Time‐varying feedback for stabilization in prescribed finite time

Abstract

Summary This paper provides a time‐varying feedback alternative to control of finite‐time systems, which is referred to as “prescribed‐time control,” exhibiting several superior features: (i) such time‐varying gain–based prescribed‐time control is built upon regular state feedback rather than fractional‐power state feedback, thus resulting in smooth (Cm) control action everywhere during the entire operation of the system; (ii) the prescribed‐time control is characterized with uniformly prespecifiable convergence time that can be preassigned as needed within the physically allowable range, making it literally different from not only the traditional finite‐time control (where the finite settling time is determined by a system initial condition and a number of design parameters) but also the fixed‐time control (where the settling time is subject to certain constraints and thus can only be specified within the corresponding range); and (iii) the prescribed‐time control relies only on regular Lyapunov differential inequality instead of fractional Lyapunov differential inequality for stability analysis and thus avoids the difficulty in controller design and stability analysis encountered in the traditional finite‐time control for high‐order systems.

Enhancing the settling time estimation of a class of fixed‐time stable systems

“…A similar argument is done for (4), proposed in the work of Zuo and Tie 22 for the case of k = 1 and 0 < < 1, where is given in (3). Let > 0, = and = 1 ; then, the upper bound for the settling time T(x 0 ) in (4) becomes…”

“…These restrictions can be relaxed using the fixed‐time stability concept. It is an extension of global finite‐time stability and guarantees the convergence (settling) time to be globally uniformly bounded, ie, the bound does not depend on the initial state of the system . To this end, the class of systems where x is a scalar state variable and the real numbers α , β , p , q , k >0 are system parameters, which satisfy the constraints kp <1, and kq >1, was proposed in the works of Polyakov and Andrieu et al and has been extensively used.…”

On the design of nonautonomous fixed‐time controllers with a predefined upper bound of the settling time

“…Thus, the controller is not realizable in practice. These are drawbacks also present in the nonautonomous controllers proposed by Song et al 23,37 and Pal et al; 22 and as stated by Song et al, 23 also the finite‐horizon optimal control approach with a terminal constraint, inevitably yields gains that go to infinity. However, unlike such methods, our methodology allows us to design controllers with bounded time‐varying gains and the application to perturbed systems, as we illustrate in our next case.…”

“…First, in methodologies like those proposed by Basin et al 32 and Tian et al, 28,29 which are based on the homogeneity property, 3 the UBST is unknown.Second, autonomous controllers derived based on Lyapunov analysis 1,16,26,34-36 may provide non‐conservative estimates of the UBST for the scalar case (see, eg, the work from Sanchez‐Torres et al 2 and Aldana‐López et al 34 ), but the estimate of the UBST becomes very conservative in high‐order systems (see, eg, section 5 in Zimenko et al 15 ). Third, in nonautonomous controllers based on time‐varying gains, 22,23,37 the origin is reached exactly at the desired time, but the time‐varying gain tends to infinity as the time approaches the desired convergence time.…”

“…Our methodology consists of redesigning known autonomous controllers by adding time‐varying gains, constructed from time‐base generators 38 . However, unlike existing methods based on time‐base generators, 22,23,37 we provide sufficient conditions such that our time‐varying gains remain bounded. Moreover, contrary to Becerra et al, 39 no initial state is explicitly used in the feedback law; and unlike Song et al 23 and Becerra et al, 39 predefined‐time convergence is guaranteed even in the presence of external disturbances.…”