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If we distinguish between the three possible cases based on the reference input states: null, constant or
persistently excited, and as mentioned earlier, the third case’s application highlights the difference between
the proposed modification and the aforementioned σ-modification. When the σ-modification is used to
adjust the control parameter θ, in the presence of the disturbance v, we can set the error equations as in
Equation (23),

where it has been shown that three stable equilibrium states exist in case x = 0, none of them is the origin,
m
and has a single equilibrium state whose distance from the origin decreases as the amplitude of x increases,
m
and that is in case x is a constant. Which clearly highlights the addition of the ϵ-modification.
m
2.5. Summary
Basically, the approaches discussed above reduce the effects of parameter adaptation when the measure
error is not due to an error in the parameter estimates. They contribute to either parameter error, noise
error, high-frequency unmodelled dynamics error, or disturbances, which consist of anything undescribed
by the three previous groups . A brief comparison of all the aforementioned modification techniques is
[45]
shown in Table 1.

Considering the robustness problem, one can see that the disturbance is generated internally, which makes
it dependable in the actual plant’s input-output signals. Particularly, the disturbance will grow unboundedly
if the adaptive system is unstable and the input-output signals are growing without bound. Videlicet, the
stability problem becomes internal and signal dependant. Thus, the boundedness of the disturbance should
not be presumed, which proves that, despite the intrinsic results shown in the previous literature, the
aforementioned approaches do not necessarily solve the robustness problem in the presence of bounded
[46]
disturbances .

Over the years, the adaptive controllers have proven themselves effective, especially in the process that can
be modeled linearly with slowly time-varying parameters relative to the system’s dynamics. The 1980s were
the peak of theoretical research on this case. On the other hand, many practical examples can be found in
these research [8,47-52] .

An overview of some practical examples of adaptive control applications in two different fields, thermal
exchange and robotics, is given in Table 2 [53-57] . We would also like to refer the readers to a very concise
survey written by Åström in 1983 for more practical examples of the applications of adaptive control. In
[8]
addition, adaptive controllers are extremely practical and fruitful when it comes to servo systems that have
large disturbances, like load changes, or uncertainties, like frictions, and that have measurable states. The
number one practical field in that era was robotics [53-57] .

Obviously, adaptive controllers are not the “perfect” solution to all control problems. For instance, they do
not provide stability for systems where parameter dynamics are at least the same magnitude as the system’s
dynamics. The controller robustness can be improved by employing artificial intelligence (AI) techniques,
such as fuzzy logic and neural networks [58-60] . Essentially, these methods approximate a nonlinear function
and provide a good representation of the nonlinear unknown plant , although it is typically used as a
[61]
model-free controller. The plant is treated as a “black box”, with input and output data gathered and trained