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Page 46 Harib et al. Intell Robot 2022;2(1):37-71 https://dx.doi.org/10.20517/ir.2021.19
where a is an unknown constant. The reference model is defined in Equation (18),
p
where a > 0 and r is a bounded piecewise continuous function. Equation (19) shows how the aim of the
m
adaptive control is to choose the control input, u, such that the output of the plant approaches that of the
model,
where θ is the control parameter. Therefore, we deduct the error equations in Equation (20),
and hence, based on Equation (21), the proposed adaptive law can be defined,
where ϵ is playing a double role since it attempts to decrease the magnitude of the output error while
keeping the parameter θ or the parameter error φ bounded. The choice of the Lyapunov function, in
Equation (22), gives the time derivative of V,
If we define a set D,
we then can deduct that V ≤ 0 inside the set D. The modification, which is synthesized by the additional
term -|ϵ|θ in the adaptive law, shows that the set D is compact, which allows us to apply LaSalle’s theorem
[44]
and prove that all solutions of the error equations are bounded.
