There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Convex Analysisの二,三の進展について

Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Holder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.

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Finite-Time Stability of Continuous Autonomous Systems

Two types of nonlinear control algorithms are presented for uncertain linear plants. Controllers of the first type are stabilizing polynomial feedbacks that allow to adjust a guaranteed convergence time of system trajectories into a prespecified neighborhood of the origin independently on initial conditions. The control design procedure uses block control principles and finite-time attractivity properties of polynomial feedbacks. Controllers of the second type are modifications of the second order sliding mode control algorithms. They provide global finite-time stability of the closed-loop system and allow to adjust a guaranteed settling time independently on initial conditions. Control algorithms are presented for both single-input and multi-input systems. Theoretical results are supported by numerical simulations.

https://hal.inria.fr/hal-00757561/document

Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control Systems

This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.

Coverage control for mobile sensing networks

Nonlinear controllability theory is applied to the time-varying attitude dynamics of a magnetically actuated spacecraft in a Keplerian orbit in the geomagnetic field. First, sufficient conditions for accessibility, strong accessibility and controllability of a general time-varying system are presented. These conditions involve application of Lie-algebraic rank conditions to the autonomous extended system obtained by augmenting the state of the original time-varying system by the time variable, and require the rank conditions to be checked only on the complement of a finite union of level sets of a finite number of smooth functions. At each point of each level set, it is sufficient to verify escape conditions involving Lie derivatives of the functions defining the level sets along linear combinations over smooth functions of vector fields in the accessibility algebra. These sufficient conditions are used to show that the attitude dynamics of a spacecraft actuated by three magnetic actuators and subjected to a general time-varying magnetic field are strongly accessible if the magnetic field and its time derivative are linearly independent at every instant. In addition, if the magnetic field is periodic in time, then the attitude dynamics of the spacecraft are controllable. These results are used to show that the attitude dynamics of a spacecraft actuated by three magnetic actuators in a closed Keplerian orbit in a nonrotating dipole approximation of the geomagnetic field are strongly accessible and controllable if the orbital plane does not coincide with the geomagnetic equatorial plane.

Controllability of nonlinear time-varying systems: applications to spacecraft attitude control using magnetic actuation

This technical note describes an application of nonlinear controllability theory to the problem of spacecraft attitude control using control moment gyroscopes (CMGs). Nonlinear controllability theory is used to show that a spacecraft carrying one or more CMGs is controllable on every angular momentum level set in spite of the presence of singular CMG configurations, that is, given any two states having the same angular momentum, any one of them can be reached from the other using suitably chosen motions of the CMG gimbals. This result is used to obtain sufficient conditions on the momentum volume of the CMG array that guarantee the existence of gimbal motions which steer the spacecraft to a desired spin state or rest attitude.

Controllability of spacecraft attitude using control moment gyroscopes

Necessary and sufficient conditions in terms of Lyapunov functions are derived for the finite-time stability of equilibria of systems of differential equations with continuous but non-Lipschitzian right hand sides.

Lyapunov analysis of finite-time differential equations

Semistability is the property whereby the solutions of a system converge to stable equilibrium points determined by the initial conditions. Important applications of this notion of stability include lateral aircraft dynamics and the dynamics of chemical reactions. A notion central to semistability theory is that of convergence in which every solution converges to a limit point that may depend upon the initial condition. We give sufficient conditions for convergence and semistability of nonlinear systems. By way of illustration, we apply these results to study the semistability of linear systems and some nonlinear systems.

Lyapunov analysis of semistability

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

Sanjay P. Bhat

Papers

Controllability of nonlinear time-varying systems: applications to spacecraft attitude control using magnetic actuation

Controllability of spacecraft attitude using control moment gyroscopes

Lyapunov analysis of finite-time differential equations

Lyapunov analysis of semistability

Lyapunov stability, semistability, and asymptotic stability of matrix second-order systems