Switching between periodic orbits in impact oscillator by time-delayed feedback methods

Acknowledgements The authors also acknowledge the financial support from Coordenação de Aperfeiçoamento do Pessoal de Nivel Superior (CAPES), under the Grant Number 88881.189487/2018-01 and FAPERJ.Peer reviewedPublisher PD

Finding unstable periodic orbits: A hybrid approach with polynomial optimization

We present a novel method to compute unstable periodic orbits (UPOs) that optimize the infinite-time average of a given quantity for polynomial ODE systems. The UPO search procedure relies on polynomial optimization to construct nonnegative polynomials whose sublevel sets approximately localize parts of the optimal UPO, and that can be used to implement a simple yet effective control strategy to reduce the UPO's instability. Precisely, we construct a family of controlled ODE systems, parameterized by a scalar k, such that the original ODE system is recovered for k=0 and such that the optimal orbit is less unstable, or even stabilized, for k>0. Periodic orbits for the controlled system can often be more easily converged with traditional methods, and numerical continuation in k allows one to recover optimal UPOs for the original system. The effectiveness of this approach is illustrated on three low-dimensional ODE systems with chaotic dynamics

Adaptive Detection of Instabilities: An Experimental Feasibility Study

We present an example of the practical implementation of a protocol for experimental bifurcation detection based on on-line identification and feedback control ideas. The idea is to couple the experiment with an on-line computer-assisted identification/feedback protocol so that the closed-loop system will converge to the open-loop bifurcation points. We demonstrate the applicability of this instability detection method by real-time, computer-assisted detection of period doubling bifurcations of an electronic circuit; the circuit implements an analog realization of the Roessler system. The method succeeds in locating the bifurcation points even in the presence of modest experimental uncertainties, noise and limited resolution. The results presented here include bifurcation detection experiments that rely on measurements of a single state variable and delay-based phase space reconstruction, as well as an example of tracing entire segments of a codimension-1 bifurcation boundary in two parameter space.Comment: 29 pages, Latex 2.09, 10 figures in encapsulated postscript format (eps), need psfig macro to include them. Submitted to Physica

On The Dynamics and Control Strategy of Time-Delayed Vibro-Impact Oscillators

Being able to control nonlinear oscillators, which are ubiquitous, has significant engineering implications in process development and product sustainability design. The fundamental characteristics of a vibro-impact oscillator, a non-autonomous time-delayed feedback oscillator, and a time-delayed vibro-impact oscillator are studied. Their being stochastic, nonstationary, non-smooth, and dynamically complex render the mitigation of their behaviors in response to linear and stationary inputs very difficult if not entirely impossible. A novel nonlinear control concept featuring simultaneous control of vibration amplitude in the time-domain and spectral response in the frequency-domain is developed and subsequently incorporated to maintain dynamic stability in these nonlinear oscillators by denying bifurcation and route-to-chaos from coming to pass. Convergence of the controller is formulated to be inherently unconditional with the optimization step size being self-adaptive to system identification and control force input. Optimal initial filter weights are also derived to warrant fast convergence rate and short response time. These novel features impart adaptivity, intelligence, and universal applicability to the wavelet based nonlinear time-frequency control methodology. The validity of the controller design is demonstrated by evaluating its performance against PID and fuzzy logic controllers in controlling the aperiodic, broad bandwidth, discontinuous responses characteristic of the time-delayed, vibro-impact oscillator

Modeling diversity by strange attractors with application to temporal pattern recognition

This thesis belongs to the general discipline of establishing black-box models from real-word data, more precisely, from measured time-series. This is an old subject and a large amount of papers and books has been written about it. The main difficulty is to express the diversity of data that has essentially the same origin without creating confusion with data that has a different origin. Normally, the diversity of time-series is modeled by a stochastic process, such as filtered white noise. Often, it is reasonable to assume that the time series is generated by a deterministic dynamical system rather than a stochastic process. In this case, the diversity of the data is expressed by the variability of the parameters of the dynamical system. The parameter variability itself is then, once again, modeled by a stochastic process. In both cases the diversity is generated by some form of exogenous noise. In this thesis a further step has been taken. A single chaotic dynamical system is used to model the data and their diversity. Indeed, a chaotic system produces a whole family of trajectories that are different but nonetheless very similar. It is believed that chaotic dynamics not only are a convenient means to represent diversity but that in many cases the origin of diversity stems actually from chaotic dynamic. Since the approach of this thesis explores completely new grounds the most suitable kind of data is considered, namely approximately periodic signals. In nature such time-series are rather common, in particular the physiological signal of living beings, such as the electrocardiograms (ECG), parts of speech signals, electroencephalograms (EEG), etc. Since there are strong arguments in favor of the chaotic nature of these signals, they appear to be the best candidates for modeling diversity by chaos. It should be stressed however, that the modeling approach pursued in this thesis is thought to be quite general and not limited to signals produced by chaotic dynamics in nature. The intended application of the modeling effort in this thesis is temporal signal classification. The reason for this is twofold. Firstly, classification is one of the basic building block of any cognitive system. Secondly, the recently studied phenomenon of synchronization of chaotic systems suggests a way to test a signal against its chaotic model. The essential content of this work can now be formulated as follows. Thesis: The diversity of approximately periodic signals found in nature can be modeled by means of chaotic dynamics. This kind of modeling technique, together with selective properties of the synchronization of chaotic systems, can be exploited for pattern recognition purposes. This Thesis is advocated by means of the following five points. Models of randomness (Chapter 2) It is argued that the randomness observed in nature is not necessarily the result of exogenous noise, but it could be endogenally generated by deterministic chaotic dynamics. The diversity of real signals is compared with signals produced by the most common chaotic systems. Qualitative resonance (Chapter 3) The behavior of chaotic systems forced by periodic or approximately periodic input signals is studied theoretically and by numerical simulation. It is observed that the chaotic system "locks" approximately to an input signal that is related to its internal chaotic dynamic. In contrast to this, its chaotic behavior is reinforced when the input signal has nothing to do with its internal dynamics. This new phenomenon is called "qualitative resonance". Modeling and recognizing (Chapter 4) In this chapter qualitative resonance is used for pattern recognition. The core of the method is a chaotic dynamical system that is able to reproduce the class of time-series that is to be recognized. This model is excited in a suitable way by an input signal such that qualitative resonance is realized. This means that if the input signal belongs to the modeled class of time-series, the system approximately "locks" into it. If not, the trajectory of the system and the input signal remain unrelated. Automated design of the recognizer (Chapters 5 and 6) For the kind of signals considered in this thesis a systematic design method of the recognizer is presented. The model used is a system of Lur'e type, i.e. a model where the linear dynamic and nonlinear static part are separated. The identification of the model parameters from the given data proceed iteratively, adapting in turn the linear and the nonlinear part. Thus, the difficult nonlinear dynamical system identification task is decomposed into the easier problems of linear dynamical and nonlinear static system identification. The way to apply the approximately periodic input signal in order to realize qualitative resonance is chosen with the help of periodic control theory. Validation (Chapter 7) The pattern recognition method has been validated on the following examples — A synthetic example — Laboratory measurement from Colpitts oscillator — ECG — EEG — Vowels of a speech signals In the first four cases a binary classification and in the last example a classification with five classes was performed. To the best of the knowledge of the author the recognition method is original. Chaotic systems have been already used to produce pseudo-noise and to model signal diversity. Also, parameter identification of chaotic systems has been already carried out. However, the direct establishment of the model from the given data and its subsequent use for classification based on the phenomenon of qualitative resonance is entirely new

Hopf Bifurcations In A Power System Susceptible To Subsynchronous Resonance And A Novel Controller For Damping Torsional Oscillations

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2009Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2009Bu çalışmada, bir elektrik güç sistemindeki burulma salınımlarının analizi için çatallanma teorisinden yararlanılmıştır. Senkronaltı rezonans araştırmaları için geliştirilen IEEE İkinci Gösterge Modelinin birinci sistemi kullanılmıştır. Senkron makinenin amortisör sargıları doğrusal olmayan modele dahil edilmiştir. Seri kompanzasyon kapasitör tesis edilmiş olan iletim hatlarına bağlı senkron makineler, potansiyel olarak senkronaltı elektrik modunun, türbin-generatör şaft sisteminin burulma salınım modları ile etkileşimine maruz kalabilirler. Bu olay senkronaltı rezonans (SSR) olarak isimlendirilir. Modellenen elektrik güç sisteminde meydana gelen Hopf çatallanmalarının hangi tip olduğu, literatürde yaygın biçimde kullanılan Floquet çarpanları yöntemi yerine, birinci Lyapunov katsayılarının analitik olarak hesaplanması ile belirlenmiştir. Mekanik tork değeri, şebeke gerilim seviyesi ve uyartı geriliminin Hopf çatallanma noktaları ile birinci Lyapunov katsayısının değeri üzerindeki etkileri araştırılmıştır. Ek olarak, Otomatik Gerilim Düzenleyicisinin (AVR) Hopf çatallanması üzerindeki etkisi de incelenmiştir. Ayrıca, SSR sonucu ortaya çıkan kararsız burulma salınımlarının kararlı hale getirilmesi için, zaman gecikmeli geri besleme teorisine dayanan bir kontrolör tasarlanmıştır. Girdi olarak sadece senkron makine rotorunun açısal hızını kullanan kontrolörün zaman gecikme ve kazanç parametreleri için uygun değerler, sistemin dinamik cevabını değerlendiren bir performans endeksi hesaplanarak belirlenmiştir. Önerilen kontrolörün ektili sonuçlar verdiği, MATLAB-Simulink kullanılarak yapılan simülasyonlar ile gösterilmiştir. Son olarak, AVR ve kontrolör çıkış sınırlayıcılarının kararlı hale getirme performansı üzerindeki etkileri de araştırılmış ve seri kapasitör kompanzasyonun pratik değerleri için, sınırlayıcıların mevcut olduğu durumda da kontrolörün etkili olduğu gösterilmiştir.In this study, Bifurcation theory is employed for the analysis of torsional oscillations in a power system. The first system of the IEEE Second Benchmark Model for Subsynchronous Resonance studies has been used. Damper windings of the synchronous generator are included in the nonlinear model. Synchronous generators connected to transmission lines with series capacitor compensation are potentially subject to the interaction between the subsynchronous electrical mode and torsional oscillation modes of the turbine generator shaft system. This phenomenon is called Subsynchronous Resonance (SSR). Instead of employing the Floquet multipliers method reported in the literature, the first Lyapunov coefficients are computed analytically to determine the type of Hopf bifurcation existing in the power system under study. The impact of mechanical torque input, network voltage level and field voltage on the Hopf bifurcation point and the first Lyapunov coefficient is also explored. Moreover, an automatic voltage regulator (AVR) is included into the model. In addition, a novel controller based on the delayed feedback control theory has been developed in order to stabilize unstable torsional oscillations caused by SSR. The proposed Time Delay Auto-synchronization controller has two set parameters to be optimized and uses the synchronous generator rotor angular speed as the only input. The optimal values of the controller time delay and gain parameters have been determined by computing a performance index evaluating the dynamic responses in time domain. The effectiveness of the proposed controller is demonstrated via time-domain simulations in MATLAB-Simulink. Finally, the impact of AVR and TDAS controller limiters on the stabilizing performance is also investigated. It is demonstrated that the controller is effective even in the presence of limiters within the practical operating ranges of series capacitor compensation.DoktoraPh

Nonlinear Response and Bifurcations Analysis of Rotor-Fluid Film Bearing Systems

Nonlinear response, bifurcations and stability of rotor-fluid film bearing systems are studied using various numerical investigation schemes such as autonomous/non-autonomous shooting, arc-length continuation, direct numerical integrations, Poincaré sections, Lyapunov exponents, etc. Two types of hydrodynamic bearings, a floating ring bearing (FRB) and a tilting pad journal bearing (TPJB), are employed in this study. The nonlinear characteristic of each bearing is analyzed as supports of a rigid rotor system as well as a flexible rotor system. Depending on the existence of the unbalance force on the rotor/disks, autonomous (free vibration) and non-autonomous responses (mass unbalanced excitation) are both identified, and the nonlinear reaction force produced on the lubricant layer is obtained using the finite element method. In addition to isoviscosity lubricants, thermo-hydrodynamic lubricant model is developed to investigate thermal effects on rotordynamic bifurcations; in the procedure, a variable viscosity Reynolds equation and the energy equation are solved simultaneously. For computation efficiency in the analytical bifurcation study, an advanced shooting algorithm, which is combined with the deflation theory and the parallel computing strategy, is proposed for both the autonomous and the non-autonomous cases. In the study with flexible rotors, the finite element based beam models are employed and the model reduction technique such as Component Mode Synthesis is utilized to condense the system degree of freedom. This dissertation consists of four main discussions regarding: 1) nonlinear response and bifurcations of a rigid rotor supported by FRBs; 2) effects of a thermo-hydrodynamic (THD) FRB model on rotordynamic bifurcations; 3) nonlinear response and bifurcations of a rigid rotor supported by TPJBs; 4) extension of study to general, complex, multi-mass rotor beam models. In case 1), multiple coexistent solutions and bifurcation scenarios are identified, and those are depended on the ratio of floating ring length to diameter (L/D). Numerical illustrations regarding jumps between two stable limit cycles and quenching large vibrations are demonstrated, and chaos is investigated with the aid of Lyanpunov exponent. In case 2), the Hopf bifurcation onset is strongly dependent on thermal conditions, and the saddle node bifurcation points are significantly shifted compared to the isothermal model. In addition, the unbalanced responses stability and bifurcation onsets are highly reliant on the lubricant supply temperature. In case 3), loci of bifurcations are identified, and heavily loaded bearings and/or high unbalance force may induce consecutive transference of response in forms of synchronous to sub-synchronous, quasi-periodic responses and chaotic motions. The periodic doubling bifurcations, saddle node bifurcations and corresponding local stability are reliably determined by selections of pad preload, pivot offset, and lubricant viscosity sets. In case 4), two industrial applications such as a turbocharger supported by FRBs and an eight-stage centrifugal compressor supported by TPJBs are numerically analyzed. The turbocharger shows that torus appears with Neimark-Sacker bifurcation events and the motions are dominant in the high speed ranges (>60,000rpm). In the compressor, sub-/super-synchronous motions are identified other than the ×1 synchronous response, and the appearance of each harmonic is highly depended on the selection of pad preload and pivot offset