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efficiency of the control system, even when very small, leading to instability. In the years that followed, there
have been many attempts to overcome the limitations of adaptive control in the presence of bounded
disturbances. In these published papers [21-25] , it is shown that unmodelled dynamics or even very small
bounded disturbances can cause instability in most of the adaptive control algorithms.

Many efforts to design robust adaptive controllers in the case of unknown parameters have consistently
progressed along two different shapes [26-31] . In the first, the adaptive law is altered so that the overall system
has bounded solutions in the presence of bounded disturbances. The second relies on the persistent
excitation of certain relevant signals in the adaptive loop. The next subsections will present some of the
main “modifications” proposed to enforce robustness with bounded disturbances.

2.2. Dead-zone modification
In many physical devices, the output is zero until the magnitude of the input exceeds a certain value. Such
[32]
an input-output relation is called a dead-zone . In a first approach to prevent instability of the adaptive
[26]
process in the presence of bounded external disturbances, Egardt introduced a modification of the law so
that adaptation takes place when the identification error exceeds a certain threshold. The term dead-zone
[18]
was first proposed by Narendra and Peterson in 1980, where the adaptation process stops when the norm
of the state error vector becomes smaller than a prescribed value. In 1980, the study was initiated by
Narendra to determine an adaptive control law that ensures the boundedness of all signals in the presence
of bounded disturbances, in the case of continuous systems. In the study of Peterson and Narendra , they
[30]
highlight the cruciality of the proper choice of the dead zone for establishing global stability in the presence
of an external disturbance. A larger dead zone implies that adaptation will take place in shorter periods of
time, which also means larger parameter errors and larger output. One of the assumptions made in this
paper is that a bound of the disturbance can be determined even though the plant parameters are unknown.
The adaptive law shall consider that the module of the augmented error is not greater than the bound plus
an arbitrary positive constant. Hence, the only knowledge needed to calculate the size of the dead zone is
the bound of the disturbance, which can be computed . It is also worth noting that no prior knowledge of
[30]
the zeros of the plant’s transfer function is needed to find the bound.

Samson presented a brief study in 1983 based on all his previous works and the analysis of Egardt in his
[28]
book. Although his paper was only concerned with the stability analysis and not the convergence of the
adaptive control to an optimal state, he was able to efficiently introduce a new attempt to use the possible
statistical properties of the bounded disturbances. The three properties P -P should be verified by the
1
3
identification algorithm and are similar to the ones demanded for the disturbance-free cases, but less
restrictive. The first property states that the identified vector has to be uniformly bounded, which prevents
the system from diverging faster than exponentially. The second property ensures that the prediction error
remains relatively small, which indicates that the “adaptive observer” transfer function is very similar to that
of the system. Finally, the third property allows the control of the time-varying adaptive observer of the
system.

In 1983, a modified dead-zone technique was proposed in Bunich’s research and was widely used. This
[33]
modification permits a size reduction of the residual set for the error, hence, simplifying the convergence
proof. The drawback is the necessary, yet restrictive, knowledge of a bound on the disturbance in order to
appropriately determine the size of the dead-zone.

The work of Peterson and Narendra invigorated a new study by Sastry where he examined the robustness
[34]
aspects of MRAC. Sastry used the same approach to show that a suitably chosen dead-zone can also stabilize