57,309 research outputs found

    Uniform Stability Analysis for Time-Varying Systems Applying Homogeneity

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    International audienceThe uniform stability notion for a class of non-linear time-varying systems is studied using the homogeneity framework. It is assumed that the system is weighted homogeneous considering the time variable as a constant parameter, then several conditions of uniform stability for such a class of systems are formulated. The results are applied to the problem of adaptive estimation for a linear system

    Uniform stability analysis for time-varying systems applying homogeneity

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    Continuous Uniform Finite Time Stabilization of Planar Controllable Systems

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    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 C1\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

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

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    Consider the controlled system dx/dt=Ax+α(t)Budx/dt = Ax + \alpha(t)Bu where the pair (A,B)(A,B) is stabilizable and α(t)\alpha(t) takes values in [0,1][0,1] and is persistently exciting, i.e., there exist two positive constants μ,T\mu,T such that, for every t0t\geq 0, tt+Tα(s)dsμ\int_t^{t+T}\alpha(s)ds\geq \mu. In particular, when α(t)\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=Kxu=Kx, with KK only depending on (A,B)(A,B) and possibly on μ,T\mu,T, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when AA is neutrally stable and when the system is the double integrator

    On the stabilization of persistently excited linear systems

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    We consider control systems of the type x˙=Ax+α(t)bu\dot x = A x +\alpha(t)bu, where uRu\in\R, (A,b)(A,b) is a controllable pair and α\alpha is an unknown time-varying signal with values in [0,1][0,1] satisfying a persistent excitation condition i.e., \int_t^{t+T}\al(s)ds\geq \mu for every t0t\geq 0, with 0<μT0<\mu\leq T independent on tt. We prove that such a system is stabilizable with a linear feedback depending only on the pair (T,μ)(T,\mu) if the eigenvalues of AA have non-positive real part. We also show that stabilizability does not hold for arbitrary matrices AA. 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 μ/T\mu/T

    Robust exact differentiators with predefined convergence time

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    The problem of exactly differentiating a signal with bounded second derivative is considered. A class of differentiators is proposed, which converge to the derivative of such a signal within a fixed, i.e., a finite and uniformly bounded convergence time. A tuning procedure is derived that allows to assign an arbitrary, predefined upper bound for this convergence time. It is furthermore shown that this bound can be made arbitrarily tight by appropriate tuning. The usefulness of the procedure is demonstrated by applying it to the well-known uniform robust exact differentiator, which the considered class of differentiators includes as a special case
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