Continuous Uniform Finite Time Stabilization of Planar Controllable Systems

Continuous homogeneous controllers are utilized in a full state feedback setting for the uniform finite time stabilization of a perturbed double integrator in the presence of uniformly decaying piecewise continuous disturbances. Semiglobal strong $\mathcal{C}^1$ Lyapunov functions are identified to establish uniform asymptotic stability of the closed-loop planar system. Uniform finite time stability is then proved by extending the homogeneity principle of discontinuous systems to the continuous case with uniformly decaying piecewise continuous nonhomogeneous disturbances. A finite upper bound on the settling time is also computed. The results extend the existing literature on homogeneity and finite time stability by both presenting uniform finite time stabilization and dealing with a broader class of nonhomogeneous disturbances for planar controllable systems while also proposing a new class of homogeneous continuous controllers

On the stabilization of persistently excited linear systems

We consider control systems of the type $\dot x = A x +\alpha(t)bu$, where $u\in\R$, $(A,b)$ is a controllable pair and $\alpha$ is an unknown time-varying signal with values in $[0,1]$ satisfying a persistent excitation condition i.e., \int_t^{t+T}\al(s)ds\geq \mu for every $t\geq 0$, with $0<\mu\leq T$ independent on $t$. We prove that such a system is stabilizable with a linear feedback depending only on the pair $(T,\mu)$ if the eigenvalues of $A$ have non-positive real part. We also show that stabilizability does not hold for arbitrary matrices $A$. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter $\mu/T$

Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases

Consider the controlled system $dx/dt = Ax + \alpha(t)Bu$ where the pair $(A,B)$ is stabilizable and $\alpha(t)$ takes values in $[0,1]$ and is persistently exciting, i.e., there exist two positive constants $\mu,T$ such that, for every $t\geq 0$, $\int_t^{t+T}\alpha(s)ds\geq \mu$. In particular, when $\alpha(t)$ becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback $u=Kx$, with $K$ only depending on $(A,B)$ and possibly on $\mu,T$, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when $A$ is neutrally stable and when the system is the double integrator

A characterization of switched linear control systems with finite L 2 -gain

Motivated by an open problem posed by J.P. Hespanha, we extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of the L2-gain is equivalent, for a large set of switched linear control systems, to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one

Flat systems, equivalence and trajectory generation

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of a Lie-Backlund mapping, we say that two systems are equivalent if they are related by a Lie-Backlund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft

Efficient Reorientation Maneuvers for Spacecraft with Multiple Articulated Payloads

A final report is provided which describes the research program during the period 3 Mar. 1992 to 3 Jun. 1993. A summary of the technical research questions that were studied and of the main results that were obtained is given. The specific outcomes of the research program, including both educational impacts as well as research publications, are listed. The research is concerned with efficient reorientation maneuvers for spacecraft with multiple articulated payloads

Predefined-time tracking of a class of mechanical systems

In this paper the problem of predefined-time exact tracking of fully actuated and unperturbed mechanical systems is solved by means of a continuous controller. It is assumed the availability of the state and the desired trajectory as well as its two first derivatives. This is accomplished introducing the idea of second-order predefined-time stable systems, which is based on the nested application of the first-order predefined-time stabilizing function. As an example, the proposed solution is applied over a two-link planar manipulator and numerical simulations are conducted to show its performance.ITESO, A.C.CINVESTA

Underwater locomotion from oscillatory shape deformations

This paper considers underwater propulsion that is generated by variations in body shape. We summarize and extend some of the emerging approaches for the uniform modeling and control of such underactuated systems. Two examples illustrate these ideas