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Imitation: Mimicking Human Behavior
Published in Changliu Liu, Te Tang, Hsien-Chung Lin, Masayoshi Tomizuka, Designing Robot Behavior in Human-Robot Interactions, 2019
Changliu Liu, Te Tang, Hsien-Chung Lin, Masayoshi Tomizuka
should also be positive semi-definite. Therefore V˙≤0, and the closed-loop system is Lyapunov stable at (x,x˙) = 0. By further applying LaSalle’s invariance principle [106] on this autonomous system, it is found that the largest invariant set contains the only equilibrium point (x,x˙) = 0. Finally, we can conclude that the closed-loop system is asymptotically stable at (x,x˙) = 0.
Backstepping Control
Published in Bogdan M. Wilamowski, J. David Irwin, Control and Mechatronics, 2018
The stabilizing terms −h1z24 and −g1qz22 increase negativity of U˙3. LaSalle’s invariance principle now implies that the origin is asymptotically stable. The following result formalizes this.
-Control for Continuous-Time Systems
Published in M.D.S. Aliyu, Nonlinear H∞-Control, Hamiltonian Systems and Hamilton-Jacobi Equations, 2017
and therefore, the closed-loop system is locally stable. Moreover, the condition V˙≡0 for all t ≥ tc, for some tc, implies that z ≡ 0 for all t ≥ tc, and consequently, limt→∞x(t) = 0 by zero-state detectability. Finally, by the LaSalle’s invariance-principle, we conclude asymptotic-stability. This establishes (i).
Quantiser-based Hands-off control for robust [𝒦, 𝒦ℒ] sector
Published in International Journal of Control, 2023
Ankit Sachan, Sandeep Kumar Soni, Jitendra Kumar Goyal, Kranthi Kumar Deveerasetty, Xiaogong Xiong
It is to be noted that the above-discussed control laws were active throughout mentioned the presence of excessive control signal. However, such difficulty is overcome by designing a hands-off control (Nagahara et al., 2016), which allows the dynamical system to steer towards the attractive basin with the minimal amount of assistance. Herein, the attractive basin of the nonlinear system is originated from Zubov's method (Valmorbida & Anderson, 2017) by considering its boundary to approach the value 1. An alternative characterisation to show the attractive basin is via a positive-definite candidate-Lyapunov function (candidate-LF) with its proper derivative is to be negative semi-definite (Jurdjevic & Quinn, 1978). When the derivative of a candidate-LF is zero, the system trajectories remain identical to an invariant set. The trajectories converge to the largest invariant set that lies within the level set of a candidate-LF, according to the Krasovskii-LaSalle invariance principle (Merkin, 2012), and the system's trajectory remains identical everywhere except at the origin. Then, the origin is defined as the largest invariant set to ensure asymptotic stability.
Controller design to stabilization of Schrödinger equation with boundary input disturbance
Published in Applicable Analysis, 2020
Haoyue Cui, Zhongjie Han, Genqi Xu
Note that the system (3) is a nonautonomous system, due to the existence of disturbance. We can not directly use the LaSalle invariance principle to analyze its asymptotic stability. The invariance-like theorem of nonautonomous system, see H. K. Khalil's book [28, Chapter 8, Theorem 8.4] for finite dimensional system and [29] for infinite-dimensional system, gives us inspiration to discuss the stability of the system.
Local and global stability of an HCV viral dynamics model with two routes of infection and adaptive immunity
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
Marya Sadki, Sanaa Harroudi, Karam Allali
In order to discuss the global stability of the system (1.1), for all the steady state, we construct appropriate Lyapunov functional and adopting the Lyapunov-LaSalle invariance principle. Let us define a function Note that and