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Page 44 Harib et al. Intell Robot 2022;2(1):37-71 https://dx.doi.org/10.20517/ir.2021.19

the adaptive system against the effects of unmodelled dynamics. Though, the error between the plant and
the model output does not converge to zero but rather to a magnitude less than the size of the dead-zone. In
other terms, no adaptation takes place when the system is unable to distinguish between the error signal and
the disturbance.

The issue in the dead-zone modification is that it is not Lipschitz, which may cause high-frequency
oscillations and other undesirable effects when the tracking error is at or near the dead-zone boundary. In
1986, Slotine and Coetsee proposed a “smoother” version of the dead-zone modification. Unfortunately,
[35]
we were not able to get a hold of a copy of this paper, but the major idea was explained in his book in
1990 .
[32]

2.3. σ-modification
The dead-zone modification assumes a priori knowledge of an upper bound for the system disturbance. On
the other hand, the σ-modification scheme does not require any prior information about bounds for the
[36]
disturbances. This modification was proposed by Ioannou and Kokotovic in 1983, which Ioannou
referred to later as “fixed σ-modification”. The modification basically adds damping to the ideal adaptive
law. They introduced the modification by adding a decay term -σΓθ to the disturbance-free integral adaptive
law, where σ is a positive scalar to be chosen by the designer. The stability properties with the modification
were established based on the existence of a positive constant p such that, for σ > p, the solutions for the
error and adaptive law equations are bounded for any bounded initial condition. A conservative value of σ
has to be chosen in order to guarantee σ > p. It was also shown that the modification yields the local stability
of an MRAC scheme when the plant is a linear system of relative degree one and has unmodeled parasitics.

However, even though the robustness achievement is done smoothly and in a simpler way with the
σ-modification scheme, there is thepotential destruction of some of the convergence properties, since there
is no more asymptotic convergence and the fine tracking error is confined within a bounded region.
Consequently, many additional modifications have been suggested later, motivated by the aforementioned
drawback of the σ-modification. In 1986, Ioannou and Tsakalis proposed the “switching σ-modification”.
[37]
In contrast to Ioannou’s earlier work [38,39] , the switching of σ from 0 to σ is modified so that σ is a continuous
0
function of |θ(t)| [Equation (16)], since the previous modification choices forced the adaptive law to be
discontinuous, which might not guarantee the existence of a solution and would probably cause oscillations
on the switching surface during implementation. Hence, the continuous switching, as shown in Figure 2,
replaces the discontinuous one and is defined in Equation (16),

*
where M > 0, σ > 0 are design constants and M is chosen to be large enough so that M > |θ |.
0
0
0
0
In 1992, Tsakalis employed the σ-modification to target the adaptive control problem of a linear, time-
[40]
varying SISO plant. The signal boundedness for adaptive laws was guaranteed using the σ-modification,
normalization and a sufficient condition. The condition relates the speed and the range of the plant
parameter variations with the σ value and simplifies the selection of the design parameters in the adaptive
law.