Composite Observer of a Linear Time-Varying Singularly Perturbed System with Quasidifferentiable Coefficients

Abstract

For a linear time-varying singularly perturbed system with a small parameter μ for a part of derivatives and quasi-differentiable coefficients, existence conditions are established and μ-asymptotic composite full- and reduced-order observers are constructed. The error in estimating a state with an arbitrary predetermined exponential decay rate converges to an infinitesimal value of the same order of smallness as the small parameter. The observer gain vector are expressed in terms of the gain vectors of subsystems of smaller dimension than the original one and independent of the small parameter, and the parameters of the original system are subject to weaker requirements than those previously known. A constructive algorithm for calculating the gain vector of a composite observer is presented.

APPENDIX

Proof of Theorem 1. It follows from (i) that the DS (A_s, c_s) (8) is uniformly observable and for it there exists a ρ-exponential observer. Under assumption (iii), there exists a μ⁰ρ-exponential observer for the t‑family of fast subsystems (9).

We will look for a full-order observer for the LTVSPS (5) in the form of the system (7). We will look for the gain vector K(t, μ) in the form

$$K(t,\mu ) = \operatorname{diag} \left\{ {{{E}_{{{{n}_{1}}}}},\,\,\frac{1}{\mu }{{E}_{{{{n}_{2}}}}}} \right\}\left( \begin{gathered} {{k}_{1}}(t) \\ {{k}_{2}}(t) \\ \end{gathered} \right),\;\;{{k}_{1}}(t) \in {{\mathbb{R}}^{{{{n}_{1}}}}},\;\;{{k}_{2}}(t) \in {{\mathbb{R}}^{{{{n}_{2}}}}},$$

where k₁(t), k₂(t) are not yet defined. Then the observer (7) will take the form (22), and the error dynamics equations ${{\varepsilon }_{x}}(t,\mu )$ = x(t, μ) – w_x(t, μ), ε_y(t, μ) = y(t, μ) – w_y(t, μ) for the observer (22) will look like:

$$\begin{gathered} {{{\dot {\varepsilon }}}_{x}}(t) = {{{\tilde {A}}}_{1}}(t){{\varepsilon }_{x}}(t) + {{{\tilde {A}}}_{2}}(t){{\varepsilon }_{y}}(t),\quad {{\varepsilon }_{x}} \in {{\mathbb{R}}^{{{{n}_{1}}}}}, \\ \mu {{{\dot {\varepsilon }}}_{y}}(t) = {{{\tilde {A}}}_{3}}(t){{\varepsilon }_{x}}(t) + {{{\tilde {A}}}_{4}}(t){{\varepsilon }_{y}}(t),\quad {{\varepsilon }_{y}} \in {{\mathbb{R}}^{{{{n}_{2}}}}},\quad t > {{t}_{0}}. \\ \end{gathered} $$

(A.1)

Since the observation error dynamics system (A.1) has the form of the LTVSPS (1)–(2), then when the assumptions (v), (vi) of Theorem 1 for the error system (A.1) the conditions of Theorem 3.1 from ([32], p. 212) and, therefore, there is a decoupling Lyapunov transformation of the form (3.4) from ([32], p. 210) with continuously differentiable matrices $\tilde {L}$(t, μ), $\tilde {H}$(t, μ) bounded on T, which satisfy the following system (in order not to clutter the notation, we will omit the dependence of functions on the argument t in some places):

$${{\tilde {A}}_{3}} - {{\tilde {A}}_{4}}\tilde {L}(\mu ) + \mu \tilde {L}(\mu )\left( {{{{\tilde {A}}}_{1}} - {{{\tilde {A}}}_{2}}\tilde {L}(\mu )} \right) = \mu \dot {\tilde {L}}(\mu ),$$

(A.2)

$$\mu \left[ {{{{\tilde {A}}}_{1}} - {{{\tilde {A}}}_{2}}\tilde {L}(\mu )} \right]\tilde {H}(\mu ) - \tilde {H}(\mu )\left[ {{{{\tilde {A}}}_{4}} + \mu \tilde {L}(\mu ){{{\tilde {A}}}_{2}}} \right] + {{\tilde {A}}_{2}} = \mu \dot {\tilde {H}}(\mu ).$$

(A.3)

From (A.2), (A.3) taking into account (i), (vi) we have the approximation:

$$\begin{gathered} \tilde {L}(t,\,\,\mu ) = A_{4}^{{ - 1}}(t){{k}_{2}}(t){{c}_{2}}(t)\tilde {L}(t,\,\,\mu ){{{\tilde {A}}}_{3}}(t) + O(\mu ), \\ \tilde {L}(t,\,\,\mu ) = \tilde {A}_{4}^{{ - 1}}(t){{{\tilde {A}}}_{3}}(t) + O(\mu ), \\ \tilde {H}(t,\,\,\mu ) = \left( {\tilde {H}(t,\,\,\mu ){{k}_{2}}(t){{c}_{2}}(t) + {{{\tilde {A}}}_{2}}(t)} \right)A_{4}^{{ - 1}}(t) + O(\mu ), \\ \tilde {H}(t,\,\,\mu ) = {{{\tilde {A}}}_{2}}(t)\tilde {A}_{4}^{{ - 1}}(t) + O(\mu ). \\ \end{gathered} $$

(A.4)

As a result of the decoupling transformation, the error dynamics system (A.1) will take the form of a system separated by time scales:

$$\begin{gathered} {{{\dot {\varepsilon }}}_{\xi }}(t) = {{A}_{\xi }}(t,\,\,\mu ){{\varepsilon }_{\xi }}(t),\quad {{\varepsilon }_{\xi }} \in {{\mathbb{R}}^{{{{n}_{1}}}}}, \\ \mu {{{\dot {\varepsilon }}}_{\eta }}(t) = {{A}_{\eta }}(t,\,\,\mu ){{\varepsilon }_{\eta }}(t),\quad {{\varepsilon }_{\eta }} \in {{\mathbb{R}}^{{{{n}_{2}}}}},\quad t > {{t}_{0}}, \\ \end{gathered} $$

(A.5)

where

$${{A}_{\xi }}(t,\,\,\mu ) = {{\tilde {A}}_{1}}(t) - {{\tilde {A}}_{2}}(t)\tilde {L}(t,\mu ),\quad {{A}_{\eta }}(t,\,\,\mu ) = {{\tilde {A}}_{4}}(t) + \mu \tilde {L}(t,\,\,\mu ){{\tilde {A}}_{2}}(t).$$

(A.6)

Moreover, according to Statement 1, the solutions (A.1) and (A.5) satisfy the following equalities:

$$\begin{gathered} {{\varepsilon }_{\xi }}(t) = {{\varepsilon }_{x}}(t) + O(\mu ),\;{{\varepsilon }_{x}}(t) = {{\varepsilon }_{\xi }}(t) + O(\mu ), \\ {{\varepsilon }_{\eta }}(t) = \tilde {A}_{4}^{{ - 1}}(t){{{\tilde {A}}}_{3}}(t){{\varepsilon }_{x}}(t) + {{\varepsilon }_{y}}(\tau ) + O(\mu ), \\ {{\varepsilon }_{y}}(t) = - \tilde {A}_{4}^{{ - 1}}(t){{{\tilde {A}}}_{3}}(t){{\varepsilon }_{\xi }}(t) + {{\varepsilon }_{\eta }}(\tau ) + O(\mu ). \\ \end{gathered} $$

(A.7)

Let’s put

$${{k}_{2}}(t) = {{k}_{f}}(t)$$

(A.8)

and we will look for k₁(t) in the form:

$${{k}_{1}}(t) = {{k}_{s}}(t) + {{\tilde {H}}^{0}}(t){{k}_{2}}(t),\quad {{\tilde {H}}^{0}}(t) = ({{A}_{2}}(t) - {{k}_{s}}(t){{c}_{2}}(t))A_{4}^{{ - 1}}(t).$$

(A.9)

Let’s substitute (A.9), (A.8) into (A.6) and perform sequential transformations:

$${{A}_{\xi }}(\mu )\mathop = \limits^{({\text{A}}{\text{.9}})} {{A}_{1}} - \left( {{{k}_{s}} + {{{\tilde {H}}}^{0}}{{k}_{2}}} \right){{c}_{1}} - \left( {{{A}_{2}} - \left( {{{k}_{s}} + {{{\tilde {H}}}^{0}}{{k}_{2}}} \right){{c}_{2}}} \right)\tilde {L}(\mu )$$

$$ = \left( {{{A}_{1}} - {{A}_{2}}\tilde {L}(\mu )} \right) + \left( {{{k}_{s}} + {{{\tilde {H}}}^{0}}{{k}_{2}}} \right)\left( {{{c}_{1}} - {{c}_{2}}\tilde {L}(\mu )} \right)$$

$$\mathop = \limits^{{\text{(A}}{\text{.4)}}} {{A}_{1}} - {{A}_{2}}A_{4}^{{ - 1}}\left( {{{k}_{2}}{{c}_{2}}\tilde {L}(\mu ) + {{A}_{3}} - {{k}_{2}}{{c}_{1}}} \right)$$

$$ - \;\left( {{{k}_{s}} + {{{\tilde {H}}}^{0}}{{k}_{2}}} \right)\left[ {{{c}_{1}} - {{c}_{2}}A_{4}^{{ - 1}}\left( {{{k}_{2}}{{c}_{2}}\tilde {L}(\mu ) + {{A}_{3}} - {{k}_{2}}{{c}_{1}}} \right)} \right] + O(\mu )$$

$$\mathop = \limits^{(As,cs)} {{A}_{s}} - {{k}_{s}}{{c}_{s}} + \left( { - {{A}_{2}} + {{k}_{s}}{{c}_{2}} + {{{\tilde {H}}}^{0}}{{k}_{2}}{{c}_{2}}} \right)A_{4}^{{ - 1}}{{k}_{2}}{{c}_{2}}\tilde {L}(\mu )$$

$$ + \;\left( {{{A}_{2}} - {{k}_{s}}{{c}_{2}} - {{{\tilde {H}}}^{0}}{{A}_{4}}} \right)A_{4}^{{ - 1}}{{k}_{2}}{{c}_{1}} + {{\tilde {H}}^{0}}{{k}_{2}}{{c}_{2}}A_{4}^{{ - 1}}({{A}_{3}} - {{k}_{2}}{{c}_{1}}) + O(\mu )$$

$$\mathop = \limits^{{\text{(A}}{\text{.9)}}} {{A}_{s}} - {{k}_{s}}{{c}_{s}} + {{\tilde {H}}^{0}}{{k}_{2}}{{c}_{2}}A_{4}^{{ - 1}}\left( {{{A}_{4}}\tilde {L}(\mu ) + {{k}_{2}}{{c}_{2}}\tilde {L}(\mu ) - {{k}_{2}}{{c}_{1}} + {{A}_{3}}} \right) + O(\mu )$$

$$\mathop = \limits^{{\text{(A}}{\text{.4)}}} {{A}_{s}} - {{k}_{s}}{{c}_{s}} + O(\mu ).$$

Thus, for A_ξ(t, μ) with k₁(t), k₂(t) of the form (A.9), (A.8) the approximation is valid:

$${{A}_{\xi }}(t,\,\,\mu ) = {{A}_{s}}(t) - {{k}_{s}}(t){{c}_{s}}(t) + O(\mu ).$$

(A.10)

Further, from (A.6) it follows

$${{A}_{\eta }}(t,\,\,\mu ) = ({{A}_{4}}(t) - {{k}_{2}}(t){{c}_{2}}(t)) + O(\mu ).$$

(A.11)

Thus, combining (A.10) and (A.11) from (A.5) we get:

$$\begin{gathered} {{{\dot {\varepsilon }}}_{\xi }}(t) = ({{A}_{s}}(t) - {{k}_{s}}(t){{c}_{s}}(t) + O(\mu )){{\varepsilon }_{\xi }}(t), \\ \mu {{{\dot {\varepsilon }}}_{\eta }}(t) = ({{A}_{4}}(t) - {{k}_{2}}(t){{c}_{2}}(t) + O(\mu )){{\varepsilon }_{\eta }}(t),\quad t > {{t}_{0}}. \\ \end{gathered} $$

(A.12)

Since in (A.12) k_s(t), k₂(t) are the gain vectors for the observer (14) of the DS (8) and the observer (15) of the t-family of fast subsystems, then the parameters of the error system (A.12) are O(μ)-close to the parameters of the error dynamics system for the DS and the t-family of fast subsystems observers with gain vectors k_s and k_f, respectively. Therefore, due to the continuous dependence of the solution (A.12) on additive perturbations of the system coefficients, the following estimates are valid: ||ε_ξ(t)|| $\leqslant$ ${{c}_{{{{\rho }_{s}}}}}\exp$(–ρ(t – $\bar {t}$)) + O(μ), t $\geqslant$ $\bar {t}$, ||ε_η(t)|| $\leqslant$ ${{c}_{{{{\rho }_{f}}}}}\exp \left( { - {{\mu }^{0}}\rho \frac{{(t - \bar {t})}}{\mu }} \right)$ + O(μ), t $\geqslant$ $\bar {t}$, whence, taking into account (A.7), it follows that the estimates are fair

$$\begin{gathered} \left\| {{{\varepsilon }_{x}}(t)} \right\|\;\leqslant \;{{c}_{{{{\rho }_{s}}}}}\exp ( - \rho (t - \bar {t})) + O(\mu ),\quad t\; \geqslant \;\bar {t}, \hfill \\ \left\| {{{\varepsilon }_{y}}(t)} \right\|\;\leqslant \;{{c}_{{{{\rho }_{s}}}}}\left\| {\tilde {A}_{4}^{{ - 1}}(t){{{\tilde {A}}}_{3}}(t)} \right\|\exp ( - \rho (t - \bar {t})) + {{c}_{{{{\rho }_{f}}}}}\exp \left( { - {{\mu }^{0}}\rho \left( {\frac{{t - \bar {t}}}{\mu }} \right)} \right) + O(\mu ),\quad t\; \geqslant \;\bar {t}. \hfill \\ \end{gathered} $$

Let c_ρ = $\max \left\{ {{{c}_{{{{\rho }_{s}}}}},{{c}_{{{{\rho }_{f}}}}},{{c}_{{{{\rho }_{s}}}}}\left\| {\tilde {A}_{4}^{{ - 1}}(t){{{\tilde {A}}}_{3}}(t)} \right\|} \right\}$. For μ ∈ (0, μ⁰] the estimate $\exp \left( { - {{\mu }^{0}}\rho \left( {\frac{{t - \bar {t}}}{\mu }} \right)} \right)$ < exp(–ρ(t – $\bar {t}$)), t $\geqslant$ $\bar {t}$, which implies the validity of the estimates ||ε(t, μ)|| $\leqslant$ ${{c}_{\rho }}\exp$(–ρ(t – $\bar {t}$)) + O(μ), t $\geqslant$ $\bar {t}$, and, according to the Definition 7 and from the coincidence of (A.9) and (21) imply the truth of Theorem 1.