The properties of output frequencies of nonlinear volterra systems

Nonlinear systems usually have complicated output frequencies in the frequency domain. For the class of nonlinear Volterra systems, some interesting properties for system output frequencies are studied in this paper. These properties provide a novel insight into the output frequencies of Volterra systems, i.e., the periodicity of the output frequencies. They also demonstrate several novel frequency characteristics of system output spectrum such as the opposite property, and reveal clearly the nonlinear effects on system output spectrum from different nonlinearities. These new results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities, and provide an important guidance to applications of Volterra system theory in practices for analysis and design of nonlinear systems. Examples and discussions are given to illustrate these new results

Resonances and resonant frequencies for a class of nonlinear systems

Resonant phenomena for a class of nonlinear systems, which can be described by a SDOF model with a polynomial type nonlinear stiffness, are investigated using Nonlinear Output Frequency Response Functions (NOFRFs). The concepts of resonance and resonant frequencies are proposed for the first time for a class of nonlinear systems. The effects of damping on the resonances and resonant frequencies are also analyzed. These results produce a novel interpretation of energy transfer phenomena in this class of nonlinear systems and show how the damping effect influences the system resonant frequencies and amplitudes. The results are important for the design and fault diagnosis of mechanical systems and structures which can be described by the SDOF nonlinear model

Analysis of output frequencies of nonlinear systems

In this paper, an algorithm is derived for the determination of the output frequency ranges of nonlinear systems, which extends previous results on the output frequencies of nonlinear systems to a more general situation. The new results are significant for the analysis of the output frequency response of a wide class of nonlinear systems

A nonparametric learning framework for nonlinear robust output regulation

This paper proposes a nonparametric learning solution framework for a generic internal model design of nonlinear robust output regulation. The global robust output regulation problem for a class of nonlinear systems with output feedback subject to a nonlinear exosystem can be tackled by constructing a linear generic internal model, provided that a continuous nonlinear mapping exists. An explicit continuous nonlinear mapping was constructed recently in [1] under the assumption that the steady-state generator is linear in the exogenous signal. We further relax such an assumption to a relaxed assumption that the steady-state generator is polynomial in the exogenous signal. A nonparametric learning framework is proposed to solve a linear time-varying equation to make the nonlinear continuous mapping always exist. With the help of the proposed framework, the nonlinear robust output regulation problem can be converted into a robust non-adaptive stabilization problem for the augmented system with integral Input-to-State Stable (iISS) inverse dynamics. Moreover, a dynamic gain approach can adaptively raise the gain to a sufficiently large constant to achieve stabilization without requiring any a priori knowledge of the uncertainties appearing in the dynamics of the exosystem and the system. We further apply the nonparametric learning framework to globally reconstruct and estimate multiple sinusoidal signals with unknown frequencies without using adaptive techniques. An explicit nonlinear mapping can directly provide the estimated parameters, which will exponentially converge to the unknown frequencies. As a result, a feedforward control design is proposed to solve the output regulation using our nonparametric learning framework.Comment: 15 pages; Nonlinear control; iISS stability; output regulation; parameter estimation; Non-adaptive contro

On self-learning mechanism for the output regulation of second-order affine nonlinear systems

This paper studies global robust output regulation of second-order nonlinear systems with input disturbances that encompass the fully-actuated Euler-Lagrange systems. We assume the availability of relative output (w.r.t. a family of reference signals) and output derivative measurements. Based on a specific separation principle and self learning mechanism, we develop an internal model-based controller that does not require apriori knowledge of reference and disturbance signals and it only assumes that the kernels of these signals are a family of exosystems with unknown parameters (e.g., amplitudes, frequencies or time periods). The proposed control framework has a self-learning mechanism that extricates itself from requiring absolute position measurement nor precise knowledge of the feedforward kernel signals. By requiring the high-level task/trajectory planner to use the same class of kernels in constraining the trajectories, the proposed low-level controller is able to learn the desired trajectories, to suppress the disturbance signals, and to adapt itself to the uncertain plant parameters. The framework enables a plug-and-play control mechanism in both levels of control

An algorithm for determining the output frequency range of Volterra models with multiple inputs

A new algorithm for determining the output frequency range and the frequency components of Volterra models under multiple inputs is introduced for nonlinear system analysis. For a given Volterra model, the output frequency components corresponding to a multi-tone input can easily be calculated using the new algorithm

Evaluation of output frequency responses of nonlinear systems under multiple inputs

In this paper, a new method for evaluating output frequency responses of nonlinear systems under multiple inputs, defined as a sum of sinusoids of different frequencies, is developed. The method circumvents difficulties associated with the existing “frequency-mix vector” based approaches and can easily be applied to investigate nonlinear behaviors of practical systems, including electronic circuits, at the system simulation and design stages. Application of the method to the analysis of nonlinear interference and distortion effects in communication receivers is studied, and specific procedures are proposed which can be directly used in practice for this analysi

Analysis of bilinear oscillators under harmonic loading using nonlinear output frequency response functions

In this paper, the new concept of Nonlinear Output Frequency Response Functions (NOFRFs) is extended to the harmonic input case, an input-independent relationship is found between the NOFRFs and the Generalized Frequency Response Functions (GFRFs). This relationship can greatly simplify the application of the NOFRFs. Then, beginning with the demonstration that a bilinear oscillator can be approximated using a polynomial type nonlinear oscillator, the NOFRFs are used to analyze the energy transfer phenomenon of bilinear oscillators in the frequency domain. The analysis provides insight into how new frequency generation can occur using bilinear oscillators and how the sub-resonances occur for the bilinear oscillators, and reveals that it is the resonant frequencies of the NOFRFs that dominate the occurrence of this well-known nonlinear behaviour. The results are of significance for the design and fault diagnosis of mechanical systems and structures which can be described by a bilinear oscillator model

Nonlinear output frequency response functions for multi-input nonlinear volterra systems

The concept of Nonlinear Output Frequency Response Functions (NOFRFs) is extended to the nonlinear systems that can be described by a multi-input Volterra series model. A new algorithm is also developed to determine the output frequency range of nonlinear systems from the frequency range of the inputs. These results allow the concept of NOFRFs to be applied to a wide range of engineering systems. The phenomenon of the energy transfer in a two degree of freedom nonlinear system is studied using the new concepts to demonstrate the significance of the new results

Crack detection using nonlinear output frequency response functions - an experimental study

The new concept of Nonlinear Output Frequency Response Functions (NOFRFs) is introduced in this paper to detect cracks in beams using frequency domain information. The results show that the NOFRFs are a sensitive indicator of the presence of cracks providing the excitation is of an appropriate strength. The new results provide a novel and effective method for the detection of cracks in beams, with applications in structural fault diagnosis