On Adaptive Identification of Interconnected Systems

1. Introduction

Interconnected systems (ICS) used [1,2,3,4,5] to control actuators for robots and manipulators [6,7], and various technical systems [8,9]. ICS identification methods based on an adaptive approach [10], a black box concept for non-stationary multi-connected systems with uncertainty [11]. Adaptive algorithms of decentralized robust control with a reference model are proposed for ICS with a time delay [12]. The identifiability issues considering for a closed interconnected stochastic system in [13]. The system decomposition proposes into subsystems. The identifiability considers as system individual elements and individual closed circuits. Convergence sufficient conditions got almost certainly for parameter estimates. A high-modulus normalized adaptive lattice algorithm proposes for multichannel filtering in [14].

In [15], an algorithm presents to identify transfer function models for 2-D spatially interconnected systems which are semi-causal both in open and closed-loop form using neural network. In [16], a decentralized robust adaptive stabilization system considers with output feedback. adaptive nonlinear damping and a robust adaptive observer are the base for controlling laws. In [17,18], there get similar results. In [19], the identification method presents for a spatially interconnected system in a rational form.

Adaptive control of large-scale systems consisting for arbitrary of subsystems with unknown parameters, nonlinearities and limited perturbations is considered in [20]. Topological structural identification [21] considers for large-scale dynamic systems based on a small dataset. In [22], the approach proposes to the blind identification of a two-channel systems. The proposed approach bases on the data analysis in the frequency domain. Problem analysis of blind multichannel identification and some modern adaptive algorithms present in [23]. The review [24] analyzed some modern approaches based on the identification problem (IP) decomposition for systems with multiple inputs /one output (MISOS). The identification of stationary linear MISOS discusses in [13,25,26]. Correlation analysis [25] uses for the ICS identification in the frequency domain.

Transfer functions and the state space Models are the basis for analyzing processes in ICS. Adaptive procedures are used to synthesize control algorithms. Adaptive control algorithms apply under the perturbations and unmodeled dynamics action. Explain this IP state (i) incomplete information about the state and parameters of the system and (ii) the accounting complexity of relationships in the system. The identification of ICS parameters base on statistical procedures, frequency methods and neural network technologies. Apply of adaptive identification methods (AIM) study in some works. AIM use to the tuning (identification) control parameters. Parametric uncertainties that arise in this case compensate by the choice of a control algorithm. The Identification of two-channel systems has not been studied.

We propose the approach to the ICS adaptive identification based on the use of measurement results. First, two-channel systems (TCS) with cross-links (CL) are considered. The approach proposes to the TCS identifiability under various assumptions about LC. Identifiability estimates have obtained. The adaptive system stability proves. We propose the approach to ICS identification.

2. Two-channel systems

2.1. Problem Setting

Consider TCS with CL. We believe that elements in channels are identical. Transfer functions are the basis of TCS analysis [27]. The TCS description in the state space is a more adequate representation in identification problems. Let TCS have $n$ vertical layers.

(i)

the first layer

$$\begin{array}{l}\{\begin{array}{l}{\dot{X}}_{11}=A_1{X}_{11}+B_1{e}_{11}=A_1{X}_{11}+B_1{v}_{11}\hspace{0.17em},\\ y_{11}=C_1^T{X}_{11},\end{array}\\ \{\begin{array}{l}{\dot{X}}_{21}=A_1{X}_{21}+B_1{e}_{21}=A_1{X}_{21}+B_1{v}_{21},\\ y_{21}=C_1^T{X}_{21},\end{array}\end{array}$$

(1)

(ii)

$k$-layer ($1<k\le n$)

$$\begin{array}{l}\{\begin{array}{l}{\dot{X}}_{1,k}=A_k{X}_{1,k}+B_k{u}_{1,k},\\ y_{1,k}=C_k^T{X}_{1,k},\\ u_{1,k}=y_{1,k-1}+v_{1,k-1},\\ v_{1,k-1}=d_{k-1}\left(y_{2,k-1}+d_{k-2}{v}_{2,k-1}\right),\end{array}\\ \{\begin{array}{l}{\dot{X}}_{2,k}=A_k{X}_{2,k}+B_k{u}_{2,k},\\ y_{2,k}=C_k^T{X}_{1,k},\\ u_{2,k}=y_{2,k-1}+v_{2,k-1},\\ v_{2,k-1}=d_{k-1}\left(y_{1,k-1}+d_{k-2}{v}_{1,k-1}\right),\end{array}\end{array}$$

(2)

where $v_{11}=g_1-y_{1,n},\hspace{0.17em}\hspace{0.17em}{v}_{21}=g_2-y_{2,n}$, $v_{i,k-1}$ is i-channel CL output, $i=1,\hspace{0.17em}2$; $X_{ik}\in {\text{ℝ}}^q$ is the state vector of the $k$ layer of the $i$ channel, $A_k\in {\text{ℝ}}^{q_k\times q_k}$, $C_k\in {\text{ℝ}}^{q_k}$, $B_k\in {\text{ℝ}}^{q_k}$, $y_{i,k}\in \text{ℝ}$ is output $k$ layer of the $i$ channel; $v_{i,k-1}$ is CL output; $g_i(t)\in \text{ℝ}$ is input (upsetting control) the system. $A_k$ is Hurwitz matrix.

$d_{k-1}$ is an operator. Tasks solved by TCS determine the $d_{k-1}$ type. $d_{k-1}$ can be a constant, a nonlinear function, or a differential operator.

Information set

$${\mathrm{II}}_o=\left\{g_i(t),y_{i,k}(t)\hspace{0.17em}\forall \left(i=1;2\right)\&\left(k=\overline{1,n}\right),\hspace{0.17em}t\in J=\left[t_0,t_e\right]\right\}$$

(3)

where $J$ time interval.

The structure of the TCS model coincides with (1), (2). Let ${\widehat{y}}_{i,k}\in \text{ℝ}$, $i=1,\text{}\hspace{0.17em}2;\hspace{0.17em}\hspace{0.17em}k=\overline{1,n}$ is the output model.

Task: design algorithms for tuning model parameters to

$$\underset{t\to \infty }{\lim }\left|{\widehat{y}}_{i,k}-y_{i,k}\right|\le {\delta }_{i,k}$$

(4)

where ${\delta }_{i,k}\ge 0$ is a set value.

2.2. About TCS structural aspects

The system (1), (2) identification depends on the evaluating possibility of its parameters.

Introduce the model for (1)

$$\begin{array}{l}\{\begin{array}{l}{\dot{\widehat{X}}}_{11}=K_1\left({\widehat{X}}_{11}-X_{11}\right)+{\widehat{A}}_1{X}_{11}+{\widehat{B}}_1{v}_{11},\\ {\widehat{y}}_{11}=C_1^T{\widehat{X}}_{11},\end{array}\\ \{\begin{array}{l}{\dot{\widehat{X}}}_{21}=K_1\left({\widehat{X}}_{21}-X_{21}\right)+{\widehat{A}}_1{\widehat{X}}_{21}+{\widehat{B}}_1{v}_{21},\\ {\widehat{y}}_{21}=C_1^T{X}_{21},\end{array}\end{array}$$

(5)

where $K_1\in {\text{ℝ}}^{q_1\times q_1}$ is stable matrix (reference model); ${\widehat{A}}_1\in {\text{ℝ}}^{q_1\times q_1}$, ${\widehat{B}}_1\in {\text{ℝ}}^{q_1}$ are matrixes (5), ${\widehat{X}}_{i,1}$ is model state vector.

Let $E_{11}\triangleq {\widehat{X}}_{11}-X_{11}$, $E_{21}\triangleq {\widehat{X}}_{21}-X_{21}$. Then for the first layer

$$\begin{array}{l}{\dot{E}}_{11}=K_1{E}_{11}+\Delta A_1{X}_{11}+\Delta B_1{v}_{11},\\ {\dot{E}}_{21}=K_1{E}_{21}+\Delta {\widehat{A}}_1{\widehat{X}}_{21}+\Delta {\widehat{B}}_1{v}_{21},\end{array}$$

(6)

where $\Delta A_1\triangleq {\widehat{A}}_1-A_1,\hspace{0.17em}\hspace{0.17em}\Delta B_1\triangleq {\widehat{B}}_1-B_1$ are matrices of parametric residuals.

Similarly, we obtain the error equations for the remaining layers TCS

$$\begin{array}{l}{\dot{E}}_{1,k}=K_k{E}_{1,k}+\Delta A_k{X}_{1,k}+\Delta B_k{u}_{1,k},\\ {\dot{E}}_{2k}=K_k{E}_{2,k}+\Delta {\widehat{A}}_k{\widehat{X}}_{2,k}+\Delta {\widehat{B}}_k{u}_{2,k}.\end{array}$$

(7)

Let the input $g_i(t)$, $i=1;\hspace{0.17em}\hspace{0.17em}2$ satisfy the constant excitation (CE) condition

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

(8)

where ${\underset{\_}{\alpha }}_i,{\overline{\alpha }}_i$ are positive numbers, $T>0$.

Designations:

(1) if condition (8) is true, then we write $g_i(t)\in E{C}_{{\underset{\_}{\alpha }}_i,{\overline{\alpha }}_i}$;

(2) if condition (8) is not satisfied, then $g_i(t)\notin E{C}_{{\underset{\_}{\alpha }}_i,{\overline{\alpha }}_i}$.

Theorem 1. 

Let 1) $g_i(t)\in E{C}_{{\underset{\_}{\alpha }}_i,{\overline{\alpha }}_i}$, where $\left({\underset{\_}{\alpha }}_i,{\overline{\alpha }}_i\right)>0$; 2) the system (5) is stable and detectable; 3) $K_1\in {\text{ℝ}}^{q_1\times q_1}$ is Hurwitz matrix; 4) $v_{i1}(t)\in E{C}_{{\underset{\_}{\sigma }}_{i1},{\overline{\sigma }}_{i1}}$, where ${\underset{\_}{\sigma }}_{i1}>0,\hspace{0.17em}\hspace{0.17em}{\overline{\sigma }}_{i1}>0$ $i=1,\hspace{0.17em}2$; 5) $X_{i1}(t)\in E{C}_{{\underset{\_}{\alpha }}_{X_{i1}},{\overline{\alpha }}_{X_{i1}}}$, where $\left({\underset{\_}{\alpha }}_{X_{i1}},{\overline{\alpha }}_{X_{i1}}\right)>0$. Then the system (5) is identifiable if

$${\pi }_{12}{\Vert \Delta A_1\Vert }^2+0.5{\upsilon }_{12}{\Vert \Delta B_1\Vert }^2\le (\lambda -0.5)V_1$$

(9)

where ${\lambda }_1>0.5$, ${\pi }_{12}=2\max \left({\overline{\alpha }}_{X_{11}},{\overline{\alpha }}_{X_{12}}\right)$, ${\upsilon }_{12}={\overline{\sigma }}_{11}+{\overline{\sigma }}_{21}$, ${\Vert \Delta A_1\Vert }^2=\mathrm{tr}\left(\Delta A_1^T\Delta A_1\right)$, $\mathrm{tr}$ is spur of matrix, $V_1(t)=0.5E_{11}^T(t)R_1{E}_{11}(t)+0.5E_{21}^T(t)R_1{E}_{21}(t)$, $R_1=R_1^T>0$ is symmetric matrix.

The theorem 1 proof gives in the appendix A.

Definition 1. 

If condition (9) satisfies, then the system (5) is $P{I}_{1,X}$-identifiable on a set of state variables.

The subsystem layer (1) identifiability depends on properties of the TCS output.

Consider the system (7) and the Lyapunov function (LF)

$$V_k(t)=0.5E_{1,k}^T(t)R_k{E}_{1,k}(t)+0.5E_{2,k}^T(t)R_k{E}_{2,k}(t).$$

(10)

Theorem 2. 

Let (i) $K_k\in {\text{ℝ}}^{q_k\times q_k}$is Hurwitz matrix; (ii) ${\Vert X_{1,k}\Vert }^2\in E{C}_{{\underset{\_}{\alpha }}_{X_{1,k}},{\overline{\alpha }}_{X_{1,k}}}$, ${\Vert X_{2,k}\Vert }^2\in E{C}_{{\underset{\_}{\alpha }}_{X_{2,k}},{\overline{\alpha }}_{X_{2,k}}}$; ${\pi }_k=\max \left({\overline{\alpha }}_{X_{1,k}},{\overline{\alpha }}_{X_{2,k}}\hspace{0.17em}\right)$, $y_{2k-1}\in E{C}_{{\underset{\_}{\alpha }}_{y_{2,k-1}},{\overline{\alpha }}_{y_{2,k-1}}}$; (iii) the system (5) is $P{I}_{1,X}$-identifiable; (iv) the system (7) is stable and; (v) $v_{2,k-1}^2\in E{C}_{{\underset{\_}{\alpha }}_{v_{2,k-1}},{\overline{\alpha }}_{v_{2,k-1}}}$; (vi) the $d_{k-1}$ operator is constant: $d_k\le {\omega }_k\le \omega$, where $\omega$ is some number; 6) ${\lambda }_k\ge 0.5$. Then the system (7) is $P{I}_{k,X}$-identifiable if

$$.0.5{\pi }_{k,i}{\Vert \Delta A_k\Vert }^2+0.5{\Vert \Delta B_k\Vert }^2\left({\tilde{\overline{\alpha }}}_k+2{\omega }^2\left({\tilde{\overline{\alpha }}}_k+2{\omega }^2{\beta }_k\right)\right)\le \hspace{0.17em}\left({\lambda }_k-0.5\hspace{0.17em}\right)V_k$$

(11)

where ${\pi }_{k,i}=2\max \left({\overline{\alpha }}_{X_{1,k}},{\overline{\alpha }}_{X_{2,k}}\right)$, ${\tilde{\overline{\alpha }}}_k=2\max \left({\overline{\alpha }}_{y_{2,k-1}},{\overline{\alpha }}_{y_{1,k-1}}\right)$, ${\beta }_k=\underset{i}{\max }\left({\overline{\alpha }}_{y_{i,k-1}}+{\omega }^2{\overline{\alpha }}_{v_{i,k-1}}\right)$.

The theorem 2 proof gives in the Appendix B.

Remark 1. 

The conditions ${\Vert X_{i,k}\Vert }^2\in E{C}_{{\underset{\_}{\alpha }}_{X_{i,k}},{\overline{\alpha }}_{X_{i,k}}}$, $i=1;\hspace{0.17em}\hspace{0.17em}2$ followed from $v_{i1}(t)\in E{C}_{{\underset{\_}{\alpha }}_{v_{i1}},{\overline{\alpha }}_{v_{i1}}}.$

Remark 2. 

The $k$-layer (7) identifiability depends on properties of TCS previous layers and CL. Select CL parameters so that condition (11) is fulfilled.

Let $d_k$ is differentiating.

Theorem 3. 

Let Theorem 2 conditions be satisfied and (i) operator $d_k$ is differentiating, i.e., $v_{1,k-1}=d\left(y_{2,k-1}+d_{k-2}{v}_{2,k-1}\right)/dt$; (ii) the system (1), (2) is stable detectable and recoverable. Then the system (5) $P{I}_{k,X}$-identifiable if

$$0.5{\pi }_{k,i}{\Vert \Delta A_k\Vert }^2+0.5{\Vert \Delta B_k\Vert }^2\left({\tilde{\overline{\alpha }}}_{k,\dot{y}}+2\left({\overline{\alpha }}_{\dot{y}}+{\tilde{\overline{\alpha }}}_{\ddot{v}}\right)\right)\le \hspace{0.17em}\left({\lambda }_k-0.5\hspace{0.17em}\right)V_k$$

(12)

where ${\tilde{\overline{\alpha }}}_{k,\dot{y}}=\underset{i}{2\max }{\overline{\alpha }}_{{\dot{y}}_{2k-1}}$ ${\tilde{\overline{\alpha }}}_{k,\ddot{v}}=\underset{i}{2\max }{\overline{\alpha }}_{{\ddot{v}}_{i,k-1}}$, ${\pi }_{k,i}=2\max \left({\overline{\alpha }}_{X_{1,k}},{\overline{\alpha }}_{X_{2,k}}\right)$, $V_k(t)$ has the form (10).

The theorem 3 proof gives in the Appendix C.

Obtained results give estimates of the system (1), (2) identifiability based on the analysis of the TCS state vector.

Let the set (3) measured. Convert TCS to the form [28] based on the set [28]. Consider the system (1). Let $A_1$ be the Frobenius matrix with a parameter vector $A_{1,s}\in R^{q_1}$, $A_{1,s}={\left[a_{1,s,1},a_{1,s,2},\dots ,a_{1,s,q_1}\right]}^T$, $C_1={\left[1,0,\dots ,0\right]}^T,B_1={\left[0,\dots ,0,b_{1,s}\right]}^T$. In the space $\left(v_{11},y_{11}\right)$, the system (1) has a representation (see Appendix D)

$${\dot{y}}_{1,1}={\overline{A}}_{1,1}^T{P}_{1,1},$$

(13)

where ${\overline{A}}_{1,1}^T=\left[-a_{1,1,1},a_{1,1,2},\dots ,a_{1,1,q_1};b_{1,s},b_{1,2},\dots ,b_{1,q_1}\right]$, ${\overline{A}}_{1,1}\in {\text{ℝ}}^{2q_1}$, $P_{1,1}\in {\text{ℝ}}^{2q_1}$.

Considering (13), the system (1) has the form

$$\{\begin{array}{l}{\dot{y}}_{1,1}={\overline{A}}_{1,1}^T{P}_{1,1},\\ {\dot{y}}_{2,1}={\overline{A}}_{1,1}^T{P}_{2,1}.\end{array},$$

(14)

Evaluate the identifiability of the system (14) by output ($P{I}_{1,y}$-identifiability). Consider the model

$$\{\begin{array}{l}{\dot{\widehat{y}}}_{1,1}=-\chi e_{1,1}+{\widehat{\overline{A}}}_{1,1}^T{P}_{1,1},\\ {\dot{\widehat{y}}}_{2,1}=-\chi e_{2,1}+{\widehat{\overline{A}}}_{1,1}^T{P}_{2,1},\end{array}$$

(15)

where ${\chi }_1>0$, $e_{i,1}={\widehat{y}}_{i,1}-y_{i,1}$ is output $y_{i,1}$ prediction error, $i=1;\hspace{0.17em}2$. The equation for $e_{i,1}$

$${\dot{e}}_{i,1}=-{\chi }_1{e}_{i,1}+\Delta {\overline{A}}_{1,1}^T{P}_{i,1},\Delta {\overline{A}}_{1,1}={\widehat{\overline{A}}}_{1,1}-{\overline{A}}_{1,1}.$$

(16)

Consider LF $V_{1,2}(e_{1,1},e_{2,1})=0.5\left(e_{1,1}^2+e_{2,1}^2\right)$ and ${\dot{V}}_{1,2}$ has the form

$${\dot{V}}_{1,2}=-2{\chi }_1{V}_{1,2}+\Delta {\overline{A}}_{1,1}^T\left(P_{1,1}{e}_{1,1}+P_{2,1}{e}_{2,1}\right)\le -\frac{{\chi }_1}{2}{V}_{1,2}+\frac{1}{2{\chi }_1}\Delta {\overline{A}}_{1,1}^T{P}_1{P}_1^T\Delta {\overline{A}}_{1,1},$$

where $P_1{P}_1^T=P_{1,1}{P}_{1,1}^T+P_{2,1}{P}_{2,1}^T$.

It follows from (16) that $\Delta {\overline{A}}_{1,1}=0$, if the vector $P_1(t)\in E{C}_{{\underset{\_}{\alpha }}_{P_1},{\overline{\alpha }}_{P_1}}$ $(i=1;\hspace{0.17em}2)$ and subsystem (1) is parametrically $P{I}_{1,y}$-identifiable by output. The $P{I}_{1,y}$-identifiability of subsystems (2) is justified similarly.

Representation for the $k$-layer (system (2))

$$\{\begin{array}{l}{\dot{y}}_{1,k}={\overline{A}}_{1,k}^T{P}_{1,k},\\ {\dot{y}}_{2,k}={\overline{A}}_{1,k}^T{P}_{2,k}.\end{array}$$

(17)

where $P_{i,k}\in {\text{ℝ}}^{2q_k}$ is generalized input, ${\overline{A}}_{i,k}\in {\text{ℝ}}^{2q_k}$ is parameter vector.

Determining the CL type is one of the TCS identification tasks under uncertainty. Apply the following approachю Consider the $k$-layer of the system (2) with identical cross-links and $d_{k-1}=\mathrm{cons}$. Construct the ${\mathrm{S}}_{u_{i,k},y_{i,k}}$-structure described by the function $f_{u_{i,k},y_{i,k}}:u_{i,k}\to y_{i,k}$, $i=1;\hspace{0.17em}2$ for both channels of the $k$-layer. ${\mathrm{S}}_{u_{i,k},y_{i,k}}$ reflects the input-output state of the TCS. Determine the secants for ${\mathrm{S}}_{u_{i,k},y_{i,k}}$

$${\xi }_{u_{i,k},y_{i,k}}=a_{{\xi }_{0,i,k}}+a_{{\xi }_{1,i,k}}{u}_{i,k},$$

(18)

where $a_{{\gamma }_{0,i,k}},\hspace{0.17em}\hspace{0.17em}{a}_{{\gamma }_{1,i,k}}$ are parameters determined using the least squares method.

Since CL is rigid, the angle between secants ${\xi }_{u_{i,k},y_{i,k}}$ does not exceed a certain value ${\delta }_{{\xi }_{i,k}}$: $\left|a_{{\xi }_{1,1,k}}-a_{{\xi }_{1,2,k}}\right|\le {\delta }_{{\xi }_{i,k}}$. Therefore, CLs are positive. If the cross-links are asymmetric, then $\left|a_{{\xi }_{1,1,k}}-a_{{\xi }_{1,2,k}}\right|>{\delta }_{{\xi }_{i,k}}$. In this case, the signal $v_{1,k-1}$ acts in antiphase with the output $y_{1,k-1}$ previous layer of the first channel.

Remark 3. 

Proved theorems are applicable if each layer channels are non-identical.

2.3. TCS Adaptive identification

Consider the representation (13)-(15), (17). Introduce the model for (17)

$$\{\begin{array}{l}{\dot{\widehat{y}}}_{1,k}=-{\chi }_k{e}_{1,k}+{\widehat{\overline{A}}}_{1,k}^T{P}_{1,k},\\ {\dot{\widehat{y}}}_{2,k}=-{\chi }_k{e}_{2,k}+{\widehat{\overline{A}}}_{1,k}^T{P}_{2,k},\end{array}$$

(19)

where ${\chi }_k>0$, $e_{i,k}={\widehat{y}}_{i,k}-y_{i,k}$ is $i$-output prediction error of the $i$-element for the $k$-layer.

the equation for $e_{i,k}$

$${\dot{e}}_{i,k}=-{\chi }_k{e}_{i,k}+\Delta {\overline{A}}_{i,k}^T{P}_{i,k},\Delta {\overline{A}}_{1,k}={\widehat{\overline{A}}}_{1,k}-{\overline{A}}_{1,k}.$$

(20)

Adaptive parameter tuning algorithm for (19)

$$\Delta {\dot{\overline{A}}}_{1,k}={\dot{\widehat{\overline{A}}}}_{1,k}=-{\Gamma }_k{e}_{i,k}{P}_{1,k}$$

(21)

where ${\Gamma }_k\in {\text{ℝ}}^{q_k\times q_k}$ is diagonal matrix with ${\gamma }_{k,j}>0$.

Consider subsystem (14)–(16) and LF ${\tilde{V}}_1(e_{1,1})=0.5e_{1,1}^2$. From condition ${\dot{\tilde{V}}}_1\le 0$ we obtain an adaptive algorithm for adjusting vector ${\widehat{\overline{A}}}_{1,1}$

$$\Delta {\dot{\overline{A}}}_{1,1}={\dot{\widehat{\overline{A}}}}_{1,1}=-{\Gamma }_1{e}_{1,1}{P}_{1,1},$$

(22)

where ${\Gamma }_1\in {\text{ℝ}}^{q_1\times q_1}$ is diagonal matrix with ${\gamma }_{1,j}>0$, $j$ is the diagonal element number.

Remark 4. 

As TCS has identical channels, we adjust only one channel.

The feedback effect leads to unidentifiability of the system (14)–(16), (22) parameters. The equation (16) writes as

$${\dot{e}}_{1,1}=-{\chi }_1{e}_{1,1}+\Delta {\overline{A}}_{1,1}^T{P}_{1,1}+f\left({\overline{A}}_{1,1},P_{1,1}\right),$$

(23)

where $f(\cdot )$ is the uncertainty that is the result from $P_{1,1}\notin E{C}_{{\underset{\_}{\alpha }}_{P_{1,1}},{\overline{\alpha }}_{P_{1,1}}}$.

Consider LF ${\tilde{V}}_{1,e}\left(e_{1,1}\right)=0.5e_{1,1}^2$, ${\tilde{V}}_{1,\Delta }\left(\Delta {\overline{A}}_{1,1}\right)=0.5\Delta {\overline{A}}_{1,1}^T{\Gamma }_1^{-1}\Delta {\overline{A}}_{1,1}$.

Theorem 4. 

Let 1) $g_i(t)\in E{C}_{{\underset{\_}{\alpha }}_i,{\overline{\alpha }}_i}$, $P_{1,1}\notin E{C}_{{\underset{\_}{\alpha }}_{P_{1,1}},{\overline{\alpha }}_{P_{1,1}}}$; 2) $y_{i,1}\notin E{C}_{{\underset{\_}{\alpha }}_{y_{i,1}},{\overline{\alpha }}_{y_{i,1}}}$, $i=1,\hspace{0.17em}\hspace{0.17em}2;$ 3) ${\left|f\right|}^2\le {\alpha }_f,$ where ${\alpha }_f\ge 0;$ 4) there is a $\upsilon >0$ such that for $t>>t_0$ it is $e_{1,1}\Delta {\overline{A}}_{1,1}^T{P}_{1,1}=\upsilon \left(e_{1,1}^2+\Delta {\overline{A}}_{1,1}^T{P}_{1,1}{P}_{1,1}^T\Delta {\overline{A}}_{1,1}\right)$; 5) ${\lambda }_{\max }^{-1}\left({\Gamma }_1\right){\Vert \Delta {\overline{A}}_{1,1}\Vert }^2\le {\tilde{V}}_{1,\Delta }\le {\lambda }_{\min }^{-1}\left({\Gamma }_1\right){\Vert \Delta {\overline{A}}_{1,1}\Vert }^2$; 6) the system of differential inequalities is valid for the system (16), (23)

$$\left[\begin{array}{c}{\dot{\tilde{V}}}_{1,e}\\ {\dot{\tilde{V}}}_{1,\Delta }\end{array}\right]\le \underset{A_{S_1}}{\underbrace{\left[\begin{array}{cc}-\frac{{\chi }_1}{2}& 2\frac{{\pi }_{P_{1,1}}}{{\chi }_1}\\ 4\upsilon & -\frac{\upsilon \eta }{2}\end{array}\right]}}\left[\begin{array}{c}{\tilde{V}}_{1,e}\\ {\tilde{V}}_{1,\Delta }\end{array}\right]+\underset{B_{S_1}}{\underbrace{\left[\begin{array}{c}\frac{{\alpha }_f}{{\chi }_1}\\ 0\end{array}\right]}}.$$

(24)

and the comparison system ${\dot{S}}_1(t)=A_{S_1}{S}_1(t)+B_{S_1}$ for (24), where $S_1(t)={\left[s_{1,e}(t),s_{1,\Delta }(t)\right]}^T$, $s_{1,w}(t)$ $(w=e,\Delta )$ is a majority factor for ${\tilde{V}}_{1,w}(t)$ и $s_{1,w}\left(t_0\right)\ge {\tilde{V}}_{1,w}\left(t_0\right)$. Then the system (14)–(16), (23) is exponentially dissipative with the estimate

$${\left[{\tilde{V}}_{1,e}(t)\hspace{0.17em}{\tilde{V}}_{1,\Delta }(t)\right]}^T\le e^{A_{S_1}\left(t-t_0\right)}{S}_1(t_0)+{\displaystyle \underset{t_0}{\overset{t}{\int }}{e}^{A_{S_1}\left(t-\tau \right)}{B}_{S_1}d\tau }$$

where ${\chi }_1^2\eta >8{\vartheta }_{P_{1,1}}$, then $\eta ={\underset{\_}{\pi }}_{P_{1,1}}{\lambda }_{\min }^2\left({\Gamma }_1\right)$,${\underset{\_}{\pi }}_{P_{1,1}}\ge 0$, $\Vert P_{1,1}{P}_{1,1}^T\Vert \le {\vartheta }_{P_{1,1}}\left({\overline{\alpha }}_{P_{1,1}}\right)$, ${\lambda }_{\min }\left({\Gamma }_1\right)$ is the minimum eigenvalue of the matrix ${\Gamma }_1$.

The theorem 4 proof gives in the Appendix E.

As follows from Theorem 4, the adaptive identification system guarantees biased estimates for system (14)–(16) parameters.

Consider the system (17), (19), (21). Let $d_{k-1}$ in (2) is constant.

Present $P_{1,k}$ and $\Delta {\overline{A}}_{1,k}$ as: $P_{1,k}={\left[{\tilde{P}}_{1,k}^T,p_{j,v_{1,2,k-1}}\right]}^T$,$\Delta {\overline{A}}_{1,k}={\left[\Delta {\tilde{\overline{A}}}_{1,k}^T,\Delta d_{j,k}\right]}^T$ where $p_{j,v_{i,2,k-1}}$ is transformation $v_{1,2,k-1}$, $j$ is jth element of vector $\Delta {\overline{A}}_{1,k}$. Then (20)

$${\dot{e}}_{1,k}=-{\chi }_k{e}_{1,k}+\Delta {\overline{A}}_{1,k}^T{P}_{1,k}+f_{1,k}\left({\overline{A}}_{1,k}{P}_{1,k}\right),$$

(25)

where $f_{1,k}(\cdot )\in \text{ℝ}$ is uncertainty, which is $P_{1,k}\notin E{C}_{{\underset{\_}{\alpha }}_{P_{1,k}},{\overline{\alpha }}_{P_{1,k}}}$.

Consider the system (19), (20), (25) and LF ${\tilde{V}}_{k,\Delta }\left(\Delta {\overline{A}}_{1,k}\right)=0.5\Delta {\overline{A}}_{1,k}^T{\Gamma }_k^{-1}\Delta {\overline{A}}_{1,k},{\tilde{V}}_{k,e}\left(e_{1,k}\right)=0.5e_{1,k}^2.$

Theorem 5. 

Let the Theorem 4 conditions be satisfied and (i) $y_{1,k}(t)\notin E{C}_{{\underset{\_}{\alpha }}_{y_{i,k}},{\overline{\alpha }}_{y_{i,k}}}$, $P_{1,k}\notin E{C}_{{\underset{\_}{\alpha }}_{P_{1,k}},{\overline{\alpha }}_{P_{1,k}}}$; (ii) $u_{1,k}={\upsilon }_{u_{1,k}}\left(y_{1,k-1},v_{1,k-1}\right)$, $u_{1,k}\notin EC$; (iii) $f_{1,k}^2\le {\alpha }_{f_{1,k}}$, where ${\alpha }_{f_{1,k}}\ge 0;$ (iv) ${\lambda }_{\max }^{-1}\left({\Gamma }_k\right){\Vert \Delta {\overline{A}}_{1,k}\Vert }^2\le {\tilde{V}}_{k,\Delta }\le {\lambda }_{\min }^{-1}\left({\Gamma }_k\right){\Vert \Delta {\overline{A}}_{1,k}\Vert }^2$, where ${\lambda }_{\min }\left({\Gamma }_k\right)$ is the minimum eigenvalue of the matrix ${\Gamma }_k$; (v) $\Vert P_{1,k}{P}_{1,k}^T\Vert \le {\pi }_{P_{1,k}}<{\overline{\alpha }}_{P_{1,k}}$; (vi) there is a $\upsilon >0$ such that for $t>>t_0$ is fair

$$e_{1,k}\Delta {\overline{A}}_{1,k}^T{P}_{1,k}=\upsilon \left(e_{1,k}^2+\Delta {\overline{A}}_{1,k}^T{P}_{1,k}{P}_{1,k}^T\Delta {\overline{A}}_{1,k}\right)$$

(26)

(vii) the system of differential inequalities is valid for the system (21), (25)

$$\underset{{\dot{\tilde{V}}}_k}{\underbrace{\left[\begin{array}{c}{\dot{\tilde{V}}}_{k,e}\\ {\dot{\tilde{V}}}_{k,\Delta }\end{array}\right]}}\le \underset{A_{S_k}}{\underbrace{\left[\begin{array}{cc}-\frac{{\chi }_k}{2}& 2\frac{{\vartheta }_{P_{1,k}}}{{\chi }_k}\\ 4\upsilon & -\frac{\upsilon \eta }{2}\end{array}\right]}}\underset{{\tilde{V}}_k}{\underbrace{\left[\begin{array}{c}{\tilde{V}}_{k,e}\\ {\tilde{V}}_{k,\Delta }\end{array}\right]}}+\underset{B_{S_k}}{\underbrace{\left[\begin{array}{c}\frac{{\alpha }_{f_{1,k}}}{{\chi }_k}\\ 0\end{array}\right]}}$$

(27)

and the comparison system ${\dot{S}}_k(t)=A_{S_k}{S}_k(t)+B_{S_k}$ для (27), where $S_k(t)={\left[s_{k,e}(t),s_{k,\Delta }(t)\right]}^T$, $s_{k,w}(t)$ ($w=e,\Delta$) is a majority factor for ${\tilde{V}}_{k,w}(t)$, $s_{k,w}\left(t_0\right)\ge {\tilde{V}}_{k,w}\left(t_0\right)$. Then the system (19), (20), (25) is exponentially dissipative with the estimate

$${\tilde{V}}_k(t)\le e^{A_{S_k}\left(t-t_0\right)}{S}_k(t_0)+{\displaystyle \underset{t_0}{\overset{t}{\int }}{e}^{A_{S_k}\left(t-\tau \right)}{B}_{S_k}d\tau }$$

if ${\chi }_k^2\eta >32{\vartheta }_{P_{1,k}}$, ${\underset{\_}{\pi }}_{P_{1,k}}\ge 0$, $\eta ={\underset{\_}{\pi }}_{P_{1,k}}{\lambda }_{\min }\left({\Gamma }_k\right)$.

The theorem 5 proof gives in the Appendix F.

As follows from (27), adaptive identification system properties depend on the $k$-layer on cross-links.

So, we have proved the TCS identifiability by the state and output of the $k$-layer. The results confirming the convergence of the estimates for the system parameters obtained. Adaptive identification system properties depend on the CL parameters and information properties of TCS signals.

Remark 5. 

If the TCS contains non-identical channels, then it is necessary to apply the models (17) for each $k$-layer.

3. Interconnected systems

Consider a system $S_{MS}$

$$\dot{X}(t)=AX(t)+D{F}_1(X,t)+BU(t),$$

(27)

$$\mathrm{L}Y(t)=CX(t)+F_2(X,t),$$

(28)

where $X\in {ℝ}^m$ is state vector, $A\in {ℝ}^{m\times m}$ is state matrix, $D\in {ℝ}^{m\times q}$, $F_1(X,t):{ℝ}^m\to {ℝ}^q$ is nonlinear vector function, $U\in {ℝ}^k$ is input vector (control), $B\in {ℝ}^{m\times k}$, $Y\in {ℝ}^n$ is output vector, $C\in {ℝ}^{n\times m}$, $F_2(X,t):{ℝ}^m\to {ℝ}^n$ is perturbation vector (measurement errors),, $\mathrm{L}$ is operator that forms the vector $Y$. $\mathrm{L}$ can be: 1) a differential operator reflecting the dynamic properties of the measurement system; 2) as the operator describing the method of interaction between subsystems. Matrices $A,D,B$ are block-based and reflect the state of independent subsystems. The $F_2(X,t)$ vector can be a perturbation (measurement error) or a variable reflecting the influence of independent subsystems.

The observed data set

$$I_o=\left\{Y(t),U(t),t\in \left[t_0,t_N\right]\right\},t_N<\infty .$$

(29)

Assumption 1. 

Elements ${\phi }_{1,i}\left(x_j\right)\in F_1$, ${\phi }_{2,i}\left(x_j\right)\in F_2$ are smooth, one-valued functions.

Apply the model to estimate parameters of matrices $A,D,B,C$

$$\{\begin{array}{l}\dot{\widehat{X}}(t)=\widehat{A}(t)\widehat{X}(t)+\widehat{D}(t)F_1(X,t)+\widehat{B}(t)U(t),\\ \mathrm{L}\widehat{Y}(t)=C\widehat{X}(t)+F_2(\widehat{X},t),\end{array}$$

(30)

where $\widehat{A}(t)$, $\widehat{D}(t)$, are matrices with tuning parameters.

Problem: synthesize the model (30) for the system (27) satisfying the assumption 1, and find parameter tuning laws of t matrices $\widehat{A}(t)$, $\widehat{D}(t)$ и $\widehat{B}(t)$ to

$$\underset{t\to \infty }{\lim }\Vert \widehat{Y}(t)-Y(t)\Vert \le {\delta }_y,{\delta }_y\ge 0,$$

where $\Vert \cdot \Vert$ is the Euclidean norm.

Remark 6. 

Consider (28) as an equation describing the connections between subsystems for some class of ICS. The vector $F_2(X,t)$ form (see the section 2) must be evaluated in this case.

The synthesis of adaptive identification algorithms bases on the approach described in Section 2.

Remark 7. 

If the TCS contains nonlinear subsystems, then the decision on nonlinearity form bases on the construction of an interconnection graph [31] under uncertainty.

Мoдели и алгoритмы адаптации сoвпадают с уравнениями, пoлученными в предыдущем разделе. Для oценки качества рабoты адаптивнoй пoдсистемы идентификации мoжнo применить теoремы § 2.2.

4. Examples

1. The TCS for target angular tracking with identical azimuth and elevation channels and asymmetric CL considers [32].

$$\dot{x}=-a_{x}x+b_x(g-y),$$

(31)

$$\ddot{y}=-a_y\dot{y}+b_y\left(x-\left(g_1-y_1\right)-k{\left(g_1-y_1\right)}^{\prime }\right),$$

(32)

$${\dot{x}}_1=-a_x{x}_1+b_x(g_1-y_1),$$

(33)

$${\ddot{y}}_1=-a_y{\dot{y}}_1+b_y\left(x_1+\left(g-y\right)+k{\left(g-y\right)}^{\prime }\right),$$

(34)

where $a_x=T_x^{-1}$, $b_x=T_x^{-1}{k}_x$, $T_x$, $k_x$ are time constant and amplifier gain; $k$ is the cross-linking parameter; $a_y=T_y^{-1}$, $b_y=T_y^{-1}{k}_y$ are servomotor parameters; $g,\hspace{0.17em}\hspace{0.17em}{g}_1$ are inputs, ${\left(g-y\right)}^{\prime }=d\left(g-y\right)/dt$. CL represents in parentheses as a differentiating link with the parameter $k$.

Outputs $x_i(t),y_i(t)$ of links and the inputs $g_i(t)$ measure. It is necessary to evaluate the system parameters (31), (32).

Apply the transformation described in Appendix 4 to obtain the model for $y(t)$. Consider the system of filters (transformations)

$$\begin{array}{l}{\dot{p}}_y=-{\mu }_1{p}_y+y,\\ {\dot{p}}_{\vartheta }=-{\mu }_1{p}_{\vartheta }+\hspace{0.17em}\hspace{0.17em}\vartheta ,\\ \hspace{0.17em}{\dot{p}}_{\upsilon }=-{\mu }_1{p}_{\upsilon }+\hspace{0.17em}\upsilon \hspace{0.17em},\end{array}$$

(35)

where $\vartheta \triangleq x-g_1+y_1$, $\upsilon \triangleq d\left(g_1-y_1\right)/dt$, $p_i(0)=0$, $i=y,\vartheta ,\upsilon$, ${\mu }_1>0$ is a number that does not coincide with roots of the characteristic equation for the second equation in (31). The model for (31) has the form

$$\dot{\widehat{x}}=-{\chi }_x{e}_x+{\widehat{a}}_{x}x+{\widehat{b}}_x(g-y),$$

(36.1)

$$\dot{\widehat{y}}=-{\chi }_y{e}_y+{\widehat{a}}_{y}y+{\widehat{a}}_{p_y}{p}_y+{\widehat{a}}_{\vartheta }{p}_{\vartheta }+{\widehat{a}}_{\upsilon }{p}_{\upsilon },$$

(36.2)

where $e_x=\widehat{x}-x,e_y=\widehat{y}-y$, ${\chi }_x,\hspace{0.17em}\hspace{0.17em}{\chi }_y$ — positive numbers (reference model). Adaptive algorithms for tuning system parameters (36.1), (36.2)

$$\begin{array}{l}{\dot{\widehat{a}}}_x=-{\gamma }_{a_x}{e}_{x}x,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{b}}}_x=-{\gamma }_{b_x}{e}_x(g-y),\\ {\dot{\widehat{a}}}_y=-{\gamma }_{a_y}{e}_{y}y,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{a}}}_{p_y}=-{\gamma }_{a_{p_y}}{e}_y{p}_y,\\ {\dot{\widehat{a}}}_{\vartheta }=-{\gamma }_{a_{\vartheta }}{e}_y{p}_{\vartheta },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{a}}}_{\upsilon }=-{\gamma }_{a_{\upsilon }}{e}_y{p}_{\upsilon }\end{array}$$

(37)

where ${\gamma }_{a_i},\hspace{0.17em}{\gamma }_{b_i}$ ($i=x,y,a_y,p_y,\vartheta ,\upsilon$) is positive numbers guaranteeing convergence (37).

The system (31), (32) modelled with the parameters: $a_x=1.2$, $b_x=2$, $a_y=5.95$, $b_y=1$, $k=5$. Inputs $g_2(t)=1.5\sin (0.1\pi t)$, $g_2(t)=1.5\sin (0.1\pi t)$.

Figure 1 shows the structures confirming asymmetric connections in the system. The analysis of parameters $a_{{\xi }_{0,{\epsilon }_1,y}}=0.53,$ $a_{{\xi }_{0,\epsilon ,y_1}}=-0.45$ for secants ${\xi }_{{\epsilon }_1,y}$, ${\xi }_{\epsilon ,y_1}$ shows that the condition $\left|a_{{\xi }_{0,{\epsilon }_1,y}}-a_{{\xi }_{0,\epsilon ,y_1}}\right|>{\delta }_{{\xi }_0}$ fulfils with ${\delta }_{{\xi }_0}=0.2$. Therefore, CL is antisymmetric.

We consider remark 4 when evaluating the parameters of the system (31), (32). The identification system parameters.: ${\mu }_1=1.5$, ${\chi }_x=1.5$, ${\chi }_y=2$.

Adaptive identification results are shown in Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6. Adaptive identification of model parameters (36.1), (36.2) presented in Figure 2 and Figure 3.

Identification errors show in Figure 4.

We see that the tuning of layer 2 parameters depends on the layer 1 output (model (36.1)) and the CL properties. Correlations between outputs of elements (31), (33) affect the quality of parameter tunings (36.2). This influence is transmitted through CL. This conclusion confirms by structures shown in Figure 5. These results confirm statements of Theorem 5. The tuning process of model (36.1) parameters has a smooth character (Figure 6).

2. 2. Consider a pseudo-linear two-channel corrector (PLTCC) [33]

$$\{\begin{array}{l}{\dot{x}}_1=-{\alpha }_1{x}_1+{\beta }_1\left(g-x_6\right)\\ {\dot{x}}_2=-{\alpha }_2{x}_2+{\mu }_2{\left(g-x_6\right)}^{\prime }+{\beta }_2\left(g-x_6\right)\\ {\dot{x}}_6=x_7\\ {\dot{x}}_7=-{\alpha }_{01}{x}_7-{\alpha }_{02}{x}_6+{\beta }_0\left(x_{1}sign\left(x_1\right)\right)\left(sign\left(x_2\right)\right)\end{array}$$

(38)

where $g$ is setting effect, $x_6$ is output; $g-x_6$ is misalignment error; $x_1$ is amplitude channel output; $x_2$ is phase channel output; $u=\left(x_{1}sign\left(x_1\right)\right)\left(sign\left(x_2\right)\right)$ is regulator output; ${\left(g-x_6\right)}^{\prime }=d\left(g-x_6\right)/dt$; ${\alpha }_1,{\beta }_1,{\alpha }_2,{\mu }_2,{\beta }_2,{\alpha }_{01},{\alpha }_{02},{\beta }_0$ are corrector parameters.

Remark 8. 

In [33], a preliminary selection of PLTCC parameters uses.

The set ${\mathrm{II}}_o=\left\{x_1(t),x_2(t),x_6(t),t\in \left[0,t_k\right]\right\}$ is measured, where $t_k$ is known number. Apply models

$${\dot{\widehat{x}}}_1=-k_1{e}_1+{\widehat{\alpha }}_1{x}_1+{\widehat{\beta }}_1\left(g-x_6\right),$$

(39)

$${\dot{\widehat{x}}}_2=-k_2{e}_2+{\widehat{\alpha }}_2{x}_2+{\widehat{\mu }}_{2}d\left(g-x_6\right)/dt+{\widehat{\beta }}_2\left(g-x_6\right),$$

(40)

$${\dot{\widehat{x}}}_6=-k_6{e}_6+{\widehat{\alpha }}_{01}{x}_6+{\widehat{\alpha }}_{p_{x_6}}{p}_{x_6}+{\widehat{\alpha }}_{p_u}{p}_u,$$

(41)

where $k_1,\hspace{0.17em}\hspace{0.17em}{k}_2,\hspace{0.17em}\hspace{0.17em}{k}_6$ are known numbers; $e_i={\widehat{x}}_i-x_i,\hspace{0.17em}\hspace{0.17em}i=1,2,6$; ${\widehat{\alpha }}_i,{\widehat{\beta }}_i,i=1,2,\hspace{0.17em}{\widehat{\alpha }}_{p_{x_5}},\hspace{0.17em}\hspace{0.17em}{\widehat{\alpha }}_{p_{x_5}}$ are tuning parameters; ${\widehat{x}}_i\in \text{ℝ}$ ($i=1;2;6$) are model outputs; $p_{x_6},p_u$ obtain as (33).

Adaptation algorithms

$${\dot{\widehat{\alpha }}}_1=-{\gamma }_{{\alpha }_1}{e}_1{x}_1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{\beta }}}_1=-{\gamma }_{{\beta }_1}{e}_1\left(g-x_6\right),$$

$${\dot{\widehat{\alpha }}}_2=-{\gamma }_{{\alpha }_2}{e}_2{x}_2,\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{\beta }}}_2=-{\gamma }_{{\beta }_2}{e}_2\left(g-x_6\right),{\dot{\widehat{\mu }}}_2=-{\gamma }_{{\mu }_2}{e}_2{\left(g-x_6\right)}^{\prime },$$

(42)

$$\hspace{0.17em}{\dot{\widehat{\alpha }}}_{01}=-{\gamma }_{{\alpha }_{01}}{e}_6{x}_6,\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{\alpha }}}_{p_{x_6}}=-{\gamma }_{p_{x_6}}{e}_6{p}_{x_6},\hspace{0.17em}\hspace{0.17em}{\dot{\widehat{\alpha }}}_{p_u}=-{\gamma }_{p_u}{e}_6{p}_u,$$

where ${\gamma }_i>0$ is the gain factor in the corresponding parameter tuning circuit.

System (38) parameters: ${\alpha }_1=1.05,\hspace{0.17em}\hspace{0.17em}{\beta }_1=3.5,$ ${\alpha }_2=2.2,$ ${\beta }_2=2.2,$ $\hspace{0.17em}\hspace{0.17em}{\mu }_2=0.4$, ${\alpha }_{01}=3,$ ${\alpha }_{02}=5.03,\hspace{0.17em}\hspace{0.17em}{\beta }_0=5.2$ $g(t)=\sin (0.05\pi t)$. Simulation results show in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

Figure 7 represents structures (the transition process is excluded) reflecting the phase processes in the system (38). We see that the system is nonlinear. The application of results [31] shows that the system is structurally identifiable. Therefore, the input is S-synchronizing and considers the nonlinear properties of the PLTC.

The analysis shows the relationship between $x_1$ and $x_2$ (the correlation coefficient is 0.94). The density diagram (DD) (Figure 8) confirms the influence $x_1$ and $x_2$ on $x_6$. We see clear influence boundaries of variables on $x_6$. These relationships affect the convergence of adaptive algorithms.

The of adaptive model tuning process for amplitude and phase channels shows in Figure 9, Figure 10.

The tuning of model parameters for the object shows in Figure 11. The tuning process depends on the control action.

The tuning dynamics of model (41) haves a more complex form. The input influence of plays an important role. The theorem 5 statement holds for the PTLCC system.

5. Conclusion

The approach to the identification of interconnected systems proposed. First, the adaptive identification features of TCS with cross-links considered. The conditions of the TCS identifiability in the state space and the output space obtained. Structural aspects of the identification for two-channel systems are considered. WE consider the influence of input properties on the estimation of TCS parameters. Adaptive algorithms are designed for TCS identification with identical channels. Estimates and conditions confirming the estimate convergence of the TCS parameters are obtained. The properties of the adaptive identification system depend on the CL parameters and signal information properties in the TCS.

We consider the interconnected systems. We note the a priori information role in considering existing relationships for identification. It is shown that approaches used for the TCS identification applied to the interconnected system. Adaptive identification examples real systems are considered. We consider TCS with nonlinear control and complex interconnections. The paper considers only certain aspects of this multifaceted subject area. An approach that allows to evaluate the dynamics of processes in the adaptive system is proposed. It considers the influence of system elements on the quality of the parameter tuning process.