Approximation-Based Approach to Adaptive Control of Linear Time-Varying Systems

Abstract

An adaptive state-feedback control system is proposed for a class of linear time-varying systems represented in the controller canonical form. The adaptation problem is reduced to the one of Taylor series-based first approximations of the ideal controller parameters. The exponential convergence of identification and tracking errors of such an approximation to an arbitrarily small and adjustable neighbourhood of the equilibrium point is ensured if the condition of the regressor persistent excitation with a sufficiently small time period is satisfied. The obtained theoretical results are validated via numerical experiments.

APPENDIX

Proof of Proposition 1. The proof of the proposition is divided into two steps. At the first one we analyse the properties of the parametric error $\tilde {\theta }$(t), at the second one—the properties of the tracking error e_ref(t).

Step 1. Owing to Proposition 1 from [19], if i $\leqslant$ i_max < ∞, then for the differential equation

$$\dot {\tilde {\theta }}(t) = - {{\gamma }_{1}}\tilde {\theta }(t) - \dot {\theta }(t),\;\;\tilde {\theta }\left( {t_{0}^{ + }} \right) = {{\hat {\theta }}_{0}} - \theta \left( {t_{0}^{ + }} \right),$$

the following upper bound holds

$$\left\| {\tilde {\theta }(t)} \right\|\;\leqslant \;{{\beta }_{{\max }}}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}},\;\;{{\beta }_{{\max }}} > 0,$$

(A.1)

where $\dot {\theta }$(t) = $\sum\limits_{q = 1}^i {\Delta _{q}^{\theta }\delta \left( {t - t_{q}^{ + }} \right)}$, and δ : [$t_{0}^{ + }$; ∞) → {0, ∞} is the Dirac function.

Step 2. The following quadratic form is introduced:

$$\begin{gathered} {{V}_{{{{e}_{{{\text{ref}}}}}}}} = e_{{{\text{ref}}}}^{{\text{T}}}P{{e}_{{{\text{ref}}}}} + \frac{{a_{0}^{2}}}{{{{\gamma }_{1}}}}{{e}^{{ - 2{{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}},\;\;H = {\text{blockdiag}}\left\{ {P,\,\,\frac{{a_{0}^{2}}}{{{{\gamma }_{1}}}}} \right\}, \\ \underbrace {{{\lambda }_{{\min }}}(H)}_{{{\lambda }_{{\text{m}}}}}{{\left\| {{{{\bar {e}}}_{{{\text{ref}}}}}} \right\|}^{2}}\;\leqslant \;V\left( {\left\| {{{{\bar {e}}}_{{{\text{ref}}}}}} \right\|} \right)\;\leqslant \;\underbrace {{{\lambda }_{{\max }}}(H)}_{{{\lambda }_{M}}}{{\left\| {{{{\bar {e}}}_{{{\text{ref}}}}}} \right\|}^{2}}, \\ \end{gathered} $$

(A.2)

where ${{\bar {e}}_{{{\text{ref}}}}}$(t) = ${{\left[ {\begin{array}{*{20}{c}} {e_{{{\text{ref}}}}^{{\text{T}}}(t)}&{{{e}^{{ - {{\gamma }_{1}}(t - t_{0}^{ + })}}}} \end{array}} \right]}^{{\text{T}}}}$, P = P^T > 0 is the solution of the below-given Lyapunov equation in case λ_min(Q) > 2:

$$A_{{{\text{ref}}}}^{{\text{T}}}P + P{{A}_{{{\text{ref}}}}} = - Q,\;\;Q = {{Q}^{{\text{T}}}} > 0.$$

The derivative of the quadratic form (A.2) is written as:

$$\begin{matrix} {{{\dot{V}}}_{{{e}_{\text{ref}}}}}=e_{\text{ref}}^{\text{T}}\left( A_{\text{ref}}^{\text{T}}P+P{{A}_{\text{ref}}} \right){{e}_{\text{ref}}}-2a_{0}^{2}{{e}^{-2{{\gamma }_{1}}\left( t-t_{0}^{+} \right)}}+2e_{\text{ref}}^{\text{T}}P{{e}_{n}}b{{{\tilde{\theta }}}^{\text{T}}}\omega +2e_{\text{ref}}^{\text{T}}P{{e}_{n}}b\delta _{{{\theta }_{0}}}^{\text{T}}\omega \\ =-e_{\text{ref}}^{\text{T}}Q{{e}_{\text{ref}}}-2a_{0}^{2}{{e}^{-2{{\gamma }_{1}}\left( t-t_{0}^{+} \right)}}+2e_{\text{ref}}^{\text{T}}P{{e}_{n}}b{{{\tilde{\theta }}}^{\text{T}}}\left( {{\omega }_{{{e}_{\text{ref}}}}}+{{\omega }_{r}} \right)+2e_{\text{ref}}^{\text{T}}P{{e}_{n}}b\delta _{{{\theta }_{0}}}^{\text{T}}\left( {{\omega }_{{{e}_{\text{ref}}}}}+{{\omega }_{r}} \right) \\ \leqslant \ -{\kern 1pt} {{\lambda }_{\min }}(Q){{\left\| {{e}_{\text{ref}}} \right\|}^{2}}-2a_{0}^{2}{{e}^{-2{{\gamma }_{1}}\left( t-t_{0}^{+} \right)}} \\ +\ 2{{\lambda }_{\max }}(P){{b}_{\max }}{{\left\| {{e}_{\text{ref}}} \right\|}^{2}}\left\| {\tilde{\theta }} \right\|+2{{\lambda }_{\max }}(P){{{\bar{\omega }}}_{r}}{{b}_{\max }}\left\| {{e}_{\text{ref}}} \right\|\left\| {\tilde{\theta }} \right\| \\ +\ 2{{\lambda }_{\max }}(P){{b}_{\max }}{{{\dot{\mathcal{K}}}}_{\max }}T{{\left\| {{e}_{\text{ref}}} \right\|}^{2}}+2{{\lambda }_{\max }}(P){{b}_{\max }}{{{\bar{\omega }}}_{r}}{{{\dot{\mathcal{K}}}}_{\max }}T\left\| {{e}_{\text{ref}}} \right\|, \\ \end{matrix}$$

(A.3)

where

$$\left\| {\omega (t)} \right\|\;\leqslant \;\underbrace {\left\| {{\kern 1pt} \left[ {\begin{array}{*{20}{c}} {{{e}_{{{\text{ref}}}}}(t)}&0 \end{array}} \right]{\kern 1pt} } \right\|}_{\left\| {{{\omega }_{{{{e}_{{{\text{ref}}}}}}}}(t)} \right\| = \left\| {{{e}_{{{\text{ref}}}}}(t)} \right\|} + \underbrace {\left\| {{\kern 1pt} \left[ {\begin{array}{*{20}{c}} {{{x}_{{{\text{ref}}}}}(t)}&{r(t)} \end{array}} \right]{\kern 1pt} } \right\|}_{\left\| {{{\omega }_{r}}(t)} \right\|\,\leqslant \,{{{\bar {\omega }}}_{r}}}\;\leqslant \;\left\| {{{e}_{{{\text{ref}}}}}(t)} \right\| + {{\bar {\omega }}_{r}}.$$

Having applied Young’s inequality twice:

$$\begin{gathered} 2{{\lambda }_{{\max }}}(P){{{\bar {\omega }}}_{r}}{{b}_{{\max }}}\left\| {{{e}_{{{\text{ref}}}}}} \right\|\left\| {\tilde {\theta }} \right\|\;\leqslant \;{{\left\| {{{e}_{{ref}}}} \right\|}^{2}} + \lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}{{\left\| {\tilde {\theta }} \right\|}^{2}}, \\ 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}{{{\bar {\omega }}}_{r}}{{{\dot {\mathcal{K}}}}_{{\max }}}T\left\| {{{e}_{{{\text{ref}}}}}} \right\|\;\leqslant \;\lambda _{{\max }}^{2}(P)b_{{\max }}^{2}\bar {\omega }_{r}^{2}\dot {\mathcal{K}}_{{\max }}^{2}{{T}^{2}} + {{\left\| {{{e}_{{{\text{ref}}}}}} \right\|}^{2}}, \\ \end{gathered} $$

(A.4)

equation (A.3) is rewritten as:

$$\begin{gathered} {{{\dot {V}}}_{{{{e}_{{ref}}}}}}\;\leqslant \;\left[ { - {{\lambda }_{{\min }}}(Q) + 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}\left( {\left\| {\tilde {\theta }} \right\| + {{{\dot {\mathcal{K}}}}_{{\max }}}T} \right) + 2} \right]{{\left\| {{{e}_{{{\text{ref}}}}}} \right\|}^{2}} \\ - 2a_{0}^{2}{{e}^{{ - 2{{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} + \lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}{{\left\| {\tilde {\theta }} \right\|}^{2}} + \lambda _{{\max }}^{2}(P)b_{{\max }}^{2}\bar {\omega }_{r}^{2}\dot {\mathcal{K}}_{{\max }}^{2}{{T}^{2}}. \\ \end{gathered} $$

(A.5)

As the parametric error $\tilde {\theta }$(t) converges to zero exponentially (A.2), then, if λ_min(Q) > 2, then there definitely exists a time instant ${{t}_{{{{e}_{{{\text{ref}}}}}}}}\; \geqslant \;t_{0}^{ + }$ and constants T_min > 0, a₀ > λ_max(P)${{\bar {\omega }}_{r}}{{b}_{{\max }}}{{\beta }_{{\max }}}$ such that for all t $\geqslant$ ${{t}_{{{{e}_{{{\text{ref}}}}}}}}$ and 0 < T < T_min it holds that

$$\begin{gathered} - {{\lambda }_{{\min }}}(Q) + 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}\left( {{{\beta }_{{\max }}}{{e}^{{ - {{\gamma }_{1}}\left( {{{t}_{{{{e}_{{{\text{ref}}}}}}}} - t_{0}^{ + }} \right)}}} + {{{\dot {\mathcal{K}}}}_{{\max }}}T} \right) + 2 = - {{c}_{1}} < 0, \\ \lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}\beta _{{\max }}^{2} - 2a_{0}^{2} = - {{c}_{2}} < 0. \\ \end{gathered} $$

(A.6)

Then the upper bound of the derivative (A.5) for all t $\geqslant$ ${{t}_{{{{e}_{{{\text{ref}}}}}}}}$ is written as

$${{\dot {V}}_{{{{e}_{{{\text{ref}}}}}}}}\;\leqslant \; - {{\eta }_{{{{{\bar {e}}}_{{{\text{ref}}}}}}}}{{V}_{{{{e}_{{{\text{ref}}}}}}}} + \lambda _{{\max }}^{2}(P)b_{{\max }}^{2}\bar {\omega }_{r}^{2}\dot {\mathcal{K}}_{{\max }}^{2}{{T}^{2}},$$

(A.7)

where ${{\eta }_{{{{{\bar {e}}}_{{{\text{ref}}}}}}}} = \min \left\{ {\frac{{{{c}_{1}}}}{{{{\lambda }_{{\max }}}(P)}},\,\,\frac{{{{c}_{2}}{{\gamma }_{1}}}}{{a_{0}^{2}}}} \right\}$.

The solution of the differential inequality (A.7) for all t $\geqslant$ ${{t}_{{{{e}_{{{\text{ref}}}}}}}}$ is obtained as

$${{V}_{{{{e}_{{{\text{ref}}}}}}}}(t)\;\leqslant \;{{e}^{{ - {{\eta }_{{{{{\bar {e}}}_{{{\text{ref}}}}}}}}}}}\left( {t - {{t}_{{{{e}_{{{\text{ref}}}}}}}}} \right){{V}_{{{{e}_{{{\text{ref}}}}}}}}\left( {{{t}_{{{{e}_{{{\text{ref}}}}}}}}} \right) + \frac{{\lambda _{{\max }}^{2}(P)b_{{\max }}^{2}\bar {\omega }_{r}^{2}\dot {\mathcal{K}}_{{\max }}^{2}{{T}^{2}}}}{{{{\eta }_{{{{{\bar {e}}}_{{{\text{ref}}}}}}}}}}.$$

(A.8)

Tending time to infinity for (A.8) and considering expression for ${{V}_{{{{e}_{{{\text{ref}}}}}}}}$, it is concluded that (2.3) holds, which completes the proof.

Proof of Proposition 2. Owing to Assumption 2 and following (3.2), (3.3), we apply the Taylor formula (1.3) to the parameters Θ(t) to obtain:

$$\Theta (t) = \Theta \left( {t_{i}^{ + }} \right) + \overbrace {\dot {\Theta }\left( {t_{i}^{ + }} \right)\left( {t - t_{i}^{ + }} \right) + \underbrace {\int\limits_{{{t}_{i}}}^t {(t - \zeta )\ddot {\Theta }(\zeta )d\zeta } }_{{{\delta }_{1}}(t)}}^{{{\delta }_{0}}(t)},$$

(A.9)

where $\Theta \left( {t_{i}^{ + }} \right)$ = Θ_i, $\dot {\Theta }\left( {t_{i}^{ + }} \right) = {{\dot {\Theta }}_{i}}$ are the values of the system parameters Θ(t) and the rate of their change at the time instant $t_{i}^{ + }$, ||δ₁(t)|| $\leqslant$ 0.5${{\ddot {\Theta }}_{{\max }}}{{T}^{2}}$ denotes the bounded reminder of the first order (p = 1), ||δ₀(t)|| $\leqslant$ ${{\dot {\Theta }}_{{\max }}}T$ is the bounded reminder of the zeroth order (p = 0).

Equation (A.9) is rewritten in the matrix form

$$\Theta (t) = \Lambda \left( {t,\,\,t_{i}^{ + }} \right)\vartheta (t) + {{\delta }_{1}}(t),$$

(A.10)

where ϑ(t) = ${{\left[ {\begin{array}{*{20}{c}} {\Theta _{i}^{{\text{T}}}}&{\dot {\Theta }_{i}^{{\text{T}}}} \end{array}} \right]}^{{\text{T}}}} \in {{\mathbb{R}}^{{2(n + 1)}}}$.

The substitution of (A.10) into (2.1) yields

$$\dot {x}(t) = {{A}_{0}}x + {{e}_{n}}\left( {{{\Phi }^{{\text{T}}}}(t)\Lambda \left( {t,\,\,t_{i}^{ + }} \right)\vartheta (t) + {{\Phi }^{{\text{T}}}}(t){{\delta }_{1}}(t)} \right).$$

(A.11)

The expression x(t) – $l\bar {x}(t)$ is differentiated to obtain

$$\dot {x}(t) - l\dot {\bar {x}}(t) = - l(x(t) - l\bar {x}(t)) + {{A}_{0}}x + {{e}_{n}}\left( {{{\Phi }^{{\text{T}}}}(t)\Lambda \left( {t,\,\,t_{i}^{ + }} \right)\vartheta (t) + {{\Phi }^{{\text{T}}}}(t){{\delta }_{1}}(t)} \right).$$

(A.12)

The solution of (A.12) is written as

$$\begin{gathered} x(t) - l\bar {x}(t) = {{e}^{{ - l\left( {t - t_{i}^{ + }} \right)}}}x({{t}_{i}}) + {{A}_{0}}\bar {x}(t) + \int\limits_{t_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}{{e}_{n}}{{\Phi }^{{\text{T}}}}(\tau )\Lambda \left( {\tau ,\,\,t_{i}^{ + }} \right)\vartheta (\tau )d\tau } \\ + \;\int\limits_{t_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}{{e}_{n}}{{\Phi }^{{\text{T}}}}(\tau ){{\delta }_{1}}(\tau )d\tau } = {{A}_{0}}\bar {x}(t) + {{e}_{n}}\bar {\varphi }(t)\bar {\vartheta }(t) + {{e}_{n}}\underbrace {\int\limits_{t_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}{{\Phi }^{{\text{T}}}}(\tau ){{\delta }_{1}}(\tau )d\tau } }_{{{\varepsilon }_{0}}(t)}, \\ \end{gathered} $$

(A.13)

where $\bar {\vartheta }$(t) = ${{\left[ {\begin{array}{*{20}{c}} {{{\vartheta }^{{\text{T}}}}(t)}&{e_{n}^{{\text{T}}}x\left( {t_{i}^{ + }} \right)} \end{array}} \right]}^{{\text{T}}}} \in {{\mathbb{R}}^{{2n + 3}}}$, and the third equality is not violated since the reset of the filter states (4.1) and the change of parameters occur synchronously at a known time instant $t_{i}^{ + }$, i.e., $\bar {\vartheta }$(t) = const for all t ∈ $\left[ {\left. {t_{i}^{ + },\,\,t_{i}^{ + } + T} \right)} \right.$.

Equation (A.13) is substituted into (4.2) to obtain

$${{\bar {z}}_{n}}(t) = {{n}_{s}}(t)e_{n}^{{\text{T}}}[x(t) - l\bar {x}(t) - {{A}_{0}}\bar {x}(t)] = \bar {\varphi }_{n}^{{\text{T}}}(t)\bar {\vartheta }(t) + {{\bar {\varepsilon }}_{0}}(t),$$

(A.14)

where ${{\bar {z}}_{n}}(t) \in \mathbb{R}$, ${{\bar {\varphi }}_{n}}(t) \in {{\mathbb{R}}^{{2n + 3}}}$ and the perturbation ${{\bar {\varepsilon }}_{0}}(t) \in \mathbb{R}$ is bounded as follows (see definitions of Φ(t) and ${{\bar {\varphi }}_{n}}(t)$):

$$\left\| {{{{\bar {\varepsilon }}}_{0}}(t)} \right\| = \left\| {{{n}_{s}}(t)\int\limits_{t_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}{{\Phi }^{{\text{T}}}}(\tau ){{\delta }_{1}}(\tau )d\tau } } \right\|\;\leqslant \;\left\| {\bar {\varphi }_{n}^{{\text{T}}}(t)} \right\|0.5{{\ddot {\Theta }}_{{\max }}}{{T}^{2}}.$$

(A.15)

Owing to the multiplication of the regression equation (A.14) by n_s(t), the regressor $\bar {\varphi }_{n}^{{\text{T}}}(t)$. the regressand ${{\bar {z}}_{n}}(t)$ and the perturbation ${{\bar {\varepsilon }}_{0}}(t)$ are bounded. In addition, according to the upper bound (A.15), the perturbation ${{\bar {\varepsilon }}_{0}}(t)$ can be reduced by decreasing the parameter T. Therefore, further on we will use the definition ${{\bar {\varepsilon }}_{0}}(t)$ : = ${{\bar {\varepsilon }}_{0}}$(t, T) and imply that any perturbation obtained by transformation of ${{\bar {\varepsilon }}_{0}}$(t, T) can also be reduced by a reduction of T.

Having applied (4.3) and multiplicated z(t) by adj {φ(t)}, we have (commutativity of the filter (4.3a) is not violated as its reinitialization and parameters change happen synchronously at a known time instant $t_{i}^{ + }$, i.e., $\bar {\vartheta }$(t) = const for all t ∈ [$t_{i}^{ + }$, $t_{i}^{ + }$ + T))

$$\begin{gathered} Y(t)\,:\; = \operatorname{adj} \{ \varphi (t)\} z(t) = \Delta (t)\bar {\vartheta }(t) + {{{\bar {\varepsilon }}}_{1}}(t,T), \\ \operatorname{adj} \{ \varphi (t)\} \varphi (t) = \det \{ \varphi (t)\} {{I}_{{2(n + 1) + 1}}} = \Delta (t){{I}_{{2(n + 1) + 1}}}, \\ {{{\bar {\varepsilon }}}_{1}}(t,T) = \operatorname{adj} \{ \varphi (t)\} \int\limits_{t_{i}^{ + }}^t {{{e}^{{ - \sigma \left( {\tau - t_{i}^{ + }} \right)}}}{{{\bar {\varphi }}}_{n}}(\tau ){{{\bar {\varepsilon }}}_{0}}(\tau ,T)d\tau ,} \\ \end{gathered} $$

(A.16)

where Y(t) ∈ ${{\mathbb{R}}^{{2n + 3}}}$, Δ(t) ∈ $\mathbb{R}$, ${{\bar {\varepsilon }}_{1}}$(t, T) ∈ ${{\mathbb{R}}^{{2n + 3}}}$.

Owing to Δ(t) ∈ $\mathbb{R}$, the elimination (4.5) allows one to obtain the following from (A.16)

$$\begin{gathered} {{z}_{a}}(t) = {{Y}^{{\text{T}}}}(t){{\mathfrak{L}}_{a}} = \Delta (t)\vartheta _{a}^{{\text{T}}}(t) + \bar {\varepsilon }_{1}^{{\text{T}}}(t,\,\,T){{\mathfrak{L}}_{a}}, \\ {{z}_{b}}(t) = {{Y}^{{\text{T}}}}(t){{\mathfrak{L}}_{b}} = \Delta (t)\vartheta _{b}^{{\text{T}}}(t) + \bar {\varepsilon }_{1}^{{\text{T}}}(t,\,\,T){{\mathfrak{L}}_{b}}, \\ \end{gathered} $$

(A.17)

where z_a(t) ∈ ${{\mathbb{R}}^{{1 \times n}}}$, z_b(t) ∈ $\mathbb{R}$, and ϑ_a(t), ϑ_b(t) are the first order approximations of the parameters a(t) and b(t), respectively (components of the vector Θ_i).

In case Assumption 2 is met, following the definition of the signal $\mathcal{K}$(t), the first order approximations θ_x(t) and θ_r(t) of the parameters k_x(t) and k_r(t), respectively, satisfy the equations

$$a_{{{\text{ref}}}}^{{\text{T}}} - \vartheta _{a}^{{\text{T}}}(t) = {{\vartheta }_{b}}(t){{\theta }_{x}}(t),\;\;{{b}_{{{\text{ref}}}}} = {{\vartheta }_{b}}(t){{\theta }_{r}}(t),$$

(A.18)

where θ(t) = ${{\left[ {\begin{array}{*{20}{c}} {{{\theta }_{x}}(t)}&{{{\theta }_{r}}(t)} \end{array}} \right]}^{{\text{T}}}}$.

Each equation from (A.18) is multiplied by Δ(t). Equations (A.17) are substituted into the obtained result to have equation (4.6):

$$\begin{gathered} \mathcal{Y}(t) = \mathcal{M}(t)\theta (t) + d(t,\,\,T), \\ \mathcal{Y}(t)\,:\; = {{\left[ {\begin{array}{*{20}{c}} {\Delta (t)a_{{{\text{ref}}}}^{{\text{T}}} - {{z}_{a}}(t)}&{\Delta (t){{b}_{{{\text{ref}}}}}} \end{array}} \right]}^{{\text{T}}}}, \\ \mathcal{M}(t)\,:\; = {{z}_{b}}(t), \\ d(t,T)\,:\; = - {{\left[ {\begin{array}{*{20}{c}} {\bar {\varepsilon }_{1}^{{\text{T}}}(t,\,\,T){{\mathfrak{L}}_{a}} + \bar {\varepsilon }_{1}^{{\text{T}}}(t,T){{\mathfrak{L}}_{b}}{{\theta }_{r}}(t)}&{\bar {\varepsilon }_{1}^{{\text{T}}}(t,\,\,T){{\mathfrak{L}}_{b}}{{\theta }_{r}}(t)} \end{array}} \right]}^{{\text{T}}}}, \\ \end{gathered} $$

(A.19)

where $\mathcal{Y}$(t) ∈ ${{\mathbb{R}}^{{n + 1}}}$, $\mathcal{M}$(t) ∈ $\mathbb{R}$, d(t, T) ∈ ${{\mathbb{R}}^{{n + 1}}}$.

Owing to (A.19), the solution of (4.7a) is written as

$$\Upsilon (t) = \int\limits_{t_{0}^{ + }}^t {{{e}^{{\int\limits_t^\tau {kd\tau } }}}\mathcal{M}(\tau )\theta (\tau )d\tau } + \int\limits_{t_{0}^{ + }}^t {{{e}^{{\int\limits_t^\tau {kd\tau } }}}d(\tau ,\,\,T)d\tau } \pm \Omega (t)\theta (t) = \Omega (t)\theta (t) + w(t),$$

(A.20)

where

$$w(t) = \Upsilon (t) - \Omega (t)\theta (t).$$

Equation (A.20) completes the proof of the fact that equation (4.8) can be obtained via procedure (4.1)–(4.7).

In order to prove statement (a), the regressor Ω(t) is represented as:

$$\begin{gathered} \Omega (t) = {{\Omega }_{1}}(t) + {{\Omega }_{2}}(t), \hfill \\ {{{\dot {\Omega }}}_{1}}(t) = - k({{\Omega }_{1}}(t) - \Delta (t){{\vartheta }_{b}}(t)),\;\;{{\Omega }_{1}}\left( {t_{0}^{ + }} \right) = 0, \hfill \\ {{{\dot {\Omega }}}_{2}}(t) = - k\left( {{{\Omega }_{2}}(t) - \bar {\varepsilon }_{1}^{{\text{T}}}(t,\,\,T){{\mathfrak{L}}_{b}}} \right),\;\;{{\Omega }_{2}}\left( {t_{0}^{ + }} \right) = 0. \hfill \\ \end{gathered} $$

(A.21)

As k > 0 and the perturbation ${{\bar {\varepsilon }}_{1}}$(t, T) is bounded, then Ω₂(t) is bounded, moreover, for all t $\geqslant$ $t_{0}^{ + }$ the following holds

$$\left| {{{\Omega }_{2}}(t)} \right|\;\leqslant \;{{\Omega }_{{2\max }}}(T),$$

(A.22)

and there exists a limit ${{\lim }_{{T \to 0}}}{{\Omega }_{{2\max }}}(T)$ = 0 for the upper bound as, following (A.15)–(A.19), the value of ${{\bar {\varepsilon }}_{1}}$(t, T) can be arbitrarily reduced by reduction of T.

The next aim is to analyze Ω₁(t). The solution of the first differential equation from (A.21) is written for all t ∈ $\left[ {\left. {t_{i}^{ + } + {{T}_{s}},\,\,t_{{i + 1}}^{ + }} \right)} \right.$ as

$${{\Omega }_{1}}(t) = \phi \left( {t,t_{i}^{ + } + {{T}_{s}}} \right){{\Omega }_{1}}\left( {t_{i}^{ + } + {{T}_{s}}} \right) + \int\limits_{t_{i}^{ + } + {{T}_{s}}}^t {\phi (t,\tau )\Delta (\tau ){{\vartheta }_{b}}(\tau )d\tau ,} $$

(A.23)

where ϕ(t, τ) = ${{e}^{{ - \int\limits_\tau ^t {kd\tau } }}}$.

The upper bound is required for the signal Ω₁(t) over the time range under consideration. To this end, we need bounds for Δ(t), and, in its turn, the ones for φ(t).

As, according to the premises of the proposition, ${{\bar {\varphi }}_{n}}$ ∈ PE for T_s < T, then ${{\bar {\varphi }}_{n}}$ ∈ FE over $\left[ {t_{i}^{ + },\,\,t_{i}^{ + } + {{T}_{s}}} \right]$ (this fact can be validated by substitution of t = $t_{i}^{ + }$ into (1.2)). Then for all t ∈ $\left[ {\left. {t_{i}^{ + } + {{T}_{s}},\,\,t_{{i + 1}}^{ + }} \right)} \right.$ the following lower bound holds for the regressor φ(t)

$$\begin{gathered} \varphi (t) = \int\limits_{t_{i}^{ + }}^t {{{e}^{{ - \sigma \left( {\tau - t_{i}^{ + }} \right)}}}{{{\bar {\varphi }}}_{n}}(\tau )\bar {\varphi }_{n}^{{\text{T}}}(\tau )d\tau } \\ \geqslant \;\int\limits_{t_{i}^{ + }}^{t_{i}^{ + } + {{T}_{s}}} {{{e}^{{ - \sigma \left( {\tau - t_{i}^{ + }} \right)}}}{{{\bar {\varphi }}}_{n}}(\tau )\bar {\varphi }_{n}^{{\text{T}}}(\tau )d\tau } \\ \geqslant \;{{e}^{{ - \sigma \left( {t_{{i + 1}}^{ + } - t_{i}^{ + }} \right)}}}\int\limits_{t_{i}^{ + }}^{t_{i}^{ + } + {{T}_{s}}} {{{{\bar {\varphi }}}_{n}}(\tau )\bar {\varphi }_{n}^{{\text{T}}}(\tau )d\tau } \; \geqslant \;\alpha {{e}^{{ - \sigma \left( {t_{{i + 1}}^{ + } - t_{i}^{ + }} \right)}}}{{I}_{{n + 1}}}. \\ \end{gathered} $$

(A.24)

On the other hand, as ${{\left\| {{{{\bar {\varphi }}}_{n}}(t)} \right\|}^{2}}\;\leqslant \;\bar {\varphi }_{n}^{{\max }}$, then there exists an upper bound

$$\varphi (t)\;\leqslant \;\bar {\varphi }_{n}^{{\max }}\int\limits_{t_{i}^{ + }}^t {{{e}^{{ - \sigma \left( {\tau - t_{i}^{ + }} \right)}}}d\tau } \;\leqslant \;\bar {\varphi }_{n}^{{\max }}\frac{{1 - {{e}^{{ - \sigma \left( {t - t_{i}^{ + }} \right)}}}}}{\sigma }\;\leqslant \;{{\sigma }^{{ - 1}}}\bar {\varphi }_{n}^{{\max }},$$

(A.25)

and, therefore, for all t ∈ $\left[ {\left. {t_{i}^{ + } + {{T}_{s}},\,\,t_{{i + 1}}^{ + }} \right)} \right.$ it holds that Δ_UB $\geqslant$ Δ(t) $\geqslant$ Δ_LB > 0.

Taking into consideration that, following Assumptions 1 and 2, b_max $\geqslant$ |b(t)| $\geqslant$ b_min > 0, and ϑ_b(t) is the approximation of first order of b(t), then the following holds for the multiplication Δ(t)ϑ_b(t)

$$\forall t \in \left[ {\left. {t_{i}^{ + } + {{T}_{s}},\,\,t_{{i + 1}}^{ + }} \right)\;\;{{\Delta }_{{{\text{UB}}}}}{{b}_{{\max }}}\; \geqslant \;\left| {\Delta (t){{\vartheta }_{b}}(t)} \right|\; \geqslant \;{{\Delta }_{{{\text{LB}}}}}{{b}_{{\min }}} > 0.} \right.$$

(A.26)

Having applied (A.21) and (A.26) and considered that 0 $\leqslant$ ϕ(t, τ) $\leqslant$ 1, the following estimates hold for Ω₁(t)

$$\begin{gathered} \forall t \in \left[ {t_{0}^{ + },\,\,t_{0}^{ + } + {{T}_{s}}} \right]\;\;{{\Omega }_{1}}(t) \equiv 0, \\ \forall i\; \geqslant \;1\;\;\forall t \in \left[ {t_{i}^{ + } + {{T}_{s}},\,\,t_{{i + 1}}^{ + }} \right]\;\;{{\Omega }_{1}}\left( {t_{i}^{ + } + {{T}_{s}}} \right) + \left( {t_{{i + 1}}^{ + } - t_{i}^{ + } - {{T}_{s}}} \right){{\Delta }_{{{\text{UB}}}}}{{b}_{{\max }}}\; \geqslant \;{{\Omega }_{1}}(t) \\ \geqslant \;\phi \left( {t_{{i + 1}}^{ + },\,\,t_{i}^{ + } + {{T}_{s}}} \right)\left( {{{\Omega }_{1}}\left( {t_{i}^{ + } + {{T}_{s}}} \right) + \left( {t_{{i + 1}}^{ + } - t_{i}^{ + } - {{T}_{s}}} \right){{\Delta }_{{{\text{LB}}}}}{{b}_{{\min }}}} \right) > 0, \\ \end{gathered} $$

(A.27)

from which we have

$$\begin{gathered} \forall t\; \geqslant \;{{t}_{0}} + {{T}_{s}}\;\;{{\Omega }_{{1\max }}}\; \geqslant \;{{\Omega }_{1}}(t)\; \geqslant \;{{\Omega }_{{1\min }}} > 0, \\ {{\Omega }_{{1\max }}} = \mathop {\min }\limits_{\forall i\, \geqslant \,1} \left\{ {\phi \left( {t_{{i + 1}}^{ + },\,\,t_{i}^{ + } + {{T}_{s}}} \right)\left( {{{\Omega }_{1}}\left( {t_{i}^{ + } + {{T}_{s}}} \right) + \left( {t_{{i + 1}}^{ + } - t_{i}^{ + } - {{T}_{s}}} \right){{\Delta }_{{{\text{LB}}}}}{{b}_{{\min }}}} \right)} \right\}, \\ {{\Omega }_{{1\min }}} = \mathop {\max }\limits_{\forall i\, \geqslant \,1} \left\{ {{{\Omega }_{1}}\left( {t_{i}^{ + } + {{T}_{s}}} \right) + \left( {t_{{i + 1}}^{ + } - t_{i}^{ + } - {{T}_{s}}} \right){{\Delta }_{{{\text{UB}}}}}{{b}_{{\max }}}} \right\}. \\ \end{gathered} $$

(A.28)

Then, using (A.28) and (A.23), the bounds for the regressor Ω(t) are written

$$\forall t\; \geqslant \;{{t}_{0}} + {{T}_{s}}\;\,{{\Omega }_{{1\max }}} + {{\Omega }_{{2\max }}}(T)\; \geqslant \;\left| {\Omega (t)} \right|\; \geqslant \;{{\Omega }_{{1\min }}} - {{\Omega }_{{2\max }}}(T),$$

(A.29)

and, therefore, considering ${{\lim }_{{T \to 0}}}{{\Omega }_{{2\max }}}(T)$ = 0, there exists T_min > 0 such that for all 0 < T < T_min and t $\geqslant$ t₀ + T_s the following inequality holds

$${{\Omega }_{{{\text{UB}}}}}\; \geqslant \;\Omega (t)\; \geqslant \;{{\Omega }_{{{\text{LB}}}}} > 0,$$

(A.30)

which was to be proved in statement (a).

In order to prove the statement (b), the disturbance w(t) is differentiated with (A.20) and (4.7) at hand

$$\begin{aligned} \dot {w}(t) = \dot {\Upsilon }(t) - \dot {\Omega }(t)\theta (t) - \Omega (t)\dot {\theta }(t) \\ = - k(\Upsilon (t) - \mathcal{Y}(t)) + k(\Omega (t) - \mathcal{M}(t))\theta (t) - \Omega (t)\dot {\theta }(t) \\ = - k(\Upsilon (t) - \mathcal{M}(t)\theta (t) - d(t,T)) + k(\Omega (t) - \mathcal{M}(t))\theta (t) - \Omega (t)\dot {\theta }(t) \\ = - k(\Upsilon (t) - \Omega (t)\theta (t)) - \Omega (t)\dot {\theta }(t) + kd(t,\,\,T) \\ = - kw(t) - \Omega (t)\dot {\theta }(t) + kd(t,\,\,T),\;\;w\left( {t_{0}^{ + }} \right) = {{0}_{{n + 1}}}. \\ \end{aligned} $$

(A.31)

The solution of (A.31) is represented as:

$$\begin{gathered} w(t) = {{w}_{1}}(t) + {{w}_{2}}(t), \hfill \\ {{{\dot {w}}}_{1}}(t) = - k{{w}_{1}}(t) - \Omega (t)\dot {\theta }(t),\;\;{{w}_{1}}\left( {t_{0}^{ + }} \right) = {{0}_{{n + 1}}}, \hfill \\ {{{\dot {w}}}_{2}}(t) = - k{{w}_{2}}(t) + kd(t,T),\;\;{{w}_{2}}\left( {t_{0}^{ + }} \right) = {{0}_{{n + 1}}}. \hfill \\ \end{gathered} $$

(A.32)

As for the first differential equation from (A.32), in Proposition 2 from [19] it is proved (up to notation) that the following inequality holds

$$\left\| {{{w}_{1}}(t)} \right\|\;\leqslant \;{{w}_{{1\max }}}\phi \left( {t,\,\,t_{0}^{ + } + {{T}_{s}}} \right),$$

(A.33)

when i $\leqslant$ i_max < ∞.

As k > 0 and the disturbance d(t, T) is bounded, then w₂(t) is also bounded, and consequently, the following inequality holds

$$\left\| {{{w}_{2}}(t)} \right\|\;\leqslant \;{{w}_{{2\max }}}(T),$$

(A.34)

where the limit ${{\lim }_{{T \to 0}}}{{w}_{{2\max }}}(T)$ = 0 holds, as the input of the second differential equation from (A.32) depends only from the value of d(t, T), which, in its turn, according to (A.15)–(A.19), can be reduced arbitrarily by reduction of T. The combination of the inequalities (A.33) and (A.34) in accordance with (A.32) completes the proof of proposition.

Proof of Theorem 1. Proof of theorem is similar to the above-given proof of Proposition 1.

Step 1. For all t $\geqslant$ $t_{0}^{ + }$ + T_s the solution of the differential equation (4.9) is written as

$$\begin{gathered} \tilde {\theta }(t) = \phi \left( {t,\,\,t_{0}^{ + } + {{T}_{s}}} \right)\tilde {\theta }\left( {t_{0}^{ + } + {{T}_{s}}} \right) + \int\limits_{t_{0}^{ + } + {{T}_{s}}}^t {\phi (t,\tau )\frac{{{{\gamma }_{1}}w(\tau )}}{{\Omega (\tau )}}d\tau } \\ - \;\int\limits_{t_{0}^{ + } + {{T}_{s}}}^t {\phi (t,\tau )} \sum\limits_{q = 1}^i {\Delta _{q}^{\theta }\delta \left( {\tau - t_{q}^{ + }} \right)d\tau ,} \\ \end{gathered} $$

(A.35)

where ϕ(t, τ) = ${{e}^{{ - \int\limits_\tau ^t {{{\gamma }_{1}}d\tau } }}}$.

Then, following the proof of Theorem 1 from [19], if i $\leqslant$ i_max < ∞, then the boundedness of the parametric error (A.35) can be shown:

$$\begin{gathered} \left\| {\tilde {\theta }(t)} \right\|\;\leqslant \;{{\beta }_{{\max }}}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{0}}} \right)}}} + \frac{{{{\gamma }_{1}}{{w}_{{1\max }}}}}{{{{\Omega }_{{{\text{LB}}}}}}}\int\limits_{t_{0}^{ + } + {{T}_{s}}}^t {\phi (t,\tau )\phi \left( {\tau ,t_{0}^{ + } + {{T}_{s}}} \right)d\tau } \\ + \;\frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}}\int\limits_{t_{0}^{ + } + {{T}_{s}}}^t {\phi (t,\tau )d\tau } \;\leqslant \;\left( {{{\beta }_{{\max }}} + \frac{{2{{w}_{{1\max }}}}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right){{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{0}}} \right)}}} + \frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}}. \\ \end{gathered} $$

(A.36)

Step 2. The following quadratic form is introduced for all t $\geqslant$ $t_{0}^{ + }$ + T_s:

$$\begin{gathered} {{V}_{{{{e}_{{{\text{ref}}}}}}}} = e_{{{\text{ref}}}}^{{\text{T}}}P{{e}_{{{\text{ref}}}}} + \frac{{4a_{0}^{2}}}{{{{\gamma }_{1}}}}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{s}}} \right)}}},\;\;\,H = {\text{blockdiag}}\left\{ {P,\,\,\frac{{4a_{0}^{2}}}{{{{\gamma }_{1}}}}} \right\}, \\ \underbrace {{{\lambda }_{{\min }}}(H)}_{{{\lambda }_{{\text{m}}}}}{{\left\| {{{{\bar {e}}}_{{{\text{ref}}}}}} \right\|}^{2}}\;\leqslant \;V\left( {\left\| {{{{\bar {e}}}_{{{\text{ref}}}}}} \right\|} \right)\;\leqslant \;\underbrace {{{\lambda }_{{\max }}}(H)}_{{{\lambda }_{M}}}{{\left\| {{{{\bar {e}}}_{{{\text{ref}}}}}} \right\|}^{2}}, \\ {{{\bar {e}}}_{{{\text{ref}}}}}(t) = {{\left[ {\begin{array}{*{20}{c}} {e_{{{\text{ref}}}}^{{\text{T}}}(t)}&{{{e}^{{ - \frac{{{{\gamma }_{1}}}}{4}\left( {t - t_{0}^{ + } - {{T}_{s}}} \right)}}}} \end{array}} \right]}^{{\text{T}}}}. \\ \end{gathered} $$

(A.37)

Similar to proof of Proposition 1, the derivative of (A.37) is written as

$$\begin{gathered} {{{\dot {V}}}_{{{{e}_{{ref}}}}}}\;\leqslant \;\left[ { - {{\lambda }_{{\min }}}(Q) + 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}\left( {\left\| {\tilde {\theta }} \right\| + {{{\dot {\mathcal{K}}}}_{{\max }}}T} \right) + 2} \right]{{\left\| {{{e}_{{{\text{ref}}}}}} \right\|}^{2}} \\ + \;\lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}{{\left\| {\tilde {\theta }} \right\|}^{2}} + \lambda _{{\max }}^{2}(P)b_{{\max }}^{2}\bar {\omega }_{r}^{2}\dot {\mathcal{K}}_{{\max }}^{2}{{T}^{2}} - 2a_{0}^{2}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{s}}} \right)}}}. \\ \end{gathered} $$

(A.38)

As for all t $\geqslant$ $t_{0}^{ + }$ + T_s the parametric error $\tilde {\theta }$(t) meets the inequality (A.36), then, considering

$${{\left\| {\tilde {\theta }(t)} \right\|}^{2}}\;\leqslant \;{{\left( {{{\beta }_{{\max }}} + \frac{{2{{w}_{{1\max }}}}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right)}^{2}}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + } - {{T}_{0}}} \right)}}} + {{\left( {\frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right)}^{2}}$$

$$ + \;2\left( {{{\beta }_{{\max }}} + \frac{{2{{w}_{{1\max }}}}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right)\frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{0}}} \right)}}}$$

$$\leqslant \;\left( {{{\beta }_{{\max }}} + \frac{{2{{w}_{{1\max }}}}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right)\left( {{{\beta }_{{\max }}} + \frac{{2({{w}_{{1\max }}} + {{\gamma }_{1}}{{w}_{{2\max }}}(T))}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right){{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{0}}} \right)}}} + {{\left( {\frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right)}^{2}}$$

$$ = {{\bar {\beta }}_{{\max }}}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{0}}} \right)}}} + {{\left( {\frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right)}^{2}},$$

the upper bound of (A.38) is written as follows:

$$\begin{gathered} {{{\dot {V}}}_{{{{e}_{{{\text{ref}}}}}}}}\;\leqslant \;\left[ { - {{\lambda }_{{\min }}}(Q) + 2 + 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}^{{^{{^{{^{{}}}}}}}}} \right. \\ \left. { \times \;\left( {\left( {{{\beta }_{{\max }}} + \frac{{2{{w}_{{1\max }}}}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right){{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{s}}} \right)}}} + \frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}} + {{{\dot {\mathcal{K}}}}_{{\max }}}T} \right)} \right]{{\left\| {{{e}_{{{\text{ref}}}}}} \right\|}^{2}} \\ + \;\lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}{{{\bar {\beta }}}_{{\max }}}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{s}}} \right)}}} + \lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}{{\left( {\frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right)}^{2}} \\ + \;\lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}\dot {\mathcal{K}}_{{\max }}^{2}{{T}^{2}} - 2a_{0}^{2}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{s}}} \right)}}}. \\ \end{gathered} $$

(A.39)

There definitely exists a time instant ${{t}_{{{{e}_{{{\text{ref}}}}}}}}\; \geqslant \;t_{0}^{ + } + {{T}_{s}}$ and constants T → 0, a₀ > λ_max(P)${{\bar {\omega }}_{r}}{{b}_{{\max }}}\bar {\beta }_{{\max }}^{{\frac{1}{2}}}$ such that for all t $\geqslant$ ${{t}_{{{{e}_{{{\text{ref}}}}}}}}$ it holds that

$$\begin{aligned} - {{\lambda }_{{\min }}}(Q) + 2 + 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}\left( {\left( {{{\beta }_{{\max }}} + \frac{{2{{w}_{{1\max }}}}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right){{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {{{t}_{{{{e}_{{ref}}}}}} - t_{0}^{ + } - {{T}_{s}}} \right)}}}} \right. \\ \left. { + \;\frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}} + {{{\dot {\mathcal{K}}}}_{{\max }}}T} \right) = - {{c}_{1}} < 0, \\ \lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}{{{\bar {\beta }}}_{{\max }}} - 2a_{0}^{2} = - {{c}_{2}} < 0.\quad \; \\ \end{aligned} $$

(A.40)

Then the upper bound for the derivative (A.39) for all t $\geqslant$ ${{t}_{{{{e}_{{{\text{ref}}}}}}}}$ is obtained as

$${{\dot {V}}_{{{{e}_{{{\text{ref}}}}}}}}\;\leqslant \; - {\kern 1pt} {{\eta }_{{{{{\bar {e}}}_{{{\text{ref}}}}}}}}{{V}_{{{\text{ref}}}}} + \lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}{{\left( {\frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right)}^{2}} + \lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}\dot {\mathcal{K}}_{{\max }}^{2}{{T}^{2}},$$

(A.41)

where ${{\eta }_{{{{{\bar {e}}}_{{{\text{ref}}}}}}}} = \min \left\{ {\frac{{{{c}_{1}}}}{{{{\lambda }_{{\max }}}(P)}},\,\,\frac{{{{c}_{2}}{{\gamma }_{1}}}}{{4a_{0}^{2}}}} \right\}$.

The solution of the differential inequality (A.41) for all t $\geqslant$ ${{t}_{{{{e}_{{{\text{ref}}}}}}}}$ is written as

$$\begin{gathered} {{V}_{{{{e}_{{{\text{ref}}}}}}}}(t)\;\leqslant \;{{e}^{{ - {{\eta }_{{{{{\bar {e}}}_{{{\text{ref}}}}}}}}\left( {t - {{t}_{{{{e}_{{{\text{ref}}}}}}}}} \right)}}}{{V}_{{{{e}_{{{\text{ref}}}}}}}}\left( {{{t}_{{{{e}_{{{\text{ref}}}}}}}}} \right) \\ + \;\frac{1}{{{{\eta }_{{{{{\bar {e}}}_{{{\text{ref}}}}}}}}}}\left( {\lambda _{{\max }}^{2}(P)\bar {\omega }_{r}^{2}b_{{\max }}^{2}{{{\left( {\frac{{{{\gamma }_{1}}{{w}_{{2\max }}}(T)}}{{{{\Omega }_{{{\text{LB}}}}}}}} \right)}}^{2}} + \lambda _{{\max }}^{2}(P)b_{{\max }}^{2}\bar {\omega }_{r}^{2}\dot {\mathcal{K}}_{{\max }}^{2}{{T}^{2}}} \right), \\ \end{gathered} $$

(A.42)

which completes the proof of statement (ii) of theorem.

Step 3. Owing to (A.36) and (A.42), the error $\tilde {\theta }$(t) is bounded for all t $\geqslant$ $t_{0}^{ + }$ + T_s, and the error e_ref(t)—for all t $\geqslant$ ${{t}_{{{{e}_{{{\text{ref}}}}}}}}$. Then, to prove the statement (i), we need to show that $\tilde {\theta }$(t) is bounded over $\left[ {\left. {t_{0}^{ + },\,\,t_{0}^{ + } + {{T}_{s}}} \right)} \right.$, and e_ref(t) is bounded over $\left[ {\left. {t_{0}^{ + },\,\,t_{{{{e}_{{{\text{ref}}}}}}}^{{}}} \right)} \right.$.

In the conservative case, the inequality Ω(t) $\leqslant$ Ω_LB is satisfied over $\left[ {\left. {t_{0}^{ + },\,\,t_{0}^{ + } + {{T}_{s}}} \right)} \right.$, whence, owing to $\dot {\tilde {\theta }}$(t) = 0_n + 1, if Assumption 1 is met, it follows that the parametric error $\tilde {\theta }$(t) = $\hat {\theta }\left( {t_{0}^{ + }} \right)$ – θ(t) is bounded over $\left[ {\left. {t_{0}^{ + },\,\,t_{0}^{ + } + {{T}_{s}}} \right)} \right.$ and, as a consequence, for all t $\geqslant$ $t_{0}^{ + }$.

Considering the time range $\left[ {\left. {t_{0}^{ + },\,\,t_{{{{e}_{{{\text{ref}}}}}}}^{{}}} \right)} \right.$ and taking into account the notation from (A.3), (A.18), the error equation (3.1) is written in the following form:

$${{\dot {e}}_{{{\text{ref}}}}}(t) = \left( {{{A}_{{{\text{ref}}}}} + {{e}_{n}}b(t)\left( {{{{\hat {\theta }}}_{x}}(t) - {{k}_{x}}(t)} \right)} \right){{e}_{{{\text{ref}}}}}(t) + {{e}_{n}}b(t)\left( {{{{\hat {\theta }}}^{{\text{T}}}}(t) - {{\mathcal{K}}^{{\text{T}}}}(t)} \right){{\omega }_{r}}(t),$$

which, as it has been proved that $\tilde {\theta }$(t) is bounded for all t $\geqslant$ $t_{0}^{ + }$ and Assumptions 1 and 2 are met, allows one, using Theorem 3.2 from [20], to make the conclusion that (1) e_ref(t) is bounded over $\left[ {\left. {t_{0}^{ + },\,\,t_{{{{e}_{{{\text{ref}}}}}}}^{{}}} \right)} \right.$, (2) ξ(t) ∈ L_∞ for all t $\geqslant$ $t_{0}^{ + }$.