Decentralized tracking control design based on intelligent critic for an interconnected spring-mass-damper system

1. INTRODUCTION

For large-scale nonlinear interconnected systems, which are considered as nonlinear plants consisting of many interconnected subsystems, decentralized control has become a research hotspot in the last few decades ^[1–4]. Compared with the centralized control, the decentralized control has the advantages of simplifying the structure and reducing the computation burden of the controller. Besides, the local controller only depends on the information of the local subsystem. Meanwhile, with the development of science and technology, interconnected engineering applications have become increasingly complex, such as robotic systems ^[5] and power systems ^{[6, 7]}. In ^[8–10], we found that the decentralized control of the large-scale system was connected with the optimal control of the isolated subsystems, which means the optimal control method ^[11–14] can be adopted to achieve the design purpose of the decentralized controllers. However, the optimal control of the nonlinear system often needs to solve the Hamilton-Jacobi-Bellman (HJB) or Hamilton-Jacobi-Isaacs (HJI) equation, which can be solved by using the adaptive dynamic programming (ADP) method ^{[15, 16]}. Besides, in ^[13], Wang et al. investigated the latest intelligent critic framework for advanced optimal control. In ^[14], the optimal feedback stabilization problem was discussed with discounted guaranteed cost for nonlinear systems. It follows that the interconnection plays a significant role in designing the controller. Hence, it can be classified as decentralized and distributed control schemes. There is a certain distinction between decentralized control and distributed control. For decentralized control, each sub-controller only uses local information and the interconnection among subsystems can be assumed to be weak in nature. Compared with the decentralized control, the distributed control ^[17–19] can be introduced to improve the performance of the subsystems when the interconnections among subsystems become strong. In ^[20], the distributed optimal observer was devised to assess the nonlinear leader state for all followers. In ^[21], the distributed control was developed by means of online reinforcement learning for interconnected systems with exploration.

It is worth mentioning that the ADP algorithm has been extensively employed for dealing with various optimal regulation problems and tracking problems ^[22–24], which will achieve the goal, that is, the actual signal can track the reference signal under the noisy and the uncertain environment. In ^[25], Ha et al. proposed a novel cost function to explore the evaluation framework of the optimal tracking control problem. Then, aimed at complicated control systems, it is necessary to consider decentralized tracking control (DTC) problems ^[26–29]. The DTC systems can be transformed into the the nominal augmented tracking isolated subsystems (ATISs), which are composed of the tracking error and the reference signal. In ^[26], Qu et al. proposed a novel formulation consisting of a steady-state controller and a modified optimal feedback controller of the DTC strategy. Besides, the asymptotic DTC was realized by introducing two integral bounded functions in ^[27]. In ^[28], Liu et al. proposed a finite-time DTC method for a class of nonstrict feedback interconnected systems with disturbances. Moreover, the adaptive fuzzy output-feedback DTC design was investigated for switched large-scale systems in ^[29].

Game theory is a discipline that implements corresponding strategies. It contains cooperative and noncooperative types, that is, zero-sum (ZS) games and non-ZS games. In particular, ZS games have been widely applied in many fields ^[30–33]. The object of the ZS game is to derive the Nash equilibrium of nonliner systems, which makes the cost function optimized. In ^[31], the finite-horizon H-infinity state estimator design was studied for periodic neural networks over multiple fading channels. The noncooperative control problem was formulated as a two-player ZS game in ^[32]. In ^[33], Wang et al. investigated the stability of the general value iteration algorithm for ZS games. At the same time, we can also combine the ZS problem with the tracking problem to make the system more stable while achieving the trajectory tracking. In ^[34], Zhang et al. developed an online model-free integral reinforcement learning algorithm for solving the H-infinity optimal tracking problem for completely unknown systems. In ^[35], a general bounded $$ L_{2} $$ gain tracking policy was introduced with a discounted function. In ^[36], Hou et al. proposed an action-disturbance-critic neural network frame to realize the iterative dual heuristic programming algorithm.

As can be seen from the above, there are few studies that combine the DTC problem with the ZS game problem. It is necessary to take the related discounted cost function into account for the DTC system, which can transform the DTC problem into an optimal control problem with disturbances. In practice, the existence of disturbances will make an unpredictable impact on the plant. Hence, it is of vital importance to consider the stability of the DTC system. In the experimental simulation, it is a challenge to achieve the goal of effective online weight training, which is implemented under the tracking control law and the disturbance control law. Consequently, in this paper, we put forward a novel method in view of ADP to resolve the DTC problem with external disturbances for continuous-time (CT) nonlinear systems. More importantly, for the sake of overcoming the difficulty of selecting initial admissible control policies, an additional term is added during the weight updating process. Remarkably, in this paper, we introduce the discount factor for maximizing and minimizing the corresponding cost function.

The contributions of this paper are as follows: First, considering the disturbance input in the DTC system, the strategy feasibility and the system stability are discussed through theoretical proofs. It is worth noting that the discount factor is introduced to the cost function. Moreover, in the process of online weight training, we can make the DTC system reach a stable state without selecting the initial admissible control law. Additionally, we present the experimental process of the spring-mass-damper system. Besides, we derive the desired tracking error curves as well as control strategy curves, which demonstrates that they are uniformly ultimately bounded (UUB).

The whole paper is divided into six sections. The first section is the introduction of relevant background knowledges of the research content. The second section is the problem statement of basic problems about the two person ZS game and the DTC strategy. In the third section, we design the decentralized tracking controller by using the optimal control method through solving the HJI equations. Meanwhile, the relevant lemma and theorem are given to validate the establishment of the DTC strategy. In the fourth section, the design method in accordance with adaptive critic is elaborated. Most importantly, an improved critic learning rule is implemented via critic networks. In the fifth section, the practicability of this method is validated by an interconnected spring-mass-damper system. Finally, the sixth section displays conclusions and summarizes overall research content of the whole paper.

2. PROBLEM STATEMENT

Consider a CT nonlinear interconnected system with disturbances, which is composed of $$ N $$ interconnected subsystems. Its dynamic description can be expressed as

where $$ i={1, 2, \dotsc, N} $$, $$ x_{i}(t)\in{\mathbb R}^{n_{i}} $$ is the state vector of the $$ i $$th subsystem and $$ x(t) $$ denotes the partial interconnected state related to other subsystems of the large-scale system. $$ \bar{u}_{i} (x_{i}(t))\in{\mathbb R}^{m_{i}} $$ is the control input and $$ v_{i}(x_{i}(t))\in{\mathbb R}^{q_{i}} $$ is the external disturbance input. As for the $$ i $$th subsystem, we denote $$ f_{i}(x_{i}(t)) $$, $$ g_{i}(x_{i}(t)) $$, $$ h_{i}(x_{i}(t)) $$, and $$ \bar{Z}_{i}(x(t)) $$ as the nonlinear internal dynamics, the input gain matrix, the disturbance gain matrix, and the interconnected item in sequence. Besides, $$ {[{x}^{\mathit{T}}_{1}, {x}^{\mathit{T}}_{2}, \dotsc, {x}^{\mathit{T}}_{N}]}^{\mathit{T}}\in{\mathbb R}^{n} $$ denotes the whole state of the large-scale system Equation (1), where $$ n=\sum_{i=1}^{N}n_{i} $$. Accordingly, $$ {x_{1}, x_{2}, \dotsc, x_{N}} $$ are named local states and $$ \bar{u}_{1}(x_{1}), \bar{u}_{2}(x_{2}), \dotsc, \bar{u}_{N}(x_{N}) $$ are called local controllers. We let $$ R_{i}\in{\mathbb R}^{m_{i}\times m_{i}} $$ be the symmetric positive definite matrix and denote $$ Z_{i}(x(t))={R_{i}}^{{1}/{2}}\bar{Z}_{i}(x(t)) $$. In addition, $$ {Z}_{i}(x(t))\in{\mathbb R}^{m_{i}} $$ is bounded as follows:

where $$ j={1, 2, \dotsc, N} $$, $$ \alpha_{ij} $$ is the nonnegative constant, $$ \theta_{ij}(x_{j}) $$ is the positive semidefinite function. Besides, we define $$ \theta_{j}(x_{j})=\max{\{\theta_{1j}(x_{j}), \theta_{2j}(x_{j}), \dotsc, \theta_{Nj}(x_{j})\}} $$ and the element of $$ {\{\theta_{1j}(x_{j}), \theta_{2j}(x_{j}), \dotsc, \theta_{Nj}(x_{j})\}} $$ will not reach zero at the same time. For this reason, $$ \beta_{ij}\geq \alpha_{ij}\theta_{ij}(x_{j})/\theta_{j}(x_{j}) $$ holds, where $$ \beta_{ij} $$ is also the nonnegative constant.

In this paper, considering the nonlinear system Equation (1), a reference system is introduced as follows:

where $$ r_{i}(t)\in{\mathbb R}^{n_{i}} $$ denotes the desired trajectory with $$ r_{i}(0) = r_{i0} $$, the function $$ \zeta_{i} $$ is locally Lipschitz continuous satisfying $$ \zeta_{i}(0)=0 $$. For the $$ i $$th subsystem, the trajectory tracking error can be defined as $$ e_{i}(t)=x_{i}(t)-r_{i}(t) $$ with $$ e_{i}(0) = e_{i0} $$. Thus, the dynamics of the tracking error is

Noticing $$ x_{i}(t)=e_{i}(t)+r_{i}(t) $$, we define the augmented subsystem states as $$ y_{i}(t)=[{e}^{\mathit{T}}_{i}(t), {r}^{\mathit{T}}_{i}(t)]^{\mathit{T}}\in{\mathbb R}^{2n_{i}} $$ with $$ y_{i}(0)=y_{i0}=[{e}^{\mathit{T}}_{i0}, {r}^{\mathit{T}}_{i0}]^{\mathit{T}} $$. Hence, the dynamic of the $$ i $$th ATIS based on Equations (1) and (3) can be formulated as a concise form

where $$ \mathcal{F}_{i}(y_{i}(t))\in{\mathbb R}^{2n_{i}} $$, $$ \mathcal{G}_{i}(y_{i}(t))\in{\mathbb R}^{2n_{i}\times m_{i}} $$, and $$ \mathcal{H}_{i}(y_{i}(t))\in{\mathbb R}^{2n_{i}\times q_{i}} $$ respectively. Specifically, they can be expressed as

We aim to design a pair of decentralized control policies $$ \bar{u}_{1}, \bar{u}_{2}, \dotsc, \bar{u}_{N} $$ to ensure that large-scale system Equation (1) can track the desired object while being restricted by external disturbances. It means that as $$ t\to +\infty $$, $$ \|x_{i}(t)-r_{i}(t)\|\to 0 $$. Meanwhile, it is noteworthy that the control pair $$ \bar{u}_{1}, \bar{u}_{2}, \dotsc, \bar{u}_{N} $$ should be pointed out only as a corresponding controller with the local information. In what follows, it presents the DTC problem by transforming it into the optimal controller design of ATISs by considering an appropriate discounted cost function.

3. DTC DESIGN VIA OPTIMAL REGULATION

4. OPTIMAL DTC DESIGN VIA NEURAL NETWORKS

4.2. Improved critic learning rule via neural networks

Herein, an additional Lyapunov function is introduced for the purpose of improving the critic learning mechanism. Then, the following rational assumption is given.

Assumption 1 Consider the dynamic of the $$ i $$th ATIS Equation (9) with the optimal cost function Equation (14) and the closed-loop optimal control policy Equation (32). We select $$ J_{si}(y_{i}) $$ as a continuously differentiable Lyapunov function and have the following relation:

(42) $$ \begin{align} \dot{J}_{si}(y_{i})=(\nabla J_{si}(y_{i}))^{\mathit{T}}[\mathcal{F}_{i}(y_{i})+\mathcal{G}_{i}(y_{i}){u}_{i}^*(y_{i}) +\mathcal{H}_{i}(y_{i})v_{i}^*(y_{i})]<0. \end{align} $$

In other words, there exists a positive definite matrix $$ \mathscr{B} $$ such that

(43) $$ \begin{align} (\nabla J_{si}(y_{i}))^{\mathit{T}}[\mathcal{F}_{i}(y_{i})+\mathcal{G}_{i}(y_{i}){u}_{i}^*(y_{i}) +\mathcal{H}_{i}(y_{i})v_{i}^*(y_{i})] =-(\nabla J_{si}(y_{i}))^{\mathit{T}}\mathscr{B}\nabla J_{si}(y_{i})\leq -s_{mi}\|\nabla J_{si}(y_{i})\|^2, \end{align} $$

where $$ s_{mi} $$ is the minimum eigenvalue of the matrix $$ \mathscr{B} $$.

Remark 2.Herein, the motivation of selecting the cost function $$ J_{si}(y_{i}) $$ is to obtain the optimal DTC strategy, which can minimize and maximize $$ J_{si}(y_{i}) $$ under the optimal control law and the worst disturbance law. Moreover, we can discuss the stability of closed-loop systems by the constructed optimal cost function. Besides, just to be clear, $$ J_{si}(y_{i}) $$ is derived by properly selecting the quadratic polynomial in terms of the state vector. We generally choose $$ J_{si}(y_{i})=0.5{y_{i}}^{\mathit{T}}y_{i} $$.

When the condition occurs, that is, $$ (\nabla J_{si}(y_{i}))^{\mathit{T}}[\mathcal{F}_{i}(y_{i})+\mathcal{G}_{i}(y_{i}){u}_{i}^*(y_{i}) +\mathcal{H}_{i}(y_{i})v_{i}^*(y_{i})] $$$$ >0 $$, which means the system is in an unstable state under the optimal control law Equation (36). In this case, an additional term is introduced to ensure the system stability. Based on Equation (36), some processing is performed as follows:

(44) $$ \begin{align} &\frac{-\partial[(\nabla J_{si}(y_{i}))^{\mathit{T}}(\mathcal{F}_{i}(y_{i})+\mathcal{G}_{i}(y_{i}){u}_{i}^*(y_{i}) +\mathcal{H}_{i}(y_{i})v_{i}^*(y_{i}))]}{\partial \hat{w}_{ci}}\\ &=\bigg(\frac{\partial \hat u_{i}^*(y_{i})}{\partial \hat{w}_{ci}}\bigg)^{\mathit{T}}\frac{-\partial[(\nabla J_{si}(y_{i}))^{\mathit{T}}(\mathcal{F}_{i}(y_{i})+\mathcal{G}_{i}(y_{i}){u}_{i}^*(y_{i}) +\mathcal{H}_{i}(y_{i})v_{i}^*(y_{i}))]}{\partial \hat u_{i}^*(y_{i})} \\ &=\frac{1}{2}\nabla \sigma_{ci}(y_{i})\mathcal{G}_{i}(y_{i})R_{i}^{-1}{\mathcal{G}}^{\mathit{T}}_{i}(y_{i}) \nabla J_{si}(y_{i}). \end{align} $$

Thus, we describe the improved learning rule as

(45) $$ \dot{\hat{w}}_{ci}= -\eta_{ci}\frac{\phi_{i}}{(1+\phi^{\mathit{T}}_{i}\phi_{i})^2}e_{ci}+\frac{1}{2}\eta_{si}\Pi_{i}(y_{i}, \hat{u}_{i}^*, \hat{v}_{i}^*)\nabla \sigma_{ci}(y_{i})\mathcal{G}_{i}(y_{i})R_{i}^{-1}{\mathcal{G}}^{\mathit{T}}_{i}(y_{i}) \nabla J_{si}(y_{i}), $$

where $$ \eta_{si}>0 $$ represents the additional learning rate with respect to the stabilising term and $$ \Pi_{i}(y_{i}, \hat{u}_{i}^*, \hat{v}_{i}^*) $$ stands for the adaptation parameter term that tests the stability of the ATIS. The definition of $$ \Pi_{i} $$ is as follows:

(46) $$ \Pi_{i}(y_{i}, \hat{u}_{i}^*, \hat{v}_{i}^*)=\left\lbrace \begin{array}{cl} 0, & \text{if}\ \dot{J}_{si}^*(y_{i})<0, \\ 1, & \text{else}.\end{array}\right. $$

It is found that when the derivative of $$ J_{si}(y_{i}) $$ satisfies $$ \dot{J}_{si}(y_{i})<0 $$, the latter term of the weight update rule does not play its role so that the update mode is still the traditional normalized steepest descent algorithm. When $$ \dot{J}_{si}(y_{i})>0 $$, the latter term of the weight update rule starts to play its role of ensuring the stability, that is, the improved weight update method is adopted. It can be seen that the system can be adjusted to be stable under the improved weight updating criterion. Moreover, in order to clearly highlight that we have achieved the elimination of the initial admissible control law, herein, we set the initial weight vector to zero. Through the new critic learning rule, the structure of the proposed DTC strategy for ATIS is performed in Figure 1.

Figure 1. Control structure of the ATIS. ATIS: augmented tracking isolated subsystem。

In accordance to $$ \dot{\tilde{w}}_{ci}=-\dot{\hat{w}}_{ci} $$ and Equation (39), the specific form of $$ {\tilde{w}}_{ci} $$ is derived. Then, we can convert the estimated weight $$ \hat{w}_{ci} $$ into the form of the weight vector $$ {w}_{ci} $$ and the error weight vector $$ \tilde{w}_{ci} $$, which can be employed by proving the state $$ y_{i} $$ and the weight estimation error $$ \tilde{w}_{ci} $$ are UUB for the closed-loop system.

5. SIMULATION EXPERIMENT

In this section, we will introduce the common mechanical vibration system, that is, the spring-mass-damper system. The structural diagram of the mechanical system is shown in Figure 2. From it, $$ M_{1} $$ and $$ M_{2} $$ denote the mass of two objects, $$ K_{1} $$, $$ K_{2} $$, and $$ K_{3} $$ represent the stiffness constants of three springs. $$ C_{1} $$, $$ C_{2} $$, and $$ C_{3} $$ stand for the damping, respectively.

Figure 2. Simple diagram of the interconnected mass–spring–damper system.

In addition, let $$ P_{i} $$, $$ V_{i} $$, $$ F_{i} $$, and $$ f_{\mu_{i}} $$ be the position, the velocity, the force, and the friction applied to the object, where $$ i=1, 2 $$. Hence, the system dynamics for $$ M_{1} $$ and $$ M_{2} $$ are as follows:

For the object $$ M_{1} $$, we define $$ x_{11}=P_{1} $$, $$ x_{12}=V_{1} $$, $$ \bar{u}_{1}(x_{1})=F_{1} $$, and $$ {v}_{1}(x_{1})=f_{\mu_{1}} $$. In the same way, we let $$ x_{21}=P_{2} $$, $$ x_{22}=V_{2} $$, $$ \bar{u}_{2}(x_{2})=F_{2} $$, and $$ {v}_{2}(x_{2})=f_{\mu_{2}} $$ for the object $$ M_{2} $$. Next, the state-space of the spring-mass-damper system is written as

and

where $$ x_{1}=[x_{11}, x_{12}]^{\mathit{T}}\in{\mathbb R}^{2} $$ and $$ x_{2}=[x_{21}, x_{22}]^{\mathit{T}}\in{\mathbb R}^{2} $$ are system states. $$ \bar{u}_{1}(x_{1})\in{\mathbb R} $$, $$ \bar{u}_{2}(x_{2})\in{\mathbb R} $$, $$ v_{1}(x_{1})\in{\mathbb R} $$, and $$ v_{2}(x_{2})\in{\mathbb R} $$ are control inputs and disturbance inputs of the subsystem 1 and the subsystem 2, respectively. Simultaneously, $$ Z_{1}(x)=K_{2}\left(x_{21}-x_{11}\right)+C_{2}\left(x_{22}-x_{12}\right) $$ and $$ Z_{2}(x)=K_{2}\left(x_{11}-x_{21}\right)+C_{2}\left(x_{12}-x_{22}\right) $$, which indicates the spring $$ K_{2} $$ and the damping $$ C_{2} $$ play a connecting role for two subsystems. Herein, we let $$ \theta_{1}(x_{1})= ||x_{1}|| $$ and $$ \theta_{2}(x_{2})= |x_{22}| $$. Besides, we choose $$ \beta_{11}=\beta_{12}=1 $$, $$ \beta_{21}=\beta_{22}=1/2 $$, and $$ {\mu_{1}}={\mu_{2}}=1 $$. Moreover, we select $$ \lambda_{1}=\lambda_{2}=0.6 $$, $$ \varrho_{1}=\varrho_{2}=1 $$, $$ R_{1}=R_{2}=2 $$, and $$ Q_{1}=Q_{2}=2I_{4} $$, where $$ I_{4} $$ is the four-dimensional identity matrix. Above all, the desired reference trajectories $$ r_{1} $$ and $$ r_{2} $$ for two subsystems are generated by the following command system:

where $$ r_{1}=[r_{11}, r_{12}]^{\mathit{T}}\in{\mathbb R}^{2} $$ and $$ r_{2}=[r_{21}, r_{22}]^{\mathit{T}}\in{\mathbb R}^{2} $$ are reference states. Then, we define the tracking errors as $$ e_{i1}=x_{i1}-r_{i1} $$ and $$ e_{i2}=x_{i2}-r_{i2} $$. Hence, the augmented state vector can be expressed as $$ y_{i}=[y_{i1}, y_{i2}, y_{i3}, y_{i4}]^{\mathit{T}}=[e_{i1}, e_{i2}, r_{i1}, r_{i2}]^{\mathit{T}} $$, $$ i=1, 2 $$. We set practical parameters as $$ M_{1}=1 $$kg, $$ K_{1}=3 $$N/m, and $$ C_{1}=0.5 $$Ns/m for the subsystem 1. Similarly, we let $$ M_{2}=2 $$kg, $$ K_{3}=5 $$N/m, and $$ C_{3}=1 $$Ns/m for the subsystem 2. Considering Equations (51-53), the augmented system dynamics $$ \dot{y}_{1} $$ and $$ \dot{y}_{2} $$ can be obtained in the following forms:

and

Based on the online ADP algorithm, two critic networks are constructed as follows:

and

During the online learning process, we take basic learning rates and additional learning rates as $$ \eta_{c1}=0.01 $$, $$ \eta_{c2}=0.03 $$ as well as $$ \eta_{s1}=\eta_{s2}=0.01 $$. Let initial system states and reference states be $$ x_{10}=[1.5, 0]^{\mathit{T}} $$, $$ x_{20}=[1, -1]^{\mathit{T}} $$, and $$ r_{10}=r_{20}=[0.5, -0.5]^{\mathit{T}} $$, respectively. Therefore, initial states of the ATIS are $$ y_{10}=[1, 0.5, 0.5, -0.5]^{\mathit{T}} $$ and $$ y_{20}=[0.5, -0.5, 0.5, -0.5]^{\mathit{T}} $$.

Herein, two probing noises are added within the beginning 400 steps to keep the persistence of excitation condition of the ATIS. The weight convergence curves are shown in Figure 3. It can be seen that the weight has converged to a certain numerical value before turning off the excitation condition, which confirms the validity of the improved weight update algorithm. Form it, we find the initial weights are selected as zero, which indicates the initial admissible control is eliminated.

Figure 3. Weights convergence process of the critic network 1 and the critic network 2.

Next, in order to make the system achieve the purpose of the optimal tracking, feedback gains are selected as $$ k_{1}=k_{2}=1 $$. Then, the DTC strategy $$ \{k_{1}\hat u_{1}^*(y_{1}), k_{2}\hat u_{2}^*(y_{2}) \}$$ can be derived from the obtained weight vector for the spring-mass-damper interconnected system. In addition, the evolution curves are shown in Figure 4, which displays the tracking control inputs and disturbance inputs for the subsystem 1 and the subsystem 2. Then, the obtained DTC strategy is applied to the controlled system for 50 s, and its tracking error trajectory curves are displayed in Figure 5. It is obvious that the tracking error curves are eventually enforced to the origin. Taken together, this simulation result verifies the effectiveness of the proposed DTC strategy.

Figure 4. Tracking control inputs and disturbance inputs for subsystem 1 and subsystem 2.

Figure 5. Tracking error trajectories for subsystem 1 and subsystem 2.

6. CONCLUSION

In this paper, the optimal DTC strategy for CT nonlinear large-scale systems with external disturbances is proposed by employing the ADP algorithm. The approximate optimal control law of the ATISs can achieve the trajectory tracking goal. Then, the establishment of the DTC strategy is derived by adding the appropriate feedback gain, whose feasibility has been proved via the Lyapunov theory. Note that all the above-mentioned results are investigated by considering a cost function with the discount. Then, only a series of single critic networks are employed to solve HJI equations of $$ N $$ ATISs, so that we acquire the approximate optimal control law and the worst disturbance law. In addition, the stability term added in the weight updating process avoids the selection of the initial stable control policy. Furthermore, the simulation results are displayed for the spring-mass-damper system to indicate the validity of the proposed DTC method. In the future, we will utilize more advanced methods to deal with the DTC problem for nonaffine systems. Besides, we can also consider the unmatched interconnected relationship for the DTC problem, which is a considerable direction of improved research.

DECLARATIONS