We consider the leader-following consensus problem of discrete-time multiagent systems on a directed communication topology. Two types of distributed observer-based consensus protocols are considered to solve such a problem. The observers involved in the proposed protocols include full-order observer and reduced-order observer, which are used to reconstruct the state variables. Two algorithms are provided to construct the consensus protocols, which are based on the modified discrete-time algebraic Riccati equation and Sylvester equation. In light of graph and matrix theory, some consensus conditions are established. Finally, a numerical example is provided to illustrate the obtained result.

1. Introduction

In recent decades, the cooperate and control problem of distributed dynamic systems has been a challenging research field, owing to its widespread applications in many areas such as swarm of animals [1], collective motion of particles [2], schooling for underwater vehicles [3, 4], neural networks [5, 6], and distributed sensor networks [7].

The consensus problem, as one fundamental problem for coordinated control of multiagent systems, has gained significant attention from different research communities. Consensus problem considers how to design an information interaction protocol between agents and requires all agents to converge to a common value [8, 9]. Based on matrix theory, algebraic graph theory, and control theory, many researchers have acquired abundant results in studying consensus problem of multiagent systems. In [10], the authors proposed a general framework for consensus problem in fixed and switching networks and gave solution to the case with communication time delays. Olfati-Saber et al. established a general model for consensus problems of the multiagent systems and introduced Lyapunov method to reveal the contract with the connectivity of the graph theory and the stability of the system in [11]. Sometimes, it is better to consider a tracking consensus problem by adding a leader which can make all agents reach a command trajectory with any initial condition [12]. The leader-following consensus problem has been addressed in many references [13–17].

Many proposed distributed consensus protocols need to know neighbors’ state information, but it may be difficult to measure this information. To make the system achieve consensus, it often contains an observer in the control protocol, which is used to estimate those unmeasurable state variables. The distributed observer-based control laws were proposed to solve first-order and second-order multiagent consensus problems in [12, 17]. To estimate the general active leader’s unmeasurable state variables, [18] proposed a distributed algorithm for first-order agent, and [19] extended the results of [18] to the time-delay case. The distributed observer-based consensus protocols were addressed to solve multiagent consensus with general linear or linearized agent dynamics in [17, 20–24]. In [25], the author proposed an observer-type consensus protocol to the consensus problem for a class of fractional-order uncertain multiagent systems with general linear dynamics. In [26], the authors proposed distributed reduced-order observer-based protocols to solve consensus problem, which were generalized to solve leader-following consensus problem under switching topology by [27]. The observer-based consensus protocol can be viewed as a special case of the dynamic compensation method, which has been investigated by [28–30].

Discrete-time dynamic systems are commonly involved in the neural network, sampled control, signal filters, and state estimators. The discrete-time neural network was studied by [31–33]. The sampled-data discrete-time coordination of multiagent systems was investigated in [16, 34, 35]. The first-order discrete-time consensus has been investigated by [8, 9, 36–38]. In [39], the authors discussed discrete-time second-order consensus protocols for dynamics with nonuniform and time-varying communication delays under dynamically switching topology. The distributed H_∞ consensus problem was studied in [30] to solve multiagent consensus problem with discrete-time high-dimensional linear coupling dynamics subjected to external disturbances. The distributed state-feedback protocols for linear discrete-time multiagent were proposed in [40, 41]. The distributed observer-based protocol was proposed to solve leader-following consensus problem with linear discrete-time dynamics in [23, 42, 43].

Motivated by the above works, we focus our research on a group of agents with discrete-time high-dimensional linear coupling dynamics and directed interaction topology. We propose distributed observer-based protocols for leader-following multiagent systems. The full-order observer and reduced-order observer are adopted to reconstruct the state variables. Contrary to [23] and [40], the gain matrix design approach used in this paper is based on the modified discrete-time algebraic Riccati equations (MDARE) but not the normal discrete-time algebraic Riccati equations. The proposed design method must be feasible if spectral radius of system matrix is not greater than 1. Of course, the proposed design method can be used to construct the consensus protocols provided by [23] and [40]. Further, the separation principle is shown to be valid, from which we can establish consensus condition for closed-loop multiagent systems.

This paper is organized as follows. Section 2 presents the related notations and the problem formulated with graph theory. In Section 3, the distributed state feedback design is considered. In Sections 4 and 5, the distributed full-order and reduced-order observer-based consensus protocols are proposed, respectively, which are the main results of this paper. Section 6 presents a simulation example to illustrate our established results. Finally, the conclusion is given in Section 7.

2. Preliminaries and Problem Formulation

2.1. Notations and Graph Theory

Re(ξ) denotes the real part of ξ ∈ C. Let R^m×n and C^m×n be the set of m × n real matrices and complex matrices, respectively. 1_n ∈ Rⁿ is the column vector with all components equal to one. Let I be the identity matrix with compatible dimension. For a given matrix A, a_ij represents its element of ith row and jth column, A^T denotes its transpose, and A^H denotes its conjugate transpose. A matrix is said to be Schur-stable if all its eigenvalues are inside unit circle. ρ(A) represents the spectral radius of matrix A. λ_max(A) and λ_min(A) represent its maximum and minimum eigenvalues of symmetric matrix A, respectively. For symmetric matrices A and B, A > B means that A − B is positive definite, that is, A − B > 0. ⊗ denotes Kronecker product, which satisfies (A ⊗ B)(C ⊗ D) = (AC)⊗(BD).

We describe the interaction relationship among n agents by a simple weighted diagraph 𝒢 = {𝒱, ɛ, W}, where 𝒱 = {v₁, v₂, …, v_n} is the set of vertices and ɛ ⊂ 𝒱 × 𝒱 is the set of edges. If (v_i, v_j) ∈ ɛ, the vertex v_j is called a neighbor of vertex v_i, and the index set of neighbors of vertex v_i is denoted by 𝒩_i = {j∣(v_i, v_j) ∈ ɛ}. W = [w_ij] _n×n represents weighted adjacency matrix associated with graph 𝒢, where w_ij > 0 if (v_i, v_j) ∈ ɛ and w_ij = 0 otherwise. The degree matrix D = diag {d₁, d₂, …, d_n} of digraph 𝒢 is a diagonal matrix with diagonal elements . Then, the Laplacian matrix of 𝒢 is defined as L = D − W. v_i is called globally reachable node if there exists at least a directed path from every other node to node v_i in digraph 𝒢. A directed graph 𝒢 has a globally reachable node if and only if there exists a directed spanning tree in 𝒢 (see [9]).

For a multiagent system with leader (labeled as 0), the interaction topology is expressed by graph , which contains graph 𝒢 and vertex v₀ and edges from other vertices to vertex v₀. Let g_i, i = 1,2, …, n, be weight of (v_i, v₀). g_i > 0 if (v_i, v₀) is an edge of graph and g_i = 0 otherwise. Let G_d = diag {g₁, g₂, …, g_n}. The matrix L + G_d has the following property.

Lemma 1 (see [13].)Matrix L + G_d is positive stable if and only if graph has a directed spanning tree with root v₀.

2.2. Problem Formulation

Consider the multiagent system which is composed of n identical following agents and a leader. Each following agent has dynamics modeled by the discrete-time linear system:

(1)

where x_i(k) ∈ R^m, u_i(k) ∈ R^p, and y_i(k) ∈ R^q are, respectively, the state variable, control input, and measured output of agent i.

The dynamics of the leader is given as

(2)

where x₀(k) is the state and y₀(k) is the measured output of the leader. The leaderless consensus problem for multiagent system has been investigated by [26, 28, 44], which require the system matrix A to be Schur-stable. There is not such requirement to A in this paper. A, B, C are constant matrices with compatible dimensions. It is assumed that (A, B, and C) is stabilizable and detectable.

The x₀(k) is often called as “consensus reference state” and assumed to be available only to a subgroup of the followers. The main objective of leader-following consensus problem is to design distributed consensus protocol to make multiagent system achieve consensus.

Definition 2. The leader-following multiagent system is said to achieve consensus if the state variables of all following agents satisfy lim _k→∞ (x_i(k) − x₀(k)) = 0, i = 1,2, …, n for any initial state. One says that the protocol u_i(k) can solve the leader-following consensus problem if the closed-loop system achieves consensus.

2.3. Preliminary Results

In this subsection, we introduce some preliminary results which will be used to establish our main results. Consider the following MDARE:

(3)

where Q is any given positive definite matrix. Since Q is positive definite, (A, Q^1/2) must be detectable. The solvability of the MDARE is addressed by the following lemma.

Lemma 3 (see [45], [46].)If (A, Q^1/2) is detectable, (A, B) is stabilizable, then there exists a δ_c ∈ [0,1) such that the modified discrete time algebraic Riccati equation (3) has a unique positive-definite solution P for any δ_c < δ ≤ 1. Furthermore, P = lim _k→∞P_k for any initial condition P₀ ≥ 0, where P_k satisfies

(4)

Remark 4. The MDARE (3) is reduced, respectively, to the well-known discrete-time Riccati equation (DARE) and Stain equation as δ = 1 and δ = 0. The Stain equation has a unique positive-definite solution if A is Schur-stable. It is well known that DARE has a unique positive-definite solution if (A, B) is stablizable. If the involved matrix A is not Schur-stable, it is easy to see that 0 < δ_c ≤ 1. More details for issue δ_c can be referenced to [45]. Moreover, if the matrix A has no eigenvalues with magnitude larger than 1 and (A, C) is detectable, MDARE (3) has a unique positive-definite solution P for any δ satisfying 0 < δ ≤ 1.

Lemma 5. For a given δ satisfying δ_c < δ ≤ 1, let P be the unique positive-definite solution of the MDARE (3). Choose a feedback matrix K = (I + B^TPB) ⁻¹B^TPA. Then, A − sBK is Schur-stable for any .

Proof. From the MDARE (3), we have

(5)

Thus, we know that if

, A − sBK is Schur-stable.

3. Distributed State Feedback Design

In this section, we investigate the multiagent consensus via state variable feedback control, which has been addressed by [23]. Here, we also use the control protocol proposed by [23] and provide a new design approach to construct the feedback gain matrix.

The neighborhood disagreement error of agent i is defined as

(6)

Consider the following distributed state feedback protocol for agent i:

(7)

where

, c₁ is the coupling strength and K is a feedback gain matrix, which will be determined later.

Denote e_i(k) = x_i(k) − x₀(k) and

. Then, we can derive that the close loop system has the global tracking error dynamics as follows [23]

(8)

where Γ = (I + D + G_d) ⁻¹(L + G_d).

Definition 6 (see [23].)A covering circle related to matrix Γ is a closed circle in the complex plane centered at c₀ ∈ R and for all i = 1,2, …, n.

Then, we provide a new design technique to construct feedback gain matrix K, which is presented in the following theorem.

Theorem 7. For multiagent system (1) and (2), assume that the interconnection topology has a directed spanning tree with root v₀. If there exists a covering circle such that

(9)

then there must exist fitted c₁ and K such that the global tracking error dynamics (8) is asymptotically stable. Furthermore, by taking δ which satisfies

(10)

and solving the MDARE (3) to get the unique positive-definite solution P, the feedback matrix K and the coupling strength c₁ can be chosen as

(11)

Proof. From (10), we know δ > δ_c, which means that the MDARE (3) has a unique positive-definite solution P. Any λ_i satisfies |λ_i − c₀ | ≤ r₀. Thus, . According to Lemma 5, all A − c₁λ_iBK, i = 1,2, …, n are Schur-stable.

Let U be a Schur transformation matrix of Γ such that

(12)

Then, we have

(13)

Certainly, U ⊗ I is also a unitary matrix. Matrix [I_n ⊗ A − c₁Γ ⊗ (BK)] is Schur-stable if and only if all A − c₁λ_iBK, i = 1,2, …, n are Schur-stable. Now, the proof is completed.

Remark 8. From condition (9), it is required that 0 < c₀ < r₀, which means that the covering circle should be located in the open right half plane. Moreover, the small enough r₀/c₀ will guarantee that the MDARE (3) is solvable, which is the key point in the proposed design approach. The weight parameter in the feedback law (7) need not take c₁(1 + d_i + g_i) ⁻¹, which can be selected as c₁(d_i + g_i) ⁻¹, c₁, and so on as long as there exists a covering circle for the related matrix c₁Γ that satisfies the condition (9).

Next, we will discuss the covering circle of the matrix c₁Γ. Based on Gershgorin disk theorem [47], all the eigenvalues of (I + D + G_d) ⁻¹(I + W) are located in the union of n discs:

(14)

It is easy to see that this union is included in a unit circle {s : |s| ≤ 1} and the circular boundaries of the union of n discs have only one intersection with the circle at s = 1. If the interconnection topology

has a directed spanning tree with root v₀, we know that L + G_d is nonsingular, and then, Γ is nonsingular too. Noting that (I + D + G_d) ⁻¹(I + W) = I − Γ, then we know that all eigenvalues of matrix (I + D + G_d) ⁻¹(I + W) are not equal to 1. Thus, all eigenvalues of matrix Γ can be covered by circle

with r₀ < 1. On the other hand, it is necessary to assume that the interconnection topology

has a directed spanning tree with root v₀. Otherwise, there exists at least one agent which cannot get the leader’s information directly and indirectly. Certainly, if A is not Schur-stable, those agents cannot track the leader with some initial values. From this point, the assumption that the interconnection topology

has a directed spanning tree with root v₀ is necessary.

An interesting special case is that matrix A has no eigenvalues with magnitude larger than 1, that is, ρ(A) ≤ 1. The well-known second-order discrete-time multiagent system

(15)

has been addressed in many references [34, 38]. The system matrix A of second-order discrete-time multiagent system is

, which has no eigenvalues with magnitude larger than 1.

According to Theorem 7, we present the following corollary for this special case.

Corollary 9. For multiagent system (1) and (2) with ρ(A) ≤ 1, assume that the interconnection topology has a directed spanning tree with root v₀. Take and solve the MDARE (3) to get the unique positive-definite solution P. Choose K = (I + B^TPB) ⁻¹B^TPA and C₁ = 1. Then, the distributed feedback control (7) guarantees that all following agents can track leader.

Proof. According to Remark 4, we know δ_c = 0 if ρ(A) ≤ 1. Select . From above analysis, we know that δ > 0 and C(1, δ) are a covering circle. Thus, the MDARE (3) is solvable. According to Theorem 7, we can obtain the corollary directly.

4. Consensus Protocol Design with Full-Order Observer

In many applications, each agent only accesses the neighbor’s output variable. To solve leader-following consensus problem, we propose a new observer-based consensus protocol for agent i, which consists of a distributed estimation law and a feedback control law.

(i)
Local estimation law for agent i:
(16)
where z_i(k) ∈ R^m is the protocol state, is the constructed variable to estimate x_i(k), and F ∈ R^m×m, G ∈ R^m×q, and T ∈ R^m×m are the designed parameter matrices.
(ii)
Neighbor-based feedback control law for agent i:
(17)
where the neighborhood disagreement observer error η_i(k) of agent i is denoted as
(18)
and K is a given feedback gain matrix.

Next, an algorithm is provided to select the parameter matrices used in estimation law (16).

Algorithm 10. Given that (A, C) is observable. The parameter matrices F, G, and T used in estimation law (16) can be constructed as follows.

(1)
Select a Schur-stable m × m matrix F with a set of desired eigenvalues that contain no eigenvalues in common with those of A.
(2)
Select G ∈ R^m×q randomly such that (F, G) is controllable.
(3)
Solve Sylvester equation
(19)

to get a nonsingular solution T. If T is singular, select another G until T is nonsingular.

Denote and . Then, after manipulations and combining (1) and (16), we can obtain

(20)

For tracking error e_i(k) = x_i(k) − x₀(k), we have

(21)

From (20) and (21), the error dynamics of closed-loop system will be expressed as

(22)

Obviously, the error dynamics system (22) is Schur-stable if and only if I_n ⊗ A − c₁Γ ⊗ (BK) and I ⊗ F are Schur-stable. Similar to Theorem 7, we present the following theorem directly, and the proof is omitted.

Theorem 11. For multiagent system (1) and (2), assume that the interconnection topology has a directed spanning tree with root v₀. If there exists a covering circle such that

(23)

then the distributed observer-based protocols (16) and (17) can solve the discrete-time leader-following consensus problem. Furthermore, the parameter matrices F, G, and T used in observer (16) are constructed by Algorithm 10. By taking δ satisfied

(24)

and solving the MDARE (3) to get the unique positive-definite solution P, the feedback matrix K and the coupling strength c₁ can be chosen as

(25)

Remark 12. Of course, when system matrix A satisfies ρ(A) ≤ 1, we can also establish similar corollaries as Corollary 9 in this section and the next section. In [23], three different observer/controller architectures are proposed for dynamic output feedback regulator design. Besides design feedback matrix K, another key technique is to choose an observer gain matrix L which makes I_n ⊗ A − c₁Γ ⊗ (LC) Schur-stable. By using duality property, solve the following MDARE:

(26)

to get the unique positive definite solution P. Then, the observer gain matrix L is chosen as L = APC^T(I + CPC^T) ⁻¹. Thus, the proposed design method in this paper can also be applied to construct the protocols proposed by [23]. In this paper, we propose two new observer/controller architectures, which will replenish cooperative observer and regulator theory. Contrary to [23], our proposed approach must be feasible if system matrix A satisfies ρ(A) ≤ 1.

5. Consensus Protocol Design with Reduced-Order Observer

In this section, we assume that C has full row rank, that is, Rank(C) = q. The following reduced-order observer-based consensus protocol, which consists of a reduced-order estimation law and a feedback control law, is proposed for agent i.

(i)
Local reduced-order estimation law for agent i:
(27)
where v_i(k) ∈ R^m−q is the protocol state, F ∈ R^m−q×m−q, and G ∈ R^m−q×q and T ∈ R^m−q×m are parameter matrices.
(ii)
Neighbor-based feedback control law for agent i:
(28)
where the disagreement error ζ_i(k) of agent i is given as
(29)
and K is a gain matrix.

Similarly, an algorithm is presented to design the same parameter matrices used in the protocols (27) and (28).

Algorithm 13. Given that (A, C) is observable. The parameter matrices F, G, T, Q₁, and Q₂ can be constructed as follows.

(1)
Select a Schur matrix F ∈ R^{(m−q)×(m−q)} with a set of desired eigenvalues that contain no eigenvalues in common with those of A.
(2)
Select G ∈ R^(m−q)×q randomly such that (F, G) is controllable.
(3)
Solve Sylvester equation
(30)

to get the unique solution T, which satisfies that

is nonsingular. If

is singular, go back to step (2) to select another G until

is nonsingular.

(4)
Compute matrices Q₁ ∈ R^m×q and Q₂ ∈ R^m×(m−q) by .

Now, we present the result related to reduced-order observer.

Theorem 14. For multiagent system (1) and (2), assume that the interconnection topology has a directed spanning tree with root v₀. If there exists a covering circle such that

(31)

then the distributed observer-based protocols (16) and (17) can solve the discrete-time leader-following consensus problem. Furthermore, the parameter matrices F, G, T, Q₁, and Q₂ used in protocols (27) and (28) are constructed by Algorithm 13. By taking δ which satisfies

(32)

and solving the MDARE (3) to get the unique positive-definite solution P, the feedback matrix K and the coupling strength c₁ can be chosen as

(33)

Proof. To analyze convergence, denote e_i(k) = x_i(k) − x₀(k) and ɛ_i = v_i(k) − Tx₀(k). Then, the dynamics of e_i(k) and ɛ_i(k) satisfy

(34)

Let

and

. From (34), the closed-loop error dynamics can be represented as

(35)

It is easy to see that the leader-following multiagent system achieves consensus if the closed-loop error dynamics system (35) is Schur-stable.

Let , which is nonsingular, and . By step (2) of Algorithm 13, we have

(36)

The matrix

is block upper triangular matrix with diagonal block matrix entries I_n ⊗ A − c₁Γ ⊗ (BK) and F. Because F is Schur-stable, the matrix H is Schur-stable if and only if A − c₁Γ ⊗ (BK) is Schur-stable. The rest of the proof is omitted, because it is very similar to the proof of Theorem 7.

6. Simulation Example

In this section, we give an example to illustrate the effectiveness of the obtained result. The multiagent system consists of four agents and one leader, that is, n = 4. The following agents and leader are, respectively, modeled by the linear dynamics (1) and (2) with system matrices

(37)

The matrices L and G of the interaction graph

are given by

(38)

By some simple computations, it is proper to take c₀ = 0.5768, r₀ = 0.5001. Therefore, take c₁ = 1.7337. By solving MDRAE (3) with δ = 0.2482, the unique positive definite solution is

(39)

Then, the gain matrix can be chosen as

(40)

The multiagent system adopts the consensus protocols (16) and (17) with randomly initial state. The matrices F, G, and T are designed as follows:

(41)

The state tracking errors showed in Figure 1, which show all following agents can track the leader. As for the reduced-order observer case, the matrices F, G, T, Q₁, and Q₂ used in the protocols (27) and (28) can be constructed by Algorithm 13 as follows:

(42)

With consensus protocols (27) and (28), the state tracking errors showed in Figure 2, which also show all following agents, can track the leader.

Details are in the caption following the image — **Figure 1 (a)**
Open in figure viewer PowerPoint

Error trajectories of three state components with full-order observer.

7. Conclusions

This paper solves a leader-following consensus problem of discrete-time multiagent system with distributed controllers and observers. We provide a general framework for designing distributed consensus protocols by applying full state feedback information and measured output feedback information. Furthermore, we propose a reduced-order observer-based protocol to solve the leader-following consensus problem. The interconnection topology is modeled by graph, whose connectivity is a key factor to guarantee that the multiagent achieves consensus. The consensus problem is transformed into the stability problem of error dynamical system, which also preserves the property of the separation principle. The gain matrices can be designed by solving the MDARE and the Sylvester equation. Presented results could be generalized to switching and jumping interaction topology in future work.

Acknowledgments

This work was supported by the Zhejiang Provincial Natural Science Function of China under Grant no. LY13F030048 and the National Natural Science Function of China under Grant nos. 61074123, 61174063.

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Leader-Following Consensus Stability of Discrete-Time Linear Multiagent Systems with Observer-Based Protocols

Abstract

1. Introduction