only estimated a subset of these either simultaneously or individually. Therefore not only are new estimates provided for the parameters relating to the transmission and control of the classes of pathogens under study, but also information about the relationships between them. We exploit the close link between mathematical modelling, structural identifiability analysis, and parameter estimation to obtain biological insights into the system modelled.

Finite-time control for robot manipulators

Finite-time control of the robot system is studied through both state feedback and dynamic output feedback control. The effectiveness of the proposed approach is illustrated by both theoretical analysis and computer simulation. In addition to offering an alternative approach for improving the design of the robot regulator, this research also extends the study of the finite-time control problem from second-order systems to a large class of higher order nonlinear systems.

A limit to the capability of feedback

Feedback is ubiquitous and is a basic concept in the area of control, where it is used primarily for reducing internal or external uncertainties, or both. In this note, we will study the capability of feedback in dealing with both internal and external uncertainties for a class of pth order nonlinear autoregressive control systems. The size of the uncertainty is described by the Lipschitz constant (say L) of the uncertain nonlinear function in consideration. It is shown that if p and L satisfy a certain inequality, then there exists no globally stabilizing feedback for the corresponding class of uncertain systems, and thus finding a quantitative limit to the capability of the feedback mechanism in dealing with structural uncertainties.

A unified Krylov–Bogoliubov–Mitropolskii method for solving nth order nonlinear systems

A unified theory is presented for obtaining the transient response of nth order nonlinear systems with small nonlinearities by Krylov–Bogoliubov–Mitropolskii method. The method is a generalization of Bogoliubov's asymptotic method and covers all three cases when the roots of the corresponding linear equation are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the roots in the general result that the solution corresponding to each of the three cases can be obtained. The method is illustrated by examples.

The asymptotic nature of a class of second order nonlinear system

In this paper, we obtain some results on the nonoscillatory behaviour of the system (1), which contains as particular cases, some well known systems. By negation, oscillation criteria are derived for these systems. In the last section we present some examples and remarks, and various well known oscillation criteria are obtained.

Variable universe stable adaptive fuzzy control of nonlinear system

A kind of stable adaptive fuzzy control of nonlinear system is implemented using variable universe method. First of all, the basic structure of variable universe adaptive fuzzy controllers is briefly introduced. Then the contraction-expansion factor that is a key tool of variable universe method is defined by means of integral regulation idea, and a kind of adaptive fuzzy controllers is designed by using such a contraction-expansion factor. The simulation on first order nonlinear system is done. Secondly, it is proved that the variable universe adaptive fuzzy control is asymptotically stable by use of Lyapunov theory. The simulation on the second order nonlinear system shows that its simulation effect is also quite good. Finally a useful tool, called symbolic factor, is proposed, which may be of universal significance. It can greatly reduce the settling time and enhance the robustness of the system.

On Global Asymptotic Stability of Second Order Nonlinear Differential Systems

This paper investigates the global asymptotic stability of the autonomous planar systems $ \\dot {x} = p_2(y)q_2(x)y $ , $ \\dot {y} = p_3(y)q_3(x)x + p_3(y)q_4(x)y $ and $ \\dot {x} = f_1(x) + h_2(x)y $ , $ \\dot {y} = f_3(x) + h_4(x)y $ , under the assumption that all functions involved in the equations are continuous and that the origin is a unique equilibrium. We present necessary and sufficient conditions for the origin to be globally asymptotically stable.

Theory of Differential Inequalities for Two-Point Boundary Value Problems and their Applications to Three-Point B.V.Ps Associated with n th Order Non-Linear System of Differential Equations

This paper presents a theory of differential inequalities for two-point boundary value problems (B.V.Ps) associated with the system of n th order non-linear differential equations. Using these inequalities as a tool we establish the existence and uniqueness of solutions to three-point B.V.Ps associated with the system of n th order non-linear differential equations by using the idea of matching solutions.

Qualitative properties of nonlinear difference systems

This paper gives criteria for the existence and uniqueness of solutions to two and multipoint boundary value problems associated with a typical system of kth order nonlinear Lyapunov system. A variation of parameters formula for nonlinear kth order system is established. Using this formula, we also study various concepts of stability and establish sufficient conditions for the kth order difference system to be Lipschitz stable and uniform Lipschitz stable.

Neural Network Adaptive Robust Control Of Siso Nonlinear Systems In A Normal Form

ABSTRACTIn this paper, performance oriented control laws are synthesized for a class of single‐input‐single‐output (SISO) n‐th order nonlinear systems in a normal form by integrating the neural networks (NNs) techniques and the adaptive robust control (ARC) design philosophy. All unknown but repeat‐able nonlinear functions in the system are approximated by the outputs of NNs to achieve a better model compensation for an improved performance. While all NN weights are tuned on‐line, discontinuous projections with fictitious bounds are used in the tuning law to achieve a controlled learning. Robust control terms are then constructed to attenuate model uncertainties for a guaranteed output tracking transient performance and a guaranteed final tracking accuracy. Furthermore, if the unknown nonlinear functions are in the functional ranges of the NNs and the ideal NN weights fall within the fictitious bounds, asymptotic output tracking is achieved to retain the perfect learning capability of NNs. The precision motion control of a linear motor drive system is used as a case study to illustrate the proposed NNARC strategy.

A numerical identifiability test for state-space models - application to optimal experimental design

This paper describes a mathematical tool for identifiability analysis, easily applicable to high order non-linear systems modelled in state-space and implementable in simulators with a time-discrete approach. This procedure also permits a rigorous analysis of the expected estimation errors (average and maximum) in calibration experiments. The methodology is based on the recursive numerical evaluation of the information matrix during the simulation of a calibration experiment and in the setting-up of a group of information parameters based on geometric interpretations of this matrix. As an example of the utility of the proposed test, the paper presents its application to an optimal experimental design of ASM Model No. 1 calibration, in order to estimate the maximum specific growth rate microH and the concentration of heterotrophic biomass XBH.

Quasi-Norm Local Error Estimators forp-Laplacian

In this paper, we extend the quasi-norm techniques used in a priori error estimation of finite element approximation of degenerate nonlinear systems in order to carry out an improved a posteriori error analysis for the p-Laplacian. We derive quasi-norm a posteriori error estimators of residual type, which are shown to provide improved upper and lower bounds on the discretization error. For sufficiently regular solutions, these estimators are further shown to be equivalent on the discretization error in a quasi norm. Numerical results demonstrating these a posteriori estimators are also presented.

Adaptive passivity of nonlinear systems using time-varying gains

In this paper, an adaptive controller with time-varying gains is proposed to solve the problem of making a single-input single-output (SISO) nonlinear system, with explicit linear parametric uncertainty, equivalent to a passive system. Some stability issues associated to the resultant closed-loop passive system are also discussed. The results obtained are applied to two examples, a third order nonlinear system and a model of a magnetic levitation system, to show the controller methodology design.

A Positivity-Preserving Numerical Scheme for a Nonlinear Fourth Order Parabolic System

\noindent A positivity-preserving numerical scheme for a fourth order nonlinear parabolic system arising in quantum semiconductor modeling is studied. The system is numerically treated by introducing an additional nonlinear potential and a subsequent semidiscretization in time. The resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions, which is proved by an exponential transformation of variables. The stability of the scheme is shown and convergence is proved in one space dimension. Numerical results concerning the switching behavior of a resonant tunneling diode are presented.

Stability analysis of nonlinear Lyapunov systems associated withan nth order system of matrix differential equations

This paper introduces the notion of Lipschitz stability for nonlinear nth order matrix Lyapunov differential systems and gives sufficient conditions for Lipschitz stability. We develop variation of parameters formula for the solution of the nonhomogeneous nonlinear nth order matrix Lyapunov differential system. We study observability and controllability of a special system of nth order nonlinear Lyapunov systems.

Performance prediction of SRM drive systems under normal and fault operating conditions using GA-based ANN method

A method to predict the performance characteristics of switched reluctance motor (SRM) drive systems under normal and fault operating conditions is presented. The method uses a genetic algorithm (GA) based artificial neural networks (ANNs) approach which is applied for its interpolation capabilities for highly nonlinear systems in order to obtain a fast and accurate prediction of the performance of the SRM drive system.