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Sideslip angle and tire-road friction coefficient estimation simultaneously for autonomous vehicle
Published in Maksym Spiryagin, Timothy Gordon, Colin Cole, Tim McSweeney, The Dynamics of Vehicles on Roads and Tracks, 2018
Xuefeng Lin, Lu Xiong, Xin Xia
Lyapunov function is defined as V=0.5μ˜2+0.5y˜r2+0.5α˜f2, where the superscript ~ represents the error value. It can be demonstrated that the derivative of Lyapunov function is negative.
Background on Dynamic Systems
Published in F.L. Lewis, S. Jagannathan, A. Yeşildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems, 2020
F.L. Lewis, S. Jagannathan, A. Yeşildirek
In many situations the simple quadratic Lyapunov functions do not suffice; it can be extremely difficult to find a Lyapunov function for complex systems. Failure to find a Lyapunov function may be because the system is not stable, or because: the designer simply lacks insight and experience. The Lyapunov function is closely connected to the deep physical properties of the system, which can often aid in selecting a suitable candidate.
Kinematic Motion Control
Published in Xiaorui Zhu, Youngshik Kim, Mark Andrew Minor, Chunxin Qiu, Autonomous Mobile Robots in Unknown Outdoor Environments, 2017
Xiaorui Zhu, Youngshik Kim, Mark Andrew Minor, Chunxin Qiu
The discontinuity at the origin of the polar coordinates allows a time-invariant controller to be derived, and Brockett’s obstruction does not apply [71]. Due to the singularity, Lyapunov-based techniques are valid everywhere except for the origin, but linearization at the origin can be used to verify stability. Thus, for practical purposes, a stabilizing control law can be derived using Lyapunov functions [71,81].
A dynamical system for solving inverse quasi-variational inequalities
Published in Optimization, 2023
It is known that Lyapunov functions play a key role in the stability analysis of dynamical systems. Recall that a function is said to be a Lyapunov function (about ) for the dynamical system (11) if the following three properties are satisfied: V is positive definite, namely, for all and if and only if ; is negative definite along the trajectories of (11), that is, if is a trajectory of (11), then for all and for all ;V is coercive (also known as radially unbounded), that is, as .
Dissipation of energy analysis approach for vehicle plane motion stability
Published in Vehicle System Dynamics, 2022
Fanyu Meng, Shuming Shi, Minghui Bai, Boshi Zhang, Yunxia Li, Nan Lin
To study the stability of nonlinear systems, another classical method besides the phase plane analysis method, the Lyapunov function method is used [13]. Johnson and Huston [17] established two Lyapunov functions using standard methods and improved kinetic energy functions based on a 2-DOF nonlinear vehicle model, and analysed the lateral stability of the vehicle while keeping straight-line motion. However, because the tire force calculation uses a cubic term expression, the obtained stability region is a conservative elliptical region. On the basis of [17], Samsundar and Huston [14] used the same vehicle model to derive the analytical expression of the equilibrium point and used the Lyapunov function method, the tangent point method, and the trajectory reversal approach to numerically estimate the lateral stability region of the vehicle, in which the trajectory reversal approach has obtained better estimation results. Sobhan Sadri and Christine Wu [18] proposed two new Lyapunov functions based on the research in [14] and [17]. These two functions do not depend explicitly on the vehicle parameters, and compared with previous work, a larger stability region is obtained. However, there is no general method for constructing the Lyapunov function currently. For complex strongly nonlinear systems, to construct the form of the Lyapunov function, it is usually necessary to simplify the model to a certain extent. Meanwhile, the solution result of the stability region is also limited by the function expression. It is difficult to obtain the accurate stability region and generalise to a high degree of freedom systems.
Explicit criteria for moment exponential stability and instability of switching diffusions with Lévy noise
Published in International Journal of Control, 2022
As pointed out in Applebaum (2009) and Zhu et al. (2015), the Lévy noise is able to incorporate both small and big jumps, while the Markovian switching provides the structural changes of the system. In Ji et al. (2020), the authors use the Perron–Frobenius theorem to provide some verifiable conditions on the stability and instability in probability and in almost sure sense of regime-switching jump diffusion processes. In Chao et al. (2017), the authors provide sufficient conditions for pth moment stability in terms of the existence of appropriate Lyapunov functions for the whole system and also by using M-matrices. Moreover, several sufficient conditions are given based on the connection of the Lyapunov exponent and the Legendre transformation. In practice, it might be difficult to apply such criteria. For instance, it is not easy to construct Lyapunov functions or find effective matrices. Our objective is to find verifiable criteria for moment exponential stability and instability. The focus is not on a specific pth moment. Instead, we are interested in the case that a given system is pth moment exponential stability or pth moment exponential instability for some p>0.