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Tracking fundamentals
Published in Lyubomir T. Gruyitch, Control of Linear Systems, 2018
The Lyapunov stability theory does not permit nonnominal disturbances. Therefore, the Lyapunov stability properties represent special cases of the corresponding asymptotic tracking properties in the Lyapunov sense.The reachability time is infinite. This concept does not demand that the real output vector and its derivatives composing Yk(t) $ {\text{Y}}^{k} (t) $ and the desired output vector and its derivatives forming Ydk(t) $ {\text{Y}}_{d}^{k} (t) $ become equal in a finite time. It ensures the asymptotic convergence of the former to the latter only as time t escapes to infinity: t → ∞,
Nonlinear stability analysis using Lyapunov stability theory for car-trailer systems
Published in Maksym Spiryagin, Timothy Gordon, Colin Cole, Tim McSweeney, The Dynamics of Vehicles on Roads and Tracks, 2018
In this section, a nonlinear sensitivity analysis is presented to examine the nonlinear dynamic of the 6-DOF car-trailer model. The nonlinear stability analysis is performed using the bifurcation analysis and phase-plane methods, as well as Lyapunov stability theory. The nonlinear behavior of the CTS under the variation of vehicle forward velocity is discussed. To perform the bifurcation analysis, we compute all the existing equilibria of the CTS, which can be represented in bifurcation diagrams of a state variable (Fuller, AT. 1992). The stability analysis of the equilibrium point will be supported by Lyapunov stability theory.
Soft Computing Methodologies in Sliding Mode Control
Published in Bogdan M. Wilamowski, J. David Irwin, Control and Mechatronics, 2018
There are several SMC controller types seen in the literature; the choice of the type of controller to be used is dependent upon the specific problems to be dealt with. However, central to the SMC design is the use of the Lyapunov stability theory, in which the Lyapunov function of the form () V=12sTs
Stabilising PID tuning for a class of fourth-order integrating nonminimum-phase systems
Published in International Journal of Control, 2019
The Routh-Hurwitz test on the stability of polynomials is one of the most classical approaches from stability theory. The test of necessary and sufficient conditions for the stability of an LTI system is to ensure that all the roots of the characteristic polynomial have negative real parts, based on which one may be able to determine all possible stabilising regions of the given plant. For a high-order system, more limited stabilising region is often given, especially for the process with right-half plane (RHP) zeros (i.e. with inverse-response behaviour). In the present study, simple characterisation of the boundaries of stabilising PID controller parameters are determined based on the necessary and sufficient criteria of Routh stability. From the known boundaries (if they exist), it is possible to attain a better understanding on the design of PID control for a high-order complex process, e.g. fourth-order integrating system. Interestingly, the explicit ranges and limits on stabilising PID controller parameters (Kc, τI and τD) can be established systematically. The proposed method provides an alternative way to represent the stabilising PID parameter regions, which requires no plots as it gives the stabilising regions in explicit mathematical expressions. It is somehow different from those existing methods which rely quite a lot on graphical plots to represent the stability regions.