Output containment control for heterogeneous discrete-time linear descriptor multi-agent systems by a feedforward output regulation approach

Abstract

This paper studies the cooperative output containment control of the heterogeneous linear discrete-time descriptor multi-agent systems based on the output regulation theory. First, the output containment control problem is transformed into a standard cooperative output regulation problem. Second, since only a portion of the followers can directly obtain the information of the leaders, a new distributed observer is designed for each follower to estimate the combinations of the leaders’ states by using the local state containment errors. Third, two types of distributed control algorithms, as well as the corresponding sufficient conditions, are provided to ensure the achievement of the output containment control. Finally, numerical simulation results illustrate the validity of the proposed method.

Keywords

Descriptor systems heterogeneous multi-agent systems output containment control output regulation distributed observer

Introduction

Multi-agent systems (MASs) have been used in a variety of practical applications, such as path planning of small- or medium-sized unmanned aerial vehicles (UAVs) (Aggarwal and Kumar, 2020), tracking control of autonomous underwater vehicles (Zhang et al., 2020), and formation control of UAVs (Dong et al., 2015). Their widespread application has aroused the attention and interest in cooperative control problems of MASs, which aims to make all agents achieve a desired cooperative behavior, such as consensus, cooperative tracking, and formation. Containment control problem is a special cooperative tracking problem with multiple leaders, the purpose of which is to design the distributed control algorithms such that the followers could converge into the convex hull formed by the leaders eventually. The theory of containment control has been applied in many fields, such as driverless cars, urban traffic management systems, and robot formation.

In the past few years, many excellent results have been obtained for the containment control problem. In Ji et al. (2008), a hybrid Stop-Go control method was proposed, which can derive the follower-agents to a given target destination. On account of the directed and/or switching communication topology in physical systems, the results in Ji et al. (2008) were generalized to the case of fixed/switching directed communication network, where the Lyapunov method was adopted to the analysis (Cao et al., 2012). In Haghshenas et al. (2015a), the containment control problem was studied for a group of non-identical and nonlinear agents, and the distributed adaptive controllers were designed to achieve the containment motions. Furthermore, one novelty of Haghshenas et al. (2015a) lies in that a new kind of containment error was developed based on $M -$ matrix. In Wang et al. (2017), the adaptive containment control problem was further addressed under the case that the leaders have nonzero and time-varying inputs. In Zou and Xiang (2019), the event-triggered containment control problem was investigated for a group of second-order nonlinear agents, where the distributed containment protocols were designed based on the non-smooth analysis, and the event-triggered conditions were presented to exclude the Zeno behaviors. In addition, the scenario of unbounded communication delays was considered in Shen and Lam (2016), and the containment convergence was analyzed by transforming the containment problem into the stability problem of an associated error system. Recently, as a real application, a finite-time fault-tolerant containment control protocols were obtained based on the nonsingular faster terminal sliding-mode method, which can derive the states of multiple ocean bottom flying nodes converge to the sliding surface in finite time (Qin et al., 2020).

Output regulation theory can be employed to achieve asymptotic tracking and/or disturbance attenuation of a class of autonomous system (also called exosystem) (Huang, 2004). To date, there have been many results of the cooperative control for MASs under the distributed output regulation framework, such as leaderless output synchronization based on internal model principle in Wieland et al. (2011), output consensus of MASs with structural uncertainties in Li et al. (2015), event-triggered cooperative robust practical output regulation in Liu and Huang (2017), and adaptive cooperative output regulation in Liu and Huang (2018). In recent years, output regulation-based containment control has also become an important issue in MASs (relevant work can be found in Chu et al., 2015; Haghshenas et al., 2015b; Li et al., 2019; Liang et al., 2017; Yuan et al., 2018; Zou and Xiang, 2017; Zuo et al., 2017a, 2017b). In Haghshenas et al. (2015b), the containment control problem for heterogeneous linear MASs was studied by resorting to the containment error presented in Haghshenas et al. (2015a). The similar problem was also considered in Chu et al. (2015) with the adaptive design and the dynamic compensator techniques. Furthermore, Zou and Xiang (2017) proposed the event-triggered containment control protocols. In these works, the containment error was defined based on the neighbor state information and the feedforward-based output regulation method was adopted. Very recently, the output containment control problem and optimal robust output containment problem were investigated for heterogeneous linear MASs in Zuo et al. (2017a, 2017b) using internal model principle, respectively, where the agents may have different dynamics and state dimensions, and the small gain theorem was applied to ensure the global asymptotic stability. In addition, the containment control problem was formulated as several independent $H_{\infty}$ stabilization sub-problems in Yuan et al. (2018), and the distributed controller synthesis conditions were eventually obtained by convex optimization. For containment control of discrete-time MASs, Liang et al. (2017) presented a dynamic output feedback containment controller based on a distributed discrete-time compensator and a distributed internal model compensator. Moreover, Li et al. (2019) adopted the power series approach to address the complex regulator equations encountered in nonlinear discrete-time MASs.

Descriptor system is a more natural representation of the real world (Yip and Sincovec, 1981), which has attracted more and more researchers to conduct research deeply on the consensus and cooperative tracking problems of descriptor MASs (Gao et al., 2020; Lu et al., 2019; Yang and Liu, 2012, 2014). The necessary and sufficient consensus conditions for descriptor MASs were provided in Yang and Liu (2012). By introducing the dynamic compensators, the similar consensus problem of descriptor MASs was dealt with in Yang and Liu (2014). The guaranteed-performance consensus of nonlinear descriptor MASs was further considered in Gao et al. (2020). The cooperative optimal preview tracking problem of descriptor MASs was investigated in Lu et al. (2019). In addition, some tracking consensus problems and containment control problems of descriptor MASs can be converted into the cooperative output regulation problems (e.g. Cong et al., 2018; Liang et al., 2020; Ma et al., 2015; Zhang et al., 2020). In particular, the distributed containment control problems of heterogeneous descriptor MASs were addressed based on feedforward control technique and internal model principle in Cong et al. (2018) and Liang et al. (2020), respectively. Furthermore, as a development of Ma et al. (2015) and Cong et al. (2018), Zhang et al. (2020) provided two kinds of distributed containment control strategies: state feedback control and reduced-order normal observer-based output feedback control.

Notice that the works of Cong et al. (2018) and Zhang et al. (2020) concentrated on the continuous-time setting and the state containment control problems. It turns out that the regulator equation and the distributed observer play an important role in designing a distributed dynamic control strategy based on feedforward output regulation approach. We are concerned in this paper with the output containment problem for discrete-time descriptor linear MASs under a directed communication topology. Compared with state containment control of continuous-time heterogeneous singular MASs (Zhang et al., 2020), there exist two challenges for addressing the concerned problem in this paper.

First, output containment control requires more complicated generalized output regulator equation than that for state containment problem. Moreover, the existence and computation of the solution associated with the regulator equation are two important problems in the application of distributed feedforward output regulation theory to output containment control.

Second, when using feedforward output regulation approach to deal with the distributed containment problem, a distributed observer is always needed to asymptotically estimate the convex combination of the leaders’ states. In Zhang et al. (2020), the distributed observer for continuous-time singular MASs exists provided that the observer coupling gain is sufficiently large. We consider its discrete counterpart and deduce the sufficient conditions for achieving asymptotical estimation, which can be found in Remark 3. However, it can be observed that a sufficiently large observer coupling gain is not suitable for discrete-time case. Besides, the modulus of the eigenvalues corresponding to the communication matrix are greater than a certain value, which may limit the chosen of communication weights. Moreover, although previous studies have also done good jobs for designing discrete distributed observer to estimate leaders’ states (e.g. Li et al., 2019; Liang et al., 2017), the results in Li et al. (2019) were only applied for the scenario that the state matrix of the exosystem is neutrally stable, and Liang et al. (2017) required that the spectrum of communication matrix is a subset of the given region, which is similar to the results presented in Remark 3.

Corresponding to the above difficulties and challenges, this paper has two new features:

According to the restricted system equivalent relations, a sufficient condition is given to guarantee the existence of the solution, meanwhile a constructive method is also presented to calculate it.

A new discrete distributed observer is proposed for each follower, with the help of the discrete modified algebraic Riccati equation and two free parameters, it is proved that the above-mentioned limitation subjected to discrete distributed observer can be relaxed.

Notations : $R^{n \times n}$ and $C^{n \times n}$ denote the set of $n \times n$ real matrices and complex matrices, respectively. $‖ x ‖$ denotes the Euclidean norm of a vector $x \in R^{n}$ . $A \otimes B$ denotes the Kronecker product of matrices $A$ and $B$ . $1_{N}$ means a column vector with all the elements equal to $1$ . $Λ (•)$ denotes the eigenvalues set of a matrix. $Re (λ)$ represents the real part of $λ \in C$ . Moreover, let $λ (E, A) =$ ${s | s \in C, \det (sE - A) = 0}$ .

Preliminaries and problem formulation

Basic concepts of the graph

In this paper, a directed graph (digraph) is used to model the information interaction among the agents. A digraph $G$ is an ordered pair $(ν, ε)$ , that is $G = (ν, ε)$ , where $ν = {v_{1}, \dots, v_{N}}$ represents a vertex set, and $ε \subseteq ν \times ν$ means an arc set. An arc from $v_{i}$ to $v_{j}$ is denoted by $(v_{j}, v_{i})$ . For $(v_{j}, v_{i})$ , $v_{j}$ is a neighbor of $v_{i}$ . If a digraph $G$ contains a finite nonempty sequence $ξ = v_{1} e_{1} v_{2} \dots v_{k - 1} e_{k - 1} v_{k}$ without the same vertices and arcs, then $ξ$ is called a directed path from $v_{1}$ to $v_{k}$ , where $e_{i} = (v_{i}, v_{j}) \in ε$ . A vertex that exists at least one directed path to other vertices and has no neighbor in a digraph is called the root. A digraph $G$ contains a spanning tree if there exists a root in $G$ .

For a digraph $G$ with $N + M$ vertices, the adjacent matrix is defined as $A = [a_{ij}] \in R^{(N + M) \times (N + M)}$ , where $a_{ij} > 0$ if $(v_{j}, v_{i}) \in ε$ , otherwise $a_{ij} = 0$ . The degree matrix is defined as $D = [d_{ij}] \in R^{(N + M) \times (N + M)}$ with $d_{ii} = \sum_{i = 1}^{N + M} a_{ij}$ and $d_{ij} = 0$ ( $i \neq j$ ). The Laplacian matrix is defined as $Ω = D - A$ . Here, we assume that all the leaders have no neighbor; therefore, $Ω$ can be divided as follows:

Ω = [\begin{matrix} Ω_{1} & Ω_{2} \\ 0_{N \times M} & 0_{N \times N} \end{matrix}] .

(1)

Moreover, the symbols $F \overset{Δ}{=} {1, \dots, M}$ and $R \overset{Δ}{=} {M + 1, \dots, M + N}$ are used to denote the sets of followers and leaders, respectively. For the digraph $G$ , the following assumption is made.

Assumption 1. For each follower in digraph $G$ , there exists at least one leader having a directed path to that follower.

Lemma 1. (Cao et al., 2012) Let Assumption 1 hold, then, all the eigenvalues of $Ω_{1}$ have positive real parts, each element of $- Ω_{1}^{- 1} Ω_{2}$ is non-negative, and every row sum of $- Ω_{1}^{- 1} Ω_{2}$ is 1.

Basic results of descriptor linear system

Denote a discrete-time descriptor linear system as

{\begin{matrix} Ex (k + 1) = Ax (k) + Bu (k), \\ y (k) = Cx (k) \end{matrix}

(2)

where $x (k) \in R^{n}$ , $u (k) \in R^{r}$ , and $y (k) \in R^{m}$ represent the state, control input, and measurable output, respectively. $E$ is singular with $rank (E) = q < n$ .

For system (2), the following results are required.

Definition 1. (Xu and Lam, 2006)

(i) The pair $(E, A)$ is regular if $\det (sE - A)$ is not identically zero.

(ii) The pair $(E, A)$ is causal if $\deg (\det (sE - A)) = rank (E)$ .

(iii) The pair $(E, A)$ is stable if $λ (E, A) < 1$ .

(iv) The pair $(E, A)$ is admissible if it is regular, causal, and stable.

Lemma 2. (Dai, 1989) The system (1) is regular and causal if and only if the following condition holds

rank [\begin{matrix} E & 0 \\ A & E \end{matrix}] = n + rank (E) .

Lemma 3. (Dai, 1989) The necessary and sufficient conditions for the R-controllability and R-observability of system (1), respectively, are

rank [\begin{matrix} sE - A & B \end{matrix}] = n, \forall s \in C,;

rank [\begin{matrix} sE - A \\ C \end{matrix}] = n, \forall s \in C, s is finite .

Lemma 4. (Gao et al., 2016) Suppose that $(E, A)$ is regular and causal, and $(E, A, B)$ is R-controllable. Then for any given positive definite matrices $Q_{1}$ and $R_{1}$ , there exists a constant $δ_{c} \in [0, 1)$ , such that the modified discrete-time generalized algebraic Riccati equation

A^{T} P_{1} A - E^{T} P_{1} E - δ_{1} A^{T} P_{1} B (R_{1} + B^{T} P_{1} B)^{- 1} B^{T} P_{1} A + E^{T} Q_{1} E = 0

(3)

has at least one positive semi-definite solution $P_{1}$ for any given $1 \geq δ_{1} > δ_{c}$ .

Remark 1. Under the conditions given by Lemma 4, if the gain matrix is chosen as $K = (R_{1} + B^{T} P_{1} B)^{- 1} B^{T} P_{1} A$ , then there exists feedback control law $u (k) = - Kx (k)$ such that $Ex (k + 1) = (A - BK) x (k)$ is admissible.

Problem formulation

The dynamic behavior of each follower is described by the following descriptor state space equation:

{\begin{matrix} E_{i} x_{i} (k + 1) = A_{i} x_{i} (k) + B_{i} u_{i} (k), \\ y_{i} (k) = C_{i} x_{i} (k), \end{matrix} i \in F

(4)

where $E_{i}$ , $A_{i} \in R^{n \times n}$ , $B_{i} \in R^{n \times m}$ , and $C_{i} \in R^{d \times n}$ ; $x_{i} (k) \in R^{n}$ , $u_{i} (k) \in R^{m}$ , and $y_{i} (k) \in R^{d}$ represent the state, control input, and measurable output of $i$ th follower, respectively. Here, $E_{i}$ is assumed to be singular with $rank (E_{i}) = \hat{p} < n$ .

The output signal of each leader (also called exosystems) is generated by the following linear autonomous systems:

{\begin{matrix} w_{l} (k + 1) = S w_{l} (k), \\ r_{l} (k) = Γ w_{l} (k), \end{matrix} l \in ℛ

(5)

where $w_{l} (k) \in R^{p}$ and $r_{l} (k) \in R^{d}$ express the state and the output of $l$ th leader, respectively.

Definition 2. A set $C \in R^{n}$ is convex if $(1 - λ) x + λ y \in C$ , for any $x, y \in C$ and $λ \in [0, 1]$ . The convex hull of a finite set of points $W = {w_{1}, w_{2}, . . ., w_{q}}$ is defined as

Co (W) = {\sum_{i = 1}^{q} α_{i} w_{i} | w_{i} \in W, α_{i} \in R, α_{i} > = 0, \sum_{i = 1}^{q} α_{i} = 1}

which is the minimal convex set containing all the points in $W$ .

Based on convex hull, we give the definition of output containment control of descriptor MASs (4).

Definition 3. For the heterogeneous descriptor MASs (4) and the exosystems (5), find a distributed control algorithm for each follower such that the followers’ outputs converge to the convex hull formed by those of the leaders, or equivalently, the output containment control is achieved.

Since the outputs of the leaders are not available for all the followers, the local containment errors for each follower is defined as

e_{i} (k) = \sum_{j = 1}^{M} a_{ij} [y_{i} (k) - y_{j} (k)] + \sum_{l = M + 1}^{M + N} a_{il} [y_{i} (k) - r_{l} (k)]

(6)

Let

e (k) = {[e_{1}^{T} (k), \dots, e_{M}^{T} (k)]}^{T}, x (k) = {[x_{1}^{T} (k), \dots, x_{M}^{T} (k)]}^{T},

y (k) = {[y_{1}^{T} (k), \dots, y_{M}^{T} (k)]}^{T}, r (k) = {[r_{M + 1}^{T} (k), \dots, r_{M + N}^{T} (k)]}^{T}

Then, equation (6) can be organized as the following compact form:

\begin{matrix} e (k) = (Ω_{1} \otimes I_{d}) y (k) + (Ω_{2} \otimes I_{d}) r (k) \\ = (Ω_{1} \otimes I_{d}) [y (k) + (Ω_{1}^{- 1} Ω_{2} \otimes I_{d}) r (k)] \end{matrix}

(7)

It follows from Lemma 1 that $Ω_{1}$ is nonsingular under Assumption 1, therefore,

lim_{k \to + \infty} e (k) = 0 \Leftrightarrow lim_{k \to + \infty} y (k) = lim_{k \to + \infty} [- (Ω_{1}^{- 1} Ω_{2} \otimes I_{d}) r (k)]

(8)

Note that each row sum of $- Ω_{1}^{- 1} Ω_{2}$ is 1, if $lim_{k \to + \infty} e (k) = 0$ holds, then the descriptor MASs (4) can achieve the output containment control with respect to the exosystems (5).

In the following, we list some assumptions about descriptor MASs (4) and exosystems (5).

Assumption 2. $(E_{i}, A_{i})$ is regular and causal for any $i \in F$ .

Assumption 3. $(E_{i}, A_{i}, B_{i})$ is R-controllable and $(E_{i}, A_{i}, C_{i})$ is R-observable, for any $i \in F$ .

Assumption 4. All the eigenvalues of matrix $S$ are not located in the unit circle of the complex plane.

Assumption 5. For any $λ \in Λ (S)$ , the following rank condition holds

rank [\begin{matrix} A_{i} - λ E_{i} & B_{i} \\ C_{i} & 0 \end{matrix}] = n + d (full row rank) .

(9)

Remark 2. In Assumption 2, the regularity of $(E_{i}, A_{i})$ guarantees the existence of the solution of the system (3). The R-controllability of $(E_{i}, A_{i}, B_{i})$ and R-observability of $(E_{i}, A_{i}, C_{i})$ in Assumption 3 can be utilized to determine the gain matrices of the distributed controllers and distributed observers. Assumptions 4 and 5 are standard in output regulation theory of descriptor systems.

To solve the output containment problem by output regulation theory, as shown in Huang (2004), it is necessary to discuss the sufficient conditions for the generalized output regulator equations and the existence of their solutions. Based on Assumptions 2 and 5, we have the following lemma.

Lemma 5. For matrices $E_{i}$ , $A_{i}$ , $B_{i}$ , and $C_{i}$ with appropriate dimensions, if Assumptions 2 and 5 hold, the following generalized output regulator equation

{\begin{matrix} E_{i} Π_{i} S = A_{i} Π_{i} + B_{i} U_{i} \\ 0 = C_{i} Π_{i} - Γ \end{matrix}

(10)

exists solutions $(Π_{i}, U_{i})$ , $i \in F$ .

Proof . If $(E_{i}, A_{i})$ is regular and causal, then according to Dai (1989), there exist non-singular matrices ${\bar{Q}}_{i}$ and ${\bar{P}}_{i}$ satisfying

{\bar{Q}}_{i} E_{i} {\bar{P}}_{i} = [\begin{matrix} I_{\hat{p}} \\ 0 \end{matrix}], {\bar{Q}}_{i} A_{i} {\bar{P}}_{i} = [\begin{matrix} {\bar{A}}_{i} \\ I_{n - \hat{p}} \end{matrix}], {\bar{Q}}_{i} B_{i} = [\begin{matrix} B_{1 i} \\ B_{2 i} \end{matrix}], C_{i} {\bar{P}}_{i} = [\begin{matrix} C_{1 i} & C_{2 i} \end{matrix}]

(11)

First, using ${\bar{Q}}_{i}$ and ${\bar{P}}_{i}$ to perform the elementary transformation for the matrices in rank condition (9) yields

\begin{array}{l} rank ({\bar{V}}_{i}) = rank ([\begin{matrix} {\bar{Q}}_{i} & 0 \\ 0 & I \end{matrix}] [\begin{matrix} A_{i} - λ E_{i} & B_{i} \\ C_{i} & 0 \end{matrix}] [\begin{matrix} {\bar{P}}_{i} & 0 \\ 0 & I \end{matrix}]) \\ = rank [\begin{matrix} {\bar{A}}_{i} - λ I_{\hat{p}} & 0 & B_{1 i} \\ 0 & I_{n - \hat{p}} & B_{2 i} \\ C_{1 i} & C_{2 i} & 0 \end{matrix}] = n + d . \end{array}

Second, multiplying the second column of ${\bar{V}}_{i}$ by $- B_{2 i}$ and adding it to the third column gives

[\begin{matrix} {\bar{A}}_{i} - λ I_{\hat{p}} & 0 & B_{1 i} \\ 0 & I_{n - \hat{p}} & B_{2 i} \\ C_{1 i} & C_{2 i} & 0 \end{matrix}] \to [\begin{matrix} {\bar{A}}_{i} - λ I_{\hat{p}} & 0 & B_{1 i} \\ 0 & I_{n - \hat{p}} & 0 \\ C_{1 i} & C_{2 i} & - C_{2 i} B_{2 i} \end{matrix}] .

Finally, for the right matrices of the above formula, multiplying the second row by $- C_{2 i}$ and adding it to the third row gets

[\begin{matrix} {\bar{A}}_{i} - λ I_{\hat{p}} & 0 & B_{1 i} \\ 0 & I_{n - \hat{p}} & 0 \\ C_{1 i} & C_{2 i} & - C_{2 i} B_{2 i} \end{matrix}] \to [\begin{matrix} {\bar{A}}_{i} - λ I_{\hat{p}} & 0 & B_{1 i} \\ 0 & I_{n - \hat{p}} & 0 \\ C_{1 i} & 0 & - C_{2 i} B_{2 i} \end{matrix}] .

According to the property that elementary transformation of a matrix does not change the rank, therefore equation (9) is equivalent to

rank [\begin{matrix} A_{i} - λ E_{i} & B_{i} \\ C_{i} & 0 \end{matrix}] = rank [\begin{matrix} {\bar{A}}_{i} - λ I_{\hat{p}} & B_{1 i} \\ C_{1 i} & - C_{2 i} B_{2 i} \end{matrix}] + n - \hat{p},

that is

rank [\begin{matrix} {\bar{A}}_{i} - λ I_{\hat{p}} & B_{1 i} \\ C_{1 i} & - C_{2 i} B_{2 i} \end{matrix}] = \hat{p} + d (full row rank),

where $λ \in Λ (S)$ . Under the above rank condition, if Assumption 4 holds, then it follows from Huang (2004) that the following regulator equation

{\begin{matrix} Π_{1 i} S = {\bar{A}}_{i} Π_{1 i} + B_{1 i} U_{i} \\ 0 = C_{1 i} Π_{1 i} - C_{2 i} B_{2 i} U_{i} - Γ \end{matrix}

(12)

exists unique solution $(Π_{1 i}, U_{i})$ , $i \in F$ . Furthermore, equation (12) can be reorganized as

{\begin{matrix} [\begin{matrix} I_{\hat{p}} \\ 0 \end{matrix}] [\begin{matrix} Π_{1 i} \\ - B_{2 i} U_{i} \end{matrix}] S = [\begin{matrix} {\bar{A}}_{i} \\ I_{n - \hat{p}} \end{matrix}] [\begin{matrix} Π_{1 i} \\ - B_{2 i} U_{i} \end{matrix}] + [\begin{matrix} B_{1 i} \\ B_{2 i} \end{matrix}] U_{i}, \\ 0 = [\begin{matrix} C_{1 i} & C_{2 i} \end{matrix}] [\begin{matrix} Π_{1 i} \\ - B_{2 i} U_{i} \end{matrix}] - Γ \end{matrix}

(13)

Choosing $Π_{i}$ as $Π_{i} = {\bar{P}}_{i} [\begin{matrix} Π_{1 i} \\ - B_{2 i} U_{i} \end{matrix}]$ and using equation (11), then it follows from equation (13) that

{\begin{matrix} {\bar{Q}}_{i}^{- 1} [\begin{matrix} I_{\hat{p}} \\ 0 \end{matrix}] {\bar{P}}_{i}^{- 1} Π_{i} S = {\bar{Q}}_{i}^{- 1} [\begin{matrix} {\bar{A}}_{i} \\ I_{n - \hat{p}} \end{matrix}] {\bar{P}}_{i}^{- 1} Π_{i} + {\bar{Q}}_{i}^{- 1} [\begin{matrix} B_{1 i} \\ B_{2 i} \end{matrix}] U_{i}, \\ 0 = [\begin{matrix} C_{1 i} & C_{2 i} \end{matrix}] {\bar{P}}_{i}^{- 1} Π_{i} - Γ . \end{matrix}

(14)

Obviously, the equation (14) is just the generalized output regulator equation (10). The conclusion of Lemma 5 is proved.

Let $E = diag {E_{1}, \dots, E_{M}}$ , $A = diag {A_{1}, \dots, A_{M}}$ , $B = diag {B_{1}, \dots, B_{M}}$ , $C = diag {C_{1}, \dots, C_{M}}$ , $U = diag {U_{1}, \dots, U_{M}}$ , $Π = diag {Π_{1}, \dots, Π_{M}}$ , then equation (10) can be organized into the following compact form

{\begin{matrix} E Π (I_{M} \otimes S) = A Π + BU, \\ 0 = C Π - (I_{M} \otimes Γ) . \end{matrix}

(15)

Before proceeding further, we also need the following two lemmas.

Lemma 6. ( Sinopoli et al., 2004 ) If $(S, Γ)$ is detectable, $Q_{2}$ and $R_{2}$ are symmetric positive definite, $S$ is unstable, then there is a $λ_{c} \in [0, 1)$ that makes the following modified algebraic Riccati equation

P_{2} = S P_{2} S^{T} + Q_{2} - δ_{2} \bar{R}

(16)

has a positive definite matrix $P_{2}$ for a given $δ_{2} \in (λ_{c}, 1]$ , in which $\bar{R} = S P_{2} Γ^{T} (Γ P_{2} Γ^{T} + R_{2})^{- 1} Γ P_{2} S^{T}$

Lemma 7. If $(E, A)$ is admissible and $f (k)$ asymptotically converges to $0$ , then the discrete-time descriptor linear system

Ex (k + 1) = Ax (k) + f (k), x (t_{0}) = x_{0} \in R^{n}

(17)

is asymptotically convergent.

Proof . If $(E, A)$ is admissible, then there exist nonsingular matrices $\bar{Q}$ and $\bar{P}$ such that

\bar{Q} E \bar{P} = [\begin{matrix} I \\ 0 \end{matrix}], \bar{Q} A \bar{P} = [\begin{matrix} A_{1} \\ I \end{matrix}]

where $A_{1}$ is Schur (stable) (Dai, 1989). Carry out a linear nonsingular transformation $[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = {\bar{P}}^{- 1} x$ for the system (17), which is equivalent to

[\begin{matrix} I \\ 0 \end{matrix}] [\begin{matrix} x_{1} (k + 1) \\ x_{2} (k + 1) \end{matrix}] = [\begin{matrix} A_{1} \\ I \end{matrix}] [\begin{matrix} x_{1} (k) \\ x_{2} (k) \end{matrix}] + [\begin{matrix} {\hat{Q}}_{1} \\ {\hat{Q}}_{2} \end{matrix}] f (k)

where $\bar{Q} = [\begin{matrix} {\hat{Q}}_{1} \\ {\hat{Q}}_{2} \end{matrix}]$ . Expanding them gives

x_{1} (k + 1) = A_{1} x_{1} (k) + {\hat{Q}}_{1} f (k), x_{10} = [\begin{matrix} I & 0 \end{matrix}] {\bar{P}}^{- 1} x_{0}

(18)

x_{2} (k) = - {\hat{Q}}_{2} f (k), x_{20} = [\begin{matrix} 0 & I \end{matrix}] {\bar{P}}^{- 1} x_{0} = - {\hat{Q}}_{2} f (0)

(19)

Note that $A_{1}$ is Schur. Moreover, it is easy to test that ${\hat{Q}}_{1} f (k)$ $\to 0$ (asymptotically) as $k \to + \infty$ . Hence, by Lemma 1 of Liu and Huang (2018), the asymptotical stability of equation (18) can be obtained immediately. According to equation (19), $x_{2} (k)$ satisfies

‖ x_{2} ‖ = ‖ - {\hat{Q}}_{2} f ‖ \leq ‖ {\hat{Q}}_{2} ‖ ‖ f ‖ .

Since $‖ {\hat{Q}}_{2} ‖$ is a constant, the asymptotical stability of $x_{2} (k)$ will completely depend on $f (k)$ , which implies that $x_{2} (k)$ also converges to zero asymptotically.

From the asymptotical stability of $x_{1}$ and $x_{2}$ , as well as

‖ x ‖_{2} = ‖ \bar{P} [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] ‖_{2} \leq ‖ \bar{P} ‖_{2} ‖ [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] ‖_{2} \leq ‖ \bar{P} ‖_{2} ({‖ x_{1} ‖}_{2} + {‖ x_{2} ‖}_{2})

it can be seen that the zero solution of equation (17) is asymptotically stable, then the proof is completed.

Main results

The output containment control can be achieved by two steps: distributed observer design and distributed dynamic controller design.

Distributed observer

When solving the output regulation problem, the state of the exosystem is usually required to be measurable (Huang, 2004). But in the digraph $G$ , not every follower can directly obtain all the information of the leaders’ states, which requires the follower to estimate the leaders’ states based on its neighbor information. Assuming that the observer’s state of each follower is $η_{i} (k)$ , similar to the equation (6), we give the following estimation error of the leaders’ states for each follower:

{\bar{e}}_{i} (k) = \sum_{j = 1}^{M} a_{ij} [η_{i} (k) - η_{j} (k)] + \sum_{l = M + 1}^{M + N} a_{il} [η_{i} (k) - w_{l} (k)], i \in F

(20)

Denote

\begin{matrix} \bar{e} (k) = {[{\bar{e}}_{1}^{T} (k), \dots, {\bar{e}}_{M}^{T} (k)]}^{T}, η (k) = {[η_{1}^{T} (k), \dots, η_{M}^{T} (k)]}^{T}, w (k) \\ = {[w_{M + 1}^{T} (k), \dots, w_{M + N}^{T} (k)]}^{T} \end{matrix}

then the equation (20) can be rewritten as the following compact form:

\begin{matrix} e (k) = (Ω_{1} \otimes I_{p}) η (k) + (Ω_{2} \otimes I_{p}) w (k) \\ = (Ω_{1} \otimes I_{p}) [η (k) + (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k)] \\ = (Ω_{1} \otimes I_{p}) \tilde{w} (k) \end{matrix}

(21)

Let $W = {w_{M + 1} (k), \dots, w_{M + N} (k)}$ , since each row sum of $- Ω_{1}^{- 1} Ω_{2}$ is 1, the convex hull $Co (W)$ can be described by the set $W$ and all the elements in each row of the matrix $- Ω_{1}^{- 1} Ω_{2}$ . If the designed observer makes $li m_{k \to + \infty} \tilde{w} (k) = 0$ hold, then $η_{i} (k)$ can realize the estimation of a convex combination ${\bar{w}}_{i} (k) \in Co (W)$ . Furthermore, based on ${\bar{e}}_{i} (k)$ , we propose the following distributed observer

η_{i} (k + 1) = S η_{i} (k) + μ L Γ {\bar{e}}_{i} (k)

(22)

where $μ$ and $L$ are free parameter and observer gain to be designed later, respectively. Based on Lemma 6, the following conclusion is obtained.

Lemma 8. Under Assumptions 1 and 2, suppose that the eigenvalues of $Ω_{1}$ are written as $λ_{i} = Re (λ_{i}) + j Im (λ_{i})$ , where $j$ represents the units of imaginary numbers, $i \in F$ . Select $L = - S P_{2} Γ^{T} (Γ P_{2} Γ^{T} + R_{2})^{- 1}$ , if $δ_{2}$ satisfies

δ_{2} < min {min_{i \in F} (\frac{λ_{min} (Q_{2})}{λ_{max} (\bar{R})} + \frac{{Re}^{2} (λ_{i})}{{| λ_{i} |}^{2}}), 1}

(23)

and $μ$ belongs to the set

φ (μ) = ⋂_{i \in F} φ_{i} (μ)

(24a)

with

φ_{i} (μ) = {μ \in R | | u - \frac{Re (λ_{i})}{{| λ_{i} |}^{2}} | < \sqrt{\frac{{Re}^{2} (λ_{i})}{{| λ_{i} |}^{4}} + \frac{1}{{| λ_{i} |}^{2}} (\frac{λ_{min} (Q_{2})}{λ_{max} (\bar{R})} - δ_{2})}}

(24b)

then for any initial states $η_{i} (0)$ ,

lim_{k \to + \infty} (η_{i} (k) - {\bar{w}}_{i} (k)) = 0, i \in F

(25)

asymptotically. Moreover, $Q_{2}$ and $\bar{R}$ are defined in (16).

Proof . Note that the equation (22) can be expressed as

η (k + 1) = [(I_{M} \otimes S) + μ (Ω_{1} \otimes L Γ)] η (k) + μ (Ω_{2} \otimes L Γ) w (k)

(26)

According to the analysis of the equation (21), if we can prove $lim_{k \to + \infty} \tilde{w} (k) = 0$ , then the lemma follows immediately. Hence, we have

\begin{matrix} \tilde{w} (k + 1) = η (k + 1) + (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k + 1) \\ = [(I_{M} \otimes S) + μ (Ω_{1} \otimes L Γ)] η (k) + μ (Ω_{2} \otimes L Γ) w (k) \\ + (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) (I_{M} \otimes S) w (k) \\ = [(I_{M} \otimes S) + μ (Ω_{1} \otimes L Γ)] η (k) + μ (Ω_{1} \otimes L Γ) \\ (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k) \\ + (I_{M} \otimes S) (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k) \\ = [(I_{M} \otimes S) + μ (Ω_{1} \otimes L Γ)] [η (k) + (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k)] \\ = [(I_{M} \otimes S) + μ (Ω_{1} \otimes L Γ)] \tilde{w} (k) . \end{matrix}

(27)

Choosing the coordinate transformation as $ξ (k) = ({\hat{T}}^{- 1} \otimes I_{p}) \tilde{w} (k)$ , where $\hat{T}$ is a nonsingular matrix, which makes $Ω_{1}$ be similar to $J$ , where $J$ is the Jordan canonical matrix of $Ω_{1}$ . Consequently, we obtain

ξ (k + 1) = [(I_{M} \otimes S) + μ (J \otimes L Γ)] ξ (k)

(28)

Note that $I_{N} \otimes S + μ (J \otimes L Γ)$ is an upper triangular matrix; therefore, the stability of the system (28) is determined by the stability of $S + μ λ_{i} L Γ$ , where $λ_{i}$ are the eigenvalues of matrix $J$ . Thereupon, under the conditions given by the theorem, the following only needs to prove that the system

ξ_{i} (k + 1) = (S + μ λ_{i} L Γ) ξ_{i} (k), i \in F

(29)

is asymptotically stable.

Note that a matrix has the same eigenvalues as its transpose, so the system (29) has the same asymptotic stability as

ξ_{i} (k + 1) = (S + μ λ_{i} L Γ)^{T} ξ_{i} (k), i \in F

(30)

With the help of the modified algebraic Riccati equation (16) in Lemma 6, the Lyapunov function candidate can be defined as

V (ξ_{i} (k)) = ξ_{i}^{T} (k) P_{2} ξ_{i} (k), i \in F

It can be easily observed that $V (ξ_{i} (k))$ is positive definite from $P_{2} = P_{2}^{T} > 0$ . Taking the difference of $V (ξ_{i} (k))$ along the trajectory of the systems (30), we have

\begin{array}{l} Δ V (ξ_{i} (k)) = ξ_{i}^{T} (k + 1) P_{2} ξ_{i} (k + 1) - ξ_{i}^{T} (k) P_{2} ξ_{i} (k) \\ = ξ_{i}^{T} (k) [(S + μ λ_{i} L Γ) P_{2} {(S + μ λ_{i} L Γ)}^{T} - P_{2}] ξ_{i} (k) \\ = ξ_{i}^{T} (k) [- Q_{2} + δ_{2} \bar{R} - 2 Re (λ_{i}) μ \bar{R} \\ + {| λ_{i} |}^{2} μ^{2} S P_{2} Γ^{T} {(Γ P_{2} Γ^{T} + R_{2})}^{- 1} (Γ P_{2} Γ^{T} + R_{2} - R_{2}) \\ {(Γ P_{2} Γ^{T} + R_{2})}^{- 1} Γ P_{2} S^{T}] ξ_{i} (k) \\ \leq ξ_{i}^{T} (k) [({| λ_{i} |}^{2} μ^{2} - 2 Re (λ_{i}) μ + δ_{2}) \bar{R} - Q_{2}] ξ_{i} (k) \\ \leq ξ_{i}^{T} (k) {[({| λ_{i} |}^{2} μ^{2} - 2 Re (λ_{i}) μ + δ_{2}) λ_{\max} (\bar{R}) - λ_{\min} (Q_{2})] \cdot I} ξ_{i} (k) \end{array}

where $\bar{R}$ is defined in Lemma 6, $λ_{max} (\bar{R})$ and $λ_{min} (Q_{2})$ represent the maximum and minimum eigenvalues of the matrices $\bar{R}$ and $Q_{2}$ , respectively. Let

g_{i} (μ) \overset{Δ}{=} {| λ_{i} |}^{2} μ^{2} - 2 Re (λ_{i}) μ + (δ_{2} - \frac{λ_{min} (Q_{2})}{λ_{max} (\bar{R})}), i \in F

Hence, if there exists a domain of $μ$ such that all of the $g_{i} (μ) < 0$ hold, then we have $Δ V (ξ_{i} (k)) < 0$ and thereby $lim_{k \to + \infty} (η_{i} (k) - {\bar{w}}_{i} (k)) = 0$ .

For a given $i$ , observing that ${| λ_{i} |}^{2} > 0$ , by using the basic properties of quadratic function, if $δ_{2}$ satisfies the inequality (23) and $μ$ belongs to the set $φ_{i} (μ)$ defined in equation (24b), then $g_{i} (μ) < 0$ hold. Furthermore, if $μ$ belongs to the intersection $φ (μ) = ⋂_{i \in F} φ_{i} (μ)$ , then $g_{i} (μ) < 0$ will hold for any $i$ . Thereafter, the asymptotic stability of the systems (28) is ensured. Furthermore, by the non-singularity of the matrix $\hat{T}$ , it can be obtained that $η (k) \to (- Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k)$ as $k \to + \infty$ . Therefore, the conclusion is proved.

Remark 3. Zhang et al. (2020) considered a distributed observer of the following form:

\begin{matrix} {\overset{\cdot}{η}}_{i} (t) = \\ S η_{i} (t) + μ F [\sum_{j = 1}^{M} a_{ij} [η_{i} (t) - η_{j} (t)] + \sum_{j = M + 1}^{M + N} a_{ij} [η_{i} (t) - w_{j} (t)]], i \in F \end{matrix}

(31)

where $η_{i} (t)$ denotes the state of the observer, $μ > 0$ is coupling gain and $F$ is feedback gain matrix. The discrete counterpart of (31) is

\begin{matrix} η_{i} (k + 1) = \\ S η_{i} (k) + μ F [\sum_{j = 1}^{M} a_{ij} [η_{i} (k) - η_{j} (k)] + \sum_{j = M + 1}^{M + N} a_{ij} [η_{i} (k) - w_{j} (k)]], i \in F \end{matrix}

(32)

According to Lemma 6, take $F = - (P_{2} + R_{2})^{- 1} P_{2} S$ , where $P_{2}$ satisfies the algebraic Riccati equation

P_{2} = S^{T} P_{2} S + Q_{2} - \hat{R}

(33)

with $\hat{R} = S^{T} P_{2} (P_{2} + R_{2})^{- 1} P_{2} S$ , $R_{2} > 0$ , and $Q_{2} > 0$ .

Similar to the proof used in Lemma 8, we show that under Assumptions 1 and 2, if

Δ_{i} = {Re}^{2} (λ_{i}) - {| λ_{i} |}^{2} [1 - \frac{λ_{min} (Q_{2})}{λ_{max} (\hat{R})}] > 0, i \in F .

(34)

and

max {\frac{Re (λ_{i}) - \sqrt{Δ_{i}}}{{| λ_{i} |}^{2}}} < μ < min {\frac{Re (λ_{i}) + \sqrt{Δ_{i}}}{{| λ_{i} |}^{2}}},

(35)

then $η_{i} (k)$ can estimate the convex combination of the leaders’ states asymptotically.

From equation (34), an easy manipulation yields

Re (λ_{i}) > Im (λ_{i}) \sqrt{\frac{1 - a}{a}}

(36)

with $a = λ_{min} (Q_{2}) / λ_{max} (\hat{R})$ . That is to say, the distributed observer (32) works under the conditions (35) and (36). However, the inequality (36) shows that the eigenvalues of the communication matrix $Ω_{1}$ have a bound. Therefore, if condition (34) does not hold, $a_{ij}$ of the matrix $Ω$ will be readjusted, which further means communication weights among agents are limited. In contrast with the distributed observer (32), we use the detectability of $(S, Γ)$ to construct the observer (22), and with the aid of the modified algebraic Riccati equation (16), the parameter $δ_{2}$ is introduced, which adds additional design freedom to relax the limitation for communication weights. This increases the scope of the application of the new observer to a certain extent.

Distributed dynamic controllers

In this subsection, two distributed controllers are proposed based on the distributed observers (22).

First, in the case that $x_{i} (k)$ can be directly measured, the distributed dynamic state feedback controller is designed as follows

u_{i} (k) = K_{1 i} x_{i} (k) + K_{2 i} η_{i} (k) .

(37)

For any proper $K_{1 i}$ , $K_{2 i}$ is given by

K_{2 i} = U_{i} - K_{1 i} Π_{i}, i \in F,

(38)

where $(Π_{i}, U_{i})$ is the solutions of the generalized output regulator equation (10).

Second, if $x_{i} (k)$ cannot be measured directly, then the distributed controller (31) will not work. Therefore, it is necessary to design an observer for each follower to estimate $x_{i} (k)$ asymptotically. Based on the controller (37), the distributed dynamic output feedback controller is designed

{\begin{matrix} u_{i} (k) = K_{1 i} {\hat{x}}_{i} (k) + K_{2 i} η_{i} (k), \\ E_{i} {\hat{x}}_{i} (k + 1) = A_{i} {\hat{x}}_{i} (k) + B_{i} u_{i} (k) + H_{i} [C_{i} {\hat{x}}_{i} (k) - y_{i} (k)] . \end{matrix}

(39)

Theorem 1. Under Assumptions 1–5, if

(a) $K_{1 i} = - (R_{1 i} + B_{i}^{T} P_{1 i} B_{i})^{- 1} B_{i}^{T} P_{1 i} A_{i}$ , $i \in F$ , where $P_{1 i}$ is positive semi-definite solution satisfying the generalized algebraic Riccati equation

A_{i}^{T} P_{1 i} A_{i} - E_{i}^{T} P_{1 i} E_{i} - δ_{1 i} A_{i}^{T} P_{1 i} B_{i} (R_{1 i} + B_{i}^{T} P_{1 i} B_{i})^{- 1} B_{i}^{T} P_{1 i} A_{i} + E_{i}^{T} Q_{1 i} E_{i} = 0 .

(40)

Here, $Q_{1 i}$ and $R_{1 i}$ are positive definite matrices, $δ_{1 i} \in (δ_{c}, 1]$ , $δ_{c} \in [0, 1)$ ;

(b) $K_{2 i} = U_{i} - K_{1 i} Π_{i}$ , where $(Π_{i}, U_{i})$ is the solution of the generalized output regulator equation (10);

(c) Distributed observer gain $L$ is taken as $L = - S P_{2} Γ^{T} (Γ P_{2} Γ^{T} + R_{2})^{- 1}$ , where $P_{2}$ is the positive definite solution of the modified algebraic Riccati equation (16);

(d) $μ$ and $δ_{2}$ are the parameters satisfying inequality (23) and the set (24), respectively, then the distributed dynamic state feedback controllers (37) can make the descriptor MASs (4) achieve the output containment control associated with the exosystems (5).

Proof . Under Assumptions 2 and 5, the generalized output regulator equation (10) holds and exists solution $(Π_{i}, U_{i})$ . Based on condition (b), substituting the distributed dynamic state feedback controller (37) into the descriptor MASs (4) gives

\begin{matrix} E_{i} x_{i} (k + 1) = A_{i} x_{i} (k) + B_{i} u_{i} (k) \\ = A_{i} x_{i} (k) + B_{i} [K_{1 i} x_{i} (k) + K_{2 i} η_{i} (k)] \\ = A_{i} x_{i} (k) + B_{i} [K_{1 i} x_{i} (k) + (U_{i} - K_{1 i} Π_{i}) η_{i} (k)] . \end{matrix}

(41)

Let $K_{1} = diag {K_{11}, \dots, K_{1 M}}$ , then we obtain the following closed-loop system

{\begin{matrix} Ex (k + 1) = (A + B K_{1}) x (k) + B (U - K_{1} Π) η (k), \\ e (k) = (Ω_{1} \otimes I_{d}) Cx (k) + (Ω_{2} \otimes I_{d}) (I_{M} \otimes Γ) w (k) . \end{matrix}

(42)

By performing the coordinate transformation $\bar{ξ} (k) = x (k) + Π (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k)$ for the system (42), we have

\begin{matrix} E \bar{ξ} (k + 1) = (A + B K_{1}) x (k) + B (U - K_{1} Π) η (k) \\ + E Π (I_{M} \otimes S) (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k) \\ = (A + B K_{1}) \bar{ξ} (k) + B (U - K_{1} Π) η (k) \\ + [E Π (I_{M} \otimes S) - (A + B K_{1}) Π] (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k), \\ e (k) = (Ω_{1} \otimes I_{d}) Cx (k) + (Ω_{2} \otimes I_{d}) (I_{M} \otimes Γ) w (k) \\ = (Ω_{1} \otimes I_{d}) [Cx (k) + (Ω_{1}^{- 1} Ω_{2} \otimes I_{d}) (I_{M} \otimes Γ) w (k)] \\ = (Ω_{1} \otimes I_{d}) [C \bar{ξ} (k) - C Π (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k) \\ + (Ω_{1}^{- 1} Ω_{2} \otimes I_{d}) (I_{M} \otimes Γ) w (k)] \\ = (Ω_{1} \otimes I_{d}) {C \bar{ξ} (k) + [(I_{M} \otimes Γ) - C Π] (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k)} . \end{matrix}

With the equation (15) and $\tilde{w} (k) = η (k) + (Ω_{1}^{- 1} Ω_{2} \otimes I_{p}) w (k)$ , we can also rewrite them as

{\begin{matrix} E \bar{ξ} (k + 1) = (A + B K_{1}) \bar{ξ} (k) + B (U - K_{1} Π) \tilde{w} (k), \\ e (k) = (Ω_{1} \otimes I_{d}) C \bar{ξ} (k) . \end{matrix}

(43)

According to Lemma 4 and Remark 1, if condition (a) holds, then $(E_{i}, A_{i} + B_{i} K_{1 i})$ , as well as $(E, A + B K_{1})$ , is admissible. In addition, it follows from Lemma 8 that $lim_{k \to + \infty} \tilde{w} (k) = 0$ asymptotically under conditions (c) and (d). Hence, by Lemma 7, we obtain that $\bar{ξ} (k)$ converges to zero asymptotically. Note that $(Ω_{1} \otimes I_{d}) C$ is a constant matrix, then $e (k)$ will also tend to zero asymptotically. This, together with the analysis of equation (8), implies the achievement of the output containment composed of the descriptor MASs (4), the exosystems (5), and the distributed dynamic state feedback controller (37). The proof is completed.

Theorem 2. Under Assumptions 1–5, if

(a) $K_{1 i} = - (R_{1 i} + B_{i}^{T} P_{1 i} B_{i})^{- 1} B_{i}^{T} P_{1 i} A_{i}$ , $i \in F$ , where $P_{1 i}$ is the positive semi-definite solution satisfying the generalized algebraic Riccati equation (40);

(b) $H_{i} = - A_{i} P_{0 i} C_{i}^{T} (R_{0 i} + C_{i} P_{0 i} C_{i}^{T})^{- 1}$ , $i \in F$ , where $P_{0 i}$ is the positive semi-definite solution satisfying the generalized algebraic Riccati equation

A_{i} P_{i} A_{i}^{T} - E_{i} P_{0 i} E_{i}^{T} - A_{i} P_{0 i} C_{i}^{T} (R_{0 i} + C_{i} P_{0 i} C_{i}^{T})^{- 1} C_{i} P_{0 i} A_{i}^{T} + E_{i} Q_{0 i} E_{i}^{T} = 0,

(44)

and $Q_{0 i}$ and $R_{0 i}$ are definite matrices;

(c) $K_{2 i} = U_{i} - K_{1 i} Π_{i}$ , where $(Π_{i}, U_{i})$ is the solution of the generalized output regulator equation (10);

(d) Distributed observer gain $L$ is taken as $L = - S P_{2} Γ^{T} (Γ P_{2} Γ^{T} + R_{2})^{- 1}$ , where $P_{2}$ is the positive definite solution of the modified algebraic Riccati equation (16);

(e) $μ$ and $δ_{2}$ are the parameters satisfying inequality (23) and the set (24), respectively, then the distributed dynamic output feedback controller (39) can make the descriptor MASs (4) achieve the output containment control associated with the exosystems (5).

Proof . Note that the second equation in (39) represents a descriptor state observer, which is designed to estimate $x_{i} (k)$ . Let

{\tilde{x}}_{i} (k) = {\hat{x}}_{i} (k) - x_{i} (k) .

then

E_{i} {\tilde{x}}_{i} (k + 1) = (A_{i} + H_{i} C_{i}) {\tilde{x}}_{i} (k),

(45)

By dual principle, if $(E_{i}, A_{i}, C_{i})$ is $R -$ observable, then $(E_{i}^{T}, A_{i}^{T}, C_{i}^{T})$ is $R -$ controllable. Hence, under Assumptions 2 and 3, it follows from Lemma 4 that the generalized algebraic Riccati equation (44) exists positive semi-definite solution. With $H_{i}$ defined in condition (b), $(E_{i}, A_{i} + H_{i} C_{i})$ is guaranteed to be admissible, which indicates that the descriptor state observer can be utilized to realize the estimation of $x_{i} (k)$ .

Let $\tilde{x} (k) = {[{\tilde{x}}_{1}^{T} (k), {\tilde{x}}_{2}^{T} (k), \dots, {\tilde{x}}_{M}^{T} (k)]}^{T}$ . Parallel to the proof of Theorem 1, by combining equations (39) and (4) and doing a series of coordinate transformation, the following closed-loop system can be derived

{\begin{matrix} E \bar{ξ} (k + 1) = (A + B K_{1}) \bar{ξ} (k) + B (U - K_{1} Π) \tilde{w} (k) + B K_{1} \tilde{x} (k), \\ e (k) = (Ω_{1} \otimes I_{d}) C \bar{ξ} (k) . \end{matrix}

(46)

Let $f (k) = B (U - K_{1} Π) \tilde{w} (k) + B K_{1} \tilde{x} (k)$ . It can be easily verified that $f (k)$ will decay to $0$ asymptotically. Similar to the discussion of the system (43), under the given conditions, we can obtain $lim_{k \to + \infty} e (k) = 0$ , which means the achievement of the output containment control. The proof is completed.

Remark 4. According to Lemma 3 in Gao et al. (2016), with the help of the formula (11), the solution $P_{1 i}$ of the generalized algebraic Riccati equation (40) can be constructed as $P_{1 i} = {\bar{Q}}_{i}^{T} [\begin{matrix} {\tilde{P}}_{1 i} \\ 0 \end{matrix}] {\bar{Q}}_{i}$ , where ${\tilde{P}}_{1 i}$ is the positive definite solution of the following reduced-order algebraic Riccati equation

{\bar{A}}_{i}^{T} {\tilde{P}}_{1 i} {\bar{A}}_{i} - {\tilde{P}}_{1}_{i} - δ_{1 i} {\bar{A}}_{i}^{T} {\tilde{P}}_{1 i} B_{1 i} (R_{1 i} + B_{1 i}^{T} {\tilde{P}}_{1 i} B_{1 i})^{- 1} B_{1 i}^{T} {\tilde{P}}_{1 i} {\bar{A}}_{i} + {\tilde{Q}}_{1 i} = 0

(47)

with ${\tilde{Q}}_{1 i} > 0$ . Furthermore, for convenience, $δ_{1 i}$ are taken as 1 in the simulation part.

Remark 5. In Theorems 1 and 2, the feedforward gains $K_{2 i}$ depends on the solution $(Π_{i}, U_{i})$ of the generalized output regulator equation (10). The proof of Lemma 5 provides a reduced-order method to construct the solution pair $(Π_{i}, U_{i})$ . In fact, if the dimensions of the MASs (4) become high or there are more agents in the communication network, then the reduced-order output regulator equation (12) and $Π_{i} = {\bar{P}}_{i} [\begin{matrix} Π_{1 i} \\ - B_{2 i} U_{i} \end{matrix}]$ can be employed to decrease the calculation complexity significantly, where ${\bar{P}}_{i}$ is defined in equation (11). However, if not, there also exists another direct method to solve them. Rewrite the equation (10) as

[\begin{matrix} E_{i} & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} Π_{i} \\ U_{i} \end{matrix}] S = [\begin{matrix} A_{i} & B_{i} \\ C_{i} & 0 \end{matrix}] [\begin{matrix} Π_{i} \\ U_{i} \end{matrix}] - [\begin{matrix} 0 \\ Γ \end{matrix}] .

(48)

Using the property of the vectorizing operator $vec (\cdot)$ in Huang (2004), the formula (48) becomes

Φ_{i} ς_{i} = b_{i}

(49)

with

Φ_{i} = S^{T} \otimes [\begin{matrix} E_{i} & 0 \\ 0 & 0 \end{matrix}] - I_{p} \otimes [\begin{matrix} A_{i} & B_{i} \\ C_{i} & 0 \end{matrix}], ς_{i} = vec [\begin{matrix} Π_{i} \\ U_{i} \end{matrix}], b_{i} = - vec [\begin{matrix} 0 \\ Γ \end{matrix}] .

According to Assumption 5, it is not difficult to verify that $Φ_{i}$ have full row rank; therefore, the linear equation (49) has a solution $ς_{i} = Φ_{i}^{T} (Φ_{i} Φ_{i}^{T})^{- 1} b_{i}$ based on the least square algorithm.

Remark 6. In Lemma 8, we can apply the following iterative formula to obtain the solution $P_{2}$ of the modified algebraic Riccati equation (16)

P_{2} (k + 1) = S P_{2} (k) S^{T} + Q_{2} - δ_{2} S P_{2} (k) Γ^{T} (Γ P_{2} (k) Γ^{T} + R_{2})^{- 1} Γ P_{2} (k) S^{T} .

(50)

Moreover, it is better to take the value of $δ_{2}$ from the side close to $λ_{c}$ , where $λ_{c} = 1 - 1 / max | λ_{i} (S) |$ , $i = 1, 2, \dots, p$ , is defined in Sinopoli et al. (2004). By substituting the chosen $δ_{2}$ into the equation (50), we also need to check if $δ_{2}$ exceeds the given bound (23) based on the obtained solution $P_{2}$ . This is very important since it decides whether there exists the set $φ (μ)$ defined in equation (24a) or not.

Numerical simulation

In this section, the validity of the distributed controllers (31) and (33) are demonstrated by the simulation results. Consider MASs with two leaders and five followers; the communication topology associated with these agents is presented in Figure 1.

The Laplacian matrix is described as follows

Ω = [\begin{matrix} 1.25 & - 0.25 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 0.4 & 0 & 0 & - 0.4 & 0 & 0 \\ - 0.45 & 0 & 0.45 & 0 & 0 & 0 & 0 \\ 0 & - 0.4 & - 0.2 & 0.6 & 0 & 0 & 0 \\ 0 & 0 & - 0.5 & 0 & 1.5 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] .

Correspondingly, according to equation (1), $Ω_{1}$ and $Ω_{2}$ are obtained

Ω_{1} = [\begin{matrix} 1.25 & - 0.25 & 0 & 0 & 0 \\ 0 & 0.4 & 0 & 0 & - 0.4 \\ - 0.45 & 0 & 0.45 & 0 & 0 \\ 0 & - 0.4 & - 0.2 & 0.6 & 0 \\ 0 & 0 & - 0.5 & 0 & 1.5 \end{matrix}], Ω_{2} = [\begin{matrix} - 1 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & - 1 \end{matrix}] .

The coefficient matrices of the descriptor MASs (4) and the exosystems (5) are selected as

S = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}], Γ = [\begin{matrix} 0.1 & 0.9 \end{matrix}],

E_{1} = [\begin{matrix} 1.8 & 0 \\ 0 & 0 \end{matrix}], E_{2} = E_{3} = E_{4} = E_{5} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], A_{1} = [\begin{matrix} 0.5 & - 0.1 \\ - 0.8 & 0.2 \end{matrix}],

A_{2} = [\begin{matrix} 0.9 & 0 \\ 1 & 0.7 \end{matrix}], A_{3} = [\begin{matrix} 0.2 & 0.5 \\ 0.92 & - 0.6 \end{matrix}], A_{4} = [\begin{matrix} 0.9 & 0 \\ 2 & 0.7 \end{matrix}], A_{5} = [\begin{matrix} 0.3 & 0.7 \\ 0.1 & - 0.6 \end{matrix}],

B_{1} = [\begin{matrix} 0.4 \\ 0.3 \end{matrix}], B_{2} = B_{4} = [\begin{matrix} 1 \\ 0 \end{matrix}], B_{3} = [\begin{matrix} 0.9 \\ - 1 \end{matrix}], B_{5} = [\begin{matrix} 0.4 \\ 0.5 \end{matrix}],

C_{1} = [\begin{matrix} 2.1 & 0.5 \end{matrix}], C_{2} = C_{4} = [\begin{matrix} 1 & 1 \end{matrix}], C_{3} = [\begin{matrix} - 0.4 & 0.2 \end{matrix}], C_{5} = [\begin{matrix} - 0.1 & 0.2 \end{matrix}] .

By Lemmas 2 and 3, it can be verified that the system dynamics of the followers are regular, causal, R-controllable, and R-observable. In addition, the eigenvalues of $S$ satisfy Assumption 4 and further calculations show that the rank conditions in Assumption 5 are satisfied. The following is the specific simulation process.

First, selecting ${\tilde{Q}}_{1 i} = 0.3 \cdot i$ and $R_{1 i} = 1.0 \cdot i$ , solving the reduced-order algebraic Riccati equations (41) and using $P_{1 i} = {\bar{Q}}_{i}^{T} [\begin{matrix} {\tilde{P}}_{1 i} \\ 0 \end{matrix}] {\bar{Q}}_{i}$ get

P_{11} = [\begin{matrix} 0.30277 & 0.15139 \\ 0.15139 & 0.07569 \end{matrix}], P_{12} = [\begin{matrix} 1.21095 & 0 \\ 0 & 0 \end{matrix}], P_{13} = [\begin{matrix} 11.14733 & 9.28945 \\ 9.28945 & 7.74120 \end{matrix}],

P_{14} = [\begin{matrix} 2.42191 & 0 \\ 0 & 0 \end{matrix}], P_{15} = [\begin{matrix} 1.72452 & 2.01194 \\ 2.01194 & 2.34726 \end{matrix}] .

Furthermore, based on $K_{1 i} = - (R_{i} + B_{i}^{T} P_{i} B_{i})^{- 1} B_{i}^{T} P_{i} A_{i}$ , $i = 1, 2, \dots, 5$ , we have

K_{11} = [\begin{matrix} - 0.01526 & 0 \end{matrix}], K_{12} = [\begin{matrix} - 0.33942 & 0 \end{matrix}], K_{13} = [\begin{matrix} - 0.23557 & 0 \end{matrix}],

K_{14} = [\begin{matrix} - 0.33942 & 0 \end{matrix}], K_{15} = [\begin{matrix} - 0.10597 & 0 \end{matrix}] .

Second, choosing $Q_{0 i} (1, 1) = 20 \cdot i$ and $R_{0 i} = 60 \cdot i$ , where $Q_{0 i} (1, 1)$ denote the elements in the first row and first column of the matrices $Q_{0 i}$ . Then, similar to the procedure in the first step, we can obtain $H_{i}$ as follows

H_{1} = [\begin{matrix} - 0.02070 \\ 0 \end{matrix}], H_{2} = [\begin{matrix} 0.33720 \\ 0 \end{matrix}], H_{3} = [\begin{matrix} 0.31146 \\ 0 \end{matrix}],

H_{4} = [\begin{matrix} 0.30205 \\ 0 \end{matrix}], H_{5} = [\begin{matrix} 0.01118 \\ 0 \end{matrix}] .

Third, utilizing the method in Remark 5, the solution pairs $(Π_{i}, U_{i})$ are computed as

Π_{1} = [\begin{matrix} 0.05612 & 0.58241 \\ - 0.03571 & - 0.64614 \end{matrix}], U_{1} = [\begin{matrix} 0.17347 & 1.98386 \end{matrix}]

Π_{2} = [\begin{matrix} - 0.23333 & - 2.1 \\ 0.33333 & 3 \end{matrix}], U_{2} = [\begin{matrix} - 0.02333 & - 0.44333 \end{matrix}],

Π_{3} = [\begin{matrix} - 0.38462 & 3.93491 \\ - 0.26923 & 12.36982 \end{matrix}], U_{3} = [\begin{matrix} - 0.19231 & - 3.80178 \end{matrix}],

Π_{4} = [\begin{matrix} - 0.05385 & - 0.48462 \\ 0.15385 & 1.38462 \end{matrix}], U_{4} = [\begin{matrix} - 0.00538 & - 0.10231 \end{matrix}],

Π_{5} = [\begin{matrix} 3.10526 & 11.60388 \\ 2.05263 & 10.30194 \end{matrix}], U_{5} = [\begin{matrix} 1.84211 & 10.04155 \end{matrix}] .

According to the solutions $(Π_{i}, U_{i})$ and $K_{2 i} = U_{i} - K_{1 i} Π_{i},$ the feedforward gains are given as

\begin{matrix} K_{21} = [\begin{matrix} 0.17433 & 1.99275 \end{matrix}], K_{22} = [\begin{matrix} - 0.10253 & - 1.15611 \end{matrix}], \\ K_{23} = [\begin{matrix} - 0.28291 & - 2.87482 \end{matrix}], K_{24} = [\begin{matrix} - 0.02366 & - 0.26680 \end{matrix}], \\ K_{25} = [\begin{matrix} 2.17118 & 11.27124 \end{matrix}] . \end{matrix}

Finally, the eigenvalues of the matrix $S$ are 1, according to Remark 6, we have $λ_{c} = 0$ . Choosing $δ_{2} = 0.4$ , $R_{2} = 0.2$ , $Q_{2} = 5 \cdot I_{2}$ , and using the iterative scheme (50) yield

P_{2} = [\begin{matrix} 156.2995 & 10.28984 \\ 10.28984 & 14.08224 \end{matrix}] .

The eigenvalues of the matrix $Ω_{1}$ are 0.8, 0.3762, $0.9396 \pm 0.4293 i$ , 1.4946. Simple calculation gives that

min_{i \in F} (\frac{λ_{min} (Q_{2})}{λ_{max} (\bar{R})} + \frac{{Re}^{2} (λ_{i})}{{| λ_{i} |}^{2}}) = 0.8721 .

Therefore, it follows from equation (23) that the chosen $δ_{2} = 0.4$ is suitable. Based on $P_{2}$ , the distributed observer gain $L$ is obtained as

L = - S P_{2} Γ^{T} (Γ P_{2} Γ^{T} + R_{2})^{- 1} = [\begin{matrix} - 2.56919 \\ - 0.91221 \end{matrix}] .

Furthermore, utilizing $δ_{2} = 0.4$ , the intersection $φ (μ)$ can be computed as follows:

φ (μ) = {μ \in R | 0.5237 < μ < 1.2063}

The admissible initial states of the descriptor MASs (4) and the full dimensional state observers (39) are, respectively, taken as

x_{1} (0) = [\begin{matrix} 0.11 \\ 0.43 \end{matrix}], x_{2} (0) = [\begin{matrix} - 0.1 \\ - 0.2 \end{matrix}], x_{3} (0) = [\begin{matrix} 0.21 \\ 2.3 \end{matrix}], x_{4} (0) = [\begin{matrix} - 0.03 \\ - 0.04 \end{matrix}], x_{5} (0) = [\begin{matrix} 4 \\ 5 \end{matrix}] .

{\hat{x}}_{1} (0) = [\begin{matrix} 0.2 \\ 0.3 \end{matrix}], {\hat{x}}_{2} (0) = [\begin{matrix} - 0.2 \\ - 0.2 \end{matrix}], {\hat{x}}_{3} (0) = [\begin{matrix} 0.1 \\ 0.3 \end{matrix}], {\hat{x}}_{4} (0) = [\begin{matrix} 0 \\ 0 \end{matrix}], {\hat{x}}_{5} (0) = [\begin{matrix} 0.5 \\ 0.5 \end{matrix}] .

By the simulation method proposed in Liao et al. (2017), the numerical simulation is performed with $μ = 0.6$ . Figures 2 and 3 show the output containment behaviors under the distributed dynamic state feedback controller (37) and distributed dynamic output feedback controller (39), respectively.

Figures 4 and 5 depict the time evolution of the output containment errors under different distributed dynamic controller. It is straightforward to see that all the output containment errors converge to zero asymptotically, which means that two kinds of distributed dynamic controllers are able to achieve the output containment control.

Figures 6 and 7 plot the estimation errors of the distributed observer (22), where ${\tilde{w}}_{i 1}$ and ${\tilde{w}}_{i 2}$ are the components of $\tilde{w}$ . With Remark 3, we know that the asymptotical estimation of the distributed observer (32) can be achieved if

Re (λ_{i}) > Im (λ_{i}) \sqrt{\frac{1 - a}{a}}

and

max {\frac{Re (λ_{i}) - \sqrt{Δ_{i}}}{{| λ_{i} |}^{2}}} < μ < min {\frac{Re (λ_{i}) + \sqrt{Δ_{i}}}{{| λ_{i} |}^{2}}}

hold. However, with communication matrix $Ω$ , $R_{2} = 0.2 \cdot I_{2}$ , $Q_{2} = 5 \cdot I_{2}$ , the domain of $μ$ is empty by calculation, which means the distributed observer (32) cannot work for this occasion. To corroborate the theoretical advantages of the proposed distributed observer, we adjust the communication weights $a_{ij}$ according to $Re (λ_{i}) > Im (λ_{i}) \sqrt{(1 - a) / a}$ , and choose the following new communication matrix

\tilde{Ω} = [\begin{matrix} 1.25 & - 0.25 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 0.7 & 0 & 0 & - 0.7 & 0 & 0 \\ - 1.2 & 0 & 1.2 & 0 & 0 & 0 & 0 \\ 0 & - 0.8 & - 0.8 & 1.6 & 0 & 0 & 0 \\ 0 & 0 & - 0.5 & 0 & 1.1 & 0 & - 0.6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] .

Correspondingly, according to equation (1), ${\tilde{Ω}}_{1}$ and ${\tilde{Ω}}_{2}$ can be obtained as

{\tilde{Ω}}_{1} = [\begin{matrix} 1.25 & - 0.25 & 0 & 0 & 0 \\ 0 & 0.7 & 0 & 0 & - 0.7 \\ - 1.2 & 0 & 1.2 & 0 & 0 \\ 0 & - 0.8 & - 0.8 & 1.6 & 0 \\ 0 & 0 & - 0.5 & 0 & 1.1 \end{matrix}], Ω_{2} = [\begin{matrix} - 1 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & - 0.6 \end{matrix}] .

The eigenvalues of ${\tilde{Ω}}_{1}$ are 1.6000, 0.4408, 1.6647, $1.0723 \pm 0.5304 i$ .

For distributed observer (22), solving the modified algebraic Riccati equation (16) with $δ_{2} = 0.4$ , $R_{2} = 0.2$ , $Q_{2} = 5 \cdot I_{2}$ , we have

\begin{matrix} min_{i \in F} (\frac{λ_{min} (Q_{2})}{λ_{max} (\bar{R})} + \frac{{Re}^{2} (λ_{i})}{{| λ_{i} |}^{2}}) \\ = 0.8471, φ (μ) = {μ \in R | 0.4470 < μ < 1.0831} . \end{matrix}

For distributed observer (32), solving the algebraic Riccati equation (33) with $R_{2} = 0.2 \cdot I_{2}$ and $Q_{2} = 5 \cdot I_{2}$ , then according to Remark 3, we obtain

{\begin{matrix} Δ_{1} = 0.9480, Δ_{2} = 0.2612, Δ_{3} = 0.9864, \\ Δ_{4} = 0.4702, Δ_{5} = 0.4702 \end{matrix}}

μ \in (0.8973, 0.9638) .

Comparing the above two sets of results, we can find that:

Choosing $δ_{2}$ such that the condition (23) holds is much easier than adjusting $a_{ij}$ to satisfy the condition (34);

The domain of $μ$ for the distributed observer (22) is significantly larger than that for the distributed observer (32). To further verify this observation, we repeat the simulation with different $R_{2}$ and $Q_{2}$ . Tables 1 and 2 compare the domain of $μ$ for two kinds of distributed observers, which clearly illustrates that the distributed observer (22) is easier to achieve the asymptotical estimation of combination of the leaders’ states.

Table 1.

Region of $μ$ for distributed observers (22) and (32).

	$R_{2} = 0.01$ ( $0.01 \cdot I_{2}$ ) $Q_{2} = 5 \cdot I_{2}$	$R_{2} = 0.1$ ( $0.1 \cdot I_{2}$ ) $Q_{2} = 5 \cdot I_{2}$	$R_{2} = 0.5$ ( $0.5 \cdot I_{2}$ ) $Q_{2} = 5 \cdot I_{2}$	$R_{2} = 1$ ( $I_{2}$ ) $Q_{2} = 5 \cdot I_{2}$
Distributed observer (22)	(0.4454, 1.0835)	(0.4462, 1.0833)	(0.4493, 1.0824)	(0.4527, 1.0815)
Distributed observer (32)	(0.8682, 0.9715)	(0.8825, 0.9678)	(0.9370, 0.9533)	empty

Table 2.

Region of $μ$ for distributed observers (22) and (32).

	$R_{2} = 0.2$ ( $0.2 \cdot I_{2}$ ) $Q_{2} = 1.5 \cdot I_{2}$	$R_{2} = 0.2$ ( $0.2 \cdot I_{2}$ ) $Q_{2} = 15 \cdot I_{2}$	$R_{2} = 0.2$ ( $0.2 \cdot I_{2}$ ) $Q_{2} = 150 \cdot I_{2}$	$R_{2} = 0.2$ ( $0.2 \cdot I_{2}$ ) $Q_{2} = 1500 \cdot I_{2}$
Distributed observer (22)	(0.4505, 1.0821)	(0.4459, 1.0833)	(0.4454, 1.0835)	(0.4453, 1.0835)
Distributed observer (32)	empty	(0.8773, 0.9691)	(0.8677, 0.9717)	(0.8667, 0.9719)

Conclusion

The distributed output containment control problem of linear discrete-time descriptor MASs has been addressed by using the feedforward-based output regulation approach. Through introducing two free parameters, a novel distributed observer has been established for every follower to estimate the convex combination of the leaders’ states asymptotically. On this basis, two distributed dynamic controllers have also been derived to achieve the output containment control. For future research, notice that communication delays and input delays may exist in networked control systems (Chu et al., 2020, 2021a, 2021b; Deng et al., 2020); it would be interesting to extend this result to the discrete descriptor MASs with communication delays and input delays. Moreover, to investigate the problem for discrete descriptor MASs with different dimensions is also a meaningful topic.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is partially supported by the Fundamental Research Funds for the Central Universities (FRF-TP-18-101A1), the Science-Technology Foundation for Young Scientist of Gansu Province (Grant No.21JR7RA246), the National Natural Science Foundation of China (62003033, 61863026, 61751315), and the Guangdong Basic and Applied Basic Research Foundation (2019A1515111141).

ORCID iDs

Yanrong Lu

Liang Qiao

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