Dirk Aeyels

Stabilization of a class of nonlinear systems by a smooth feedback control

We consider the stabilizability problem for systems of the form x = f(x)+ ub in the neighborhood of an equilibrium point of f(x). First, by means of center manifold theory, a lower order system is introduced. If this system is stabilizable, then so is the original system. Results on the stabilization of the low order system are presented. No resort is taken to Liapunov theory. The relation between stabilizability and controllability is discussed.

A new asymptotic stability criterion for nonlinear time-variant differential equations

A new sufficient condition for asymptotic stability of ordinary differential equations is proposed. Unlike classical Lyapunov theory, the time derivative along solutions of the Lyapunov function may have positive and negative values. The classical Lyapunov approach may be regarded as an infinitesimal version of the present theorem. Verification in practical problems is harder than in the classical case; an example is included in order to indicate how the present theorem may be applied.

GENERIC OBSERVABILITY OF DIFFERENTIABLE SYSTEMS

A dynamical system consists of a smooth vectorfield defined on a differentiable manifold, and a smooth mapping from the manifold to the real numbers. The vectorfield represents the dynamics of a physical system. The mapping stands for a measuring device by which experimental information on the dynamics is made available. The information itself is modeled as a sampled version of the image of the state trajectory under the smooth mapping. In this paper the observability of this set-up is discussed from the viewpoint of genericity. First the observability property is expressed in terms of transversality conditions. Then the theory of transversal intersection is called upon to yield the desired results. It is shown that almost any measuring device will combine with a given physical system to form an observable dynamical system, if

Supremum preserving upper probabilities

(2n + 1)

Stabilization by smooth feedback of the angular velocity of a rigid body

samples are taken and not fewer, where n is the dimension of the manifold. Dually, it is shown that almost any physical system will combine with a given measuring device ...

Pole assignment for linear time-invariant systems by periodic memoryless output feedback

Abstract We study the relation between possibility measures and the theory of imprecise probabilities, and argue that possibility measures have an important part in this theory. It is shown that a possibility measure is a coherent upper probability if and only if it is normal. A detailed comparison is given between the possibilistic and natural extension of an upper probability, both in the general case and for upper probabilities defined on a class of nested sets. We prove in particular that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability, defined on a class of nested sets. We show that possibilistic extension can be interpreted in terms of natural extension. We also prove that when either the upper or the lower cumulative distribution function of a random quantity is specified, possibility measures very naturally emerge as the corresponding natural extensions. Next, we go from upper probabilities to upper previsions. We show that if a coherent upper prevision defined on the convex cone of all non-negative gambles is supremum preserving, then it must take the form of a Shilkret integral associated with a possibility measure. But at the same time, we show that such a supremum preserving upper prevision is never coherent unless it is the vacuous upper prevision with respect to a non-empty subset of the universe of discourse.

Existence of Partial Entrainment and Stability of Phase Locking Behavior of Coupled Oscillators

Abstract Stabilization by smooth feedback of the null solution of Eulers angular velocity equations is investigated. Assume there is one external torque, aligned with a principal axis. The main result is that there exists a stabilizing smooth feedback law if the moment of inertia of the rigid body along that principal axis is either larger or smaller than the remaining two. The feedback law is robust with respect to changes in the parameters defining the control law.

Brief On exponential stability of nonlinear time-varying differential equations

It is well known that the poles of a linear time-invariant controllable and observable system can be assigned arbitrarily by state feedback. When only the output is available, pole assignment is still possible by means of dynamic output feedback. In this paper the potential of time-varying memoryless output feedback is considered. It is shown that, up to some technical conditions, it is indeed possible to allocate the poles of a linear time-invariant discrete-time system by memoryless output feedback with periodic gains. The period of the gains is (n + 1) with n the order of the system. The power of the design technique is proved to be comparable to what can be achieved by the classical dynamic feedback approach.

Practical stability and stabilization

We study a network of all-to-all interconnected phase oscillators as modeled by the Kuramoto model. For coupling strengths larger than a critical value, we show the existence of a collective behavior called phase locking: the phase differences between all oscillators are constant in time. As the coupling strength increases, the distance between each pair of phases decreases. Stability of each phase locking solution is proven for general frequency distributions. There exist one unique asymptotically stable phase locking solution. Furthermore a description is given of partial entrainment, which can be regarded as the finite number analogon of partial synchronization in the infinite number case. When the network is partially entraining some phase differences possess an upper and lower bound. Partial entrainment of the three-cell network is analyzed: an estimate of the onset of partial entrainment is given and the existence of partial entrainment is proven. Furthermore, local stability of partial entrainment is proven for the three-cell network with two identical oscillators.

Feature selection for splice site prediction: A new method using EDA-based feature ranking

Within the Liapunov framework, a sufficient condition for exponential stability of ordinary differential equations is proposed. Unlike with classical Liapunov theory, the time derivative of the Liapunov function, taken along solutions of the system, may have positive and negative values. Verification of the conditions of the main theorem may be harder than in the classical case. It is shown that the proposed conditions are useful for the investigation of the exponential stability of fast time-varying systems. This sets the stability study by means of averaging in a Liapunov context. In particular, it is established that exponential stability of the averaged system implies exponential stability of the original fast time-varying system. A comparison of our work with results taken from the literature is included.