Finite-time stabilization of linear systems by bounded linear time-varying feedback

Prescribed‐time control of uncertain strict‐feedback‐like systems

“…For the strict-feedback-like system as described in (1) to admit a feasible prescribed-time control solution, the following conditions are imposed.…”

“…When the actual control gain g n (X, u, t) = g n (X, t) and the virtual control gain g i (X, u, t) = g i (X, t) (i = 1, … , n − 1) all are precisely known and the disturbances f i (X, u, t) = f i (X, t) (i = 1, … , n) are of some special forms and satisfy certain somewhat restrictive conditions, Krishnamurthy et al propose a method to achieve prescribed-time regulation by using the dynamic high gain scaling and temporal scale transformation technologies, 16 which has been extended to output feedback control recently. 17 The challenge associated with prescribed-time stabilization of such system is obvious and significant when the actual and virtual control gains are unknown yet time-varying and the uncertain nonlinear terms (involved in all the channels) do not satisfy the dominance conditions related to the control gains, which in fact makes the existing methods 1,3,13,16 inapplicable, rendering the underlying prescribed-time control an interesting open problem.…”

“…Our method is partially motivated by the existing work, 13 but the difference as compared with Reference 13 is essential. In fact, the method in Reference 13 only applies to systems in normal-form with matched uncertainties, and the control scheme only works on finite time interval, whereas the system here is allowed to be in the strict-feedback-like form as described in (1), where all the control gains g i (X, u, t) (i = 1, … , n) and disturbances f i (X, u, t) (i = 1, … , n) are unknown yet time-varying and do not necessarily need to be in special form to meet the dominance conditions. Furthermore, the system (achieving regulation in prescribed time) is expected to operate over infinite time interval.…”

Time-Varying Second-Order Sliding Mode Control for Systems Subject to External Disturbance

“…[4] proposed Lyapunov theorem on FTS of continuous autonomous systems, which provided a basic tool for analysis of FTS of nonlinear control systems. Moreover, FTS properties have been studied for various systems, such as impulsive systems [5], non-autonomous systems [6], time-varying systems [7], [8]. To avoid confusion, it should be pointed out that the FTS considered in this paper is different from another FTS concept adopted in [9], which dealt with the finite-time boundedness.…”

L2 norm-based finite-time stability and boundedness of singular distributed parameter systems with parabolic type

“…Due to its good robustness nature, some impressive results about FT stability have been presented in many papers recently. 22–29 Given the singular analysis technique and Lyapunov theory, some sufficient conditions about robust finite-time stability of singular nonlinear systems have been studied in the work by Yang et al 29 Li et al 30 considered the problems on finite-time stability and boundedness for switched nonlinear systems with unstable subsystems by virtue of Lyapunov-like functions, respectively. Using the singular value decomposition approach, the sufficient conditions for the finite-time stability of the singular nonlinear switched time-delay systems were discussed in the work by Thanh et al 28 Input–output finite-time stabilization for a class of nonlinear impulsive hybrid systems was developed in the work by Yao and Li 31 The concepts of FT stability and FT stabilization were extended to the distributed parameter systems, 20,32 and the state feedback controller was designed to guarantee the FT stabilization of the closed-loop distributed parameter systems in the sense of L 2 norm.…”