







































P. Chaubey et al. /Future Technology                                                                               February 2026| Volume 05 | Issue 01 | Pages 
148-158 

148 

 

 

 

Article 

Exploring various neural network configurations 

for the NN-based MPC in a multi-agent system 
Piyush Chaubey1,2, Anilkumar Markana1*, Dhaval Vyas1, Deepak Kumar Goyal2 

1Pandit Deendayal Energy University, Gandhinagar Gujarat, India 
2Government Engineering College, Bharatpur Rajasthan, India 

A R T I C L E   I N F O 
 

Article history: 
Received 18 August 2025  
Received in revised form 
15 October 2025 
Accepted 02 November 2025 
 
Keywords:  
Multimodal fusion, Context awareness, 
Smart kitchens, Reinforcement learning, 
Personalized recommendation 
 
*Corresponding author 
Email address: 
Anil.markana@spt.pdpu.ac.in 
 
DOI: 10.55670/fpll.futech.5.1.13 

A B S T R A C T 
 

Multi-robot cooperation, unmanned aerial vehicle (UAV) formation control, 
intelligent transport systems, and distributed sensor networks are just a few 
domains where multi-agent systems are crucial, as they require coordinated 
behavior to achieve common goals such as exploration, resource allocation, 
distributed sensing, and target tracking. This paper investigates various neural 
network configurations utilized in the NN-MPC framework for consensus 
control of multi-agent robotic systems. The NN-MPC control is applied to the 
consensus problem of a leader-follower multi-agent system, where agents 
coordinate to achieve collective behavior. In this approach, MPC is utilized to 
predict the future values of the control objective, which is optimized by 
minimizing a cost function with various neural network architectures. Different 
neural network configurations based on feed-forward, recurrent neural 
networks, Fitnet, and cascade networks are explored for the NN-MPC-based 
multi-agent systems. The analysis is performed through a simulation-based 
model of a quadrotor fleet system. Results show that the follower agents 
achieve consensus 60% faster than with RNN-MPC in comparison to the 
feedforward neural network, whereas the results are more effective when 
compared with the cascade network configuration-based MPC, where agents 
reach consensus 90% early if paired with suitable training structures.  Overall, 
the article contributes to the recent topic of research on learning-based MPC of 
the multi-agent system in achieving consensus for the leader-follower strategy. 

1. Introduction 

Multi-agent systems are gaining prominence in the area 
of applications like autonomous vehicles, smart grids, 
healthcare systems, and environmental monitoring [1]. MAS 
presents unique challenges due to the need for coordination 
and cooperation among multiple agents, often in dynamic and 
uncertain environments [2]. The consensus problem in a 
leader–follower MAS refers to the process by which a group 
of agents (followers) coordinate their states to match that of 
a designated leader through local interactions and 
information exchange. In such types of problems, the leader 
acts as a reference providing a desired trajectory or state, 
while the followers adjust their states according to their 
neighbor states and, in some cases, directly from the leader. 
The main objective is to design control protocols that ensure 
all followers asymptotically track the leader’s state despite 
challenges such as communication delays, switching 
topologies, nonlinear dynamics, or external disturbances. 
Leader–follower consensus algorithm proves helpful in 
applications like formation control of autonomous vehicles, 
cooperative robotics, sensor networks, and distributed 
decision-making systems, where achieving coordination with 

minimal communication overhead is crucial. Model Predictive 
Control has evolved as a smart control strategy for controlling 
complex systems, offering advantages such as constraint 
handling, disturbance rejection, and trajectory optimization 
[3]. The integration of learning techniques with the MPC has 
opened new avenues for enhancing the performance and 
adaptability of multi-agent systems. Integrating learning 
techniques with MPC offers the potential to improve the 
multi-agent systems' performance considerably, enabling 
adaptation to dynamic environments, learning from past 
experiences, and refining decision-making processes [4]. 
Driverless vehicle [5], power system management [6], and 
industrial control [7-9] are just a few of the control challenges 
that MPC has been employed to address. Neural network-
based learning enhances MPC by adapting the system 
dynamics, cost functions, or constraint sets based on data 
[10]. The shallow neural network involves learning intricate 
functions, presenting inherent limitations when contrasted 
with deep architectures [11]. There are various architectures 
for shallow neural networks with distinguished 
characteristics.  

 

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ISSN 2832-0379 

February 2026| Volume 05 | Issue 01 | Pages 148-158 

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P. Chaubey et al. /Future Technology                                                                               February 2026| Volume 05 | Issue 01 | Pages 148-158 

149 

 

Abbreviations  & List of Symbols 
 
NN Neural Network 
RNN Recurrent Neural Network 
MPC Model Predictive Control 
MAS Multiagent system 
SAC Soft actor–critic 
θ(t) 
 
θ(k) 
 
xi(k) 
 
x0(k) 
 
Ui(t) 
 
U(k) 
 
Pp 

Mode of communication topology in 
continuous time (t)  
Switching topology mode in discrete time 
at sample (kth) instant 
Position variable of ith follower agent in 
discrete form at (kth) sample instant 
Position variable of the leader agent in a 
discrete form at (kth) sample instant 
Input control variable in continuous time 
(t) 
Consensus control input in discretized 
form at sample (kth) instant 
Positive Definite Matrix 

Kk 

di(t) 

E(k) 
 
J(E(k)) 
 
V(E(k)) 

Optimal control input gain 
Disturbance Input 
Error between the ith follower and the 
leader agent state variable  
Predicted cost function over a future 
horizon h 
Quadratic cost function associated with 
the error state 

 

The FitNet neural network topology is the most basic 
feed-forward neural network; it has no feedback connections 
within or between layers and propagates activity 
unidirectionally from the input to the output stage. Feed-
forward networks schematically stack perceptron layers on 
top of one another, allowing for the approximation of complex 
non-linear functions through the composition of simple linear 
transformations and non-linear activation functions [12]. 
Another neural network structure is the cascade-forward 
network, in which each layer receives input from all previous 
layers [13]. In the cascade network structure, unlike standard 
feed-forward networks, where only the first layer directly 
receives the input, every layer receives the input directly, 
facilitating the learning of more intricate and hierarchical 
representations of the input data [14]. Recurrent neural 
networks incorporate feedback connections, allowing them to 
model systems with memory and temporal dependencies 
[15]. The use of internal memory enables RNNs to retrieve 
data from past history, enabling a loop from the hidden node 
to itself [16]. The key contributions of this article are as 
follows:  
• This work compares different neural network 

architectures within the NN-MPC framework for achieving 
optimal leader–follower consensus in multi-agent systems. 

• As the prediction is done with MPC and optimization is 
carried out by the neural network-based architectures, the 
computational burden on the MPC is minimized, which 
helps in the improvement of system performance. 

• The results are validated using mean square error (MSE) 
for all the structures and outcomes, with the best training 
function presented with a trade-off between fast response 
and least error performance.  

• Results are compared with previously published findings 
on event-triggered control, and it has been shown that the 
results of RNN-based MPC follower agents achieve 
consensus faster than the previous work suggested. 

2. Literature review 

This section describes the recent trends in the control 
techniques for the leader-follower multi-agent system 
consensus problem. This includes event-triggered-based 
control strategy, MPC-based strategy, and learning based 
strategy. Finally, discussed the research gap in the present 
literature and the future scope for improvement in 
overcoming these research gaps. 

2.1 Event-triggered-based strategy 
Event-triggered control to reduce communication 

overhead and handle faults very effectively. Over the past few 
decades, event-triggered leader–follower consensus control 
strategy has gained significant importance for improving 
communication efficiency and robustness in multi-agent 
systems. Chen and Peng [17] proposed an event-triggered 
impulsive control scheme capable of handling packet loss in 
leader–follower networks, achieving the consensus using the 
Lyapunov stability theory, where sufficient criteria are 
identified to realize leader–follower quasi-consensus, 
ensuring reliable consensus under intermittent 
communication links. Similar work is explained by Zhi et al. 
[18], where a finite-time consensus control scheme 
employing an observer is proposed for second-order systems 
under velocity unknown, thereby achieving reduced 
communication updates through terminal sliding mode 
control. Also, Wu et al. [19] developed a fixed-time event-
triggered consensus approach that guarantees convergence 
within a predetermined time despite delays and disturbances. 

2.2 MPC-based strategy 
The application of Model Predictive Control (MPC) to 

multi-agent systems is well established, as it effectively 
facilitates leader–follower consensus by predicting future 
trajectories, managing system constraints, and optimizing 
controls in real time. Kuriki et al. [20] explained how to 
combine consensus-based control with a decentralized MPC 
technique for multi-UAV formation, allowing for collision 
avoidance while preserving formation goals. In order to 
improve scalability and safety, Dubay and Pan [21] have 
extended this concept by proposing a distributed MPC 
framework for multiple quadcopters. In this framework, each 
agent computes its control action locally to reach consensus 
while avoiding collisions. By resetting the MPC optimization 
under specific circumstances, Saeednia  and Khayatian [22] 
presented a reset MPC-based control technique for the MAS 
with fast convergence speed and robustness. Collectively, 
these research investigations demonstrate that MPC is ideal 
for applications like autonomous vehicles, cooperative 
robotics, and UAV swarms because it not only guarantees 
precise leader tracking but also offers a methodical approach 
to integrating safety, constraints, and optimal performance 
into leader–follower consensus control. 

2.3 Learning based strategy 
Learning-based controllers identify unknown dynamics 

while maintaining synchronization. Reinforcement learning 
frameworks effectively coordinate follower behavior without 
prior knowledge of the leader’s state. Applying learning-
based techniques to the leader-follower system has advanced 
significantly in recent times, especially for situations with 
non-linearities, uncertainties, and communication 
limitations. The aim of recent developments in learning-based 
control for multi-agent systems (MAS) is to reduce 
communication requirements, robustness to uncertainty, and 
model-free adaptation. For nonlinear MAS, Filiberto et al. [23] 
suggested a distributed control method based on Gaussian 



P. Chaubey et al. /Future Technology                                                                               February 2026| Volume 05 | Issue 01 | Pages 148-158 

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Process regression, which ensures Lyapunov stability and 
allows for precise leader-follower agreement without the 
need for explicit system models. In order to enable accurate 
formation tracking with constrained residual errors, Yang et 
al. [24] presented a dynamical neural network-based control 
method that approximates unknown nonlinearities and 
disturbances. By combining radial basis function neural 
networks with fixed, relative, and switch triggering strategies, 
Wang et al. [25] built on communication efficiency to provide 
an adaptive event-triggered leader–follower control 
framework that prevents Zeno behavior and maintains 
consensus. Lastly, Li et al. [26] designed an adaptive 
distributed formation control method using a recurrent SAC 
reinforcement learning algorithm, enabling agents to achieve 
formation tracking in dynamic and uncertain environments 
with improved stability and adaptability. These works 
collectively reflect an ongoing shift toward data-driven, 
adaptive, and communication-effective solutions for the 
leader-follower consensus, enabling MAS to perform reliably 
in complex, uncertain, and resource-constrained 
environments. Table 1 presents a comparative analysis of 
event-triggered-based, MPC-based, and learning-based 
strategies. Various advantages and limitations of these 
strategies can be easily identified, and research gaps can be 
identified to further improve performance, such as fast 
convergence, reduced computational cost, stability, and 
constraint handling. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.4 Research gap 
From Table 1, it can be observed that even with great 

advancements in this research area, there are still 
considerable research gaps in applying event-triggered, MPC, 
and learning-based approaches to the leader-follower 
consensus problem. Designing asynchronous triggers for 
event-triggered control that ensure stability and performance 
regardless of packet failures, communication delays, and 
heterogeneous agent dynamics, while completely avoiding 
Zeno behavior, continues to be a challenge. Despite its 
effectiveness in managing restrictions and maximizing 
performance, the MPC-based approaches have limitations in 
terms of scalability for large networks, real-time viability on 
platforms with limited resources, and tolerance to 
nonlinearity and uncertainty. Although learning-based 
techniques like neural networks and reinforcement learning 
provide flexibility in unpredictable situations, they frequently 
lack formal stability guarantees, have significant data 
requirements, and present difficulties for safe real-world 
implementation. Moreover, integrated frameworks 
combining these methods remain underexplored, particularly 
in developing hybrid schemes that balance communication 
efficiency, scalability issues, robustness, and fast convergence 
for leader–follower consensus in dynamic and uncertain 
multi-agent environments. Table 2 shows a wide scope for 
improvement across various aspects of addressing the 
consensus problem in multi-agent systems, including model 
dependencies, constraint handling, and the computational 
burden imposed by the MPC strategy.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table1. Comparison of related past work for the consensus problem of multi-agent systems  

Strategy Related 

Works 

Key Features / Working Principle Advantages Limitations 

2.1 

Event-

triggered 

based 

strategy 

[17-19] • Control actions and communications 

occur only when specific events or 

thresholds are triggered, reducing 

communication load. 

• Uses Lyapunov-based conditions for 

stability and convergence. 

• Reduces unnecessary 

communication and energy 

consumption. 

• Handles packet loss and 

intermittent communication 

effectively. 

• Provides finite-time or 

fixed-time convergence. 

• Requires careful design of 

triggering conditions to avoid 

Zeno behavior. 

• Performance may degrade with 

high network delays or noise. 

• Limited scalability for very large 

networks. 

2.2 MPC-

based 

strategy 

[20-22]  • Uses Model Predictive Control to 

predict future trajectories and optimize 

control inputs under constraints. 

• Each agent solves an optimization 

problem locally to achieve consensus 

while respecting safety and collision 

avoidance. 

• Systematic handling of 

constraints (safety, collision 

avoidance). 

• Provides optimal and 

coordinated performance. 

• Enables scalability and 

robustness for multi-agent 

systems. 

• High computational cost due to 

online optimization. 

• Requires accurate system 

models and prediction horizons. 

• Limited applicability in real-

time or highly dynamic 

environments with 

communication delays. 

2.3 

Learning-

based 

strategy 

[23-26] • Employs data-driven or reinforcement 

learning techniques to handle unknown 

dynamics and uncertainties. 

• Uses neural networks or Gaussian 

Process regression for adaptive control 

without explicit models. 

• Model-free and adaptive—

suitable for uncertain and 

nonlinear systems. 

• Reduces dependency on 

accurate modeling and prior 

knowledge. 

• Capable of learning 

optimal coordination 

policies over time. 

• Training requires extensive data 

and computation. 

• Stability and convergence 

proofs are complex. 

• May suffer from poor 

generalization or instability 

under unseen conditions. 

• Implementation in real-time 

may be challenging. 

 



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Also, communication failures encountered by event-

triggered strategies, which work effectively only under 
excellent trigger conditions, make the control strategy 
unreliable. Lastly, the independently used learning-based 
strategies also lack convergence to the optimal solution and 
are ineffective at handling constraints in the problem.  Among 
the control strategies used in the past literature, the NN-based 
MPC strategy (Figure 1) can be shown to be helpful in 
combining the advantages of learning-based and MPC 
strategies and eliminating limitations such as constraint 
handling, high computational cost, and model-based 
dependencies, as it allows operation in a model-free 
environment. NN- MPC also reduces the chances of 
communication failure faced by event-based strategies as it 
provides distributed control through proper communication 
between leader–follower and follower-follower with graph 
theory and switching topology using Markovian switching. 
Overall, these advantages help achieve consensus in the 
minimum time, as discussed in the results section.  

3. Methodology 

This section details the framework and analytical 
formulation used to develop and validate a NN-MPC system 
for a multi-agent robotic system. The section details the 
method for optimizing the consensus-based objective 
problem in multi-agent systems, where future states are 
predicted using model predictive control over the prediction 
horizon.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
This integration of a neural network-based strategy with 

MPC shows remarkable improvement in achieving consensus 
for the multi-agent systems. 

3.1 Preliminary 
The problem formulation and mathematical analysis of 

the control strategy implemented are discussed in this 
section. This includes details about the consensus among 
followers and the leader. It also provides a brief on graph 
theory and switching topology, which provides the basic 
information on communication among agents. 
Consensus tracking of multiple agents: This research work 
addressed the consensus problem in a leader-follower multi-
agent system (MAS), where the goal is for all agents to achieve 
a desired state that is common to all through local 
interactions over a communication network [29–31]. For a 
network of N followers connected through a communication 
system, where the followers Xi(t), where i = 1, 2…. N aims to 
track the trajectory of a leader X0(t). Consensus tracking is 
achieved if, for any initial conditions, the states of all followers 
reach the consensus, Lt t→∞ ∣Xi(t)−X0(t)∣=0. 
Graph theory: The basic structure used for communication 
among agents in the graph theory is modeled using a 
weighted graph G = {V, E}. The vertex set V = {v1, v2, …, vN} 
represents the placement of agents in the communication 
network, while the edge set E ⊆ V×V, denotes the 
communication links. The weighted adjacency matrix is given 
by A=[aij] ∈ N×N of the graph G is defined such that aij > 0 if 
(vj,vi) ∈ E and aij=0 otherwise. This graph connection is the 

Table2. Improvement of AI-based MPC over traditional leader–follower consensus strategies 

Aspect Event-triggered 

Control 

Classical MPC Learning-based 

Control 

AI-based MPC – Improvements 

Model 

Dependence 

Relies on known system 

dynamics and triggering 

conditions for stability. 

Requires accurate 

mathematical models for 

prediction and 

optimization. 

Model-free but lacks 

constraint 

interpretability. 

AI learns or approximates system 

dynamics online, reducing 

dependency on exact models 

while retaining MPC’s structure. 

Adaptability Limited adaptability to 

nonlinear or time-

varying systems. 

Struggles with strong 

nonlinearities unless 

extended (e.g., nonlinear 

MPC). 

Highly adaptive but 

sometimes unstable. 

AI enables online adaptation 

using reinforcement or continual 

learning, improving robustness 

to dynamic environments. 

Computational 

Efficiency 

Reduces communication 

but not computation. 

Computationally heavy 

due to repeated 

optimization. 

High training cost, 

sometimes offline only. 

AI-based surrogates or neural 

approximators replace solvers, 

improving real-time efficiency. 

Communication 

Efficiency 

Excellent – triggers only 

on events. 

No inherent 

communication saving. 

Sometimes includes 

communication-

efficient frameworks. 

AI-based MPC can learn optimal 

communication schedules, 

combining event-triggering with 

adaptive learning. 

Constraint 

Handling 

Heuristic or limited. Strong theoretical 

constraint handling. 

Weak or implicit 

constraint 

management. 

Maintains MPC’s explicit 

constraint satisfaction while 

learning new constraints 

adaptively. 

Convergence 

and Stability 

Strong analytical 

guarantees under known 

models. 

Stable if model is 

accurate. 

Difficult to guarantee 

convergence formally. 

Combines safe RL and Lyapunov-

based design for provable 

stability under learned models. 

Application 

Scope 

Suitable for resource-

constrained or periodic 

update systems. 

Best for structured and 

well-modeled systems. 

Effective for uncertain 

and nonlinear systems. 

Unified approach—robust, 

adaptive, and safe; ideal for 

autonomous vehicles, UAVs, and 

robotics. 

 



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152 

 

basic communication network of the leader-follower agents, 
where followers communicate with other follower agents and 
with the leader. A directed graph is used in this work. 
Switching leader-follower connection: The proposed work of 
neural network-based MPC works with a switching topology 
of the follower agents with a leader connection. The switching 
is performed on the basis of the switching system and the 

Laplacian matrix as lii(θ(k)) = ∑ aij(θ(k))N
j=1,j≠i  and 

lij(θ(k)) = −aij(θ(k)) for j ≠ i. Here, θ(k) is the switching 

topology mode. This switching is followed by a probability 
matrix based on a Markovian chain, which is available in the 
simulation study section of this paper.  

3.2 Problem formulation 
Each agent is modeled with a first-order integrator 

system for the multi-agent system: 

ẋi(t) = ui(t)                 (1) 

where xi(t) and ui(t) are the state and control variables, 

respectively. 

The consensus control input in discretized form is defined as:  

u(k) = (L(θ(k) ∗Kk) x(k)                (2) 

where Kk represents the control gain and x(k) is the collective 

state variables of all followers. 

To achieve the control objective given in equation (6), we 

consider an NN-MPC control strategy as illustrated in Fig. 1. 

The MPC layer predicts system behavior and computes a cost 

function, while the NN layer learns to optimize the cost by 

adjusting Kk using inputs x(k) and θ(k). In a more general 

setting, the agent dynamics can include non-linearity and 

unknown disturbance: 

ẋi(t) = Axi(t) +Bui(t) + f (t, xi(t)) + Bd di(t)                (3) 

where f (t, xi(t)) models nonlinear internal dynamics. 

The proposed approach is simulated using MATLAB 2022, 

and the performance is analyzed for various NN training 

algorithms to achieve consensus in a fleet of quadrotors [32]. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.  Block diagram of NN-Based MPC for multi-agent systems 

This section outlines the integration of prediction with 
MPC and optimization using a neural network for achieving 
consensus in the quadrotor fleet multi-agent system as 
proposed in reference [16]. Consider the first-order multi-
agent system as given in equation (1), the objective of the 
follower agent is to track the leader state position. The 
consensus is shown as an error between the ith follower and 
the leader agent state variable: 

E(k) = xi(k) − x0(k)                 (4) 

where xi(k) and x0(k) are the state variables of the ith follower 

and leader agent. 

To guide the system towards the consensus, we define a 

quadratic cost function associated with the error state:  

V (E(t), θ(t)) = ET(t)Pp E(t),                                 (5) 

The predicted cost function is given by 

 J(E(k)  =  ∑ 𝐸k|k [V (E(t), θ(t))]𝐾+ℎ+1
𝑡=𝑘+1                (6) 

where J(E(k) is the predicted cost function over a future 

horizon h. 

To minimize this cost, the optimal control input gain is 

computed by: 

Kk = arg min
             Kk,,….  Kk+h

J(E(k))                  (7) 

where Kk is the optimal control gain for the optimization 

function J(E(k)). 

This optimization is handled by an NN-based learning 

mechanism that updates the gain Kk by training the network 

to reduce the predicted cost J(E(k)) over iterations. This 

combined NN-MPC framework enables the multi-agent 

system to reach consensus effectively in the dynamic network 

environment. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



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3.3 Neural network architectures for NN training 
This section describes various neural network 

architectures used for training are explained in this section. 
The basic training function used in each architecture is 
Levenberg-Marquardt. The L-M training algorithm uses a 
second-order approximation for the performance evaluation 
as a sum of squares instead of computing the actual Hessian 
matrix. Figure 2 shows the architecture of the Fit Net neural 
network. This is the basic network with one hidden layer [27]. 
It finds application in problems involving function 
approximation and regression analysis. The output equation 
of the Fit Net architecture is given by: 

J = y = f (W2 · f (W1.x + b1) + b2)               (8) 

J = y is the optimized output from the trained network, where 
the network is trained by adjusting the weight matrices W1 
and W2 during the training. Here, the sigmoid activation 
function f(.) is used for training the weighted inputs, whereas 
b1 and b2 are the bias vectors. The node h is a neuron layer 
between input x and output y where input weights are 
adjusted for the network during training. Once the training is 
completed, the inputs are applied to the network and an 
optimal solution y is obtained, which is the minimum of errors 
of the state matrices between the leader and follower states. 
The same is applied as control gain Kk according to equation 
(7) to the control gain matrix given by equation (2), which 
further updates the state given by the system model equation 
(1) at each iteration, and the process repeats until consensus 
is achieved. Figure 3 shows the architecture of a feedforward 
neural network. This network includes two or more hidden 
layers [27], useful for learning complex features. The output 
equation of the feedforward network architecture is given by: 

J = y = f (W3 · f (W2 · f (W1.x + b1) + b2) + b3)                 (9) 

 
Figure 2. Architecture of the Fit Net neural network training set 

 

 
Figure 3. Architecture Of Feedforward Neural Network Training Set 

This architecture is similar to the Fitnet, except it can 
have more hidden layers, which can improve the network's 
accuracy, but at the same time make the network more 
complex. Figure 4 shows the architecture of a recurrent 
neural network. A Cascade-forward network allows 
connections from all layers in parallel, including input to the 
output directly [28], enhancing learning flexibility. Equation 
(10) gives the output of the cascade network configuration.  

J = y = f (W3 · f (W2 · f (W1.x + b1) + W4.x + b2) + b3              (10) 

The uniqueness of this configuration is that it includes a 
direct connection between the inputs and outputs, also during 
the network learning. The same activation function is used in 
this architecture as is used in the Fitnet and feedforward 
network configuration. The weights are adjusted during 
training and learning of the network. Once the network is 
trained, the optimal solution is obtained and applied to the 
control gain matrix Kk, which further updates the system. 
Process repeats until consensus. Figure 5 shows the RNNs 
that include loops to retain memory over time steps [16, 28]. 
They are ideal for sequence-based tasks. The output equation 
for the recurrent neural network is given by Eq (11) and Eq 
(12). 

ht = f (Wx xt + Whht−1 + b)                            (11) 

J = yt = f (Wyht + c)                          (12) 

The above output J = yt is the optimized output from the 
trained recurrent. Sigmoid activation function f(.) is used for 
tuning the input weights for RNNs. 

The hidden layer has a loop between ht and ht-1, where 
hidden weights Wh are adjusted for the previous hidden state 
during training. Once the training is completed, the inputs are 
applied to the network, and the optimal solution y is obtained. 
The input x(t) to the neural network represented by Eq (13) 
is as follows:   

x(t) = [1 K11. . . K1n .... Km1. . . Kpq]’             (13) 

The algorithm used for the optimization of the objective 
cost function is shown in Figure 6, which explains how the 
optimization is achieved with the various neural network 
architectures for NN-based predictive control. At the 
beginning of the simulation, the control gain KK is assigned an 
initial value of zero to ensure a safe starting point for the 
learning process.  

 
Figure 4. Architecture of cascade neural network training set 

 

 

 



P. Chaubey et al. /Future Technology                                                                               February 2026| Volume 05 | Issue 01 | Pages 148-158 

154 

 

 
Figure 5. Architecture of the recurrent neural network training set 

 

 

Figure 6. Flow chart of the algorithm for the NN-MPC-based multi-
agent system 

Figure 6 illustrates the algorithm for the NN-based MPC 
for a leader-follower system with various neural network 
configurations. The network training starts with initialization 
of the values x(k), θ (k), and Kk. The neural network is trained 
for different cases based on Eqs (8-12). Then, the initial 
optimization is done using the trained network. The value of 
the Kk is used for the control gain u(k) calculation as given by 
equation (2). Then, the optimal cost is predicted using 
equations (4-6) with MPC, based on the future values of errors 
in the follower and leader states. After every iteration, the 
criteria for optimization are checked J< ץ, if the condition is 
satisfactory, then the iteration stops; otherwise, it continues 
with updating the values of J and K. 

 

 

4. Results and discussion  

4.1 Simulation study 
The system model used for the simulation purpose is the 

same as that used in the reference [32], and the data is taken 
from [16] and [29]. For the system model given by equation 
(3), f (t, xi(t)) = 0.01 sin(xi(t)), and the initial values are the 
same as those considered in [16]. A, B, and Bd matrices are 
given as:  

A = [
0 1
0 −0.5

],  B = [
0.8
1.2

] ,  Bd = [
0 1
1 0

] .  

The Laplacian matrices are considered as follows, 

L(1) = [

    1     0 0 0
 −1     1 0 0
    0
  −1

−1
   0

1
0

0
1

],  

 L(2) = [

    1  −1 0 0
     0   1 0 0
   −1
      0

 0
  0

1
−1

0
1

] 

The probability matrix is given by:  π = [
0.95 0.05
0.02 0.98

] 

The directed graph is used in this work as shown in Figure 7. 
and switching of the graph is carried out between L(1) and 
L(2) based on the probability matrix π. Sampling time for the 
discretization is Ts = 0.01. 

 

                   

(a) L(1) Directed Graph      (b) L(1) Directed Graph 

 

 Figure 7. Directed graph topology 

The simulation is carried out using N = 10 neurons. To 
achieve optimal results for all the cases, the prediction 
horizon is taken as 100. This horizon is deemed sufficient for 
reaching the optimal point. Convergence is obtained for λ = 
0.01. For validation, the chosen performance metric, the MSE 
is defined as: 

MSE =  
1

kf

∑ |eri(k)|2kf
k=0          (14) 

Where, eri(k) denotes the error values ei(k).  
Figure 8 reflects that for the NN-MPC with the Fit Net 
architecture. All the agents achieve consensus at 3 sec in 
‘position variable’ while it reaches consensus after 4 sec for 
the ‘velocity variable’. Figure 9 also reflects that NN-MPC has 
the feed-forward network configuration. It reflects that all the 
agents achieve consensus at 3.5 sec in the ‘position variable’ 
and 4 seconds in the ‘velocity variable’ respectively. The 
agents achieve consensus 16.6% faster with the Fit Net 
architecture-based NN in comparison to the feedforward-
based NN architecture for multi-agent systems.  

 

0 

2 1 

3 4 

0 

2 1 

3 4 



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155 

 

Figure 8. Position and velocity variable response for the followers 
with the Fit Net architecture 

 
Figure 9. Position and velocity variable response for the followers 
with a feed-forward net architecture 

Figure 10 reflects RNN-MPC with the recurrent neural 
network architecture. It reflects that all the agents reach 
consensus at 2 sec for the ‘position state’ while just after 3 sec 
for the ‘velocity state’. NN-MPC with the cascade network 
architecture is shown in Figure 11. It reflects that all the 
agents reach the leader position at 4 sec for ‘position state’ 
and at 5 sec for the ‘velocity state’. With RNN-MPC, control 
agents can achieve consensus almost 100 % faster in 
comparison to the cascade-forward-based NN architecture 
for multi-agent systems.  

 

 

 

 

 

 

 

 

 

 

Table 3 shows the comparison of mean squared errors 
for the ‘position and velocity variable’ of various neural 
network architectures for achieving consensus of follower 
agents. It can be observed that the least error is found with 
three cases as Fit Net, RNN, and CFN for ‘position state’ and 
‘velocity state’.  

Figure 10. Position and velocity variable response for the followers 
with a recurrent neural network architecture 

 
Figure 11. Position and velocity variable response for the followers 
with a cascade forward net architecture 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table3. Mean squared error comparison of various neural network architectures for NN-MPC based multi-agent system 

(ith agent, rth state) FITNET 

Feed forward 

(FFN) 

Cascade 

Forward Net 

(CFN) 

Recurrent 

Neural Network 

(RNN) 

Gao et al. 

Least MSE 

(1,1) 0.0866 0.0947 0.1028 0.0671 0.3395 RNN 

(1,2) 0.0967 0.0943 0.0740 0.1278 0.0781 CFN 

(2,1) 0.0010 0.0019 0.0022 0.0022 0.0588 FITNET 

(2,2) 0.0204 0.0230 0.0237 0.0232 0.0194 Gao et al. 

(3,1) 0.1972 0.0305 0.0149 0.0134 0.0824 RNN 

(3,2) 0.3387 0.2198 0.1978 0.1948 0.1029 Gao et al. 

(4,1) 0.0826 0.0664 0.0377 0.0644 0.4142 CFN 

(4,2) 0.3825 0.3663 0.3324 0.4014 0.1823 CFN 

 



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156 

 

MSE is least with RNN architecture for the ‘position state’ 
of follower ‘1’ and both ‘position and velocity state’ of follower 
‘3’. Whereas, MSE is least for both ‘position state’ and ‘velocity 
state’ error for the followers ‘2’ if a Fit Net architecture is 
used, a neural network structure. In other cases, like ‘Position 
state’ of follower ‘4’ and ‘velocity state’ of follower ‘1’ and ‘4’, 
the MSE is minimum with the CNN architecture. It can be 
observed that no single architecture can give the least MSE for 
all the follower agents possible, but a compromise can be 
possible for a configuration that can give a better response 
(fast response) with the least MSE required for the consensus 
of the agents.  From the above observations, we can say that 
RNN-MPC reaches the consensus in the minimum time, 
thereby showing the least MSE for the ‘position state’ of 
followers ‘1’ and ‘3’. Results of Gao et al. show minimum MSE 
for the ‘velocity state’ of followers ‘2’ and ‘3’. However, there 
is no single training configuration that provides the least MSE 
for all agents for the ‘position state’ and ‘velocity state’. 

4.2 Comparison to the related work 
If the comparison is made with the previous work for 

achieving consensus of follower agents presented by Gao et 
al. [32], it can be said that the RNN-MPC gives desired results 
in minimum time thereby follower agents reaching to the 
consensus in the minimum time at 2 sec thereby showing 
much better performance in achieving consensus than the 
event triggered based strategy where consensus is achieved 
in more than 10 seconds. However, there can be more chances 
of improvement as far as MSE is concerned for the position 
and velocity variables of the followers. The proposed NN-MPC 
approach differs from conventional MPC and existing 
learning-based MPC strategies by incorporating neural 
networks as FITNET, feedforwardnet, cascadenetwork, and 
recurrent neural network within the control framework. 
Traditional MPC predicts the future states and optimizes the 
objective cost by itself, which limits its performance in 
nonlinear or uncertain environments. In contrast, the 
proposed NN-MPC takes the optimization burden of the MPC 
with various neural networks—such as feedforward, 
recurrent, Fitnet, and cascade architectures—that learn in the 
complex system dynamics and inter-agent interactions. This 
integration allows the controller to predict future and 
optimize the error between the state trajectories of the 
leader-follower agents more accurately and adapt to changing 
conditions in real time.  In the future, the existing learning-
based MPCs that typically utilize a single neural network 
trained offline can be replaced by the proposed framework 
systematically, which can compare multiple neural 
architectures to determine the most effective configuration 
for achieving consensus in leader–follower multi-agent 
systems. Consequently, the NN-MPC enhances adaptability, 
robustness, and coordination performance while preserving 
the optimization and constraint-handling advantages of the 
MPC structure. 

5. Conclusion 

Conclusions drawn from the above results suggest that 
the RNN-MPC-based configuration of the neural network 
gives a fast response to the leader-follower system in 
achieving consensus in minimum time over other neural 
network-based architectures like FITNET, feedforward 
network, and cascade network. Also, there is a considerable 
reduction in the MSE for the ‘position variable’ of followers ‘1 
and 3’ that validates the effectiveness of the recurrent neural 
network-based training network. However, there is no 
architecture found that provides the least MSE for all agents 

for the ‘position state’ and ‘velocity state’. Future research 
efforts should be focused on finding such a learning-based 
MPC that can provide fast response as well as the least MSE 
for most of the agents to achieve the consensus of the multi-
agent quadrotor fleet system.  

Ethical issue 
The authors are aware of and comply with best practices in 
publication ethics, specifically regarding authorship 
(avoidance of guest authorship), dual submission, 
manipulation of figures, competing interests, and compliance 
with research ethics policies. The authors adhere to 
publication requirements that the submitted work is original 
and has not been published elsewhere. 

Data availability statement 
The manuscript contains all the data. However, more data will 

be available upon request from the authors. 

Conflict of interest 

The authors declare no potential conflict of interest. 

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