On local asymptotic stabilization of the nonlinear systems with time-varying perturbations by state-feedback control

Abstract

In this paper, we are interested in the relation between the solutions of the control systemẋ = f (x, u) and the solutions of its (potentially unknown) perturbationẋ = f (x, u) + w(x, t). Under the assumption that the linear part of the unperturbed system at the point (0, 0) is controllable and that disturbance w(x, t) is sufficiently small, there exists a state-feedback controller of the form u = −Kx such that the perturbed system preserves the local asymptotic stability of the zero solution of unperturbed system. The main result of this paper gives the sufficient conditions, more specifically, the relations between the important parameters of the system, to ensure this property and at the same time provides the method for calculating the lower bound of region of attraction. Moreover, we obtain a nontrivial extension of the classical result of H. K. Khalil regarding asymptotic behavior of the (uncontrolled) perturbed systems whose nominal part is exponentially asymptotically stable at the origin x = 0.

Finite-Time H∞ Control for Time-Delay Markovian Jump Systems with Partially Unknown Transition Rate via General Controllers

“…From ( 8 ) and ( 13 ), it is easy to obtain where , so Integrating both sides of ( 14 ) from 0 to t yields Taking the mathematical expectation on both sides of ( 15 ), the following is concluded i.e., Applying Lemma 1 or the Gronwall–Bellman-type inequality for the three functions [ 55 ] to ( 16 ) yields Set and . Together with ( 10 ), we have From conditions ( 17 ) to ( 20 ), it is derived For all holds, which is obtained by which is ( 11 ).…”

Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems

“…On the other side, the analysis of the robustness of uncontrolled systems often merges with the mathematical theory of dynamical systems. As is traditional in perturbation theory of linear and nonlinear dynamical systems, the behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system [19][20][21][22][23]. In principle, to answer the question regarding behavior of the solutions of perturbed systems as t → ∞, it usually makes a difference whether the origin remains an equilibrium for the perturbed system or not.…”