Adaptive Observer of State and Disturbances for Linear Overparameterized Systems

Abstract

The problem of state reconstruction is considered for a class of linear systems with time-invariant unknown parameters and overparameterization that are affected by external perturbations generated by a known exosystem with unknown initial conditions. An extended adaptive observer is proposed, which, in contrast to existing approaches, solves state and perturbation adaptive estimation problems for systems that are not represented in the observer canonical form. The obtained theoretical results are validated via mathematical modeling.

APPENDIX

Proof of Lemma 1. The parameterization (3.3) is obtained as a combination of the results from [12, 24] with the dynamic regressor extension and mixing procedure from [14, 19]. The proof of Lemma 1 is derived on the basis of Lemma 1 and Theorem 2 from [24]. To make it easier to understand the adopted notation and ensure that the results of the paper are self-contained, we next present the proof of this lemma in accordance with the one in [24]. In contrast to the results [24], in this paper, owing to Assumption 2, β is known, which allows one not to avoid overparameterization in (3.3) (see (A.23)).

Step 1. The following error is considered:

$$\tilde {\xi }(t) = \xi (t) - z(t) - \Omega (t){{\psi }_{a}}(\theta ) - P(t){{\psi }_{b}}(\theta ).$$

(A.1)

The time derivative of (A.1) is written:

$$\begin{gathered} \dot {\tilde {\xi }}(t) = {{A}_{0}}\xi (t) + {{\psi }_{a}}(\theta )y(t) + {{\psi }_{b}}(\theta )u(t) + {{\psi }_{d}}(\theta )\delta (t) - {{A}_{K}}z(t) \\ - \;Ky(t) - ({{A}_{K}}\Omega (t) + {{I}_{n}}y(t)){{\psi }_{a}}(\theta ) - ({{A}_{K}}P(t) + {{I}_{n}}u(t)){{\psi }_{b}}(\theta ) \\ = {{A}_{0}}\xi (t) - {{A}_{K}}z(t) - Ky(t) - {{A}_{K}}\Omega (t){{\psi }_{a}}(\theta ) - {{A}_{K}}P(t){{\psi }_{b}}(\theta ) + {{\psi }_{d}}(\theta )\delta (t) \\ = {{A}_{K}}\tilde {\xi }(t) + {{\psi }_{d}}(\theta )\delta (t). \\ \end{gathered} $$

(A.2)

The solution of Eq. (A.2) is obtained as

$$\tilde {\xi }(t) = {{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}) + \bar {\delta }(t),$$

(A.3)

where the external perturbation $\bar {\delta }(t)$ is described as a set of equations

$$\left\{ \begin{gathered} \dot {\bar {\delta }}(t) = {{A}_{K}}\bar {\delta }(t) + {{\psi }_{d}}(\theta )\delta (t) \hfill \\ {{{v}}_{f}}(t) = C_{0}^{{\text{T}}}\bar {\delta }(t). \hfill \\ \end{gathered} \right.$$

(A.4)

Having substituted (A.3) into (A.1), it is written:

$$\begin{gathered} {{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}) + \bar {\delta }(t) = \xi (t) - z(t) - \Omega (t){{\psi }_{a}}(\theta ) - P(t){{\psi }_{b}}(\theta ), \\ \Updownarrow \\ \xi (t) = {{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}) + \bar {\delta }(t) + z(t) + \Omega (t){{\psi }_{a}}(\theta ) + P(t){{\psi }_{b}}(\theta ). \\ \end{gathered} $$

(A.5)

Equation (A.5) is multiplied by $C_{0}^{{\text{T}}}$ to obtain:

$$y(t) = C_{0}^{{\text{T}}}\xi (t) = C_{0}^{{\text{T}}}z(t) + C_{0}^{{\text{T}}}\Omega (t){{\psi }_{a}}(\theta ) + C_{0}^{{\text{T}}}P(t){{\psi }_{b}}(\theta ) + {{{v}}_{f}}(t) + C_{0}^{{\text{T}}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}).$$

(A.6)

Considering (A.6), the function $\bar {q}$ = y(t) – $C_{0}^{{\text{T}}}z(t)$ is differentiated:

$$\dot {\bar {q}}(t) = C_{0}^{{\text{T}}}\dot {\Omega }(t){{\psi }_{a}}(\theta ) + C_{0}^{{\text{T}}}\dot {P}(t){{\psi }_{b}}(\theta ) + {{{\dot {v}}}_{f}}(t) + C_{0}^{{\text{T}}}{{A}_{K}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}).$$

(A.7)

Step 2. The next aim is to parametrize the term ${{{\dot {v}}}_{f}}(t)$ of Eq. (A.7) as a linear regression equation with a measurable regressor. For this purpose, the system (A.4) is rewritten as a transfer function:

$${{{v}}_{f}}(t) = C_{0}^{{\text{T}}}{{(s{{I}_{n}} - {{A}_{K}})}^{{ - 1}}}{{\psi }_{d}}(\theta )\delta (t) = {{W}_{f}}[\delta (t)].$$

(A.8)

The derivative of the perturbation δ(t) is represented as:

$$\dot {\delta }(t) = h_{\delta }^{{\text{T}}}{{\mathcal{A}}_{\delta }}{{x}_{\delta }}(t) + \delta ({{t}_{0}}){{D}_{\delta }}(t),$$

(A.9)

where D_δ(t) is a Dirac delta function.

A virtual signal δ_d(t) = $h_{\delta }^{{\text{T}}}{{\mathcal{A}}_{\delta }}{{x}_{\delta }}(t)$ is introduced into consideration. Then the following equalities hold

$$\begin{gathered} {{{\dot {x}}}_{\delta }}(t) = {{\mathcal{A}}_{\delta }}{{x}_{\delta }}(t), \hfill \\ {{\delta }_{d}}(t) = \bar {h}_{\delta }^{{\text{T}}}{{x}_{\delta }}(t),\quad \bar {h}_{\delta }^{{\text{T}}} = h_{\delta }^{{\text{T}}}{{\mathcal{A}}_{\delta }}. \hfill \\ \end{gathered} $$

(A.10)

Equation (A.8) is differentiated, and then (A.9), (A.10) are substituted into the obtained result to write:

$${{{\dot {v}}}_{f}} = s{{W}_{f}}[\delta (t)] = {{W}_{f}}[\mathop \delta \limits^. (t)] = {{W}_{f}}\left[ {h_{\delta }^{{\text{T}}}{{A}_{\delta }}{{x}_{\delta }}(t) + \delta ({{t}_{0}}){{D}_{\delta }}(t)} \right] = \underbrace {{{W}_{f}}[{{\delta }_{d}}(t)]}_{{{{v}}_{f}}(t)} + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)].$$

(A.11)

Thus, owing to the fact that the matrix A_K is a Hurwitz one, it is sufficient to parametrize ${{{v}}_{f}}(t)$ to parameterize ${{{\dot {v}}}_{f}}(t)$. For this purpose, an auxiliary signal ζ(t) = ${{M}_{\delta }}{{x}_{\delta }}(t)$ is considered, where the transformation matrix M_δ is a solution of the Sylvester equation

$${{M}_{\delta }}{{\mathcal{A}}_{\delta }} - G{{M}_{\delta }} = l\bar {h}_{\delta }^{{\text{T}}},$$

(A.12)

which has a unique solution [15, 16, 24] as, owing to Assumption 2, the pair $\left( {h_{\delta }^{{\text{T}}},{{\mathcal{A}}_{\delta }}} \right)$ is observable and, following the premises of this lemma, (G, l) is controllable and σ{${{\mathcal{A}}_{\delta }}$} ∩ σ{G} = 0.

We differentiate ζ(t) to obtain:

$$\mathop \zeta \limits^. (t) = {{M}_{\delta }}{{\mathcal{A}}_{\delta }}{{x}_{\delta }}(t) = G{{M}_{\delta }}{{x}_{\delta }}(t) + l\bar {h}_{\delta }^{{\text{T}}}{{x}_{\delta }}(t) = G\zeta (t) + l{{\delta }_{d}}(t),$$

(A.13)

form which, considering x_δ(t) = $M_{\delta }^{{ - 1}}\zeta (t)$, it follows that

$${{\delta }_{d}}(t) = \bar {h}_{\delta }^{{\text{T}}}M_{\delta }^{{ - 1}}\zeta = {{\beta }^{{\text{T}}}}\zeta ,\quad \beta = \bar {h}_{\delta }^{{\text{T}}}M_{\delta }^{{ - 1}}.$$

(A.14)

Taking into account (A.14), Eq. (A.11) is rewritten as:

$${{{\dot {v}}}_{f}}(t) = {{W}_{f}}\left[ {{{\beta }^{{\text{T}}}}\zeta (t)} \right] + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)] = {{\beta }^{{\text{T}}}}{{W}_{f}}[\zeta (t)] + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)] = {{\beta }^{{\text{T}}}}{{\zeta }_{w}}(t) + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)].$$

(A.15)

The signal ${{{v}}_{f}}(t)$ is filtered via (A.13) instead of δ_d(t):

$${{\zeta }_{f}}(t) = {{(sI - G)}^{{ - 1}}}l[{{{v}}_{f}}(t)] + {{e}^{{G(t - {{t}_{0}})}}}{{\zeta }_{f}}({{t}_{0}}),$$

(A.16)

then, owing to ζ(t) = (sI – G)^–1l[δ_d(t)] + ${{e}^{{G(t - {{t}_{0}})}}}\zeta ({{t}_{0}})$, the following equality holds:

$$\begin{gathered} {{\zeta }_{w}}(t) = {{W}_{f}}[\zeta (t)] = {{W}_{f}}\left[ {{{{(sI - G)}}^{{ - 1}}}l[{{\delta }_{d}}(t)] + {{e}^{{G(t - {{t}_{0}})}}}\zeta ({{t}_{0}})} \right] \\ = {{(sI - G)}^{{ - 1}}}l{{W}_{f}}[{{\delta }_{d}}(t)] + {{W}_{f}}\left[ {{{e}^{{G(t - {{t}_{0}})}}}\zeta ({{t}_{0}})} \right] \\ = {{(sI - G)}^{{ - 1}}}l{{{v}}_{f}} + {{W}_{f}}\left[ {{{e}^{{G(t - {{t}_{0}})}}}\zeta ({{t}_{0}})} \right] \\ = {{\zeta }_{f}}(t) - {{e}^{{G(t - {{t}_{0}})}}}{{\zeta }_{f}}({{t}_{0}}) + {{W}_{f}}\left[ {{{e}^{{G(t - {{t}_{0}})}}}\zeta ({{t}_{0}})} \right]. \\ \end{gathered} $$

(A.17)

Having substituted (A.17) into (A.15), it is written:

$${{{\dot {v}}}_{f}}(t) = {{\beta }^{{\text{T}}}}{{\zeta }_{f}}(t) - {{\beta }^{{\text{T}}}}{{e}^{{G(t - {{t}_{0}})}}}{{\zeta }_{f}}({{t}_{0}}) + {{\beta }^{{\text{T}}}}{{W}_{f}}\left[ {{{e}^{{G(t - {{t}_{0}})}}}\xi ({{t}_{0}})} \right] + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)].$$

(A.18)

The following observer of state ζ_f(t) is introduced:

$${{\hat {\zeta }}_{f}}(t) = F(t) + H(t){{\psi }_{b}}(\theta ) + N(t){{\psi }_{a}}(\theta ) + ly(t).$$

(A.19)

Considering equations (A.7), (A.11), (A.16), (A.19), the error is differentiated ${{\tilde {\zeta }}_{f}}(t)$ = ${{\zeta }_{f}}(t)$ – ${{\hat {\zeta }}_{f}}(t)$ to obtain:

$${{\dot {\tilde {\zeta }}}_{f}} = G{{\zeta }_{f}}(t) + l{{{v}}_{f}}(t) - GF(t) - Gly(t) + lC_{0}^{{\text{T}}}\dot {z}(t)$$

$$ - \;\left( {GH(t) - lC_{0}^{{\text{T}}}\dot {P}(t)} \right){{\psi }_{b}}(\theta ) - \left( {GN(t) - lC_{0}^{{\text{T}}}\dot {\Omega }(t)} \right){{\psi }_{a}}(\theta )$$

$$ - \;lC_{0}^{{\text{T}}}\dot {z}(t) - lC_{0}^{{\text{T}}}\dot {\Omega }(t){{\psi }_{a}}(\theta ) - lC_{0}^{{\text{T}}}\dot {P}(t){{\psi }_{b}}(\theta )$$

$$ - \;l({{{v}}_{f}}(t) + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)]) - lC_{0}^{{\text{T}}}{{A}_{K}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}})$$

(A.20)

$$ = G{{\zeta }_{f}}(t) - \underbrace {GF(t) - Gly(t) - GH(t){{\psi }_{b}}(\theta ) - GN(t){{\psi }_{a}}(\theta )}_{G{{{\hat {\zeta }}}_{f}}(t)}$$

$$ - \;l{{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)] - lC_{0}^{{\text{T}}}{{A}_{K}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}})$$

$$ = G{{\tilde {\zeta }}_{f}} - lC_{0}^{{\text{T}}}{{A}_{K}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}) - l{{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)].$$

The set of equations (A.20) is solved:

$${{\tilde {\zeta }}_{f}}(t) = {{\zeta }_{f}}(t) - {{\hat {\zeta }}_{f}}(t) = {{e}^{{G(t - {{t}_{0}})}}}{{\tilde {\zeta }}_{f}}({{t}_{0}}) - \mathfrak{H}\left[ {C_{0}^{{\text{T}}}{{A}_{K}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}) + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)]} \right],$$

(A.21)

which allows one to rewrite (A.18) as follows:

$${{{\dot {v}}}_{f}}(t) = {{\beta }^{{\text{T}}}}{{\hat {\zeta }}_{f}}(t) + {{\beta }^{{\text{T}}}}{{e}^{{G(t - {{t}_{0}})}}}{{\tilde {\zeta }}_{f}}({{t}_{0}})$$

$$ - \;{{\beta }^{{\text{T}}}}\mathfrak{H}\left[ {C_{0}^{{\text{T}}}{{A}_{K}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}) + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)]} \right]$$

$$\begin{gathered} - \;{{\beta }^{{\text{T}}}}{{e}^{{G(t - {{t}_{0}})}}}{{\zeta }_{f}}({{t}_{0}}) + {{\beta }^{{\text{T}}}}{{W}_{f}}\left[ {{{e}^{{G(t - {{t}_{0}})}}}\xi ({{t}_{0}})} \right] + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)] \\ = {{\beta }^{{\text{T}}}}(F(t) + ly(t)) + {{\beta }^{{\text{T}}}}H(t){{\psi }_{b}}(\theta ) + {{\beta }^{{\text{T}}}}N(t){{\psi }_{a}}(\theta ) \\ \end{gathered} $$

(A.22)

$$ + \;{{\beta }^{{\text{T}}}}{{e}^{{G(t - {{t}_{0}})}}}{{\tilde {\zeta }}_{f}}({{t}_{0}}) - {{\beta }^{{\text{T}}}}\mathfrak{H}\left[ {C_{0}^{{\text{T}}}{{A}_{K}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}) + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)]} \right]$$

$$ - \;{{\beta }^{{\text{T}}}}{{e}^{{G(t - {{t}_{0}})}}}{{\zeta }_{f}}({{t}_{0}}) + {{\beta }^{{\text{T}}}}{{W}_{f}}\left[ {{{e}^{{G(t - {{t}_{0}})}}}\xi ({{t}_{0}})} \right] + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)],$$

where $\mathfrak{H}[.]$ = ${{(s{{I}_{{{{n}_{\delta }}}}} - G)}^{{ - 1}}}l{\kern 1pt} [.]$.

Equation (A.22) is substituted into (A.7) to obtain:

$$\dot {\bar {q}}(t) = C_{0}^{{\text{T}}}\dot {\Omega }(t){{\psi }_{a}}(\theta ) + C_{0}^{{\text{T}}}\dot {P}(t){{\psi }_{b}}(\theta )$$

$$ + \;{{\beta }^{{\text{T}}}}(F(t) + ly(t)) + {{\beta }^{{\text{T}}}}H(t){{\psi }_{b}}(\theta ) + {{\beta }^{{\text{T}}}}N(t){{\psi }_{a}}(\theta )$$

$$\begin{gathered} + \;{{\beta }^{{\text{T}}}}{{e}^{{G(t - {{t}_{0}})}}}{{{\tilde {\zeta }}}_{f}}({{t}_{0}}) - {{\beta }^{{\text{T}}}}\mathfrak{H}\left[ {C_{0}^{{\text{T}}}{{A}_{K}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}}) + {{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)]} \right] \\ - \;{{\beta }^{{\text{T}}}}{{e}^{{G(t - {{t}_{0}})}}}{{\zeta }_{f}}({{t}_{0}}) + {{\beta }^{{\text{T}}}}{{W}_{f}}\left[ {{{e}^{{G(t - {{t}_{0}})}}}\xi ({{t}_{0}})} \right] \\ \end{gathered} $$

(A.23)

$$ + \;{{W}_{f}}[\delta ({{t}_{0}}){{D}_{\delta }}(t)] + C_{0}^{{\text{T}}}{{A}_{K}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}\tilde {\xi }({{t}_{0}})$$

$$ = {{\bar {\varphi }}^{{\text{T}}}}(t)\eta (\theta ) + {{\beta }^{{\text{T}}}}(F(t) + ly(t)) + \bar {\varepsilon }(t),$$

where $\bar {\varepsilon }(t)$ are aggregated exponentially vanishing functions.

Step 3. The next aim is to transform the regression equation (A.23) into the form of (3.3) via application of the dynamic regressor extension and mixing procedure. For this purpose, considering (A.23), (3.5), the signal χ(t) = $\bar {q}(t)$ – ${{k}_{1}}{{\bar {q}}_{f}}(t)$ is differentiated to obtain:

$$\begin{gathered} \dot {\chi }(t) = {{{\bar {\varphi }}}^{{\text{T}}}}(t)\eta (\theta ) + {{\beta }^{{\text{T}}}}(F(t) + ly(t)) + \bar {\varepsilon }(t) - {{k}_{1}}\left( { - {{k}_{1}}{{{\bar {q}}}_{f}}(t) + \bar {q}(t)} \right) \\ = - {{k}_{1}}\chi (t) + {{{\bar {\varphi }}}^{{\text{T}}}}(t)\eta (\theta ) + {{\beta }^{{\text{T}}}}(F(t) + ly(t)) + \bar {\varepsilon }(t). \\ \end{gathered} $$

(A.24)

The solution of the differential equation (A.24) allows one to write:

$$\bar {q}(t) - {{k}_{1}}{{\bar {q}}_{f}}(t) - {{\beta }^{{\text{T}}}}({{F}_{f}}(t) + l{{y}_{f}}(t)) = {{e}^{{ - {{k}_{1}}(t - {{t}_{0}})}}}\bar {q}({{t}_{0}}) + \bar {\varphi }_{f}^{{\text{T}}}(t)\eta (\theta ) + {{\bar {\varepsilon }}_{f}}(t),$$

(A.25)

where ${{\dot {\bar {\varepsilon }}}_{f}}(t)$ = $- {{k}_{1}}{{\bar {\varepsilon }}_{f}}(t)$ + ${{k}_{1}}\bar {\varepsilon }(t)$, ${{\bar {\varepsilon }}_{f}}({{t}_{0}})$ = 0.

Owing to (A.25), the solution of the first differential equation from (3.4) satisfies the following equation

$$q(t) = \varphi (t)\eta (\theta ) + \varepsilon (t),$$

(A.26)

where $\mathop \varepsilon \limits^. (t) = - {{k}_{2}}\varepsilon (t)$ + ${{\bar {\varphi }}_{f}}(t)\left( {{{{\bar {\varepsilon }}}_{f}}(t) + {{e}^{{ - {{k}_{1}}(t - {{t}_{0}})}}}\bar {q}({{t}_{0}})} \right)$, ε(t₀) = 0_2n.

Having multiplied Eq. (A.26) by k(t)adj{φ(t)} and applied the property

$$\operatorname{adj} \{ \varphi (t)\} \varphi (t) = \det \{ \varphi (t)\} {{I}_{{2n}}},$$

Eq. (3.3) is obtained with $\epsilon (t)$ = k(t)adj{φ(t)}ε(t).

In accordance with Lemma 6.8 from [6], when $\bar {\varphi }(t)$ ∈ PE, it also holds that ${{\bar {\varphi }}_{f}}(t)$ ∈ PE. Following Proposition 1, when ${{\bar {\varphi }}_{f}}(t)$ ∈ PE, then it holds that Δ(t) $\geqslant$ Δ_min > 0. Since the signals y(t), u(t) are bounded by Assumption 1, owing to the stability of the filters (3.4)–(3.6), the inequality Δ_max $\geqslant$ Δ(t) holds for all t $\geqslant$ t₀. Then for all t $\geqslant$ t₀ + T it holds that Δ_max $\geqslant$ Δ(t) $\geqslant$ Δ_min > 0, which completes the proof of Lemma.

Proof of Lemma 2. According to Definition 1 and Hypothesis 1 and owing to

$${{\Xi }_{\mathcal{S}}}(\Delta ) = {{\bar {\Xi }}_{\mathcal{S}}}(\Delta )\Delta (t),\quad {{\Xi }_{\mathcal{G}}}(\Delta ) = {{\bar {\Xi }}_{\mathcal{G}}}(\Delta )\Delta (t),$$

$${{\mathcal{Y}}_{{ab}}}(t) = {{\mathcal{L}}_{{ab}}}\mathcal{Y}(t) = \Delta (t){{\mathcal{L}}_{{ab}}}\eta (\theta ) = \Delta (t){{\psi }_{{ab}}}(\theta ),$$

$${{\bar {\Xi }}_{\mathcal{S}}}(\Delta )\Delta (t){{\psi }_{{ab}}}(\theta ) = {{\bar {\Xi }}_{\mathcal{S}}}(\Delta ){{\mathcal{Y}}_{{ab}}}(t),$$

$${{\bar {\Xi }}_{\mathcal{G}}}(\Delta )\Delta (t){{\psi }_{{ab}}}(\theta ) = {{\bar {\Xi }}_{\mathcal{G}}}(\Delta ){{\mathcal{Y}}_{{ab}}}(t),$$

it follows from (3.9) that

$${{\mathcal{T}}_{\mathcal{S}}}\left( {{{{\bar {\Xi }}}_{\mathcal{S}}}(\Delta ){{\mathcal{Y}}_{{ab}}}} \right) = {{\mathcal{T}}_{\mathcal{G}}}\left( {{{{\bar {\Xi }}}_{\mathcal{G}}}(\Delta ){{\mathcal{Y}}_{{ab}}}} \right)\theta .$$

(A.27)

Then, having multiplied (A.27) by adj$\left\{ {{{\mathcal{T}}_{\mathcal{G}}}\left( {{{{\bar {\Xi }}}_{\mathcal{G}}}(\Delta ){{\mathcal{Y}}_{{ab}}}} \right)} \right\}$, the following regression equation is obtained

$${{\mathcal{Y}}_{\theta }}(t) = {{\mathcal{M}}_{\theta }}(t)\theta ,$$

(A.28)

which is used together with (3.8) to write:

$${{\mathcal{T}}_{\mathcal{Z}}}\left( {{{{\bar {\Xi }}}_{\mathcal{Z}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right) = {{\mathcal{T}}_{\mathcal{X}}}\left( {{{{\bar {\Xi }}}_{\mathcal{X}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right){{\Theta }_{{AB}}}(\theta ),$$

(A.29a)

$${{\mathcal{T}}_{\mathcal{W}}}\left( {{{{\bar {\Xi }}}_{\mathcal{W}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right) = {{\mathcal{T}}_{\mathcal{R}}}\left( {{{{\bar {\Xi }}}_{\mathcal{R}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right){{\psi }_{d}}(\theta ).$$

(A.29b)

Having multiplied (A.29a) by adj$\left\{ {{{\mathcal{T}}_{\mathcal{X}}}\left( {{{{\bar {\Xi }}}_{\mathcal{X}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right)} \right\}$, the regression equation ${{\mathcal{Y}}_{{AB}}}(t)$ = ${{\mathcal{M}}_{{AB}}}(t){{\Theta }_{{AB}}}(\theta )$ is obtained.

The next aim it to parametrize equation with respect to L(θ). If Assumption 2 is met, then, following the generalized pole placement theory [15, 16], the vector L(θ) can be obtained as a solution of the following set of equations

$$\left\{ \begin{gathered} {{A}^{{\text{T}}}}(\theta )M - M\Gamma = C{{B}^{{\text{T}}}}(\theta ) \hfill \\ {{B}^{{\text{T}}}}(\theta ) = {{L}^{{\text{T}}}}(\theta )M, \hfill \\ \end{gathered} \right.$$

(A.30)

which has a unique solution [15, 16], as, following Assumption 3, the pair $\left( {{{A}^{{\text{T}}}}(\theta ),C} \right)$ is controllable, the pair $\left( {{{B}^{{\text{T}}}}(\theta ),\Gamma } \right)$ is observable and $\sigma \left\{ {{{A}^{{\text{T}}}}(\theta )} \right\}$ ∩ σ{Γ} = 0.

Having vectorized the first equation from (A.30) and considered the property vec(AB) = (I ⊗ A)vec(B) = $\left( {{{B}^{{\text{T}}}} \otimes I} \right)vec(A)$, it is obtained that:

$$\left( {{{I}_{n}} \otimes {{A}^{{\text{T}}}}(\theta ) - {{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right)vec(M) = vec\left( {C{{B}^{{\text{T}}}}(\theta )} \right).$$

(A.31)

As Eqs. (A.30), (A.31) have unique solutions, then

$$\det \left\{ {{{I}_{n}} \otimes {{A}^{{\text{T}}}}(\theta ) - {{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\} \ne 0,$$

and therefore, having multiplied (A.31) by an adjoint matrix adj$\left\{ {{{I}_{n}} \otimes {{A}^{{\text{T}}}}(\theta ) - {{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\}$, it is written:

$$\begin{gathered} \det \left\{ {{{I}_{n}} \otimes {{A}^{{\text{T}}}}(\theta ) - {{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\}vec(M) \\ = \operatorname{adj} \left\{ {{{I}_{n}} \otimes {{A}^{{\text{T}}}}(\theta ) - {{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\}vec\left( {C{{B}^{{\text{T}}}}(\theta )} \right). \\ \end{gathered} $$

(A.32)

The obtained result is devectorized (vec^–1{.}) and substituted into the second equation of (A.30):

$$\begin{gathered} \underbrace {\det \left\{ {{{I}_{n}} \otimes {{A}^{{\text{T}}}}(\theta ) - {{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\}B(\theta )}_{\mathcal{Q}({{\Theta }_{{AB}}})} \\ = \underbrace {ve{{c}^{{ - 1}}}{{{\left\{ {\operatorname{adj} \left\{ {{{I}_{n}} \otimes {{A}^{{\text{T}}}}(\theta ) - {{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\}vec\left( {C{{B}^{{\text{T}}}}(\theta )} \right)} \right\}}}^{{\text{T}}}}}_{\mathcal{P}({{\Theta }_{{AB}}})}L(\theta ), \\ \end{gathered} $$

(A.33)

where det{$\mathcal{P}$(Θ_AB)} ≠ 0.

The following equalities are introduced:

$$\begin{gathered} {{\mathcal{M}}_{{AB}}}(t){{A}^{{\text{T}}}}(\theta ) = ve{{c}^{{ - 1}}}\left( {{{\mathcal{L}}_{{{{A}^{{\text{T}}}}}}}{{\mathcal{D}}_{\Phi }}{{\mathcal{Y}}_{{AB}}}(t)} \right), \\ {{\mathcal{M}}_{{AB}}}(t){{B}^{{\text{T}}}}(\theta ) = {{\left[ {{{\mathcal{L}}_{B}}{{\mathcal{D}}_{\Phi }}{{\mathcal{Y}}_{{AB}}}(t)} \right]}^{{\text{T}}}}, \\ {{\mathcal{M}}_{{AB}}}(t)B(\theta ) = {{\mathcal{L}}_{B}}{{\mathcal{D}}_{\Phi }}{{\mathcal{Y}}_{{AB}}}(t). \\ \end{gathered} $$

(A.34)

Having multiplied (A.33) by Π_L(${{\mathcal{M}}_{{AB}}}$) = $\mathcal{M}_{{AB}}^{{{{n}^{2}} + 1}}{{I}_{n}}$, used the properties ${{c}^{n}}\det \{ A\}$ = det{cA}, ${{c}^{{n - 1}}}\operatorname{adj} \{ A\}$ = adj{cA}, A ∈ ${{\mathbb{R}}^{{n \times n}}}$ and substituted (A.34), it is obtained:

$${{\mathcal{T}}_{\mathcal{P}}}({{\Xi }_{\mathcal{P}}}({{\mathcal{M}}_{{AB}}}){{\Theta }_{{AB}}}) = {{\Pi }_{L}}({{\mathcal{M}}_{{AB}}})\mathcal{P}({{\Theta }_{{AB}}}) = \mathcal{M}_{{AB}}^{{{{n}^{2}} + 1}}\mathcal{P}({{\Theta }_{{AB}}})$$

$$ = \mathcal{M}_{{AB}}^{{{{n}^{2}} + 1}}ve{{c}^{{ - 1}}}{{\left\{ {\operatorname{adj} \left\{ {{{I}_{n}} \otimes {{A}^{{\text{T}}}}(\theta ) - {{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\}vec\left( {C{{B}^{{\text{T}}}}(\theta )} \right)} \right\}}^{{\text{T}}}}$$

$$\begin{gathered} = ve{{c}^{{ - 1}}}{{\left\{ {{{\mathcal{M}}_{{AB}}}\operatorname{adj} \left\{ {{{I}_{n}} \otimes ve{{c}^{{ - 1}}}\left( {{{\mathcal{L}}_{{{{A}^{{\text{T}}}}}}}{{\mathcal{D}}_{\Phi }}{{\mathcal{Y}}_{{AB}}}} \right) - {{\mathcal{M}}_{{AB}}}{{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\}vec\left( {C{{{({{\mathcal{L}}_{B}}{{\mathcal{D}}_{\Phi }}{{\mathcal{Y}}_{{AB}}})}}^{{\text{T}}}}} \right)} \right\}}^{{\text{T}}}}, \\ {{\mathcal{T}}_{\mathcal{Q}}}({{\Xi }_{\mathcal{Q}}}({{\mathcal{M}}_{{AB}}}){{\Theta }_{{AB}}}) = {{\Pi }_{L}}({{\mathcal{M}}_{{AB}}})\mathcal{Q}({{\Theta }_{{AB}}}) = \mathcal{M}_{{AB}}^{{{{n}^{2}} + 1}}\mathcal{Q}({{\Theta }_{{AB}}}) \\ \end{gathered} $$

(A.35)

$$ = \mathcal{M}_{{AB}}^{{{{n}^{2}} + 1}}\det \left\{ {{{I}_{n}} \otimes {{A}^{{\text{T}}}}(\theta ) - {{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\}B(\theta )$$

$$ = \det \left\{ {{{I}_{n}} \otimes ve{{c}^{{ - 1}}}\left( {{{\mathcal{L}}_{{{{A}^{{\text{T}}}}}}}{{\mathcal{D}}_{\Phi }}{{\mathcal{Y}}_{{AB}}}(t)} \right) - {{\mathcal{M}}_{{AB}}}(t){{\Gamma }^{{\text{T}}}} \otimes {{I}_{n}}} \right\}({{\mathcal{L}}_{B}}{{\mathcal{D}}_{\Phi }}{{\mathcal{Y}}_{{AB}}}(t)),$$

where ${{\Xi }_{\mathcal{P}}}({{\mathcal{M}}_{{AB}}}) = {{\Xi }_{\mathcal{Q}}}({{\mathcal{M}}_{{AB}}}) = {{\mathcal{M}}_{{AB}}}(t)$.

The following regression equation is written on the basis of Eqs. (A.33) and (A.35):

$${{\mathcal{T}}_{\mathcal{Q}}}\left( {{{{\bar {\Xi }}}_{\mathcal{Q}}}({{\mathcal{M}}_{{AB}}}){{\mathcal{Y}}_{{AB}}}} \right) = {{\mathcal{T}}_{\mathcal{P}}}\left( {{{{\bar {\Xi }}}_{\mathcal{P}}}({{\mathcal{M}}_{{AB}}}){{\mathcal{Y}}_{{AB}}}} \right)L(\theta ),$$

(A.36)

where ${{\bar {\Xi }}_{\mathcal{P}}}({{\mathcal{M}}_{{AB}}}) = {{\bar {\Xi }}_{\mathcal{Q}}}({{\mathcal{M}}_{{AB}}})$ = 1.

Having multiplied (A.36) by adj$\left\{ {{{\mathcal{T}}_{\mathcal{P}}}\left( {{{{\bar {\Xi }}}_{\mathcal{P}}}({{\mathcal{M}}_{{AB}}}){{\mathcal{Y}}_{{AB}}}} \right)} \right\}$, the regression equation ${{\mathcal{Y}}_{L}}(t)$ = ${{\mathcal{M}}_{L}}(t)L(\theta )$ is obtained.

The next aim is to derive the regression equation with respect to x_δ0. Using the properties of the vectorization operation

$$vec\left( {{{\psi }_{d}}(\theta )h_{\delta }^{{\text{T}}}{{\Phi }_{\delta }}(t){{x}_{{\delta 0}}}} \right) = \underbrace {\left( {x_{{\delta 0}}^{{\text{T}}} \otimes {{\psi }_{d}}(\theta )} \right)}_{n \times {{n}_{\delta }}}\underbrace {vec\left( {h_{\delta }^{{\text{T}}}{{\Phi }_{\delta }}} \right)}_{{{n}_{\delta }}},$$

$$vec\left( {\left( {x_{{x0}}^{{\text{T}}} \otimes {{\psi }_{d}}(\theta )} \right)vec\left( {h_{\delta }^{{\text{T}}}{{\Phi }_{\delta }}(t)} \right)} \right) = \underbrace {\left( {h_{\delta }^{{\text{T}}}{{\Phi }_{\delta }}(t) \otimes {{I}_{n}}} \right)}_{n \times n{{n}_{\delta }}}\underbrace {vec\left( {x_{{\delta 0}}^{{\text{T}}} \otimes {{\psi }_{d}}(\theta )} \right)}_{n{{n}_{\delta }}},$$

Eq. (3.1) is rewritten as follows:

$$\begin{gathered} \dot {\xi }(t) = {{A}_{0}}\xi (t) + {{\psi }_{a}}(\theta )y(t) + {{\psi }_{b}}(\theta )u(t) \\ + \;\left( {h_{\delta }^{{\text{T}}}{{\Phi }_{\delta }}(t) \otimes {{I}_{n}}} \right)vec\left( {x_{{\delta 0}}^{{\text{T}}} \otimes {{\psi }_{d}}(\theta )} \right). \\ \end{gathered} $$

(A.37)

The following error is introduced:

$$e(t) = \xi (t) - z(t) - \Omega (t){{\psi }_{a}}(\theta ) - P(t){{\psi }_{b}}(\theta ) - V(t)vec\left( {x_{{\delta 0}}^{{\text{T}}} \otimes {{\psi }_{d}}(\theta )} \right).$$

(A.38)

Having differentiated (A.38), Eq. $\dot {e}(t) = {{A}_{K}}e(t)$ is obtained in a similar way as (A.2). Then, having multiplied (A.38) by $C_{0}^{{\text{T}}}$, it is written:

$$\begin{gathered} \bar {q}(t) = C_{0}^{{\text{T}}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}e({{t}_{0}}) + C_{0}^{{\text{T}}}\Omega (t){{\psi }_{a}}(\theta ) \\ + \;C_{0}^{{\text{T}}}P(t){{\psi }_{b}}(\theta ) + C_{0}^{{\text{T}}}V(t)vec\left( {x_{{\delta 0}}^{{\text{T}}} \otimes {{\psi }_{d}}(\theta )} \right). \\ \end{gathered} $$

(A.39)

Using the properties

$$\begin{gathered} x_{{\delta 0}}^{{\text{T}}} \otimes {{\psi }_{d}}(\theta ) = {{\psi }_{d}}(\theta ){{x}_{{\delta 0}}}, \\ vec({{\psi }_{d}}(\theta ){{x}_{{\delta 0}}}) = \underbrace {({{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta ))}_{n{{n}_{\delta }} \times {{n}_{\delta }}}{{x}_{{\delta 0}}}, \\ \end{gathered} $$

Eq. (A.39) is transformed into

$$\begin{gathered} \bar {q}(t) = C_{0}^{{\text{T}}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}e({{t}_{0}}) + C_{0}^{{\text{T}}}\Omega (t){{\psi }_{a}}(\theta ) \\ + \;C_{0}^{{\text{T}}}P(t){{\psi }_{b}}(\theta ) + C_{0}^{{\text{T}}}V(t)vec\left( {x_{{\delta 0}}^{{\text{T}}} \otimes {{\psi }_{d}}(\theta )} \right). \\ \end{gathered} $$

(A.40)

To compensate for the unknown terms $C_{0}^{{\text{T}}}\Omega (t){{\psi }_{a}}(\theta )$ + $C_{0}^{{\text{T}}}P(t){{\psi }_{b}}(\theta )$, the following auxiliary signal is introduced

$$\begin{gathered} {{{\bar {p}}}_{e}}(t) = \Delta (t)C_{0}^{{\text{T}}}\Omega (t){{\psi }_{a}}(\theta ) + \Delta (t)C_{0}^{{\text{T}}}P(t){{\psi }_{b}}(\theta ) \\ = C_{0}^{{\text{T}}}\Omega (t){{\mathcal{L}}_{a}}\mathcal{Y}(t) + C_{0}^{{\text{T}}}P(t){{\mathcal{L}}_{b}}\mathcal{Y}(t), \\ \end{gathered} $$

(A.41)

where

$$\begin{gathered} {{\mathcal{L}}_{a}}\mathcal{Y}(t) = {{\mathcal{L}}_{a}}\Delta (t)\eta (\theta ) = \Delta (t){{\mathcal{L}}_{a}}\eta (\theta ) = \Delta (t){{\psi }_{a}}(\theta ), \\ {{\mathcal{L}}_{b}}\mathcal{Y}(t) = \Delta (t){{\mathcal{L}}_{b}}\eta (\theta ) = \Delta (t){{\psi }_{b}}(\theta ). \\ \end{gathered} $$

Having multiplied (A.40) by Δ(t) and subtracted (A.41) from the obtained result, it is written:

$$\begin{gathered} p(t) = \Delta (t)\bar {q}(t) - {{{\bar {p}}}_{e}}(t) \\ = \Delta (t)C_{0}^{{\text{T}}}V(t)\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right){{x}_{{\delta 0}}} + \Delta (t)C_{0}^{{\text{T}}}{{e}^{{{{A}_{K}}(t - {{t}_{0}})}}}e({{t}_{0}}). \\ \end{gathered} $$

(A.42)

To implement the multiplier ψ_d(θ) indirectly, Eq. (A.29b) is multiplied by adj $\left\{ {{{\mathcal{T}}_{\mathcal{R}}}\left( {{{{\bar {\Xi }}}_{\mathcal{R}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right)} \right\}$:

$$\begin{gathered} {{\mathcal{Y}}_{{{{\psi }_{d}}}}}(t) = {{\mathcal{M}}_{{{{\psi }_{d}}}}}(t){{\psi }_{d}}(\theta ), \\ {{\mathcal{Y}}_{{{{\psi }_{d}}}}}(t) = \operatorname{adj} \left\{ {{{\mathcal{T}}_{\mathcal{R}}}\left( {{{{\bar {\Xi }}}_{\mathcal{R}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right)} \right\}{{\mathcal{T}}_{\mathcal{W}}}\left( {{{{\bar {\Xi }}}_{\mathcal{W}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right), \\ {{\mathcal{M}}_{{{{\psi }_{d}}}}}(t) = \det \left\{ {{{\mathcal{T}}_{\mathcal{R}}}\left( {{{{\bar {\Xi }}}_{\mathcal{R}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right)} \right\}. \\ \end{gathered} $$

(A.43)

The multiplication of (A.42) by ${{\mathcal{M}}_{{{{\psi }_{d}}}}}(t)$ and substitution of (A.43) into the obtained result allow one to write:

$$\begin{gathered} {{\mathcal{M}}_{{{{\psi }_{d}}}}}(t)p(t) = {{\mathcal{M}}_{{{{\psi }_{d}}}}}(t)\Delta (t)C_{0}^{{\text{T}}}V(t)\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right){{x}_{{\delta 0}}} \\ = \Delta (t)C_{0}^{{\text{T}}}V(t)\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\mathcal{Y}}_{{{{\psi }_{d}}}}}(t)} \right){{x}_{{\delta 0}}}. \\ \end{gathered} $$

(A.44)

Having filtered (A.44) via (4.3) and multiplied the obtained result by adj{V_f(t)}, the regression equation ${{\mathcal{Y}}_{{{{x}_{{\delta 0}}}}}}(t)$ = ${{\mathcal{M}}_{{{{x}_{{\delta 0}}}}}}(t){{x}_{{\delta 0}}}$ is obtained, which completes proof of statement that Eqs. (4.2) can be formed on the basis of the measurable signals.

Following Lemma 1, if $\bar {\varphi }(t)$ ∈ PE, then for all t $\geqslant$ t₀ + T it holds that Δ(t) $\geqslant$ Δ_min > 0, and, owing to Hypotheses 1–3 and proved inequalities:

$${{\det }^{2}}\{ \mathcal{X}(\theta )\} > 0,\quad {{\det }^{2}}\{ \mathcal{R}(\theta )\} > 0,$$

$${{\det }^{2}}\{ \mathcal{G}({{\psi }_{{ab}}})\} > 0,\quad {{\det }^{2}}\{ \mathcal{P}({{\Theta }_{{AB}}})\} > 0,$$

$$\det \{ {{\Pi }_{\theta }}(\Delta )\} \; \geqslant \;{{\Delta }^{{{{\ell }_{\theta }}}}}(t),\quad \det \{ {{\Pi }_{\Theta }}({{\mathcal{M}}_{\theta }})\} \; \geqslant \;\mathcal{M}_{\theta }^{{{{\ell }_{\Theta }}}}(t),$$

$$\det \{ {{\Pi }_{{{{\psi }_{d}}}}}({{\mathcal{M}}_{\theta }})\} \; \geqslant \;\mathcal{M}_{\theta }^{{{{\ell }_{{{{\psi }_{d}}}}}}}(t),\quad \det \{ {{\Pi }_{L}}({{\mathcal{M}}_{{AB}}})\} \; \geqslant \;\mathcal{M}_{{AB}}^{{{{n}^{3}} + n}}(t),$$

we have that, if $\bar {\varphi }(t)$ ∈ PE, then for all t $\geqslant$ t₀ + T the following holds:

$$\left| {{{\mathcal{M}}_{\theta }}(t)} \right| = \left| {\det \left\{ {{{\mathcal{T}}_{\mathcal{G}}}\left( {{{{\bar {\Xi }}}_{\mathcal{G}}}(\Delta ){{\mathcal{Y}}_{{ab}}}} \right)} \right\}} \right| = \left| {\det \{ {{\Pi }_{\theta }}(\Delta )\} \det \{ \mathcal{G}({{\psi }_{{ab}}})\} } \right|$$

$$ \geqslant \;\left| {\det \{ \mathcal{G}({{\psi }_{{ab}}})\} } \right|\Delta _{{\min }}^{{{{\ell }_{\theta }}}} = \underline {{{\mathcal{M}}_{\theta }}} > 0,$$

$$\left| {{{\mathcal{M}}_{{AB}}}(t)} \right| = \left| {\det \left\{ {{{\mathcal{T}}_{\mathcal{X}}}\left( {{{{\bar {\Xi }}}_{\mathcal{X}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right)} \right\}} \right| = \left| {\det \{ {{\Pi }_{\Theta }}({{\mathcal{M}}_{\theta }})\} \det \{ \mathcal{X}(\theta )\} } \right|$$

$$ \geqslant \;\left| {{{{\det }}^{{{{\ell }_{\Theta }}}}}\{ \mathcal{G}({{\psi }_{{ab}}})\} } \right|\left| {\det \{ \mathcal{X}(\theta )\} } \right|\Delta _{{\min }}^{{{{\ell }_{\theta }}{{\ell }_{\Theta }}}} = \underline {{{\mathcal{M}}_{{AB}}}} > 0,$$

$$\left| {{{\mathcal{M}}_{{{{\psi }_{d}}}}}(t)} \right| = \left| {\det \left\{ {{{\mathcal{T}}_{\mathcal{R}}}\left( {{{{\bar {\Xi }}}_{\mathcal{R}}}({{\mathcal{M}}_{\theta }}){{\mathcal{Y}}_{\theta }}} \right)} \right\}} \right| = \left| {\det \{ {{\Pi }_{{{{\psi }_{d}}}}}({{\mathcal{M}}_{\theta }})\} \det \{ \mathcal{R}(\theta )\} } \right|$$

$$ \geqslant \;\left| {{{{\det }}^{{{{\ell }_{{{{\psi }_{d}}}}}}}}\{ \mathcal{G}({{\psi }_{{ab}}})\} } \right|\left| {\det \{ \mathcal{R}(\theta )\} } \right|\Delta _{{\min }}^{{{{\ell }_{\theta }}{{\ell }_{{{{\psi }_{d}}}}}}} = \underline {{{\mathcal{M}}_{{{{\psi }_{d}}}}}} > 0,$$

$$\left| {{{\mathcal{M}}_{L}}(t)} \right| = \left| {\det \left\{ {{{\mathcal{T}}_{\mathcal{P}}}\left( {{{{\bar {\Xi }}}_{\mathcal{P}}}({{\mathcal{M}}_{{AB}}}){{\mathcal{Y}}_{{AB}}}} \right)} \right\}} \right| = \left| {\det \{ {{\Pi }_{L}}({{\mathcal{M}}_{{AB}}})\} \det \{ \mathcal{P}({{\Theta }_{{AB}}})\} } \right|$$

$$ \geqslant \;\left| {\det \{ \mathcal{P}({{\Theta }_{{AB}}})\} } \right|\mathcal{M}_{{AB}}^{{{{n}^{3}} + n}}\; \geqslant \;\left| {\det \{ \mathcal{P}({{\Theta }_{{AB}}})\} } \right|\underline {\mathcal{M}_{{AB}}^{{{{n}^{3}} + n}}} = \underline {{{\mathcal{M}}_{L}}} > 0.$$

To obtain the lower bound for the regressor ${{\mathcal{M}}_{{{{x}_{{\delta 0}}}}}}(t)$, first of all, such bound needs to be derived for the solution of the differential equation for V_f(t) in case $\bar {\varphi }(t)$ ∈ PE and $\left( {h_{\delta }^{{\text{T}}}{{\Phi }_{\delta }}(t) \otimes {{I}_{n}}} \right)$ ∈ PE:

$${{V}_{f}}(t) = \int\limits_{{{t}_{0}}}^t {{{e}^{{ - {{k}_{2}}(t - \tau )}}}{{\Delta }^{2}}(\tau ){{{\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\mathcal{Y}}_{{{{\psi }_{d}}}}}(\tau )} \right)}}^{{\text{T}}}}{{V}^{{\text{T}}}}(\tau ){{C}_{0}}C_{0}^{{\text{T}}}V(\tau )\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\mathcal{Y}}_{{{{\psi }_{d}}}}}(\tau )} \right)d\tau } $$

$$ = {{\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)}^{{\text{T}}}}\int\limits_{{{t}_{0}}}^t {{{e}^{{ - {{k}_{2}}(t - \tau )}}}\mathcal{M}_{{{{\psi }_{d}}}}^{2}(\tau ){{\Delta }^{2}}(\tau ){{V}^{{\text{T}}}}(\tau ){{C}_{0}}C_{0}^{{\text{T}}}V(\tau )d\tau \left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)} $$

$$ \geqslant \;\underline {\mathcal{M}_{{{{\psi }_{d}}}}^{2}} \Delta _{{\min }}^{2}{{\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)}^{{\text{T}}}}\int\limits_{{{t}_{0}}}^t {{{e}^{{ - {{k}_{2}}(t - \tau )}}}{{V}^{{\text{T}}}}(\tau ){{C}_{0}}C_{0}^{{\text{T}}}V(\tau )d\tau \left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)} $$

$$ \geqslant \;\underline {\mathcal{M}_{{{{\psi }_{d}}}}^{2}} \Delta _{{\min }}^{2}{{\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)}^{{\text{T}}}}\left[ {\int\limits_{{{t}_{0}}}^{t - \bar {k}T} {{{e}^{{ - {{k}_{2}}(t - \tau )}}}{{V}^{{\text{T}}}}(\tau ){{C}_{0}}C_{0}^{{\text{T}}}V(\tau )d\tau } } \right.$$

$$\left. { + \;\sum\limits_{k = 1}^{\bar {k}} {\int\limits_{t - kT}^{t - kT + T} {{{e}^{{ - {{k}_{2}}(t - \tau )}}}{{V}^{{\text{T}}}}(\tau ){{C}_{0}}C_{0}^{{\text{T}}}V(\tau )d\tau } } } \right]\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)$$

$$ \geqslant \;\underline {\mathcal{M}_{{{{\psi }_{d}}}}^{2}} \Delta _{{\min }}^{2}{{e}^{{ - {{k}_{2}}t}}}{{\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)}^{{\text{T}}}}\sum\limits_{k = 1}^{\bar {k}} {\int\limits_{t - kT}^{t - kT + T} {{{e}^{{{{k}_{2}}\tau }}}{{V}^{{\text{T}}}}(\tau ){{C}_{0}}C_{0}^{{\text{T}}}V(\tau )d\tau \left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)} } $$

$$ \geqslant \;\underline {\mathcal{M}_{{{{\psi }_{d}}}}^{2}} \Delta _{{\min }}^{2}{{\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)}^{{\text{T}}}}\sum\limits_{k = 1}^{\bar {k}} {{{e}^{{ - {{k}_{2}}kT}}}\int\limits_{t - kT}^{t - kT + T} {{{V}^{{\text{T}}}}(\tau ){{C}_{0}}C_{0}^{{\text{T}}}V(\tau )d\tau \left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)} } ,$$

where $\bar {k}\; \geqslant \;k\; \geqslant \;1$ are integers.

In accordance with Lemma 6.8 from [6], if $\left( {h_{\delta }^{{\text{T}}}{{\Phi }_{\delta }}(t) \otimes {{I}_{n}}} \right)$ ∈ PE, then the following inequality holds

$$\int\limits_t^{t + T} {{{V}^{{\text{T}}}}(\tau ){{C}_{0}}C_{0}^{{\text{T}}}V(\tau )d\tau } \; \geqslant \;\alpha {{I}_{{n{{n}_{\delta }}}}}$$

(A.45)

and, using the properties of the Kronecker product, it is obtained that:

$$\begin{gathered} {{\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)}^{{\text{T}}}}\underbrace {\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right)}_{n{{n}_{\delta }} \times {{n}_{\delta }}} = \left( {I_{{{{n}_{\delta }}}}^{{\text{T}}} \otimes \psi _{d}^{{\text{T}}}(\theta )} \right)\left( {{{I}_{{{{n}_{\delta }}}}} \otimes {{\psi }_{d}}(\theta )} \right) \\ = {{I}_{{{{n}_{\delta }}}}} \otimes \psi _{d}^{{\text{T}}}(\theta ){{\psi }_{d}}(\theta ) = \underbrace {\psi _{d}^{{\text{T}}}(\theta ){{\psi }_{d}}(\theta )}_{ > 0}{{I}_{{{{n}_{\delta }}}}}. \\ \end{gathered} $$

(A.46)

Then for all t $\geqslant$ t₀ + T it holds that:

$${{V}_{f}}(t)\; \geqslant \;\underbrace {\underline {\mathcal{M}_{{{{\psi }_{d}}}}^{2}} \Delta _{{\min }}^{2}\alpha \sum\limits_{k = 1}^{\bar {k}} {{{e}^{{ - {{k}_{2}}kT}}}\psi _{d}^{{\text{T}}}(\theta ){{\psi }_{d}}(\theta )} }_{ > 0}{{I}_{{{{n}_{\delta }}}}}\; \geqslant \;\sqrt[{{{n}_{\delta }}}]{{{{\mathcal{M}}_{{{{x}_{{\delta 0}}}}}}}}{{I}_{{{{n}_{\delta }}}}},$$

(A.47)

from which for all t $\geqslant$ t₀ + T we have ${{\mathcal{M}}_{{{{x}_{{\delta 0}}}}}}$ $\geqslant$ $\underline {{{\mathcal{M}}_{{{{x}_{{\delta 0}}}}}}}$ > 0, which allows one to obtain:

$$\forall t\; \geqslant \;{{t}_{0}} + T\left| {{{\mathcal{M}}_{\kappa }}(t)} \right| = \left| {\mathcal{M}_{{AB}}^{{{{n}_{\Theta }}}}(t)\mathcal{M}_{L}^{n}(t)\mathcal{M}_{{{{x}_{{\delta 0}}}}}^{{{{n}_{\delta }}}}(t)} \right|\; \geqslant \;\underline {\mathcal{M}_{{AB}}^{{{{n}_{\Theta }}}}} {\kern 1pt} \underline {\mathcal{M}_{L}^{n}} {\kern 1pt} \underline {\mathcal{M}_{{{{x}_{{\delta 0}}}}}^{{{{n}_{\delta }}}}} = \underline {{{\mathcal{M}}_{\kappa }}} > 0.$$

(A.48)

This completes proof of Lemma 2.