Average variation L2-stability criteria for time-varying feedback systems-A unified approach

The problem of developing L2-stability criteria for feedback systems with a single time-varying gain, which impose average variation constraints on the gain is treated. A unified approach is presented which facilitates the development of such average variation criteria for both linear and nonlinear systems. The stability criteria derived here are shown to be more general than the existing results.

Regularization of descriptor systems

Implicit dynamic-algebraic equations, known in control theory as descriptor systems, arise naturally in many applications. Such systems may not be regular (often referred to as singular). In that case the equations may not have unique solutions for consistent initial conditions and arbitrary inputs and the system may not be controllable or observable. Many control systems can be regularized by proportional and/or derivative feedback.We present an overview of mathematical theory and numerical techniques for regularizing descriptor systems using feedback controls. The aim is to provide stable numerical techniques for analyzing and constructing regular control and state estimation systems and for ensuring that these systems are robust. State and output feedback designs for regularizing linear time-invariant systems are described, including methods for disturbance decoupling and mixed output problems. Extensions of these techniques to time-varying linear and nonlinear systems are discussed in the final section.

PRACTICAL RESOLUTION OF DIFFERENTIAL SYSTEMS

3. PRACTICAL RESOLUTION OF DIFFERENTIAL SYSTEMS by Marilia Pires, University of Évora, Portugal This practice presents the main features of a free software to solve mathematical equations derived from concrete problems: i.- Presentation of Scilab (or python) ii.- Basics (number, characters, function) iii.- Graphics iv.- Linear and nonlinear systems v.- Differential equations

Optimal controllers with complex order derivatives

This paper studies the optimization of complex-order algorithms for the discrete-time control of linear and nonlinear systems. The fundamentals of fractional systems and genetic algorithms are introduced. Based on these concepts, complexorder control schemes and their implementation are evaluated in the perspective of evolutionary optimization. The results demonstrate not only that complex-order derivatives constitute a valuable alternative for deriving control algorithms, but also the feasibility of the adopted optimization strategy.

Robust multi-sensor data fusion

This research obtains the optimal estimation and data fusion for linear and nonlinear systems suffering from uncertain observations (missing measurements). The noise from the different data sources are considered to correlated. The derivation of the robust Kalman filter for systems subject to aditional uncertainties in the modelling parameters is presented.

Time-Frequency Analysis of Nonlinear Systems: The Skeleton Linear Model and the Skeleton Curves

The joint time-frequency analysis method is adopted to study the nonlinear behavior varying with the instantaneous response for a class of S.D.O.F nonlinear system. A time-frequency masking operator, together with the conception of effective time-frequency region of the asymptotic signal are defined here. Based on these mathematical foundations, a so-called skeleton linear model (SLM) is constructed which has similar nonlinear characteristics with the nonlinear system. Two skeleton curves are deduced which can indicate the stiffness and damping in the nonlinear system. The relationship between the SLM and the nonlinear system, both parameters and solutions, is clarified. Based on this work a new identification technique of nonlinear systems using the nonstationary vibration data will be proposed through time-frequency filtering technique and wavelet transform in the following paper.

Development and Evaluation of Blind Identification Techniques for Nonlinear Systems

Identification and Control of Non‐linear dynamical systems are challenging problems to the control engineers.The topic is equally relevant in communication,weather prediction ,bio medical systems and even in social systems,where nonlinearity is an integral part of the system behavior.Most of the real world systems are nonlinear in nature and wide applications are there for nonlinear system identification/modeling.The basic approach in analyzing the nonlinear systems is to build a model from known behavior manifest in the form of system output.The problem of modeling boils down to computing a suitably parameterized model,representing the process.The parameters of the model are adjusted to optimize a performanace function,based on error between the given process output and identified process/model output.While the linear system identification is well established with many classical approaches,most of those methods cannot be directly applied for nonlinear system identification.The problem becomes more complex if the system is completely unknown but only the output time series is available.Blind recognition problem is the direct consequence of such a situation.The thesis concentrates on such problems.Capability of Artificial Neural Networks to approximate many nonlinear input-output maps makes it predominantly suitable for building a function for the identification of nonlinear systems,where only the time series is available.The literature is rich with a variety of algorithms to train the Neural Network model.A comprehensive study of the computation of the model parameters,using the different algorithms and the comparison among them to choose the best technique is still a demanding requirement from practical system designers,which is not available in a concise form in the literature.The thesis is thus an attempt to develop and evaluate some of the well known algorithms and propose some new techniques,in the context of Blind recognition of nonlinear systems.It also attempts to establish the relative merits and demerits of the different approaches.comprehensiveness is achieved in utilizing the benefits of well known evaluation techniques from statistics. The study concludes by providing the results of implementation of the currently available and modified versions and newly introduced techniques for nonlinear blind system modeling followed by a comparison of their performance.It is expected that,such comprehensive study and the comparison process can be of great relevance in many fields including chemical,electrical,biological,financial and weather data analysis.Further the results reported would be of immense help for practical system designers and analysts in selecting the most appropriate method based on the goodness of the model for the particular context.

Identification of Nonlinear Systems Through Time-Frequency Filtering Technique

In the previous paper, a class of nonlinear system is mapped to a so-called skeleton linear model (SLM) based on the joint time-frequency analysis method. Behavior of the nonlinear system may be indicated quantitatively by the variance of the coefficients of SLM versus its response. Using this model we propose an identification method for nonlinear systems based on nonstationary vibration data in this paper. The key technique in the identification procedure is a time-frequency filtering method by which solution of the SLM is extracted from the response data of the corresponding nonlinear system. Two time-frequency filtering methods are discussed here. One is based on the quadratic time-frequency distribution and its inverse transform, the other is based on the quadratic time-frequency distribution and the wavelet transform. Both numerical examples and an experimental application are given to illustrate the validity of the technique.

Application of Volterra series in nonlinear mechanical system identification and in structural health monitoring problems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

New methods for the estimation of Takagi-Sugeno model based extended Kalman filter and its applications to optimal control for nonlinear systems

This paper describes new approaches to improve the local and global approximation (matching) and modeling capability of Takagi–Sugeno (T-S) fuzzy model. The main aim is obtaining high function approximation accuracy and fast convergence. The main problem encountered is that T-S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the application of the T-S method because this type of membership function has been widely used during the last 2 decades in the stability, controller design of fuzzy systems and is popular in industrial control applications. The approach developed here can be considered as a generalized version of T-S identification method with optimized performance in approximating nonlinear functions. We propose a noniterative method through weighting of parameters approach and an iterative algorithm by applying the extended Kalman filter, based on the same idea of parameters’ weighting. We show that the Kalman filter is an effective tool in the identification of T-S fuzzy model. A fuzzy controller based linear quadratic regulator is proposed in order to show the effectiveness of the estimation method developed here in control applications. An illustrative example of an inverted pendulum is chosen to evaluate the robustness and remarkable performance of the proposed method locally and globally in comparison with the original T-S model. Simulation results indicate the potential, simplicity, and generality of the algorithm. An illustrative example is chosen to evaluate the robustness. In this paper, we prove that these algorithms converge very fast, thereby making them very practical to use.

A new approach to fuzzy estimation of Takagi-Sugeno model and its applications to optimal control for nonlinear systems

An efficient approach is presented to improve the local and global approximation and modelling capability of Takagi-Sugeno (T-S) fuzzy model. The main aim is obtaining high function approximation accuracy. The main problem is that T-S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the use of the T-S method because this type of membership function has been widely used during the last two decades in the stability, controller design and are popular in industrial control applications. The approach developed here can be considered as a generalized version of T-S method with optimized performance in approximating nonlinear functions. A simple approach with few computational effort, based on the well known parameters' weighting method is suggested for tuning T-S parameters to improve the choice of the performance index and minimize it. A global fuzzy controller (FC) based Linear Quadratic Regulator (LQR) is proposed in order to show the effectiveness of the estimation method developed here in control applications. Illustrative examples of an inverted pendulum and Van der Pol system are chosen to evaluate the robustness and remarkable performance of the proposed method and the high accuracy obtained in approximating nonlinear and unstable systems locally and globally in comparison with the original T-S model. Simulation results indicate the potential, simplicity and generality of the algorithm.

Relative entropy rate based model selection for linear hybrid system filters of uncertain nonlinear systems

Hybrid system representations have been exploited in a number of challenging modelling situations, including situations where the original nonlinear dynamics are too complex (or too imprecisely known) to be directly filtered. Unfortunately, the question of how to best design suitable hybrid system models has not yet been fully addressed, particularly in the situations involving model uncertainty. This paper proposes a novel joint state-measurement relative entropy rate based approach for design of hybrid system filters in the presence of (parameterised) model uncertainty. We also present a design approach suitable for suboptimal hybrid system filters. The benefits of our proposed approaches are illustrated through design examples and simulation studies.

Linear and weakly nonlinear stability analyses of cooperative car-following models

Stability analyses have been widely used to better understand the mechanism of traffic jam formation. In this paper, we consider the impact of cooperative systems (a.k.a. connected vehicles) on traffic dynamics and, more precisely, on flow stability. Cooperative systems are emerging technologies enabling communication between vehicles and/or with the infrastructure. In a distributed communication framework, equipped vehicles are able to send and receive information to/from other equipped vehicles. Here, the effects of cooperative traffic are modeled through a general bilateral multianticipative car-following law that improves cooperative drivers' perception of their surrounding traffic conditions within a given communication range. Linear stability analyses are performed for a broad class of car-following models. They point out different stability conditions in both multianticipative and nonmultianticipative situations. To better understand what happens in unstable conditions, information on the shock wave structure is studied in the weakly nonlinear regime by the mean of the reductive perturbation method. The shock wave equation is obtained for generic car-following models by deriving the Korteweg de Vries equations. We then derive traffic-state-dependent conditions for the sign of the solitary wave (soliton) amplitude. This analytical result is verified through simulations. Simulation results confirm the validity of the speed estimate. The variation of the soliton amplitude as a function of the communication range is provided. The performed linear and weakly nonlinear analyses help justify the potential benefits of vehicle-integrated communication systems and provide new insights supporting the future implementation of cooperative systems.