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Page 42 Harib et al. Intell Robot 2022;2(1):37-71 https://dx.doi.org/10.20517/ir.2021.19

T
T
where Γ = Γ > 0 represents constant rates of adaptation, and P = P > 0 is the unique symmetric positive
K
K
definite solution of the algebraic Lyapunov equation,

with Q = Q > 0. The time derivative of V, along the trajectories of Equation (7),
T

Applying the trace identity,

yields,

Using the following adaptive law yields,

then,

and, consequently, V < 0 outside of the set,

Hence, trajectories [e(t),ΔK(t)], of the error dynamics in Equation (7) coupled with the adaptive law in
Equation (13), enter the set E in finite time and stay there for all future times. However, the set E is not
0
0
compact in the (e,ΔK) space. Moreover, it is unbounded since ΔK is not restricted. Inside the set E , V can
0
become positive and consequently, the parameter errors can grow unbounded, even though the tracking
error norm remains less then e at all times. This phenomenon is caused by the disturbance term ξ(t). It
0
shows that the adaptive law in Equation (13) is not robust to bounded disturbances, no matter how small
the latter is.

In the 1980s, several studies analyzed the stability of adaptive control systems and many of them
concentrated on linear disturbance-free systems [16-20] . The results, however, are not completely satisfactory,
since they do not consider the cases where disturbances are present, which could completely change the