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Proportional integral derivative plus control for nonlinear discrete-time state-dependent parameter. . .
methodologies that do not necessitate a known by NFTSMC. One of the main factors contribut-
regression matrix. One such approach involves ing to TDE’s prominence is its ease of use and
the utilization of neural networks to approxi- ability to estimate the dynamics of unknown sys-
mate the unknown dynamics of robotic systems. 17 tems. Thus, TDE is a good substitute for model-
These neural network-based adaptive controllers based control approaches.
have demonstrated promising outcomes in man- The following are the primary contributions
aging complex, high-degree-of-freedom robotic of this work:
manipulators. 18 However, the implementation of
• Using MRAC and TDE, a model-free con-
these advanced control techniques frequently re-
trol is designed to avoid the regression ma-
quires substantial computational resources and trix and provide robust, accurate, and ex-
meticulous tuning of hyperparameters. 19
act tracking.
In the literature, various notable model-free • The unknown dynamics of the robotic
control types of research have been proposed us- manipulator are estimated using TDE,
ing the time delay estimation (TDE) scheme. 20 and the state gains are updated using
The TDE approach has offered an appropriate MRAC, and the TDE error is compen-
and highly efficient estimation method for un- sated using NN.
known uncertain dynamical systems with exter- • The Lyapunov stability criterion is used
nal disturbances and provides an easy implemen- to establish and carry out the closed-loop
tation of the model-free approach. 21 system’s overall asymptotic stability anal-
ysis.
Thanks to the TDE scheme, model-free con-
• For uncertain robotic manipulators with
trol can be developed. The general TDE con-
unknown external disturbances, MRAC
cept is to estimate the unknown dynamical sys- using NNTDE is a novel approach.
tems by inserting a constant delay. 22 Moreover,
The other sections are arranged as follows. In
TDE has been extensively integrated with nu-
Section 2, the preliminary is presented. Section
merous controllers, such as sliding mode con-
3 presents the design of the proposed MRAC-
trol (SMC), adaptive control, fuzzy control, NN,
NNTDE method and the stability analysis of the
and PID control schemes, to obtain accurate es-
timation and robustness precisely. 23–27 As stated, overall system. To show the efficacy of the pro-
posed method, simulation results are depicted in
TDE provides robust estimation and offers sat-
isfactory performance, accurately and efficiently Section 4, and their discussion is given in Section
in terms of joint position tracking, better conver- 5. Finally, this work concludes in Section 6.
gence, and small steady-state error.
2. Preliminaries
Control development in the context of TDE
with adaptive control has shown great interest
by researchers in exploiting the advantages of 2.1. Definition 1 33
both techniques. 28,29 Therefore, several model-
independent control techniques related to TDE The perturbed dynamical system with function
with adaptive control have been proposed for ro- z(t) is defined as
bot manipulators where the unknown dynamics
are estimated using TDE, and the estimation er- ˙ z(t) = g(z) + h(z)u(t) + p(t). (1)
ror is dealt with using adaptive gain. 30,31 To name
a few, bounded nonlinearities have been taken where g(z) is the unknown function, h(z) is a dis-
into consideration; Jin et al. proposed a controller tribution matrix, p(t) shows unknown external
that combines TDE and adaptive control using disturbance, and u(t) represents the control in-
ideal velocity feedback to drive the robot manipu- put. When known and unknown functions are
lators toward their desired trajectory. 32 Aiming to separated, Equation (1) can be expressed as
control the time-varying dynamics of robotic ma- ψ(z, t) = g(z) + p(t) = ˙ z(t) − h(z)u(t). (2)
nipulators, an SMC-based TDE was given to con-
trol time-varying dynamics to observe the desired One can calculate the TDE estimation of un-
trajectory and knowledge of unknown dynam- known dynamics as
ics known by TDE. In, 29 fault-tolerant control ˆ ∆ ∆
ψ(z, t) = ˆ g(z) + ˆ p(t) = g(z) (t−d) + p (t−d)
using TDE and nonsingular fast terminal SMC (3)
u
= ˙ z (t−d) − h(z) (t−d) (t−d) .
(NFTSMC) has been proposed to estimate the
ˆ
unknown dynamics under actuator faults, and ro- where d used as constant time delay and ψ(z, t)
bustness and fast convergence have been obtained denoted the estimated unknown dynamics.
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