Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Vibroacoustics of the piano soundboard: (Non)linearity and modal properties in the low- and mid-frequency ranges

Research highlights: - Nonlinearity characterization of a structure independently from its exciting device. - Experimental modal identification, including damping, in the mid-frequency range. - Boundary conditions, low-and high-frequency regimes clarified in the piano soundboard. - Good match between experimental observations and FEM results at low frequencies. - FEM of a non-regularly ribbed soundboard reveals high-frequency localization in piano.International audienceThe piano soundboard transforms the string vibration into sound and therefore, its vibrations are of primary importance for the sound characteristics of the instrument. An original vibro-acoustical method is presented to isolate the soundboard nonlinearity from that of the exciting device (here: a loudspeaker) and to measure it. The nonlinear part of the soundboard response to an external excitation is quantitatively estimated for the first time, at ≈ -40 dB below the linear part at the ff nuance. Given this essentially linear response, a modal identification is performed up to 3 kHz by means of a novel high resolution modal analysis technique (Ege et al., High-resolution modal analysis, JSV, 325(4-5), 2009). Modal dampings (which, so far, were unknown for the piano in this frequency range) are determined in the midfrequency domain where FFT-based methods fail to evaluate them with an acceptable precision. They turn out to be close to those imposed by wood. A finite-element modelling of the soundboard is also presented. The low-order modal shapes and the comparison between the corresponding experimental and numerical modal frequencies suggest that the boundary conditions can be considered as blocked, except at very low frequencies. The frequency-dependency of the modal density and the observation of modal shapes reveal two well-separated regimes. Below ≈ 1 kHz, the soundboard vibrates more or less like a homogeneous plate. Above that limit, the structural waves are confined by ribs, as already noticed by several authors, and localised in restricted areas (one or a few inter-rib spaces), presumably due to a slightly irregular spacing of the ribs across the soundboard

Networked Control System Design and Parameter Estimation

Networked control systems (NCSs) are a kind of distributed control systems in which the data between control components are exchanged via communication networks. Because of the attractive advantages of NCSs such as reduced system wiring, low weight, and ease of system diagnosis and maintenance, the research on NCSs has received much attention in recent years. The first part (Chapter 2 - Chapter 4) of the thesis is devoted to designing new controllers for NCSs by incorporating the network-induced delays. The thesis also conducts research on filtering of multirate systems and identification of Hammerstein systems in the second part (Chapter 5 - Chapter 6). Network-induced delays exist in both sensor-to-controller (S-C) and controller-to-actuator (C-A) links. A novel two-mode-dependent control scheme is proposed, in which the to-be-designed controller depends on both S-C and C-A delays. The resulting closed-loop system is a special jump linear system. Then, the conditions for stochastic stability are obtained in terms of a set of linear matrix inequalities (LMIs) with nonconvex constraints, which can be efficiently solved by a sequential LMI optimization algorithm. Further, the control synthesis problem for the NCSs is considered. The definitions of H₂ and H∞ norms for the special system are first proposed. Also, the plant uncertainties are considered in the design. Finally, the robust mixed H₂/H∞ control problem is solved under the framework of LMIs. To compensate for both S-C and C-A delays modeled by Markov chains, the generalized predictive control method is modified to choose certain predicted future control signal as the current control effort on the actuator node, whenever the control signal is delayed. Further, stability criteria in terms of LMIs are provided to check the system stability. The proposed method is also tested on an experimental hydraulic position control system. Multirate systems exist in many practical applications where different sampling rates co-exist in the same system. The l₂-l∞ filtering problem for multirate systems is considered in the thesis. By using the lifting technique, the system is first transformed to a linear time-invariant one, and then the filter design is formulated as an optimization problem which can be solved by using LMI techniques. Hammerstein model consists of a static nonlinear block followed in series by a linear dynamic system, which can find many applications in different areas. New switching sequences to handle the two-segment nonlinearities are proposed in this thesis. This leads to less parameters to be estimated and thus reduces the computational cost. Further, a stochastic gradient algorithm based on the idea of replacing the unmeasurable terms with their estimates is developed to identify the Hammerstein model with two-segment nonlinearities. Finally, several open problems are listed as the future research directions

Linear and nonlinear parametric hydrodynamic models for wave energy converters identified from recorded data

Ocean waves represent an important resource of renewable energy, which can provide a significant support to the development of more sustainable energy solutions and to the reduction ofCO2 emissions. The amount of extracted energy from the ocean waves can be increased by optimizing the geometry and the control strategy of the wave energy converter (WEC), which both require mathematical hydrodynamic models, able to correctly describe the WEC-fluid interaction. In general, the construction of a model is based on physical laws describing the system under investigation. The hydrodynamic laws are the foundation for a complete description of the WEC-fluid interaction, but their solution represents a very complex and challenging problem. Different approaches to hydrodynamic WEC-fluid interaction modelling, such as computational fluid dynamics (CFD) and linear potential theory (LPT), lead to different mathematical models, each one characterised by different accuracy and computational speed. Fully nonlinear CFD models are able to describe the full range of hydrodynamic effects, but are very computationally expensive. On the other hand, LPT is based on the strong assumptions of inviscid fluid, irrotational flow, small waves and small body motion, which completely remove the hydrodynamic nonlinearity of the WEC-fluid interaction. Linear models have good computational speed, but are not able to properly describe nonlinear hydrodynamic effects, which are relevant in some WEC power production conditions, since WECs are designed to operate over a wide range of wave amplitudes, experience large motions, and generate viscous drag and vortex shedding. The main objective of this thesis is to propose and investigate an alternative pragmatic framework, for hydrodynamic model construction, based on system identification methodologies. The goal is to obtain models which are between the CFD and LPT extremes, a good compromise able to describe the most important nonlinearities of the physical system, without requiring excessively computational time. The identified models remain sufficiently fast and simple to run in real-time. System identification techniques can ‘inject’ into the model only the information contained in the identification data; therefore, the models obtained from LPT data are not able to describe nonlinear hydrodynamic effects. In this thesis, instead of traditional LPT data, experimental wave tank data (both numerical wave tank (NWT), implemented with a CFD software package, and real wave tank (RWT)) are proposed for hydrodynamic model identification, since CFD-NWT and RWT data can contain the full range of nonlinear hydrodynamic effects. In this thesis, different typologies of wave tank experiments and excitation signals are investigated in order to generate informative data and reduce the experiment duration. Indeed, the reduction of the experiment duration represents an important advantage since, in the case of a CFD-NWT, the amount of computation time can become unsustainable whereas, in the case of a RWT, a set of long tank experiments corresponds to an increase of the facility renting costs

Compensation of nonlinear distortion in RF amplifiers for mobile communications

Compensation of nonlinear distortion of power amplifiers in mobile communications is an important requirement for improving power consumption performance while maintaining efficiency, since mobile phone became an essential accessory for everyone nowadays. This problem demands a good power amplifier model, in order to develop an effective predistortion system. Current researches are focused on modelling and predistortion of power amplifiers with memory, as well as memoryless ones. Different methods for modelling are used, as the Volterra series, polynomial models, look-up tables, the Hammerstein models, the Wiener models, and artificial intelligence systems. For predistortion feedback, feedforward and digital predistortion techniques are used. Among digital predistortion methods there are artificial intelligence systems, used in this thesis for linearization of power amplifier. This thesis presents developed robust method for modelling power amplifiers without memory effects and gives a comparison of proposed method with least squares method. Also, this research presents two novel techniques based on artificial intelligence systems for modelling and predistortion of highly nonlinear power amplifier with memory. The first approach is based on artificial neural networks, while the second one uses adaptive fuzzy logic systems. Forward and inverse models of power amplifier are created with both proposed methods. Superiority of artificial intelligence systems over partial least squares method is presented. Developed models are employed in a cascade to make a linearized system. Verification of proposed methods is carried out through the signal performance parameters and spectra of measured signal and signal from predistortion system. The feasibility and performances of the proposed digital predistortions are examined by simulations and experiments. The comparison of proposed methods is given to present advantages/disadvantages of both methods. The achieved distortion suppression from 72.2% to 93.6% and spectral regrowth improvement from 11.4 dB to 16.2 dB prove that the proposed methods have great ability to compensate the nonlinear distortion in power amplifier

Tree-Structured Nonlinear Adaptive Signal Processing

In communication systems, nonlinear adaptive filtering has become increasingly popular in a variety of applications such as channel equalization, echo cancellation and speech coding. However, existing nonlinear adaptive filters such as polynomial (truncated Volterra series) filters and multilayer perceptrons suffer from a number of problems. First, although high Order polynomials can approximate complex nonlinearities, they also train very slowly. Second, there is no systematic and efficient way to select their structure. As for multilayer perceptrons, they have a very complicated structure and train extremely slowly Motivated by the success of classification and regression trees on difficult nonlinear and nonparametfic problems, we propose the idea of a tree-structured piecewise linear adaptive filter. In the proposed method each node in a tree is associated with a linear filter restricted to a polygonal domain, and this is done in such a way that each pruned subtree is associated with a piecewise linear filter. A training sequence is used to adaptively update the filter coefficients and domains at each node, and to select the best pruned subtree and the corresponding piecewise linear filter. The tree structured approach offers several advantages. First, it makes use of standard linear adaptive filtering techniques at each node to find the corresponding Conditional linear filter. Second, it allows for efficient selection of the subtree and the corresponding piecewise linear filter of appropriate complexity. Overall, the approach is computationally efficient and conceptually simple. The tree-structured piecewise linear adaptive filter bears some similarity to classification and regression trees. But it is actually quite different from a classification and regression tree. Here the terminal nodes are not just assigned a region and a class label or a regression value, but rather represent: a linear filter with restricted domain, It is also different in that classification and regression trees are determined in a batch mode offline, whereas the tree-structured adaptive filter is determined recursively in real-time. We first develop the specific structure of a tree-structured piecewise linear adaptive filter and derive a stochastic gradient-based training algorithm. We then carry out a rigorous convergence analysis of the proposed training algorithm for the tree-structured filter. Here we show the mean-square convergence of the adaptively trained tree-structured piecewise linear filter to the optimal tree-structured piecewise linear filter. Same new techniques are developed for analyzing stochastic gradient algorithms with fixed gains and (nonstandard) dependent data. Finally, numerical experiments are performed to show the computational and performance advantages of the tree-structured piecewise linear filter over linear and polynomial filters for equalization of high frequency channels with severe intersymbol interference, echo cancellation in telephone networks and predictive coding of speech signals