Third-order Nonlinear Systems Research Articles

Advances on the extended stochastic rayleigh quotient estimation theory

A new extended stochastic Rayleigh quotient estimation theory is developed for the identification of the unknown feedback matrix and nonlinear function parameters of a proposed multivariable plant. Systems tractable to this approach encompass a wide class of nonlinear closed-loop time-variant control models that are observed at two localities in a statistically-known white Gaussian noisy environment. Phases of the estimation problem via a partitioning frame technique are given that yield pragmatical computable solutions. An optimal modified-predictor—corrector maximum-likelihood scheme is delineated for solving the state estimation problem, and its invariance to a priori statistics is investigated. In addition, this article presents the analysis of extended stochastic Rayleigh quotient algorithms, extended SRQA's , for the evaluation of the unknown parameters. Nonlinear programming formulations are treated for the algorithms' commencement. Moreover, a noncyclic adaptive computational procedure is depicted to ensure the pointwise convergence of the extended SRQA's in the mean-square sense. Finally, applicability of the devised theory to a nonlinear third-order system is demonstrated as well as a comparison between different suggested methods.

The analysis of a third-order system

An example of third-order nonlinear feedback system is considered. Its previously known sector for global asymptotic stability is (0, 1]. In the present letter, several sectors for global asymptotic stability are obtained.

Existence and stability of periodic solutions of a third-order non-linear autonomous system simulating immune response in animals

SynopsisA 3 × 3 autonomous, non-linear system of ordinary differential equations modelling the immune response in animals to invasion by active self-replicating antigens has been introduced by G. I. Bell and studied by G. H. Pimbley Jr. Using Hopf's theorem on bifurcating periodic solutions and a stability criterion of Hsu and Kazarinoff, we obtain existence of a family of unstable periodic solutions bifurcating from one steady state of a reduced 2×2 form of the 3×3 system. We show that no periodic solutions bifurcate from the other steady state. We also prove existence and exhibit a stability criterion for families of periodic solutions of the full 3×3 system. We provide two numerical examples. The second shows existence of orbitally stable families of periodic solutions of the 3×3 system.

Correspondence: Analytical determination of limit cycles for a class of third-order non-linear system

Correspondence: Analytical determination of limit cycles for a class of third-order non-linear system

Study of third-order non-linear systems—Variation of parameters approach

In this paper, the transient response of a third-order non-linear system is obtained by first reducing the given third-order equation to three first-order equations by applying the method of variation of parameters. On the assumption that the variations of amplitude and phase are small, the functions are expanded in ultraspherical polynomials. The expansion is restricted to the constant term. The resulting equations are solved to obtain the response of the given third-order system. A numerical example is considered to illustrate the method. The results show that the agreement between the approximate and digital solution is good thus vindicating the approximation.

Ultraspherical polynomials approach to the study of third-order non-linear systems

In this study, the Krylov-Bogoliubov-Mitropolskii-Popov asymptotic method is used to determine the transient response of third-order non-linear systems. Instead of averaging the non-linear functions over a cycle, they are expanded in ultraspherical polynomials and the constant term is retained. The resulting equations are solved to obtain the approximate solution. A numerical example is considered and the approximate solution is compared with the digital solution. The results show that there is good agreement between the two values.

Application of ultraspherical polynomials to forced oscillations of a third-order non-linear system

In this paper the method of ultraspherical polynomial approximation is applied to study the steady-state response in forced oscillations of a third-order non-linear system. The non-linear function is expanded in ultraspherical polynomials and the expansion is restricted to the linear term. The equation for the response curve is obtained by using the linearized equation and the results are presented graphically. The agreement between the approximate solution and the analog computer solution is satisfactory. The problem of stability is not dealt with in this paper.

Analytic determination of limit cycle for a class of third-order non-linear system

The purpose of this paper is to show a new criterion on the stability of the limit cycle for a general third-order non-linear system : that the linear-part transfer function can be either stable or unstable and the non-linear element has a real describing function. The presented method is purely analytic. Combining with an analytic method for predicting the amplitudes and frequencies of limit cycles, an analytic process for determining the stable limit cycles of a general third-order non-linear system is completed. Many examples are given.

A mathematical model of the vagus-heart period system in the cat.

The dynamic relation between changes in heart period and changes in the frequency of stimulation applied to the nerves involved in slowing the heart (vagus nerves) in chloralosed vagotomized cats pretreated with propranolol was investigated in ten cats using time-domain techniques. The strength of the stimuli was such that a sinus rhythm was always maintained. The typical response of heart period to step changes in the frequency of stimulation (range 4-50 Hz) was found to be an increase in heart period to a steady value preceded by an overshoot, and followed, after the stimulus was removed, by an undershoot (postvagal tachycardia). The observed response was accurately described by a dynamic model consisting of a third-order nonlinear system with time delay. The postvagal tachycardia was accounted for by including in the model a negative feedback loop; a portion of the overshoot was accounted for by the feedback loop and the remainder by including in the model a negative coefficient that varied as a negative exponential function of frequency. For small variations in the frequency of stimulation, it was shown in agreement with previous results, that the model could be approximated by a first-order system with time delay. In addition, it was found that provided a sinus rhythm was maintained, the steady-state value of heart period during vagal stimulation was neither a hyperbolic nor a logarithmic function of frequency of stimulation as suggested by previous investigators, but rather exhibited a maximum in the neighborhood of the middle of the frequency range studied.

The transient response of certain third-ordernon-linear systems

In this paper a method of solving certain third-order non-linear systems by using themethod of ultraspherical polynomial approximation is proposed. By using the method of variation of parameters the third-order equation is reduced to three partial differential equations. Instead of being averaged over a cycle, the non-linear functions are expanded in ultraspherical polynomials and with only the constant term retained, the equations are solved. The results of the procedure are compared with the numerical solutions obtained on a digital computer. A degenerate third-order system is also considered and results obtained for the above system are compared with numerical results obtained on the digital computer. There is good agreement between the results obtained by the proposed method and the numerical solution obtained on digital computer.

Equivalent non-linear and linear third-order systems

A method of establishing equivalent linear systems for classes of third-order nonlinear systems is presented along with the sufficient conditions on the non-linearities of the systems which will permit the determination of such equivalent systems. This equivalence is established through the use of differential transformation techniques involving transformations of both the dependent and independent variables.

Subharmonics and jump resonance in a third-order nonlinear system

A third-order nonlinear system was simulated on an analog computer and examined for subharmonic frequency generation and jump resonance. The results were compared to existing knowledge of the same phenomena in certain second-order non-linear systems. The results indicate that the systems exhibit comparable subharmonic and jump resonance characteristics.

Reply to “ A note on ‘ Graphical analysis of a class of third-order non-linear systems ’ by R. Luus ”

Reply to “ A note on ‘ Graphical analysis of a class of third-order non-linear systems ’ by R. Luus ”

A note on ‘Graphical analysis of a class of third-order non-linear systems’

A note on ‘Graphical analysis of a class of third-order non-linear systems’

Graphical analysis of a class of third-order non-linear systems†

In this paper the concept of phase plane analysis, which is well developed for second-order systems, is extended to a class of third-order non-linear systems. The method of analysis presented is applicable to the unity feedback closed-loop systems, having a third-order linear memory system and a single-valued zero-memory system in cascade combination in the forward path. The technique involves plotting a (x2, x3) master plane having all possible trajectories for x1 = 0. The variation and the initial condition of variable x1 are incorporated by adding an origin shift scale on to the master plane. Using this master plane, the required (x1, x2) and (x1, x3) trajectories can be obtained for any set of initial conditions, following a simple procedure illustrated by typical examples. This technique is also applicable for the analysis of such systems for step inputs.

A Numerical Method for the Location of Limit Cycles of Higher-order Nonlinear Autonomous Systems

A numerical method for the determination of the limit cycles of higher-order nonlinear autonomous systems of arbitrary order and investigation of their stability is presented. The method is based on Andronov's point-transformation method and Liapunov stability theory. Several illustrative examples of second- and third-order nonlinear autonomous systems are given.

Limit Cycles of Higher Order Nonlinear Autonomous Systems

An analytic criterion for the establishment of the existence of the limit cycles in higher order nonlinear autonomous systems of soft-excitation type is presented. The same analytical criterion can be used to obtain the necessary and sufficient conditions in the parameter space for the asymptotic stability of the equilibrium point of these systems. Nonlinear autonomous systems of soft-excitation type are defined and a conjecture to identify these systems is given. Several illustrative examples of second- and third-order nonlinear autonomous systems are presented.

A new approach for the study of limit-cycle behaviour for a class of third-order non-linear control systems†

An analytical expression for the condition of existence of limit-cycle behaviour for a class of third-order non-linear control systems with memory-type non-linearity is derived. The criteria for the stability of the limit cycle are also included. A typical example is taken for the purpose of illustrating the method of finding the limit-cycle parameters. The divergent, convergent and semistable equilibrium points on the F(a,b,α,β)/E plane are clearly defined.

Comments on "Class of Nonlinear Third-Order Systems Reducible to Equivalent Linear Systems"

Comments on "Class of Nonlinear Third-Order Systems Reducible to Equivalent Linear Systems"

Noise Loading Analysis of a Third-Order Nonlinear System with Memory

This paper presents the noise loading analysis of a third-order nonlinear system characterized by Volterra kernels. The analysis is of practical interest since intermodulation distortion is often a limiting consideration in the design of solid-state long-haul broad-band frequency-division multiplexed transmission systems, and since speech on such systems is best represented by zero mean Gaussian noise. The analysis technique is amenable to digital computer calculation; as such, quantitive results are readily obtained. The analysis was applied to a single-stage transistor amplifier that was represented by a frequency-dependent nonlinear model. An excellent agreement between measured and calculated results was obtained. Portions of the program written for this example are readily applicable to the calculation of intermodulation distortion for any thirdorder nonlinear system (e.g., amplifier, repeater, or cascade of repeaters and cable sections) given the appropriate Volterra kernels.