Distributed Adaptive Fixed-Time Tracking Consensus Control for Multiple Uncertain Nonlinear Strict-Feedback Systems under a Directed Graph
In this brief, we study the distributed adaptive fixed-time tracking consensus control problem for multiple strict-feedback systems with uncertain nonlinearities under a directed graph topology. It is assumed that the leader’s output is time varying and has been accessed by only a small fraction of followers in a group. The distributed fixed-time tracking consensus control is proposed to design local consensus controllers in order to guarantee the consensus tracking between the followers and the leader and ensure the error convergence time is independent of the systems’ initial state. The function approximation technique using radial basis function neural networks (RBFNNs) is employed to compensate for unknown nonlinear terms induced from the controller design procedure. From the Lyapunov stability theorem and graph theory, it is shown that, by using the proposed fixed-time control strategy, all signals in the closed-loop system and the consensus tracking errors are cooperatively semiglobally uniformly bounded and the errors converge to a neighborhood of the origin within a fixed time. Finally, the effectiveness of the proposed control strategy has been proved by rigorous stability analysis and two simulation examples.
In the past decade, the finite-time consensus control of the multiagent has become one of the hot topics, since the finite-time control has a faster convergence rate, higher precision, and more robustness. However, there is a limitation for finite-time control that the convergence time increases accordingly when the initial state of the system is far from the equilibrium point. To overcome this limitation, fixed-time stability control  was proposed in 2012 by Polyakov et al.
In 1965, finite-time control was first proposed, and many related research results emerged in the following decades. Due to the different agent dynamics in multiagent systems (MASs), there are various finite-time control developed [2–7]. To list a few, in [8, 9], by utilizing the principle of homogeneity , the authors proposed finite-time consensus control protocols for first-order and second-order MASs, respectively. In , power integrator technique was employed to solve the problem of finite-time consensus control for MASs with double integrator dynamics. Furthermore, in , by combining the homogeneous domination method with the power integrator method, the authors proposed two classes of finite-time consensus protocols for MASs composed of first-order and second-order integrator agents. Thereafter, by using the Lyapunov-function-based method, several finite-time consensus control strategies were proposed for high-order uncertain MASs [13–15]. Even so, there is a limitation for the finite-time consensus control results that the convergence time is seriously dependent on the MASs’ initial states. That is to say, once the initial state is far away from the equilibrium point, the convergence time will increase as a result. In practice, it is much more reasonable for a predicable convergence time.
To overcome the limitation of finite-time control, a serial of research results [16–26] on fixed-time consensus control for MASs has been carried out since its setting time can be bounded and predefined. Note that most of the fixed-time control results [16–20] were designed for MASs with first order or second order. Only a few results [21–26] solved the fixed-time consensus problems for high-order MASs. On the contrary, strict-feedback or lower triangular systems have been widely regarded as a control target due to the description of various physical systems . In fact, the strict-feedback systems require the recursive and systematic control design procedure since the control input is not matched with system nonlinearities. In , the adaptive backstepping design was proposed to satisfy this requirement. Furthermore, in [29, 30], the adaptive neural network backstepping controllers were proposed for a class of strict-feedback nonlinear systems. Nevertheless, there is a problem with using backstepping to design a controller for the strict-feedback systems; it can cause “explosion of complexity.” To overcome this problem, in , a dynamic surface design technique was proposed for strict-feedback systems. Furthermore, in , neural network-based adaptive dynamic surface control was proposed. Subsequently, this method was extended to MASs in [33–35], and a distributed adaptive dynamic surface design technique was developed.
Motivated by the above observations, in this paper, a distributed adaptive fixed-time tracking consensus control is proposed for multiple strict-feedback systems with unknown nonlinearities under a directed graph topology. In the design, we employed RBF neural networks to compensate for unknown nonlinear terms and some functions that are difficult to calculate induced from the controller design procedure. From Lyapunov stability theorem and fixed-time control theory, by using distributed coordinate conversion, virtual controllers and actual controllers were designed. Finally, a novel distributed fixed-time consensus control protocol is designed for the considered strict-feedback nonlinear MASs, which can guarantee that all signals in the total closed-loop systems are bounded and the tracking errors can quickly converge to the neighborhood of the origin within the fixed time. The principal contributions of this paper are as follows. (1) A novel method to solve distributed adaptive fixed-time tracking consensus control for strict-feedback nonlinear MASs has been proposed. Compared with some existing results [16–26], our control strategy, by using adaptive RBF neural network control algorithm, is applicable to high-order MASs with different unknown nonlinear functions and order of dynamics. (2) In order to solve the “complexity explosion” problem caused by the repeated differentiation in the controller design process, it is different from the traditional dynamic surface technique; this paper constructs a smooth function derived from Lyapunov stability to compensate the part of in the differential of the virtual controller , while the other part of the differential is approximated by RBF neural networks. This method greatly simplifies the design process of the controller while ensuring Lyapunov stability. (3) In [36–39], their controller designed have a power function similar to , . However, incorrect selection of will lead to singularity. In this paper, a novel fixed-time controller is proposed to solve this problem. (4). RBF neural networks are introduced to approximate the unknown functions in MASs to make our designed fixed-time control can suitable more systems with complex dynamics.
The organization for the remaining part of this paper is given below. Section 2 presents preliminaries and problem description. Sections 3 and 4 give the detailed process of distributed fixed-time control protocol and stability analysis, respectively. In Section 5, the proposed control scheme is proved to be effective through a simulation experiment. Finally, conclusions are summarized in Section 6.
2. Preliminaries and Problem Description
2.1. Graph Theory
Let be a directed graph with the set of nodes or vertices and the set of edges or arcs . An edge means that agent can obtain information from agent j, but agent j cannot obtain agent ’s. The set of neighbors of a node is , which is the set of nodes with edges incoming to node . The weighted adjacency matrix is defined as if and otherwise. Self-loop is not allowed, i.e., . The Laplacian matrix is defined as , where ; is the diagonal element of the degree matrix .
A directed path from node to node is a sequence of edges of the form in a directed graph. A directed tree is a directed graph where every node has exactly one parent except for the root and the root has directed paths to every other node. A directed graph has a directed spanning tree if there exists at least a node having a directed path to all the other nodes.
An RBF neural network  is applied in this paper to approximate arbitrary continuous functions. The RBF neural network is defined as follows:where is the weight vector, is the number of nodes of the neural network, is the input of the RBF neural network, is the input dimension, is the basis vector function, and is the output of the th neural node. A Gaussian function is always chosen as , i.e., , , where is the width of the base function and is the center of the basis function. With a sufficient number of neural nodes selected, an RBF neural network can approximate arbitrary continuous function in a compact set with arbitrary accuracy :where is the approximation error with and is the given ideal constant weight vector, which is defined as follows:
In this paper, let with , where are the estimates of the unknown constants , is the ideal weight vector of the RBF neural network, is a positive design parameter, and is the norm.
Assumption 1. (see ). There are unknown constants that make , .
2.3. Fixed Time
Definition 1. Consider the following nonlinear system:where and , and assume that the origin is an equilibrium point.
Lemma 2. (see ). If there exist design constants , , , and such thatwhere is a continuous differentiable positive definite function; system (5) is global fixed-time stable, and the fixed convergence time satisfies
Remark 2. The sufficient conditions and convergence time for finite-time and fixed-time consensus control schemes are shown in Table 1.Where , , , and are positive design parameters, , and . From convergence time, it can be seen that the finite-time control is related to the initial state , while the fixed-time control is not.
Lemma 3. (see ). If there exist design constants , , , , , and such thatThen, the origin of system (5) is practical fixed-time stable and the fixed-time can be estimated byThe residual set of the solution of system (5) is given by
Lemma 4. (see ). Let . Then,
Lemma 5. (see ). For any variable and any positive constant , the following relationship holds:
Lemma 6. For , , and any positive constant , then satisfying
2.4. Problem Statement
Suppose that there exist one leader agent, labeled as , and M follower agents, labeled as to , under a directed communication graph topology. The dynamic models of followers in the strict-feedback form are considered as follows:where , , ; with , , and are the state vector and the control input of the th follower, respectively, and is the output of the th follower. Are nonlinear function vectors. It is assumed that the leader’s motion is independent of the followers’ motion.
The communication topology for the agents is described by a directed graph with , To represent the communications among followers, we define a subgraph as with . The adjacency matrix of the subgraph is ; if , otherwise, and . Then, the Laplacian matrix is defined aswhere , with if the leader and otherwise, which denotes the communication weight from the leader to followers, , and with is the Laplacian matrix of the subgraph denoting the communication among followers.
Remark 3. For the simplicity of analysis, in the following, we only consider the case where . The analysis and main results still hold for any dimension by using the Kronecker product.
Remark 4. If the directed graph has a spanning tree, rank . Then, rank from , where is an vector of all ones. Therefore, is invertible.
Assumption 2. The nonlinear functions are unknown on a directed graph .
Assumption 3. The leader output signal is an -order differentiable and bounded function and available for the followers satisfying , .
Definition 2. (see ). The distributed consensus tracking errors for nonlinear followers (15) under the communication graph are said to be cooperatively semiglobally uniformly ultimately bounded (CSUUB) if there exist adjustable constants and , and the bounds and , independent of , and for every and , there is a time , independent of , such that and for all , where , , and .
The objective of this brief is to design RBF-neural networks-based distributed fixed-time consensus control laws for followers (15) with unknown nonlinearities so that, under the directed graph, the follower outputs synchronize to the dynamic leader output within fixed time while all signals in the total closed-loop systems are bounded.
Remark 5. The strict-feedback system (15) can describe many state-space models of nonlinear systems, i.e., various physical systems, such as flight systems, biochemical process, jet engine, and robotic systems . Therefore, system (15) under a graph topology can represent multiagent systems consisting of several practical applications with different dynamics.
Remark 6. Notice that followers (15) can have various forms. That is, a group of the followers with different nonlinear functions and order of the dynamics can be considered in this brief.
Remark 7. Compared with the previous consensus works, this brief considers the consensus problem of a group of agents consisting of nonlinear followers with nonlinearities unmatched in the control input. Besides, the nonlinearities are unknown under the total communication topology, and the tracking convergence time independent of the initial state of system (15).
3. Distributed Fixed-Time Consensus Controller Design
The design procedure on the follower contains steps. The distributed backstepping design coordinate transformation is as follows:where and and is the virtual controller of the subsystem of follower.
An RBF neural network is used in this paper to approximate the unknown functions :and the inequalities involved in the following text are as below:where with , is the input vector, and , , , , , , and are positive design parameters.
Step 1. : according to (15), (17), and (18), we havewhere .
Construct a Lyapunov function asTime differentiation of yieldsThe virtual controller is defined aswhere , , , and are positive design parameters.
In (27), and are defined aswhere , and is a positive design parameter, and coefficients , are calculated using the following equation:where , , , and .
Remark 8. One of the main contributions of this paper is to design suitable virtual controllers so that nonlinear strict-feedback systems meet the requirement of fixed-time control. In [36-39], virtual controllers designed exhibit similar power functions , where . If is not appropriately selected, it will make unsolvable at the origin and in the negative domain. As , therefore, power exponents , which leads to the possibility of negative power exponents. For example, if , is unsolvable at , that is, does not exist. Suppose , where is an even number, then is unsolvable at the negative domain. For example, if , is unsolvable at the negative domain, that is, does not exist. The controller designed in this paper overcomes the abovementioned defect and promotes the application of fixed-time control in more common nonlinear systems.
Substituting (27) into (26) yieldsAccording to (19) and inequalities (20)-(21), we havewhere .
The adaptive law is then defined aswhere with are positive design parameters.
Combining with Assumption 1 and substituting (33) into (32), we havewhere , , and .
According to (22), (23), (28), and (29), if , then (34) is rewritten asAccording to (22), (23), (28), and (29), if , then (34) is rewritten as
Remark 9. (see ). Based on (28) and (29), when , there is an additional term in (36): . Note that if , this additional term is obviously limited by some smaller constant , so the structure of (35) is retained, while the constant term only slightly increases. Owing to page limitations and to avoid repetitive discussions, we will omit this part in the rest of the analysis.
Remark 10. is a smooth function that is used to overcome the design difficulty of .
The virtual controller is defined aswhere , , , and are positive design parameters.
Choose the adaptive law asCombining (19)–(21) and Assumption 1 and substituting (41) and (42) into (38) and adopting the same design method as in Step 1 yieldswhere , , , and .
According to (22), (23), and Remark 9, we consider only . Then, (43) can be written asIt can be seen from (44) that defining the design smooth function to overcome the design difficulty of is one of the difficulties of designing the controllers in this paper.
From Lemma 1, Lemma 5, Lemma 6, and (33), it follows thatTherefore, can be defined aswith the result thatSubstituting (47) into (44) yieldsStep : according to (15) and (18), we havewhere .
Construct a Lyapunov function asThe derivative of is written aswhereDefining virtual controller aswhere , , , and are positive design parameters.
Choose the adaptive law asCombining (19)–(21) and Assumption 1 and substituting (53) and (54) into (51) yieldswhere , , , and .
According to (22), (23), and Remark 9, we consider only . Then, (55) can be written as