Identification of Decentralised Control Systems

1. Introduction

Decentralised control systems (DCS) are widely used to solve various tasks. Ensuring the DS stability and quality is the principal goal of control. The system works under incomplete a priori information. Thus, an adaptive robust DCS with a reference model is proposed for interconnected time-delayed systems in [1]. the system asymptotic stability is proved. A similar problem of stabilising the DS output using feedback is considered in [2]. Control laws are based on the application of nonlinear damping, adaptive state observer and Lyapunov functions. Various variants of the adaptive control problem under uncertainty are studied in [3, 4]. In [5], a design method is proposed for adaptive decentralised regulators based on an identifier and a reference model. Recurrent neural networks [6] are used to control large-scale systems under uncertainty. Algorithms are used to control a flat robot with two degrees of freedom.

Robust DS control of a nonlinear multidimensional object is proposed in [7]. The system identification is based on the frequency approach. The model approach [8] recommends for the control unknown large-scale DS. In [9], correlation analysis and the least squares method were used to identify DS. Correlation analysis and the least squares method [9] are the basis for the DS identification. Stochastic procedures for the DS identification are proposed in [10, 11]. The identification of DS with feedback [12] is based on the analysis of transient characteristics. Adaptive control of nonlinear large-scale systems (LSS) with limited perturbations is considered in [13]. The asymptotic tracking issue for LSS based on nonlinear output feedback considers in [14, 15]. Adaptive algorithms guarantee compensation disturbances.

We see that various identification procedures and methods are used in the DS. Parametric uncertainties are compensated by adjusting the parameters of the adaptive control law. Applied methods of retrospective identification do not always consider the current state of the system. The properties of the proposed algorithms, the system identifiability, and the influence of connections in the system are studied. These difficulties are compensated using multistep identification procedures.

We consider the adaptive identification problem of DS with nonlinearities (NDS) for which the quadratic condition is satisfied (Section 2). Section 3 contains a solution to the parametric identifiability (PI) problem for NDS. We study the influence of the information space on PI. The approach to the synthesis of adaptive identification algorithms based on the second Lyapunov method is proposed. We analyse parametric and signalling algorithms. Properties of the identification system are studied. We proof the exponential dissipativity of the adaptive identification system (AS I).

2. Problem Statement

Consider the system comprising $m$ interconnected subsystems

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

where $X_i\in {ℝ}^{n_i}$, $Y_i\in {ℝ}^{q_i}$ are vectors of the state and output of $S_i$-subsystem, $u_i\in ℝ$ is a control, $i=\overline{1,m}$, ${\displaystyle \sum _{i=1}^m{n}_i=n}$. Parameters 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 $S_j$ subsystem. $F_i\left(X_i\right)\in {ℝ}^{n_i}$ consider the nonlinear state of the $S_i$-subsystem, and $A_i$ is Hurwitz matrix (stable).

Assumption 1. $F_i\left(X_i\right)$belongs to the class

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

and satisfies the quadratic condition

${\left({\pi }_{2}X-F\left(X\right)\right)}^T\left(F\left(X\right)-{\pi }_{1}X\right)\ge 0$, (3)

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

Information set of measurements for subsystems${\mathrm{II}}_{o,i}=\left\{X_i(t),\hspace{0.17em}\hspace{0.17em}{u}_i(t),\hspace{0.17em}\hspace{0.17em}{X}_j(t),\hspace{0.17em}\hspace{0.17em}t\in J=\left[t_0,t_k\right]\right\}$ .

Mathematical model for (1)

${\dot{\widehat{X}}}_i=K_i\left({\widehat{X}}_i-X_i\right)+{\widehat{A}}_i{X}_i+{\widehat{B}}_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m{\widehat{\overline{A}}}_{ij}{X}_j+}{\widehat{F}}_i\left(X_i\right)$, (4)

where $K_i\in H$ is Hurwitz matrix with known parameters (reference model); ${\widehat{A}}_i,$ ${\widehat{B}}_i$, ${\widehat{\overline{A}}}_{ij}$ are tuning matrices of corresponding dimensions, ${\widehat{F}}_i$ is a priori given nonlinear vector function.

Problem: find such algorithms for estimating model (4) parameters based on the analysis of the set${\mathrm{II}}_{o,i}$ and the fulfilment assumption 1, so

$\underset{t\to \infty }{\lim }‖{\widehat{X}}_i(t)-X_i(t)‖\le {\delta }_i$,

where ${\delta }_i\ge 0$.

3. On identifiability $S_i$-Subsystem

Identifiability is the basis for estimating $S_i$-subsystem parameters. It knows that fulfilment the constant excitation (CE) condition for${\mathrm{II}}_{o,i}$ guarantees identifying DS parameters.

The CE condition

$E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}:{\underset{¯}{\alpha }}_{u_i}\le u_i^2(t)\le {\overline{\alpha }}_{u_i}\hspace{0.17em}\hspace{0.17em}\forall t\in \left[t_0,t_0+T\right]\hspace{0.17em}$, (5)

where ${\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}$ are positive numbers, $T>0$. Next, condition (5) will be written as $u_i(t)\in E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}$. If $u_i(t)$ does not have the CE property, then we will write $u_i(t)\notin E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}$ or $u_i(t)\notin EC$.

Remark 1. Condition (5) requires modification, considering S-synchronisation for NDS [16]. We write property (5) as:

$E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}^S:\hspace{0.33em}\hspace{0.33em}\left({\underset{¯}{\alpha }}_{u_i}\le u_i^2(t)\le {\overline{\alpha }}_{u_i}\hspace{0.17em}\right)\&\left({\Omega }_{u_i}(\omega )\subseteq {\Omega }_{\mathrm{S}}(\omega )\right)$,

where ${\Omega }_{u_i}(\omega )$ is the set of frequencies for $u_i$; ${\Omega }_{\mathrm{S}}(\omega )$ is the set of acceptable frequencies for $u_i$, guaranteeing S-synchronizability. Next, we will denote the CE property as $E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}$, assuming, that it guarantees $E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}^S$.

To obtain the conditions of identifiability, consider the model (4). The error equation:

${\dot{E}}_i=K_i{E}_i+\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^M\Delta {\overline{A}}_{ij}{X}_j+}\Delta F_i\left(X_i\right)$, (6)

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

Lemma 1. If the nonlinearity $F(X)\in {ℝ}^n$, $X\in {ℝ}^n$ belongs to the class $N_F\left({\pi }_1,{\pi }_2\right)$ and

${\pi }_{1}X\le F\left(X\right)\le {\pi }_{2}X,$ (7)

then

${‖F\left(X\right)‖}^2\le \eta {\overline{\alpha }}_X$, (8)

where ${\pi }_1>0,\hspace{0.17em}{\pi }_2>0,\hspace{0.17em}$ $\eta =\eta \left({\pi }_1,{\pi }_2\right)>0,{\overline{\alpha }}_X={\overline{\alpha }}_X\left(X\right)>0$.

Lemma 1 proof is presented in Appendix A.

Lemma 2. If Lemma 1 conditions are satisfied, then the estimate is valid for $\Delta F\left(X\right)$$\Delta F^T\Delta F\le 2\eta {\overline{\alpha }}_X+{\delta }_F$,

where $\eta =2\overline{\pi }+{\pi }^2$, $\overline{\pi }={\pi }_1{\pi }_2$, $\pi ={\pi }_1+{\pi }_2$, ${\delta }_F>0$.

Lemma 2 proof is presented in Appendix B.

Consider the system (6) and Lyapunov function (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 $‖\Delta A_i‖=\sqrt{\mathrm{Sp}(\Delta A_i^T\Delta A_i)}$, $‖\Delta {\overline{A}}_{ij}‖=\sqrt{\mathrm{Sp}(\Delta A_{ij}^T\Delta A_{ij})}$ is

Let $‖\Delta A_i‖=\sqrt{\mathrm{Sp}(\Delta A_i^T\Delta A_i)}$, $‖\Delta {\overline{A}}_{ij}‖=\sqrt{\mathrm{Sp}(\Delta A_{ij}^T\Delta A_{ij})}$ are matrix norms $\Delta A_i$, $\Delta {\overline{A}}_{ij}$.

Theorem 1. Let 1) $A_i\in H$; 2) $X_i(t)\in P{E}_{{\underset{¯}{\alpha }}_{X_i},{\overline{\alpha }}_{X_i}}$,$X_j(t)\in P{E}_{{\underset{¯}{\alpha }}_{X_j},{\overline{\alpha }}_{X_j}}$ $u_i(t)\in P{E}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}$; 3) conditions of Lemma 1, 2 are satisfied for $F_i\left(X_i\right)$. Then subsystem (1) is identifiable on the set${\mathrm{II}}_{o,i}$ if

$2\left({\overline{\alpha }}_{X_i}{‖\Delta A_i‖}^2+{\overline{\alpha }}_{u_i}{‖\Delta B_i‖}^2+{\displaystyle \sum _{j=1,j\ne i}^m{\overline{\alpha }}_{X_j}{‖\Delta {\overline{A}}_{ij}‖}^2+}2\eta {\overline{\alpha }}_{X_i}+{\delta }_{F_i}\right)\le {\overline{\lambda }}_i{V}_i$, (9)

where ${\overline{\lambda }}_i={\lambda }_i-k_i$,${\lambda }_i>0$is the minimum eigenvalue of the matrix$Q_i$,$k_i>0$, $K_i{R}_i+K_i^T{R}_i=-Q_i$,$Q_i$is positive symmetric matrix,$\eta =2\overline{\pi }+{\pi }^2$,$\pi ={\pi }_1+{\pi }_2$,$\overline{\pi }={\pi }_1{\pi }_2$,${\delta }_{F_i}\ge 0$.

Theorem 1 proof is presented in Appendix C.

If Theorem 1conditions are fulfilled, then the subsystem $S_i$ is identifiable on the set${\mathrm{II}}_{o,i}$ or $P{I}_{X_i}$-identifiable.

Consider the identifiability of the $S_i$-subsystem on the set${\mathrm{II}}_{o,Y_i}=\left\{Y_i(t),\hspace{0.17em}\hspace{0.17em}{u}_i(t),\hspace{0.17em}\hspace{0.17em}{Y}_j(t),\hspace{0.17em}\hspace{0.17em}t\in J=\left[t_0,t_k\right]\right\}$.

Representation of the $S_i$-subsystem on${\mathrm{II}}_{o,Y_i}$${\dot{Y}}_i=A_{\#,i}{Y}_i+B_{\#,i}{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m{\overline{A}}_{\#,ij}{Y}_j+}{\tilde{C}}_i^{\#}{F}_i\left({\tilde{C}}_i{Y}_i\right),$ (10)

where ${\tilde{C}}_i={\left(C_i^T{C}_i\right)}^{\#}{C}_i^T$, $A_{\#,i}={\tilde{C}}_i^{\#}{A}_i{\tilde{C}}_i,\hspace{0.17em}\hspace{0.17em}{B}_{\#,i}={\tilde{C}}_i^{\#}{B}_i,\hspace{0.17em}\hspace{0.17em}{\overline{A}}_{\#,ij}={\tilde{C}}_i^{\#}{\overline{A}}_{ij}{\tilde{C}}_j$, # is a sign of a pseudo-inreversal of the matrix.

The model for (10) has a similar structure. Introduce the error $E_{\#,i}={\widehat{Y}}_i-Y_i$ and LF $V_{\#,i}\left(E_{\#,i}\right)=0.5E_{\#,i}^T{R}_i{E}_{\#,i}$.

Theorem 2. Let 1) $A_{\#,i}\in H$; 2) $Y_i(t)\in E{C}_{{\underset{¯}{\alpha }}_{Y_i},{\overline{\alpha }}_{Y_i}}$, $Y_j(t)\in E{C}_{{\underset{¯}{\alpha }}_{Y_j},{\overline{\alpha }}_{Y_j}}$ $u_i(t)\in E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}$; 3) $F_i\left(X_i\right)$ satisfies conditions of Lemmas 1, 2; 4) the $S_i$ subsystem is observable. Then subsystem (1) is identifiable on the set${\mathrm{II}}_{o,Y_i}$ if

$2\left({\overline{\alpha }}_{Y_i}{‖\Delta A_{\#,i}‖}^2+{\overline{\alpha }}_{u_i}{‖\Delta B_{\#.i}‖}^2+{\displaystyle \sum _{j=1,j\ne i}^m{\overline{\alpha }}_{Y_j}{‖\Delta {\overline{A}}_{\#,ij}‖}^2+}2\eta {\overline{\alpha }}_{Y_i}+{\delta }_{F_i}\right)\le {\overline{\lambda }}_{\#,i}{V}_{\#,i}$,

where$\Delta A_{\#,i}={\widehat{A}}_{\#,i}-A_{\#,i},\hspace{0.17em}\Delta B_{\#,i}={\widehat{B}}_{\#,i}-B_{\#,i},\hspace{0.17em}\Delta {\overline{A}}_{\#,ij}={\widehat{\overline{A}}}_{\#,ij}-{\overline{A}}_{\#,ij},\hspace{0.17em}\Delta F_i={\widehat{F}}_i-F_i$, ${\overline{\lambda }}_{\#,i}>0$.

The proof of Theorem 2 coincides with the proof of Theorem 1

4. Synthesis of Adaptation Algorithms

Consider LF $V_i\left(E_i\right)=0.5E_i^T{R}_i{E}_i$ and

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

We require that the functional constraint be satisfied for all $\forall t\ge t_0$${\dot{V}}_i\le -\chi \left(\Delta A_i,\Delta {\tilde{A}}_i,\Delta F_i\right)$,

where

$\chi \left(\Delta A_i,\Delta {\overline{A}}_i,\Delta B_i,\Delta F_i\right)=0.5\left({\phi }_{A_i}(t){‖\Delta A_i(t)‖}^2+{\phi }_{{\tilde{A}}_i}(t){‖\Delta {\overline{A}}_i(t)‖}^2+{\phi }_{B_i}(t){‖\Delta B_i‖}^2+{\phi }_{F_i}(t){‖\Delta F_i(t)‖}^2\right)$,

${\phi }_{A_i}(t)$, ${\phi }_{{\tilde{A}}_i}(t)$, ${\phi }_{B_i}(t)$, ${\phi }_{F_i}(t)$ are limited non-negative functions. Then:

${\dot{V}}_i=-E_i^T{Q}_i{E}_i+\chi \left(\Delta A_i,\Delta {\overline{A}}_i,\Delta F_i\right)+E_i^T{R}_i\left(\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m\Delta {\overline{A}}_{ij}{X}_j+}\Delta F_i\left(X_i\right)\right)$ (11)

From (11), we obtain adaptive algorithms

$\begin{array}{l}\Delta {\dot{A}}_i=-{\Gamma }_{A_i}\left({\phi }_{A_i}(t)\Delta A_i+E_i^T{R}_i{X}_i\right),\\ \Delta {\dot{\overline{A}}}_{ij}=-{\Gamma }_{{\overline{A}}_{ij}}\left({\phi }_{{\tilde{A}}_{ij}}(t)\Delta {\overline{A}}_{ij}+E_i^T{R}_i{X}_j\right),\\ \Delta {\dot{B}}_i=-{\Gamma }_{B_i}\left({\phi }_{B_{ij}}(t)\Delta B_i+E_i^T{R}_i{u}_i\right),\end{array}$ (12)

where ${\Gamma }_{A_i}$, ${\Gamma }_{{\tilde{A}}_{ij}}$, ${\Gamma }_{B_i}$ are diagonal matrices of corresponding dimensions with positive diagonal elements, ensuring the stability of adaptation processes.

4.1. Parametric Algorithm for $F_i$

Parametric and signal algorithms can be used to evaluating $F_i$. Consider the parametric approach [17].

Assumption 2. The function $F_i(X_i)$ is given on the set$\begin{array}{c}{F}_i\in {\mathrm{F}}_{F_i}=\left\{F_i\right.\in N_F\left({\pi }_1,{\pi }_2\right):F_i(X_i)={\tilde{F}}_i^T(X_i,N_{i,1})N_{i,2},\\ \left.N_i={\left[N_{i,1}^T,N_{i,2}^T\right]}^T,\hspace{0.17em}\hspace{0.33em}{N}_i\in {\mathbb{N}}_a\right\},\end{array}$ (13)

where ${\mathbb{N}}_{i,a}=\left\{N_i\in {ℝ}^n:\hspace{0.33em}\underset{¯}{N_i}\le N_i\le {\overline{N}}_i\right\}$ is a posteriori generated parametric domain for $F_i$; $\underset{¯}{N},\overline{N}$ are vector boundaries for $N$, understood as ${\underset{¯}{n}}_i\in {\underset{¯}{N}}_i,{\overline{n}}_i\in {\overline{N}}_i$; $N_{i,1}$ is a priori set vector of nonlinearity parameters, $N_{i,2}$ is a priori, an unknown set of parameters, which we consider as a vector to be evaluated. Some elements of ${\underset{¯}{N}}_i,\overline{N_i}$may be unknown. The ${\tilde{F}}_i\left(X_i,N_{i,1}\right)$ structure is formed a priori considering the known vector $N_{i,1}$.

As follows from (13), the estimation for function $F_i\left(X_i\right)$ is defined in the form:

${\widehat{F}}_i(X_i)={\tilde{F}}_i^T\left(X_i,{\widehat{N}}_{i,1}\right){\widehat{N}}_{i,2}$, (14)

where ${\widehat{N}}_{i,1}\in {ℝ}^{n_{i,1}}$ is a priori estimation of known parameters, ${\widehat{N}}_{i,2}\in {ℝ}^{n_{i,2}}$ is the vector of tuning parameters.

We believe ${\mathbb{N}}_i={\mathbb{N}}_{i,1}\cup {\mathbb{N}}_{i,2}$. The set ${\mathbb{N}}_{i,1}\subset {ℝ}^{n_{i,1}}$ $\left(N_{i,1}\in {\mathbb{N}}_{i,1}\right)$ contains elements that are not available for adjusting. We get estimates of elements $N_{i,2}\in {\mathbb{N}}_{i,2}\subset {ℝ}^{n_{i,2}}$ at the identification stage. The matrix ${\tilde{F}}_i\left(y,{\widehat{N}}_{i,1}\right)$ is formed at the stage of structural synthesis (analysis) of the system. Representation (14) is a consequence of the proposed parametric concept for $F_i\left(X_i\right)$.

Remark 2. The vector ${\widehat{N}}_{i,1}$ can be adjusted iteratively based on the coercion algorithm [17].

As $\Delta F={\tilde{F}}_i^T\left(y,{\widehat{N}}_{i,1}\right){\widehat{N}}_{i,2}-F_i(X_i)$, then we get an adaptive algorithm for ${\widehat{N}}_{i,2}$ from the condition ${\dot{V}}_i\le 0$${\dot{\widehat{N}}}_{i,2}=-{\Gamma }_{F_i}\left({\phi }_{F_i}{\dot{\widehat{N}}}_{i,2}+{\tilde{F}}_i^T{R}_i{E}_i\left(X_i,{\widehat{N}}_{i,1}\right)\right)$, (15)

where ${\Gamma }_{F_i}$ is a diagonal matrix with positive diagonal elements. Designate the system (6), (11), (15) as $A{S}_{A{F}_i}$.

4.2. Signal Algorithm

Consider the model

${\dot{\widehat{X}}}_i=K_i\left({\widehat{X}}_i-X_i\right)+{\widehat{A}}_i{X}_i+{\widehat{B}}_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m{\widehat{\overline{A}}}_{ij}{X}_j}+U_i$ (16)

and the equation for the error

${\dot{E}}_i=K_i{E}_i+\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m\Delta {\overline{A}}_{ij}{X}_j}+U_i\left(X_i\right)-F_i\left(X_i\right)$. (17)

Then

${\dot{V}}_i=-E_i^T{Q}_i{E}_i+E_i^T{R}_i\left(\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m\Delta {\overline{A}}_{ij}{X}_j}+U_i-F_i\right)$. (18)

Choose the algorithm for $U_i$ in the form:

$U_i=-D_i{R}_i{E}_i$, (19)

where $D_i\in {ℝ}^{n_i\times n_i}$ is a diagonal matrix with positive diagonal elements. As $F_i\left(X_i\right)\in N_F\left({\pi }_1,{\pi }_2\right)$, then Lemma 1 is valid for $F_i\left(X_i\right)$.

Apply the approach [18]. Select matrix $D_i$ elements from the condition $‖D_i‖\ge d_i\ge \eta {\overline{\alpha }}_{X_i}$. Then:

${\dot{V}}_i=-E_i^T{Q}_i{E}_i+E_i^T{R}_i\left(\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m\Delta {\overline{A}}_{ij}{X}_j}-D_i{R}_i{E}_i-F_i\right)$. (20)

As $E_i^T{R}_i{D}_i{R}_i{E}_i\ge 2\eta {\underset{¯}{\alpha }}_{X_i}{\underset{¯}{\lambda }}_{R_i}{V}_i$ then

${\dot{V}}_i\le -\sigma V_i+E_i^T{R}_i\left(\Delta A_i{X}_i+\Delta B_i{u}_i+{\displaystyle \sum _{j=1,j\ne i}^m\Delta {\tilde{A}}_{ij}{X}_j}\right)$, (21)

where ${\underset{¯}{\lambda }}_{R_i}$ is the smallest eigenvalue of the matrix $R_i,$ $\sigma ={\lambda }_{Q_i}+2\eta {\overline{\alpha }}_{X_i}{\underset{¯}{\lambda }}_{R_i}$. We apply the approach outlined in the proof of Theorem 1, and get

${\dot{V}}_i\le -\sigma V_i+2\left({\overline{\alpha }}_{X_i}{‖\Delta A_i‖}^2+{\overline{\alpha }}_{u_i}{‖\Delta B_i‖}^2+{\displaystyle \sum _{j=1,j\ne i}^m{\overline{\alpha }}_{X_j}{‖\Delta {\overline{A}}_{ij}‖}^2}\right)$.

If

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

then the system (16) is parametrically identifiable on the set $\left\{u_i(t),\hspace{0.17em}{X}_i(t),\hspace{0.17em}\hspace{0.17em}{X}_j(t)\right\}$ on the algorithms (12), (19) class, if $\left(u_i(t),\hspace{0.17em}{X}_i(t),\hspace{0.17em}\hspace{0.17em}{X}_j(t)\right)\in EC$.

Designate the system (6), (12), (19) as $A{S}_{A{S}_i}$.

5. Functional Restriction and Synthesis of Adaptive Algorithms

The described synthesis method of adaptive algorithms (AA) is typical for the adaptive identification. Another approach is based on accounting of the limitations that are imposed on ASI. This approach requires some knowledge and does not always provide workable algorithms. We propose the method based on consideration of requirements for ASI. Show the algorithm synthesis method using the example of the $\Delta A_i$ matrix.

Consider of LF

$V_i^{\Delta }\left(E_i,\Delta A_i,\Delta {\dot{A}}_i\right)=0.5E_i^T{R}_i{E}_i+0.5\mathrm{Sp}\left(\Delta A_i^T\Delta {\dot{A}}_i\right)$.

Let

${\dot{V}}_i^{\Delta }\le -{\chi }_{\Delta }=-{\alpha }_{\Delta }\mathrm{Sp}\left(\Delta A_i^T\Delta A_i\right)$, ${\alpha }_{\Delta }\ge 0$. (22)

Denote ${\eta }_{\Delta }={\dot{V}}_i^{\Delta }+{\chi }_{\Delta }$ and obtain:

${\eta }_{\Delta }=-E_i^T{Q}_i{E}_i+E_i^T{R}_i\Delta A_i{X}_i+\mathrm{Sp}\left(\Delta {\dot{A}}_i^T\Delta {\dot{A}}_i\right)+\mathrm{Sp}\left(\Delta A_i^T\Delta {\ddot{A}}_i\right)+{\alpha }_{\Delta }\mathrm{Sp}\left(\Delta A_i^T\Delta A_i\right).$Adaptive algorithm for $\Delta A_i$$\Delta {\ddot{A}}_i=-\Delta {\dot{A}}_i-{\alpha }_{\Delta }\Delta A_i-{\Gamma }_{A_i}{E}_i{R}_i{X}_i^T$. (23)

Let $\Delta A_i=Z_1$. Then:

$S_{AA}:\left\{\begin{array}{l}{\dot{Z}}_1=Z_2,\\ {\dot{Z}}_2=-{\alpha }_{\Delta }{Z}_1-Z_2-{\Gamma }_{A_i}{E}_i{R}_i{X}_i^T.\end{array}\right.$ (24)

So, if the functional restriction ${\chi }_{\Delta }\ge 0$ is imposed on ASI, then AA is described by the system $S_{AA}.$ The $S_{AA}$-algorithm use is associated with difficulties of application. Therefore, using the $S_{AA}$ requires their modification.

Let ${\chi }_{e,\Delta }={\alpha }_{e,\Delta }\mathrm{Sp}\left(\Delta A_i^{T}D\left(\left|E\right|\right)\Delta {\dot{A}}_i\right)$ and ${\eta }_{e,\Delta }={\dot{V}}_i+{\chi }_{e,\Delta }$. Then ${\mathrm{A}}_M$-algorithm is presented as:

$\Delta {\dot{A}}_i=-{\Gamma }_{A_i}{E}_i{R}_i{X}_i^T-{\alpha }_{e,\Delta }{\Gamma }_{{\dot{A}}_i}D\left(\left|E_i\right|\right)\Delta {\dot{A}}_i$ (25)

or

$\Delta {\dot{A}}_i=-M^{-1}{\Gamma }_{A_i}{E}_i{R}_i{X}_i^T,$ (26)

where $\hspace{0.17em}M=I_{n\times n}+{\alpha }_{e,\Delta }{\Gamma }_{{\dot{A}}_i}D\left(\left|E\right|\right)$, $D\left(\left|E\right|\right)$ – диагoнальная матрица oт вектoра $\left|E\right|$, $I_{n\times n}$ – единична матрица, ${\Gamma }_{{\dot{A}}_i}={\Gamma }_{{\dot{A}}_i}^T\hspace{0.17em}\hspace{0.17em}>0$.

We describe ${\mathrm{A}}_M$-algorithm (25) as:

$\Delta {\dot{A}}_i(t)=-{\Gamma }_{A_i}{E}_i(t)R_i{X}_i^T(t)-{\omega }_e{\Gamma }_{{\dot{A}}_i}D\left(\left|E\right|\right)\left(\Delta A_i(t)-\Delta A_i(t-\tau )\right)$, (27)

where ${\omega }_{e,\Delta }={\tilde{\alpha }}_{e,\Delta }\hspace{0.17em}{\tau }^{-1}$. It is difficult to evaluate the properties of the algorithm (27). If the matrix D2 is unique, then

$\Delta {\dot{A}}_i(t)=-{\Gamma }_{A_i}{E}_i(t)R_i{X}_i^T(t)-{\omega }_e{\Gamma }_{{\dot{A}}_i}\left(\Delta A_i(t)-\Delta A_i(t-\tau )\right)$. (27a)

Convergence conditions of estimates for algorithm (27a) at ${\omega }_{e,\Delta }=1$ and the matrix ${\Gamma }_{A_i}$is diagonal.

Theorem 3. Let 1) $A_{\#,i}\in H$; 2) $X_i(t)\in P{E}_{{\underset{¯}{\alpha }}_{X_i},{\overline{\alpha }}_{X_i}}$, $X_j(t)\in P{E}_{{\underset{¯}{\alpha }}_{X_j},{\overline{\alpha }}_{X_j}}$ $u_i(t)\in E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}$; 3) $F_i\left(X_i\right)$ satisfies conditions of Lemmas 1, 2; 4) $V_{\Delta ,i}=\mathrm{Sp}\left(\Delta A_i^T{\Gamma }_{{\dot{A}}_i}^{-1}\Delta A_i(t)\right)$; 5) there exists υ > 0 such that

$\mathrm{Sp}\left(\Delta A_i^T{\Gamma }_{{\dot{A}}_i}^{-1}{\Gamma }_{A_i}{E}_i{R}_i{X}_i^T\right)=\upsilon \left[\left(\mathrm{Sp}\left(\Delta A_i^T{\Gamma }_{{\dot{A}}_i}^{-1}{\Gamma }_{A_i}\Delta A_i\right)+E_i^T{R}_i^2{E}_i{‖X_i‖}^2\right)\right]$is valid at. Then algorithm (27a) estimates are bounded if

${‖\Delta A_i(t-\tau )‖}^2\le \eta V_{\Delta ,\mathrm{i}}-\upsilon {\displaystyle \frac{2{\sigma }_e}{3{\sigma }_{\Delta }}}{V}_i$$\forall t>t_0$,

where$\eta ={\underset{¯}{\lambda }}_{{\Gamma }_{{\dot{A}}_i}}\left(1+{\displaystyle \frac{3}{4}}\upsilon {\sigma }_{\Delta }\right)$, ${\sigma }_e={\underset{¯}{\alpha }}_{X_i}2{\underset{¯}{\lambda }}_{R_i}$${\sigma }_{\Delta }={\underset{¯}{\lambda }}_{{\Gamma }_{{\dot{A}}_i}}{\underset{¯}{\lambda }}_{{\Gamma }_{A_i}}$, ${\underset{¯}{\lambda }}_{{\Gamma }_{{\dot{A}}_i}}$ is the minimum eigenvalue of the matrix ${\Gamma }_{{\dot{A}}_i}$Theorem 3 proof is presented in Appendix D.

Remark 3. Algorithm (27) is a differential equation with an aftereffect. The equation (27) discrete analogues are proposed by various authors for regression models. They are based on the intuition of the researcher.

6. Properties of Adaptive System

6.1. System $A{S}_{A{F}_i}$

Consider systems $A{S}_{A{F}_i}$, $A{S}_{A{F}_j}$ и LF $V_i\left(E_i\right)=0.5E_i^T{R}_i{E}_i$,

$V_{\Delta ,i}=0.5\cdot \mathrm{Sp}\left(\Delta A_i^T{\Gamma }_i^{-1}\Delta A_i\right)+0.5{\displaystyle \sum _{j=1}^m\mathrm{Sp}\left(\Delta {\overline{A}}_{ij}^T{\Gamma }_{ij}^{-1}\Delta {\overline{A}}_{ij}\right)}+0.5\Delta B_i^T{\Gamma }_i^{-1}\Delta B_i+0.5\Delta N_{i,2}^T{\Gamma }_{N_{i,2}}^{-1}\Delta N_{N_{i,2}}$, (28)

where $\mathrm{Sp}(\cdot )$ is the spur of matrix. We believe that the interference matrix ${\overline{A}}_{ij}$ ensures the stability of the $S_i$-subsystem.

Theorem 4. Let (i) Lyapunov functions $V_i(t)$, $V_{\Delta ,i}(t),$ admit an infinitesimal upper limit; (ii) $A_i\in H$; (iii) ${\overline{A}}_{ij}$ ensures the stability of the subsystem $S_i$; (iv) $X_i(t)\in E{C}_{{\underset{¯}{\alpha }}_{X_i},{\overline{\alpha }}_{X_i}}$,$X_j(t)\in E{C}_{{\underset{¯}{\alpha }}_{X_j},{\overline{\alpha }}_{X_j}}$ $u_i(t)\in E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}$; (v$F_i\in {\mathrm{F}}_{F_i}$) ; (vi) the system of inequalities

$\left[\begin{array}{l}{\dot{V}}_i\\ {\dot{V}}_{\Delta ,i}\end{array}\right]\le \underset{A_{W_i}}{\underbrace{\left[\begin{array}{cc}-{\mu }_i& {\displaystyle \frac{2}{{\mu }_i}}{\kappa }_i\\ {\vartheta }_{\chi \alpha ,i}{\rho }_i& -{\beta }_{\lambda \chi ,i}\end{array}\right]}}\left[\begin{array}{l}V\\ V_{\Delta ,i}\end{array}\right]+\underset{L_i}{\underbrace{\left[\begin{array}{l}{\displaystyle \frac{2}{{\mu }_i}}{\overline{\upsilon }}_{{\tilde{F}}_i}\\ 0.5{\delta }_{N,i}\end{array}\right]}}$ (29)

is valid for the Lyapunov vector function $W_i={\left[V_i,\hspace{0.17em}{V}_{\Delta ,i}\right]}^T$, where${\mu }_i,\hspace{0.17em}{\kappa }_i,\hspace{0.17em}{\vartheta }_{\chi \alpha ,i}{\rho }_i,\hspace{0.17em}{\beta }_{\lambda \chi ,i},\hspace{0.17em}{\overline{\upsilon }}_{{\tilde{F}}_i},\hspace{0.17em}{\delta }_{N,i}$ are positive numbers depending on the subsystem$S_i$ parameters and set${\mathrm{II}}_{o,i}$ properties; (vi) the upper solution for $W_i$ satisfies the system of equation ${\dot{S}}_{W_i}=A_{W_i}{S}_{W_i}+L_i$ if

$w_{\rho }(t)\le s_{\rho }(t)$ $\forall \left(t\ge t_0\right)\&\left(w_{\rho }\left(t_0\right)\le s_{\rho }\left(t_0\right)\right)$,

$w_{\rho }\in W_i$, $s_{\rho }\in S_{W_i}$,$\rho =e,i;\Delta ,i$. Then the$A{S}_{F,i}$-system is exponentially dissipative with the estimate:

$W_i(t)\le e^{A_{W_i}\left(t-t_0\right)}{S}_{W_i}(t_0)+{\displaystyle \underset{t_0}{\overset{t}{\int }}{e}^{A_{W_i}\left(t-\tau \right)}{L}_{i}d\tau }$, (30)

if

${\mu }_i^2{\beta }_{\lambda \chi ,i}\ge 2{\kappa }_i{\vartheta }_{\chi \alpha ,i}{\rho }_i$. (31)

As follows from (30), the $A{S}_{F,i}$ limiting properties are determined by vector $L_i$ elements. If the vector ${\widehat{N}}_{i,1}$ structure and parameters are known, then the $A{S}_{F,i}$-subsystem is exponentially stable if ${\tilde{F}}_i\in EC$.

Theorem 4 proof is presented in Appendix E.

Consider the system $A{S}_{F,i,j}$ with subsystems $A{S}_{F,i}$ and $A{S}_{F,j}$. We have the system of inequalities for $A{S}_{F,i,j}$$\left[\begin{array}{l}{\dot{W}}_i\\ {\dot{W}}_j\end{array}\right]\le \left[\begin{array}{cc}{A}_{W_i}& 0\\ 0& A_{W_j}\end{array}\right]\left[\begin{array}{l}{W}_i\\ W_j\end{array}\right]+\left[\begin{array}{l}{L}_i\\ L_j\end{array}\right]$, (32)

where $A_{W_i}$and $L_i$ have the form (29). Exponential dissipative conditions

${\mu }_i^2{\beta }_{\lambda \chi ,i}\ge 2{\kappa }_i{\vartheta }_{\chi \alpha ,i}{\rho }_i$, ${\mu }_i^2{\beta }_{\lambda \chi ,j}\ge 2{\kappa }_j{\vartheta }_{\chi \alpha ,j}{\rho }_j$.

6.2. System $A{S}_{A{S}_i}$

Consider LF $V_i(t)$ and

$\begin{array}{c}{V}_{{\Delta }_S,i}=0.5\cdot \mathrm{Sp}\left(\Delta A_i^T{\Gamma }_i^{-1}\Delta A_i\right)+0.5{\displaystyle \sum _{j=1}^m\mathrm{Sp}\left(\Delta {\overline{A}}_{ij}^T{\Gamma }_{ij}^{-1}\Delta {\overline{A}}_{ij}\right)}+0.5\Delta B_i^T{\Gamma }_i^{-1}\Delta B_i+\\ +0.5{\displaystyle \underset{t_0}{\overset{t}{\int }}{\Delta }_{U_i}{F}_i^T(\tau ){\Delta }_{U_i}{F}_i(\tau )d\tau },\end{array}$

where ${\Delta }_{U_i}{F}_i=U_i-F_i$.

Theorem 5. Let (i) Lyapunov functions $V_i(t)$ and $V_{{\Delta }_S,i}(t)$ to have an infinitesimal upper limit; (ii) $A_i\in H$; (iii) ${\overline{A}}_{ij}$ ensures the stability of the subsystem $S_i$; (iv) $F_i\left(X_i\right)\in N_F\left({\pi }_1,{\pi }_2\right)$; (v) $u_i(t)\in E{C}_{{\underset{¯}{\alpha }}_{u_i},{\overline{\alpha }}_{u_i}}$, $X_i(t)\in E{C}_{{\underset{¯}{\alpha }}_{X_i},{\overline{\alpha }}_{X_i}}$, $X_j(t)\in E{C}_{{\underset{¯}{\alpha }}_{X_j},{\overline{\alpha }}_{X_j}}$ ; (v) exist ${\nu }_i>0$ such that the condition $-F_i^T{\Delta }_{U_i}{F}_i=-{\nu }_i\left({‖F_i‖}^2+{‖{\Delta }_{U_i}{F}_i‖}^2\right)$ satisfy at $t>>t_0$ in some area of the origin; (vi) the system of inequalities

$\underset{{\dot{W}}_{S_i}}{\underbrace{\left[\begin{array}{l}{\dot{V}}_i\\ {\dot{V}}_{{\Delta }_{S,i}}\end{array}\right]}}\le \underset{A_{W_{S_i}}}{\underbrace{\left[\begin{array}{cc}-{\mu }_i& {\displaystyle \frac{2}{{\mu }_i}}{\kappa }_{{\Delta }_{S,i}}\\ {\vartheta }_{{\Delta }_{S,i}}{\rho }_i& -{\beta }_{{\Delta }_{S,i}}\end{array}\right]}}\underset{W_{S_i}}{\underbrace{\left[\begin{array}{l}{V}_i\\ V_{{\Delta }_{S,i}}\end{array}\right]}}+\underset{L_{S,i}}{\underbrace{\left[\begin{array}{c}0\\ \sqrt{0.125}{v}_i{\eta }_i{\overline{\alpha }}_{X_i}\end{array}\right]}}$ (33)

is valid for the Lyapunov vector function $W_{S,i}={\left[V_i,\hspace{0.17em}{V}_{{\Delta }_S,i}\right]}^T$, where ${\mu }_i,{\kappa }_{{\Delta }_S,i},{\vartheta }_{\Delta .i},{\rho }_i,{\beta }_{\Delta ,i},\hspace{0.17em}{v}_i,{\eta }_i$ are positive numbers depending on the subsystem$S_i$ parameters and set${\mathrm{II}}_{o,i}$ properties; (vii) the upper solution for $W_i$ satisfies the system of equation ${\dot{S}}_{W_i}=A_{W_i}{S}_{W_i}+L_i$ if

$w_{\rho }(t)\le s_{\rho }(t)$ $\forall \left(t\ge t_0\right)\&\left(w_{\rho }\left(t_0\right)\le s_{\rho }\left(t_0\right)\right)$,

$\rho =e,i;{\Delta }_{S,i}$for elements$W_{S_i}$,$w_{\rho }\in W_{S_i}$, $s_{\rho }\in S_{W_{S,i}}$. Then the$A{S}_{A{S}_i}$-system is exponentially dissipative with an estimate

$W_{S_i}(t)\le e^{A_{W_{S_i}}\left(t-t_0\right)}{S}_{W_{S_i}}(t_0)+{\displaystyle \underset{t_0}{\overset{t}{\int }}{e}^{A_{W_{S_i}}\left(t-\sigma \right)}{L}_{S,i}d\sigma }$, (34)

if${\mu }_i^2{\beta }_{\Delta ,i}\ge 2{\vartheta }_{\Delta .i}{\rho }_i{\kappa }_{{\Delta }_S,i}$.

As follows from Theorem 4, the $A{S}_{A{S}_i}$-system application gives biased estimates for the parameters of the $S_i$-subsystem.

Theorem 5 proof is presented in Appendix F.

Remark 4. Signalling algorithms (SA) are widely used in adaptive control systems (see review [19]). The rationale SA is based on ensuring on non-positivity derivative LF. This is a feature of using quadratic LF, which does not fully reflect the specifics of the processes in the system $A{S}_{A{S}_i}$. The Lyapunov function $V_{{\Delta }_S,i}$ proposed in the paper allows to prove the properties of the adaptive system.

Remark 5. Algorithm (19) is a compensating control. Therefore, the term "signal adaptation" reflects only the gain factor in (19). In identification systems, the SA use depends on the quality requirements of the identification system.

Remark 6. The analysis of the properties of algorithms (12) with ${\phi }_i=0$ is based on the results got in [20].

7. Example

Consider the system

$\begin{array}{l}{S}_1:\left\{\begin{array}{l}\hspace{0.17em}\left[\begin{array}{l}{\dot{x}}_{11}\\ {\dot{x}}_{12}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -a_{21}& -a_{22}\end{array}\right]\left[\begin{array}{l}{x}_{11}\\ x_{12}\end{array}\right]+\left[\begin{array}{l}0\\ {\overline{a}}_1\end{array}\right]x_2+\left[\begin{array}{l}0\\ b_1\end{array}\right]u_1+\left[\begin{array}{l}0\\ c_1\end{array}\right]f_1\left(x_{11}\right),\\ y_1=x_{11},\end{array}\right.\hspace{0.17em}\\ \hspace{0.17em}{S}_2:\hspace{0.17em}\left\{\begin{array}{l}{\dot{x}}_2=-a_2{x}_2+{\overline{a}}_2{x}_{11}+b_2{u}_2+c_2{f}_1\left(x_{11}\right),\\ y_2=x_2,\end{array}\right.\end{array}$ (35)

where $X_1={\left[x_{11}\hspace{0.17em}\hspace{0.17em}{x}_{12}\right]}^T$, $y_1$ is the state vector and output of the subsystem $S_1$; $u_1$ is input (control)); $f_1\left(x_{11}\right)=\mathrm{sat}\left(x_{11}\right)$ is saturation function; $f_2\left(x_2\right)=\mathrm{sign}\left(x_2\right)$ is sign function; $y_2$ is subsystem $S_2$ output. System parameters (35): $b_1=1$, $a_{21}=2,\hspace{0.17em}\hspace{0.17em}{a}_{22}=3$, ${\overline{a}}_1=1.5$, $b_1=1$, $c_1=1$, $a_2=1.25,$ ${\overline{a}}_2=0.2$, $b_2=1$, $c_2=0.25$. Inputs $u_i(t)$ are sinusoidal.

Since the variable $x_{12}$ is not measured, the subsystem $S_1$ is converted to a form where only observable variables are used [20]. Subsystem $S_1$ has the form in the input-output space:

${\dot{y}}_1=-{\alpha }_1{y}_1+{\alpha }_2{p}_{y_1}+{\beta }_{12}{p}_{x_2}+b_1{p}_{u_1}+c_1{p}_{f_1}$, (36)

where ${\alpha }_1$, ${\alpha }_2$,${\beta }_{12}$, $b_1,\hspace{0.17em}\hspace{0.17em}{c}_1$ is unknown coefficients; $\mu >0$,

$\begin{array}{c}{\dot{p}}_{y_1}=-\mu p_{y_1}+y_1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{p}}_{x_2}=-\mu p_{x_2}+x_2,\hspace{0.17em}\hspace{0.17em}\\ {\dot{p}}_{u_1}=-\mu p_{u_1}+u_1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{p}}_{f_1}=-\mu p_{f_1}+f_1.\end{array}$ (37)

We present the phase portrait for S₁ in Figure 1. Processes in $S_1$ are nonlinear. There is a relationship between $y_1$ and $y_2$ (the determination coefficient is 75%). This reflects in properties of the subsystem $S_1$ (see Figure 1). In particular, $y_2$ effects on S-synchronizability and parameter estimation. Apply the approach [16] and get that the $S_1$ subsystem is structurally identifiable.

Models for subsystems $S_1$ и $S_2$${\dot{\widehat{y}}}_1=-k_1{e}_1+{\widehat{a}}_{11}{y}_1+{\widehat{a}}_{12}{p}_{y_1}+{\widehat{\beta }}_{12}{p}_{x_2}+{\widehat{b}}_1{p}_{u_1}+{\widehat{c}}_1{p}_{f_1}$, (38)

${\dot{\widehat{y}}}_2=-k_2{e}_2+{\widehat{a}}_2{y}_2+{\widehat{\overline{a}}}_2{y}_1+{\widehat{b}}_2{u}_2+{\widehat{c}}_1{f}_2$, (39)

where $k_1,k_2$ are a priori set positive numbers (reference model); $e_1={\widehat{y}}_1-y_1$, $e_2={\widehat{y}}_2-y_2$ are identification errors; ${\widehat{a}}_i,\hspace{0.17em}{\widehat{\overline{a}}}_i,\hspace{0.17em}{\widehat{b}}_i,\hspace{0.17em}{\widehat{c}}_i$ are tuning parameters.

Apply algorithms (12) with ${\phi }_i=0$:

$\begin{array}{c}{\dot{\widehat{a}}}_{11}=-{\gamma }_{a_{11}}{e}_1{y}_1,\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{a}}}_{12}=-{\gamma }_{a_{12}}{e}_1{p}_{y_1},\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{\beta }}}_{12}=-{\gamma }_{{\beta }_{12}}{e}_1{p}_{x_2},\\ {\widehat{b}}_1=-{\gamma }_{b_1}{e}_1{p}_{u_1},\hspace{0.17em}\hspace{0.17em}{\widehat{c}}_1=-{\gamma }_{c_1}{e}_1{p}_{f_1},\end{array}$ (40)

$\begin{array}{c}\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{a}}}_2=-{\gamma }_{a_2}{e}_2{y}_2,\hspace{0.17em}{\dot{\widehat{\overline{a}}}}_2=-{\gamma }_{{\overline{a}}_2}{e}_2{y}_1,\\ {\widehat{b}}_2=-{\gamma }_{b_2}{e}_2{u}_1,\hspace{0.17em}\hspace{0.17em}{\widehat{c}}_2=-{\gamma }_{c_2}{e}_2{f}_2,\end{array}$ (41)

where ${\gamma }_{a_{ij}}>0,{\gamma }_{{\overline{a}}_{ij}}>0,\hspace{0.17em}\hspace{0.17em}{\gamma }_{{\beta }_{ij}}>0,\hspace{0.17em}{\gamma }_{b_i}>0,\hspace{0.17em}{\gamma }_{c_i}>0$ are gain factors of the adaptation subsystem.

Figure 2 shows tuning parameters of the model (38) for $S_1$.

Show the adequacy estimation of the of models (31), (32) in Figure 3.

Figure 4 shows the tuning process dynamics of the model (32) parameters depending on $e_2$. We see that the processes in the adaptive identification system (ASI) for $S_2$ are nonlinear. The tuning process is more regular in the ASI for the $S_1$ subsystem.

Models (38) and (39) adaptation processes have different speeds (Figure 5).

Consider an ASI with signal adaptation for $S_2$. Apply the model

${\dot{\widehat{y}}}_2=-k_2{e}_2+{\widehat{a}}_2{y}_2+{\widehat{\overline{a}}}_2{y}_1+{\widehat{b}}_2{u}_2+u_{s,2}$, (39а)

where $u_{s,2}=-d_2{e}_2{x}_2$.

We show results for the ASI in Figure 6, Figure 7, Figure 8, Figure 9. Show the adaptation of the model parameters (38) and the adequacy of the models in the output space in Fig. 6, 7. Fig. 8 reflects the dynamics in ASI and the change in SA as $e_2$ function for the subsystem $S_2$.

We see that the $S_2$-subsystem output with SA effects on the ASI adaptation of the $S_1$-subsystem. Therefore, ASI with SA should be applied considering the quality requirements of the identification process. Despite the compensating properties of CA, CA can lead to more complicated processes in ASI.

8. Conclusions

We consider a class of nonlinear decentralised control systems for which the quadratic condition is valid. The problem of identifiability S1-subsystem of DS is studied. We note the constant excitation condition role in the parametric identifiable analysis of decentralised systems. Quadratic estimates are got for the nonlinear part of the $S_i$-subsystem. Parametric identifiability conditions are got. Algorithms of parametric and signal adaptation are synthesised, and identification system properties are studied. The exponential dissipativity of the adaptive identification system is proved. We present the simulation results confirming the proposed approach efficiency. Appendix contains proof results.