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Exponential stability of nonlinear non-autonomous multivariable systems

100%
Gil' M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2015
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tom 35
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nr 1
89-100
EN
We consider nonlinear non-autonomous multivariable systems governed by differential equations with differentiable linear parts. Explicit conditions for the exponential stability are established. These conditions are formulated in terms of the norms of the derivatives and eigenvalues of the variable matrices, and certain scalar functions characterizing the nonlinearity. Moreover, an estimate for the solutions is derived. It gives us a bound for the region of attraction of the steady state. As a particular case we obtain absolute stability conditions. Our approach is based on a combined usage of the properties of the "frozen" Lyapunov equation, and recent norm estimates for matrix functions. An illustrative example is given.
2
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Extended lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems

88%
Zhai G., Xu X., Lin H., Liu D.
International Journal of Applied Mathematics and Computer Science
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2007
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tom 17
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nr 4
447-454
EN
We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.
3
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Observer-based controller design of time-delay systems with an interval time-varying delay

88%
Thuan M., Phat V., Trinh H.
International Journal of Applied Mathematics and Computer Science
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2012
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tom 22
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nr 4
921-927
EN
This paper considers the problem of designing an observer-based output feedback controller to exponentially stabilize a class of linear systems with an interval time-varying delay in the state vector. The delay is assumed to vary within an interval with known lower and upper bounds. The time-varying delay is not required to be differentiable, nor should its lower bound be zero. By constructing a set of Lyapunov-Krasovskii functionals and utilizing the Newton-Leibniz formula, a delay-dependent stabilizability condition which is expressed in terms of Linear Matrix Inequalities (LMIs) is derived to ensure the closed-loop system is exponentially stable with a prescribed α-convergence rate. The design of an observerbased output feedback controller can be carried out in a systematic and computationally efficient manner via the use of an LMI-based algorithm. A numerical example is given to illustrate the design procedure.
4
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Local attractivity in nonautonomous semilinear evolution equations

88%
Blot J., Buşe C., Cieutat P.
Nonautonomous Dynamical Systems
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2014
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tom 1
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nr 1
EN
We study the local attractivity of mild solutions of equations in the form u’(t) = A(t)u(t) + f (t, u(t)), where A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at infinity
5
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A variable structure observer for the control of robot manipulators

88%
Abdessameud A., Khelfi M. F.
International Journal of Applied Mathematics and Computer Science
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2006
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tom 16
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nr 2
189-196
EN
This paper deals with the application of a variable structure observer developed for a class of nonlinear systems to solve the trajectory tracking problem for rigid robot manipulators. The analyzed approach to observer design proposes a simple design methodology for systems having completely observable linear parts and bounded nonlinearities andor uncertainties. This observer is basically the conventional Luenberger observer with an additional switching term that is used to guarantee robustness against modeling errors and system uncertainties. To solve the tracking problem, we use a control law developed for robot manipulators in the full information case. The closed loop system is shown to be globally asymptotically stable based on Lyapunov arguments. Simulation results on a 3-DOF robot manipulator show the asymptotic convergence of the vectors of observation and tracking errors.
6
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A multi-model approach to Saint-Venant equations: A stability study by LMIs

75%
Dos Santos Martins V., Rodrigues M., Diagne M.
International Journal of Applied Mathematics and Computer Science
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2012
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tom 22
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nr 3
539-550
EN
This paper deals with the stability study of the nonlinear Saint-Venant Partial Differential Equation (PDE). The proposed approach is based on the multi-model concept which takes into account some Linear Time Invariant (LTI) models defined around a set of operating points. This method allows describing the dynamics of this nonlinear system in an infinite dimensional space over a wide operating range. A stability analysis of the nonlinear Saint-Venant PDE is proposed both by using Linear Matrix Inequalities (LMIs) and an Internal Model Boundary Control (IMBC) structure. The method is applied both in simulations and real experiments through a microchannel, illustrating thus the theoretical results developed in the paper.
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