Distributed nonlinear model predictive control for a quadrotor UAV

Abstract

A Distributed Nonlinear Model Predictive Control (DNMPC) approach is proposed to control the simplified decoupled dynamics of a quadrotor UAV. The performance of DNMPC is compared, in terms of tracking and execution time, to that of standard control configurations based on centralized MPC and PID control. The aim is to show the suitability of each configuration in terms of performance and practicality in real-time applications. The results show the advantage of using DNMPC in terms of ease of tuning and computational cost over more centralized feedback control approaches. For extra realism, wind disturbances and sensor noise are incorporated into the simulations.

Keywords

Nonlinear MPC distributed control quadrotor flight control UAV

Introduction

In the last decades, different control techniques have been used in the aerospace industry, these range from basic classical control to modern artificial intelligence techniques used in many other fields. Classical controllers such as Proportional–Integral–Derivative PID,¹ one of the best-known controllers, successfully stabilises a wide range of systems.² Aerospace systems, such as aircraft or spacecraft, are nonlinear systems that could benefit from being controlled by advanced control systems enforcing strict stability and reliability requirements.

MPC is an advanced control approach in which the control law is internally determined according to the forecasted response of the dynamic model to be controlled.² An open-loop optimal control problem (OLOCP) is solved within a specified duration called the Prediction Horizon to produce an optimal manipulated input trajectory for the optimal anticipated state trajectory. Out of the determined manipulated input trajectory, only the first control input action is applied to the system, and the whole process is repeated at every time sample. Furthermore, one of the significant features of MPC is that it can handle multiple outputs and multiple inputs, or MIMO systems,³ making it perfect for multivariable feedback control since the 1990s.⁴ In recent years, significant progress in enhancing MPC capabilities has been made. For instance, learning-based MPC integrates machine learning techniques to improve model accuracy and adaptability in dynamic environments.⁵ Robust and stochastic MPC formulations have also been widely developed to account for model uncertainties and external disturbances explicitly, ensuring reliable performance even under challenging conditions.⁶ Furthermore, real-time optimization improvements and parallel computing strategies have expanded the feasibility of implementing computationally intensive MPC schemes on embedded systems.^7,8 The testbed for this research is a Quadrotor, an Unmanned Aerial Vehicle (UAV) that features exceptional manoeuvrability, hovering, vertical take-off and landing capabilities. MPC is extensively used in the literature as a flight controller for quadrotors. Identifying a suitable dynamic model for a quadrotor is crucial for MPC. MPC has been applied to many model types, including linear time-invariant, linear time-varying, piecewise affine, and nonlinear systems. Since the mathematical model is the centrepiece of MPC, the model type is pivotal in how efficiently the optimization solver will solve the OLOCP. Complex dynamics representations may lead to a computational burden when solving the OLOCP.

Linear dynamics were considered in designing a linear model predictive control (LMPC) for the quadrotor in Islam et al.,⁹ resulting in a good tracking performance and disturbance rejection for different trajectories. Alexies et al.¹⁰ used a switching type MPC in controlling the quadrotor but with a linearized piecewise affine model around multiple operating points to track a trajectory with the presence of disturbances. The authors decoupled the dynamics to form a “dual control scheme.” A separate MPC was used to control the translational motion using the translational augmented dynamics, and another MPC was used to control the attitude of the quadrotor. As a matter of fact, the attitude dynamics (angular motion) of the quadrotor are faster in nature compared to the translational dynamics.¹⁰ Thus, separating the controller for the translational motion and attitude is possible when the dynamics are decoupled.

A centralized nonlinear MPC was investigated in Abdolhosseini et al.¹¹ based on a reduced nonlinear dynamic model of a quadrotor. It was shown that using a reduced nonlinear model led to a simpler MPC solution that was easier to implement and produced a similar performance to that of an MPC solution based on a full-order model. Reducing the computational complexity has been addressed effectively in Li et al.¹² using machine learning to learn the control law from well-designed MPC flight data. However, this approach replaces the MPC with a clone that is limited by the conditions of the dataset used in the training stage.

Also, many researchers used the nominal MPC in conjunction with another technique. The authors in Cheng and Yang¹³ proposed a two-loop controller in which the outer loop is MPC-based to control the translational motion and a faster inner loop based on PID to control the attitude of the quadrotor. Although they have proved in simulation tests the effectiveness of using MPC to respect the constraints of the actuators and track the trajectory accurately, the disturbance effect was not introduced in their proposal. Hence, robustness concerns could arise, and the constraints may be violated if the controller is not robust enough when the quadrotor is subjected to a disturbance. Moreover, comparative studies were conducted to quantify the performance and the control effort of the linear MPC versus PID, PD, and Linear Quadratic Regulator (LQR) in the Okasha et al.¹⁴ The authors claim that MPC offers better tracking performance than other techniques. Their study adopted the centralized LMPC and tested on a smooth trajectory without sharp manoeuvres. The results are convincing, but when it comes to fast and very sharp manoeuvres, the results need to be verified.

The configuration of MPC formulation and how it handles the system dynamics could be varied according to the system complexity and the performance requirements.¹⁵ In general, there are three different MPC configurations or architectures to control the system, namely, Centralized, Decentralized, and Distributed MPC. In centralized MPC, large-scale systems are controlled using a single MPC with one optimal problem. Although this architecture is the greatest for performance, it is considered impracticable for large systems due to its computational requirements.¹⁶ The decentralized MPC architecture features a separate MPC for each dynamics subsystem. Usually, the subsystem dynamics are decoupled from each other, the mutual interaction is ignored, and each MPC has its optimal problem.^15,16 On the other hand, the distributed model predictive control (DMPC) structure shown in Figure 1, is introduced when the decentralized configuration is associated with a communication ability between the individual MPC agents to facilitate exchanging information between them.¹⁶ In this generalized setup of $N_{a}$ subsystems or agents, each subsystem $i$ is associated with its own MPC agent, where $x_{i}$ and $u_{i}$ represent the state and control input of subsystem $i$ , and $y_{i}$ is the corresponding output. As illustrated in Figure 1, the distributed MPC architecture allows communication between different agents, enabling the exchange of state and control information ( $x_{j}$ , $u_{j}$ ) among the MPCs. The dashed colored arrows in the figure highlight these communication links, facilitating coordination between multiple subsystems in the networked control scheme.

Figure 1.

Generalized configuration of distributed MPC. Each subsystem $i$ ( $i = 1, 2,$ … $, N_{a}$ ) is controlled by its corresponding model predictive controller (MPC). The variables $x_{i}$ denote the states of subsystem $i$ , $u_{i}$ is the control input generated by MPC $i$ , and $y_{i}$ is the subsystem output. The solid black arrows represent the physical interconnection between each controller and its subsystem. The dashed arrows indicate the exchange of information, specifically, state and control data, between different MPC agents, where each color represents communication from a specific subsystem, as shown in the legend.

The theory of DMPC and its stability has been extensively studied in the literature.^17,18 Although DMPC aims to control large-scale systems with multiple objectives, it has been applied successfully in controlling AC/AC converters in power applications to reduce the computational burden when centralized MPC is used Tarisciotti et al.¹⁹ The previous statement brings us to the contributions of the current research. To the best of our knowledge, no study was conducted to analyze the performance of different MPC configurations, mainly when the nonlinear model is used. Furthermore, adopting DNMPC for quadrotors has gained less interest in the UAV control literature. Therefore, the contributions of this research are twofold:

Introducing Distributed Nonlinear MPC for controlling the quadrotor by decomposing the centralized MPC into multiple agents whilst reducing the computation complexity of the overall controller.

Evaluate various MPC configurations in terms of performance and computational complexity.

The remainder of this paper is organised as follows: Section “Quadrotor dynamic model” focuses on deriving the quadrotor UAV model and its dynamics as a part of the control system. Section “Problem formulation” highlights the problem formulation and theory underlying the Distributed Nonlinear MPC. Section “DNMPC for quadrotor UAV” presents the design of the DNMP for the quadrotor, Section “Numerical tests & results” demonstrates the various tests applied to the proposed DNMPC and other configurations, along with their results. Section “Discussion” provides a detailed discussion of the results, while Section “Conclusion” concludes the findings of the paper.

Notation

In this paper, vectors are represented in bold lowercase letters (e.g., $x$ ), and the uppercase bolded letters are used for matrices (e.g., $A$ ). Other symbols represent scalars regardless of the letter case. The Euclidean $Q$ -weighted norm is denoted by $‖ x ‖_{Q}^{2} = x^{T} Q x$ where $Q$ is a positive definite real symmetric matrix.

Quadrotor dynamic model

Defining the coordinate systems is crucial to modelling the dynamics of the quadrotor and describing its states. The Earth frame, fixed on the Earth, denoted $F_{E}$ , will be assumed to be inertial with $x$ -axis directed to the north, $y$ -axis directed to the east and $z$ -axis directed down toward the Earth’s centre. The frame $F_{B}$ is the body frame attached to the quadrotor as shown in Figure 2. The attitudes of the quadrotor are defined by the Euler angles: roll angle $ϕ$ , pitch angle $θ$ , and yaw angle $ψ$ . The equations of motion are developed in the body frame FB as shown in $F_{B}$ .²⁰ The absolute position of the quadrotor is $ξ = [x, y, z]^{T}$ and the attitude denoted by $η = [ϕ, θ, ψ]^{T}$ . The model in this paper follows the derivations made in Bolandi et al.²¹ and Suresh et al.²² The thrust $T_{i}$ is aligned with the rotor axis $i$ which rotates at angular velocity $ω_{i}$ and is given by:

T_{i} = b ω_{i}^{2}, i \in 1, 2, 3, 4

(1)

in which

b

is the lift constant equal to

C_{T} ρ A_{p} r_{p}^{2}

, where

ρ

is the density of the air that surrounds the propeller of surface area

A_{p}

and radius of

r_{p}

and where

C_{T}

is the thrust factor. Hence, the thrust along the z-direction of the quadrotor is given by:

T = b \sum_{i = 0}^{4} ω_{i}^{2} = b (ω_{1}^{2} + ω_{2}^{2} + ω_{3}^{2} + ω_{4}^{2})

(2)

On the other hand,

M_{ϕ}

and

M_{θ}

are the pitching and rolling moments due to the thrust differences between the rotors. The net drag force due to the rotors is causing the heading moment

M_{ψ}

. The moments are given by:

M_{ϕ} = l (T_{4} - T_{2})

(3a)

M_{θ} = l (T_{3} - T_{1})

(3b)

M_{ψ} = d ({- ω}_{1}^{2} + ω_{2}^{2} - ω_{3}^{2} + ω_{4}^{2})

(3c)

where

l

is the distance from the quadrotor’s CoM to the rotor axis,

d

is the drag constant given by

C_{P} ρ A_{p} r_{p}^{3}

, and

C_{P}

is the torque coefficient of the motor. If quadrotor translational and angular velocity vectors are

\dot{ξ} = [u, v, w]^{T}

and

\dot{η} = [p, q, r]^{T}

respectively, the translational and angular motion equations are given by Sanca et al.²³:

[\begin{matrix} \dot{u} \\ \dot{v} \\ \dot{w} \\ \dot{p} \\ \dot{q} \\ \dot{r} \end{matrix}] = [\begin{matrix} (\sin ϕ \sin ψ + \cos ϕ \sin θ \cos ψ) \frac{T}{m} - \frac{A_{x}}{m} u \\ (- \sin ϕ \cos ψ + \cos ϕ \sin θ \sin ψ) \frac{T}{m} - \frac{A_{y}}{m} v \\ - g + (\cos ϕ \cos θ) \frac{T}{m} - \frac{A_{z}}{m} w \\ \frac{1}{I_{x x}} {(I_{y y} - I_{z z}) q r - J_{r} ω_{T} q + M_{ϕ} - A_{r} p} \\ \frac{1}{I_{y y}} {(I_{z z} - I_{x x}) p r - J_{r} ω_{T} p + M_{θ} - A_{r} q} \\ \frac{1}{I_{z z}} {(I_{x x} - I_{y y}) p q + M_{ψ} - A_{r} R} \end{matrix}]

(4)

Figure 2.

The quadrotor’s frames coordinate systems.

where $ω_{T} = - ω_{1} + ω_{2} - ω_{3} + ω_{4}$ , $J_{r}$ is the motor’s rotor inertia, $A_{r}$ is the rotational aerodynamic drag coefficient and $A_{x}$ , $A_{y}$ and $A_{z}$ are the linear aerodynamic drag coefficients in the $x$ , $y$ , and $z$ directions respectively. Also, $I_{x x}$ , $I_{y y}$ , and $I_{z z}$ denote the time-invariant inertia of the quadrotor about the body axes in $F_{B}$ . As discussed before, the only variable that can be controlled is the angular velocity of the motors. Before designing a control system, it is necessary to map the control signals to state equations. Thrust and moments are related to the motor’s angular velocities thanks to an allocation matrix, as follows:

[\begin{matrix} T \\ M_{ϕ} \\ M_{θ} \\ M_{ψ} \end{matrix}] = [\begin{matrix} b & b & b & b \\ 0 & - l b & 0 & l b \\ - l b & 0 & l b & 0 \\ - d & d & - d & d \end{matrix}] [\begin{matrix} ω_{1}^{2} \\ ω_{2}^{2} \\ ω_{3}^{2} \\ ω_{4}^{2} \end{matrix}]

(5)

The parameters of the quadrotor UAV used in this paper are adopted from Sanca et al.²³ and given in Table 1.

Table 1.

Quadrotor UAV parameters.

Symbol	Description	Value (Units)
$g$	Acceleration due to gravity	9.806 ( $m / s^{2}$ )
$A_{r}$	Rotational aerodynamic drag coefficient	10 $\times 10^{- 2}$ ( $N m . s / r a d$ )
$m$	Total mass of quadcopter UAV	0.65 ( $k g$ )
$l$	distance from the quadrotor’s COM motor	0.232 ( $m$ )
$A_{x}, A_{y}, A_{z}$	Linear aerodynamic drag coefficient	10 $\times 10^{- 2}$ ( $N . s / m$ )
$J_{r}$	Rotor inertia	4 $\times 10^{- 4}$ ( $k g . m^{2}$ )
$I_{x x}$	Moment of Inertia along x-axis	7.5 $\times 10^{- 3}$ ( $N m . s^{2} / r a d$ )
$I_{y y}$	Moment of Inertia along y-axis	7.5 $\times 10^{- 3}$ ( $N m . s^{2} / r a d$ )
$I_{z z}$	Moment of Inertia along z-axis	1.3 $\times 10^{- 2}$ ( $N m . s^{2} / r a d$ )
$ρ$	Air Density	1.293 ( $k g / m^{3}$ )
$r_{p}$	Propellers Radius	0.15 ( $m$ )
$C_{T}$	Thrust Coefficient	0.055
$C_{P}$	Torque Coefficient	0.024

Problem formulation

Distributed Nonlinear MPC (DNMPC) comprises $N_{a}$ agents, each of which can be viewed as a local model predictive controller. For agent $i$ the nonlinear discrete-time model is Shanbi et al.¹⁸:

x_{k + 1}^{i} = f_{i} (x_{k}^{i}, u_{k}^{i}), i \in {1, \dots, N_{a}}

(6)

Where, $x_{k}^{i} \in R^{n_{i}}$ and $u_{k}^{i} \in R^{m_{i}}$ are the state and the control input of agent $i$ respectively, $n$ , and $m$ representing the local state, and input vector dimensions, respectively. $x_{k}^{i}$ is the successor or the next state and $f_{i} : R^{n_{i}} \times R^{m_{i}} \to R^{n_{i}}$ represents the sub-system dynamics. The sets of constraints that represent the safety limits and physical requirements for the local state and control inputs are denoted $X^{i} \in R^{n_{i}}$ and $U^{i} \in R^{m_{i}}$ respectively. They also represent the feasible state and input of the agent $i$ . Therefore

x_{k}^{i} \in X^{i}, u_{k}^{i} \in U^{i}, k \geq 0, i \in {1, \dots, N_{a}}

(7)

At time step $k$ , the current measure state $x_{k}^{i}$ is used to solve an OLOCP based on the local system model (6). Whatever the formulation of OLOCP, its solution is an optimal state trajectory that stabilizes the system and an optimal control input trajectory that satisfies the anticipated state trajectory. Specifying the prediction horizon is vital to formulating the OLOCP problem and selecting the appropriate optimization solver. According to Eren et al.,¹⁶ the OLOCP problem is infinite or finite. In this paper, the upper approximate finite OLOCP version is adopted with the Terminal Cost function $V_{f_{i}}$ , and Terminal Constraint $X_{f_{i}} \in R^{n_{i}}$ such that:

x_{N}^{i} \in X_{f_{i}}, i \in {1, \dots, N_{a}}

(8)

Where

N

is the prediction horizon. The control objective is to use the receding horizon control approach to stabilize the

N_{a}

agents cooperatively and converge the state of each agent to the equilibrium point

(x_{e}^{i}, u_{e}^{i})

by minimizing the following OLOCP

min_{u} J_{k} (x_{k}, u_{k}) = \sum_{k = 0}^{N - 1} ℓ (x_{k}, u_{k}) + V_{f} (x_{N})

(9a)

s.t. k \in {0, \dots, N}

(9b)

x_{k} = {[(x_{k}^{1})^{T}, \dots, (x_{k}^{N_{a}})^{T}]}^{T}, x_{k} \in X

(9c)

u_{k} = {[(u_{k}^{1})^{T}, \dots, (u_{k}^{N_{a}})^{T}]}^{T}, u_{k} \in U

(9d)

x_{N} = {[(x_{N}^{1})^{T}, \dots, (x_{N}^{N_{a}})^{T}]}^{T}, x_{N} \in X_{f}

(9e)

V_{f} (x_{N}) = \sum_{i = 1}^{N_{a}} V_{f_{i}}

(9f)

Where, $V_{f}$ is the termianl cost function of the OLOCP in (9). Hence, the overall dynamics can be expressed as

x_{k + 1} = f (x_{k}, u_{k})

(10)

where

f = [f_{1}, \dots, f_{N_{a}}]^{T}

f : R^{n} \times R^{m} \to R^{n}

and the global equilibrium point for all agents is

(x_{e}, u_{e})

. The OLOCP in (9) can be used in centralized MPC and is subjected to

{X = X}^{1} \times \dots \times X^{N_{a}}

{U = U}^{1} \times \dots \times U^{N_{a}}

, and

X_{f} {= X}_{f_{1}} \times \dots \times X_{f_{N_{a}}}

. Although this formulation slightly increases the computation cost when it comes to solving the optimization problem, the terminal cost and terminal constraints assure the stability and feasibility of the controlled system.^16,24 There is no specific rule to determine the terminal cost. However, it captures the cost beyond

N

up to

\infty

to guarantee the full covering of the horizon.²⁵

The objective of each agent’s optimization problem is to minimize the tracking error between the local state $x_{k}^{i}$ and its reference trajectory ${\bar{x}}^{i}$ . Furthermore, to decompose the centralized MPC into a distributed MPC, the following two assumptions are made:

Assumption 1.

We will assume that the models in (6) are decoupled and that communication between agents is sufficient and available as needed by any agent.

Accordingly, in a distributed style, problem (9) is divided into multiple OLOCPs given by

\begin{aligned} min_{u^{i}} & J_{k}^{i} (u_{k}^{i}, u_{k}^{i}, {\bar{x}}_{k}^{i}) = \sum_{k = 0}^{N - 1} ℓ_{i} (x_{k}^{i}, u_{k}^{i}, {\bar{x}}_{k}^{i}) + V_{f_{i}} (x_{N}^{i}) \end{aligned}

(11)

s.t. (6), (7), (8) $x_{0}^{i} = x_{i}^{k}$

Where the stage cost function $ℓ_{i} (x_{k}^{i}, u_{k}^{i})$ and terminal cost function $V_{f_{i}} (x_{N}^{i})$ are defined as:

ℓ_{i} (x_{k}^{i}, u_{k}^{i}, {\bar{x}}_{k}^{i}) = ‖ x_{k}^{i} - {\bar{x}}_{k}^{i} ‖_{Q_{i}}^{2} + ‖ u_{k}^{i} ‖_{R_{i}}^{2}

(12a)

V_{f_{i}} (x_{N}^{i}) = ‖ x_{k}^{i} - {\bar{x}}_{N}^{i} ‖_{P_{i}}^{2}

(12b)

where

Q_{i}

R_{i}

, and

P_{i}

are positive definite weighting matrices for penalising the states, manipulated inputs, and terminal state, respectively for agent

i

. For each agent

i

, the control law at the end of each prediction iteration is

u_{k}^{i} = u_{0}^{i} * (x_{k}^{i})

(13)

The superscript * indicates the optimal value and

u_{0}^{i}

is the first control input action in the optimal control input trajectory obtained from the OLOCP solution that is applied to the system.

DNMPC for Quadrotor UAV

When designing a controller for quadrotors, different effects could be considered in the dynamics model; the reader is referred to Ghazbi et al.²⁶ for a list of these effects. Including all effects with a full dynamics representation for the plant will intuitively increase computational complexity. Therefore, a simplified version of the dynamics, such as Abdolhosseini et al.,¹¹ is used to achieve a faster solution during each prediction iteration. However, contrary to Cheng and Yang,¹³ the linear and rotational aerodynamic drag forces are considered in this paper. The resultant dynamics are driven by (4), decoupled and reduced based on the following assumptions:

Assumption 2.

The gyroscopic effects of the motors we will be ignored and will assume that the pitch, roll and yaw axes are decoupled.

Assumption 3.

The heading of the quadrotor is assumed not controlled, and the related torque is zero.

If $T_{s}$ is the sampling time, the subscript $k$ denotes the value of the variables at time $k T_{s}$ , and by using the forward Euler discretization, the simplified decoupled dynamics are given as follows

x_{k + 1} = [\begin{matrix} x_{k} \\ u_{k} \\ y_{k} \\ v_{k} \\ z_{k} \\ w_{k} \\ ϕ_{k} \\ p_{k} \\ θ_{k} \\ q_{k} \end{matrix}] + T_{s} [\begin{matrix} u_{k} \\ m^{- 1} (\sin θ_{k} T_{k} - A_{x} u_{k}) \\ v_{k} \\ m^{- 1} (- \sin ϕ_{k} T_{k} - A_{y} v_{k}) \\ w_{k} \\ - g + m^{- 1} (T_{k} - A_{z} w_{k}) \\ p_{k} \\ {I_{x x}}^{- 1} {M_{ϕ}}_{k} - A_{r} p_{k} \\ q_{k} \\ {I_{y y}}^{- 1} {M_{θ}}_{k} - A_{r} q_{k} \end{matrix}]

(14)

Although the nonlinearity has been significantly reduced and weakened, the above dynamics, without considering the linear and rotational aerodynamic drag forces, are validated in Abdolhosseini et al.¹¹ when the pitching and rolling angles are limited to $\pm 15^{\circ}$ . The limit of $\pm 15^{\circ}$ for the pitch and roll angles represents the maximum acceptable angular deviation for the validity of the simplified decoupled dynamic model, as reported in Abdolhosseini et al.¹¹

This threshold marks the boundary of the flight envelope within which the simplifying assumptions (such as negligible cross-coupling and small-angle approximations) hold. Exceeding this angular limit leads to dynamics no longer accurately captured by the reduced-order model, which can compromise controller performance. Therefore, for the controller design and validation in this work, the roll and pitch angles are constrained within $\pm 15^{\circ}$ rad to ensure the model remains valid throughout the flight. Considering the linear and rotational aerodynamic drag forces, it is expected to perform better and act more accurately compared to what was adopted in Abdolhosseini et al.¹¹ with a bit of additional computation cost.

DNMPC controller

Inspired by Tarisciotti et al.,¹⁹ with assumptions (1-3), we considered the Quadrotor UAV dynamics as a large-scale system in this paper, thanks to dynamics decoupling, (14) represented as five interconnected subsystems where each subsystem is controlled by its own MPC agent shown in Figure 3 to form the DNMPC with $N_{a} = 5$ . Z-MPC, X-MPC, and Y-MPC form the first MPC agents’ group for controlling the translational motion, z-axis, x-axis, and y-axis, respectively. On the other hand, Rolling MPC and Pitching MPC form the second MPC agents’ group for controlling the angular position of $ϕ$ and $θ$ respectively. The choice of $N_{a} = 5$ means we split the quadrotor control problem into five connected subsystems, each controlled by its own MPC agent. Three agents (Z-MPC, X-MPC, Y-MPC) are dedicated to translational motion along the vertical, longitudinal, and lateral axes, while two agents (Rolling MPC and Pitching MPC) control the roll and pitch angles. This setup works well with the decoupled dynamics and helps manage the control tasks more efficiently. Even though the full details of the quadrotor setup are not given, this structure matches the known physical degrees of freedom of the system. The number $N_{a} = 5$ affects the results by balancing how much detail we have in control and how much the agents need to coordinate. Using more agents would make coordination more complex, and using fewer agents might reduce control performance. Therefore, $N_{a} = 5$ gives a good balance between control quality and computational cost.

Figure 3.

Block diagram for Distributed nonlinear MPC for Quadrotor UAV.

The reference set-point is provided for each corresponding MPC agent in the first group. The role of the Z-MPC is to determine the optimal thrust force $T$ , then feed it to the allocation matrix, X-MPC and Y-MPC as a cooperative input within the first group. The proposed design utilized X-MPC and Y-MPC to generate the attitude trajectory for the second MPC agents’ group. Thus, the two manipulated inputs $ϕ$ and $θ$ were the optimal inputs resultant from the Y-MPC and X-MPC, respectively and are provided to the second group as reference trajectory. Finally, the second group of agents, namely, Rolling MPC and Pitching MPC, are designed to determine the optimal rotational torques, ${M_{ϕ}}_{k}$ , and ${M_{θ}}_{k}$ , respectively and feed them to the allocation matrix. The local subsystem’s states and manipulated inputs for each MPC agent are summarized in the equations set (15). It is noteworthy to mention that according to Assumption 3, no controller was used for controlling the heading of the Quadrotor UAV.

x_{k}^{1} = [\begin{matrix} z_{k} \\ w_{k} \end{matrix}], u_{k}^{1} = T_{k}

(15a)

x_{k + 1}^{1} = x_{k}^{1} + T_{s} [\begin{matrix} \begin{matrix} w_{k} \\ - g + m^{- 1} (T_{k} - A_{z} w_{k}) \end{matrix} \end{matrix}]

(15b)

x_{k}^{2} = [\begin{matrix} x_{k} \\ u_{k} \end{matrix}], u_{k}^{2} = θ_{k}

(15c)

x_{k + 1}^{2} = x_{k}^{2} + T_{s} [\begin{matrix} \begin{matrix} u_{k} \\ m^{- 1} (\sin θ_{k} T_{k} - A_{x} u_{k}) \end{matrix} \end{matrix}]

(15d)

x_{k}^{3} = [\begin{matrix} y_{k} \\ v_{k} \end{matrix}], u_{k}^{3} = ϕ_{k}

(15e)

x_{k + 1}^{3} = x_{k}^{3} + T_{s} [\begin{matrix} \begin{matrix} v_{k} \\ m^{- 1} (- \sin ϕ_{k} T_{k} - A_{y} v_{k}) \end{matrix} \end{matrix}]

(15f)

x_{k}^{4} = [\begin{matrix} ϕ_{k} \\ p_{k} \end{matrix}], u_{k}^{4} = {M_{ϕ}}_{k}

(15g)

x_{k + 1}^{4} = x_{k}^{4} + T_{s} [\begin{matrix} p_{k} \\ {I_{x x}}^{- 1} {M_{ϕ}}_{k} - A_{r} p_{k} \end{matrix}]

(15h)

x_{k}^{5} = [\begin{matrix} θ_{k} \\ q_{k} \end{matrix}], u_{k}^{5} = {M_{θ}}_{k}

(15i)

x_{k + 1}^{5} = x_{k}^{5} + T_{s} [\begin{matrix} q_{k} \\ {I_{y y}}^{- 1} {M_{θ}}_{k} - A_{r} q_{k} \end{matrix}]

(15j)

Handling constraint

To unify the framework of the analysis and introduce a fair comparison between the various configurations, unified constraints have been applied. The constraints set for the forces (5) are calculated based on the technical parameters of the brushless motors and their propellers. Assuming that all propulsion brushless motors are identical, the minimum lifting force is equal to $m g$ , whereas the maximum lift force is given in (2). Also, by recalling (3), the maximum and the minimum value of the rotational moment can be calculated by substituting the parameters of Table 1 in (3).

Numerical tests & results

This section explains the simulation environment under which the proposed DNMPC controller and the other three configurations are tested. The other configurations are the Centralized Nonlinear MPC with full dynamics (CNMPC-F) built with dynamics (4), the Centralized Nonlinear MPC with reduced decoupled dynamics (CNMPC-R) built with dynamics (14), and Nonlinear MPC with PID (MPC-PID) in Cheng and Yang.¹³

Simulation setup

The modelling of the quadrotor UAV has been done in MATLAB/Simulink, while the model predictive controllers in all configurations have been carried out in MATLAB/Simulink by integrating CasADi, an open-source optimization toolkit with a MATLAB interface.²⁷ The nonlinear OLOCP has been solved using the Interior Point Optimizer (Ipopt),²⁸ natively available in CasADi. The weighting matrices for each configuration, prediction horizon, control horizon, and sampling time are summarized in Table 2.

Table 2.

Controllers parameters.

Parameter	Description	Value
$T_{s}$	CNMPC Sampling Time	0.05 s
$T_{s}$	DNMPC Translation Group	0.1 s
$T_{s}$	DNMPC Attitude Group	0.05 s
$T_{s}$	MPC in MPC-PID	0.05 s
$T_{s}$	PID Sampling time	0.01 s
$N$	Prediction horizon, all MPCs	40
$Q, P$	Weighting matrices of	$d i a g (15, 1, 15, 1, 150, 10)$
$R$	CNMPC	$d i a g (1.5, 5, 5, 8)$
$Q_{1}, P_{1}$		$d i a g (10, 2)$
$R_{1}$	Z-MPC in DNMPC	$0.1$
$Q_{2}, Q_{3}, P_{2}, P_{3}$	Weighting matrices of	$d i a g (35, 13)$
$R_{2}, R_{3}$	X,Y-MPC in DNMPC	$50$
$Q_{4}, Q_{5}, P_{4}, P_{5}$	Weighting matrices of Rolling	$d i a g (200, 1)$
$R_{3}$	and Pitching MPCs in DNMPC	$20$
$Q, P$	Weighting matrices	$d i a g (5, 1, 5, 1, 100, 10)$
$R$	of MPC in MPC-PID	$d i a g (0.1, 7, 7)$
${K P}_{θ}, {K P}_{θ}, {K P}_{ψ}$		$25$
${K I}_{θ}, {K I}_{θ}, {K I}_{ψ}$	PID gain parameters	$0.1$
${K D}_{θ}, {K D}_{θ}, {K D}_{ψ}$		$15$

The weighting matrices $Q$ , $P$ , and $R$ play an essential role in tuning the performance of each MPC agent. Specifically, $Q$ penalizes the deviation of the states from the reference trajectory, $P$ is the terminal cost matrix that improves stability and performance near the end of the prediction horizon, and $R$ penalizes the control effort, helping to avoid aggressive or impractical control actions. By adjusting these matrices, we can balance tracking performance against effort and smoothness of the control inputs, which directly affects the real behavior of the quadrotor in terms of stability, responsiveness, and energy efficiency. The PID parameters in Table 2 refer to the proportional $K P$ , integral $K I$ , and derivative $K D$ gains, which are tuned to achieve stable and satisfactory tracking performance. These gains influence how quickly and accurately the PID controller reacts to errors. The values for all these parameters were selected through systematic tuning and simulation trials to ensure the quadrotor achieves the best tracking performance and stability under the tested scenarios.

Noise modelling

To assess the robustness of the DNMPC algorithm for the quadrotor, Gaussian white noise was introduced to the state measurements, simulating realistic sensor inaccuracies. The noise was applied to the position $(x, y, z)$ , velocity $(u, v, w)$ , and orientation angles $(ϕ, θ, ψ)$ using Simulink’s Band-Limited White Noise block, with a sampling time of ( $T_{s} = 0.05$ s). Table 3 summarizes the noise parameters, including standard deviation $(σ)$ , noise power, and noise power in decibels (dB), calculated relative to a reference power of $1$ in the respective units. Detailed equations and derivations for the noise model are provided in Appendix A.

Table 3.

Noise parameters used in quadrotor simulation.

State	$σ$	Noise Power	Noise Power (dB)
Position ( $x, y, z$ )	$0.01$ m	$5 \times 10^{- 6}$ $m^{2} s$	$- 53.01$ dB
Velocity ( $u, v, w$ )	$0.05$ m/s	$1.25 \times 10^{- 4}$ $(m/s)^{2} s$	$- 39.03$ dB
Angles ( $ϕ, θ, ψ$ )	$0.01$ rad	$5 \times 10^{- 6}$ ${rad}^{2} s$	$- 53.01$ dB

Step response results

All configurations were tested by applying a step input of $5$ m on the x-axis while the quadrotor is at a hovering point $(0, 0, 4)$ m. The step response characteristics for all configurations are shown in Figure 4, and the response performance for each configuration has been summarised in Table 4.

Figure 4.

Step response of DNMPC, CNMPC-F, CNMPC-R, and MPC-PID controllers for x-axis position change from $0$ m to $5$ m under noisy conditions.

Table 4.

Step response performance summary of DNMPC, CNMPC-F, CNMPC-R, and MPC-PID controllers for x-axis position change from $0$ m to $5$ m under noisy conditions.

	CNMPC-F	CNMPC-R	DNMPC	MPC-PID
Rise Time	1.2576	1.471	1.5451	2.3697
Transient Time	3.1631	3.0213	3.5529	3.6456
Settling Time	3.1653	3.0227	3.5541	4.1543
Settling Min	4.5285	4.5753	4.596	4.511
Settling Max	5.2047	5.1243	5.2757	5.0722
Overshoot	4.2862	2.7352	5.5051	2.0306
Peak	5.2047	5.1243	5.2757	5.0722
Peak Time	2.65	2.9	3.1	4.15

Tracking results

Two 3D trajectories were used to test the tracking performance of each control configuration. Figures 5 and 6 illustrate the 3D tracking response for the infinity shape and the trajectory of the multiple short rectangles, respectively. Also, Figures 7 and 8 illustrate the tracking response for both trajectory in $x$ -, $y$ -, and $z$ -axes.The corresponding motors’ angular velocities for each tracking response are illustrated in Figures 9 and 10. For reproducibility, the trajectory generation logic used in this study is detailed in Appendix B.

Figure 5.

Trajectory tracking performance of DNMPC, CNMPC-F, CNMPC-R, and MPC-PID configurations for a 3D infinite shape trajectory under noisy conditions.

Figure 6.

Trajectory tracking performance of DNMPC, CNMPC-F, CNMPC-R, and MPC-PID configurations for a 3D multiple short-rectangles trajectory under noisy conditions.

Figure 7.

Tracking performance of DNMPC, CNMPC-F, CNMPC-R, and MPC-PID configurations in $x$ -, $y$ -, and $z$ -axes for the multiple short rectangles trajectory under noisy conditions.

Figure 8.

Trajectory tracking performance of DNMPC, CNMPC-F, CNMPC-R, and MPC-PID configurations in $x$ -, $y$ -, and $z$ -axes for the multiple short rectangles trajectory under noisy conditions.

Figure 9.

Comparison of motor angular velocities for each configuration for the infinity shape trajectory under noisy conditions. The DNMPC controller produces smoother inputs, as quantified by the standard deviation in Table 5.

Figure 10.

Comparison of motor angular velocities for each configuration for the multiple short rectangles under noisy conditions. The DNMPC controller produces smoother inputs, as quantified by the standard deviation in Table 5.

Table 5.

Tracking performance for the tested trajectories, including the RMSE and the solution time for each configuration.

Infinity Shape Trajectory (40 s)
	CNMPC-F	CNMPC-R	DNMPC	MPC-PID
RMSE $x$ -axis (m)	0.29178	0.21072	0.38225	0.31654
RMSE $y$ -axis (m)	0.2905	0.21093	0.3761	0.30063
RMSE $z$ -axis (m)	0.46434	0.46428	0.60878	0.40506
Motor Velocity Std. Dev. (rad/s)	13.7442	13.7534	11.0473	49.7316
Average Solving time (ms)	25.1	20.2	7.4	13.7

Multiple Short Rectangles’ Trajectory (26 s)
	CNMPC-F	CNMPC-R	DNMPC	MPC-PID
RMSE $x$ -axis (m)	0.956	1.0181	1.0715	1.1226
RMSE $y$ -axis (m)	0.64136	0.67598	0.71711	0.7648
RMSE $z$ -axis (m)	0.46561	0.46616	0.55761	0.42907
Motor Velocity Std. Dev. (rad/s)	25.3393	24.1329	15.9229	52.0653
Average Solving time (ms)	39.5	35.4	8	19.6

To investigate the practicability and performance of each configuration, the solution time for the OLOCP in each prediction cycle has been monitored for each trajectory tracking case and illustrated in Figures 11 and 12. In addition, the root mean square error RMSE has been determined for all trajectories. Table 5 shows each configuration’s solution time and RMSE for the tested trajectories.

Figure 11.

The solution time for each configuration’s OLOCP prediction cycle when tracking the infinity shape trajectory under noisy conditions.

Figure 12.

The solution time for each configuration’s OLOCP prediction cycle when tracking the multiple short-rectangle trajectories under noisy conditions.

Moreover, to quantify this smoothness, the standard deviation of the motor angular velocities was calculated for each controller and trajectory. For each case, the time-series data of all four motors were combined into a single vector, and the standard deviation was computed, expressed in rad/s, in Table 5.

Performance under wind disturbances

To assess the robustness of the controllers under realistic external influences, the quadrotor was subjected to a hovering task at a constant altitude of $10$ m throughout $50$ s while exposed to wind disturbances. These disturbances were applied in both the $x$ - and $y$ -axes with a magnitude of $2$ m/s during two intervals: from $12$ s to $23$ s and from $34$ s to $39$ s, simulating environmental wind gusts. Additionally, sensor noise was introduced to all state measurements as outlined in Table 3, to emulate realistic onboard sensor conditions. Figures 13 and 14 depict the tracking performance of all tested configurations under these conditions. Figure 13 illustrates the 2D projections, highlighting the deviation from the hovering point due to wind. Figure 14 presents the corresponding position states and the wind disturbance profile. Quantitative evaluation of the tracking error is shown in Table 6, where the RMSE is reported for each axis over the full 50-second simulation.

Figure 13.

Performance for DNMPC, CNMPC-F, CNMPC-R, and MPC-PID controllers at hovering under wind disturbances ( $2$ m/s) in $x$ –axes from $12$ s to $23$ s and in $y$ -axes from $34$ s to $39$ s. The figure shows tracking in 2D projections in the $x y$ - and $z y$ -planes.

Figure 14.

Comparison of the four controller configurations under wind disturbances: tracking performance along the $x$ -, $y$ -, and $z$ -axes, as well as the corresponding velocity responses in the $x$ - and $y$ -axes alongside the wind disturbance profiles.

Table 6.

RMSE of the infinity shape trajectory tracking for each control configuration.

	$x$ -axis ( $m$ )	$y$ -axis ( $m$ )	$z$ -axis ( $m$ )
DNMPC	0.28318	0.19831	0.035294
CNMPC-F	0.30235	0.2602	0.095848
CNMPC-R	0.21609	0.15692	0.04746
MPC-PID	0.44152	0.345	0.43604

Discussion

This section discusses the performance of four different controller configurations, CNMPC-F, CNMPC-R, DNMPC, and MPC-PID, evaluated over two trajectories (Infinity Shape and Multiple Short Rectangles) under noisy and disturbed conditions. A total of eight test cases were conducted to analyze various trade-offs, including tracking accuracy, robustness, computational efficiency, and actuation smoothness. It is seen from Table 5 that decomposing the dynamics and reducing their complexity in the DNMPC led to a significant computation time reduction. This reduction is expected, as computational complexity and thus computation cost can be reduced when the problem dimension is scaled down. For instance, as described in Nocedal and Wright,²⁹ the factorization in Newton-type solvers typically requires $\frac{2}{3} n^{3}$ flops (floating-point operations) for a system with a state dimension $n$ . Therefore, scaling down dynamics (4) used in the CNMPC-F to dimension 2 in the individual state equations of (15) reduces the number of flops drastically (where $n$ is divided by 6).

Concerning tuning, all configurations were tuned using a simple approach where weighting parameters were initialized at one and iteratively adjusted until a best step response was achieved. Tuning the centralized MPCs proved challenging, while the DNMPC configuration was more flexible due to the independence of each agent’s cost function and penalty weights. In terms of the step response analysis, a quantitative analysis under sensor noise conditions is provided in Table 4 to evaluate transient performance. The CNMPC-F controller demonstrates the fastest rise time ( $1.26$ s), while CNMPC-R achieves the shortest settling time ( $3.02$ s) and lowest overshoot ( $2.74 %$ ). In contrast, DNMPC exhibits a longer rise and settling time and the highest overshoot ( $5.5 %$ ), though still within acceptable limits. MPC-PID shows the slowest response (rise time $2.37$ s, settling time $4.15$ s) and the lowest overshoot ( $2.03 %$ ), suggesting a more conservative behavior.

On the other hand, the tracking performance across the two trajectories further distinguishes the controllers. When tracking the Infinity Shape Trajectory, CNMPC-R achieves the lowest RMSE on all axes ( $0.21$ m for both $x$ and $y$ ), while DNMPC shows slightly higher errors ( $0.38$ m) but the shortest solver time ( $7.4$ ms). MPC-PID offers moderate accuracy ( $0.30$ m RMSE) and the highest motor variability ( $49.73$ rad/s), suggesting potential actuation instability. CNMPC-F yields the best tracking accuracy for the Multiple Short Rectangles Trajectory on the $x$ -axis ( $0.96$ m) and $y$ -axis ( $0.68$ m). In comparison, DNMPC offers a favorable trade-off with acceptable accuracy ( $1.07$ m) and the lowest computation time ( $8.0$ ms). MPC-PID remains computationally efficient ( $19.6$ ms) but again results in the highest motor variability ( $52.07$ rad/s), which may not be desirable in hardware-sensitive scenarios.

Further insights from Table 5 reveal that DNMPC consistently achieves the lowest motor velocity variability ( $15.29$ rad/s for Infinity Shape and $11.04$ rad/s for Multiple Rectangles), indicating superior smoothness compared to CNMPC-F ( $25.33$ rad/s, $13.74$ rad/s), CNMPC-R ( $24.13$ rad/s, $13.75$ rad/s), and MPC-PID ( $52.06$ rad/s, $49.73$ rad/s). This suggests DNMPC effectively mitigates noise-induced variability, enhancing control stability and actuator longevity. Moreover, the results presented in Table 6 indicate that the DNMPC configuration outperforms all other controllers in terms of RMSE in the $z$ -axis and maintains acceptable performance in the $x$ - and $y$ -axes, despite the disturbances being applied only in those directions. This robustness is largely attributed to the decoupled dynamics used in the DNMPC design, which isolate the $z$ -axis control from lateral disturbances. Similarly, CNMPC-R benefits from a decoupled formulation and shows competitive tracking accuracy, particularly in the lateral axes. In contrast, the centralized CNMPC-F and MPC-PID configurations exhibit higher tracking errors, especially in the $z$ -axis, where coupling effects appear to degrade performance. These findings underscore the advantage of using decoupled models not only for computational efficiency but also for improved disturbance rejection in specific axes.

It is noteworthy to mention that this work adopts the assumption of seamless communication between agents Assumption 1, which simplifies theoretical analysis and supports the design of the distributed MPC scheme. In our implementation, this assumption holds, as the entire multi-agent system is simulated and executed on a single computer, eliminating delays and packet losses. However, in practical networked control systems, communication is rarely ideal. Real-world deployments are subject to transmission delays, packet losses, and limited bandwidth, which may degrade performance or compromise stability. For instance, unreliable information from neighboring agents can lead to infeasible or suboptimal control actions.

Overall, the discussion includes eight test cases: four controllers evaluated on two trajectories under noisy and disturbance conditions. The results show that while CNMPC-R demonstrates strong tracking accuracy, DNMPC provides the best balance between robustness, smoothness, and computational efficiency, effectively mitigating noise and disturbances. MPC-PID continues to offer real-time performance but at the cost of reduced accuracy and increased control variability.

Conclusion

This paper explores the possible configurations for designing MPC-based flight controllers for a quadrotor UAV. The dynamics of the quadrotor were modelled using its equations of motion and employed as the internal model for the MPC. Four control configurations were studied based on two architectural approaches: centralized and distributed. A nonlinear flight controller based on Distributed MPC (DNMPC) was proposed, in which five separate MPC agents were used to stabilize the quadrotor. All configurations were tested with step inputs and by tracking two different trajectories. The simulation results highlight the strong performance of the DNMPC controller, especially in scenarios involving noisy conditions, where it maintained the smoothest motor velocity profiles indicated by the lowest standard deviation of motor speeds (Table 5). This characteristic is expected to enhance energy efficiency by reducing torque oscillations and associated power losses. Although CNMPC-R achieved the best tracking accuracy under wind disturbances (Table 6), the DNMPC controller maintained a competitive performance while offering notable computational advantages and smoother control actions. These results suggest that DNMPC provides a favorable trade-off between accuracy, computational load, and control stability, making it particularly suitable for real-time and resource-constrained applications. In conclusion, the DNMPC architecture presents a viable and computationally efficient approach to quadrotor trajectory tracking in realistic operating environments. Its distributed design simplifies tuning, supports real-time feasibility, and offers robustness to noise. Future work could involve implementing the controller on embedded hardware and extending the framework to adaptive or learning-based control strategies.

Footnotes

Acknowledgements

This research is funded by the Higher Committee for Education Development in Iraq (HCED) and supported by Sulaimani Polytechnic University, Sulaymaniyah, Iraq.

ORCID iD

Bilal Mubdir

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

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