Normal Form Of Nonlinear Systems Research Articles

A direct approach for simplifying nonlinear systems with external periodic excitation using normal forms

In this article, we present a straightforward methodology to obtain the normal forms of nonlinear systems subjected to external periodic excitation. Moreover, the nonlinear systems are not mandated to be minimally excited and may be parametrically excited or possess constant coefficients. This methodology applies an intuitive state augmentation scheme which serves to liberate the analysis from traditional approaches that require special strategies such as ‘book-keeping’ parameters, detuning parameters, ad hoc new unsolved differential equations and variables. Because our technique affiliates the excitation frequency terms with the augmented states in a direct, consistent and explicit manner, this approach is applicable to a broad range of nonlinear systems with single or multiple periodic excitations. We performed analysis of the forced Duffing and the Mathieu–Duffing equations via normal forms computed by our methodology. The analysis scrutinized the systems amplitude variations in time and frequency domains. Observed conformity between the normal forms results and the numerically integrated results validated the reliability of our unified approach to accurately construct normal forms of nonlinear systems with external periodic excitation.

Nonlinear Backstepping Control Design for Coupled Nonlinear Systems under External Disturbances

A nonlinear backstepping control is proposed for the coupled normal form of nonlinear systems. The proposed method is designed by combining the sliding‐mode control and backstepping control with a disturbance observer (DOB). The key idea behind the proposed method is that the linear terms of state variables of the second subsystem are lumped into the virtual input in the first subsystem. A DOB is developed to estimate the external disturbances. Auxiliary state variables are used to avoid amplification of the measurement noise in the DOB. For output tracking and unmatched disturbance cancellation in the first subsystem, the desired virtual input is derived via the backstepping procedure. The actual input in the second subsystem is developed to guarantee the convergence of the virtual input to the desired virtual input by using a sliding‐mode control. The stability of the closed‐loop is verified by using the input‐to‐state stable (ISS) property. The performance of the proposed method is validated via numerical simulations and an application to a vehicle system based on CarSim platform.

Control of Hopf bifurcations of nonlinear infinite-dimensional systems: application to axial flow engine compressor

We study the local feedback stabilization of Hopf bifurcations for nonlinear systems of infinite dimensions in the case where the linearized vector field has a pair of simple nonzero imaginary eigenvalues and all its other eigenvalues lie strictly in the left half-plane. Discussing the normal form of nonlinear systems obtained by making use of the integral averaging method, we obtain sufficient and necessary condition for controlling the stability of the systems even if the critical modes are uncontrollable. As an application, we apply the obtained results to the control of axial flow engine compressor.

On Lyapunov stability and normal forms of nonlinear systems with a nonsemisimple critical mode. I. Zero eigenvalue

This work evolved from an endeavor to derive stability criteria and Poincare normal forms for nonlinear systems associated with a nonsemisimple zero (in Part I) or a pair of imaginary eigenvalues (in Part II). The stability criteria are given in terms of the noninteracting vector restoring and restraining forces, which are motivated from the Lienard equation for nonlinear mass-damper-spring system models. Lyapunov functions are constructed explicitly to fulfill the La Salle invariant principle for local or global stability assertion. It turned out that the Lyapunov functions thus constructed apply to a wide variety of linear stability scenarios. By introducing the notions of restoring and restraining forces, how the Lyapunov functions, the stability criteria and the system dynamics interplay are also exhibited. Two distinct classes of nonlinearities which we referred to as being arithmetical and transcendental, emerged. In some sense, such systems carry nonlinear lags coexisting with the linear lead. In particular, a characteristic of the nonlinear dynamics, a staircase structure, is discovered. Further extension is also made to incorporate nondestabilizing perturbation, which bears important bifurcational implications.

On Lyapunov stability and normal forms of nonlinear systems with a nonsemisimple critical mode. II. Imaginary eigenvalues pair

For pt. I see ibid., vol. 47, no. 6 (June 2000). This paper studies nonlinear systems associated with or linked to a nonsemisimple (NSS) pair of imaginary eigenvalues. Stability criteria and Poincare normal forms (PNF's) are derived via explicit construction of Lyapunov functions. It is a counterpart and continuation of Part I, which considered the case with an NSS zero eigenvalue. While originating from stability analysis for a nonsemisimple critical mode, the development extends to coexisting (semi-) simple critical modes and to stable modes, as in Part I. The present NSS imaginary pairs (NSSIP) case is shown to possess certain characteristics common to NSS zero (NSSZ), such as the nonlinear lag and the staircase structure, but retains its own identities as well. These include the less surprising, the submodular variable-pair circles which emerge from the units of the potential generator and the more remarkable stricter specification in the dynamics, which is possibly attributable to the inherited conjugacy. An elementary partial differential equation traversing the development shows potential for nontrivial generalization.