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Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
then, existence of a Lyapunov function V(x) is guaranteed by the converse Lyapunov theorem. On the other hand, in solving the feedback stabilization problem, it is of primary importance to recognize a priori whether the choice of a candidate Lyapunov function is appropriate or not, in the sense that a corresponding stabilizing law u=α(x) exists for a given V(x). A candidate Lyapunov function is a control Lyapunov function (CLF) if LfV(x)<0
Rate Control Design
Published in Christos N. Houmkozlis, George A. Rovithakis, End-to-End Adaptive Congestion Control in TCP/IP Networks, 2017
Christos N. Houmkozlis, George A. Rovithakis
where V(e) is the previously defined robust control Lyapunov function. The parameter errors Wi˜, i = 1, 2 are defined as Wi˜=W−W*, i = 1, 2.
Mathematical Preliminaries
Published in Edgar N. Sanchez, Fernando Ornelas-Tellez, Discrete-Time Inverse Optimal Control for Nonlinear Systems, 2017
Edgar N. Sanchez, Fernando Ornelas-Tellez
Note that if V(xk) is an ISS–control Lyapunov function for (2.22), then V(xk) is a control Lyapunov function for the 0-disturbance system xk+1=f(xk)+g(xk)uk.
A hybrid control design for stabilisation of nonlinear systems from the null controllable region*
Published in International Journal of Control, 2023
Maaz Mahmood, Tyler Homer, Prashant Mhaskar
The control and operation of process systems needs to grapple with several challenges, including nonlinearity and constraints. One of the limitations imposed by the presence of input constraints is to limit the set of initial conditions from where the system can be stabilised. This set has been termed the null controllable region (NCR) (Hu et al., 2002). The recognition of the NCR also provides a natural benchmark for control design – in terms of whether or not a particular control design can stabilise the system from the entire NCR. Control designs tackling this objective have often taken a Lyapunov-based approach. In particular, the notion of Lyapunov functions has been generalised to the problem of control analysis and design in the form of control Lyapunov function (CLF) (Artstein, 1983; Sontag, 1983), enabling estimating the controllability/stability region. While constructive procedures for CLFs exist, most of the procedures inherently do not recognise the presence of input constraints.
Global finite-time control for a class of switched nonlinear systems with different powers via output feedback
Published in International Journal of Systems Science, 2018
Jun-yong Zhai, Zhibao Song, Hamid Reza Karimi
Switched systems as an important class of hybrid systems have been attracted much attention in the past decades. The studies on switched systems mainly focused on stability analysis (Chatterjee & Liberzon, 2007; He, Ai, Ren, Dong, & Liu, 2018; Li, Xiang, & Karimi, 2013; Liberzon & Morse, 1999; Liu, Kao, Karimi, & Gao, 2016; Mancilla-Aguilar, 2000; Ye, Michel, & Hou, 1998; Zhao & Dimirovski, 2004) and the references therein. The control Lyapunov function approach is considered to be a powerful method for designing feedback laws for nonlinear systems, see, e.g. (Artstein, 1983; Sontag, 1983,1989). Using the backstepping method, the switched nonlinear systems in lower-triangular form can be globally asymptotically stable by a continuous state feedback controller (Wu, 2009). It is a desirable property of switched systems under arbitrary switchings. It was shown in Liberzon (2003) that a necessary and sufficient condition for switched systems to be asymptotically stable under arbitrary switchings is the existence of a common Lyapunov function for all subsystems. The work (Ma & Zhao, 2010) discussed the global stabilisation problem for switched nonlinear systems in strict-feedback form under arbitrary switchings. The work (Ma, Liu, Zhao, Wang, & Zong, 2015) studied global stabilisation problem for switched power integrator triangular systems with different powers. A continuous stirred tank reactor (CSTR) with two modes feed stream as a practical application was shown in Li, Long, and Zhao (2015). The work (Yang & Zhao, 2017) considered output tracking control problem for a class of switched LPV systems and gave its application to an aero-engine model. The work (Niu, Ahn, Li, & Liu, 2017) investigated adaptive control issue for switched non-lower triangular nonlinear systems and provided its application to a one-link manipulator. Up to now, the aforementioned works have discussed the global asymptotic stability problems for switched nonlinear systems.