Proof of Theorem 1. To prove the theorem,
we substitute the expression (3.3) into
Eq. (3.13). Then, under the condition
\(\theta = \mathrm{const}\), we have the equation
$$ \dot {\tilde \theta } = - \Gamma \omega {\omega ^\mathrm
{T}}\tilde \theta .$$
(A.1)
We select a candidate for the Lyapunov function as the quadratic form
$$ \begin {gathered} V = {{\tilde \theta }^\mathrm
{T}}{\Gamma ^{ - 1}}\tilde \theta ,\\ {\lambda _{\min }}\left ( {{\Gamma ^{ - 1}}}
\right ){\left \| {\tilde \theta } \right \|^2} \leqslant V \leqslant {\lambda
_{\max }}\left ( {{\Gamma ^{ - 1}}} \right ){\left \| {\tilde \theta } \right
\|^2}, \end {gathered}$$
(A.2)
where
\(\lambda _{\min }(.) \) and
\(\lambda _{\max }(.) \) are the minimum and maximum eigenvalues of a
matrix.
In view of Eqs. (3.9) and
(3.13), the derivative of the quadratic form
(A.2) along the trajectories of Eq. (A.1) is as follows:
$$ \begin {aligned} \dot V &= 2{{\tilde \theta }^{^\mathrm
{T}}}{\Gamma ^{ - 1}}\dot {\tilde \theta } + {{\tilde \theta }^\mathrm {T}}{{\dot
\Gamma }^{ - 1}}\tilde \theta = - 2{{\tilde \theta }^{^\mathrm {T}}}{\Gamma ^{ -
1}}\left [ {\Gamma \omega {\omega ^\mathrm {T}}\tilde \theta } \right ] + {{\tilde
\theta }^\mathrm {T}}\left [ {\omega {\omega ^\mathrm {T}} - \lambda {\Gamma ^{ -
1}}} \right ]\tilde \theta \\ &= - {{\tilde \theta }^\mathrm {T}}\omega {\omega
^\mathrm {T}}\tilde \theta - \lambda {{\tilde \theta }^\mathrm {T}}{\Gamma ^{ -
1}}\tilde \theta = - \left ( {B_\mathrm {ref\,}^\dag \varepsilon } \right ){\left
( {B_\mathrm {ref\,}^\dag \varepsilon } \right )^\mathrm {T}} - \lambda {{\tilde
\theta }^\mathrm {T}}{\Gamma ^{ - 1}}\tilde \theta \\ &\leqslant - {\left \|
{B_\mathrm {ref\,}^\dag } \right \|^2}{\left \| \varepsilon \right \|^2} - \lambda
{\lambda _{\min }}\left ( {{\Gamma ^{ - 1}}} \right ){\left \| {\tilde \theta }
\right \|^2}. \end {aligned}$$
(A.3)
The derivative (A.3) of the
positive definite quadratic form (A.2) is a
negative semidefinite function, and therefore, the parameter error is \(\tilde \theta \in {L_\infty } \), the generalized error is \(\varepsilon \in {L_\infty } \), and Eq. (A.2) is a Lyapunov function for system (A.1). At the same time, the Lyapunov function (A.2) has a finite limit as \(t\to \infty \),
$$ \begin {gathered} V\left ( {\tilde \theta \left ( {t \to
\infty } \right )} \right ) = V\left ( {\tilde \theta \left ( {{t_0}} \right )}
\right ) + \displaystyle \int \limits _{{t_0}}^\infty {\dot Vd} t = V\left (
{\tilde \theta \left ( {{t_0}} \right )} \right ) - \displaystyle \int \limits
_{{t_0}}^\infty {\left [ {\left ( {B_\mathrm {ref\,}^\dag \varepsilon } \right
){{\left ( {B_\mathrm {ref\,}^\dag \varepsilon } \right )}^\mathrm {T}} + \lambda
\left ( {{{\tilde \theta }^\mathrm {T}}{\Gamma ^{ - 1}}\tilde \theta } \right )}
\right ]d} t\\ \Rightarrow \displaystyle \int \limits _{{t_0}}^\infty {\left [
{{{\left \| {B_\mathrm {ref\,}^\dag } \right \|}^2}{{\left \| \varepsilon \right
\|}^2} + \lambda {\lambda _{\min }}\left ( {{\Gamma ^{ - 1}}} \right ){{\left \|
{\tilde \theta } \right \|}^2}} \right ]d} t {}= V\left ( {\tilde \theta \left (
{{t_0}} \right )} \right ) - V\left ( {\tilde \theta \left ( {t \to \infty }
\right )} \right ) < \infty , \end {gathered}$$
and
then
\(\tilde \theta \in {L_2} \cap {L_\infty }\) and
\(\omega \in {L_\infty }\) (as a result of the fact that
\(\varepsilon \in {L_2} \cap {L_\infty }\)).
We have thus proved the first part of Theorem 1. To prove the second part of Theorem 1, we find the second derivative of the Lyapunov function
(A.2),
$$ \begin {aligned} \ddot V &= - 2\left ( {B_\mathrm
{ref\,}^\dag \dot \varepsilon } \right ){\left ( {B_\mathrm {ref\,}^\dag
\varepsilon } \right )^\mathrm {T}} - \lambda \left ( {2{{\tilde \theta }^\mathrm
{T}}{\Gamma ^{ - 1}}\dot {\tilde \theta } + {{\tilde \theta }^\mathrm {T}}{{\dot
\Gamma }^{ - 1}}\tilde \theta } \right ) \\ &= - 2\left ( {B_\mathrm {ref\,}^\dag
\dot \varepsilon } \right ){\left ( {B_\mathrm {ref\,}^\dag \varepsilon } \right
)^\mathrm {T}} - \lambda \left ( {2{{\tilde \theta }^\mathrm {T}}\left [ { -
\omega {\omega ^\mathrm {T}}\tilde \theta } \right ] + {{\tilde \theta }^\mathrm
{T}}\left [ {\omega {\omega ^\mathrm {T}} - \lambda {\Gamma ^{ - 1}}} \right
]\tilde \theta } \right ) \\ &= - 2\left ( {B_\mathrm {ref\,}^\dag \dot
\varepsilon } \right ){\left ( {B_\mathrm {ref\,}^\dag \varepsilon } \right
)^\mathrm {T}} + \lambda \left ( {2{{\tilde \theta }^\mathrm {T}}\omega {\omega
^\mathrm {T}}\tilde \theta - {{\tilde \theta }^\mathrm {T}}\left [ {\omega {\omega
^\mathrm {T}} - \lambda {\Gamma ^{ - 1}}} \right ]\tilde \theta } \right ). \end
{aligned}$$
(A.4)
Based on the expression (A.4), it
is difficult to draw a conclusion on the boundedness of the second derivative of the function
(A.2); therefore, in view of the relation
\(\dot {\tilde \theta } = \dot {\hat \theta }\), we find the
derivative of the generalized parameter error (3.2),
$$ \dot \varepsilon = B_\mathrm {ref\,}\left [ \dot {\tilde
\theta }^\mathrm {T}\omega + {\tilde \theta }^\mathrm {T}\dot \omega \right ] =
B_\mathrm {ref\,}\left [ - \Gamma \omega \omega ^\mathrm {T}\tilde \theta \omega +
{\tilde \theta }^\mathrm {T}\dot \omega \right ].$$
(A.5)
Taking into account the expression (A.5), for calculation we rewrite Eq. (A.4) as
$$ \ddot V = - 2\left [ { - \Gamma \omega {\omega ^\mathrm
{T}}\tilde \theta \omega + {{\tilde \theta }^\mathrm {T}}\dot \omega } \right
]{\left ( {B_\mathrm {ref\,}^\dag \varepsilon } \right )^\mathrm {T}} + \lambda
\left ( {2{{\tilde \theta }^\mathrm {T}}\omega {\omega ^\mathrm {T}}\tilde \theta
- {{\tilde \theta }^\mathrm {T}}\left [ {\omega {\omega ^\mathrm {T}} - \lambda
{\Gamma ^{ - 1}}} \right ]\tilde \theta } \right ).
$$
According to what has been proved, we have \(\tilde \theta \in {L_2} \cap {L_\infty }\), \(\varepsilon \in {L_2} \cap {L_\infty }\), and \(\omega \in {L_\infty } \), and by the statement of Theorem 1, \(\,\dot \omega {\,\in \,}{L_\infty } \). Then, to conclude that \(\ddot V {\,\in \,}{L_\infty } \), it remains to prove the \(L_\infty \)-boundedness of the matrices \(\Gamma \) and \(\Gamma ^{-1} \). To this end, we obtain a solution of the
differential equation (3.9),
$$ {\Gamma ^{ - 1}}\left ( t
\right ) = {\Gamma ^{ - 1}}\left ( 0 \right ){e^{ - \lambda t}} + \displaystyle
\int \limits _0^{t} {{e^{ - \lambda \left ( {t - \tau } \right )}}\omega \left (
\tau \right ){\omega ^\mathrm {T}}\left ( \tau \right )d} \tau .
$$
Under the persistent excitation condition (2.13), it can readily be shown that for all \(t \geqslant T \) the value of \(\Gamma ^{-1} \) is bounded below by the expression
$$ \begin {aligned} {\Gamma ^{ - 1}}\left ( t \right )
&\geqslant \displaystyle \int \limits _0^{t} {{e^{ - \lambda \left ( {t - \tau }
\right )}}\omega \left ( \tau \right ){\omega ^\mathrm {T}}\left ( \tau \right )d}
\tau \\ &= \displaystyle \int \limits _{t - T}^{t} {{e^{ - \lambda \left ( {t -
\tau } \right )}}\omega \left ( \tau \right ){\omega ^\mathrm {T}}\left ( \tau
\right )d} \tau + \displaystyle \int \limits _0^{t - T} {{e^{ - \lambda \left ( {t
- \tau } \right )}}\omega \left ( \tau \right ){\omega ^\mathrm {T}}\left ( \tau
\right )d} \tau . \end {aligned}$$
(A.6)
Now we obtain upper bounds for each of the two integrals on the right-hand side
in (A.6) by the mean value theorem. To this end,
we rewrite the persistent excitation condition (2.13) in the equivalent form
$$ \displaystyle \int \limits _{t - T}^{t} {\omega \left (
\tau \right ){\omega ^\mathrm {T}}\left ( \tau \right )d} \tau \geqslant \alpha I.
$$
(A.7)
Then, in view of the expression (A.7), the lower bound for the first integral has the form
$$ \displaystyle \int \limits _{t - T}^{t} {{e^{ - \lambda
\left ( {t - \tau } \right )}}\omega \left ( \tau \right ){\omega ^\mathrm
{T}}\left ( \tau \right )d} \tau \geqslant {e^{ - \lambda T}}\displaystyle \int
\limits _{t - T}^{t} {\omega \left ( \tau \right ){\omega ^\mathrm {T}}\left (
\tau \right )d} \tau \geqslant {e^{ - \lambda T}}\alpha I.
$$
(A.8)
In a similar manner, we produce a lower bound for the second integral,
$$ \displaystyle \int \limits _0^{t - T} {{e^{ - \lambda
\left ( {t - \tau } \right )}}\omega \left ( \tau \right ){\omega ^\mathrm
{T}}\left ( \tau \right )d} \tau \geqslant {e^{ - \lambda T}}\displaystyle \int
\limits _0^{t - T} {\omega \left ( \tau \right ){\omega ^\mathrm {T}}\left ( \tau
\right )d} \tau \geqslant 0.$$
(A.9)
Adding (A.8) and (A.9), we obtain a lower bound for the entire matrix
\(\Gamma ^{-1} \),
$$ {\Gamma ^{ - 1}}\left ( t \right ) \geqslant {e^{ -
\lambda T}}\alpha I.$$
(A.10)
Now we obtain a lower bound for the matrix \(\Gamma ^{-1} \) \(\forall t\leqslant T \),
$$ {\Gamma ^{ - 1}}\left ( t \right ) \geqslant {\Gamma ^{ -
1}}\left ( 0 \right ){e^{ - \lambda T}} \geqslant {\lambda _{\min }}\left (
{{\Gamma ^{ - 1}}\left ( 0 \right )} \right ){e^{ - \lambda T}}I.
$$
(A.11)
Then, in view of the estimates (A.10) and (A.11), the lower bound for the matrix \(\Gamma ^{-1} \) for all \(t \geqslant 0 \) has the form
$$ {\Gamma ^{ - 1}}\left ( t \right ) \geqslant {\min }\left
\{ {{\lambda _{\min }}\left ( {{\Gamma ^{ - 1}}\left ( 0 \right )} \right ),\alpha
} \right \}{e^{ - \lambda T}}I.$$
(A.12)
Since \(\omega \in {L_\infty } \) by what has been proved, it follows that the
expression \( \omega \omega ^\mathrm {T}\) satisfies the
inequality
$$ {\lambda _{\min }}\left ( {\omega {\omega ^\mathrm {T}}}
\right ) \leqslant \omega {\omega ^\mathrm {T}} \leqslant {\lambda _{\max }}\left
( {\omega {\omega ^\mathrm {T}}} \right ).$$
(A.13)
Taking into account inequality (A.13), we obtain an upper bound for the matrix
\(\Gamma ^{-1} \),
$$ \Gamma ^{-1} \left (t\right ) \leqslant \Gamma ^{-1}
\left (0\right )+\lambda _{\max } \left (\omega \omega ^\mathrm {T}\right )
\displaystyle \int \limits _0^t e^{-\lambda \left (t-\tau \right )}\,d\tau I
\leqslant \lambda _{\max }\left (\Gamma ^{-1}\left (0\right )\right )I + \dfrac
{\lambda _{\max }\left (\omega \omega ^\mathrm {T}\right )}{\lambda } I.
$$
(A.14)
By combining the expressions (A.12) and (A.14), we obtain inequalities for \(\Gamma \) and \(\Gamma ^{-1} \),
$$ \begin {aligned} {\min }\Big \{\lambda _{\min }\left
(\Gamma ^{-1}\left (0\right )\right ),\alpha \Big \} e^{-\lambda T}I &\leqslant
\Gamma ^{-1}\left ( t \right )\leqslant \lambda _{\max }\left (\Gamma ^{-1}\left
(0\right )\right )I + \dfrac {\lambda _{\max }\left (\omega \omega ^\mathrm
{T}\right )}{\lambda }I, \\[.3em] \left (\lambda _{\max }\left (\Gamma ^{-1}\left
(0\right )\right ) + \dfrac {\lambda _{\max }\left (\omega \omega ^\mathrm
{T}\right )}{\lambda }\right )^{-1}I &\leqslant \Gamma \left (t\right ) \leqslant
\max \Big \{\lambda _{\min }^{-1}\left (\Gamma ^{-1}\left (0\right )\right
),\alpha ^{-1} \Big \}e^{\lambda T}I. \end {aligned}
$$
(A.15)
It clearly follows from the expressions (A.15) that \(\Gamma \in L_\infty \), \(\Gamma ^{-1}\in L_\infty \), and hence \(\ddot V \in {L_\infty } \). Then the derivative (A.3) of the Lyapunov function (A.2) is uniformly continuous, and \(\dot V \to 0 \) by Barbalat’s lemma. Accordingly, we achieve the
convergence \( \tilde \theta \to 0\) as \(t \to \infty \).
To find an estimate for the convergence rate of the error \(\tilde \theta \) to zero, we obtain an upper bound for the
derivative (A.3) with allowance for inequality
(A.13),
$$ \dot V = - {\tilde \theta ^\mathrm {T}}\omega {\omega
^\mathrm {T}}\tilde \theta - \lambda {\tilde \theta ^\mathrm {T}}{\Gamma ^{ -
1}}\tilde \theta \leqslant - {\lambda _{\min }}\left ( {\omega {\omega ^\mathrm
{T}}} \right ){\left \| {\tilde \theta } \right \|^2} - \lambda {\lambda _{\min
}}\left ( {{\Gamma ^{ - 1}}} \right ){\left \| {\tilde \theta } \right \|^2}.
$$
(A.16)
Further, to determine the minimum convergence rate, we proceed from the lower
and upper bounds (A.15) for the matrix
\(\Gamma ^{-1} \) to an expression for the lower and upper bounds for
its norm,
$$ \begin {aligned} \left \| {{\Gamma ^{ - 1}}} \right \|
&\geqslant \underbrace {\sqrt {n + 1} \left [ {{\min }\left \{ {{\lambda _{\min
}}\left ( {{\Gamma ^{ - 1}}\left ( 0 \right )} \right ),\alpha } \right \}{e^{ -
\lambda T}}} \right ]}_{{\lambda _{\min }}\left ( {{\Gamma ^{ - 1}}} \right )} ,\\
\left \| {{\Gamma ^{ - 1}}} \right \| &\leqslant \underbrace {\sqrt {n + 1} \left
[ {{\lambda _{\max }}\left ( {{\Gamma ^{ - 1}}\left ( 0 \right )} \right ) +
\dfrac {{{\lambda _{\max }}\left ( {\omega {\omega ^\mathrm {T}}} \right
)}}{\lambda }} \right ]}_{{\lambda _{\max }}\left ( {{\Gamma ^{ - 1}}} \right )}.
\end {aligned}$$
(A.17)
Taking into account the expression (A.17), we rewrite the upper bound (A.16) as
$$ \begin {aligned} \dot V &\leqslant - {\lambda _{\min
}}\left ( {\omega {\omega ^\mathrm {T}}} \right ){\left \| {\tilde \theta } \right
\|^2} - \lambda \sqrt {n + 1} \left [ {{\min }\left \{ {{\lambda _{\min }}\left (
{{\Gamma ^{ - 1}}\left ( 0 \right )} \right ),\alpha } \right \}{e^{ - \lambda
T}}} \right ]{\left \| {\tilde \theta } \right \|^2} \\ &\leqslant - \left [
{\dfrac {{\lambda {\lambda _{\min }}\left ( {\omega {\omega ^\mathrm {T}}} \right
) + {\lambda ^2}\sqrt {n + 1} \left [ {{\min }\left \{ {{\lambda _{\min }}(
{{\Gamma ^{ - 1}}( 0 )} ),\alpha } \right \}{e^{ - \lambda T}}} \right ]}}{{\sqrt
{n + 1} \left [ {\lambda {\lambda _{\max }}\left ( {{\Gamma ^{ - 1}}\left ( 0
\right )} \right ) + {\lambda _{\max }}\left ( {\omega {\omega ^\mathrm {T}}}
\right )} \right ]}}} \right ]{\lambda _{\max }}( {{\Gamma ^{ - 1}}} ) {\left \|
{\tilde \theta } \right \|^2} \leqslant - \kappa V. \end {aligned}
$$
Let us solve the resulting differential inequality substituting the lower bound
$$ \left \| {\tilde \theta } \right \| \leqslant \sqrt
{\lambda _{\min }^{ - 1}\left ( {{\Gamma ^{ - 1}}} \right ){e^{ - \kappa \cdot
t}}V\left ( 0 \right )}$$
(A.18)
for the Lyapunov function into
the left-hand side of the solution.
It follows from the expression (A.18) that the error \(\tilde \theta \) decays exponentially at a rate faster than
\(\kappa \), which is exactly what is claimed in the second
part of Theorem 1. \(\quad \blacksquare \)
Proof of Theorem 2. A candidate for the
Lyapunov function in the study of the stability of the closed-loop system (2.12) can be selected in the form of the sum of two quadratic
forms,
$$ \begin {gathered} V = {\xi ^\mathrm {T}}H\xi = e_\mathrm
{ref\,}^\mathrm {T}P{e_\mathrm {ref\,}} + {{\tilde \theta }^\mathrm {T}}{\Gamma ^{
- 1}}\tilde \theta ,\\[.03em] H = \mathrm {blockdiag} \left \{ {\begin
{array}{*{20}{c}} P&{{\Gamma ^{ - 1}}} \end {array}} \right \},\\[.03em] {\lambda
_{\min }}\left ( H \right ){\left \| \xi \right \|^2} \leqslant V \leqslant
{\lambda _{\max }}\left ( H \right ){\left \| \xi \right \|^2}. \end {gathered}
$$
(A.19)
In view of the relation \(\dot {\tilde \theta } = \dot {\hat \theta } \) and Eq. (3.3), the derivative of the quadratic form (A.19) according to the deviation equation (2.12) and the adaptation loop equations (4.1) acquire the form
$$ \begin {aligned} \dot V &= \dot e_\mathrm {ref\,}^\mathrm
{T}P{e_\mathrm {ref\,}} + e_\mathrm {ref\,}^\mathrm {T}P{{\dot e}_\mathrm {ref\,}}
+ 2{{\tilde \theta }^\mathrm {T}}{\Gamma ^{ - 1}}\dot {\tilde \theta } + {{\tilde
\theta }^\mathrm {T}}{{\dot \Gamma }^{ - 1}}\tilde \theta \\ &= e_\mathrm
{ref\,}^\mathrm {T}\left [ {A_\mathrm {ref\,}^\mathrm {T}P + P{A_\mathrm {ref\,}}}
\right ]{e_\mathrm {ref\,}} + 2e_\mathrm {ref\,}^\mathrm {T}P{B_\mathrm
{ref\,}}{{\tilde \theta }^\mathrm {T}}\omega - 2{{\tilde \theta }^\mathrm
{T}}\omega {\left [ {B_\mathrm {ref\,}^\dag \varepsilon + B_\mathrm
{ref\,}^\mathrm {T}P{e_\mathrm {ref\,}}} \right ]^\mathrm {T}} + {{\tilde \theta
}^\mathrm {T}}{{\dot \Gamma }^{ - 1}}\tilde \theta \\ &= - e_\mathrm
{ref\,}^\mathrm {T}Q{e_\mathrm {ref\,}} - 2{{\tilde \theta }^\mathrm {T}}\omega
{\omega ^\mathrm {T}}\tilde \theta + {{\tilde \theta }^\mathrm {T}}\left [
{2\omega {\omega ^\mathrm {T}} - \lambda {\Gamma ^{ - 1}}} \right ]\tilde \theta
\\ &= - e_\mathrm {ref\,}^\mathrm {T}Q{e_\mathrm {ref\,}} - \lambda {{\tilde
\theta }^\mathrm {T}}{\Gamma ^{ - 1}}\tilde \theta \leqslant - {\lambda _{\min
}}\left ( Q \right ){\left \| {{e_\mathrm {ref\,}}} \right \|^2} - \lambda
{\lambda _{\min }}\left ( {{\Gamma ^{ - 1}}} \right ){\left \| {\tilde \theta }
\right \|^2}. \end {aligned}$$
(A.20)
The derivative (A.20) of the
positive definite quadratic form (A.19) is a
negative semidefinite function; therefore, the error is \( \xi {\, \in \,} {L_\infty } \), and Eq. (A.19) is a Lyapunov function for system (2.12). At the same time, the Lyapunov function (A.19) has a finite limit as \(t \to \infty \),
$$ \begin {aligned} &V\left ( {\tilde \theta \left ( {t \to
\infty } \right )} \right ) = V\left ( {\tilde \theta \left ( {{t_0}} \right )}
\right ) + \displaystyle \int \limits _{{t_0}}^\infty {\dot Vd} t = V\left (
{\tilde \theta \left ( {{t_0}} \right )} \right ) - \displaystyle \int \limits
_{{t_0}}^\infty {\left [ {e_\mathrm {ref\,}^\mathrm {T}Q{e_\mathrm {ref\,}} +
\lambda \left ( {{{\tilde \theta }^\mathrm {T}}{\Gamma ^{ - 1}}\tilde \theta }
\right )} \right ]d} t\\ &\qquad {}\Rightarrow \displaystyle \int \limits
_{{t_0}}^\infty {\left [ {{\lambda _{\min }}\left ( Q \right ){{\left \|
{{e_\mathrm {ref\,}}} \right \|}^2} + \lambda {\lambda _{\min }}\left ( {{\Gamma
^{ - 1}}} \right ){{\left \| {\tilde \theta } \right \|}^2}} \right ]d} t = V\left
( {\tilde \theta \left ( {{t_0}} \right )} \right ) - V\left ( {\tilde \theta
\left ( {t \to \infty } \right )} \right ) < \infty , \end {aligned}
$$
and then
\(\xi \in {L_2} \cap {L_\infty } \) and
\(\omega \in {L_\infty } \) (because
\({e_\mathrm {ref\,}} \in {L_2} \cap {L_\infty }\)).
We have thus proved the first part of Theorem 2. To prove the second part of Theorem 2, we find the second derivative of the Lyapunov function
(A.19) taking into account Eq. (3.3),
$$ \begin {aligned} \ddot V &= - \dot e_\mathrm
{ref\,}^\mathrm {T}Qe_\mathrm {ref\,} - e_\mathrm {ref\,}^\mathrm {T}Q{\dot
e}_\mathrm {ref\,} - \lambda \left ( {\tilde \theta }^\mathrm {T}{\dot \Gamma }^{
- 1}\tilde \theta + 2{\tilde \theta }^\mathrm {T}\Gamma ^{ - 1}\dot {\tilde \theta
} \right ) \\ &= - 2e_\mathrm {ref\,}^\mathrm {T}Q\left [ A_\mathrm
{ref\,}e_\mathrm {ref\,} + B_\mathrm {ref\,}{\tilde \theta }^\mathrm {T}\omega
\right ] + 2\lambda {\tilde \theta }^\mathrm {T}\omega \left [ B_\mathrm
{ref\,}^\dag \varepsilon + B_\mathrm {ref\,}^\mathrm {T}Pe_\mathrm {ref\,} \right
]^\mathrm {T} \\ &\qquad {}- \lambda \left ( {\tilde \theta }^\mathrm {T}\left [
2\omega \omega ^\mathrm {T} - \lambda \Gamma ^{ - 1} \right ]\tilde \theta \right
) = - 2e_\mathrm {ref\,}^\mathrm {T}Q\left [ A_\mathrm {ref\,}e_\mathrm {ref\,}+
B_\mathrm {ref\,}{\tilde \theta }^\mathrm {T}\omega \right ] \\ &\qquad {} +
2\lambda {\tilde \theta }^\mathrm {T}\omega e_\mathrm {ref\,}^\mathrm
{T}PB_\mathrm {ref\,} + \lambda ^2\left ( {\tilde \theta }^\mathrm {T}\Gamma ^{ -
1}\tilde \theta \right ). \end {aligned}$$
Since it has been proved that \(\tilde \theta \in {L_2} \cap {L_\infty } \), \({e_\mathrm {ref\,}} \in {L_2} \cap {L_\infty }\), and \(\omega \in {L_\infty } \), it follows under the persistent excitation condition
that \(\Gamma \in L_{\infty }\) and \(\Gamma ^{-1} \in L_{\infty } \) (the proof is similar to (A.6)–(A.15) in
the proof of Theorem 1) and hence
\(\ddot V \in {L_\infty }\) as well. In this case, the derivative
(A.20) of the Lyapunov function (A.19) is uniformly continuous, and \(\dot V \to 0 \) by Barbalat’s lemma; accordingly, the convergence
\(\xi \to 0 \) as \(t \to \infty \) is achieved.
To determine an estimate for the rate of convergence of the error
\(\xi \) to zero, we rewrite the upper bound for the
derivative (A.20) as
$$ \dot V\leqslant - \dfrac {\lambda _{\min }\left ( Q
\right )}{\lambda _{\max }\left ( P \right )}\lambda _{\max }\left ( P \right )
\left \|e_\mathrm {ref\,}\right \|^2- \dfrac {\lambda {}\lambda _{\min }\left
(\Gamma ^{-1}\right )} {\lambda _{\max }\left (\Gamma ^{-1}\right )} \lambda
_{\max }\left (\Gamma ^{-1}\right )\left \|\tilde \theta \right \|^2.
$$
(A.21)
Further, to determine the minimum convergence rate using the results obtained
when proving Theorem 1, we write the lower and
upper bounds for the norm \(\Gamma ^{-1}\) for the
adjustment law \( \Gamma \) in the adaptation loop (4.1),
$$ \begin {aligned} \left \| {{\Gamma ^{ - 1}}} \right \|
&\geqslant \underbrace {\sqrt {n + 1} \Big [ {{\min }\left \{ {{\lambda _{\min
}}\left ( {{\Gamma ^{ - 1}}\left ( 0 \right )} \right ),2\alpha } \right \}{e^{ -
\lambda T}}} \Big ]}_{{\lambda _{\min }}\left ( {{\Gamma ^{ - 1}}} \right )} ,\\
\left \| {{\Gamma ^{ - 1}}} \right \| &\leqslant \underbrace {\sqrt {n + 1} \left
[ {{\lambda _{\max }}\left ( {{\Gamma ^{ - 1}}\left ( 0 \right )} \right ) +
\dfrac {{2{\lambda _{\max }}\left ( {\omega {\omega ^\mathrm {T}}} \right
)}}{\lambda }} \right ]}_{{\lambda _{\max }}\left ( {{\Gamma ^{ - 1}}} \right )}.
\end {aligned}$$
(A.22)
Taking into account (A.22), we
rewrite the upper bound for the derivative (A.21)
as
$$ \begin {aligned} \dot V &\leqslant -\frac {\lambda _{\min
}(Q)}{\lambda _{\max }(P)} \lambda _{\max }(P)\left \|e_\mathrm {ref\,}\right \|^2
\\ &\qquad {}- \frac {\lambda ^2\min \left \{\lambda _{\min }\left (\Gamma
^{-1}(0)\right ),2\alpha \right \}e^{-\lambda T}} {\lambda {}\lambda _{\max }\left
(\Gamma ^{-1}(0)\right )+2\lambda _{\max }\left (\omega {}\omega ^\mathrm {T}
\right )} \lambda _{\max }\left (\Gamma ^{-1}\right )\left \|\tilde \theta \right
\|^2 \leqslant -\eta _{\min }V, \\ {\eta _{\min }} &= {\min }\left \{ {\frac
{{{\lambda _{\min }}\left ( Q \right )}}{{{\lambda _{\max }}\left ( P \right
)}}{\rm {; }}\dfrac {{{\lambda ^2}{\min }\left \{ {{\lambda _{\min }}\left (
{{\Gamma ^{ - 1}}\left ( 0 \right )} \right ),2\alpha } \right \}{e^{ - \lambda
T}}}}{{\lambda {}\lambda _{\max }\left ( {{\Gamma ^{ - 1}}\left ( 0 \right )}
\right ) + 2{\lambda _{\max }}\left ( {\omega {\omega ^\mathrm {T}}} \right )}}}
\right \}. \end {aligned}$$
(A.23)
Let us solve the resulting differential inequality while substituting the lower bound
for the Lyapunov function into the left-hand side of the solution,
$$ \left \| \xi \right \| \leqslant \sqrt {\lambda _{\min
}^{ - 1}\left ( H \right ){e^{ - {\eta _{\min }} \cdot t}}V\left ( 0 \right )}.
$$
(A.24)
It follows from the majorant (A.24) that the error \(\xi \) decays exponentially at a rate faster than
\(\eta _{\min } \); this is exactly what is claimed in the second part
of Theorem 2.
To prove the third part of Theorem 2, we write a lower bound for the derivative (A.20),
$$ \begin {gathered} \dot V \geqslant - \dfrac {\lambda
_{\max }\left ( Q \right )} {\lambda _{\max }\left (P\right )}{\lambda _{\max
}\left (P\right )\left \| e_\mathrm {ref\,} \right \|^2} - \lambda {\lambda _{\max
}}\left ( {{\Gamma ^{ - 1}}} \right ){\left \| {\tilde \theta } \right \|^2}
\geqslant - {\eta _{\max }}V,\\ {\eta _{\max }} = {\max }\left \{\dfrac {\lambda
_{\max }\left ( Q \right )}{\lambda _{\max }\left ( P \right )};\lambda \right \}.
\end {gathered}$$
(A.25)
We solve the differential inequality (A.25) while substituting the upper bound for the Lyapunov
function into the left-hand side of the solution,
$$ \left \| \xi \right \| \geqslant \sqrt {\lambda _{\max
}^{ - 1}\left ( H \right ){e^{ - {\eta _{\max }} \cdot t}}V\left ( 0 \right )}.
$$
(A.26)
It follows from the definition of \(\eta _{\max }\) in
(A.25) and the minorant (A.26) that by increasing the parameter \(\lambda \), one can make the maximum rate of convergence of
the error \(\xi \) arbitrarily large; this is what is claimed in the
third part of Theorem 2. \(\quad \blacksquare \)
Remark.
As
\(\lambda \to \infty \), the maximum convergence rate satisfies
\(\eta _{\max }\to \infty \), but the minimum convergence rate
satisfies \(\eta _{\min }\to 0\). Since \(\lambda T\to \infty \) in (A.23), this leads to a considerable increase in the distance
between the majorant (A.24) and the
minorant (A.26); this, in turn, leads to
oscillations with respect to \(\xi \). Therefore, in
practice, it makes little sense to use values of \(\lambda \) exceeding
\(\lambda _{\max }=T^{-1}\).
