Stability Verification of Nearly Periodic Impulsive Linear Systems using Reachability Analysis

International audienceThe paper provides stability analysis to certain classes of hybrid systems, more precisely impulsive linear systems. This analysis is conducted using the notion of reachable set. The main contribution in this work is the derivation of theoretical necessary and sufficient conditions for impulsive linear systems with nearly periodic resets subject to timing contracts. This characterization serves as the basis of a computational method for the stability verification of the considered class of systems. In addition, we show how this work handles the problem of timing contract synthesis for the considered class and we generalize our approach to verify stability of impulsive linear systems with stochastic reset instants. Applications on sampled-data control systems and comparisons with existing results are then discussed, showing the effectiveness of our approach

Spacecraft Trajectory Planning for Optimal Observability using Angles-Only Navigation

This work leverages existing techniques in angles-only navigation to develop optimal range observability maneuvers and trajectory planning methods for spacecraft under constrained relative motion. The resulting contribution is a guidance method for impulsive rendezvous and proximity operations valid for elliptic orbits of arbitrary eccentricity. The system dynamics describe the relative motion of an arbitrary number of maneuvering (chaser) spacecraft about a single non-cooperative resident-space-object (RSO). The chaser spacecraft motion is constrained in terms of the 1) collision bounds of the RSO, 2) maximum fuel usage, 3) eclipse avoidance, and 4) optical sensor field of view restrictions. When more than one chaser is present, additional constraints include 1) collision avoidance between formation members, and 2) formation longevity via fuel usage balancing. Depending on the type of planetary orbit, quasi-circular or elliptic, the relative motion dynamics are approximated using a linear time-invariant or a linear time-varying system, respectively. The proposed method uses two distinct parameterizations corresponding to each system type to reduce the optimization problem from 12 to 2 variables in Cartesian space, thus simplifying an otherwise intractable optimization problem

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

Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area

Invariant Measures, Geometry, and Control of Hybrid and Nonholonomic Dynamical Systems

Constraints are ubiquitous when studying mechanical systems and fall into two main categories: hybrid (1-sided, unilateral) and nonholonomic/holonomic (2-sided, bilateral) constraints. A hybrid constraint takes the form h(x)≥0. An example of a constraint of this nature is requiring a billiard ball to remain within the confines of a table-top. The notable feature of these constraints is that when the ball reaches the boundary of the table-top (i.e. when h(x)=0), an impact occurs; this is a discontinuous jump in the dynamics. Dynamical systems that have this phenomenon generally fall under the domain of hybrid dynamical systems. On the other hand, nonholonomic constraints take the form h(x)=0. Generally, h will depend on both the positions and velocities and cannot be integrated to only depend on the positions (when it can be integrated, the constraint is called holonomic). An example of a nonholonomic constraint is an ice skate: motion is not allowed perpendicular to the direction of the skate. It is common that these systems are studied using tools from differential geometry. This thesis studies both hybrid and nonholonomic constraints together using the language of differential (specifically symplectic) geometry. However, due to the exotic nature of hybrid dynamics, some auxiliary results are found that pertain to the asymptotic nature of these systems. These include the idea of a hybrid limit-set, Floquet theory, and a Poincaré-Bendixson theorem for planar systems. The bulk of this work focuses on finding (smooth) invariant measures for both nonholonomic and hybrid systems (as well as systems involving both types of constraints). Necessary and sufficient conditions are found which guarantee the existence of an invariant measure for nonholonomic systems in which the density depends only on the configuration variables. Extending this idea to hybrid nonholonomic systems requires that the impact preserves the measure as well. To build towards this, relatively simple conditions to test whether or not a differential form is hybrid-invariant are derived. In the cases where the density depends on only the configuration variables, the measure is still invariant under the hybrid dynamics independent of the choice of impacts. The billiard problem with a vertical rolling disk as the billiard ball is one such system and is therefore recurrent for any choice of compact table-top. This thesis concludes with optimal control of hybrid systems. First, Hamilton-Jacobi is extended to the hybrid setting (nonholonomic constraints are not considered here) and the idea of completely integrable hybrid systems is introduced. It is shown that the usual billiard problem on a circular table is completely integrable. Finally, the hybrid Hamilton-Jacobi theory is extended to a hybrid Hamilton-Jacobi-Bellman theory which allows for the study of optimal control problems.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163071/1/wiclark_1.pd