Adaptive output‐feedback stabilization in prescribed time for nonlinear systems with unknown parameters coupled with unmeasured states

Abstract

Summary The prescribed‐time output‐feedback stabilization (ie, regulation of the state and control input to zero within a “prescribed” time picked by the control designer irrespective of the initial state) of a general class of uncertain nonlinear strict‐feedback‐like systems is considered. Unlike prior results, the class of systems considered in this article allows crossproducts of unknown parameters (without any required magnitude bounds on unknown parameters) and unmeasured state variables in uncertain state‐dependent nonlinear functions throughout the system dynamics. We show that prescribed‐time output‐feedback stabilization (ie, both prescribed‐time state estimation and prescribed‐time regulation) is achieved through a novel output‐feedback control design involving specially designed dynamics of an adaptation state variable and a high‐gain scaling parameter in combination with a temporal transformation and a dual high‐gain scaling based observer and controller design. While standard dynamic adaptation techniques cannot be applied due to crossproducts of unknown parameters and unmeasured states, we show that instead, the dynamics of the high‐gain scaling parameter and adaptation parameter can be designed with temporal forcing terms to ensure that unknown parameters in system dynamics are dominated by a particular fractional power of the high‐gain scaling parameter and the adaptation parameter after a subinterval (of unknown length) of the prescribed time interval. We show that the control law can be designed such that the system state and input are regulated to zero in the remaining subinterval of the prescribed time interval.

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

“…(1) Aiming at a more general yet more complex class of nonlinear systems, we establish a method for prescribed-time regulation. Different from the finite/fixed-time control that is based on fractional power of state feedback, our result provides a solution with settling time being fully independent of initial conditions; (2) Unlike the prescribed performance control method, [18][19][20] our method makes all the system states converge to the origin precisely within a preset time without the need for barrier Lyapunov function or any other state transformation; (3) In contrast to most existing results 13,14,16,17 that are only valid for t ∈ [0, t f ), our control scheme, featured with simplicity and elegance, is fully functional for t ∈ [0, + ∞) in that it makes each system state converge to zero (within user-chosen settling time) and remain zero thereafter, allowing for nonstop running of the system beyond the settling time t f .…”

“…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.…”

Adaptive finite‐time stabilization with prescribed output convergence via an integrated scheme

“…Whereas this article only realizes the prescribed performance of the system output, one may explore whether it is possible to integrate more powerful schemes to realize prescribed performance of other system signals. Irrespective of prescribed performance, merely finite‐time stabilization can be established in this article, while prescribed‐time stabilization has been achieved recently (see e.g., References 29 and 30). Hence, how to achieve prescribed‐time stabilization on the premise of prescribed performance deserves further investigation.…”

An α‐variable model‐free prescribed‐time control for nonlinear system with uncertainties and disturbances

“…18 Adaptive output-feedback stabilization in prescribed time was achieved by introducing reasonable adaptation parameter. 19 To avoid unnecessary high precision and excessive control input of prescribed-time controller in practice, an adaptive fault-tolerant prescribed-time controller was proposed with position error constraints for teleoperation systems. 20 For broadening the application of prescribed-time control, lower-triangular nonlinear systems, 21 p-normal form system 22 and switched nonlinear systems in p-normal form 23 have been gradually considered.…”

“…Adaptive output‐feedback stabilization in prescribed time was achieved by introducing reasonable adaptation parameter 19 . To avoid unnecessary high precision and excessive control input of prescribed‐time controller in practice, an adaptive fault‐tolerant prescribed‐time controller was proposed with position error constraints for teleoperation systems 20 …”