Dynamical problems and phase transitions

Issued as Financial status report, Technical reports [nos. 1-12], and Final report, Project B-06-68

Control of chaos in nonlinear circuits and systems

Nonlinear circuits and systems, such as electronic circuits (Chapter 5), power converters (Chapter 6), human brains (Chapter 7), phase lock loops (Chapter 8), sigma delta modulators (Chapter 9), etc, are found almost everywhere. Understanding nonlinear behaviours as well as control of these circuits and systems are important for real practical engineering applications. Control theories for linear circuits and systems are well developed and almost complete. However, different nonlinear circuits and systems could exhibit very different behaviours. Hence, it is difficult to unify a general control theory for general nonlinear circuits and systems. Up to now, control theories for nonlinear circuits and systems are still very limited. The objective of this book is to review the state of the art chaos control methods for some common nonlinear circuits and systems, such as those listed in the above, and stimulate further research and development in chaos control for nonlinear circuits and systems. This book consists of three parts. The first part of the book consists of reviews on general chaos control methods. In particular, a time-delayed approach written by H. Huang and G. Feng is reviewed in Chapter 1. A master slave synchronization problem for chaotic Lur’e systems is considered. A delay independent and delay dependent synchronization criteria are derived based on the H performance. The design of the time delayed feedback controller can be accomplished by means of the feasibility of linear matrix inequalities. In Chapter 2, a fuzzy model based approach written by H.K. Lam and F.H.F. Leung is reviewed. The synchronization of chaotic systems subject to parameter uncertainties is considered. A chaotic system is first represented by the fuzzy model. A switching controller is then employed to synchronize the systems. The stability conditions in terms of linear matrix inequalities are derived based on the Lyapunov stability theory. The tracking performance and parameter design of the controller are formulated as a generalized eigenvalue minimization problem which is solved numerically via some convex programming techniques. In Chapter 3, a sliding mode control approach written by Y. Feng and X. Yu is reviewed. Three kinds of sliding mode control methods, traditional sliding mode control, terminal sliding mode control and non-singular terminal sliding mode control, are employed for the control of a chaotic system to realize two different control objectives, namely to force the system states to converge to zero or to track desired trajectories. Observer based chaos synchronizations for chaotic systems with single nonlinearity and multi-nonlinearities are also presented. In Chapter 4, an optimal control approach written by C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao is reviewed. Systems with nonparametric regression with jump points are considered. The rough locations of all the possible jump points are identified using existing kernel methods. A smooth spline function is used to approximate each segment of the regression function. A time scaling transformation is derived so as to map the undecided jump points to fixed points. The approximation problem is formulated as an optimization problem and solved via existing optimization tools. The second part of the book consists of reviews on general chaos controls for continuous-time systems. In particular, chaos controls for Chua’s circuits written by L.A.B. Tôrres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes are discussed in Chapter 5. An inductorless Chua’s circuit realization is presented, as well as some practical issues, such as data analysis, mathematical modelling and dynamical characterization, are discussed. The tradeoff among the control objective, the control energy and the model complexity is derived. In Chapter 6, chaos controls for pulse width modulation current mode single phase H-bridge inverters written by B. Robert, M. Feki and H.H.C. Iu are discussed. A time delayed feedback controller is used in conjunction with the proportional controller in its simple form as well as in its extended form to stabilize the desired periodic orbit for larger values of the proportional controller gain. This method is very robust and easy to implement. In Chapter 7, chaos controls for epileptiform bursting in the brain written by M.W. Slutzky, P. Cvitanovic and D.J. Mogul are discussed. Chaos analysis and chaos control algorithms for manipulating the seizure like behaviour in a brain slice model are discussed. The techniques provide a nonlinear control pathway for terminating or potentially preventing epileptic seizures in the whole brain. The third part of the book consists of reviews on general chaos controls for discrete-time systems. In particular, chaos controls for phase lock loops written by A.M. Harb and B.A. Harb are discussed in Chapter 8. A nonlinear controller based on the theory of backstepping is designed so that the phase lock loops will not be out of lock. Also, the phase lock loops will not exhibit Hopf bifurcation and chaotic behaviours. In Chapter 9, chaos controls for sigma delta modulators written by B.W.K. Ling, C.Y.F. Ho and J.D. Reiss are discussed. A fuzzy impulsive control approach is employed for the control of the sigma delta modulators. The local stability criterion and the condition for the occurrence of limit cycle behaviours are derived. Based on the derived conditions, a fuzzy impulsive control law is formulated so that the occurrence of the limit cycle behaviours, the effect of the audio clicks and the distance between the state vectors and an invariant set are minimized supposing that the invariant set is nonempty. The state vectors can be bounded within any arbitrary nonempty region no matter what the input step size, the initial condition and the filter parameters are. The editors are much indebted to the editor of the World Scientific Series on Nonlinear Science, Prof. Leon Chua, and to Senior Editor Miss Lakshmi Narayan for their help and congenial processing of the edition

Cell Modeling

The Air Force is currently developing new products that incorporate a variety of chemicals which may come in contact with product users. To define which chemicals are dangerous to the user, toxicity studies have been performed. However, analysis of toxicity ultimately requires models of the exposed cellular systems. This thesis provides an introduction of how to model and analyze small and large cellular systems. Understanding the underlying behavior of small models and their relation to large systems will lead to a better understanding of how the Air Force should construct intracellular models to assist in future toxicology studies. Developing analysis techniques to include steady state analysis through linearization, and then considering small reaction systems, such as the Brusselator and Schnackenberg models, led to a basic understanding of model behavior. This knowledge was applied to create new models in an effort to begin a transition from previously created models to the construction of models unique to the Air Force. Sensitivity analyses performed on existing systems furthered research efforts by developing knowledge of how systems behave under various initial conditions and perturbations of uncertain constant parameters. Analysis displayed great sensitivity within some models. This analysis was applied to a new model to look for interesting behavior such as oscillatory convergence. The new model was then incorporated into a larger model to determine how its behavior changed with respect to changes in the larger model. This knowledge of how small systems behave in relation to larger systems should help the Air Force to develop and analyze intracellular toxicology models

Stability analysis of the boiling water reactor : methods and advanced designs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Nuclear Science and Engineering, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 278-285).Density Wave Oscillations (DWOs) are known to be possible when a coolant undergoes considerable density reduction while passing through a heated channel. In the development of boiling water reactors (BWRs), there has been considerable concern about the effects of such oscillations when coupled with neutronic feedback. The current trend of increasing reactor power density and relying more extensively on natural circulation for core cooling may have consequences for the stability characteristics of new BWR designs. This work addresses a wide range of issues associated with the BWR stability: 1) flashing-induced instability and natural circulation BWR startup; 2) stability of the BWRs with advanced designs involving high power :densities; 3) modeling assumptions in stability analysis methods; and 4) the fuel clad performance during power and flow oscillations. To capture the effect of flashing on density wave oscillations during low pressure startup conditions, a code named FISTAB has been developed in the frequency domain. The code is based on a single channel thermal-hydraulic model of the balance of the water/steam circulation loop, and incorporates the pressure dependent water/steam thermodynamic properties, from which the evaporation due to flashing is captured. The functionality of the FISTAB code is confirmed by testing the experimental results at SIRIUS-N facility. Both stationary and perturbation results agree well with the experimental results. The proposed ESBWR start-up procedure under natural convection conditions has been examined by the FISTAB code. It is confirmed that the examined operating points along the ESBWR start-up trajectory from TRACG simulation will be stable. To avoid the instability resulting from the transition from single-phase natural circulation to two-phase circulation, a simple criterion is proposed for the natural convection BWR start-up when the steam dome pressure is still low. Using the frequency domain code STAB developed at MIT, stability analyses of some proposed advanced BWRs have been conducted, including the high power density BWR core designs using the Large Assembly with Small Pins (LASP) or Cross Shape Twisted (CST) fuel designs developed at MIT, and the Hitachi's RBWR cores utilizing a hard neutron spectrum and even higher power density cores. The STAB code is the predecessor of the FISTAB code, and thermodynamic properties of the coolant are only dependent on system pressure in STAB. It is concluded that good stability performance of the LASP core and the CST core can be maintained at nominal conditions, even though they have 20% higher reactor thermal power than the reference core. Power uprate does not seem to have significant effects on thermal-hydraulic stability performance when the power-to-flow ratio is maintained. Also, both the RBWR-AC and RBWR-TB2 designs are found viable from a stability performance point of view, even though the core exit qualities are almost 3 times those of a traditional BWR. The stability of the RBWRs is enhanced through the fast transient response of the shorter core, more flat power and power-to-flow ratio distributions, less negative void feedback coefficient, and the core inlet orifice design. To examine the capability of coupled 3D thermal-hydraulics and neutronics codes for stability analysis, USNRC's latest system analysis code, TRACE, is chosen in this work. Its validation for stability analysis and comparison with the frequency domain approach, have been performed against the Ringhals 1 stability tests. Comprehensive assessment of modeling choices on TRACE stability analysis has been made, including effects of timespatial discretization, numerical schemes, thermal-hydraulic channel grouping, neutronics modeling, and control system modeling. The predictions from both the TRACE and STAB codes are found in reasonably good agreement with the Ringhals 1 test results. The biases for the predicted global decay ratio are about 0.07 in TRACE results, and -0.04 in STAB results. However, the standard deviations of decay ratios are both large, around 0.1, indicating large uncertainties in both analyses. Although the TRACE code uses more sophisticated neutronic and thermal hydraulic models, the modeling uncertainty is not less than that of the STAB code. The benchmark results of both codes for the Ringhals stability test are at the same level of accuracy. The fuel cladding integrity during power oscillations without reactor scram is examined by using the FRAPTRAN code, with consideration of both the stress-strain criterion and thermal fatigue. Under the assumed power oscillation conditions for high burn-up fuel, the cladding can satisfy the stress-strain criteria in the ASME Code. Also, the equivalent alternating stress is below the fatigue threshold stress, thus the fatigue limit is not violated. It can be concluded that under a large amount of the undamped power oscillation cycles, the cladding would not fail, and the fuel integrity is not compromised.by Rui Hu.Ph.D