Neuro-adaptive augmented distributed nonlinear dynamic inversion for consensus of nonlinear agents with unknown external disturbance

This paper presents a novel neuro-adaptive augmented distributed nonlinear dynamic inversion (N-DNDI) controller for consensus of nonlinear multi-agent systems in the presence of unknown external disturbance. N-DNDI is a blending of neural network and distributed nonlinear dynamic inversion (DNDI), a new consensus control technique that inherits the features of Nonlinear Dynamic Inversion (NDI) and is capable of handling the unknown external disturbance. The implementation of NDI based consensus control along with neural networks is unique in the context of multi-agent consensus. The mathematical details provided in this paper show the solid theoretical base, and simulation results prove the effectiveness of the proposed scheme.

www.nature.com/scientificreports/ These papers implemented a variety of nonlinear controllers (e.g. feedback linearization, Lyapunov function, sliding mode, backstepping etc.) and a neural network approximation for uncertainty and unknown disturbances. In this paper, we have presented a neuro-adaptive augmented distributed controller, which is designed based on Distributed Nonlinear Dynamic Inversion (DNDI) 13 . We named it N-Distributed NDI (N-DNDI). It can be mentioned that the adaptive control expression in the papers mentioned earlier contains a linear or nonlinear error feedback term, and an adaptive term is added to it. However, N-DNDI is a new neuro-adaptive structure augmented in the DNDI frame. The primary reasons for selecting NDI are given as follows.
• The NDI is an effective way to design a controller for plants with nonlinear dynamics. The nonlinearities in the plant are eliminated by using feedback linearization theory. Moreover, the response of the closed-loop plant is similar to a stable linear system. • The NDI controller has many advantages. Examples of these advantages include 1. simple and closed-form control expression, 2. easily implementable, global exponential stability of the tracking error, 3. use of nonlinear kinematics in the plant inversion, 4. minimize the need for individual gain tuning, etc.
Many researchers have used NDI to solve their research problems. Enns et al. 14 implemented NDI to design a flight controller. Singh et al. 15 developed a controller for autonomous landing of a UAV. Padhi et al. 16 described reactive obstacle avoidance schemes for UAVs in a Partial Integrated Guidance and Control (PIGC) framework using neuro-adaptive augmented dynamic inversion. Mondal et al. 17 applied NDI to propose a formation flying scheme. They presented how the NDI is implemented for tracking the leader's commands in terms of coordinate, velocity, and orientation. Caverly et al. 18 used NDI to control the attitude of a flexible aircraft. Horn et al. 19 designed a controller of rotorcraft using Dynamic Inversion. Lombaerts et al. 20 proposed NDI-based attitude control of a hovering quad tilt-rotor eVTOL Vehicle. The contribution is given as follows.
• In this paper, a novel neuro-adaptive Distributed NDI (N-DNDI) is proposed to achieve the consensus among a class of nonlinear agents in the presence of unknown external disturbance. It can be mentioned that DNDI is a new consensus protocol 13 and augmentation of the neural network with DNDI is a new formulation. Hence, this is new in the context of MASs and not reported in the literature. • The main advantage of N-DNDI is it inherits the features of NDI. Moreover, the augmentation of the neural network provides a very good approximation of the unknown external disturbances. Therefore, N-DNDI is a perfect combination for designing consensus controllers for nonlinear agents. The realistic simulation study justifies the effectiveness of blending DNDI and neural networks. • The formulation to accommodate the neuro-adaptive structure in the DNDI framework is a significant contribution. Moreover, the mathematical details for convergence are provided to show the solid theoretical base of this new controller.
The rest of the paper is organized as follows. In section "Preliminaries", preliminaries are given. Section "Problem formulation" presents the problem definition. The mathematical details of the DNDI are provided in section "Nominal distributed nonlinear dynamic inversion (DNDI) controller". The mathematical details of N-DNDI are given in section "Neuro-adaptive augmented DNDI for consensus". The simulation study is presented in section "Simulation results". The conclusion is given in section "Conclusion".

Preliminaries
The topics which are relevant to the problem considered in this paper are given in this section.

Consensus of multiple agents. The consensus of MASs on communication network is discussed in this
section. The definition of the consensus is given as follows.
Definition 1 Let us consider a MASs with N agents, where X i , (i = 1, 2, 3, ..., N) denotes the states of the ith agent. The MASs will achieve the consensus if � X i − X j �→ 0, ∀i � = j as t → +∞.
The consensus protocol aims to minimize the error in similar states of the individual agent with their neighbour by sharing information over the communication network, which is generally described using graph theory.
There is no self loop in the graph. This fact is expressed by selecting the diagonal elements of the adjacency matrix A as zero, i.e., i ∈ V , a ii = 0 . The degree matrix is denoted by D ∈ R N×N = diag{d 1 d 2 . . . d N } , where d i = j∈N i a ij . The Laplacian matrix is written as L = D − A . A graph with the property that a ij = a ji is said to be undirected graph. If any two nodes v i , v j ∈ V , there exists a path from v i to v j , then the graph is called a connected graph. In this paper, we suppose that the topology G of the network is undirected and connected. ral networks are widely implemented to approximate unknown functions and the radial basis function neural network (RBFNN) is a good candidate 21 . A continuous unknown nonlinear function ζ(X) : R n → R m can be approximated by where X ∈ R n is input vector, W NN ∈ R q×m is the weights of RBFs, �(X) = [φ 1 (X) . . . φ q (X)] T denotes the basis function vector. 'q' denotes the number of neurons. ǫ X ∈ R m is the approximation error. The ith basis function φ i is given by where µ i ∈ R n is the center of the receptors and ψ i is width of the ith gaussian function.
Useful lemma. The useful lemmas used in this paper are given as follows.
Lemma 1 22 The Laplacian matrix L in an undirected graph is semi-positive definite, it has a simple zero eigenvalue and all the other eigenvalues are positive if and only if the graph is connected. Therefore, L is symmetric and it has N non-negative, real-valued eigenvalues 0 Lemma 2 23 Let ψ 1 (t), ψ 2 (t) ∈ R m be continuous positive vector functions, by Cauchy inequality and Young's inequality, there exists the following inequality: where Lemma 3 24 Let R(t) ∈ R be a continuous positive function with bounded initial R(0). If the inequality holds Ṙ (t) ≤ −βR(t) + η where, β > 0, η > 0 , then the following inequality holds.

Problem formulation
In this section, the problem definition is given. The objective is to design a neuro-adaptive consensus protocol that enables a class of nonlinear agents to achieve the consensus in the presence of external disturbance. Let us consider a group of N nonlinear agents. They are connected by the undirected and connected network topology. All the agents are homogeneous, i.e., they have similar dynamics. The dynamics of ith agent is given by Eqs. where, X i ∈ R n , U i ∈ R n are states and control respectively. f is a continuously differentiable vector-valued function representing the nonlinear dynamics. D i (X i ) ∈ R n is the unknown bounded and smooth external disturbance term with ∀t ≥ 0.

Nominal distributed nonlinear dynamic inversion (DNDI) controller
It is relevant to get an overview of the DNDI controller 13 and its convergence behaviour before augmenting neuro-adaptive structure is explained.
Brief overview of DNDI. A brief overview of DNDI controller is presented here. The block diagram of the consensus control scheme with nominal DNDI is shown in the Fig. 1.
The nominal dynamics of ith agent is given as follows.
(3) www.nature.com/scientificreports/ where, X i ∈ R n , U id ∈ R n . e i denotes the consensus error of ith agent given by Convergence of DNDI. Convergence study of DNDI is presented here. Let us consider a smooth scalar function given by L ⊗ I n can be written as  www.nature.com/scientificreports/ Remark 1 It can be observed from Eqs. (14) and (16) that Let us design the gain K i as follows.

Neuro-adaptive augmented DNDI for consensus
Before going to the main derivation of Neuro-adaptive DNDI, we present the philosophy of neuro-adaptive control design 25 . Philosophy of neuro-adaptive control. The sole objective of the design is to drive the actual state X to desired state X d . The scheme adopted is to make actual state X to track the desired or nominal state X d through the virtual state X a as shown in Fig. 2.
The tracking of X to X a and X a to X d is achieved by enforcing error dynamics to obtain the control considering nonlinear plant dynamics. We use the same philosophy to design the Neuro-adaptive distributed NDI controller in the next section.

Mathematical details of neuro-adaptive augmented DNDI (N-DNDI). Neuro-adaptive augmented
DNDI is a blending of neuro-adaptive control and DNDI. The block diagram of the control scheme is shown in Fig. 3. The portion of the diagram inside the blue border is the proposed design of neuro-adaptive controller.
In case of neuro-adaptive augmented DNDI, the consensus error of ith agent is defined such that, the virtual state of ith agent, i.e., X ai ∈ R n reach consensus with the neighbours. Therefore, the consensus error of ith agent is given by The expression of control U iN can be obtained by simplifying Eq. (40) as follows.   www.nature.com/scientificreports/ Therefore, Eq. (51) is written as follows.
Applying Lemma 3 we can write Lemma 4 24 Consider the positive function given by where e(t) = x(t) − x d (t) and W =Ŵ − W * . If the following inequality holds: then, given any initial compact set defined by we can conclude that

the states and weights will eventually converge to the compact sets defined by where constants
We will present the Uniformly Ultimate Boundedness (UUB) here using conclusion 2. Using Eqs. (52), (53), and (56) we can write www.nature.com/scientificreports/ If V i (0) = ζ i δ i then for a given µ E ai > µ * E ai there exist a T E > 0 such that ∀t > T E , we get � E ai �≤ µ E ai Therefore, we conclude In a similar fashion, we can conclude Therefore, according to conclusion 2, the proposed controller is able to make the approximation error to converge in the compact set defined by s .

Simulation results
Simulation results are presented here. The simulation study is performed on PC with AMD Ryzen 5 processor and 8 Gb RAM.
Agent dynamics. The agent dynamics are given as follows.
where X i = X i 1 X i 2 T . Equations (71) and (72) give and and The values of the parameters used in this simulation study are given as follows.
The le ar ning rate γ i = 30 . We have s ele c te d RBF NN b asis f unc t ions g iven by The value of σ i is chosen as 0.12. The disturbance added is given by which is unknown to the controller. The state trajectories of all the agents are shown as X 1 and X 2 , where, X 1 = [X 1 1 X 2 1 . . . X 10 1 ] and X 2 = [X 1 2 X 2 2 . . . X 10 2 ] . Similarly, the controls for the agents are shown by U 1 = [U 1 1 U 2 1 . . . U 10 1 ] , and U 2 = [U 1 2 U 2 2 . . . U 10 2 ] . Also, the virtual states are given by X a 1 = [X a1 1 X a2 1 . . . X a10 1 ] , and X a 2 = [X a1 2 X a2 2 . . . X a10 2 ] . The initial values of the states of all the agents are given in Table 1.
The adjacency matrix is given by www.nature.com/scientificreports/ The unknown external disturbance is approximated by a neuro-adaptive controller. The approximated and real disturbance is shown in Fig. 4a and the approximation error is shown in Fig. 4b. It can be observed that the approximation is very good, which can be confirmed using the approximation error plot. Consequently, the states of the agents achieved the consensus in a few seconds. The state trajectories of all the agents, i.e., X 1 and X 2 , are shown in Fig. 5a and 5b respectively. The states of the agents reach the consensus in finite time.
The consensus is achieved by neuro-adaptive consensus controls U 1 and U 2 which are shown in Fig. 6a and 6b respectively.
The convergence of the states is shown by the consensus errors E di in state X 1 and X 2 . They are shown in Fig. 7a and 7b respectively. The errors converged in a few seconds. This means the virtual states X a1 and X a2 successfully reach the consensus.
The virtual states X a 1 and X a 2 are shown in Fig. 8a and 8b respectively. It can be observed that the consensus value of the virtual state and the actual states are the same. Therefore, the actual states tracked the virtual states accurately. The proof of the tracking can be given by virtual errors.
The virtual errors E ai in states X 1 and X 2 are shown in Fig. 9a and 9b respectively. They have converged in finite time.  Table 1. Initial conditions of the states of the agents. www.nature.com/scientificreports/

Conclusion
The augmentation of neuro-adaptive structure to distributed nonlinear dynamic inversion (DNDI) frame produces a unique adaptive controller (N-DNDI) that efficiently handles the external disturbance. The N-DNDI inherits the features of the NDI technique and handles the unknown external disturbance. The convergence study provided in this paper explains the correctness of the design. The simulation results show that the neural network embedded in the controller approximates the unknown external function and the DNDI controller computes the consensus control signal accordingly. Consequently, the consensus is achieved in finite time. Hence, the proposed N-DNDI is a deserving candidate for consensus control in the presence of unknown external disturbances. We consider the heterogeneous agents along with communication issues as part of our future research plan. Also, we will present a comparison study of the proposed controller with the existing controllers.