Discrete-Time Adaptive Control for Systems With Input Time-Delay and Non-Sector Bounded Nonlinear Functions

Abstract

This paper presents a discrete-time adaptive control approach for nonlinear systems with input delay. The nonlinearity is assumed to be non-sector bounded, resulting in the key technical lemma being inapplicable. The main aim of this paper is to present a general implementation inspired from Kanellakopoulos and Fu, et al. for uncertain scalar and multivariable input delay systems with uncertain parameters as well as uncertain input gain. While it has been shown by Kanellakopoulos and Fu, et al. that it is possible to design adaptive control laws that compensate for the growth of the nonlinearity for single parameter scalar systems, a rigorous analysis of multiple parameter systems is not shown. In this paper, it is shown that an adaptive controller design that compensates for the growth of the nonlinearity is possible for both multiple parameter scalar and multivariable systems with input delay. Rigorous stability proofs and simulations are presented to verify the validity of the approach. INDEX TERMS Adaptive control, discrete-time systems, nonlinear control, time-delay systems.

Position and stiffness control of an antagonistic variable stiffness actuator with input delay using super-twisting sliding mode control

“…It turns out from (8) that relative degrees of the link position and stiffness errors, are 4 and 2 respectively. It is remarkable that all nonlinear functions φ, ϕ and ψ should be replaced from (2). Since the original system model is affine with respect to control inputs, the transformed model ( 8) is also affine in control inputs and hence, new control input variables u 1 (t−τ ) and u 2 (t−τ ) can be decoupled with respect to the original control inputs τ α (t − τ ) and τ β (t − τ ) of the system.…”

“…More specifically, the approximate prediction vector appears under integral in (17) whereas the prediction vector of the original system is used under integral of new prediction vector in [30,31]. Current definition avoid limiting assumptions on system nonlinearities such as Lipschitz condition assumed in [30,31], sector bounded condition in [17] and some customized conditions used in [15,2].…”

Discrete‐time adaptive control of uncertain sampled‐data systems with uncertain input delay: a reduction

“…is the estimate of the parameter vector θ respectively. The purpose of the adaptive estimator (8) is to facilitate in the computation of the control law which would otherwise be difficult due to the uncertain parameters in the system (6).…”

“…This suggests that a control law should predict the future plant state in order to compensate for it. Examples of predictor-based control laws for linear and nonlinear plants are [5] and [6] respectively.…”

Discrete-time Adaptive Regulation of Systems with Uncertain Upper-bounded Input Delay: A State Substitution Approach

“…x k+d+1 = f (x k+d , u k ), it becomes obvious that in order to affect the state at k + d + 1 it would be desirable to have knowledge of the future state at k + d. A rigorous argument for the case of linear time invariant plants with a single input delay is found in (Mirkin and Raskin, 2003). Examples of predictor-based control laws for linear and nonlinear plants are (Manitius and Olbrot, 1979) (Abidi, 2014) and (Abidi and Postlethwaite, 2019) respectively.…”