A new convergence analysis for the Volterra series representation of nonlinear systems

The convergence of the Volterra series representation of nonlinear systems is the fundamental requirement for the analysis of nonlinear systems in the frequency domain. In the present study, a new criterion is derived to determine the convergence of the Volterra series representation of nonlinear systems described by a NARX (Nonlinear Auto Regressive with eXegenous input) model. The analysis is performed based on a new function known as Generalized Output Bound Characteristic Function (GOBCF), which is defined in terms of the input, output and parameters of the NARX model of nonlinear systems. Compared to the existing results, the new criterion provides a much more rigorous and effective approach to the analysis of the convergence conditions and properties of the Volterra series representation of nonlinear systems. Two case studies have been used to demonstrate the effectiveness of the new convergence analysis criterion and the advantages of the new analysis over those produced by existing approaches

The parametric characteristics of frequency response functions for nonlinear systems

The characteristics of the frequency response functions of nonlinear systems can be revealed and analyzed through the analysis of the parametric characteristics of these functions. To achieve these objectives, a new operator is defined, and several fundamental and important results about the parametric characteristics of the frequency response functions of nonlinear systems are developed. These theoretical results provide a significant and novel insight into the frequency domain characteristics of nonlinear systems and circumvent a large amount of complicated integral and symbolic calculations which have previously been required to perform nonlinear system frequency domain analysis. Several new results for the analysis and synthesis of nonlinear systems are also developed. Examples are included to illustrate potential applications of the new results

Analysis and design of nonlinear systems in the frequency domain

Nonlinear system analyses have been widely applied in engineering practice, where the frequency domain approaches have been developed to satisfy the requirement of the analysis and design of nonlinear systems. However, there exist many problems with current techniques including the challenges with the nonlinear system representation using physically meaningful models, and difficulties with the evaluation of the frequency properties of nonlinear systems. In the present work, some new approaches, that have potential to be used to systematically address these problems, are developed based on the NDE (Nonlinear Differential Equation) model and the NARX (Nonlinear Auto Regressive with eXegenous input) model of nonlinear systems. In this thesis, the background of the frequency domain analysis and design of nonlinear systems is introduced in Chapter 1, and the existing approaches are reviewed in Chapter 2. In general, the frequency analysis of nonlinear systems is conducted based on the Volterra series representation of nonlinear systems, and as basic issues, the evaluation of the Volterra series representation and its convergence are discussed in Chapters 3 and 4, respectively. An extension of the existing frequency analysis and design techniques is discussed in Chapter 5 to facilitate the analysis of the effects of both linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems. An experimental study is conducted in Chapter 6 to show how a nonlinear component can benefit the engineering system, such to emphasis the significance of developing the analysis and design approaches of nonlinear systems. The main contributions are summarized as below. (1) The GALEs is proposed that can accurately evaluate the system Volterra series representation. By using the GALEs, the solution to the NDE model or the NARX model of nonlinear systems can be obtained by simply dealing with a series of linear differential or difference equations, which can facilitate a wide range of nonlinear system analyses and associated practical applications. (2) A new criterion is derived to determine the convergence of the Volterra series representation of nonlinear systems described by a NARX model. The analysis is performed based on a new function known as Generalized Output Bound Characteristic Function (GOBCF), which is defined in terms of the input, output and parameters of the NARX model of nonlinear systems. Compared to the existing results, the new criterion provides a much more rigorous and effective approach to the analysis of the convergence conditions and properties of the Volterra series representation of nonlinear systems. (3) The Output Frequency Response Function (OFRF) in terms of physical parameters of concern is introduced for the NARX Model with parameters of interest for Design (NARX-M-for-D). Moreover, a new concept known as the Associated Output Frequency Response Function (AOFRF) is introduced to facilitate the analysis of the effects of both linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems. (4) Nonlinear damping can achieve desired isolation performance of a system over both low and high frequency regions and the optimal nonlinear damping force can be realized by closed loop controlled semi-active dampers. Both simulation and laboratory experiments are studied, demonstrating the advantages of the proposed nonlinear damping technologies over both traditional linear damping and more advanced Linear-Quadratic Gaussian (LQG) feedback control which have been used in practice to address building isolation system design and implementation problems

The Effects of Linear and Nonlinear Characteristic Parameters on the Output Frequency Responses of Nonlinear Systems: The Associated Output Frequency Response Function

In the present study, a new concept known as the Associated Output Frequency Response Function (AOFRF) is introduced to facilitate the analysis of the effects of both linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems. Based on the AOFRF concept, the study has shown, for the first time, that the output frequency responses of a wide class of nonlinear systems that are described by the NARX (Nonlinear Auto Regressive with eXegenous input) model can be represented by a polynomial function of both the system linear and nonlinear characteristic parameters of interests to the system analysis. Moreover, an efficient algorithm is derived to determine the structure and coefficients of the AOFRF based representation for system output frequency responses. Finally, a case study is provided to demonstrate the effectiveness and advantages of the new AOFRF based representation and the implication of the result to the analysis and design of nonlinear systems in the frequency domain

Frequency domain theory of nonlinear Volterra systems based on parametric characteristic analysis.

The frequency domain methods tor linear systems are well accepted by engineers and have been widely applied in engineering practice because the transfer function of linear systems can always provide a coordinate-free and equivalent description for system characteristics and are convenient to be used for the system analysis and design. Although the analysis and design of linear systems in the frequency domain have been well established and the frequency domain methods for nonlinear systems have aheady been investigated for many years, the frequency domain analysis for nonlinear systems is far from being fully developed

Nonlinear interactions in the thalamocortical loop in essential tremor: A model-based frequency domain analysis.

There is increasing evidence to suggest that essential tremor has a central origin. Different structures appear to be part of the central tremorogenic network, including the motor cortex, the thalamus and the cerebellum. Some studies using electroencephalogram (EEG) and magnetoencephalography (MEG) show linear association in the tremor frequency between the motor cortex and the contralateral tremor electromyography (EMG). Additionally, high thalamomuscular coherence is found with the use of thalamic local field potential (LFP) recordings and tremulous EMG in patients undergoing surgery for deep brain stimulation (DBS). Despite a well-established reciprocal anatomical connection between the thalamus and cortex, the functional association between the two structures during "tremor-on" periods remains elusive. Thalamic (Vim) LFPs, ipsilateral scalp EEG from the sensorimotor cortex and contralateral tremor arm EMG recordings were obtained from two patients with essential tremor who had undergone successful surgery for DBS. Coherence analysis shows a strong linear association between thalamic LFPs and contralateral tremor EMG, but the relationship between the EEG and the thalamus is much less clear. These measurements were then analyzed by constructing a novel parametric nonlinear autoregressive with exogenous input (NARX) model. This new approach uncovered two distinct and not overlapping frequency "channels" of communication between Vim thalamus and the ipsilateral motor cortex, defining robustly "tremor-on" versus "tremor-off" states. The associated estimated nonlinear time lags also showed non-overlapping values between the two states, with longer corticothalamic lags (exceeding 50ms) in the tremor active state, suggesting involvement of an indirect multisynaptic loop. The results reveal the importance of the nonlinear interactions between cortical and subcortical areas in the central motor network of essential tremor. This work is important because it demonstrates for the first time that in essential tremor the functional interrelationships between the cortex and thalamus should not be sought exclusively within individual frequencies but more importantly between cross-frequency nonlinear interactions. Should our results be successfully reproduced on a bigger cohort of patients with essential tremor, our approach could be used to create an on-demand closed-loop DBS device, able to automatically activate when the tremor is on

Bayesian Non-linear System Identification and Frequency Response Analysis with Application to Soft Smart Actuators

Newly emerging classes of next generation soft-smart actuators are set to have a huge impact on the fields of robotics, orthotics and prosthetics due to their lightweight, high-strain and muscle-like properties. Like muscle, these actuators can be used in multiple roles, e.g. both as actuators and brakes, due their variable compliance. One important class of soft actuator is the dielectric elastomer actuator (DEA). However, DEAs are extremely difficult to control due to their non-linear and time varying dynamics. A crucial step in the advancement of this technology is the development of techniques for systems level modelling and analysis, which is the focus of this thesis. In the first part of the thesis, a set of DEAs are identified and analysed using standard methods from the field of system identification, obtaining non-linear autoregressive with exogenous input (NARX) models. These provide a benchmark against which later methods are evaluated. The key novelty in this part is the development of NARX models of DEAs for use in non-linear frequency-domain analysis. This result provides insight for the first time into how a set of similarly fabricated DEAs vary in different ways. A further aspect of DEA behaviour is their unexplained time varying behaviour. The system identification approach used to identify NARX models of DEAs is in a convenient form such that it can be easily extended to cater for this time varying behaviour. There are however very few available methods for the frequency domain analysis of time varying systems. A novel method for time varying frequency domain analysis of NARX systems is developed in this work and applied to the DEAs. The analysis procedure is used to provide insight on how the dynamic behaviour of DEAs change over time. In the second part of the thesis a novel approach to the joint structure detection and parameter estimation of NARX models is developed using a sparse Bayesian method. The Bayesian framework allows for the estimation of posterior distributions over model parameters, characterising the model uncertainty. Analytic solutions are found that describe model uncertainty in the frequency-domain as confidence bounds on both linear and higher order frequency response functions. The sparse Bayesian identification algorithm is applied to the DEA data sets and is used to give the first non-linear dynamic model of DEAs with uncertainty bounds plus the first description of DEA dynamics in the frequency-domain, again with uncertainty bounds

Modeling of electrical circuit with recurrent neural networks

In this dissertation, a circuit modeling methodology using recurrent neural networks (RNNs) is developed. The methodology covers model structure selection, data generation, training, and model implementation for circuit simulation. Several different RNN structures are investigated and their capabilities in circuit modeling are compared. The stability of RNN in the context of circuit modeling is defined and methods to guarantee stability for some RNN structures are developed. The modeling methodology is supported by test cases showing the accuracy and efficiency of RNN models

Application of non-linear system identification approaches to modelling, analysis, and control of fluid flows.

Flow control has become a topic of great importance for several applications, ranging from commercial aircraft, to intercontinental pipes and skyscrapers. In these applications, and many more, the interaction with a fluid flow can have a significant influence on the performance of the system. In many cases the fluids encountered are turbulent and detrimental to the latter. Several attempts have been made to solve this problem. However, due to the non-linearity and infinite dimensionality of fluid flows and their governing equations, a complete understanding of turbulent behaviour and a feasible control approach has not been obtained. In this thesis, model reduction approaches that exploit non-linear system identification are applied using data obtained from numerical simulations of turbulent three-dimensional channel flow, and two-dimensional flow over the backward facing step. A multiple-input multiple-output model, consisting of 27 sub-structures, is obtained for the fluctuations of the velocity components of the channel flow. A single-input single-output model for fluctuations of the pressure coefficient, and two multiple-input single-output models for fluctuations of the velocity magnitude are obtained in flow over the BFS. A non-linear model predictive control strategy is designed using identified one- and multi-step ahead predictors, with the inclusion of integral action for robustness. The proposed control approach incorporates a non-linear model without the need for expensive non-linear optimizations. Finally, a frequency domain analysis of unmanipulated turbulent flow is perfumed using five systems. Higher order generalized frequency response functions (GFRF) are computed to study the non-linear energy transfer phenomena. A more detailed investigation is performed using the output FRF (OFRF), which can elucidate the contribution of the n-th order frequency response to the output frequency response

Robust Empirical Model-Based Algorithms for Nonlinear Processes

This research work proposes two robust empirical model-based predictive control algorithms for nonlinear processes. Chemical process are generally highly nonlinear thus predictive control algorithms that explicitly account for the nonlinearity of the process are expected to provide better closed-loop performance as compared to algorithms based on linear models. Two types of models can be considered for control: first-principles and empirical. Empirical models were chosen for the proposed algorithms for the following reasons: (i) they are less complex for on-line optimization, (ii) they are easy to identify from input-output data and (iii) their structure is suitable for the formulation of robustness tests. One of the key problems of every model that is used for prediction within a control strategy is that some model parameters cannot be known accurately due to measurement noise and/or error in the structure of the assumed model. In the robust control approach it is assumed that processes can be represented by models with parameters' values that are assumed to lie between a lower and upper bound or equivalently, that these parameters can be represented by a nominal value plus uncertainty. When this uncertainty in control parameters is not considered by the controller the control actions might be insufficient to effectively control the process and in some extreme cases the closed-loop may become unstable. Accordingly, the two robust control algorithms proposed in the current work explicitly account for the effect of uncertainty on stability and closed-loop performance. The first proposed controller is a robust gain-scheduling model predictive controller (MPC). In this case the process is represented within each operating region by a state-affine model obtained from input-output data. The state-affine model matrices are used to obtain a state-space based MPC for every operating region. By combining the state-affine, disturbance and controller equations a closed-loop representation was obtained. Then, the resulting mathematical representation was tested for robustness with linear matrix inequalities (LMI's) based on a test where the vertices of the parameter box were obtained by an iterative procedure. The result of the LMI's test gives a measure of performance referred to as γ that relates the effect of the disturbances on the process outputs. Finally, for the gain-scheduling part of the algorithm a set of rules was proposed to switch between the available controllers according to the current process conditions. Since every combination of the controller tuning parameters results in a different value of γ, an optimization problem was proposed to minimize γ with respect to the tuning parameters. Accordingly, for the proposed controller it was ensured that the effect of the disturbances on the output variables was kept to its minimum. A bioreactor case study was presented to show the benefits of the proposed algorithm. For comparison purposes a non-robust linear MPC was also designed. The results show that the proposed algorithm has a clear advantage in terms of performance as compared to non-robust linear MPC techniques. The second controller proposed in this work is a robust nonlinear model predictive controller (NMPC) based on an empirical Volterra series model. The benefit of using a Volterra series model for this case is that its structure can be split in two sections that account for the nominal and uncertain parameter values. Similar to the previously proposed gain-scheduled controller the model parameters were obtained from input-output data. After identifying the Volterra model, an interconnection matrix and its corresponding uncertainty description were found. The interconnection matrix relates the process inputs and outputs and is built according to the type of cost function that the controller uses. Based on the interconnection representing the system a robustness test was proposed based on a structured singular value norm calculation (SSV). The test is based on a min-max formulation where the worst possible closed-loop error is minimized with respect to the manipulated variables. Additional factors that were considered in the cost function were: manipulated variables weighting, manipulated variables restrictions and a terminal condition. To show the benefits of this controller two case studies were considered, a single-input-single-output (SISO) and a multiple-input-multiple-output (MIMO) process. Both case studies show that the proposed controller is able to control the process. The results showed that the controller could efficiently track set-points in the presence of disturbances while complying with the saturation limits imposed on the manipulated variables. This controller was also compared against a non-robust linear MPC, non-robust NMPC and non-robust first-principles NMPC. These comparisons were performed for different levels of uncertainty and for different values of the suppression or control actions weights. It was shown through these comparisons that a tradeoff exists between nominal performance and robustness to model error. Thus, for larger weights the controller is less aggressive resulting in more sluggish performance but less sensitivity to model error thus resulting in smaller differences between the robust and non-robust schemes. On the other hand when these weights are smaller the controller is more aggressive resulting in better performance at the nominal operating conditions but also leading to larger sensitivity to model error when the system is operated away from nominal conditions. In this case, as a result of this increased sensitivity to model error, the robust controller is found to be significantly better than the non-robust one