Coupling effect and pole assignment in trajectory regulation of multi-agent systems

This paper revisits a well studied leader-following consensus problem of linear multi-agent systems, while aiming at follower nodes' transient performance. Conventionally, when not all follower nodes have access to the leader's state information, distributed observers are designed to estimate the leader's state, and the observers are coupled via communication network. Then each follower node only needs to track its observer's state independently, without interacting with its neighbors. This paper deliberately introduces certain coupling effect among follower nodes, such that the follower nodes tend to converge to each other cooperatively on the way they converge to the leader. Moreover, by suitably designing the control law, the poles of follower nodes can be assigned as desired, and thus transient tracking performance can also be adjusted.


Introduction
Cooperative control of multi-agent systems (MASs) has been extensively investigated for the past two decades. It remains to gain increasing attention in the control community for its widespread applications in the areas of microgrids, unmanned aerial vehicles (UAVs), unmanned ground vehicles (UGVs), social networks, and so on. For a comprehensive literature review, readers are referred to some survey papers [4,6,8] and references therein.
Among various problem formulations of multi-agent systems, a fundamental one is the cooperative tracking control problem, also known as leader-following consensus problem or trajectory regulation problem. In this scenario, there is one leader node and a group of follower nodes, and all the follower nodes are driven to track the trajectory of the leader node [9,14]. A salient feature of this problem is that only part of the follower nodes can or need to have access to the leader's information due to limited communication capability, and thus a distributed control algorithm is required.
It is also worth mentioning that almost all the existing works on distributed control of multi-agent systems focus on steady state collective behaviors, without considering transient performance of the group. An exception is a line of research called prescribed performance control [2,13] or funnel control [10] of multi-agent systems. Both prescribed performance control and funnel control provide simple control laws that can restrict the profiles of the synchronization error within the prescribed error bounds. These control laws use neither the system dynamics information nor the graph topology information, and can be applied to linear systems for sure. However, their control laws consist of certain time functions, generating the prescribed performance bound profiles. This will lead to non-autonomous nonlinear closed-loop systems, whose performance depends on the initial time. Moreover, the initial values must be restricted within the prescribed bounds. Also, it is well known that both transient and steady state performance of a linear system relies on the location of its poles. Then an interesting question is naturally raised that whether we can design a distributed control law such that the poles of all follower agents can be assigned as desired. This will provides an insight into the transient and steady state performance of multi-agent systems. To the best of our knowledge, this problem is still open.
Motivated by the above-mentioned statements, this paper aims to propose a novel distributed control law, which not only solves the trajectory regulation problem, but also considers the coupling effect between agents as well as desired pole assignment. More specifically, we intend to deliberately introduce coupling effect for cooperative regulation, while avoiding over-coupling that may cause different issues, or even violation of system stability. Technically, compared with [15], this paper gives a quantitative analysis of this trade-off in terms of stability, dominant pole assignment, and fully system decoupling and hence pole assignment. It is noted that a relevant work [12] also introduces coupling effect between follower nodes, but the design relies on the solution to an linear matrix inequality and each follower node must design its own coupling effect, and pole assignment is not considered.
The rest of this paper is organized as follows. The problem is formulated in Section 2. Rigorous analysis of stability and pole assignment is provided in Section 3. A numerical example in Section 4 illustrates the effectiveness of the proposed algorithm. Section 5 concludes the paper. Technical lemmas are summarized in Appendix.
Notations: The sets of real and complex numbers are denoted by R and C, respectively. For a matrix A, A T represents it transpose and A * its conjugate transpose. For a symmetric real matrix A, A > 0 (A ≥ 0) means A is positive definite (positive semi-definite). The determinant of a matrix A is denoted as det(A). The real part of a complex number m is denoted as Re [m]. The set N is defined as {1, 2, . . . , N } and 1 N is the N -dimensional column vector whose elements are 1. The Kronecker product is denoted by the operator ⊗. For vectors x 1 , · · · , x n , col(x 1 , · · · , x n ) = [x T 1 , · · · , x T n ] T represents the stacked vector.

Problem Formulation
The paper is concerned with control of a group of linear homogenous agents of the dynamics described bẏ where x i (t) ∈ R n is the state vector of the i-th agent and u i (t) ∈ R m the control input vector. The control task is to regulate every state trajectory x i (t) to the desired trajectory x 0 (t) ∈ R n described by the leader nodė that is, lim The task becomes complicated when x 0 (t) is unaccessible for some agents. Nevertheless, such a task has been well accomplished by using the framework of consensus of observers and trajectory regulation. More specifically, an observer is established for each agent as followṡ where e = col(e 1 , · · · , e N ) with e i =x 0i − x 0 and the consensus protocol function γ i with γ i (0) = 0 is designed such that lim t→∞ e(t) = 0. Note that the design of γ i has been well studied in literature, and is not an issue considered in this paper. See the following remark for one choice of γ i .

Remark 1
For homogenous MASs, the design of consensus tracking protocol has been well studied [11,?], e.g., in a directed graph G represented by the network adjacent weightᾱ ij ≥ 0, i, j ∈ N , with µ sufficiently large [11] (or µ can be designed in a decoupled manner µ = cK as in [14]) and g i = 0 for i ∈ S R , g i = 0 for i / ∈ S R , where S R ⊂ N is the set of nodes that have direct access to the reference signal x 0 (t). In this setting, if every follower node i ∈ N is reachable in G from the leader node, then one has lim t→∞ e(t) = 0.
The time variable (t) is ignored in (4) and the sequel for conciseness. Next, let η = col(η 1 , · · · , η N ) with A trajectory regulator with a feedback matrix F ∈ R m×n and a scalar gain As the closed-loop system iṡ it suffices to pick F and d i such that A−d i BF is Hurwitz. From above, the task (3) is accomplished by combining lim t→∞ e(t) = 0 and lim t→∞ η(t) = 0. Now, the main focus of this paper is on the design of regulator for lim t→∞ η(t) = 0. It is obvious that the simple regulator (6) is designed separately for each individual agent. Its simplicity has independent interest in many scenarios. However, in some other scenarios, researchers are also interested in more sophisticated interactions among agents when they obey the trajectory regulation protocol. For instance, individual regulation does not work when the agents intend to converge to each other before to the desired trajectory. For this purpose, we deliberately introduce coupling effect a ij among agents in the physical layer (see Fig. 1(b)), which results in where a ij ≥ 0, i, j ∈ N represents the coupling weight between agent i and j in the physical layer, with a ii = 0. Note thatᾱ ij denotes the coupling among observers in the cyber layer (see Fig. 1). It is worth mentioning that the leader model (2) has the role in the design of the observer network in (5), but it has no direct influence on the deliberate coupling network in (7). In other words, the main results given in this paper, based on the analysis of (7), may apply not only to a leader-following topology, but also to more general leaderless topologies.
With the above development, this paper aims to analyze the coupling effect of A, in comparison with D. First of all, it should be noted that (i) The diagonal entries of M are represented by D and the off-diagonal entries by −A. If d i = N j=1 a ij , i ∈ N , the matrix M becomes a Laplacian. But in general, the matrix M is adjusted as a diagonally dominant matrix for our purpose with a more significant D. (ii) The graph for the adjacency matrix A is not necessarily connected for the purpose of consensus. Especially, the consensus can be achieved by the controller with A = 0, i.e., (6).
In this paper, on one hand, we intend to introduce more significant coupling effect a ij on the physical layer for cooperative regulation. On the other hand, over-coupling may cause different issues even violation of system stability. So, the main objective of this paper is to conduct more specific analysis of the coupling effect of A relative to D from the following three aspects, which apply different upper boundary conditions on a ij . These conditions can be regarded as design criteria in selecting the strength of A for (8) or (9).
(1) The condition on A relative to D for stability of (8) or (9).
(2) A less significant A and a more significant D allow dominant pole assignment of (8) or (9). (3) The condition on A relative to D for a fully decoupled system matrix and hence pole assignment.

Main Results
The three aspects listed in the aforementioned main objective are addressed in this section in order. In the subsequent presentation, we will present the results on (9).

Stability Condition
Each agent uses its own feedback through d i and others via a ij . Intuitively, it needs more significant feedback from itself for stability. To explicitly characterize the relative significance, we introduce the condition under which M + M T is positive definite. This property plays an important role in proving the stability of (9).   (10), all the Geršgorin discs are strictly shifted to the right, not going through the origin any more, illustrated by the dotted circle.
Note that the stability of the system (9) can equivalently represented by a feedback structure illustrated in Fig. 3 in frequency domain. In other words, (9) can be rewritten as follows, with v = 0, Here v is an auxiliary input introduced for the convenience of presentation later. Now, the main result is stated below. Theorem 1 Under Assumption 1 and the condition (10), there exists F such that the system (9) is stable.
Proof: A constructive proof is given as follows. First of all, Lemma 1 shows that M + M T > 0 under the condition (10). There exists a scalar k > 0 such that M + M T > 2kI. Then, Z + Z T > 0 for Z = (M − kI)/k. That is, M = k(Z + I). Now, the system (11) can be rewritten asξ Under Assumption 1, there exists a solution X C > 0 to the control algebraic Riccati equation with Q C > 0. Let F = −kB T X C . By Lemma 3, the transfer function matrix T F (s) = F (sI −A+kBF ) −1 kB is positive real, so is T F (s) = I N ⊗T F (s). In other words, It is easy to see that the transfer function from v to y is whose poles are determined by det(I + kT F (s)Z) = 0. One has det[I + kT F (s)Z] = 0, ∀ Re[s] ≥ 0, due to (12) and Z + Z T > 0, using Lemma 4. It means that all the poles of (13) have negative real parts. The stability of (9) is thus proved.
Remark 3 A uniform selection of d i , i.e., can be used to satisfy the condition (10). It simplifies the protocol (7) with less parameters. Condition (10) requires sufficiently large d i . However, high gain of a controller may lead to some practical issues. To prevent the high gain issue, in practice, we can choose d i to the desired level and then scale a ij down to match the condition (10).

Pole Assignment
In this subsection, we aim to show that the system poles can be assigned when D is sufficiently large and A sufficiently small. In particular, the transient characteristics of the regulation behavior can be adjusted by pole assignment. We will use the inverse optimal linear quadratic regulator (LQR) technique to assign poles.
Here we assume rank(B) = m and the matrices A, B take the following form where A 22 and B 2 have the same dimension. Define the LQR probleṁ with Q = Q T ≥ 0 and R = R T > 0 for the convenience of presentation. Now, the main result is stated in the following theorem.
Theorem 2 Consider the system (9) of the structure (14) under Assumption 1. Let where σ 1 , · · · , σ m > 0 and V, Σ, K are selected such that w = −F z is a solution to the LQR problem (15). In particular, the eigenvalues of A 11 −A 12 K has the specified stable eigenvalues {s 1 , · · · , s n−m }, none of which is an eigenvalue of A 11 . If the system (9) has the following eigenvalue distribution: (i) there are (n−m)N eigenvalues λ j i of the form λ j i → s j , j = 1, · · · , n − m, i ∈ N ; and (ii) there are mN eigenvalues λ j+n−m i of the form Proof: First of all, it is noted that the system (9) can be rewritten asξ For the eigenvalues of a matrix continuously depend on its parameter variation, the eigenvalues λ j i , j = 1, · · · , n, i ∈ N , approach those ofĀ −BDF as A → 0. Furthermore, the eigenvalues ofĀ −BDF are those of A − d i BF .

Remark 4
The theorem shows that the eigenvalues of the closed-loop system approach arbitrarily specified stable poles s j for j = 1, . . . , n − m and other stable poles −d i σ j when all d i are sufficiently large and all a ij are sufficiently small. In particular, as all d i are sufficiently large, Re[s j ] −d i σ j . In other words, all specified s j are the dominant poles, enforcing the plant the specified transient characteristics.

Fully Decoupling Condition
In the previous subsection, it is proved that the dominant poles of the closed-loop system can be placed as desired such that the transient characteristics can be satisfied, provided that A → 0. It obviously contradicts to the main motivation of adding coupling effect to regulation, if we simply remove the coupling by letting A = 0. In this subsection, we will further study an explicit condition on the size of a ij under which the system can be fully decoupled and the pole assignment technique can be applied without the assumption of A → 0.
We first give a lemma for diagonalization of the coupling matrix M, followed by the main theorem.
where r i = N j=1 a ij , then M is diagonalizable. Proof: The diagonal entry of the i-th row of M is d i and the sum of the absolute values of the off-diagonal entries in the i-th row is r i = N j=1 a ij . As a result, one can define a Geršgorin disc centred at d i with radius r i , denoted D(d i , r i ). Under the condition (17), the Geršgorin discs are D(d κi , r κi ) in the order of i = 1, · · · , N , from left to right, and they do not intersect (see Fig. 4). Therefore, by Geršgorin theorem, all eigenvalues of M are distinct positive real numbers. Thus M is diagonalizable. Proof: By Lemma 2, there exists a nonsingular matrix T such that P = T MT −1 , P = diag(p 1 , · · · , p N ) With ζ = (T ⊗ I m )ξ, the system (9)   In the simulation, let F = [−0.3660, 0.9306] as in Theorem 2, and pick D = diag{2, 1.95, 2.5, 2.9, 3.4}ρ and A = [εa ij ] with a 12 = a 23 = a 34 = a 45 = a 51 = 1 and a ij = 0 otherwise, for two parameters ρ, ε > 0. The results are demonstrated in terms of x i = ξ i + x 0 with x 0 (t) governed by (2) and ξ(t) by (9), where the specific behavior of e(t) approaching zero is ignored.
x 51 ] T as the vector consisting of the first elements of the five agent states. The signal φ = max{|ξ 11 |, · · · , |ξ 51 |} represents the regulation error and ψ = max{ξ 11 , · · · , ξ 51 } − min{ξ 11 , · · · , ξ 51 } the difference among the agents. In the first case, we select ρ = 0.12 and ε = 0. As shown in Fig. 5, the agents reach consensus while their transient processes are independent and do not influence each other. In other words, the agents do not demonstrate a cooperative behavior. The profiles of the agent difference and the regulation error are shown in Fig. 6.
In the second case, we select ρ = 0.2 and ε = 0.2 as com-parison. All the agent states also converge a consensus on the same reference signal. With ε = 0.2, the agents have cooperation before achieving consensus. In the simulation, we pick the parameters such that the regulation error achieves zero with the same transient performance.
As shown in the bottom graph of Fig. 6, the error reduces from 3 to 0.3 (by 90%) at t = 15 for both two cases. It is obvious that, the difference among agents in the case with ε = 0.2 is significantly less than that in the case with ε = 0. More specifically, it takes t = 4.5 for the difference among agents to reduce below 0.5 in the former case while it takes t = 7.0 in the latter case, as shown in the top graph of Fig. 6.

Conclusion
This paper has studied a consensus tracking problem of linear multi-agent systems. Compared with the conventional observer based control approach, where each observer estimates the leader's information via neighborhood communication, this paper features itself in two aspects: first, coupling effect between follower nodes in the physical layer is deliberately introduced to take into account the cooperative behavior between all follower nodes before they converge to the leader node; second, dominant poles of follower nodes can be adjusted to obtain a desired transient performance.
The following lemma is adopted from [5] with slight modification (cf. Theorem 4.1 and Proposition 4.1 of [5]). where σ 1 , · · · , σ m > 0 and V, Σ, K are selected such that w = −F z is a solution to the LQR problem. In particular, the eigenvalues of A 11 −A 12 K has the specified stable eigenvalues {s 1 , · · · , s n−m }, none of which is an eigenvalue of A 11 . If d → ∞, then matrix A − dBF has the following eigenvalue distribution: (1) there are n − m eigenvalues λ j of the form λ j → s j , j = 1, · · · , n − m; and (2) there are m eigenvalues λ j+n−m of the form λ j+n−m → −dσ j , j = 1, · · · , m.