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Harib et al. Intell Robot 2022;2(1):37-71 https://dx.doi.org/10.20517/ir.2021.19 Page 41

n
n
where ξ(t) ∈ R is a bounded time-dependant disturbance, x ∈ R is the extended system state vector, y ∈
n
m
N
R is the controlled system output, u ∈ R is the control input and Ψ ∈ R is the known N-dimensional
m
regressor vector. We assume (A , B, B , C ) are known and A is Hurwitz. y ∈ R in this case is a
ref
ref
ref
ref
cmd
N×m
bounded command for y. Λ ∈ R m×m is a diagonal positive definite matrix and Κ ∈ R is a constant matrix,
where both matrices represent the matched uncertainties of the system. In addition, we assume that,
and that the disturbance upper bound ξ ≥ 0 is known and constant.
max
The control goal is bounded tracking of the reference model dynamics,

m
driven by a bounded time-dependant command y ∈ R .
cmd
Based on Equation (1), the control input is selected as,

where K ∈ R N×m is the matrix of adaptive parameters. If we substitute Equation (4) into Equation (1), we
get,

where,

is the matrix of estimation errors. The tracking error is e = x - x . Subtracting the reference model dynamics
ref
in Equation (3) from that of Equation (1) yields the tracking error dynamics,

The Lyapunov function candidate is selected,