Practical stability and boundedness criteria of impulsive differential system with initial time difference

In this paper, an impulsive differential system is investigated for the first time for practical stability and boundedness criteria with respect to initial time difference. The investigations are carried out by perturbing Lyapunov functions and by using comparison results. A generalized Lyapunov function has been used for the investigation. The present results indicate that the stability criteria significantly depend on the moment of impulses.Publisher's Versio

Stability in Terms of Two Measures for Nonlinear Impulsive Systems on Time Scales

We investigate some stability problems in terms of two measures for nonlinear dynamic systems on time scales with fixed moments of impulsive effects. Sufficient conditions for (uniform) stability, (uniform) asymptotic stability, and instability in terms of two measures are derived by using the method of Lyapunov functions. Our results include the existing results as special cases when the time scale reduces to the set of real numbers. Particularly, our results provide stability criteria for impulsive discrete systems in terms of two measures, which have not been investigated extensively. Two examples are presented to illustrate the efficiency of the proposed results

Stability in Terms of Two Measures for Nonlinear Impulsive Systems on Time Scales

We investigate some stability problems in terms of two measures for nonlinear dynamic systems on time scales with fixed moments of impulsive effects. Sufficient conditions for (uniform) stability, (uniform) asymptotic stability, and instability in terms of two measures are derived by using the method of Lyapunov functions. Our results include the existing results as special cases when the time scale reduces to the set of real numbers. Particularly, our results provide stability criteria for impulsive discrete systems in terms of two measures, which have not been investigated extensively. Two examples are presented to illustrate the efficiency of the proposed results

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

Minimum-Fuel Trajectory Design in Multiple Dynamical Environments Utilizing Direct Transcription Methods and Particle Swarm Optimization

Particle swarm optimization is used to generate an initial guess for designing fuel-optimal trajectories in multiple dynamical environments. Trajectories designed in the vicinity of Earth use continuous or finite low-thrust burning and transfer from an inclined or equatorial circular low-Earth-orbit to a geostationary orbit. In addition, a trajectory from near-Earth to a periodic orbit about the cislunar Lagrange point with minimized impulsive burn costs is designed within a multi-body dynamical environment. Direct transcription is used in conjunction with a nonlinear optimizer to find locally-optimal trajectories given the initial guess. The near-Earth transfers are propagated at low-level thrust where neither the very-low-thrust spiral solution nor the impulsive transfer is an acceptable starting point. The very-high-altitude transfer is designed in a multi-body dynamical environment lacking a closed-form analytical solution. Swarming algorithms excel given a small number of design parameters.When continuous control time histories are needed, employing a polynomial parameterization facilitates the generation of feasible solutions. For design in a circular restricted three-body system, particle swarm optimization gains utility due to a more global search for the solution, but may be more sensitive to boundary constraints. Computation time and constraint weighting are areas where a swarming algorithm is weaker than other approaches

Stretching Directions in Cislunar Space: Stationkeeping and an Application to Transfer Trajectory Design

The orbits of interest for potential missions are stable or nearly stable to maintain long-term presence for conducting scientific studies and to reduce the possibility of rapid departure. Near Rectilinear Halo Orbits (NRHOs) offer such stable or nearly stable orbits that are defined as part of the L1 and L2 halo orbit families in the circular restricted three-body problem. Within the Earth-Moon regime, the L1 and L2 NRHOs are proposed as long-horizon trajectories for cislunar exploration missions, including NASA's upcoming Gateway mission. These stable or nearly stable orbits do not possess well-distinguished unstable and stable manifold structures. As a consequence, existing tools for stationkeeping and transfer trajectory design that exploit such underlying manifold structures are not reliable for orbits that are linearly stable. The current investigation focuses on leveraging stretching direction as an alternative for visualizing the flow of perturbations in the neighborhood of a reference trajectory. The information supplemented by the stretching directions are utilized to investigate the impact of maneuvers for two contrasting applications; the stationkeeping problem, where the goal is to maintain a spacecraft near a reference trajectory for a long period of time, and the transfer trajectory design application, where rapid departure and/or insertion is of concern. Particularly, for the stationkeeping problem, a spacecraft incurs continuous deviations due to unmodeled forces and orbit determination errors in the complex multi-body dynamical regime. The flow dynamics in the region, using stretching directions, are utilized to identify appropriate maneuver and target locations to support a long lasting presence for the spacecraft near the desired path. The investigation reflects the impact of various factors on maneuver cost and boundedness. For orbits that are particularly sensitive to epoch time and possess distinct characteristics in the higher-fidelity ephemeris model compared to their CR3BP counterpart, an additional feedback control is applied for appropriate phasing. The effect of constraining maneuvers in a particular direction is also investigated for the 9:2 synodic resonant southern L2 NRHO, the current baseline for the Gateway mission. The stationkeeping strategy is applied to a range of L1 and L2 NRHOs, and validated in the higher-fidelity ephemeris model. For missions with potential human presence, a rapid transfer between orbits of interest is a priority. The magnitude of the state variations along the maximum stretching direction is expected to grow rapidly and, therefore, offers information to depart from the orbit. Similarly, the maximum stretching in reverse time, enables arrival with a minimal maneuver magnitude. The impact of maneuvers in such sensitive directions is investigated. Further, enabling transfer design options to connect between two stable orbits. The transfer design strategy developed in this investigation is not restricted to a particular orbit but applicable to a broad range of stable and nearly stable orbits in the cislunar space, including the Distant Retrograde Orbit (DROs) and the Lunar Orbits (LO) that are considered for potential missions. Examples for transfers linking a southern and a northern NRHO, a southern NRHO to a planar DRO, and a southern NRHO to a planar LO are demonstrated