Geometric homogeneity with applications to finite-time stability

By Sanjay P. Bhat, Dennis S. Bernstein. Math. Control. Signals Syst., 2005

1,626 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dcd538cf75c35aa9b3412faa0748b9a7bff74a66

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Citations in the corpus

Robust Finite-Time Consensus Tracking Algorithm for Multirobot Systems (2009, methodology): “Here, we recall some Lyapunov theorem for ﬁnite-time stability of nonlinear systems, which was discussed previously in [20] and [21].”
Fixed-time stabilization and consensus of nonholonomic systems (2016, positive): “[40] The origin of system (1) is a globally finite-time equilibrium if there is a function T : Rn 7→ R+ such that for all x0 ∈ Rn, the solution x(t,x0) of system (1) is defined and x(t,x0) ∈ Rn for t ∈ [0,T (x0)) and limt→T (x0) x(t,x0) = 0.”
On robust control algorithms for nonlinear network consensus protocols (2008, background): “If ν(0) = {0}, then the semi-Euler vector field becomes the Euler vector field given in [10].”
Finite-time stabilization of nonlinear impulsive dynamical systems (2007, methodology): “equilibrium for continuous-time systems, that is, finite-time stability, was rigorously studied in [11], [16] using Hölder continuous Lyapunov functions.”
Fixed-Time Attitude Control for Rigid Spacecraft With Actuator Saturation and Faults (2016, background): “Existing finite-time control methods can be broadly classified into two categories: 1) the homogeneous domination approach [8], [9] and 2) the Lyapunov-based approach [10].”
Oscillations Conditions in Homogenous Systems (2010, neutral): “, 2008; Bacciotti and Rosier, 2001; Hermes, 1991a; 1991b; Rosier, 1992), systems approximation (Hermes, 1991a), stabilization (Bhat and Bernstein, 2005; Grüne, 2000; Kawski, 1991; Moulay and Perruquetti, 2006; Sepulchre and Aeyels, 1996), and observation (Andrieu, et al.”
Adaptive Prescribed-Time Control for a Class of Uncertain Nonlinear Systems (2022, neutral): “The homogeneous approach was proposed in [14] and”
Adaptive continuous higher order sliding mode control (2016, background): “8 Evolution of the control function ( )u t ( )t ( )w t 2a ( )t ( )w t 2a 2a 2a Figure 8: Evolutions of u(t) From Proposition 8.1 presented in Bhat & Bernstein (2005), there exists an ϵb ∈ (0, 1) such that for every ᾱ ∈ (1 − ϵb, 1) the origin σ, σ̇ . . . , σ(n) = 0 is a finite time stable…”
Multivariable continuous fixed-time second-order sliding mode control: design and convergence time estimation (2017, background): “Using a non-linear scalar control law designed in [2], a linear system state can be stabilised at the origin exactly in finite time.”
Stability analysis for a second-order continuous finite-time control system subject to a disturbance (2009, background): “In recent works [5, 6 , 13], finite-time Lyapunov stability results have been presented.”
Semi-global finite-time observers for nonlinear systems (2008, background): “The continuity property and its importance in finite-time stability are realized in Bhat and Bernstein (2000, 2005).”
Prescribed-Time Control and Its Latest Developments (2022, background): “Proposition 1 [78]: For system ẋ = f (x, t), if there exists a C1 function V(x) ≥ 0 such that V̇(x) ≤ −kVq(x), where k > 0, 0 < q < 1, then the closed-loop system is FT stable and the settling time is calculated by”
Homogeneous Approximation, Recursive Observer Design, and Output Feedback (2008, background): “It is well known that when the origin of the homogeneous approximation in the 0-limit is globally asymptotically stable and with a strictly negative degree then solutions converge to the origin in finite time (see [3]).”
Finite-Time Observers: Application to Secure Communication (2008, background): “The interested reader can find more details on finite-time stability in [29]–[34].”