Prescribed-instant consensus tracking of high-order multi-agent systems

Abstract

This paper addresses the prescribed-instant consensus (PSIC) tracking problem of the leader-follower high-order multi-agent systems (MASs). Both cases of directed graphs and undirected connected graphs are considered. A Lyapunov-based sufficient condition for prescribed-instant stability of general single-loop systems is proposed. Based on the backstepping method, distributed control approaches are proposed to ensure consensus at the prescribed time instant. Consensus analyses are provided on the basis of Lyapunov stability theory and algebraic graph theory. Two numerical simulations are carried out which illustrates the validity of the proposed methods at the end.

Keywords

prescribed-instant consensus high-order multi-agent systems backstepping directed graphs

1. Introduction

Over the past few decades, there has been significant research into the coordination control of MASs, driven by its broad potential applications such as in the optimal leader allocation of autonomous vehicles (Richert and Cortés, 2013), the formation control of spacecraft (Wang et al., 2019a), and local path planning of multi-robots (Xu et al., 2015). The consensus problem in MASs is the most fundamental issue, which refers to the fact that by designing a control protocol relying on the exchange of local data, all agents can reach an agreement under specified conditions (Pei and Liang, 2025; Shahvali et al., 2018; Shahvali and Shojaei, 2016).

In the consensus problem, the settling time has consistently been a central focus, because the faster convergence rate usually leads to the better system performance (Karimoddini et al., 2013). Since finite-time stability method was formally introduced in Orlov (2004), it has been widely applied to MASs, yielding many fruitful results concerning the finite-time consensus of the MASs in Zhao et al. (2013); Shahvali and Askari (2016); Shahvali et al. (2018b); and Pei (2023). How quickly finite-time consensus is achieved depends on the initial states of the agents, which poses a challenge in determining it if these states are unknown initially (Cao and Ren, 2011). Additionally, the actual settling time will increase as the difference of the initial states becomes larger (Ni et al., 2021). Polyakov (2011) presented a groundbreaking theory of fixed-time stability to resolve this issue. Soon it was applied to MASs, which ensures that the settling time is a constant irrespective of the initial states. This not only simplifies the estimation of the settling time but also streamlines the design of control protocols. It should be emphasized that the estimated boundary is generally much larger than the actual settling time.

To resolve the concern, a concept referred to as prescribed/predefined-time stability was presented in the literature. A key characteristic is that neither initial states nor other parameters exert an influence on the settling time, which can thus be set in advance. This method has been widely utilized in numerous studies on the consensus issue of MASs because of this benefit. For example, in Wang et al. (2019b), a kind of prescribed-time distributed control approach based on the time-scaling function was proposed for MASs. What’s more, a predefined-time consensus tracking method based on a distributed observer was provided for the second-order MASs in Ni et al. (2021). Using a novel nonsingular terminal sliding mode surface and distributed observer, the method enabled the followers to track the leader within the predefined time. For the first-order MASs, Chen et al. (2022) introduced a prescribed-time event-triggered control scheme, making use of the time-varying scaling functions. By constructing control laws and triggering conditions that eliminate the need for continuous neighborhood communication, this scheme ensured bipartite consensus within the prescribed time. The challenge of obtaining leader-following consensus within a prescribed time in input-constrained linear MASs under generally directed graphs was addressed in Zhang et al. (2024), where a bounded linear time-varying control protocol was designed based on the parametric Lyapunov equation. However, the above articles focused on ensuring consensus before the prescribed/predefined time, without attempting to specify the precise moment when the consensus is achieved.

Most recently, the notion of prescribed-instant stability (PSIS) was first put forward in Kuang et al. (2023), in that a precise value of the settling time was obtained rather than its upper bound. Moreover, based on the Lyapunov method and backstepping technique, a control strategy for high-order integrator systems was presented in Kuang et al. (2024), which achieved precise convergence of the system at an arbitrarily specified time designated by the user. But it’s worth noting that research on the prescribed-instant stability of dynamical systems is just emerging, and there is still a lot of work to be extended and improved. For the high-order MASs under undirected graphs, Gu et al. (2023) successfully solved the problem of the prescribed-instant leaderless consensus. In Kuang et al. (2022), a prescribed-instant containment control scheme was developed for the first-order MASs under directed graphs. However, research on prescribed-instant consensus tracking for leader-follower high-order multi-agent systems (MASs) remains relatively scarce, despite the fact that this methodology is applicable to scenarios requiring strictly time-fixed convergence with high precision and reliability, such as coordinated aerial refueling of UAVs and simultaneous target interception by multi-robot teams.

Motivated by the aforementioned discussions, for the high-order MASs under directed graphs we will further study the prescribed-instant consensus tracking problem and extend it to undirected graphs. The primary contributions of this paper are outlined as follows:

(1) In order to solve the prescribed-instant consensus tracking problems, a distributed control strategy is presented based on the backstepping method.

(2) Different from the undirected connected graphs considered in Gu et al. (2023), we consider both undirected connected graphs and directed ones containing a spanning tree. The latter one is more general, and the corresponding analysis is more challenging.

(3) A general sufficient condition for prescribed-instant stability is provided, which not only plays a role in the prescribed-instant consensus analysis in this paper but also can be employed in corresponding problems in other systems.

The rest of this paper is structured as follows: Essential preliminaries, including graph theory, problem formulation, and key definitions for prescribed-instant consensus, are introduced in Section II. Section III presents the main theoretical contributions, encompassing the design of distributed controllers and the rigorous Lyapunov-based stability analysis. The simulated numerical results are given in Section IV. In the end, a summary of the main results is provided in Section V.

2. Preliminaries

2.1. Graph theory

Consider an undirected graph G = {V, E, A}. In this graph, V = {v₁, …, v_n} denotes the vertex set, E ⊆ V × V represents the edge set, and $A = [a_{i j}] \in R_{+}^{N \times N}$ is defined as the adjacency matrix. The edge (v_i, v_j) indicates that vertices v_i and v_j can communicate with each other. If (v_i, v_j) ∈ E, vertices v_i and v_j are called adjacent vertices to each other. The set of all neighbors of vertex v_j is denoted as n_j = {i: (v_i, v_j) ∈ E}. The elements of the matrix A = [a_ij] are all non-negative, that is, a_ij = a_ji ≥ 0. When (v_i, v_j)∉E, a_ij = 0; otherwise, a_ij > 0. Moreover, for i = 1, …, N, the diagonal elements satisfy a_ii = 0.

Compared with the undirected graph, node pairs in the directed graph are ordered. For example, the edge (v_i, v_j) indicates that the i-th agent is able to directly transmit information to the j-th agent, but the opposite direction is not assured. Let $D = diag (d_{1}, d_{2}, \dots, d_{N}) \in R_{+}^{N \times N}$ be a diagonal matrix, where $d_{i} = \sum_{j = 1}^{N} a_{i j}$ with i = 1, 2, …, N. The Laplacian matrix for the graph is represented as $L = D - A \in R_{+}^{N \times N}$ .

In the leader-follower MASs, an extended graph $\bar{G} = G \cup {v_{0}}$ is defined to model communication topology, where node v₀ is used to signify the leader. Additionally, the communication relationships between the leader and the followers is represented by a diagonal matrix B = diag{b₁, …, b_N}. If the information of the leader is able to reach the i-th follower, b_i = 1; otherwise, b_i = 0. The following are some lemmas and assumptions related to the Laplacian matrix and graph theory.

\begin{matrix} Q = P (L + B) + {(L + B)}^{T} P \\ P = diag {p_{i}}, p = {[p_{1}, \dots, p_{N}]}^{T} = {(L + B)}^{- T} 1_{N} \end{matrix}

(1)

where P > 0, Q >0, and 1_N denotes an N × 1 column vector with all elements equal to one.

Lemma 1

(Royle and Godsil, 2001): The Laplacian matrix L of an undirected connected graph G is positive semi-definite, and its eigenvalues can be reordered as 0 = λ₁(L) < λ₂(L) ≤ ⋯ ≤ λ_N(L).

Definition 1

(Ren and Beard, 2008): A directed graph G = {V, E, A} contains a rooted spanning tree only if there exists at least one vertex in V from which a directed path extends to every other vertex in V.

Lemma 2

(Li et al., 2014): If $\bar{G}$ includes a spanning tree, then L + B is a nonsingular M-matrix. Additionally, a positive diagonal matrix P can be found such that

2.2. Problem statement

We focus on a MASs that comprises a leader and N followers. For the i-th follower (i = 1, …, N), its dynamic model is defined by

\{\begin{cases} {\dot{x}}_{i, 1} = x_{i, 2} \\ {\dot{x}}_{i, 2} = x_{i, 3} \\ \dots \\ {\dot{x}}_{i, n} = u_{i}, \end{cases}

(2)

where x_i,j represents the j-th state of the i-th follower and u_i is the control input of the i-th follower. The dynamics of the leader is described by

\{\begin{cases} {\dot{x}}_{0,1} = x_{0,2} \\ {\dot{x}}_{0,2} = x_{0,3} \\ \dots \\ {\dot{x}}_{0, n} = u_{0}, \end{cases}

(3)

where x_n,0 is the n-th state of the leader, u₀ is the control input of the leader. Define

{\bar{x}}_{i} = {[x_{i, 1}, x_{i, 2}, \dots, x_{i, n}]}^{T} \in R^{n}

and

x_{0} = {[x_{0,1}, x_{0,2} \dots, x_{0, n}]}^{T} \in R^{n}

as the state vectors of the followers and the leader, respectively.

\dot{x} = g (x, u) .

(4)

Assumption 1

The leader’s input is restricted, which means there is a positive constant λ ∈ R⁺ ensuring that |u₀| ≤ λ.

Definition 2

(Kuang et al., 2023): Consider a system described as

For any initial states x (t₀) ∈ Rⁿ, the origin of system (4) is termed prescribed-instant stable, if there is a control input that allows the settling time to be exactly set as t_f which is achievable positive.

Based on the above definition of prescribed-instant stability, we aim to study the prescribed-instant consensus (PSIC) tracking problem for the leader-follower high-order MASs. Specifically, by designing distributed control inputs u_i, the states of the followers ${\bar{x}}_{i}$ are expected to track the states of the leader x₀. In detail, at the prescribed instant, it holds that ${‖ {\bar{x}}_{i} - x_{0} ‖}_{2} = 0$ , where t_f is an actual settling time instead of an upper bound. In addition, we need to ensure that ${‖ {\bar{x}}_{i} - x_{0} ‖}_{2} \equiv 0$ when t > t_f.

3. Main results

With the goal of solving the prescribed-instant consensus tracking problem, we first introduce a kind of time-varying function ϖ(t), which possesses the following properties.

(1) ϖ(t₀) > 0, where t₀ ≥ 0 denotes the initial time;

(2) ϖ(t) is continuous, monotonically increasing, and differentiable on the interval [t₀, t_f) where t_f is the prescribed time instant; for t ∈ [t_f, ∞), ϖ(t) maintains a constant value and $\dot{ϖ} (t) = 0$ ;

(3) For any q > 0, ϖ(t)^−q is monotonically decreasing with t ∈ [t₀, t_f) and $\lim_{t \to t_{f}^{-}} ϖ {(t)}^{- q} = 0$ ;

\{\begin{cases} \dot{V} (t) \leq - γ_{1} (γ + c \frac{\dot{ϖ}}{ϖ}) V (t), t \in [t_{0}, t_{f}) (a) \\ \dot{V} (t) \geq - γ_{2} (γ + c \frac{\dot{ϖ}}{ϖ}) V (t), t \in [t_{0}, t_{f}) (b) \end{cases}

(5)

and

\dot{V} (t) \leq - γ_{3} V (t), t \in [t_{f}, \infty),

(6)

where c > 0, γ > 0, γ₂ > γ₁ > 0, γ₃ > γγ₁, then the origin of the system (4) can be prescribed-instant stable.

Lemma 3

For system (4), if a differentiable positive definite function V(t) exists that satisfies the following condition

Proof

From (5) (a), it is straightforward to deduce that $\dot{V} (t) \leq 0$ . In addition, by solving the differential inequality (5) (a), we can obtain $V (t) \leq ϖ {(t)}^{- c γ_{1}} e^{- γ_{1} γ (t - t_{0})} V (t_{0})$ . According to the properties of ϖ(t), it can be inferred that V(t) will become zero before or at the time instant t = t_f, meaning that x(t) will approach zero before or at the time instant t = t_f.

Similarly, by solving the differential inequality (5) (b), we can obtain $V (t) \geq ϖ {(t)}^{- c γ_{2}} e^{- γ_{2} γ (t - t_{0})} V (t_{0})$ . Consequently, we can draw the conclusion that V(t) cannot become zero before the instant t = t_f, which means that x(t) cannot converge to zero before t_f.

Combining the above two conclusions, we can know that V(t) approaches zero at t_f, which implies that x(t) achieves the origin at t_f. Furthermore, from (6), we have that $\dot{V} (t) < 0$ on $t \in [t_{f}, \infty)$ and thus 0 ≤ V(t) ≤ V (t_f) = 0 on [t_f, ∞), which yields V(t) ≡ 0 when t ∈ [t_f, ∞). Consensus can be maintained continuously and the conclusion is established.

3.1. PSIC of high-order MASs under directed graphs

In the following, we suppose that the augmented graphs $\bar{G}$ , corresponding to the communication topology of one leader and n followers, contains a spanning tree with the leader as the root node. For the purpose of achieving prescribed-instant consensus tracking in high-order MASs, we adopt the backstepping method for the design and analysis, which will be presented in a recursive form.

Before presenting the detailed results, we first provide the definitions of some variables. For the MASs dynamics (2) and (3), define the tracking errors as

\{\begin{cases} δ_{i, 1} = x_{i, 1} - φ_{i, 1 d} \\ δ_{i, 2} = x_{i, 2} - φ_{i, 2 d} \\ \dots \\ δ_{i, n} = x_{i, n} - φ_{i, n d}, \end{cases}

(7)

where φ_i,kd is the virtual control input with i = 1, 2, …, n and k = 1, 2, …, n. Moreover, the first virtual control input φ_i,1d is defined as φ_i,1d = x_0,1. The derivatives of the tracking errors are

\{\begin{cases} {\dot{δ}}_{i, 1} = δ_{i, 2} + φ_{i, 1 d} - {\dot{φ}}_{i, 1 d} \\ {\dot{δ}}_{i, 2} = δ_{i, 3} + φ_{i, 3 d} - {\dot{φ}}_{i, 2 d} \\ \dots \\ {\dot{δ}}_{i, n} = u_{i} - {\dot{φ}}_{i, n d} . \end{cases}

(8)

Since the followers have the ability to obtain both their own states and the states of their neighbors, we define the first-order consensus error z_i,1 for each follower as

z_{i, 1} = \sum_{j = 1}^{n} a_{i j} (x_{i, 1} - x_{j, 1}) + b_{i} (x_{i, 1} - x_{0,1}) .

(9)

Substituting (7) into (9), we obtain

\begin{matrix} z_{i, 1} = & \sum_{j = 1}^{n} a_{i j} ((x_{i, 1} - φ_{i, 1 d}) - (x_{j, 1} - φ_{i, 1 d})) \\ + b_{i} (x_{i, 1} - φ_{i, 1 d}) \\ = & \sum_{j = 1}^{n} a_{i j} (δ_{i, 1} - δ_{j, 1}) + b_{i} δ_{i, 1} . \end{matrix}

(10)

Denote $x_{i} = {[x_{1, i}, x_{2, i}, \dots, x_{N, i}]}^{T} \in R^{n}$ , $z_{1} = [z_{1,1}, z_{2,1}$ , ${\dots, z_{N, 1}]}^{T} \in R^{N}, δ_{i} = {[δ_{1, i}, δ_{2, i}, \dots, δ_{N, i}]}^{T} \in R^{N}, x_{0 i} = {[x_{0, i}, x_{0, i}, \dots, x_{0, i}]}^{T} \in R^{N}, φ_{1 d} = x_{01}, φ_{i d} = [φ_{1, i d}, φ_{2, i d}$ , ${\dots, φ_{N, i d}]}^{T} \in R^{N}$ (i > 2), $u_{n} = {[u_{1}, u_{2}, \dots, u_{N}]}^{T} \in R^{N}$ and i = 1, 2, …, N. Then, (10) can be vectorized as z₁ = (L + B)δ₁.

The backstepping method consists of n steps in total. At each step, we will determine φ_id to ensure that x_i can track φ_id at the prescribed time instant. After completing all n steps, the prescribed-instant consensus of the high-order MASs will be achieved.

Step 1: Let V₁ be the defined Lyapunov function

V_{1} = \frac{1}{2} δ_{1}^{T} P δ_{1},

(11)

where P is derived from Lemma 2. And its derivative is

\begin{matrix} {\dot{V}}_{1} = & δ_{1}^{T} P {\dot{δ}}_{1} \\ = & δ_{1}^{T} P (δ_{2} + φ_{2 d} - {\dot{φ}}_{1 d}), \end{matrix}

where φ_2d, the virtual input of x₂, is designed as

φ_{2 d} = - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{1} + {\dot{φ}}_{1 d},

where γ, c are positive constants, ϖ and

\dot{ϖ}

are the time-varying functions with certain properties mentioned earlier. Substituting φ_2d into

{\dot{V}}_{1}

yields

\begin{matrix} {\dot{V}}_{1} = & δ_{1}^{T} P (δ_{2} - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{1}) \\ = & δ_{1}^{T} P δ_{2} - \frac{1}{2} (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{1}^{T} P δ_{1} . \end{matrix}

It should be noted that the virtual input φ_2d in this case does not represent the actual controller itself but serves as an intermediate variable in the design process, which will ultimately be translated into the actual controller.

Step 2: The objective of this step is to design φ_3d, ensuring the real state x₂ tracks the virtual input φ_2d at the prescribed time instant. Define the candidate Lyapunov function as

V_{2} = V_{1} + \frac{1}{2} δ_{2}^{T} P δ_{2} .

(12)

Its derivative is

\begin{matrix} {\dot{V}}_{2} = & {\dot{V}}_{1} + δ_{2}^{T} P {\dot{δ}}_{2} \\ = & δ_{1}^{T} P δ_{2} - \frac{1}{2} (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{1}^{T} P δ_{1} \\ + δ_{2}^{T} P (δ_{3} + φ_{3 d} - {\dot{φ}}_{2 d}) . \end{matrix}

Choose the virtual input φ_3d as

φ_{3 d} = - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{2} + {\dot{φ}}_{2 d} - δ_{1} .

Furthermore, it leads to

\begin{matrix} {\dot{V}}_{2} = & \frac{1}{2} (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{1}^{T} P δ_{1} + δ_{2}^{T} P (δ_{3} - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{2}) \\ = & - (γ + c \frac{\dot{ϖ}}{ϖ}) (\frac{1}{2} δ_{1}^{T} P δ_{1} + δ_{2}^{T} P δ_{2}) + δ_{2}^{T} P δ_{3} . \end{matrix}

Step i (i = 2, 3, …, n-1): Choose the following candidate Lyapunov function

V_{i} = V_{i - 1} + \frac{1}{2} δ_{i}^{T} P δ_{i} .

(13)

The derivative of it is derived as

\begin{matrix} {\dot{V}}_{i} = & {\dot{V}}_{i - 1} + δ_{i}^{T} P {\dot{δ}}_{i} \\ = & - (γ + c \frac{\dot{ϖ}}{ϖ}) \sum_{l = 1}^{i - 1} δ_{l}^{T} P δ_{l} + δ_{i - 1}^{T} P δ_{i} \\ + δ_{i}^{T} P (δ_{i + 1} + φ_{i + 1 d} - {\dot{φ}}_{i d}) . \end{matrix}

The virtual input φ_i+1d is selected as

φ_{i + 1 d} = - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{i} + {\dot{φ}}_{i d} - δ_{i - 1} .

Taking φ_id into ${\dot{V}}_{i}$ , we have

\begin{matrix} {\dot{V}}_{i} = & - (γ + c \frac{\dot{ϖ}}{ϖ}) \sum_{l = 1}^{i - 1} δ_{l}^{T} P δ_{l} \\ + δ_{i}^{T} P (δ_{i + 1} - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{i}) \\ = & - (γ + c \frac{\dot{ϖ}}{ϖ}) \sum_{l = 1}^{i} δ_{l}^{T} P δ_{l} + δ_{i}^{T} P δ_{i + 1} . \end{matrix}

Step n: The candidate Lyapunov function is defined as

V_{n} = V_{i} + \frac{1}{2} δ_{n}^{T} P δ_{n} .

(14)

Its derivative is

\begin{matrix} {\dot{V}}_{n} = & {\dot{V}}_{i} + δ_{n}^{T} P {\dot{δ}}_{n} \\ = & - (γ + c \frac{\dot{ϖ}}{ϖ}) \sum_{l = 1}^{n - 1} δ_{l}^{T} P δ_{l} + δ_{n - 1}^{T} P δ_{n} \\ + δ_{n}^{T} P (u_{n} - {\dot{φ}}_{n d}) . \end{matrix}

Following the recursive backstepping analysis discussed above, the subsequent theorem will be further presented.

u_{N} = - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{n} + {\dot{φ}}_{n d} - δ_{n - 1} .

Theorem 1

Under directed graphs containing a spanning tree with the leader as the root node, for the multiple follower systems (2) and the leader system (3), if the control input of the i-th agent is designed as

u_{i} = - (γ + c \frac{\dot{ϖ}}{ϖ}) (x_{i, n} - φ_{i, n d}) + {\dot{φ}}_{i, n d} - (x_{i, n - 1} - φ_{i, n - 1 d}),

(15)

then the prescribed-instant consensus tracking problem of the systems (2)-(3) can be solved.

Proof

Based on (15), the control input u_n can be written as

Substituting u_n into ${\dot{V}}_{n}$ yields

{\dot{V}}_{n} = - (γ + c \frac{\dot{ϖ}}{ϖ}) \sum_{l = 1}^{n} δ_{l}^{T} P δ_{l} .

(16)

From the above analysis, P are positive definite matrices. Therefore, that it leads to

\begin{matrix} {\dot{V}}_{n} \leq & - (γ + c \frac{\dot{ϖ}}{ϖ}) (λ_{\min} (P)) \sum_{l = 1}^{n} {‖ δ_{l} ‖}^{2} \\ {\dot{V}}_{n} \geq & - (γ + c \frac{\dot{ϖ}}{ϖ}) (λ_{\max} (P)) \sum_{l = 1}^{n} {‖ δ_{l} ‖}^{2} . \end{matrix}

(17)

Denote $ζ = [‖ δ_{1} ‖$ , ${‖ δ_{2} ‖, \dots, ‖ δ_{n} ‖]}^{T}$ , then (17) turns to

\begin{matrix} - λ_{\max} (P) (γ + c \frac{\dot{ϖ}}{ϖ}) {‖ ζ ‖}^{2} \leq {\dot{V}}_{n} \leq - λ_{\min} (P) (γ + c \frac{\dot{ϖ}}{ϖ}) {‖ ζ ‖}^{2} . \end{matrix}

Because of

\begin{matrix} \frac{2}{λ_{\max} (P)} V_{n} \leq {‖ ζ ‖}^{2} \leq \frac{2}{λ_{\min} (P)} V_{n}, \end{matrix}

(18)

we can obtain the following inequality

\begin{matrix} - 2 \frac{λ_{\max} (P)}{λ_{\min} (P)} (γ + c \frac{\dot{ϖ}}{ϖ}) V_{n} \leq {\dot{V}}_{n} \leq - 2 \frac{λ_{\min} (P)}{λ_{\max} (P)} (γ + c \frac{\dot{ϖ}}{ϖ}) V_{n} . \end{matrix}

(19)

From Lemma 3, we can get the conclusion that V_n will converge to zero at t_f. Then, we will get z₁ = δ_i = 0_N where 0_N denotes an N × 1 column vector with all elements equal to zero. Moreover, at the prescribed time instant t_f, the virtual controllers and control input u_n will take the following form

\{\begin{matrix} φ_{2 d} = {\dot{φ}}_{1 d} - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{1} = {\dot{φ}}_{1 d} = x_{02} \\ φ_{3 d} = {\dot{φ}}_{2 d} - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{2} - δ_{1} = {\dot{φ}}_{2 d} = x_{03} \\ \dots \\ φ_{i + 1 d} = {\dot{φ}}_{i d} - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{i} - δ_{i - 1} = {\dot{φ}}_{i d} = x_{0 i + 1} \\ \dots \\ u_{n} = {\dot{φ}}_{n d} - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{n} - δ_{n - 1} = {\dot{φ}}_{n d} = {\dot{x}}_{0 n} . \end{matrix}

(20)

Furthermore, because the tracking errors equal to zero at t_f, we can infer that

\{\begin{matrix} x_{i, 1} = φ_{i, 1 d} = x_{0,1} \\ x_{i, 2} = φ_{i, 2 d} = x_{0,2} \\ \dots \\ x_{i, n} = φ_{i, n d} = x_{0, n} . \end{matrix}

(21)

As a result, the consensus tracking is reached at the prescribed time instant t_f. In addition, according to the tracking errors δ₁ = 0, the first-order consensus error z₁ = (L + B)δ₁ = 0 at the prescribed time instant t_f.

When t ∈ [t_f, ∞), following similar analyses as above and considering $\dot{ϖ} = 0$ , we can obtain that

\begin{matrix} {\dot{V}}_{n} \leq - 2 γ \frac{λ_{\min} (P)}{λ_{\max} (P)} V_{n} \leq 0, t \in [t_{f}, \infty) . \end{matrix}

(22)

Because V(t) is a continuous function and V (t_f) = 0, it follows that

\begin{matrix} 0 \leq V_{n} (t) \leq V_{n} (t_{f}) = 0, t \in [t_{f}, \infty) . \end{matrix}

(23)

That is, V_n(t) ≡ 0 over the interval [t_f, ∞). Therefore, δ_i ≡0_N and z₁ ≡0_N on [t_f, ∞) then we can draw the conclusion that consensus tracking will continue to be maintained. This completes the proof.

Remark 1

From (15) and the expressions of each virtual control input, the control input for every agent is determined solely by its own states and the states of its neighboring agents, thereby eliminating the need for any global information. Therefore, the method introduced in this paper is a fully distributed control approach.

Remark 2

Kuang et al. (2022) investigated the prescribed-instant consensus for first-order multi-agent systems. In contrast, the paper proposes a controller via the backstepping method to solve the prescribed-instant consensus problem of leader-following high-order multi-agent systems under directed graphs. By comparison, the result proposed in the paper is of greater general significance.

3.2. PSIC of high-oder MASs under undirected graphs

Building on the research on the leader-follower PSIC of MASs under directed graphs, we further investigate the case of the leader-follower PSIC of MASs under undirected graphs. The connected undirected graphs will be considered.

Following the variables and vectors defined in the previous section, the virtual control inputs and the control input u_n are designed as

\{\begin{matrix} φ_{2 d} = - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{1} + {\dot{φ}}_{1 d} \\ φ_{3 d} = - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{2} + {\dot{φ}}_{2 d} - δ_{1} \\ \dots \\ φ_{i + 1 d} = - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{i} + {\dot{φ}}_{i d} - δ_{i - 1} \\ \dots \\ u_{n} = - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{n} + {\dot{φ}}_{n d} - δ_{n - 1} . \end{matrix}

(24)

From u_n in (24), it indicates that the control input for the i-th agent is

u_{i} = - (γ + c \frac{\dot{ϖ}}{ϖ}) δ_{i, n} + {\dot{φ}}_{i, n - 1 d} - δ_{i, n - 1} .

(25)

V_{n} = \frac{1}{2} δ_{1}^{T} P δ_{1} + \frac{1}{2} δ_{2}^{T} P δ_{2} + \dots + \frac{1}{2} δ_{n}^{T} P δ_{n} .

(26)

Theorem 2

Under undirected connected graphs, for the multiple follower systems (2) and the leader system (3), if the control input of the i-th agent is designed as (25), then the prescribed-instant consensus tracking problem of systems (2)-(3) can be achieved.

Proof

Define the candidate Lyapunov function as

Its derivative is

\begin{matrix} {\dot{V}}_{n} = & δ_{1}^{T} P {\dot{δ}}_{1} + δ_{2}^{T} P {\dot{δ}}_{2} + \dots + δ_{n}^{T} P {\dot{δ}}_{n} \\ = & δ_{1}^{T} P (δ_{2} + φ_{2 d} - {\dot{φ}}_{1 d}) \\ + δ_{2}^{T} P (δ_{3} + φ_{3 d} - \dot{φ_{2 d}}) + \dots + δ_{n}^{T} P (u_{i, n} - {\dot{φ}}_{n d}) . \end{matrix}

(27)

Substituting the respective virtual control input and u_n into (28), we have

\begin{matrix} {\dot{V}}_{n} = - (γ + c \frac{\dot{ϖ}}{ϖ}) \sum_{l = 1}^{n} δ_{l}^{T} P δ_{l} . \end{matrix}

Then, we can obtain the following inequalities

\begin{matrix} - 2 \frac{λ_{\max} (P)}{λ_{\min} (P)} (γ + c \frac{\dot{ϖ}}{ϖ}) V_{n} \leq {\dot{V}}_{n} \leq - 2 \frac{λ_{\min} (P)}{λ_{\max} (P)} (γ + c \frac{\dot{ϖ}}{ϖ}) V_{n} . \end{matrix}

(28)

According to Lemma 3, it can be inferred that V_n will converge to zero at time t_f. This implies that at the prescribed time instant t_f, z₁ = δ_i = 0_N. Since z₁ = 0_N, then x₁ = φ_1d = x₀₁ and $φ_{2 d} = {\dot{φ}}_{1 d} = x_{02}$ . Because of δ₂ = 0_N, we can get x₂ = φ_2d = x₀₂. Following this step, we can eventually obtain that ${‖ {\bar{x}}_{i} - x_{0} ‖}_{2} = 0$ at the prescribed time instant t_f.

Subsequently, for the time interval after t_f, we can obtain that

\begin{matrix} {\dot{V}}_{n} \leq - 2 γ \frac{λ_{\min} (P)}{λ_{\max} (P)} V_{n} \leq 0, t \in [t_{f}, \infty) . \end{matrix}

(29)

Therefore,

\begin{matrix} 0 \leq V_{n} (t) \leq V_{n} (t_{f}) = 0, t \in [t_{f}, \infty) . \end{matrix}

(30)

Furthermore, V_n(t) ≡ 0 on [t_f, ∞), and δ_i ≡0_N and z₁ ≡0_N on [t_f, ∞). Therefore, we have demonstrated that tracking consensus is maintained over the interval [t_f, ∞). The proof is thus completed.

Remark 3

Compared with other control methods, the prescribed-instant consensus control method proposed in this paper neither suffers from the issue that the settling time depends on initial states (as in finite-time consensus control methods) nor the problem that the settling time relies on fixed parameters (as in fixed-time consensus control methods), nor does it have the drawback that the settling time is merely an upper bound (as in prescribed/predefined-time consensus control schemes).

Remark 4

In contrast to Gu et al. (2023) and Kuang et al. (2022), we propose a more generalized time-varying function to solve the prescribed-instant consensus problem of high-order multi-agent systems rather than mandatorily requiring the presence of t − t_f in the denominator of ϖ(t).

4. Simulations

Two examples will be contained in this section to illustrate the accuracy of our proposed methods.

ϖ (t) = \{\begin{matrix} \frac{T^{h}}{{(T + t_{0} - t)}^{h}} & t \in [t_{0}, t_{f}) \\ 1 & t \in [t_{f}, \infty), \end{matrix}

where h is a positive constant, t_f = t₀ + T, and T ≥ T_s > 0 with T_s representing the duration needed for processing signals and transmitting information. It is evident that the selection of ϖ(t) satisfies the conditions enumerated in Section III.

Example 1

(high-order MASs under directed graphs) Before presenting the simulation details, we first provide the specific expression form of ϖ(t):

We investigate a multi-agent system that includes a leader and five followers, where each agent is of the third order. The directed graph is shown in Figure 1. The parameters are set as T = 4s, t₀ = 0s, t_f = 4s, h = 1, γ = 1.5, c = 0.5. The initial states of the followers are set to x₁ (0) = [−1,0,1]^T, x₂ (0) = [−1.6,0.4,2.5]^T, x₃ (0) = [−2.4,−3,3]^T, $x_{4} (0) = [3.1, 1$ , ${- 4.1]}^{T}$ , x₅ (0) = [1.8,−3.6,−2.3]^T. The initial states of the leader are chosen as x₀ (0) = [0,2,0]^T with the input u₀ =-cos(t) + 0.3. Figures 2 –4 demonstrate the realization process of the prescribed-instant tracking consensus. From the dynamic trajectories of each agent provided in Figures 2 –4, it can be observed that the each follower’s states at three orders converge to the leader’s corresponding states at the prescribed time instant 4s and continue to maintain this consensus thereafter.

ϖ (t) = \{\begin{matrix} e^{\frac{h}{T + t_{0} - t}} & t \in [t_{0}, t_{f}) \\ 1 & t \in [t_{f}, \infty) . \end{matrix}

Figure 1.

The directed graph of Example 1.

Figure 2.

The tracking trajectories of the first-order states under directed graph.

Figure 3.

The tracking trajectories of the second-order states under directed graph.

Figure 4.

The tracking trajectories of the third-order states under directed graph.

Example 2

(high-order MASs under undirected graphs) In the simulation of undirected graphs, we choose another form of ϖ(t) as

Consider a multi-agent system consisting of one leader and nine followers, where each agent is of the third order, and Figure 5 shows the undirected graph. The parameters are set as T = 2.5s, t₀ = 0 s, t_f = 2.5s, h = 1.5, γ = 1, c = 0.5. The initial states of the followers are set to $x_{1} (0) = {[2, - 1,0.5]}^{T}, x_{2} (0) = {[- 0.8, 3, - 2]}^{T}, x_{3} (0) = [1.2$ , ${- 1.5, 4]}^{T}$ , x₄ (0) = [−3,2.2,−0.7]^T, x₅ (0) = [0.6,−4,1.8]^T, $x_{6} (0) = {[- 2.5, 1, - 3.2]}^{T}, x_{7} (0) = {[3.3, - 0.5, 2.7]}^{T}, x_{8} (0) = [- 1.1$ , ${2.8, - 1.3]}^{T}$ , x₉ (0) = [4,−2.1,0.9]^T. The initial states of the leader are chosen as x₀ (0) = [0,0,1]^T with the input u₀ = 3t. Figures 6 –8 show that the states of each follower achieve consensus at the prescribed time instant t = 2.5s. All followers successfully track the leader and sustain this tracking consensus moving forward.

Figure 5.

The undirected graph of Example 2.

Figure 6.

The tracking trajectories of the first-order states under undirected graph.

Figure 7.

The tracking trajectories of the second-order states under undirected graph.

Figure 8.

The tracking trajectories of the third-order states under undirected graph.

5. Conclusion

The PSIC tracking problem of the leader-follower high-order MASs has been addressed in this paper. With the help of a kind of time-varying functions with specific properties, a sufficient condition for the prescribed-instant stability of general single-loop systems was firstly proposed. After that, relying on the backstepping method, a distributed control input was developed to realize PSIC tracking of MASs under directed graphs that include a directed spanning tree. Consensus analyses were carried out based on the sufficient condition for prescribed-instant stability as well as the algebraic graphs theory. Moreover, the case of undirected connected graphs was taken into account. To verify the proposed methods, numerical examples were shown in the simulation. In future work, we intend to investigate the prescribed-instant consensus tracking problem for multi-agent systems in the presence of nonlinear uncertainties.

Footnotes

ORCID iDs

Yongkang Su

Na Li

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grant 61703225 and Shandong Provincial Natural Science Foundation under Grant ZR2022MF297.

Declaration of conflicting interests

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

References

Cao

Ren

(2011) Distributed coordinated tracking with reduced interaction via a variable structure approach. IEEE Transactions on Automatic Control 57(1): 33–48. https://doi.org/10.1109/tac.2010.2049517

Chen

Hao

(2022) Prescribed-time event-triggered bipartite consensus of multiagent systems. IEEE Transactions on Cybernetics 52(4): 2589–2598. https://doi.org/10.1109/TCYB.2020.3004572

Kuang

, et al. (2023) A prescribed-instant convergence approach to consensus of high-order multiagent systems. In: 2023 China Automation Congress. CAC, pp. 4498–4503.

Karimoddini

Lin

Chen

, et al. (2013) Hybrid three-dimensional formation control for unmanned helicopters. Automatica 49(2): 424–433. https://doi.org/10.1016/j.automatica.2012.10.008

Kuang

Zhang

Gao

, et al. (2022) Leader-follower multiagent systems containment with prescribed instant. In: IECON 2022-48th Annual Conference of the. IEEE Industrial Electronics Society, 1–6.

Kuang

Gao

Chen

, et al. (2023) Stabilization with prescribed instant for high-order integrator systems. IEEE Transactions on Cybernetics 53(11): 7275–7284. https://doi.org/10.1109/TCYB.2022.3212409

Kuang

Gao

Sun

, et al. (2024) Stabilization with prescribed instant via lyapunov method. IEEE/CAA Journal of Automatica Sinica 11(2): 557–559. https://doi.org/10.1109/jas.2023.123801

Wen

Duan

, et al. (2014) Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs. IEEE Transactions on Automatic Control 60(4): 1152–1157. https://doi.org/10.1109/tac.2014.2350391

Liu

Tang

, et al. (2021) Predefined-time consensus tracking of second-order multiagent systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems 51(4): 2550–2560. https://doi.org/10.1109/tsmc.2019.2916257

10.

Orlov

(2004) Finite time stability and robust control synthesis of uncertain switched systems. SIAM Journal on Control and Optimization 43(4): 1253–1271. https://doi.org/10.1137/s0363012903425593

11.

Pei

(2023) Group consensus of multi-agent systems with hybrid characteristics and directed topological networks. ISA Transactions 138: 311–317. https://doi.org/10.1016/j.isatra.2023.03.030

12.

Pei

Liang

(2025) Observer-based event-triggered group consensus tracking of hybrid multi-agent systems. Transactions of the Institute of Measurement and Control. Available at: https://doi.org/10.1177/01423312251358122, In press.

13.

Polyakov

(2011) Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Transactions on Automatic Control 57(8): 2106–2110. https://doi.org/10.1109/tac.2011.2179869

14.

Ren

Beard

(2008) Distributed Consensus in multi-vehicle Cooperative Control. Springer.

15.

Richert

Cortés

(2013) Optimal leader allocation in UAV formation pairs ensuring cooperation. Automatica 49(11): 3189–3198. https://doi.org/10.1016/j.automatica.2013.07.030

16.

Royle

Godsil

(2001) Algebraic Graph Theory. Springer.

17.

Shahvali

Askari

(2016) Cooperative adaptive neural partial tracking errors constrained control for nonlinear multi-agent systems. International Journal of Adaptive Control and Signal Processing 30(7): 1019–1042. https://doi.org/10.1002/acs.2657

18.

Shahvali

Shojaei

(2016) Distributed adaptive neural control of nonlinear multi-agent systems with unknown control directions. Nonlinear Dynamics 83(4): 2213–2228. https://doi.org/10.1007/s11071-015-2476-4

19.

Shahvali

Pariz

Akbariyan

(2018a) Distributed finite-time control for arbitrary switched nonlinear multi-agent systems: an observer-based approach. Nonlinear Dynamics 94(3): 2127–2142. https://doi.org/10.1007/s11071-018-4479-4

20.

Shahvali

Naghibi-Sistani

Askari

(2018b) Adaptive output-feedback bipartite consensus for nonstrict-feedback nonlinear multi-agent systems: a finite-time approach. Neurocomputing 318: 7–17. https://doi.org/10.1016/j.neucom.2018.07.039

21.

Wang

Sun

, et al. (2019a) Distributed coordinated attitude tracking control for spacecraft formation with communication delays. ISA Transactions 85: 97–106. https://doi.org/10.1016/j.isatra.2018.10.028

22.

Wang

Song

Hill

, et al. (2019b) Prescribed-time consensus and containment control of networked multiagent systems. IEEE Transactions on Cybernetics 49(4): 1138–1147. https://doi.org/10.1109/TCYB.2017.2788874

23.

Liu

Chen

, et al. (2015) Improved artificial moment method for decentralized local path planning of multirobots. IEEE Transactions on Control Systems Technology 23(6): 2383–2390. https://doi.org/10.1109/tcst.2015.2402231

24.

Zhang

Zhou

Yang

, et al. (2024) Prescribed-time leader-following consensus of linear multi-agent systems by bounded linear time-varying protocols. Science China Information Sciences 67(1): 112201. https://doi.org/10.1007/s11432-022-3685-3

25.

Zhao

Duan

Wen

, et al. (2013) Distributed finite-time tracking control for multi-agent systems: an observer-based approach. Systems & Control Letters 62(1): 22–28. https://doi.org/10.1016/j.sysconle.2012.10.012