Exponentially Stable Adaptive Control. Part II. Switched Systems

Abstract

An adaptive state-feedback control system for a class of linear systems with piecewise-constant unknown parameters is proposed. The solution ensures a global exponential stability of a closed-loop system under condition that a regressor is finitely exciting after each parameters switch, and does not require neither any knowledge of a plant input matrix, nor the switching time instants. The obtained theoretical results are corroborated by numerical simulations.

APPENDIX

Proof of Proposition 1. The proof of ξ(t) exponential stability is divided into two steps. The first one is to show that $\tilde {\theta }$(t) exponentially converges to zero without regard to boundedness of e_ref(t) and ω(t). Using the obtained result, the second step is to show convergence of e_ref(t).

Step 1. The equation for $\tilde {\theta }$(t) = $\hat {\theta }$(t) – θ(t) obtained from (3.1) is solved:

$$\tilde {\theta }(t) = \phi \left( {t,\,\,t_{0}^{ + }} \right)\tilde {\theta }\left( {t_{0}^{ + }} \right) - \int\limits_{t_{0}^{ + }}^t {\phi (t,\,\,\tau )\sum\limits_{q = 1}^i {\Delta _{q}^{\theta }\delta \left( {\tau - t_{q}^{ + }} \right)d\tau ,} } $$

(A.1)

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

Using the sifting property of the Dirac function:

$$\int\limits_{t_{0}^{ + }}^t {f(\tau )\delta \left( {\tau - t_{q}^{ + }} \right)d\tau = f\left( {t_{q}^{ + }} \right)h\left( {t - t_{q}^{ + }} \right),\,\,\forall f(t),} $$

(A.2)

it is obtained from (A.1):

$$\begin{gathered} \left\| {\tilde {\theta }(t)} \right\|\;\leqslant \;\phi \left( {t,\,\,t_{0}^{ + }} \right)\left\| {\tilde {\theta }\left( {t_{0}^{ + }} \right)} \right\| + \sum\limits_{q = 1}^i {\phi \left( {t,\,\,t_{q}^{ + }} \right)\left\| {\Delta _{q}^{\theta }} \right\|h\left( {t - t_{q}^{ + }} \right)} \\ = \underbrace {\left( {\left\| {\tilde {\theta }\left( {t_{0}^{ + }} \right)} \right\| + \sum\limits_{q = 1}^i {\phi \left( {t_{0}^{ + },\,\,t_{q}^{ + }} \right)\left\| {\Delta _{q}^{\theta }} \right\|h\left( {t - t_{q}^{ + }} \right)} } \right)}_{\beta (t)}\phi \left( {t,\,\,t_{0}^{ + }} \right), \\ \end{gathered} $$

(A.3)

where ϕ($t_{0}^{ + }$, $t_{q}^{ + }$) = ϕ^–1($t_{q}^{ + }$, $t_{0}^{ + }$) = ϕ^–1(t, $t_{0}^{ + }$)ϕ(t, $t_{q}^{ + }$) = ϕ($t_{0}^{ + }$, t)ϕ(t, $t_{q}^{ + }$).

To prove the exponential stability of $\tilde {\theta }$(t) it remains to show that β(t) is bounded. If the number of parameters switches is finite: i $\leqslant$ i_max < ∞, then as:

(a) when i is finite, the time instants $t_{i}^{ + }$ are also finite (we do not consider the case of switches at infinite time: $\forall$i $t_{i}^{ + }$ ≠ ∞),

(b) ϕ($t_{0}^{ + }$, $t_{q}^{ + }$) is bounded in case $t_{q}^{ + }$ is finite, we have the following upper bounds:

$$\beta (t)\;\leqslant \;\left\| {\tilde {\theta }\left( {t_{0}^{ + }} \right)} \right\| + \sum\limits_{q = 1}^{{{i}_{{\max }}}} {\phi \left( {t_{0}^{ + },\,\,t_{q}^{ + }} \right)\left\| {\Delta _{q}^{\theta }} \right\|h\left( {t - t_{q}^{ + }} \right) = {{\beta }_{{\max }}}.} $$

(A.4)

If  $\forall$q ∈ $\mathbb{N}$ ||$\Delta _{q}^{\theta }$|| $\leqslant$ c_qϕ($t_{q}^{ + }$, $t_{0}^{ + }$)), c_q > c_{q + 1}, then even in case of unbounded i it holds that:

$$\beta (t)\;\leqslant \;\left\| {\tilde {\theta }\left( {t_{0}^{ + }} \right)} \right\| + \sum\limits_{q = 1}^i {{{c}_{q}}h\left( {t - t_{q}^{ + }} \right) = {{\beta }_{{\max }}}.} $$

(A.5)

The series from (A.5) is constant sign one, and all its subsums are bounded owing to monotonicity of 0 < c_{q + 1} < c_q, therefore, $\sum\nolimits_{q = 1}^\infty {{{c}_{q}}}$h(t – $t_{q}^{ + }$) < ∞, which results in β(t) $\leqslant$ β_max.

It immediately follows from the boundedness of (A.4) or (A.5) that:

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

(A.6)

The next aim is to analyze the behavior of the tracking error e_ref(t).

Step 2. The following quadratic form is introduced:

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

(A.7)

where $\bar {e}_{{{\text{ref}}}}^{{}}$ = $\left[ {e_{{{\text{ref}}}}^{{\text{T}}}{{{(t)}}_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \right.$ ${{\left. {{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}(t - t_{0}^{ + })}}}} \right]}^{{\text{T}}}}$, a₀ > 0, and P is a solution of the below-given set of equations when K = I_n×n:

$$A_{{{\text{ref}}}}^{{\text{T}}}P + P{{A}_{{{\text{ref}}}}} = - Q{{Q}^{{\text{T}}}} - \mu P,\quad P{{I}_{{n \times n}}} = QK,$$

$${{K}^{{\text{T}}}}K = D + {{D}^{{\text{T}}}},$$

which is equivalent to the Riccati equation $A_{{{\text{ref}}}}^{{\text{T}}}$P + PA_ref + PP ^T + μP = 0_n×n.

The derivative of (A.7) is written as:

$$\begin{gathered} {{{\dot {V}}}_{{{{e}_{{{\text{ref}}}}}}}} = e_{{{\text{ref}}}}^{{\text{T}}}\left( {A_{{{\text{ref}}}}^{{\text{T}}}P + P{{A}_{{{\text{ref}}}}}} \right){{e}_{{{\text{ref}}}}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} + 2e_{{{\text{ref}}}}^{{\text{T}}}P{{I}_{n}}{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega \\ = - \mu e_{{{\text{ref}}}}^{{\text{T}}}P{{e}_{{{\text{ref}}}}} - e_{{{\text{ref}}}}^{{\text{T}}}Q{{Q}^{{\text{T}}}}{{e}_{{{\text{ref}}}}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} + {\text{tr}}\left( {2{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega e_{{{\text{ref}}}}^{{\text{T}}}QK} \right). \\ \end{gathered} $$

(A.8)

As KK  ^T = K  ^TK = I_{n × n}, Eq. (A.8) is rewritten as:

$$\begin{gathered} {{{\dot {V}}}_{{{{e}_{{{\text{ref}}}}}}}} = - \mu e_{{{\text{ref}}}}^{{\text{T}}}P{{e}_{{{\text{ref}}}}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} - e_{{{\text{ref}}}}^{{\text{T}}}QK{{K}^{{\text{T}}}}{{Q}^{{\text{T}}}}{{e}_{{{\text{ref}}}}} + {\text{tr}}\left( {2{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega e_{{{\text{ref}}}}^{{\text{T}}}QK} \right) \\ = - \mu e_{{{\text{ref}}}}^{{\text{T}}}P{{e}_{{{\text{ref}}}}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} + {\text{tr}}\left( { - {{K}^{{\text{T}}}}{{Q}^{{\text{T}}}}{{e}_{{{\text{ref}}}}}e_{{{\text{ref}}}}^{{\text{T}}}QK + 2{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega e_{{{\text{ref}}}}^{{\text{T}}}QK} \right). \\ \end{gathered} $$

(A.9)

Completing the square

$$\begin{gathered} {{K}^{{\text{T}}}}{{Q}^{{\text{T}}}}{{e}_{{{\text{ref}}}}}e_{{{\text{ref}}}}^{{\text{T}}}QK - 2{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega e_{{{\text{ref}}}}^{{\text{T}}}QK + {{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega {{\omega }^{{\text{T}}}}\tilde {\theta }B_{i}^{{\text{T}}} \\ = \left( {{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega - {{K}^{{\text{T}}}}{{Q}^{{\text{T}}}}{{e}_{{{\text{ref}}}}}} \right){{\left( {{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega - {{K}^{{\text{T}}}}{{Q}^{{\text{T}}}}{{e}_{{{\text{ref}}}}}} \right)}^{{\text{T}}}}\; \geqslant \;0, \\ \end{gathered} $$

(A.10)

we have:

$$\begin{gathered} {{{\dot {V}}}_{{{{e}_{{{\text{ref}}}}}}}} = - \mu e_{{{\text{ref}}}}^{{\text{T}}}P{{e}_{{{\text{ref}}}}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} \\ + \,\,{\text{tr}}\left( { - {{K}^{{\text{T}}}}{{Q}^{{\text{T}}}}{{e}_{{{\text{ref}}}}}e_{{{\text{ref}}}}^{{\text{T}}}QK + 2{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega e_{{{\text{ref}}}}^{{\text{T}}}QK \pm {{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega {{\omega }^{{\text{T}}}}\tilde {\theta }B_{i}^{{\text{T}}}} \right) \\ \leqslant - {\kern 1pt} \mu e_{{{\text{ref}}}}^{{\text{T}}}P{{e}_{{{\text{ref}}}}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} + {\text{tr}}\left( {{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega {{\omega }^{{\text{T}}}}\tilde {\theta }B_{i}^{{\text{T}}}} \right) \\ \leqslant - {\kern 1pt} \mu {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} + b_{{\max }}^{2}{{\lambda }_{{\max }}}(\omega {{\omega }^{{\text{T}}}}){\text{||}}\tilde {\theta }{\text{|}}{{{\text{|}}}^{2}} \\ \leqslant - {\kern 1pt} \mu {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} + b_{{\max }}^{2}\beta _{{\max }}^{2}{{\lambda }_{{\max }}}(\omega {{\omega }^{{\text{T}}}}){{\phi }^{2}}\left( {t,t_{0}^{ + }} \right) \\ \leqslant - {\kern 1pt} \mu {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} + b_{{\max }}^{2}\beta _{{\max }}^{2}{{\lambda }_{{\max }}}(\omega {{\omega }^{{\text{T}}}}){{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}}, \\ \end{gathered} $$

(A.11)

where $\forall$i ∈ $\mathbb{N}$ ||B_i|| $\leqslant$ b_max follows from the fact that the pair (A_i, B_i) is controllable.

The exponential vanishing of the third term of (A.11) is required to unsure the exponential stability of the tracking error e_ref(t), which, in its turn, requires:

$$\chi (t) = {{\lambda }_{{\max }}}\left( {\omega (t){{\omega }^{{\text{T}}}}(t)} \right){{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}}\,\leqslant \,{{\chi }_{{{\text{UB}}}}},$$

(A.12)

where χ_UB > 0.

The growth rate of  λ_max(ω(t)ω^T(t)) is estimated via introduction of ${{L}_{{{{e}_{{{\text{ref}}}}}}}}$ = $e_{{{\text{ref}}}}^{{\text{T}}}$Pe_ref:

$$\begin{gathered} {{{\dot {L}}}_{{{{e}_{{{\text{ref}}}}}}}} = e_{{{\text{ref}}}}^{{\text{T}}}(A_{{{\text{ref}}}}^{{\text{T}}}P + P{{A}_{{{\text{ref}}}}}){{e}_{{{\text{ref}}}}} + 2e_{{{\text{ref}}}}^{{\text{T}}}P{{B}_{i}}{{{\tilde {\theta }}}^{{\text{T}}}}\omega \hfill \\ \,\,\,\,\,\,\,\,\leqslant - {\kern 1pt} \mu e_{{{\text{ref}}}}^{{\text{T}}}P{{e}_{{{\text{ref}}}}} + 2e_{{{\text{ref}}}}^{{\text{T}}}P{{B}_{i}}{{{\tilde {K}}}_{x}}x + 2e_{{{\text{ref}}}}^{{\text{T}}}P{{B}_{i}}{{{\tilde {K}}}_{r}}r \hfill \\ \,\,\,\,\,\,\,\,\leqslant - {\kern 1pt} \mu {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} + 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}{\text{||}}{{e}_{{{\text{ref}}}}}{\text{||}}\,{\text{||}}\tilde {\theta }{\text{||}}\,{\text{||}}x{\text{||}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\kern 1pt} {\text{ + 2}}{{\lambda }_{{\max }}}(P){{b}_{{\max }}}{\text{||}}{{e}_{{{\text{ref}}}}}{\text{||}}\,{\text{||}}\tilde {\theta }{\text{||}}{{r}_{{\max }}} \hfill \\ \,\,\,\,\,\,\,\,\leqslant - {\kern 1pt} \mu {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} + 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}{\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}}{\text{||}}\tilde {\theta }{\text{||}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\kern 1pt} {\text{ + 2}}{{\lambda }_{{\max }}}(P){{b}_{{\max }}}(x_{{{\text{ref}}}}^{{{\text{UB}}}} + {{r}_{{\max }}}){\text{||}}{{e}_{{{\text{ref}}}}}{\text{||}}\,{\text{||}}\tilde {\theta }{\text{||}} \hfill \\ \,\,\,\,\,\,\,\,\leqslant \,( - \mu {{\lambda }_{{\max }}}(P) + 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}{\text{||}}\tilde {\theta }{\text{||}}){\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\kern 1pt} + {\text{ }}2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}(x_{{{\text{ref}}}}^{{{\text{UB}}}} + {{r}_{{\max }}}){\text{||}}{{e}_{{{\text{ref}}}}}{\text{||}}\,{\text{||}}\tilde {\theta }{\text{||}}, \hfill \\ \end{gathered} $$

(A.13)

where ||x_ref(t)|| $\leqslant$ $x_{{{\text{ref}}}}^{{{\text{UB}}}}$ is an upper bound of the reference model states norm.

The error $\tilde {\theta }$(t) is bounded, then, considering the conservative case, it is obtained from (A.13) that:

$${{\dot {L}}_{{{{e}_{{{\text{ref}}}}}}}}\;\leqslant \;{{c}_{1}}{\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} + 2{{c}_{2}}{\text{||}}{{e}_{{{\text{ref}}}}}{\text{||}},$$

(A.14)

where

$${{c}_{1}} = - \mu {{\lambda }_{{\min }}}(P) + 2{{\lambda }_{{\max }}}(P){{b}_{{\max }}}{{\beta }_{{\max }}} > 0,$$

$${{c}_{2}} = {{\lambda }_{{\max }}}(P){{b}_{{\max }}}{{\beta }_{{\max }}}(x_{{{\text{ref}}}}^{{{\text{UB}}}} + {{r}_{{\max }}}).$$

Applying the Young’s inequality ab $\leqslant$ $\frac{1}{2}$a² + $\frac{1}{2}$b², we have from (A.14) that:

$${{\dot {L}}_{{{{e}_{{{\text{ref}}}}}}}}\;\leqslant \;\left( {{{c}_{1}} + 2c_{2}^{2}} \right){\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} + 0.5\;\leqslant \;\left( {{{c}_{1}} + 2c_{2}^{2}} \right){\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} + 1 = \frac{{{{c}_{1}} + 2c_{2}^{2}}}{{{{\lambda }_{{\max }}}(P)}}{{L}_{{{{e}_{{{\text{ref}}}}}}}} + 1.$$

(A.15)

Equation (A.15) is solved using

$$\begin{gathered} {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}(t){\text{|}}{{{\text{|}}}^{2}}\;\leqslant \;{{L}_{{{{e}_{{{\text{ref}}}}}}}}(t),\quad {{L}_{{{{e}_{{{\text{ref}}}}}}}}(t)\;\leqslant \;{{\lambda }_{{\max }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}(t){\text{|}}{{{\text{|}}}^{2}}: \\ {\text{||}}{{e}_{{{\text{ref}}}}}(t){\text{||}}\;\leqslant \;\sqrt {\frac{{{{\lambda }_{{\max }}}(P)}}{{{{\lambda }_{{\min }}}(P)}}} {{e}^{{\frac{{{{c}_{1}} + 2c_{2}^{2}}}{{2{{\lambda }_{{\max }}}(P)}}\left( {t - t_{0}^{ + }} \right)}}}\left\| {{{e}_{{{\text{ref}}}}}\left( {t_{0}^{ + }} \right)} \right\| + \sqrt {\frac{{{{\lambda }_{{\max }}}(P){{e}^{{\frac{{{{c}_{1}} + 2c_{2}^{2}}}{{{{\lambda }_{{\max }}}(P)}}\left( {t - t_{0}^{ + }} \right)}}}}}{{{{\lambda }_{{\min }}}(P)({{c}_{1}} + 2c_{2}^{2})}}} . \\ \end{gathered} $$

(A.16)

Therefore, the growth rate of x(t) does not exceed exponential one, and thus, as r(t) is bounded, it holds that:

$$\begin{gathered} {{\lambda }_{{\max }}}\left( {\omega (t){{\omega }^{{\text{T}}}}(t)} \right) = {\text{tr}}\left( {\omega (t){{\omega }^{{\text{T}}}}(t)} \right) = \sum\limits_{i = 1}^n {x_{i}^{2}(t)} \\ + \,\sum\limits_{i = 1}^m {r_{i}^{2}(t)\;\leqslant \;{{{\bar {c}}}_{0}}} {{e}^{{{{{\bar {c}}}_{1}}\left( {t - t_{0}^{ + }} \right)}}},\quad {{{\bar {c}}}_{0}} > 0,\quad {{{\bar {c}}}_{1}} > 0. \\ \end{gathered} $$

(A.17)

The estimate (A.17) is substituted into (A.12) to obtain that (A.12) holds if γ₁ > 0 is sufficiently large. Equation (A.12) is used in (A.11) to have:

$${{\dot {V}}_{{{{e}_{{{\text{ref}}}}}}}}\;\leqslant \; - \mu {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}{\text{|}}{{{\text{|}}}^{2}} - 2a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} + a_{0}^{2}{{e}^{{ - {{\gamma }_{1}}\left( {t - t_{0}^{ + }} \right)}}} = - {{\bar {\eta }}_{{{{e}_{{{\text{ref}}}}}}}}{{V}_{{{{e}_{{{\text{ref}}}}}}}},$$

(A.18)

where

$$a_{0}^{2} = b_{{\max }}^{2}\beta _{{\max }}^{2}{{\chi }_{{{\text{UB}}}}},\quad {{\bar {\eta }}_{{{{e}_{{{\text{ref}}}}}}}} = \min \left\{ {\frac{{\mu {{\lambda }_{{\min }}}(P)}}{{{{\lambda }_{{\max }}}(P)}},\,\,\frac{{{{\gamma }_{1}}}}{2}} \right\}.$$

The differential inequality (A.18) is solved to write:

$${{V}_{{{{e}_{{{\text{ref}}}}}}}}(t)\;\leqslant \;{{e}^{{ - {{{\bar {\eta }}}_{{{{e}_{{{\text{ref}}}}}}}}\left( {t - t_{0}^{ + }} \right)}}}{{V}_{{{{e}_{{{\text{ref}}}}}}}}\left( {t_{0}^{ + }} \right).$$

(A.19)

Therefore, the tracking error e_ref(t) exponentially convergences to zero:

$${\text{||}}{{e}_{{{\text{ref}}}}}(t){\text{||}}\;\leqslant \;\sqrt {\frac{{{{\lambda }_{M}}}}{{{{\lambda }_{m}}}}} \left\| {{{e}_{{{\text{ref}}}}}\left( {t_{0}^{ + }} \right)} \right\|{{e}^{{ - {{\eta }_{{{{e}_{{{\text{ref}}}}}}}}\left( {t - t_{0}^{ + }} \right)}}},$$

(A.20)

where

$${{\eta }_{{{{e}_{{{\text{ref}}}}}}}} = \frac{1}{2}{{\bar {\eta }}_{{{{e}_{{{\text{ref}}}}}}}}.$$

Having combined (A.20) and (A.6), it is written:

$${\text{||}}\xi (t){\text{||}}\;\leqslant \;\max \left\{ {\sqrt {\frac{{{{\lambda }_{M}}}}{{{{\lambda }_{m}}}}} \left\| {{{e}_{{{\text{ref}}}}}\left( {t_{0}^{ + }} \right)} \right\|,{{\beta }_{{\max }}}} \right\}{{e}^{{ - {{\eta }_{{{{e}_{{{\text{ref}}}}}}}}\left( {t - t_{0}^{ + }} \right)}}},$$

(A.21)

which completes the proof of Proposition 1.

Proof of Proposition 2. The expression x(t) – l$\bar {x}$(t) is differentiated:

$$\dot {x}(t) - l\,\dot {\bar {x}}(t) = - l(x(t) - l\,\bar {x}(t)) + {{\vartheta }^{{\text{T}}}}(t)\Phi (t).$$

(A.22)

The differential Eq. (A.22) is solved to obtain:

$$\begin{gathered} x(t) - l\,\bar {x}(t) = {{e}^{{ - l\left( {t - \hat {t}_{i}^{ + }} \right)}}}x\left( {\hat {t}_{i}^{ + }} \right) + \int\limits_{\hat {t}_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}{{\vartheta }^{{\text{T}}}}(\tau )\Phi (\tau )d\tau \pm {{\vartheta }^{{\text{T}}}}(t)\bar {\Phi }(t)} \\ = {{{\bar {\vartheta }}}^{{\text{T}}}}(t)\bar {\varphi }(t) + \int\limits_{\hat {t}_{i}^{ + }}^t {{{e}^{{ - k(t - \tau )}}}{{\vartheta }^{{\text{T}}}}(\tau )\Phi (\tau )d\tau - {{\vartheta }^{{\text{T}}}}(t)\bar {\Phi }(t),} \\ \end{gathered} $$

(A.23)

where ${{\bar {\vartheta }}^{{\text{T}}}}$(t) = [A_i B_i x($\hat {t}_{i}^{ + }$)] ∈ R^{n × (n + m + 1)}.

Having applied (4.2) to the left- and right-hand parts of (A.23), it is obtained:

$$\begin{gathered} \forall t\; \geqslant \;t_{0}^{ + }\,\,\,{{{\bar {z}}}_{n}}(t) = {{n}_{s}}(t)\left[ {x(t) - l\bar {x}(t)} \right] = {{{\bar {\vartheta }}}^{{\text{T}}}}(t){{{\bar {\varphi }}}_{n}}(t) + {{{\bar {\varepsilon }}}_{0}}(t), \\ {{{\bar {\varepsilon }}}_{0}}(t) = {{n}_{s}}(t)\left( {\int\limits_{\hat {t}_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}} {{\vartheta }^{{\text{T}}}}(\tau )\Phi (\tau )d\tau - {{\vartheta }^{{\text{T}}}}(t)\bar {\Phi }(t)} \right), \\ \end{gathered} $$

(A.24)

where ${{\bar {z}}_{n}}$(t) ∈ Rⁿ, ${{\bar {\varphi }}_{n}}$(t) ∈ R^n+m+1, ${{\bar {\varepsilon }}_{0}}$(t) ∈ Rⁿ.

Considering (4.4), z(t) is multiplied by adj {φ(t)} to write:

$$\begin{gathered} \,\,\,\,\,\,\,\,\,Y(t){\text{:}}\, = {\text{adj}}{\kern 1pt} \{ \varphi (t)\} \left( {z(t) \pm \varphi (t)\bar {\vartheta }(t)} \right) = \Delta (t)\bar {\vartheta }(t) + {{{\bar {\varepsilon }}}_{1}}(t), \\ {\text{adj}}{\kern 1pt} \{ \varphi (t)\} \varphi (t) = \det \{ \varphi (t)\} {{I}_{{(n + m + 1) \times (n + m + 1)}}} = \Delta (t){{I}_{{(n + m + 1) \times (n + m + 1)}}}, \\ {{{\bar {\varepsilon }}}_{1}}(t) = {\text{adj}}{\kern 1pt} \{ \varphi (t)\} \left( {z(t) - \varphi (t)\bar {\vartheta }(t)} \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{gathered} $$

(A.25)

where Y(t) ∈ R^(n+m+1)×n, Δ(t) ∈ R, ${{\bar {\varepsilon }}_{1}}$(t) ∈ R^(n+m+1)×n.

Owing to Δ(t) ∈ R, the elimination (4.5) allow one to obtain from (A.25) that:

$$\begin{gathered} {{z}_{A}}(t) = {{Y}^{{\text{T}}}}(t)\mathfrak{L} = \Delta (t){{A}_{i}} + \bar {\varepsilon }_{1}^{{\text{T}}}(t)\mathfrak{L}, \\ {{z}_{B}}(t) = {{Y}^{{\text{T}}}}(t){{\mathfrak{e}}_{{n + m + 1}}} = \Delta (t){{B}_{i}} + \bar {\varepsilon }_{1}^{{\text{T}}}(t){{\mathfrak{e}}_{{n + m + 1}}}, \\ \mathfrak{L} = {{\left[ {\begin{array}{*{20}{c}} {{{I}_{{n \times n}}}}&{{{0}_{{n \times (m + 1)}}}} \end{array}} \right]}^{{\text{T}}}} \in {{R}^{{(n + m + 1) \times n}}}, \\ {{\mathfrak{e}}_{{n + m + 1}}} = {{\left[ {\begin{array}{*{20}{c}} {{{0}_{{m \times n}}}}&{{{I}_{{m \times m}}}}&{{{0}_{{m \times 1}}}} \end{array}} \right]}^{{\text{T}}}} \in {{R}^{{(n + m + 1) \times m}}}, \\ \end{gathered} $$

(A.26)

where z_A(t) ∈ R^n×n, z_B(t) ∈ R^n×m.

Each equation from (2.7) is left-multiplied by adj {$z_{B}^{{\text{T}}}$(t)z_B(t)}$z_{B}^{{\text{T}}}$(t)Δ(t). Considering (A.26), Eqs. (4.5) are substituted into the result of multiplication, then the obtained equations are combined to have:

$$\begin{gathered} \mathcal{Y}(t) = \mathcal{M}(t)\theta (t) + d(t), \\ \mathcal{Y}(t){\kern 1pt} :\, = \left[ \begin{gathered} {\text{adj}}\left\{ {z_{B}^{{\text{T}}}(t){{z}_{B}}(t)} \right\}z_{B}^{{\text{T}}}(t)(\Delta (t){{A}_{{{\text{ref}}}}} - {{z}_{A}}(t)) \\ {\text{adj}}\left\{ {z_{B}^{{\text{T}}}(t){{z}_{B}}(t)} \right\}z_{B}^{{\text{T}}}(t)\Delta (t){{B}_{{{\text{ref}}}}} \\ \end{gathered} \right], \\ \end{gathered} $$

$${\text{adj}}\left\{ {z_{B}^{{\text{T}}}(t){{z}_{B}}(t)} \right\}z_{B}^{{\text{T}}}(t){{z}_{B}}(t) = \det \left\{ {z_{B}^{{\text{T}}}(t){{z}_{B}}(t)} \right\}{{I}_{{m \times m}}} = \,\mathcal{M}(t){{I}_{{m \times m}}},$$

(A.27)

$$d(t){\kern 1pt} \,\,: = \left[ \begin{gathered} {\text{adj}}\left\{ {z_{B}^{{\text{T}}}(t){{z}_{B}}(t)} \right\}z_{B}^{{\text{T}}}(t)\left( {\bar {\varepsilon }_{1}^{{\text{T}}}(t)\mathfrak{L} + \bar {\varepsilon }_{1}^{{\text{T}}}(t){{\mathfrak{e}}_{{n + m + 1}}}K_{i}^{x}} \right) \\ {\text{adj}}\left\{ {z_{B}^{{\text{T}}}(t){{z}_{B}}(t)} \right\}z_{B}^{{\text{T}}}(t)\bar {\varepsilon }_{1}^{{\text{T}}}(t){{\mathfrak{e}}_{{n + m + 1}}}K_{i}^{r} \\ \end{gathered} \right],$$

where $\mathcal{Y}$(t) ∈ R^{(n + m) × n}, $\mathcal{M}$(t) ∈ R, d(t) ∈ R^{(n + m) × n}.

Considering (A.27), Eq. (4.7a) is solved to have the following expression:

$$\begin{gathered} \Upsilon (t) = \int\limits_{t_{0}^{ + }}^t {{{e}^{{\int_{t_{0}^{ + }}^\tau {kd\tau } }}}} \mathcal{M}(\tau )\theta (\tau )d\tau + \int\limits_{t_{0}^{ + }}^t {{{e}^{{\int_{t_{0}^{ + }}^\tau {kd\tau } }}}} d(\tau )d\tau \pm \Omega (t)\theta (t) = \Omega (t)\theta (t) + w(t), \\ w(t) = \Upsilon (t) - \Omega (t)\theta (t), \\ \end{gathered} $$

(A.28)

which proves that (4.8) can be obtained using the procedures (4.1)–(4.7).

To prove the statement (a), Eq. (4.7b) is solved over both [$\hat {t}_{i}^{ + }$; $t_{i}^{ + }$ + T_i] and [$t_{i}^{ + }$ + T_i; $\hat {t}_{{i + 1}}^{ + }$]:

$$\begin{gathered} \forall t \in \left[ {\hat {t}_{i}^{ + };t_{i}^{ + } + {{T}_{i}}} \right]\quad \Omega (t) = {{\phi }^{{{{k}_{0}}}}}\left( {t,\,\,t_{i}^{ + }} \right)\Omega \left( {\hat {t}_{i}^{ + }} \right) + \int\limits_{\hat {t}_{i}^{ + }}^t {{{\phi }^{{{{k}_{0}}}}}(t,\tau )\mathcal{M}(\tau )d\tau ,} \\ \forall t \in \left[ {t_{i}^{ + } + {{T}_{i}};\hat {t}_{{i + 1}}^{ + }} \right]\quad \Omega (t) = {{\phi }^{{{{k}_{0}}}}}\left( {t,\,\,t_{i}^{ + } + {{T}_{i}}} \right)\Omega \left( {t_{i}^{ + } + {{T}_{i}}} \right) + \int\limits_{t_{i}^{ + } + {{T}_{i}}}^t {{{\phi }^{{{{k}_{0}}}}}(t,\tau )\mathcal{M}(\tau )d\tau .} \\ \end{gathered} $$

(A.29)

It is up to notation proved in [26] that if Φ(t) ∈ FE, $\hat {t}_{i}^{ + }$ $\geqslant$ $t_{i}^{ + }$, then $\forall$t ∈ [$t_{i}^{ + }$ + T_i; $\hat {t}_{{i + 1}}^{ + }$) it holds that Δ_UB $\geqslant$ Δ(t) $\geqslant$ Δ_LB > 0. Then the following holds for the regressor $\mathcal{M}$(t) over the time ranges considered in (A.29):

$$\begin{gathered} \forall t \in \left[ {\hat {t}_{i}^{ + };t_{i}^{ + } + {{T}_{i}}} \right]\,\,\,\mathcal{M}(t) = \det \left\{ {z_{B}^{{\text{T}}}(t){{z}_{B}}(t)} \right\} = {{\Delta }^{m}}(t)\det \left\{ {B_{i}^{{\text{T}}}{{B}_{i}}} \right\} \equiv 0, \hfill \\ \forall t \in \left[ {t_{i}^{ + } + {{T}_{i}};\hat {t}_{{i + 1}}^{ + }} \right]\,\,\,\Delta _{{{\text{UB}}}}^{m}\det \left\{ {B_{i}^{{\text{T}}}{{B}_{i}}} \right\}\; \geqslant \;\mathcal{M}(t)\; \geqslant \;\Delta _{{LB}}^{m}\det \left\{ {B_{i}^{{\text{T}}}{{B}_{i}}} \right\} \equiv 0. \hfill \\ \end{gathered} $$

(A.30)

Having substituted (A.30) into (A.29) and considered 0 $\leqslant$ ϕ(t, τ) $\leqslant$ 1, the bounds for Ω(t) are obtained:

$$\begin{gathered} \forall t \in \left[ {\hat {t}_{0}^{ + };\,\,t_{0}^{ + } + {{T}_{0}}} \right]\quad \Omega (t) \equiv 0, \\ \forall i\; \geqslant \;1\,\,\forall t \in \left[ {\hat {t}_{i}^{ + };\,\,t_{i}^{ + } + {{T}_{i}}} \right]\quad \Omega \left( {\hat {t}_{i}^{ + }} \right)\; \geqslant \;\Omega (t)\; \geqslant \;{{\phi }^{{{{k}_{0}}}}}\left( {t_{i}^{ + } + {{T}_{i}},\,\,\hat {t}_{i}^{ + }} \right)\Omega \left( {\hat {t}_{i}^{ + }} \right) > 0, \\ \forall t \in \left[ {t_{i}^{ + } + {{T}_{i}};\,\,\hat {t}_{{i + 1}}^{ + }} \right]\quad \Omega \left( {t_{i}^{ + } + {{T}_{i}}} \right) + \left( {\hat {t}_{{i + 1}}^{ + } - t_{i}^{ + } - {{T}_{i}}} \right)\Delta _{{{\text{UB}}}}^{m}{\text{det}}\left\{ {B_{i}^{{\text{T}}}{{B}_{i}}} \right\}\,\,\,\,\,\,\,\, \\ \geqslant \;\Omega (t)\; \geqslant \;{{\phi }^{{{{k}_{0}}}}}\left( {\hat {t}_{{i + 1}}^{ + },\,\,t_{i}^{ + } + {{T}_{i}}} \right)\left( {\Omega \left( {t_{i}^{ + } + {{T}_{i}}} \right)} \right. \\ \,\left. { + \left( {\hat {t}_{{i + 1}}^{ + } - t_{i}^{ + } - {{T}_{i}}} \right)\Delta _{{LB}}^{m}\det \left\{ {B_{i}^{{\text{T}}}{{B}_{i}}} \right\}} \right) > 0. \\ \end{gathered} $$

(A.31)

From which we have:

$$\forall t\; \geqslant \;t_{0}^{ + } + {{T}_{0}}\quad {{\Omega }_{{{\text{UB}}}}}\; \geqslant \;\Omega (t)\; \geqslant \;{{\Omega }_{{{\text{LB}}}}} > 0,$$

$${{\Omega }_{{{\text{LB}}}}} = \mathop {\max }\limits_{\forall i\; \geqslant \;1} \left\{ \begin{gathered} {{\phi }^{{{{k}_{0}}}}}\left( {\hat {t}_{{i + 1}}^{ + },\,\,t_{i}^{ + } + {{T}_{i}}} \right)\left( {\Omega \left( {t_{i}^{ + } + {{T}_{i}}} \right)} \right. \hfill \\ \left. { + \left( {\hat {t}_{{i + 1}}^{ + } - t_{i}^{ + } - {{T}_{i}}} \right)\Delta _{{{\text{LB}}}}^{m}\det \left\{ {B_{i}^{{\text{T}}}{{B}_{i}}} \right\}} \right), \hfill \\ {{\phi }^{{{{k}_{0}}}}}\left( {t_{i}^{ + } + {{T}_{i}},\,\,\hat {t}_{i}^{ + }} \right)\Omega \left( {\hat {t}_{i}^{ + }} \right) \hfill \\ \end{gathered} \right\},$$

(A.32)

$${{\Omega }_{{{\text{UB}}}}} = \mathop {\max }\limits_{\forall i\; \geqslant \;1} \left\{ {\Omega \left( {\hat {t}_{i}^{ + }} \right),\,\,\Omega \left( {t_{i}^{ + } + {{T}_{i}}} \right) + \left( {\hat {t}_{{i + 1}}^{ + } - t_{i}^{ + } - {{T}_{i}}} \right)\Delta _{{{\text{UB}}}}^{m}\det \left\{ {B_{i}^{{\text{T}}}{{B}_{i}}} \right\}} \right\},$$

which completes the proof of the statement (a).

To prove the statement (b) the disturbance w(t) is differentiated:

$$\begin{gathered} \dot {w}(t) = \dot {\Upsilon }(t) - \dot {\Omega }(t)\theta (t) - \Omega (t)\dot {\theta }(t) \hfill \\ \,\,\,\,\,\,\,\, = - k(\Upsilon (t) - \mathcal{Y}(t)) + k(\Omega (t) - \mathcal{M}(t))\theta (t) - \Omega (t)\dot {\theta }(t) \hfill \\ \,\,\,\,\,\,\,\, = - k(\Upsilon (t) - \mathcal{M}(t)\theta (t) - d(t)) + k(\Omega (t) - \mathcal{M}(t))\theta (t) - \Omega (t)\dot {\theta }(t) \hfill \\ \,\,\,\,\,\,\,\, = - k(\Upsilon (t) - \Omega (t)\theta (t)) - \Omega (t)\dot {\theta }(t) + kd(t) \hfill \\ \,\,\,\,\,\,\,\, = - kw(t) - \Omega (t)\dot {\theta }(t) + kd(w),\quad w\left( {t_{0}^{ + }} \right) = {{0}_{{(n + m) \times m}}}. \hfill \\ \end{gathered} $$

(A.33)

The next aim is to show that the identical equality d(t) ≡ 0 holds when $\tilde {t}_{i}^{ + }$ = 0. It follows from the definition (A.27) that ${{\bar {\varepsilon }}_{1}}$(t) ≡ 0 $\Leftrightarrow$ d(t) ≡ 0. Let it be assumed that $\forall$i ∈ $\mathbb{N}$  $\hat {t}_{i}^{ + }$ $\geqslant$ $t_{i}^{ + }$, then the definition of  ${{\bar {\varepsilon }}_{1}}$(t) is obtained over the time ranges [$\hat {t}_{i}^{ + }$;  $t_{{i + 1}}^{ + }$) and [$t_{i}^{ + }$;  $\hat {t}_{i}^{ + }$):

$$\begin{gathered} \forall t \in \left[ {\left. {\hat {t}_{i}^{ + };\,\,t_{{i + 1}}^{ + }} \right)} \right.\quad \vartheta (t) = {{\vartheta }_{i}} \\ \Updownarrow \\ {{{\bar {\varepsilon }}}_{1}}(t) = {\text{adj}}{\kern 1pt} \{ \varphi (t)\} \int\limits_{\hat {t}_{i}^{ + }}^t {{{e}^{{ - \int\limits_{\hat {t}_{i}^{ + }}^\tau {\sigma ds} }}}{{{\bar {\varphi }}}_{n}}(\tau )\bar {z}_{n}^{{\text{T}}}(\tau )d\tau - \Delta (t){{{\bar {\vartheta }}}_{i}}} \\ = {\text{adj}}\{ \varphi (t)\} \left( {\int\limits_{\hat {t}_{i}^{ + }}^t {{{e}^{{ - \int\limits_{\hat {t}_{i}^{ + }}^\tau {\sigma ds} }}}{{{\bar {\varphi }}}_{n}}(\tau )\bar {\varphi }_{n}^{{\text{T}}}(\tau )d\tau {{{\bar {\vartheta }}}_{i}} + \int\limits_{\hat {t}_{i}^{ + }}^t {{{e}^{{ - \int\limits_{\hat {t}_{i}^{ + }}^\tau {\sigma ds} }}}{{{\bar {\varphi }}}_{n}}(\tau )\bar {\varepsilon }_{0}^{{\text{T}}}(\tau )d\tau } } } \right) \\ - \,\Delta (t){{{\bar {\vartheta }}}_{i}} = \Delta (t){{{\bar {\vartheta }}}_{i}} - \Delta (t){{{\bar {\vartheta }}}_{i}} + \int\limits_{\hat {t}_{i}^{ + }}^t {{{e}^{{ - \int\limits_{\hat {t}_{i}^{ + }}^\tau {\sigma ds} }}}{{{\bar {\varphi }}}_{n}}(\tau )\bar {\varepsilon }_{0}^{{\text{T}}}(\tau )d\tau = {{0}_{{(n + m + 1) \times n}}}.} \\ \end{gathered} $$

(A.34)

At the same time:

$$\begin{matrix} \forall t\in \left[ \left. t_{i-1}^{+};\,\,t_{i}^{+} \right) \right.\quad \vartheta (t)={{\vartheta }_{i-1}};\,\,\forall t\in \left[ \left. t_{i}^{+};\,\,\hat{t}_{i}^{+} \right) \right.\,\,\,\,\vartheta (t)={{\vartheta }_{i}} \\ \Updownarrow \\ \forall t\in \left[ \left. t_{i}^{+};\,\,\hat{t}_{i}^{+} \right) \right.,{{{\bar{\varepsilon }}}_{1}}(t)=\text{adj}\{\varphi (t)\}\int\limits_{\hat{t}_{i}^{+}}^{t}{{{e}^{-\int\limits_{\hat{t}_{i}^{+}}^{\tau }{\sigma ds}}}{{{\bar{\varphi }}}_{n}}(\tau )\bar{z}_{n}^{\text{T}}(\tau )d\tau -\Delta (t){{{\bar{\vartheta }}}_{i}}} \\ =\text{adj}\{\varphi (t)\}\left( \int\limits_{\hat{t}_{i-1}^{+}}^{t_{i}^{+}}{{{e}^{-\int\limits_{\hat{t}_{i-1}^{+}}^{\tau }{\sigma ds}}}{{{\bar{\varphi }}}_{n}}(\tau )\bar{\varphi }_{n}^{\text{T}}(\tau )d\tau {{{\bar{\vartheta }}}_{i-1}}+\int\limits_{\hat{t}_{i-1}^{+}}^{t}{{{e}^{-\int\limits_{\hat{t}_{i-1}^{+}}^{\tau }{\sigma ds}}}{{{\bar{\varphi }}}_{n}}(\tau )\bar{\varepsilon }_{0}^{\text{T}}(\tau )d\tau {{{\bar{\vartheta }}}_{i}}}} \right) \\ =\text{adj}\{\varphi (t)\}\left( \pm \int\limits_{\hat{t}_{i-1}^{+}}^{t_{i}^{+}}{{{e}^{-\int\limits_{\hat{t}_{i-1}^{+}}^{\tau }{\sigma ds}}}{{{\bar{\varphi }}}_{n}}(\tau )\bar{\varphi }_{n}^{\text{T}}(\tau )d\tau {{{\bar{\vartheta }}}_{i}}+\int\limits_{\hat{t}_{i-1}^{+}}^{t}{{{e}^{-\int\limits_{\hat{t}_{i-1}^{+}}^{\tau }{\sigma ds}}}{{{\bar{\varphi }}}_{n}}(\tau )\bar{\varepsilon }_{0}^{\text{T}}(\tau )d\tau }} \right)-\Delta (t){{{\bar{\vartheta }}}_{i}} \\ =\text{adj}\{\varphi (t)\}\left( \int\limits_{\hat{t}_{i-1}^{+}}^{t_{i}^{+}}{{{e}^{-\int\limits_{\hat{t}_{i-1}^{+}}^{\tau }{\sigma ds}}}{{{\bar{\varphi }}}_{n}}(\tau )\bar{\varphi }_{n}^{\text{T}}(\tau )d\tau \left( {{{\bar{\vartheta }}}_{i-1}}-{{{\bar{\vartheta }}}_{i}} \right)+\int\limits_{\hat{t}_{i-1}^{+}}^{t}{{{e}^{-\int\limits_{\hat{t}_{i-1}^{+}}^{\tau }{\sigma ds}}}{{{\bar{\varphi }}}_{n}}(\tau )\bar{\varepsilon }_{0}^{\text{T}}(\tau t)d\tau }} \right). \\ \end{matrix}$$

(A.35)

Having combined (A.34) and (A.35), it is written that:

$${{\bar {\varepsilon }}_{1}}(t){\text{:}}\, = \left\{ \begin{gathered} {\text{adj}}{\kern 1pt} \{ \varphi (t)\} \left( {\int\limits_{\hat {t}_{{i - 1}}^{ + }}^{t_{i}^{ + }} {{{e}^{{ - \int\limits_{\hat {t}_{{i - 1}}^{ + }}^\tau {\sigma ds} }}}{{{\bar {\varphi }}}_{n}}(\tau )\bar {\varphi }_{n}^{{\text{T}}}(\tau )d\tau \left( {{{{\bar {\vartheta }}}_{{i - 1}}} - {{{\bar {\vartheta }}}_{i}}} \right)} } \right. \hfill \\ \left. { + \int\limits_{\hat {t}_{{i - 1}}^{ + }}^t {{{e}^{{ - \int\limits_{\hat {t}_{{i - 1}}^{ + }}^\tau {\sigma ds} }}}{{{\bar {\varphi }}}_{n}}(\tau )\bar {\varepsilon }_{0}^{{\text{T}}}(\tau )d\tau } } \right),\quad i > 0,\quad \forall t \in \left[ {t_{i}^{ + };\,\,\hat {t}_{i}^{ + }} \right) \hfill \\ {{0}_{{(n + m + 1) \times n}}},\quad \forall t \in \left[ {\hat {t}_{i}^{ + };\,\,t_{{i + 1}}^{ + }} \right), \hfill \\ \end{gathered} \right.$$

(A.36)

from which it follows that ${{\bar {\varepsilon }}_{1}}$(t) ≡ 0 when $t_{i}^{ + }$ = 0, and consequently that d(t) ≡ 0.

Using (A.2) and considering d(t) ≡ 0, Eq. (A.33) is solved:

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

(A.37)

It should be noted that, owing to Assumption 2, there are no switches over [$t_{0}^{ + }$; $t_{0}^{ + }$ + T₀), so the summation in (A.37) is from q = 1 to i.

If the number of switches is finite: i $\leqslant$ i_max < ∞, then, as:

(a) finite i means that time instants $t_{i}^{ + }$ are also finite (we do not consider the case of switches at infinite time: $\forall$i $t_{i}^{ + }$ ≠ ∞);

(b) $\forall$q ∈ $\mathbb{N}$ ${{\phi }^{{{{k}_{0}}}}}$($t_{0}^{ + }$ + T₀, $t_{q}^{ + }$) is finite in case $t_{q}^{ + }$ is finite,

(c) k₀ $\geqslant$ 1,

the following upper bound holds:

$$\begin{gathered} {\text{||}}w(t{\text{)||}}\;\leqslant \;\phi \left( {t,\,\,t_{0}^{ + } + {{T}_{0}}} \right)\sum\limits_{q = 1}^{{{i}_{{\max }}}} {{{\phi }^{{{{k}_{0}}}}}\left( {t_{0}^{ + } + {{T}_{0}},\,\,t_{q}^{ + }} \right){{\Omega }_{{{\text{UB}}}}}\left\| {\Delta _{q}^{\theta }} \right\|h\left( {t - t_{q}^{ + }} \right)} \\ = {{w}_{{\max }}}\phi \left( {t,\,\,t_{0}^{ + } + {{T}_{0}}} \right)\;\leqslant \;{{w}_{{\max }}}. \\ \end{gathered} $$

(A.38)

If $\forall$q ∈ $\mathbb{N}$   ||$\Delta _{q}^{\theta }$|| $\leqslant$ c_q${{\phi }^{{{{k}_{0}}}}}$($t_{q}^{ + }$, $t_{0}^{ + }$), c_q > c_q+1, then we have from (A.37) that:

$${\text{||}}w(t){\text{||}}\;\leqslant \;{{\phi }^{{{{k}_{0}}}}}\left( {t,\,\,t_{0}^{ + } + {{T}_{0}}} \right){{\Omega }_{{{\text{UB}}}}}{{\phi }^{{{{k}_{0}}}}}\left( {t_{0}^{ + } + {{T}_{0}},\,\,t_{0}^{ + }} \right)\sum\limits_{q = 1}^i {{{c}_{q}}h\left( {t - t_{q}^{ + }} \right)} .$$

(A.39)

All subsums of positive terms series from (A.39) are bounded, so $\sum\nolimits_{q = 1}^i {{{c}_{q}}}$h(t – $t_{q}^{ + }$) < ∞, and even if the number of switches is infinite, the following holds:

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

(A.40)

which completes the proof of Proposition 2.

Remark 2. The disturbance d(t), which reflects the difference between the real perturbation w(t) and the estimate (A.40), occurs in the proposed parametrization when $\tilde {t}_{i}^{ + }$ > 0 over the finite time intervals [$t_{i}^{ + }$; $\hat {t}_{i}^{ + }$], and $\forall$t $\geqslant$ $\hat {t}_{i}^{ + }$ its contribution into w(t) is an exponentially vanishing function. Thus d(t) affects only the transient quality of  $\tilde {\theta }$(t) and e_ref(t), but not the global properties of the tracking error ξ(t). The effect of d(t) can be reduced by improvement of the parameter σ (detailed analysis on that matter is given in Proposition 4 in [26]).

Proof of Proposition 3. According to the results of [26], the algorithm (4.10) ensures that $\tilde {t}_{i}^{ + }$ = Δ_pr $\leqslant$ T_i holds if the function $\epsilon$(t) is an indicator of the system parameters switch:

$$\forall t \in \left[ {t_{i}^{ + };\,\,\hat {t}_{i}^{ + }} \right)\,\,f(t) \ne 0,\quad \forall t \in \left[ {\hat {t}_{i}^{ + };\,\,t_{{i + 1}}^{ + }} \right)\,\,f(t) \ne 0,$$

(A.41)

i.e., it is non-zero only over the time range [$t_{i}^{ + }$; $\hat {t}_{i}^{ + }$).

Equations (A.25) and (A.24) are substituted into (4.9) to obtain:

$$\begin{gathered} \epsilon (t) = \Delta (t){{{\bar {\varphi }}}_{n}}(t)\bar {z}_{n}^{{\text{T}}}(t) - {{{\bar {\varphi }}}_{n}}(t)\bar {\varphi }_{n}^{{\text{T}}}(t)Y(t) = \Delta (t){{{\bar {\varphi }}}_{n}}(t)\bar {\varphi }_{n}^{{\text{T}}}(t)\bar {\vartheta }(t) \\ + \,\,\Delta (t){{{\bar {\varphi }}}_{n}}(t)\bar {\varepsilon }_{0}^{{\text{T}}}(t) - \Delta (t){{{\bar {\varphi }}}_{n}}(t)\bar {\varphi }_{n}^{{\text{T}}}(t)\bar {\vartheta }(t) - {{{\bar {\varphi }}}_{n}}(t)\bar {\varphi }_{n}^{{\text{T}}}(t){{{\bar {\varepsilon }}}_{1}}(t) \\ = \Delta (t){{{\bar {\varphi }}}_{n}}(t)\bar {\varepsilon }_{0}^{{\text{T}}}(t) - {{{\bar {\varphi }}}_{n}}(t)\bar {\varphi }_{n}^{{\text{T}}}(t){{{\bar {\varepsilon }}}_{1}}(t). \\ \end{gathered} $$

(A.42)

The error $\epsilon$(t) satisfies the definition (A.41) if $\bar {\varepsilon }_{0}^{{\text{T}}}$(t) and ${{\bar {\varepsilon }}_{1}}$(t) meet (A.41). Using the results of Proposition 2 (see (A.36)), the function ${{\bar {\varepsilon }}_{1}}$(t) is an indicator of the system parameters switch. Then now we need to prove the same thesis for $\bar {\varepsilon }_{0}^{{\text{T}}}$(t). Let it be assumed that $\forall$i ∈ $\mathbb{N}$ $\hat {t}_{i}^{ + }$ $\geqslant$ $t_{i}^{ + }$, then:

$$\begin{gathered} \forall t \in \left[ {\left. {\hat {t}_{i}^{ + };\,\,t_{{i + 1}}^{ + }} \right)} \right.\quad \vartheta (t) = {{\vartheta }_{i}} \\ \Updownarrow \\ \forall t \in \left[ {\left. {\hat {t}_{i}^{ + };\,\,t_{{i + 1}}^{ + }} \right)} \right.\quad {{{\bar {\varepsilon }}}_{0}}(t) = {{n}_{s}}(t)\left( {\int\limits_{\hat {t}_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}} \dot {x}(\tau )d\tau - \vartheta _{i}^{{\text{T}}}\bar {\Phi }(t)} \right) \\ = {{n}_{s}}(t)\left( {\vartheta _{i}^{{\text{T}}}\int\limits_{\hat {t}_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}} \Phi (\tau )d\tau - \vartheta _{i}^{{\text{T}}}\bar {\Phi }(t)} \right) = {{n}_{s}}(t)\,\,\left( {\vartheta _{i}^{{\text{T}}}\bar {\Phi }(t) - \vartheta _{i}^{{\text{T}}}\bar {\Phi }(t)} \right) = 0. \\ \end{gathered} $$

(A.43)

At the same time:

$$\begin{gathered} \forall t \in \left[ {\left. {t_{{i - 1}}^{ + };\,\,t_{i}^{ + }} \right)} \right.\quad \vartheta (t) = {{\vartheta }_{{i - 1}}};\,\,\forall t \in \left[ {\left. {t_{i}^{ + };\,\,\hat {t}_{i}^{ + }} \right)} \right.\,\,\,\,\vartheta (t) = {{\vartheta }_{i}}, \\ \Updownarrow \\ \forall t \in \left[ {\left. {t_{i}^{ + };\,\,\hat {t}_{i}^{ + }} \right)} \right.,\quad {{{\bar {\varepsilon }}}_{0}}(t) = {{n}_{s}}(t)\left( {\int\limits_{\hat {t}_{{i - 1}}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}\dot {x}(\tau )d\tau - \vartheta _{i}^{{\text{T}}}\bar {\Phi }(t)} } \right) \\ = {{n}_{s}}(t)\left( {{{e}^{{ - l(t - t_{i}^{ + })}}}\int\limits_{\hat {t}_{{i - 1}}^{ + }}^{t_{i}^{ + }} {{{e}^{{ - l(t_{i}^{ + } - \tau )}}}\vartheta _{{i - 1}}^{{\text{T}}}\Phi (\tau )d\tau + \int\limits_{t_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}\vartheta _{i}^{{\text{T}}}\Phi (\tau )d\tau } } } \right. \\ \left. { - \,\vartheta _{i}^{{\text{T}}}\left( {{{e}^{{ - l(t - t_{i}^{ + })}}}\int\limits_{\hat {t}_{{i - 1}}^{ + }}^{t_{i}^{ + }} {{{e}^{{ - l(t_{i}^{ + } - \tau )}}}\Phi (\tau )d\tau + \int\limits_{t_{i}^{ + }}^t {{{e}^{{ - l(t - \tau )}}}\Phi (\tau )d\tau } } } \right)} \right) \\ = {{n}_{s}}(t){{e}^{{ - l(t - t_{i}^{ + })}}}\left( {\vartheta _{{i - 1}}^{{\text{T}}} - \vartheta _{i}^{{\text{T}}}} \right)\int\limits_{\hat {t}_{{i - 1}}^{ + }}^{t_{i}^{ + }} {{{e}^{{ - l(t_{i}^{ + } - \tau )}}}} \Phi (\tau )d\tau . \\ \end{gathered} $$

(A.44)

Having combined (A.43) and (A.44), it is obtained:

$${{\bar {\varepsilon }}_{0}}(t)\,\,{\text{: = }}\,\,\left\{ \begin{gathered} {{n}_{s}}(t){{e}^{{ - l\left( {t - t_{i}^{ + }} \right)}}}\left( {\vartheta _{{i - 1}}^{{\text{T}}} - \vartheta _{i}^{{\text{T}}}} \right)\int\limits_{\hat {t}_{{i - 1}}^{ + }}^{t_{i}^{ + }} {{{e}^{{ - l(t_{i}^{ + } - \tau )}}}\Phi (\tau )d\tau ,\quad i > 0,\quad \forall t \in \left[ {t_{i}^{ + };\,\,\hat {t}_{i}^{ + }} \right)} \hfill \\ {{0}_{n}},\,\,\,\forall t \in \left[ {\hat {t}_{i}^{ + };\,\,t_{{i + 1}}^{ + }} \right), \hfill \\ \end{gathered} \right.$$

(A.45)

which, considering (A.36), allows one to write:

$$\forall i \in \mathbb{N},\quad \epsilon (t)\,\,{\text{: = }}\left\{ \begin{gathered} \Delta (t){{{\bar {\varphi }}}_{n}}(t)\bar {\varepsilon }_{0}^{{\text{T}}}(t) - {{{\bar {\varphi }}}_{n}}(t)\bar {\varphi }_{n}^{{\text{T}}}(t){{{\bar {\varepsilon }}}_{1}}(t),\quad i > 0,\quad \forall t \in \left[ {t_{i}^{ + };\,\,\hat {t}_{i}^{ + }} \right) \hfill \\ {{0}_{{(n + m + 1) \times n}}},\quad \forall t \in \left[ {\hat {t}_{i}^{ + };\,\,t_{{i + 1}}^{ + }} \right), \hfill \\ \end{gathered} \right.$$

(A.46)

from which $\epsilon$(t) is an indicator of the system parameters switch, and, following the results from [26], when Δ(t) ∈ FE and ${{\bar {\varphi }}_{n}}$(t) ∈ FE over [$\hat {t}_{i}^{ + }$; $t_{i}^{ + }$ + T_i] (which holds as Assumptions 2 and 3 are met), then $\tilde {t}_{i}^{ + }$ = Δ_pr $\leqslant$ T_i.

Proof of Theorem 1. The proof of theorem is arranged in the same way as the one of Proposition 1.

Two time ranges are considered: [$t_{0}^{ + }$; $t_{0}^{ + }$ + T₀) and [$t_{0}^{ + }$ + T₀; ∞). As for [$t_{0}^{ + }$; $t_{0}^{ + }$ + T₀), it holds that Ω(t) $\leqslant$ Ω_LB in the conservative case, so $\dot {\tilde {\theta }}$(t) = 0_(n+m)×m $\Rightarrow$ $\tilde {\theta }$(t) = $\tilde {\theta }$($t_{0}^{ + }$) (as there are no switches over [$t_{0}^{ + }$; $t_{0}^{ + }$ + T₀) according to Assumption 2). Then, taking the proof of Proposition 1 into consideration (see (A.13)–(A.17)), the exponential growth rate of e_ref(t) follows from the boundedness of $\tilde {\theta }$(t), and, as a result, as well the boundedness of e_ref(t) by its finite value at the right-hand border of the time interval in question: $\forall$t ∈ [$t_{0}^{ + }$; $t_{0}^{ + }$ + T₀) e_ref(t) $\leqslant$ e_ref[$t_{0}^{ + }$ + T₀). Therefore, ξ(t) is bounded over the time range [$t_{0}^{ + }$; $t_{0}^{ + }$ + T₀).

The next aim is to consider the interval [$t_{0}^{ + }$ + T₀; ∞).

Step 1. The exponential convergence of $\tilde {\theta }$(t) $\forall$t $\geqslant$ $t_{0}^{ + }$ + T₀ is to be proved.

Taking into consideration (A.38) or (A.40) and the boundedness of Ω(t) $\geqslant$ Ω_LB, the solution of Eq. (4.11)$\forall$t $\geqslant$ $t_{0}^{ + }$ + T₀ meets the inequality:

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

(A.47)

As at least one of the following conditions is met:

(1) i $\leqslant$ i_max $\leqslant$ ∞,

(2) $\forall$q ∈ $\mathbb{N}$   ||$\Delta _{q}^{\theta }$|| $\leqslant$ c_q${{\phi }^{{{{k}_{0}}}}}$($t_{q}^{ + }$, $t_{0}^{ + }$) $\leqslant$ c_qϕ($t_{q}^{ + }$, $t_{0}^{ + }$), c_q > c_q+1,

then, by the analogy with (A.3)–(A.5), the following upper bound is obtained from (A.47):

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

(A.48)

where χ₁(t) is a time-varying parameter:

$${{\chi }_{1}}(t) = {{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}(t - t_{0}^{ + } - {{T}_{0}})}}}\left( {t - t_{0}^{ + } - {{T}_{0}}} \right),\quad {{\chi }_{1}}\left( {t_{0}^{ + } + {{T}_{0}}} \right) = 0,$$

and β(t) for both cases under consideration is defined as:

$$\beta (t)\;\leqslant \;\left\| {\tilde {\theta }\left( {t_{0}^{ + } + {{T}_{0}}} \right)} \right\| + \sum\limits_{q = 1}^{{{i}_{{\max }}}} {\phi \left( {t_{0}^{ + } + {{T}_{0}},t_{q}^{ + }} \right)\left\| {\Delta _{q}^{\theta }} \right\|h\left( {t - t_{q}^{ + }} \right) = {{\beta }_{{\max }}},} $$

(A.49)

$$\begin{gathered} \beta (t)\;\leqslant \;\left\| {\tilde {\theta }\left( {t_{0}^{ + } + {{T}_{0}}} \right)} \right\| + \sum\limits_{q = 1}^i {\phi \left( {t_{0}^{ + } + {{T}_{0}},\,\,t_{q}^{ + }} \right)\phi \left( {t_{q}^{ + },\,\,t_{0}^{ + }} \right){{c}_{q}}h\left( {t - t_{q}^{ + }} \right)} \hfill \\ \,\,\,\,\,\,\,\, = \left\| {\tilde {\theta }\left( {t_{0}^{ + } + {{T}_{0}}} \right)} \right\| + \sum\limits_{q = 1}^i {\phi \left( {t_{0}^{ + } + {{T}_{0}},\,\,t_{0}^{ + }} \right){{c}_{q}}h\left( {t - t_{q}^{ + }} \right)} = {{\beta }_{{\max }}}. \hfill \\ \end{gathered} $$

(A.50)

If the parameter χ₁(t) is bounded, then it holds for $\tilde {\theta }$(t) that:

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

(A.51)

Then |χ₁(t)| $\leqslant$ $\chi _{1}^{{{\text{UB}}}}$ is to be proved. We differentiate χ₁(t) with respect to time:

$${{\dot {\chi }}_{1}}(t) = - \frac{{{{\gamma }_{1}}}}{2}{{\chi }_{1}}(t) + {{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{0}}} \right)}}}.$$

(A.52)

The upper bound of the solution of (A.52) is written as:

$${\text{|}}{{\chi }_{1}}(t){\text{|}}\;\leqslant \;\left| {\int\limits_{t_{0}^{ + } + {{T}_{0}}}^t {{{e}^{{ - \int_\tau ^t {\frac{{{{\gamma }_{1}}}}{2}d\tau } }}}{{e}^{{\frac{{ - {{\gamma }_{1}}}}{2}\left( {\tau - t_{0}^{ + } - {{T}_{0}}} \right)}}}d\tau } } \right|\;\leqslant \;\left| {\int\limits_{t_{0}^{ + } + {{T}_{0}}}^t {{{e}^{{\frac{{ - {{\gamma }_{1}}}}{2}\left( {\tau - t_{0}^{ + } - {{T}_{0}}} \right)}}}d\tau } } \right|\;\leqslant \;\frac{2}{{{{\gamma }_{1}}}},$$

(A.53)

which proves the required boundedness |χ₁(t)| $\leqslant$ $\chi _{1}^{{{\text{UB}}}}$.

The exponential convergence (A.51) immediately follows from boundedness (A.53), which was to be proved at Step 1.

Step 2. The exponential convergence of the error ξ(t) $\forall$t $\geqslant$ $t_{0}^{ + }$ + T₀ is to be proved.

To prove the convergence of ξ(t) $\forall$t $\geqslant$ $t_{0}^{ + }$ + T₀, owing to the estimate (A.51), it remains to prove the convergence of the tracking error e_ref(t) $\forall$t $\geqslant$ $t_{0}^{ + }$ + T₀.

The following quadratic form is introduced:

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

(A.54)

$${{\bar {e}}_{{{\text{ref}}}}}(t) = {{\left[ {e_{{{\text{ref}}}}^{{\text{T}}}(t)\,\,{{e}^{{ - \frac{{{{\gamma }_{1}}}}{4}\left( {t - t_{0}^{ + } - T_{0}^{ + }} \right)}}}} \right]}^{{\text{T}}}}.$$

By analogy with the proof of Proposition 1, $\forall$t $\geqslant$ $t_{0}^{ + }$ + T₀ the derivative of (A.54) is written as:

$${{\dot {V}}_{{{{e}_{{{\text{ref}}}}}}}}(t)\;\leqslant \; - \mu {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}(t){\text{|}}{{{\text{|}}}^{2}} - 2a_{0}^{2}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - T_{0}^{ + }} \right)}}} + b_{{\max }}^{2}{{\lambda }_{{\max }}}\left( {\omega (t){{\omega }^{{\text{T}}}}} \right){{\left\| {\tilde {\theta }(t)} \right\|}^{2}}.$$

(A.55)

Using (A.55), the following upper bound is introduced for $b_{{\max }}^{2}$||$\tilde {\theta }$(t)||²:

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

(A.56)

Equation (A.56) is substituted into (A.55):

$$\begin{gathered} {{{\dot {V}}}_{{{{e}_{{{\text{ref}}}}}}}}(t)\;\leqslant \; - \mu {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}(t){\text{|}}{{{\text{|}}}^{2}} - 2a_{0}^{2}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - T_{0}^{ + }} \right)}}} \\ + \,\,b_{{\max }}^{2}{{\left( {{{\beta }_{{\max }}} + \frac{{{{\gamma }_{1}}{{w}_{{\max }}}}}{{{{\Omega }_{{LB}}}}}\chi _{1}^{{{\text{UB}}}}} \right)}^{2}}{{\lambda }_{{\max }}}\left( {\omega (t){{\omega }^{{\text{T}}}}(t)} \right){{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - T_{0}^{{}}} \right)}}}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + } - {{T}_{0}}} \right)}}}. \\ \end{gathered} $$

(A.57)

The exponential stability of e_ref(t) requires the third term of (A.57) to be exponentially vanishing, which demands:

$$\chi (t) = {{\lambda }_{{\max }}}\left( {\omega (t){{\omega }^{{\text{T}}}}(t)} \right){{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}\left( {t - t_{0}^{ + }} \right)}}}\;\leqslant \;{{\chi }_{{{\text{UB}}}}},$$

(A.58)

where χ_UB > 0.

The error $\tilde {\theta }$(t) is bounded (A.51). In such case, following results of Proposition 1, the growth rate of λ_max(ω(t)ω^T(t)) does not exceed the exponential one (A.17). So, when γ₁ > 0 is sufficiently large, then the estimate (A.58) holds.

Equation (A.58) is substituted into (A.57) to obtain:

$${{\dot {V}}_{{{{{\text{e}}}_{{{\text{ref}}}}}}}}(t)\;\leqslant \; - {\kern 1pt} \mu {{\lambda }_{{\min }}}(P){\text{||}}{{e}_{{{\text{ref}}}}}(t){\text{|}}{{{\text{|}}}^{2}} - 2a_{0}^{2}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}(t - t_{0}^{ + } - T_{0}^{ + })}}} + a_{0}^{2}{{e}^{{ - \frac{{{{\gamma }_{1}}}}{2}(t - t_{0}^{ + } - T_{0}^{{}})}}}\;\leqslant \; - {\kern 1pt} {{\bar {\eta }}_{{{{e}_{{{\text{ref}}}}}}}}{{V}_{{{{e}_{{{\text{ref}}}}}}}}(t),$$

(A.59)

where

$$a_{0}^{2} = b_{{\max }}^{2}{{\left( {{{\beta }_{{\max }}} + \frac{{{{\gamma }_{1}}{{w}_{{\max }}}}}{{{{\Omega }_{{LB}}}}}\chi _{1}^{{{\text{UB}}}}} \right)}^{2}}{{\chi }_{{{\text{UB}}}}},\quad {{\bar {\eta }}_{{{{e}_{{{\text{ref}}}}}}}} = \min \left\{ {\frac{{\mu {{\lambda }_{{\min }}}(P)}}{{{{\lambda }_{{\max }}}(P)}},\,\,\frac{{{{\gamma }_{1}}}}{4}} \right\}.$$

The differential inequality (A.59) is solved to obtain:

$${{V}_{{{{e}_{{{\text{ref}}}}}}}}(t)\;\leqslant \;{{\epsilon }^{{ - {{{\bar {\eta }}}_{{{{e}_{{{\text{ref}}}}}}}}(t - t_{0}^{ + } - {{T}_{0}})}}}{{V}_{{{{e}_{{{\text{ref}}}}}}}}(t_{0}^{ + } + {{T}_{0}}),$$

(A.60)

from which we have the exponential convergence of the tracking error e_ref(t) to zero:

$${\text{||}}{{e}_{{{\text{ref}}}}}(t){\text{||}}\;\leqslant \;\sqrt {\frac{{{{\lambda }_{M}}}}{{{{\lambda }_{m}}}}} {\text{||}}{{e}_{{{\text{ref}}}}}(t_{0}^{ + } + {{T}_{0}}){\text{||}}{{e}^{{ - {{\eta }_{{{{e}_{{{\text{ref}}}}}}}}(t - t_{0}^{ + } - {{T}_{0}})}}},$$

(A.61)

where

$${{\eta }_{{{{e}_{{{\text{ref}}}}}}}} = \frac{1}{2}{{\bar {\eta }}_{{{{e}_{{{\text{ref}}}}}}}}.$$

Having combined (A.61) and (A.51), it is obtained:

$${\text{||}}\xi (t){\text{||}}\;\leqslant \;\max \left\{ {\sqrt {\frac{{{{\lambda }_{M}}}}{{{{\lambda }_{m}}}}} {\text{||}}{{e}_{{{\text{ref}}}}}(t_{0}^{ + } + {{T}_{0}}){\text{||}},\,\,{{\beta }_{{\max }}} + \frac{{{{\gamma }_{1}}{{w}_{{\max }}}}}{{{{\Omega }_{{LB}}}}}\chi _{1}^{{{\text{UB}}}}} \right\}{{e}^{{ - {{\gamma }_{{{{e}_{{{\text{ref}}}}}}}}(t - t_{0}^{ + } - {{T}_{0}})}}},$$

(A.62)

which, taking into consideration that ξ(t) is bounded over [$t_{0}^{ + }$;  $t_{0}^{ + }$ + T₀], allows one to make conclusions of both global boundedness of ξ(t) ∈ L_∞ and the exponential convergence of ξ(t) to zero $\forall$t $\geqslant$ $t_{0}^{ + }$ + T₀. The proof of Theorem 1 is complete.