Liapunov functions and stability in control theory

By A. Bacciotti, Lionel Rosier. 2001

1,075 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/5ac67746582563d73c446eb863155742f2b76991

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In Geometry and Topology

Citations in the corpus

Homogeneous Approximation, Recursive Observer Design, and Output Feedback (2008, background): “This notion of local approximation of a function or of a vector field can be found in [9, 29, 2, 10].”
A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems (2005, neutral): “We recall the definitions of homogeneous systems with weighted dilation (refer to [11], [9], [1] for details).”
Discontinuous dynamical systems (2008, background): “Even control systems whose inputs appear linearly (see Example 21) may be subject to obstructions that preclude the existence of continuous statedependent stabilizing feedback [12] [13] [14].”
Robustness of Control Barrier Functions for Safety Critical Control (2016, background): “1k, on system (1). 1While Theorem 2.8 requires the function Vto be smooth, can always be smoothed as shown in Proposition 4.2 in [12]. 5 The rst result in Proposition 2 corresponds to Theorem 2.8 in [13]. A disturbance satisfying the inequality kg1(x)k˙(kxkC) is called a vanishing perturbation since”
A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback (2007, methodology): “Of particular importance in this paper, concerning homogeneity, are the definitions of homogeneous systems with weighted dilation and the homogeneous norm [15, 19, 20, 23], where the particular work of Qian [15] is the inspiration for the methodology discussed in this paper.”
Sparse Coding via Thresholding and Local Competition in Neural Circuits (2008, methodology): “Speciﬁcally, a system is said to be locally asymptotically stable (Bacciotti and Rosier, 2001) at an equilibrium point u ∗ if one can specify ǫ > 0 such that Previous research (Cohen and Grossberg, 1983; Yang and Dillon, 1994; Li et al., 1988) has used the tools of Lyapunov functions (Khalil, 2002)…”
Finite-Time Observers: Application to Secure Communication (2008, methodology): “Let us recall the notion of finite time stability involving the settling-time function given in [24, Definition 2.2] and [ 25 ].”
Homogeneity approach to high-order sliding mode design (2005, neutral): “( Bacciotti & Rosier, 2001 ), where m1 ,...,m n are some positive numbers (weights), if for any > 0 the identity f( x)= −q f( dx) holds (respectively, F( x)= −q d−1 F( dx) ,o r f( x)=−q d−1 f( dx)).”
Finite-Time Semistability and Consensus for Nonlinear Dynamical Networks (2008, neutral): “extends some of the results in [16], [23], [33], [24]–[26].”
Uniform Robust Exact Differentiator (2010, neutral): “The homogeneous norm [16] is defined as”
Oscillations Conditions in Homogenous Systems (2010, methodology): “, 2008) this definition has been introduced for 0 0 λ = and 0 λ = + ∞ (the function g is called homogeneous in the bi-limit if it is simultaneously ( 0 r , 0 , 0 g )-homogeneous and ( ∞ r , +∞ , g∞ )-homogeneous), the case 0 0 λ = has been also treated in (Bacciotti and Rosier, 2001; Hermes, 1991b; ”
Control Barrier Function Based Quadratic Programs for Safety Critical Systems (2016, neutral): “Therefore, existing robustness results in the literature (such as [47], [48]) can be used to characterize the extent to which forward invariance of the set C is robust with respect to different perturbations on the dynamics (1).”