Parametric Identifiability of Dynamic Systems Based on Adaptive Systems

3. Approach to PI Estimation

The representation is valid for the system (1) in space $(u,\hspace{0.17em}y)$:

$$\dot{y}=A^{T}P,$$

(4)

where $A\in {ℝ}^{2n}$ is the vector of parameters, $P\in {ℝ}^{2n}$ is the generalised input vector, which is got based on the processing $(u,\hspace{0.17em}y)$ by a system of auxiliary filters.

To evaluate elements of vector $A$, introduce the model based on the set$t=\left\{u(t),\hspace{0.17em}\hspace{0.17em}y(t)\right\}$ for each $t$:

$$\dot{\hat{y}}=-k(\hat{y}-y)+{\hat{A}}^{T}P,$$

(5)

where $\hat{A}\in {ℝ}^{2n}$ is the vector of model parameters, $k>0$ is the parameter setting the properties of the model.

Equation for identification error (prediction) $e=\hat{y}-y$:

$$\dot{e}=-\kappa e+\Delta A^{T}P,$$

(6)

where $\Delta A=\hat{A}-A$.

Let the elements of the vector $p_i\in P$ by constantly excited (CE):

$$\mathrm{C}{\mathrm{E}}_{p_i}:{\underset{¯}{\alpha }}_i\le p_i^2(t)\le {\overline{\alpha }}_i,$$

(7)

where ${\underset{¯}{\alpha }}_i>0,\hspace{0.17em}\hspace{0.17em}{\overline{\alpha }}_i>0$.

Notation:

(i) ${\mathrm{S}}_X(\overline{A})$ is a class of systems (1);

(ii) ${\mathrm{S}}_y(A)$ is a congruent representation (4) on the set$t=\left\{u(u),\hspace{0.17em}\hspace{0.17em}y(t),\hspace{0.17em}\hspace{0.17em}t>t_0\right\}$ ;

(iii) ${\Omega }_i$ is the frequency spectrum of the element $p_i\in P$;

(iv) $p_i\in \mathrm{C}{\mathrm{E}}_{p_i}$ or $p_i\in \mathrm{CE}$;

(v) $p_i\notin \mathrm{CE}$ if the CE condition is not held true.

(vi) the variable is $u(t)\in \mathrm{CE}$ if it has a non-degenerate frequency spectrum for all$t\in$ .

Definition 1. The system (1) of class ${\mathrm{S}}_X(\overline{A})$ is locally parametrically identifiable on the set$t$ if the condition$A\in A=\left\{A\in {ℝ}^{2n}:\hspace{0.17em}‖A-A^{*}‖\le {\epsilon }_A\hspace{0.17em}\hspace{0.17em}\forall t\ge t^{*}>t_0\hspace{0.17em}\hspace{0.17em}\&t\right\}$,

is fulfilled for its representation (4) in class ${\mathrm{S}}_y(A)$, where $A^{*}$ is some reference vector of system parameters (4), ${\epsilon }_A\ge 0$.

We see if there is an identification algorithm vector $\hat{A}$ of the system (5) on$(t)=\left\{t,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t>t_0\right\}$ for some $\left\{A\left(t_{0,i}\right)\right\}$, then starting from the moment $t^{*}$, the condition (8) will be fulfilled for the estimates of vector $A$.

Consider the Lyapunov function $V_e(t)=0.5e^2(t)$.

Theorem 1. Let 1) assumption 1 holds, i.e. the matrix in (1) $\overline{A}\in \mathrm{H}$; 2) the system (1) represents how (4) to set$(t)$ ; 3) the identification system is described by equation (6); 4) $u\in \mathrm{C}{\mathrm{E}}_u$; $y\in \mathrm{C}{\mathrm{E}}_y$,$P\in \mathrm{C}{\mathrm{E}}_P$. Then the system (4) is locally parametrically identifiable in the region$A$ if $\Delta A=0$ follows from $\Delta A^{T}P=0$ and the condition is satisfied:

$${‖\Delta A(t)‖}^2\le {\displaystyle \frac{2k^2}{{\overline{\alpha }}_P}}{V}_e(t),$$

(9)

where $P(t)P^T(t)\le {\overline{\alpha }}_P{I}_n,\hspace{0.17em}\hspace{0.17em}{\overline{\alpha }}_P>0$

, $I_n\in {ℝ}^{n\times n}$

is the unit matrix.

The proof of Theorem 1 is given in Appendix A.

Remark 1. The vector $X(t)$reconstruction in (1), based on (4) and schemes proposed in the literature, gives estimates that do not correspond to components $X(t)$. This follows directly from (4). Therefore, adaptive control laws based on the use of vector $P$ elements are applied in control systems. Estimation $x_2=\dot{y}$, $x_2\in X$ can be obtained directly from (4). The remaining components $X(t)$ are determined based on the symbolic differentiation operation.

Corollary from Theorem 1.If the adaptive algorithm

$$\Delta \dot{A}=-\Gamma eP,$$

(10)

is used to evaluate the vector $A$in (4), then the local parametric identifiability of the system (1) follows from the estimation

$$V(t)\le V(t_0)-2k{\displaystyle \underset{t_0}{\overset{t}{\int }}{V}_e(\tau )d\tau },$$

where $V(t)=V_e(t)+V_{\Delta }(t)$

, $V_{\Delta }(t)=0.5\Delta A^T{\Gamma }^{-1}\Delta A$

, $\Gamma ={\Gamma }^T>0$

is a diagonal matrix.

Let vector $P(t)$ elements be measurable for each $t$. Here, the system (4) is detectable. Then observability, detectability and recoverability of the system (1) follow from properties of system (4).

The proof corollary from Theorem 1 is given in Appendix B.

Consider the Lyapunov (LF) function $V_E=0.5{Е}^{T}RЕ$, where $R=R^T>0$.

Structures of classes ${\mathrm{S}}_X(\overline{A})$ and ${\mathrm{S}}_y(A)$ are congruent, so the following statement is valid for system (1).

Theorem 2.Let 1) the conditions of Theorem 1 are fulfilled for the system (4) of class${\mathrm{S}}_y(A)$; 2) classes ${\mathrm{S}}_X(\overline{A})$

and${\mathrm{S}}_y(A)$

are congruent; 3) $u\in \mathrm{C}{\mathrm{E}}_u$

, $X(t)\in \mathrm{C}{\mathrm{E}}_X$

. Then system (1) is locally parametrically identifiable on class ${\mathrm{S}}_X(\overline{A})$

if

$${‖\Delta \overline{A}‖}^2{\overline{\alpha }}_X+{‖\Delta B‖}^2{\overline{\alpha }}_u\le 4{\mu }^2{V}_E,$$

where${Е}^{T}QЕ\ge \mu {Е}^{T}RЕ$, $\mu >0$, $RK+K^{T}R=-Q$, $Q=Q^T>0$, $‖\Delta \overline{A}‖=\mathrm{tr}\left(\Delta {\overline{A}}^T\Delta \overline{A}\right)$, $\mathrm{tr}$ is the matrix trace.

The proof of Theorem 2 is given in Appendix C.

Consider the adaptive model

$$\dot{\hat{X}}=-KE+\hat{\overline{A}}X+\hat{B}u,$$

(11)

and apply the integral algorithm class

$$\begin{array}{l}\dot{\hat{\overline{A}}}=-{\Gamma }_{\overline{A}}ER{X}^T\\ \Delta \dot{B}=-{\Gamma }_{B}REu\end{array}$$

(12)

to tuning of matrix$\hat{\overline{A}},\hspace{0.17em}\hspace{0.17em}\hat{B}$.

The identification system is described by the equation:

$$\dot{E}=KE+\Delta AX+\Delta Bu,$$

(13)

where $E=\hat{X}-X$, $\hat{X}\in {ℝ}^n$ is model (11) state vector, $K\in \mathrm{H}$ is the matrix of dimension $n\times n$, $\hat{\overline{A}},\hat{B}$ are tuning matrices.

Corollary from Theorem 2. If the conditions of Theorem 2 are fulfilled, and the class of algorithms (12) is used to tuning the model (11) parameters, then the local parametric identifiability of the system (1) follows from the estimation

$$V_W(t)\le V_W(t_0)-2\mu {\displaystyle \underset{t_0}{\overset{t}{\int }}{V}_E(\tau )d\tau }.$$

(14)

The proof corollary from Theorem 1 is given in Appendix D.

We see that the local PI depends on the choice of initial conditions, and fulfilment of the requirements for variables and system input.

Remark 2. Presented results differ from the results [4] based on the application of AI methods. If the decision is made based on experimental data, then various statistics [4] are used. In this paper, we apply the approach to the PI analysis based on the current data analysis. This approach has not been used in PI tasks.

If conditions of Theorem 2 are fulfilled, then the class of algorithms (12) will be called locally identifying.

In the future, for the convenience of reference, the adaptive algorithm (10) will be related to class ${\mathrm{A}}_{IA}^y$, and the law (12) to class ${\mathrm{A}}_{IA}^X$.

Definition 2. A system (1) of class ${\mathrm{S}}_X(\overline{A})$ is extremely locally parametrically identifiable (ELPI) on the set$t$ if the condition$A\in {\text{˜}}_A=\left\{A\in {ℝ}^{2n}:\hspace{0.17em}\hspace{0.17em}\underset{t\to \infty }{\lim }‖A-A^{*}‖\to \tilde{\epsilon },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\tilde{\epsilon }\in (0)\hspace{0.17em}\hspace{0.17em}\forall t\ge t^{*}>t_0\hspace{0.17em}\hspace{0.17em}\&t\right\}$,

is satisfied for its representation (4) in class ${\mathrm{S}}_y(A)$, where $A^{*}$ is some reference vector of system (5) parameters$(0)$, is an area of zero.

Here, the vector $A$ identifiability is understood as the limit proximity to $A^{*}$. Under certain conditions, the global PI of the vector $A$ follows from (15).

Consider again the system (6) and LF $V_{\Delta }(t)=\Delta A^T(t){\Gamma }^{-1}\Delta A(t)$.

Theorem 3. Let the conditions of Theorem 1 be fulfilled and (i) there is a Lyapunov function $V_{\Delta }$ admitting an infinitesimal upper limit; (ii) there is $\vartheta >0$ such that the condition $e\Delta A^{T}P=\vartheta \left({‖\Delta A‖}^2+e^2\right)$ is satisfied for sufficiently large $t$ in some area of zero; (iii) $P\in \mathrm{C}{\mathrm{E}}_P$ with parameters${\underset{¯}{\alpha }}_P,{\overline{\alpha }}_P$; (iv) The inequality

$${\dot{V}}_{\Delta }\le -{\displaystyle \frac{3}{4}}\vartheta {\underset{¯}{\alpha }}_P{\underset{¯}{\lambda }}_{\Gamma }{V}_{\Delta }+{\displaystyle \frac{4}{3{\underset{¯}{\alpha }}_P}}\vartheta {\overline{a}}_P{V}_e,$$

(16)

is valid for the trajectories of the adaptive system (6) and (10), where ${\underset{¯}{\lambda }}_{\Gamma }$ is the minimum eigenvalue of the matrix $\Gamma$. Then the system (6), (10) is locally parametrically identifiable on the set$t$ with estimating $V_{\Delta }(t)\le S_{\Delta }(t)$ if the functional condition

$${\displaystyle \frac{16}{9{\underset{¯}{\alpha }}_P^2{\underset{¯}{\lambda }}_{\Gamma }}}{\overline{a}}_P{V}_e\le V_{\Delta },$$

(17)

is satisfied, where:

4. On ELPI

The ELPI fulfilment guarantees the transition to global PI (GPI). For static procedures (least squares method, maximum likelihood method), ELPI is ensured by the properties of the information matrix. For methods based on the class ${\mathrm{A}}_{IA}$, properties of the information matrix are not directly applicable, because the processes are complex in ASI.

With GPI, we understand the condition fulfilment$A\in {\text{⌢}}_A=\left\{A\in {ℝ}^{2n}:\hspace{0.17em}‖A-A^{*}‖=0\hspace{0.17em}\hspace{0.17em}\forall t\ge t^{*}>t_0\hspace{0.17em}\hspace{0.17em}\&t\right\}$:

The proposed interpretation of GPI as belonging the parameters of model (6) to the set${\text{⌢}}_A$ is linked to the absolute stability of an adaptive system.

Global parametric identifiability follows from Theorem 4 for systems of class ${\mathrm{S}}_y(A)$.

Theorem 4.Let: 1) the conditions of Theorems 1 and 3 are fulfilled; 2) The system of inequalities is valid for processes in the system (6), (10)

$$\left[\begin{array}{l}{\dot{V}}_e\\ {\dot{V}}_{\Delta }\end{array}\right]\le \underset{A_G}{\underbrace{\left[\begin{array}{cc}-k& {\displaystyle \frac{1}{k}}{\overline{\alpha }}_P{\overline{\lambda }}_{\Gamma }\\ {\displaystyle \frac{4}{3{\underset{¯}{\alpha }}_P}}\vartheta {\overline{a}}_P& -{\displaystyle \frac{3}{4}}\vartheta {\underset{¯}{\alpha }}_P{\underset{¯}{\lambda }}_{\Gamma }\end{array}\right]}}\underset{W_G}{\underbrace{\left[\begin{array}{l}{V}_e\\ V_{\Delta }\end{array}\right]}},$$

(20)

where${\overline{\lambda }}_{\Gamma },\hspace{0.17em}{\underset{¯}{\lambda }}_{\Gamma }$are the maximum and minimum eigenvalues of the matrix $\Gamma$

. Then (a) the system (4) is globally parametrically identifiable on the class${\mathrm{S}}_y(A)$, (b) the system (6), (10) is exponentially stable with the estimate:

$$W_G(t)\le e^{A_G\left(t-t_0\right)}{S}_G(t_0)$$

(21)

,

if

where$S_G\in {ℝ}^2$is the state vector of the comparison system${\dot{S}}_G(t)=A_G{S}_G(t)$, $S_G\left(t_0\right)\ge W_G\left(t_0\right)$

.

The proof of Theorem 4 is given in Appendix F.

From Theorem 4, we obtain GPI on the set of initial conditions and ELPI. Since the systems are congruent, this condition is also valid for systems (1) of class ${\mathrm{S}}_X(\overline{A})$. To substantiate this statement, apply the approach [22].

$$9k^2{\underset{¯}{\alpha }}_P^2{\underset{¯}{\lambda }}_{\Gamma }\ge 16{\overline{a}}_P^2{\overline{\lambda }}_{\Gamma },$$

(22)

5. About $m$

6. Lyapunov Exponents in PI Problem

6.1. Stationary System of Class ${\mathrm{S}}_X\left(\overline{A}\right)$

Lyapunov characteristic exponents (LE) are the characteristic of a dynamical system. LE is an indirect PI estimate of the system. This approach to PI has not been considered in the literature. The LE application has its own peculiarities in the proposed paradigm of PI. In particular, it is necessary to consider the issue of detectability, recoverability and identifiability of LE based on the information set of the system. Identifiability is understood as the detectability of Lyapunov exponents. Known approaches allow us to estimate only the maximum (largest) LE [24]. A more promising approach is based on the analysis of geometric frameworks (GF) reflecting the change in LE [24]. Issues of LE detectability based on GF analysis are presented in [24]. Therefore, they are not considered here. Detectability is the important issue for evaluating LE.

In [24], the criteria for $\mathrm{LP}$-detectability of Lyapunov exponents are presented. $\mathrm{LP}$-detectability and recoverability we understood as the ability to the LE estimate. $\mathrm{LP}$-detectability imposes certain requirements on experimental data. The approach allows us to obtain the full range LE.

Let $m=n-1-\upsilon$, where $\upsilon$ is the number of non-recoverable LE.

Definition 4. The system (1) is called $m$-detectable with a $\upsilon$-non-recoverability level if the $\upsilon$ lineal (LE) has an insignificant level.

As follows from definition 4, that if the system of class ${\mathrm{S}}_X\left(\overline{A}\right)$ is $m$-detectable with a level of $\upsilon$-non-recoverability, then this is a sufficient condition for $m$-parametric identifiability of the system. The CE requirement plays an important role, as it guarantees the S-synchronizability and structural identifiability of the nonlinear system.

Remark 5. The definition 4 provides sufficient conditions for evaluating PI systems of class ${\mathrm{S}}_X\left(\overline{A}\right)$. This issue requires further study. Note that LE (for the classes ${\mathrm{S}}_X\left(\overline{A}\right)$ under consideration, the Lyapunov exponents are the eigenvalues of the matrix) depend on system parameters.

Analysing nonstationary (periodic) systems is more difficult, since it is difficult to isolate the parametric space here.

Using LE translates the PI problem into the space of Lyapunov exponent [25] for periodic systems (PS).

6.2. PS for Class ${\mathrm{S}}_X\left(\overline{A}\right)$

Consider the system (1) with the matrix $\overline{A}(t)$. For convenience, the system will be denoted by $S_{per}$.

Assumptions.

A1. $\overline{A}(t)$ is a bounded continuous Frobenius matrix

$$‖\overline{A}(t)‖\le {\alpha }_A,$$

(27)

where ${\alpha }_A>0$, $‖\cdot ‖$ is matrix norm.

A2. $\overline{A}(t)$ is almost periodic, i.e. a subsequence can be selected from any sequence

$$\overline{A}(t)=\overline{A}(t-{\tau }_i)$$

(28)

converging uniformly along the entire axis to some almost periodic matrix $\overline{A}(t)$.

A3. $\overline{A}(t)$ is the Hurwitz matrix for almost all $t\ge 0$Let ${\mathrm{R}}_{\overline{A}}=\left\{{\chi }_i(t),\hspace{0.17em}i=\overline{1,n},\hspace{0.17em}\hspace{0.17em}t>t_0\right\}$is a spectrum of LE ${\chi }_i$ $\left(i=\overline{1,n}\right)$.

Definition 5. The function ${\chi }_i(t)$ is almost periodic in the Bohr sense [26] or the $\mathrm{BF}$- function [25], if such a positive number $l=l(\delta )$ exists, that any segment $\left[a,a+l\right]$ contains at least one number $T_f$, for which it is hold

$$\left|f(x+T_f)-f(x)\right|<\delta .$$

(29)

If ${\chi }_i(t)$ is a $\mathrm{BF}$-function, then it is $\alpha \pi$-almost periodic [26], where $\alpha ,\hspace{0.17em}\hspace{0.17em}\pi$ are positive numbers.

Let the order of the system $S_{per}$ be known. Apply the geometric structure $S{K}_{\Delta {k^{\prime }}_{s,\rho }}$ to decide on the spectrum ${\mathrm{R}}_{\overline{A}}$ [26]. Here $k_s(t,\rho )=\rho \left({\hat{y}}_g\right)/t$, $\rho \left({\hat{y}}_g\right)={\rho }_g=\ln \left|{\hat{y}}_g(t)\right|$, ${\hat{y}}_g(t)$ is an evaluation of the general solution of the system (1).

$S{K}_{\Delta {k^{\prime }}_{s,\rho }}$ described by the function $f_{sk}(t):k_s\to \Delta k_s$, where $\Delta k_s$. $k_s(t)$ is $\mathrm{B}{\mathrm{F}}_{\alpha \pi }$-function, $f_{sk}(t)$ contains areas ${\mathrm{D}}_{sk}$, where a drastic change is taking place.

Theorem 6.If the system$S_{per}$is stable and recoverable, and the function$f_{sk}(t)$contains at the interval$\left[t_0,t^{*}\right]\subset {\text{¯}}_g$$\left(t^{*}\le \overline{t}\right)$at least regions${\mathrm{D}}_{sk}$, then the system$S_{per}$has an order$m$and is$\mathrm{LP}$-identifiable ($\mathrm{LP}$-detectable).

In terms of Theorem 6${\text{¯}}_g$, is a time interval in which an estimate of the general solution of the system $S_{per}$ are obtained.

It follows from Theorem 6 that the system $S_{per}$ is identifiable on the set $\left\{S{K}_{k_{s,\rho }^i}\right\}$. As shown in [25], the location of local minima on $S{K}_{k_{s,\rho }^i}$ coincides with regions ${\mathrm{D}}_{sk}$ of the structure $S{K}_{\Delta {k^{\prime }}_{s,\rho }}$. This result allows us to obtain the set $M_{LE}$ containing estimates LE of the system$S_{per}$. Cardinal $M_{LE}$ may not match the LE number of the system. $M_{LE}$ characterizes the set of system $S_{per}$ lineals.

The detectability (identifiability) of the periodic system (1) with the matrix $\overline{A}(t)$ follows from [25].

Theorem 7. Let 1) assumptions A1-A3 are fulfilled for the system; 2) the system $S_{per}$is recoverable; 3) $u(t)\in \mathrm{C}{\mathrm{E}}_u$; 4) set $\left\{k_{s,\rho }^i(t)\right\}$ elements are $\mathrm{B}{\mathrm{F}}_{\alpha \pi }$-functions; 5) the structure $S{K}_{\Delta k_{s,\rho }^i}$ contains at least $v$ regions of ${\mathrm{D}}_{sk}^v$, which to local minima correspond to the structure$S{K}_{k_{s,\rho }^i}$. Then the set $M_{LE}$ is $\mathrm{LP}$-detectable or fully detectable.

Corollary from Theorem 7.If structures$S{L}_{\Delta k_{s,\rho }^i}$contain only$m$of regions${\mathrm{D}}_{sk}^v$, which to local minima correspond on$S{K}_{k_{s,\rho }^i}$, then the system$S_{per}$is$m$-detectable with an$\upsilon$-non-recoverable level.

Remark 6. Eigenvalues ${\lambda }_i(t)$ of the matrix $\overline{A}(t)$ are periodic functions of time. Therefore, lineals $L^i(t)$ and $L^{i+1}(t)$ corresponding to these functions may overlap. This can generate an infinite range of LE. Determine the acceptable range for $M_{LE}$ and the number that determines the mobility of the largest Lyapunov exponent. The set $M_{LE}$ upper bound is determined by the allowable mobility limit of the largest LE ${\chi }_1$. The estimate is fair for ${\chi }_1$

where${k_{s,\rho }^i}_1$ is the interval of the change of the ith indicator $k_{s,\rho }^i$. The region $M_{LE}$ lower boundary is bounded by the smallest LE ${\kappa }_n$ [25].

Definition 6. The system $S_{per}$ of class ${\mathrm{S}}_X\left(\overline{A}\right)$ with matrix $\overline{A}(t)$ satisfying assumptions A1–A3 is locally $M_{LE}$-identifiable on the set${\text{˜}}_t=\left\{X(t),\hspace{0.17em}\hspace{0.17em}u(t),\hspace{0.17em}\hspace{0.17em}t>t_0\right\}$ , if spectrum ${\mathrm{R}}_{\overline{A}}$ of LE belonging to the class $\mathrm{BF}$-function exists such that${\chi }_i\left(\overline{A}\right)\in M_{LE}=\left\{{\chi }_i(t)\in {\mathrm{R}}_{\overline{A}}:\left|{\chi }_i-{\chi }_i^{*}\right|\le {\epsilon }_{\chi }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall t\in \left[t^{*},\hspace{0.17em}\hspace{0.17em}{t}^{**}\right],\hspace{0.17em}\hspace{0.17em}{t}^{*}>t_0\hspace{0.17em}\hspace{0.17em}\&(t)\right\}$

$$, i=\overline{1,n},$$

(31)

where ${\epsilon }_{\chi }\ge 0$.

The problem of assessing the LE adequacy has its own specifics. Let ${\mathrm{S}}_{{\hat{y}}_g,{\hat{\dot{y}}}_g}$is phase portrait of the system $S_{per}$.

Definition 7 [25]. Estimates of Lyapunov exponents ${\chi }_i$ are $\chi$-adequate in the $ℝ$ space, if areas of their definition coincide with $\alpha \pi$-almost-periodicity regions of structure $S_{{\hat{y}}_g,{\hat{\dot{y}}}_g}$.

Theorem 8 [25].Let:(i) the$S_{per}$-system is stable and recoverable; (ii) the set$M_{LE}$is$\mathrm{LP}$-detectable; (iii) definition regions${\mathrm{D}}_{sl}^j$on the$S{L}_{\Delta k_{s,\rho }^i}$structure coincide with$\alpha \pi$-almost-periodicity regions for the G5 structure$S_{{\hat{y}}_g,{\hat{\dot{y}}}_g}$. Then estimates of elements for the set$M_{LE}$are$\chi$-adequate to the regions$\alpha \pi$-almost periodicity$S_{{\hat{y}}_g,{\hat{\dot{y}}}_g}$.

Remark 8. We have considered only one approach to assessing the PI of periodic systems. The PS can be considered as a system with an interval parametric domain and identifiability can be estimated within the specified limits. Here, the approaches described above in sections 4 and 5 are applied.

So, the problem of estimating LPI is reduced to a more adequate task of estimating LE for these systems. The PI conditions in a special space are got and the methods of its estimation are given. The adequacy concept of LE estimates is introduced and the area for the LE location is highlighted. We have shown the existence of a LE set for the system $S_{per}$.

7. About LPI for Decentralized Systems

Consider a decentralized system (DS)

$$S_i:\left\{\begin{array}{l}{\dot{X}}_i=A_i{X}_i+B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m{\overline{A}}_{ij}{X}_j+}{F}_i\left(X_i\right),\\ Y_i={С}_i{X}_i,\end{array}\right.$$

(32)

where $X_i\in {ℝ}^{n_i}$, $Y_i\in {ℝ}^{q_i}$ are state and output vectors of the $S_i$-subsystem, $u_i\in ℝ$ is control, $i=\overline{1,m}$, ${\displaystyle \sum _{i=1}^m{n}_i=n}$. The elements of the matrices $A_i\in {ℝ}^{n_i\times n_i},\hspace{0.17em}{B}_i\in {ℝ}^{n_i},{\overline{A}}_{ij}\in {ℝ}^{n_i\times n_j}$ are unknown; $C_i\in {ℝ}^{q_i\times n_i}$. The matrix ${\overline{A}}_{ij}$ reflects the mutual influence of the subsystem $S_j$. $F_i\left(X_i\right)\in {ℝ}^{n_i}$ considers the nonlinear state of subsystem 2$S_i$, and the $A_i\in H$ is the Hurwitz matrix (stable).

Assumption 2. $F_i\left(X_i\right)$

belongs to the class

$$N_F\left({\pi }_1,{\pi }_2\right)=\left\{F(X)\in {ℝ}^n:{\pi }_{1}X\le F(X)\le {\pi }_{2}X,\hspace{0.17em}\hspace{0.17em}F(0)=0\right\}$$

(33)

and satisfies the quadratic condition

$${\left({\pi }_{2}X-F\left(X\right)\right)}^T\left(F\left(X\right)-{\pi }_{1}X\right)\ge 0,$$

(34)

where ${\pi }_1>0,\hspace{0.17em}\hspace{0.17em}{\pi }_2>0$.

The information set of measurements for the $S_i$-subsystem has the form$o,i=\left\{X_i(t),\hspace{0.17em}\hspace{0.17em}{u}_i(t),\hspace{0.17em}\hspace{0.17em}{X}_j(t),\hspace{0.17em}\hspace{0.17em}t\in =\left[t_0,t_k\right]\right\}$

Mathematical model

$${\dot{\hat{X}}}_i=K_i\left({\hat{X}}_i-X_i\right)+{\hat{A}}_i{X}_i+{\hat{B}}_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m{\hat{\overline{A}}}_{ij}{X}_j+}{\hat{F}}_i\left(X_i\right),$$

(35)

where $K_i\in H$ is a matrix with known elements; ${\hat{A}}_i,$ ${\hat{B}}_i$, ${\hat{\overline{A}}}_{ij}$ are tuning matrices, ${\hat{F}}_i$ is a priori defined nonlinear vector function.

Problem. Obtain PI estimates for the system (32) based on the set$o,i$ analysis.

DS (32) is nonlinear, so the condition CE (7) is represented as:

$$E{C}_{{\underset{¯}{\alpha }}_{u_{\iota }},{\overline{\alpha }}_{u_{\iota }}}^{\Sigma }:\left({\underset{¯}{\alpha }}_{u_{\iota }}\le u_{\iota }^2(\tau )\le {\overline{\alpha }}_{u_{\iota }}\hspace{0.17em}\right)\&\left({\Omega }_{u_{\iota }}(\omega )\subseteq {\Omega }_S(\omega )\right),$$

Where ${\Omega }_{u_i}(\omega )$ is the set of frequencies for $u_i$; ${\Omega }_{\mathrm{S}}(\omega )$ is the set of acceptable frequencies of input $u_i$, ensuring S-synchronizability of the system.

Get the equation for the error $E_i={\hat{X}}_i-X_i$:

$${\dot{E}}_i=K_i{E}_i+\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^M\Delta {\overline{A}}_{ij}{X}_j+}\Delta F_i\left(X_i\right),$$

(37)

where $\Delta A_i={\hat{A}}_i-A_i,\hspace{0.17em}\Delta B_i={\hat{B}}_i-B_i,\hspace{0.17em}\Delta {\overline{A}}_{ij}={\hat{\overline{A}}}_{ij}-{\overline{A}}_{ij},\hspace{0.17em}\Delta F_i={\hat{F}}_i-F_i$ are parametric residuals.

Consider the system (37) and LF $V_i\left(E_i\right)=0.5E_i^T{R}_i{E}_i$, where $R_i=R_i^T>0$ is a positive symmetric matrix.

Let $‖\Delta A_i‖=\sqrt{\mathrm{tr}\left(\Delta A_i^T\Delta A_i\right)}$, $‖\Delta {\overline{A}}_{ij}‖=\sqrt{\mathrm{tr}\left(\Delta A_{ij}^T\Delta A_{ij}\right)}$ are the norm of matrixes $\Delta A_i$, $\Delta {\overline{A}}_{ij}$, $\mathrm{tr}(\cdot )$ is the trace of the matrix.

The following modification of Theorem 1 [27] is true.

Theorem 9.Let: 1) the matrix$A_i\in H$; 2)$X_i(t)\in \mathrm{C}{\mathrm{E}}_{{\underset{¯}{\alpha }}_{X_i},{\overline{\alpha }}_{X_i}}$,$X_j(t)\in P{E}_{{\underset{¯}{\alpha }}_{X_j},{\overline{\alpha }}_{X_j}}$$u_i(t)\in P{E}_{{\underset{¯}{\mathsf{\alpha }}}_{u_i},{ \mathsf{\alpha } \text{¯}}_{u_i}}$; 3)$F_i\left(X_i\right)\in N_F\left({\pi }_1,{\pi }_2\right)$and

$${‖F_i\left(X_i\right)‖}^2\le \eta {\overline{\alpha }}_{X_i}$$

where ${\pi }_1>0,\hspace{0.17em}{\pi }_2>0,\hspace{0.17em}$

$\eta =\eta \left({\pi }_1,{\pi }_2\right)>0,\hspace{0.17em}\hspace{0.17em}{\overline{\alpha }}_{\Xi }={\overline{\alpha }}_{X_i}\left(X_i\right)>0$

, $\eta =2\overline{\pi }+{\pi }^2$

, $\overline{\pi }={\pi }_1{\pi }_2$

, $\pi ={\pi }_1+{\pi }_2$

, ${\delta }_{F_i}>0$

. Then subsystem (32) is locally parametrically identifiable on the set$o,i$

if

$${\mathrm{S}}_{X_i}\left({\pi }_1,{\pi }_2\right)=\left\{X_i\in {ℝ}^{n_i}:{\pi }_1{X}_i\le F_i(X_i)\le {\pi }_2{X}_i\right\},$$

$$\hspace{0.17em}{‖\Delta F_i‖}^2\le 0.25{\overline{\lambda }}_i{V}_i+z_i,$$

(39)

$$\begin{array}{l}\Delta {\dot{A}}_i=-{\Gamma }_{A_i}{E}_i{R}_i{X}_i^T,\\ \Delta {\dot{\overline{A}}}_{ij}=-{\Gamma }_{{\overline{A}}_{ij}}{E}_j{R}_i{X}_j^T,\\ \Delta {\dot{B}}_i=-{\Gamma }_{B_i}{E}_i^T{R}_i{u}_i,\end{array}$$

(40)

$$V_{\Delta ,i}=0.5\mathrm{tr}\left(\Delta A_i^T{\Gamma }_{A_i}^{-1}\Delta A_i\right)+0.5{\displaystyle \sum _{j=1,j\ne i}^m\mathrm{tr}\left(\Delta {\overline{A}}_{ij}^T{\Gamma }_{A_{ij}}^{-1}\Delta {\overline{A}}_{ij}\right)}+0.5\Delta B_i^T{\Gamma }_{B_i}^{-1}\Delta B_i.$$

$${\sigma }_i{\chi }_i\left(t-t_0\right)\le 2{\tilde{\mu }}_i{\sigma }_i{\displaystyle \underset{t_0}{\overset{t}{\int }}V(\tau )}d\tau ,$$

(41)

$$W_{S_i}(t)\le W_{S_i}(t_0)-2{\tilde{\mu }}_i{\sigma }_i{\displaystyle \underset{t_0}{\overset{t}{\int }}V(\tau )}d\tau +{\sigma }_i{\chi }_i\left(t-t_0\right).$$

(42)

$${\hat{F}}_i(X_i)={\tilde{F}}_i^T\left(X_i,{\hat{N}}_{i,1}\right){\hat{N}}_{i,2},$$

(43)

$${\dot{\hat{N}}}_{i,2}=-{\Gamma }_{F_i}{\tilde{F}}_i^T{R}_i{E}_i\left(X_i,{\hat{N}}_{i,1}\right),$$

(44)

8. Examples

1. Consider an engine control system with the Bouc–Wen hysteresis

$$m\ddot{x}+c\dot{x}+F(x,z,t)=f(t),$$

(45)

$$F(x,z,t)=\alpha kx(t)+(1-\alpha )kdz(t),$$

(46)

$$\dot{z}=d^{-1}\left(a\dot{x}-\beta \left|\dot{x}\right|{\left|z\right|}^{n}sign(z)-\gamma \dot{x}{\left|z\right|}^n\right),$$

(47)

where $m>0$ is mass, $c>0$ is damp, $F(x,z,t)$ is the recovering force, $d>0$, $n>0$, $k>0$, $\alpha \in (0,1)$, $f(t)$ is exciting force,$a,\beta ,\gamma$ are some numbers. Set of experimental data <!-- MathType@Translator@5@5@MathML2 (no namespace).tdl@MathML 2.0 (no namespace)@ -->

$$2\left({\overline{\alpha }}_{X_i}{‖\Delta A_i‖}^2+{\overline{\alpha }}_{u_i}{‖\Delta B_i‖}^2+{\displaystyle \sum _{j=1,j\ne i}^m{\overline{\alpha }}_{X_j}{‖\Delta {\overline{A}}_{ij}‖}^2+}2\eta {\overline{\alpha }}_{X_i}+{\delta }_{F_i}\right)\le {\overline{\lambda }}_{Q_i}{V}_i$$

(38)

,

where ${\overline{\lambda }}_{Q_i}={\lambda }_{Q_i}-k_i$

, ${\lambda }_i>0$

is the minimum eigenvalue of the matrix $Q_i$

;$k_i>0$,$K_i{R}_i+K_i^T{R}_i=-Q_i$

,$Q_i=Q_i^T>0$.

The proof of Theorem 9 is given in Appendix H.

Corollary 1 of Theorem 9. Let conditions of Theorem 9 be fulfilled. Then the nonlinearity$F_i\left(X_i\right)$ is locally structurally identifiable in the parametric sector

if

where$z_i\ge 0$.

The proof corollary 1 from Theorem 9 is given in Appendix I.

Consider the system (37) and class ${\mathrm{A}}_{IA}^{S_i}$ algorithms to tune its parameters:

where ${\Gamma }_{A_i}$, ${\Gamma }_{{\overline{A}}_{ij}}$, ${\Gamma }_{B_i}$ are diagonal matrices of the corresponding dimensions.

Lyapunov function for analysis of system (37) and (4): $W_{S_i}=V_i+V_{\Delta ,i}$:

Corollary 2 of Theorem 9. Let 1) conditions of Theorem 9 be fulfilled; 2) the class ${\mathrm{A}}_{IA}^{S_i}$ of algorithms is used to tuning parameters of the model (35). Then the system (32) is locally parametrically identifiable if

where ${\chi }_i=2\eta {\overline{\alpha }}_{X_i}+{\delta }_{F_i}$

,${\gamma }_i^{-1}=\min \left(1,2{\underset{¯}{\lambda }}_{R_i}\right),\sigma ={\mu }_i-{\overline{\lambda }}_{R_i}>0$; ${\underset{¯}{\lambda }}_{R_i},\hspace{0.17em}\hspace{0.17em}{\overline{\lambda }}_{R_i}$are The minimum and maximum eigenvalues of the matrix $R_i$

, and the estimate is held

The proof corollary 2 from Theorem 9 is given in Appendix J.

As follows from Theorem 9, system (32) is LPI and structurally identifiable with nonlinearity $F_i\left(X_i\right)$ in the parametric sector ${\mathrm{S}}_{X_i}\left({\pi }_1,{\pi }_2\right)$ on the set of initial conditions and$o,i$ .

If we perform nonlinearity factorization (see, for example, [27])

where ${\hat{N}}_{i,1}\in {ℝ}^{n_{i,1}}$ is a priori estimation of known parameters, ${\hat{N}}_{i,2}\in {ℝ}^{n_{i,2}}$ is vector of tuning parameters, the structure ${\tilde{F}}_i\left(X_i,N_{i,1}\right)$is formed a priori considering the known vector $N_{i,1}$, and apply the algorithm

where ${\Gamma }_{F_i}$ is a diagonal matrix with positive diagonal elements, then we obtain the conditions for global parametric identifiability for DS on the class of algorithms ${\mathrm{A}}_{IA}^{S_i}$ and (44). They are based on the modernization of results [27].

$$\dot{e}=-k_{x}e+\Delta a_{1}x+\Delta a_2{p}_x+\Delta a_3{p}_z+\Delta b{p}_f$$

(50)

,

where $\Delta a_1={\hat{a}}_1(t)-a_1$, $\Delta a_2={\hat{a}}_2(t)-a_2,\hspace{0.17em}\hspace{0.17em}\Delta a_3={\hat{a}}_3(t)-a_3,\hspace{0.17em}\hspace{0.17em}\Delta b=\hat{b}(t)-b$.

The variable $z$ is not measured. Apply the model to estimate $z$:

and introduce a residual ${\epsilon }_z=x-{\hat{x}}_{\overline{z}}$. Let ${\epsilon }_z$ is current estimate $z$. Then we get the model to evaluate $z$

$$\dot{\hat{z}}=-k_z\left(\hat{z}-{\epsilon }_z\right)+\tilde{\dot{x}}-\hat{\beta }\left|\tilde{\dot{x}}\right|{\left|\hat{z}\right|}^{n}sign(\hat{z})-\hat{\gamma }\tilde{\dot{x}}{\left|\hat{z}\right|}^n$$

(52)

where $k_z>0$; $\hat{\beta }$, $\hat{\gamma }$ are estimates of hysteresis parameters (47); $\tilde{\dot{x}}=\left(x(t+\tau )-x(t)\right)/\tau$, $\tau$ is the integration step.

Introduce a residual $\epsilon =\hat{z}-{\epsilon }_z$, satisfying equations

where $\Delta \dot{x}=\tilde{\dot{x}}-\dot{x},\hspace{0.17em}\hspace{0.17em}\Delta \beta =\beta -\hat{\beta },\hspace{0.17em}\hspace{0.17em}\Delta \gamma =\gamma -\hat{\gamma }$, $\Delta \beta =\beta -\hat{\beta }$, $\Delta \gamma =\gamma -\hat{\gamma }$. Present (49) as:

$$\dot{\hat{x}}=-k_x(\hat{x}-x)+{\hat{a}}_{1}x+{\hat{a}}_2{p}_x+{\hat{a}}_3{p}_{\hat{z}}+\hat{b}{p}_f$$

(54)

,

$${\dot{p}}_{\hat{z}}=-\mu p_{\hat{z}}+\hat{z}$$

(55)

,

and (50) is written as:

$$\dot{e}=-k_{x}e+\Delta a_{1}x+\Delta a_2{p}_x+\Delta a_3{p}_{\hat{z}}+\Delta b{p}_f$$

(56)

.

Evaluate the identification quality using the Lyapunov function. $V_{\epsilon }(t)=0.5{\epsilon }^2(t)$. Get adaptive algorithms from ${\dot{V}}_{\epsilon }<0$:

where ${\chi }_{\beta }>0,{\chi }_{\gamma }>0$ are parameters ensuring the stability of algorithms (57).

Consider functionals:

Figure 1 and Figure 2 represent PI evaluations of the system (45) – (47). The ASI has two loops: the main one (variable $e$) and the auxiliary one (variable $\epsilon$). Figure 1 shows the structure ${\mathrm{J}}_{\left|e\right|,\Delta D_a}$ described by the function ${\phi }_a:\left|e\right|\to \Delta D_a$, and Figure 2 presents the structure ${\mathrm{J}}_{\left|\epsilon \right|,\Delta D_{\gamma ,\beta }}$ described by the function ${\phi }_{\gamma ,\beta }:\left|\epsilon \right|\to \Delta D_{\gamma ,\beta }$.

Presented structures confirm the fulfilment of the Theorem 1 conditions, since trajectories of the system for sufficiently large $t$ get into the region

.

$o=\left\{f(t),y(t),t\in \right\}$. Vector of parameters $A={\left[m,c,a,k,\alpha ,\beta ,\gamma ,n\right]}^T$.

To estimate PI on the set$o$ , equation (45) is transformed to the form [28]

$$\dot{x}=a_{1}x+a_2{p}_x+a_3{p}_z+b{p}_f,$$

(48)

$${\dot{p}}_x=-\mu p_x+x,{\dot{p}}_f=-\mu p_f+f,{\dot{p}}_z=-\mu p_z+z,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mu >0,$$

where $a_1=-(c-\mu m)/m$, $a_2=-\left(\alpha k-\mu \left(c-\mu m\right)\right)/m$, $a_3=-\left(\left(1-\alpha \right)k\right)/m$.

Model for system identification (48)

$$\dot{\hat{x}}=-k_x\left(\hat{x}-x\right)+{\hat{a}}_{1}x+{\hat{a}}_2{p}_x+{\hat{a}}_3{p}_z+\hat{b}{p}_f,$$

(49)

where $k_x>0$; ${\hat{a}}_i(t)$, $i=1,2,3$, $\hat{b}(t)$ are adjustable model parameters. Let $e=\hat{x}-x$. From (48) and (49), we obtain the equation for the identification error:

$${\dot{\hat{x}}}_{\overline{z}}=-k_x\left({\hat{x}}_{\overline{z}}-x\right)+{\hat{a}}_{1}x+{\hat{a}}_2{p}_x+\hat{b}{p}_f.$$

(51)

$$\dot{\epsilon }=-k_z\epsilon +\Delta \dot{x}+\Delta \beta \left|\tilde{\dot{x}}\right|{\left|\hat{z}\right|}^{n}sign\left(\hat{z}\right)+\beta {\eta }_{\beta }+\Delta \gamma \tilde{\dot{x}}{\left|\hat{z}\right|}^n+\gamma {\eta }_{\gamma },$$

(53)

$${\eta }_{\beta }=\left|\dot{x}\right|{\left|z\right|}^{n}sign\left(z\right)-\left|\tilde{\dot{x}}\right|{\left|\hat{z}\right|}^{n}sign\left(\hat{z}\right),$$

$$\Delta \dot{\beta }=-{\chi }_{\beta }\epsilon \left|\tilde{\dot{x}}\right|{\left|\hat{z}\right|}^{n}sign\left(\hat{z}\right),\hspace{0.17em}\hspace{0.17em}\Delta \dot{\gamma }=-{\chi }_{\gamma }\epsilon \tilde{\dot{x}}{\left|\hat{z}\right|}^n,$$

(57)

$$\Delta D_a(t)={\left(\Delta a_1^2(t)+\Delta a_2^2(t)+\Delta a_3^2(t)+\Delta b^2(t)\right)}^{0.5}.$$

(58)

2. Consider the system, the phase portrait of which is shown in Figure 3. The set of experimental data$(t)$

. Figure 3 shows the presence of oscillations in the system, the frequency of which differs from the frequency of the input. Therefore, the system is the system with periodic coefficients.

To determine LE, we apply the approach [25] and obtain estimates of the general solution and its derivative.

$${\hat{y}}_q(t)=\left[0.75;0.07;-0.22\right]{\left[1u(t)\dot{u}(t)\right]}^T$$

(59)

The coefficients of determination for these models are equal 0.99. Next, we determine estimates of the free movement for the system. The evaluation of order for the system follows from Figure 4.

Figure 4. LE set

Figure 5. Estimate of $\chi$-adequacy of LE

It follows from $S{K}_{\Delta k_{s,\rho }^1}$ that the system has a third order. From $S{K}_{\Delta k_{s,\rho }^1}$ and $S{K}_{\Delta k_{s,\rho }^2}$, we get the set LE (Figure 4).

The upper estimate for ${\kappa }_m$ is $-2.04$. Mobility limit for ${\chi }_1$ is –0.8. $\chi$-adequacy confirmation of LE estimates is shown in Figure 5. The eigenvalues of the state matrix of the system $S_{per}$ are:

$$M_{LE}=\left\{-2.04;-1.842;-1.77;-1.167;-0.878\right\}.$$

So, we see that the set $M_{LE}$ is $\mathrm{LP}$-detectable, and the system has the third order. Since the elements of the set $M_{LE}$ are recoverable and detectable, the system $S_{per}$ is LPI.

9. Conclusions

The problem of estimating parametric identifiability based on current experimental data is considered. Methods of the priori identifiability based on the analysis of the information matrix are not applicable in this case. We consider the approach based on the application of the second Lyapunov method to the PI study. LPI conditions are got based on the adaptive identification application to the linear dynamical system. We analyse data on state vector and current information on input and output in the problem PI. Conditions and estimates have been obtained that guarantee PI and LPI.

The $m$-parametric identifiability case is considered when the condition of constant excitation is not fulfilled. PI estimates are got for decentralised nonlinear systems and systems with periodic parameters. We show that Lyapunov exponents should be used to PI analyses of the system with periodic parameters.

$${\dot{V}}_e\le -kV+{\displaystyle \frac{1}{2k}}{\overline{\alpha }}_P{‖\Delta A‖}^2$$

,

(A.2)

where ${\overline{\alpha }}_P\le P{P}^T\le {\underset{¯}{\alpha }}_P$, ${\underset{¯}{\alpha }}_P>0,\hspace{0.17em}\hspace{0.17em}{\overline{\alpha }}_P>0$.

Let $u(t)$ and $y(t)$ correspond to Fourier series with multiple frequencies ${\Omega }_u,\hspace{0.17em}\hspace{0.17em}{\Omega }_y$, where ${\Omega }_y$ depends on the spectrum $u(t)$. System (1) is a frequency filter and $A\in \mathrm{H}$. Therefore, the frequency spectra of the elements of the vector $P$ will vary. Therefore, sets ${\Omega }_{p_i}$, where $p_i\in P$, will not have common areas. Then $\Delta A=0$ follows from the identifiability condition $\Delta A^{T}P=0$.

So, condition 4 is necessary for the local identifiability of the system (6). As condition 4 is fulfilled, for the limited trajectories (identifiability) of the system (4), it is necessary that:

$$\begin{array}{l}{\dot{V}}_e\le -2k{V}_e-2{\dot{V}}_{\Delta },\\ 0.5\dot{V}\le -k{V}_e,\end{array}$$

(B.2)

where from

$$V(t)\le V(t_0)-2k{\displaystyle \underset{t_0}{\overset{t}{\int }}{V}_e(\tau )d\tau }$$

(B.3)

or $V(t)\le V(t_0)$.■

$${\dot{V}}_E\le -\mu {Е}^{T}RЕ+\left|{Е}^{T}R\left(\Delta AX+\Delta Bu\right)\right|$$

,

(C.2)

where ${Е}^{T}QЕ\ge \mu {Е}^{T}RЕ$, $\mu >0$, $Q=Q^T>0$ is a positive definite matrix satisfying the equation $RK+K^{T}R=-Q$, $R=R^T>0$. Then (C.2)

$${\dot{V}}_E\le -\mu {Е}^{T}RЕ+\left|{Е}^{T}R\left(\Delta AX+\Delta Bu\right)\right|\le -2\mu V_E+0.5{\mu }^{-1}\left({\overline{\alpha }}_X{‖\Delta A‖}^2+{\overline{\alpha }}_u{‖\Delta B‖}^2\right)$$

,

(C.3)

where ${‖\Delta A‖}^2=\mathrm{tr}\left(\Delta A^T\Delta A\right)$, ${‖\Delta B‖}^2=\Delta B^T\Delta B$, ${\underset{\rightharpoondown }{\alpha }}_X{I}_n\le X(t)X^T(t)\le {\overline{\alpha }}_X{I}_n$, $I_n$ is the identity matrix.

From (C.3), we obtain the condition of LPI:

$$S_{\Delta }(t)=e^{-\sigma (t-t_0)}{S}_{\Delta }\left(t_0\right)+{\displaystyle \frac{4}{3{\underset{¯}{\alpha }}_P}}\vartheta {\overline{a}}_P{\displaystyle \underset{t_0}{\overset{t}{\int }}{e}^{-\sigma (t-\tau )}{V}_e(\tau )d}\tau$$

$\sigma =0.75\vartheta {\underset{¯}{\alpha }}_P{\underset{¯}{\lambda }}_{\Gamma }$, $S(t)$ is the upper solution of the comparison system $\dot{S}=-\sigma S+\pi V_e$ for $V_{\Delta }(t)$ (E.5), if $S_{\Delta }\left(t_0\right)\ge V_{\Delta }\left(t_0\right)$,$\pi ={\displaystyle \frac{4}{3{\underset{¯}{\alpha }}_P}}\vartheta {\overline{a}}_P$.■

$$0.5{\upsilon }_P{‖\Delta \tilde{A}‖}^2+0.5{\epsilon }_{\omega }^2\le 2k_m{V}_e$$

.

(G.3)

Represent (G.1) in the form

Then (G.4)

Transform (G.5)

Let $\eta =\min \left(1,2{\underset{¯}{\lambda }}_{\Gamma }\right)$, $V_v=V_e+{\tilde{V}}_{\Delta }$ and $k_m=k-1$. Then:

$${\dot{V}}_v\le -2k_m{\eta }^{-1}{V}_e+0.5{\eta }^{-1}{\epsilon }_{\omega }^2$$

.

(G.7)

The estimate for $V_v$ (see (26)) follows from (G.7). ■

$$\begin{array}{l}{\dot{V}}_i\le \underset{-\lambda V_i}{\underbrace{-E_i^T{Q}_i{E}_i}}+\underset{\sqrt{k_i{V}_i}}{\underbrace{\left|E_i^T{R}_i\right|}}\left|\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m\Delta {\overline{A}}_{ij}{X}_j+}\Delta F_i\left(X_i\right)\right|\le \\ \le -{\lambda }_i{V}_i+k_i{V}_i+0.5{\left(\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m\Delta {\overline{A}}_{ij}{X}_j+}\Delta F_i\left(X_i\right)\right)}^2,\end{array}$$

(H.2)

where ${\lambda }_i>0$ is the minimum eigenvalue of the matrix $Q_i$.

Apply the Cauchy-Bunyakovsky-Schwarz inequality and Titu's lemma to the last term in (H.2) and get

Consider condition 1) of Theorems 9 and ${\dot{V}}_i$ write as:

where ${\overline{\lambda }}_i={\lambda }_i-k_i$. Apply Lemmas 1, 2 [26] and get for ${‖\Delta F_i‖}^2$

$$\Delta F_i^T\Delta F_i={‖\Delta F_i‖}^2\le 2\eta {\overline{\alpha }}_{X_i}+{\delta }_{F_i}$$

,

(H.5)

where $\eta =2\overline{\pi }+{\pi }^2$, $\pi ={\pi }_1+{\pi }_2$, $\overline{\pi }={\pi }_1{\pi }_2$, ${\delta }_{F_i}\ge 0$.

Then (H.4)

If state variables are CE and the condition (38) is fulfilled, then the system (32) is the LPI on the set$o,i$ .■

$$\begin{array}{c}{\dot{V}}_i=-E_i^T{Q}_i{E}_i+E_i^T{R}_i\left(\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m\Delta {\overline{A}}_{ij}{X}_j+}\Delta F_i\left(X_i\right)+\Delta F_i\left(X_i\right)\right)=\\ =-E_i^T{Q}_i{E}_i-\mathrm{tr}\left(\Delta {\dot{A}}_i^T{\Gamma }_{A_i}^{-1}{R}_i\Delta A_i\right)-{\displaystyle \sum _{j=1,j\ne i}^m\mathrm{tr}\left(\Delta {\dot{\overline{A}}}_{ij}^T{\Gamma }_{{\overline{A}}_{ij}}^{-1}{R}_i\Delta {\overline{A}}_{ij}\right)}-\\ -\Delta {\dot{B}}_i^T{\Gamma }_{B_i}^{-1}{R}_i\Delta B_i+E_i^T{R}_i\Delta F_i\left(X_i\right).\end{array}$$

(J.1)

Let $E_i^T{Q}_i{E}_i\ge {\mu }_i{E}_i^T{R}_i{E}_i\ge 2{\mu }_i{V}_i$, where ${\mu }_i\ge 0$. Then:

where ${\underset{¯}{\lambda }}_{R_i},\hspace{0.17em}\hspace{0.17em}{\overline{\lambda }}_{R_i}$ are minimum and maximum eigenvalues of the matrix $R_i$.

The estimate (H.5) is fair for$\Delta F_i$. Therefore, (J.2) is represented as:

where ${\chi }_i=2\eta {\overline{\alpha }}_{X_i}+{\delta }_{F_i}$. Let${\gamma }_i^{-1}=\min \left(1,2{\underset{¯}{\lambda }}_{R_i}\right),\sigma ={\mu }_i-{\overline{\lambda }}_{R_i}>0$. Transform (J.3):

$${\dot{W}}_{S_i}\le -2{\sigma }_i{\gamma }_i{V}_i+{\gamma }_i{\chi }_i,$$

(J.4)

Then

if ${\sigma }_i{\chi }_i\left(t-t_0\right)\le 2{\tilde{\mu }}_i{\sigma }_i{\displaystyle \underset{t_0}{\overset{t}{\int }}V(\tau )}d\tau$.■

Modelling examples are presented that confirm the efficiency of the proposed approach.