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Page 48                              Harib et al. Intell Robot 2022;2(1):37-71  https://dx.doi.org/10.20517/ir.2021.19

               Table 1. Stability analysis of each modification technique
                Dead-zone modification  σ-modification             ϵ-modification
                • Developed based on adaptation   • Adding a damping term to the   • Adding an error dependent leakage term to the law:
                                                                        T
                hibernation principle.  adaptation law:            K = -Γ e BP(Ψ - ϵK), where ϵ > 0
                                              T                        K
                                        K = Γ (Ψe PB - σK), where σ > 0
                                           K
                • Stops adaptation when the error   • Takes different forms depending on the   • Reduces the unbounded behavior of the adaptive law
                touches the boundary of a compact   choice of sigma
                set β :
                   d
                       T
                           n
                β  = {(e,ΔK ), e∈R , ΔK∈R N×m  ||e|| ≤
                d
                e }
                d
                • Adaptation will be disabled once   • The Lyapunov function derivative is   • Following the same argument as in sigma modification: the
                reaches e               negative under some conditions that   Lyapunov function derivative is negative under certain
                     d
                • Stability is guaranteed outside of β   define a compact set β :   conditions that define a compact set β :
                                                                                           ϵ
                                      d
                                                      σ
                                                    n
                                               T
                                                                          T
                                                                               n
                • The adaptive law is defined in both  β  = {(e,ΔK ), e∈R , ΔK∈R N×m  ||e|| ≤ e  ∧  β  = {(e,ΔK ), e∈R , ΔK∈R N×m  ||e|| ≤ e  ∧ (||ΔK||  ≤ ΔK )}
                                         σ                      σ   ϵ                      ϵ      F  ϵ
                conditions as:          (||ΔK||  ≤ ΔK )}
                                                σ
                                            F
                Drawbacks: a prior knowledge about   • Error UUB is guaranteed and   • Error UUB is guaranteed and boundedness of all adaptive
                the upper bound of the disturbance is   boundedness of all adaptive gains is also   gains is also guaranteed
                required                guaranteed
                                        Drawbacks: the damping term addition   • The upper bound of the set  is determined by the upper
                                        may not be convenient in some situations  bound of the disturbance
               Table 2. Practical examples of adaptive control implementation
                Approach              Employed by…
                Robotic manipulators
                MRAC                  Dubowsky and DesForges [53]  (1979) and Nicosia and Tomei [55]  (1984)
                                               [56]
                STAC                  Koivo and Guo   (1983)
                Adaptive algorithm    Dubowsky [54]  (1981) and Horowitz and Tomizuka [57]  (1986)
                Other applications
                MRAC                  Harrell et al. [49]  (1987) and Davidson [47]  (2021)
                                              [48]                 [52]
                STAC                  Davison et al.   (1980) and Harris and Billings   (1981)
                Direct AC             Zhang and Tomizuka [50]  (1985)
                                                [51]
                Function Blocks       Lukas and Kaya   (1983)
               MRACL Model Reference Adaptive Control; STAC: self-tuning adaptive control.
               on. The AI framework addresses the plant’s model after the training phase, and can handle the plant with
               practically no need for a mathematical model. It is feasible to build the complete algorithm using AI
               techniques, or to merge the analytical and AI approaches such that some functions are done analytically and
               the remainder are performed using AI techniques .
                                                         [62]
               3. NEURAL NETWORKS FOR DYNAMIC SYSTEMS
               The sophisticated adaptive control techniques that have been created complement computer technology
               and offer significant potential in the field of applications where systems must be regulated in the face of
               uncertainty. In the 1980s, there was explosive growth in pure and applied research related to NN. As a
               result, MLN and RNN have emerged as key components that have shown to be exceptionally effective in
               pattern recognition and optimization challenges [63-68] . These networks may be thought of as components that
               can be employed efficiently in complicated nonlinear systems from a system-theoretic standpoint.

               The topic of regulating an unknown nonlinear dynamical system has been approached from a variety of
               perspectives, including direct and indirect adaptive control structures, as well as multiple NN models.
               Because NN may arbitrarily simulate static and dynamic, highly nonlinear systems, the unknown system is
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