Planar curved path following controller for a small fixed-wing unmanned aerial vehicle with constrained parameters optimized by nonlinear model predictive control

Abstract

Path following presents a pivotal challenge within the realm of small fixed-wing unmanned aerial vehicles. Firstly, a Lyapunov-stable path guidance law was formulated to follow specific planar curved paths. To ensure differentiability of the guidance law, a modified, smooth saturation function was derived. Secondly, an analysis was conducted to ascertain the interrelationship between control parameters and input constraints, thereby identifying the relevant parameter domains. Thirdly, the nonlinear model predictive control technique was harnessed to optimize both guidance law parameters, enhancing the unmanned aerial vehicle’s capacity to achieve optimal performance in both straight-line and circular path following, hereafter referred to as PFC_NMPC. By leveraging Lyapunov stability arguments for switched systems, the stability of the corresponding nonlinear switched system was guaranteed. In this study, square and circular paths were generated to assess the path-following control of a simulated fixed-wing unmanned aerial vehicle. The performance of various guidance laws, including those with fixed parameters (PFC), those with parameters tuned using fuzzy logic (PFC_FL), PFC_NMPC, vector field, and pure pursuit with line-of-sight, was compared. Notably, the proposed PFC_NMPC method exhibited the ability to expedite the unmanned aerial vehicle’s convergence to the desired path while maximizing the effective flight path length.

Keywords

Curved path following small fixed-wing UAV Lyapunov stability parameters optimization NMPC

Introduction

Small fixed-wing unmanned aerial vehicles (UAVs) have garnered substantial attention from researchers and users in recent times. These aircraft are extensively studied and employed to validate advanced control methodologies and address real-world challenges.^1,2 Notably, within the realm of UAV control technology development, the availability of stable path following control methods assumes a pivotal role. These methods constitute a fundamental component in realizing autonomous UAV flight capabilities.

Given the inherent nonlinearities in the dynamics and kinematics of fixed-wing UAVs, the application of nonlinear control technology is a logical choice to achieve stable path following control of these vehicles. Notably, among the various nonlinear control methods, those grounded in Lyapunov stability control theory are prominently featured in a substantial body of literature addressing UAV path following control.

Building upon the pure pursuit-based path following approach, a nonlinear path-following guidance technique, referred to as the L ₁ guidance algorithm, has been developed, notable for its stability that remains unaffected by variations in velocity.³ This L ₁ guidance algorithm is capable of generating acceleration commands for the UAV in both horizontal and vertical planes, as well as corresponding bank angle commands through command transformation, thus enabling the tracking of 3D curved paths.⁴ To facilitate path following control, especially in scenarios involving wind disturbances, including those with unknown wind disturbance information, an L ₁ adaptive control method has been devised. This method exhibits robustness against time-varying wind conditions, achieved through the utilization of a linearized model dynamics.⁵

Another approach employed for path following is the vector field method, which furnishes the desired course angles for each point along the specified path. Typically, the sliding mode technique is utilized to achieve zero cross-track error. In their work, the authors have introduced a standard vector field approach for both straight and circular path following controllers.⁶ To address challenges posed by unknown wind conditions or unmodeled course angle dynamics, an adaptive vector field-based guidance law has been formulated.⁷ Furthermore, in consideration of minimum turning radius constraints for the UAV, a Gaussian filter has been employed to smooth the discrete vector fields generated by the proposed algorithm.⁸ Zhao et al. harnessed the input-to-state stable properties to design control laws, with the error kinematic model being grounded in the vector field approach.⁹ For achieving precise 3D Dubins-style instructive path following, even in the presence of disturbances, the authors have introduced an enhanced vector field path following method.¹⁰

Drawing from the concept of tracking a virtual target along a predefined straight-line path, a straightforward guidance law was devised, taking into account the aircraft’s maneuvering capabilities.¹¹ The integral light-of-sight guidance logic was implemented to compute the vehicle’s course angle, with the optimization of lookahead distances achieved through the application of the nonlinear model predictive control (NMPC) technique.¹² In scenarios where the guidance law must account for the dynamics of the heading rate or bank angle, the backstepping technique proves valuable. This technique permits the systematic development of the desired control law within the framework of the Lyapunov theory, guiding the design process in a step-by-step manner.^13,14

In certain studies, the incorporation of input constraints has been a key consideration when designing guidance laws. Among the nonlinear design methodologies, the nested saturation technique emerges as a viable approach for addressing such issues. For instance, in the development of a 3D path following controller, the nested saturation technique was employed to account for constraints on roll and flight path angles.¹⁵ To address the challenge of 2D curved path following for fixed-wing UAVs under conditions involving both wind and control constraints, the nested saturation theory was effectively applied.¹⁶

The efficacy of the aforementioned guidance methods, founded on Lyapunov stability theory, has been corroborated through simulations or experiments. It is important to note that in these controllers, the majority of control parameters remain constant to uphold the stability of the associated closed-loop nonlinear systems. However, when the aircraft confronts external wind disturbances during route transitions or in-flight, the fixed controller may not assure consistent optimal performance in path following control. Therefore, it is prudent to contemplate the real-time adjustment of controller parameters while maintaining system stability to attain optimal path following control performance.

In prior studies,^17,18 a PD-like control law for straight path following, rooted in the kinematic model, was devised. Notably, the derivative-like (D-like) parameter was fine-tuned using manually crafted fuzzy rules to enhance path following performance. Nevertheless, it is essential to recognize that artificially designed rules may not encompass all potential scenarios encountered by the vehicle during its flight. Furthermore, the proportional-like (P-like) parameter remained fixed, which presents inherent limitations in addressing diverse and dynamic conditions.

The NMPC technique, renowned for its predictive capabilities and constraint handling, has found application in addressing nonlinear guidance challenges for fixed-wing UAVs.^19,20 These applications encompass both complete dynamic models and simplified kinematic models. While some researchers have formulated guidance laws using the open-loop NMPC approach, the stability of the resulting closed-loop systems remains uncertain.^21,22 In contrast, when employing NMPC with closed-loop system stability, it becomes imperative to establish certain sufficient conditions.^23,24 It is worth noting that the closed-loop stable NMPC method demands significantly greater computational resources compared to the open-loop NMPC approach.

Building upon the insights gleaned from the preceding discussions, this study introduces a PD-like smooth planar path following control law, which is an extension of the straight path following control law.¹⁷ In this approach, the NMPC technique is harnessed to optimize both the P- and D-like parameters. This optimization process is aimed at ensuring that the vehicle consistently delivers optimal path following performance throughout its flight. Furthermore, to guarantee the stability of the associated closed-loop nonlinear systems, the study leverages Lyapunov stability arguments within the context of switched systems. This contributes to the overall robustness and reliability of the proposed control methodology.

This study presents significant contributions: Firstly, it leverages Barbalat’s lemma to formulate a guidance law for following smooth planar paths, involving the derivation of a modified smooth saturation function. Input constraints are thoughtfully accounted for, and the domains of control parameters are determined through a comprehensive analysis of their relationship with these constraints. Secondly, the study pioneers the application of open-loop NMPC technology to optimize all control parameters within the guidance laws. The stability of the corresponding nonlinear switched system is rigorously ensured through the adept use of Lyapunov stability arguments concerning switched systems. The effectiveness of the proposed method was verified via a six-degrees of freedom (6-DOFs) simulation of a fixed-wing UAV. Thirdly, the efficacy and exceptional path following performance of the proposed PFC_NMPC method are rigorously examined through simulations using the Aerosonde UAV model. These results are meticulously compared with other methods, including fixed parameters (PFC), parameters tuned using fuzzy logic (PFC_FL), vector field (VF),⁶ and pure pursuit with line-of-sight (PLOS).²⁵ These contributions collectively enhance our understanding and application of path following control methodologies for unmanned aerial vehicles.

To the best of our knowledge, this proposed method represents a pioneering approach. The remainder of this paper is structured as follows. The second section presents the problem formulation. In the third section, we derive the guidance law for planar smooth path following, featuring fixed control parameters. The fourth section outlines the comprehensive process of optimizing control parameters for both straight and circular path following guidance laws using the open-loop NMPC technique. It also delves into the relationship between control parameters and constrained commands. In addition, we provide a stability analysis of the corresponding switched nonlinear system. The fifth section offers simulation results and discussions pertaining to square and circular path following. Finally, in the sixth section, we draw conclusions and present remarks on the study.

Problem description

Assuming that the fixed-wing UAV is equipped with an autopilot capable of precisely adjusting to the desired altitude and airspeed, the aircraft can effectively maintain horizontal flight. When operating within a horizontal plane, the kinematic model of the aircraft can be described as follows²⁶

{\begin{matrix} \dot{x} = V_{a} cos (ψ) + W_{x} = V_{g} cos (χ) \\ \dot{y} = V_{a} sin (ψ) + W_{y} = V_{g} sin (χ) \end{matrix}

where (x, y) is the center of mass of the vehicle in the inertial horizontal coordinate system, V_a , V_g , $ψ$ , and $χ$ denote the airspeed, groundspeed, heading, and course, respectively. W_x and W_y are the x and y components of the wind speed W, and the relationship among the above variables can be shown in Figure 1.

Figure 1.

The schematic diagram of UAV following the desired horizontal path. UAV: unmanned aerial vehicle.

To design the guidance law, the course rate control $\dot{χ} = u_{c m d}$ was introduced. Then, the model for designing the guidance law can be given as

{\begin{matrix} \dot{x} = V_{g} cos (χ) \\ \dot{y} = V_{g} sin (χ) \\ \dot{χ} = u_{cmd} \end{matrix}

When the UAV is flying, the assumptions that are appropriate for the actual situation include:

Assumption 1

The groundspeed V_g and its derivative $\dot{V_{g}}$ are bounded, and $V_{p 1} \geq V_{g} \geq V_{p 2} > 0$ , where $V_{p 1}$ and $V_{p 2}$ are all positive constants.

Assumption 2

The control $u_{cmd}$ is bounded with ${u_{cmd} | | u_{cmd} | \leq u_{max}}$ , where $u_{max}$ is a positive constant.

This article considers the planar smooth paths that can be expressed by quadratic differentiable functions as

f (x, y) = 0

The value of $f (x, y)$ contains the position information of the UAV relative to the target path, which can represent the generalized distance information. If $f (x, y) = 0$ , the UAV is right on the path.²⁷

To acquire the azimuth information of the UAV relative to the target position, it is essential to compute the first-order partial derivative and gradient magnitude of the function $f (x, y)$ at the UAV’s current position in real time. These values are expressed as follows

{\begin{matrix} f_{x} = \frac{\partial f}{\partial x} \\ f_{y} = \frac{\partial f}{\partial y} \\ ∥ f ∥ = \sqrt{f_{x}^{2} + f_{y}^{2}} \end{matrix}

For the target path, if there are places in the horizontal plane where the gradient modulus of $f (x, y)$ have a small value, the corresponding control laws for these positions may cause the aircraft to run into a danger of losing control. Thus, we artificially define a safe flight area as $D_{s} = {(x, y) | x, y \in R, ∥ \nabla f ∥ \geq λ}$ , where $λ$ is a positive constant.^26,28

Assumption 3

The first- and second-order partial derivatives of the function $f (x, y)$ , that is, f_x , f_y , $f_{x x}$ , $f_{x y}$ and $f_{y y}$ are bounded in any bounded domain $D \subset R^{2}$ .

Path following guidance law with fixed control parameters

The purpose of the guidance law is to enable the UAV to follow the target path in the correct direction. As show in Figure 1, the desired orientation $χ_{d}$ can be given as

χ_{d} = {\begin{matrix} {tan}^{- 1} (- \frac{f_{x}}{f_{y}}), i f f_{y} \neq 0 \\ {cot}^{- 1} (- \frac{f_{y}}{f_{x}}), i f f_{x} \neq 0 \end{matrix}

Introducing the virtual distance error e_d and course angle error $e_{χ}$ as

{\begin{matrix} e_{d} = f (x, y) \\ e_{χ} = χ - χ_{d} \end{matrix}

The time derivatives of e_d and $e_{χ}$ can be derived as

{\begin{matrix} {\dot{e}}_{d} = \frac{d}{d t} f (x, y) = f_{x} \dot{x} + f_{y} \dot{y} \\ = f_{x} V_{g} cos (χ) + f_{y} V_{g} sin (χ) \\ \begin{matrix} = V_{g} ∥ \nabla f ∥ sin (e_{χ}) \\ {\dot{e}}_{χ} = u_{cmd} - \frac{d}{d t} {tan}^{- 1} (- \frac{f_{x}}{f_{y}}) \end{matrix} \end{matrix}

Theorem 1

For the error model shown as equation (7), the control law

u_{cmd} = - K_{1} ∥ \nabla f ∥ V_{g} f_{sat} (e_{d}) - K_{2} V_{g} \frac{d e_{d}}{d t} + \frac{d χ_{d}}{d t}

can asymptotically drives e_d and $e_{χ}$ toward zero.

In equation (8), the two control parameters K ₁ and K ₂ are positive constants. The saturation function $f_{s a t} (x)$ is chosen as

f_{sat} (x) = \frac{2 x_{0}}{π} {tan}^{- 1} (\begin{matrix} \frac{π x}{2 x_{0}} + 0.7884 {(\frac{x}{x_{0}})}^{3} + \\ 3.9562 {(\frac{x}{x_{0}})}^{5} \end{matrix})

where x ₀ is an arbitrary given positive constant.

The function $f_{sat} (x)$ is used to constrain the range of the first part of the control law (8) by limiting e_d . Moreover, it is a strictly monotone increasing bounded function with its range equals $(- x_{0}, x_{0})$ while $x \in (- \infty, + \infty)$ .

It should be noted that the saturation function $f_{sat} (x)$ chosen here is C ². It is different from the C ⁰ saturation function $f_{sat 1} (x)$ shown as (10), which is commonly used in other literatures. It can be found that the function $f_{sat 1} (x)$ is not differentiable at points $(x_{0}, x_{0})$ and $(- x_{0}, - x_{0})$

f_{sat1} (x) = {\begin{matrix} x_{0}, x > x_{0} \\ x, - x_{0} \leq x \leq x_{0} \\ - x_{0}, x < x_{0} \end{matrix}

Define the relative difference between $f_{sat} (x)$ and $f_{sat 1} (x)$ as

h (x) = {\begin{matrix} 0, if x = 0 \\ \frac{f_{sat 1} (x) - f_{sat} (x)}{f_{sat} (x)}, else \end{matrix}

It can be found that the maximum absolute value of $h (x)$ is 0.1 when $x = \pm x_{0}$ . Furthermore, $| h (x) |$ is less than 0.02 while x changes its value in $[- 0.8 x_{0}, 0.8 x_{0}]$ . Figure 2 depicts the comparison of the two functions $f_{sat} (x)$ and $f_{sat1} (x)$ .

Figure 2.

The comparison of the two functions $f_{sat} (x)$ and $f_{sat 1} (x)$ .

Thus, the function $f_{sat 1} (x)$ can ensure that the distance error e_d does not change much as $e_{d} \in [- 0.8 x_{0}, 0.8 x_{0}]$ . In addition, the control law (8) with $f_{sat} (x)$ is differentiable so that the differential calculation involved in the subsequent optimization process can be performed.

Proof

Substitute $u_{cmd}$ shown as (8) into (7), then

{\begin{matrix} {\dot{e}}_{d} = V_{g} ∥ \nabla f ∥ sin (e_{χ}) \\ {\dot{e}}_{χ} = - K_{1} V_{g} ∥ \nabla f ∥ f_{sat} (e_{d}) - K_{2} V_{g}^{2} ∥ \nabla f ∥ sin (e_{χ}) \end{matrix}

Define the state vector $x = {[e_{d}, e_{χ}]}^{T}$ , and the domain of (12) as $D_{1} = {(e_{d}, e_{χ}) | | e_{d} | 〈 e_{d 1}, | e_{χ} | < π}$ , where $e_{d 1}$ is a positive constant. Then, (12) can be expressed as

\dot{x} = g (x, t)

where

g (x, t) = [\begin{matrix} V_{g} ∥ \nabla f ∥ sin (e_{χ}) \\ - K_{1} V_{g} ∥ \nabla f ∥ f_{sat} (e_{d}) - K_{2} V_{g}^{2} ∥ \nabla f ∥ sin (e_{χ}) \end{matrix}]

Under Assumptions 1 and 3, $g : D_{1} \to R^{2}$ is a locally Lipschitz map. It can be found that ${[0, 0]}^{T}$ is the only equilibrium point of the system (13) in D ₁. Choose the following candidate Lyapunov function

V (x) = K_{1} \int_{0}^{e_{d}} f_{sat} (y) d y + (1 - cos (e_{χ}))

It has the fact that $V (0) = 0$ . Moreover, if $e_{d} \neq 0$ , $K_{1} \int_{0}^{e_{d}} f_{sat} (y) d y > 0$ . If $e_{χ} \neq 0$ , $(1 - cos (e_{χ})) > 0$ . It derives that $V (x)$ is positive in D ₁. The time derivative of $V (x)$ along any trajectory of the closed-loop system (13) is given by

\begin{matrix} \dot{V} (x) = K_{1} {\dot{e}}_{d} f_{sat} (e_{d}) + {\dot{e}}_{χ} sin (e_{χ}) \\ \begin{matrix} = K_{1} V_{g} ∥ \nabla f ∥ f_{sat} (e_{d}) sin (e_{χ}) \end{matrix} \\ \begin{matrix} + (- K_{1} V_{g} ∥ \nabla f ∥ V f_{sat} (e_{d}) - \\ \begin{matrix} K_{2} V_{g}^{2} ∥ \nabla f ∥ sin (e_{χ})) sin (e_{χ}) \\ = - K_{2} V_{g}^{2} ∥ \nabla f ∥ {sin}^{2} (e_{χ}) \end{matrix} \end{matrix} \end{matrix}

Thus, $\dot{V} (x)$ is negative semidefinite. With Assumptions 1 and 3 adopted, the time derivative of $\nabla f$ can be derived and it is bounded. Because $V (x)$ is positive in D ₁, $V (x)$ is lower bounded. Since V_g , $\nabla f$ , $d V_{g} / d t$ , and $d \nabla f / d t$ are all bounded, it can be derived from (16) that $\dot{V} (x)$ is uniformly continuous in time. By Barbalat’s lemma,²⁹ $\dot{V} (x) \to 0$ as $t \to \infty$ . Thus, $sin (e_{χ}) \to 0$ as $t \to \infty$ , that is, $e_{χ} \to 0$ as $t \to \infty$ . From the first equation of (12), the fact that $sin (e_{χ}) \to 0$ as $t \to \infty$ derives ${\dot{e}}_{d} \to 0$ as $t \to \infty$ . Because e_d is bounded, it derives that $lim_{t \to \infty} e_{d} = {\bar{e}}_{d}$ . Because $lim_{t \to \infty} e_{χ} = 0$ , and ${\dot{e}}_{χ}$ is uniformly continuous as shown in (12), by Barbalat’s lemma, it derives that ${\dot{e}}_{χ} \to 0$ as $t \to \infty$ . From the second equation of (12), it gets that $lim_{t \to \infty} f_{sat} (e_{d}) = 0$ . Thus, ${\bar{e}}_{d} = 0$ . Finally, it derives that the equilibrium point of (13) is asymptotically stable. □

Path following controller with parameters optimized by NMPC (PFC_NMPC)

Under any given initial condition, the magnitude of the two parameters K ₁ and K ₂ have a significant impact on the response characteristics of the system (12).²⁷ Due to the fact that any smooth path on a plane can be approximated by a combination of several straight or circular segments, this article only considers the path following control of straight and circular paths and the optimal design of their parameters. When appropriate expressions are used for the desired straight line and circular paths, the gradient modulus of these functional expressions are equal to 1,²⁶ which is very convenient for subsequent optimization calculations. The functional expressions for the desired lines and circles are $a x + b y + c = 0$ with $\sqrt{a^{2} + b^{2}} = 1$ and $\sqrt{{(x - C_{x})}^{2} + {(y - C_{y})}^{2}} - C_{R} = 0$ , respectively. In the following section, when using the NMPC to obtain the optimized control parameters $K = (K_{1}, K_{2})$ in (8), it can reduce the amount of calculation.

Straight-line path following controller with parameters optimized by NMPC

The relationship between K_i and $u_{cmd}$ with constraint

During the optimization process, the control parameters $(K_{1}, K_{2})$ should be positive, and the control law $u_{cmd}$ is bounded and within the range $[- u_{max}, u_{max}]$ . By analyzing the relationship among parameters $(K_{1}, K_{2})$ , the states $(e_{d}, {\dot{e}}_{d})$ , and the control law $u_{cmd}$ based on (8), we can obtain a set of relative control parameters $P S_{l}$ as

P S_{l} = {(K_{1}, K_{2}) | \begin{matrix} | K_{1} f_{sat} (e_{d}) + K_{2} {\dot{e}}_{d} | \leq u_{max} \\ 0 \leq K_{1, min} \leq K_{1} \leq K_{1,max} \\ 0 \leq K_{2, min} \leq K_{2} \leq K_{2,max} \end{matrix}}

Because $- x_{0} \leq f_{sat} (e_{d}) \leq x_{0}$ , by choosing $x_{0} = V_{g}$ , it derives that $- V_{g} \leq f_{sat} (e_{d}) \leq V_{g}$ . Furthermore, ${\dot{e}}_{d} = V_{g} sin (e_{χ})$ , it gets $- V_{g} \leq {\dot{e}}_{d} \leq V_{g}$ . By choosing $K_{1,min} = K_{2, min} = \frac{u_{max}}{α V_{g}}$ with $α > 2$ , it derives that

| K_{1} f_{sat} (e_{d}) + K_{2} {\dot{e}}_{d} | \leq \frac{u_{max}}{α V_{g}} (f_{sat} (e_{d}) + {\dot{e}}_{d}) \leq \frac{u_{max}}{α V_{g}} 2 V_{g} = \frac{2 u_{max}}{α} < u_{max}

The proposed path following control method with control parameters optimized by NMPC was presented as Figure 3. The inner-loop control module stabilizes the dynamics and tracks the reference control signals. The pitch angle command achieves the altitude hold. The path generator module generates the desired flight paths.

Figure 3.

The structure of the proposed controller.

In this study, we set $α = 100$ , $K_{1,max} = K_{2, max} = 10$ . Then, the set $P S$ can be guaranteed to be non-empty. If $e_{d} \times {\dot{e}}_{d} \neq 0$ , the shaded portion shown in Figures 4 and 5 describes the specific range of the parameter set $P S_{l}$ .

Figure 4.

The parameters set $P S_{l}$ with given $(e_{d}, {\dot{e}}_{d})$ and $e_{d} \times {\dot{e}}_{d} > 0$ .

Figure 5.

The parameters set $P S_{l}$ with given $(e_{d}, {\dot{e}}_{d})$ and $e_{d} \times {\dot{e}}_{d} < 0$ .

During optimization calculation, if the currently obtained parameter solution $(K_{1}, K_{2})$ is within the set $P S_{l}$ , the current solution is retained. Otherwise, the nearest projection point from the current solution to the set, that is, $(K_{1}^{'}, K_{2}^{'})$ is taken as the current solution

min_{K_{1}^{'}, K_{2}^{'} \in P S_{l}} (| | K_{1}^{'} - K_{1}, K_{2}^{'} - K_{2} | |)

For the points P ₁ and P ₂ shown in Figures 4 and 5, the solutions to (19) are $P_{1}^{′}$ and A, respectively.

The parameters optimization algorithm for straight-line path following

Assume a sampling interval of duration $Δ t$ . The state vector $x_{k} = {(x_{k}, y_{k}, χ_{k})}^{T}$ and u_k is defined as the variables x, y, $χ$ , and u in (2) at time t_k . Then, the dynamics (2) can be approximated in discrete form as

{\begin{matrix} {\dot{e}}_{d, i} = V_{g} ∥ \nabla f ∥ sin (e_{χ, i}) \\ {\dot{e}}_{χ, i} = - K_{1, i} V_{g} ∥ \nabla f ∥ f_{sat} (e_{d, i}) - K_{2, i} V_{g}^{2} ∥ \nabla f ∥ sin (e_{χ, i}) \end{matrix}

where $u_{k} = u_{c m d, k}$ . Typically, the goal of NMPC is to find a control input sequence $(u_{0}, \dots, u_{N - 1})$ that minimizes the cost function³⁰

J = ϕ (x_{N}) + \sum_{i = 0}^{N - 1} L (x_{i}, u_{i})

In the case of straight-line path, the terms $ϕ (x_{N})$ and $L (x_{i}, u_{i})$ in (21) are denoted as

\begin{matrix} ϕ (x_{N}) = S_{D} {(a x_{N} + b y_{N} + c)}^{2} + S_{χ} {(χ_{N} - χ_{d})}^{2} \\ L (x_{i}, u_{i}) = Q_{D} {(a x_{i} + b y_{i} + c)}^{2} + Q_{χ} {(χ_{i} - χ_{d})}^{2} \\ + R u_{i}^{2} \end{matrix}

where S_D , $S_{χ}$ , Q_D , $Q_{χ}$ , and $R$ are weights, and N is the number of prediction steps.

To optimize the control parameters $K = (K_{1}, K_{2})$ , the control u_i in (21) is substituted by u( $K_{i}$ ) shown as

\begin{matrix} u (K_{i}) = - K_{1, i} V_{g} f_{s a t} (a x_{i} + b y_{i} + c) \\ - K_{2, i} V_{g} (a V_{g} cos (χ_{i}) + b V_{g} sin (χ_{i})) \end{matrix}

Then, $L (x_{i}, u_{i})$ becomes as

\begin{matrix} L (x_{i}, u_{i}) = L (x_{i}, K_{i}) = Q_{D} {(a x_{i} + b y_{i} + c)}^{2} + \\ + Q_{χ} {(χ_{i} - χ_{d})}^{2} + R u^{2} (K_{i}) \end{matrix}

The relative equality constraint (20) becomes as

\begin{matrix} x_{k + 1} = f (x_{k}, K_{k}) = \\ (\begin{matrix} x_{k} + Δ t V_{g} cos (χ_{k}) \\ y_{k} + Δ t V_{g} sin (χ_{k}) \\ χ_{k} + Δ t V_{g} (\begin{matrix} - K_{1, k} f_{s a t} (a x_{k} + b y_{k} + c) \\ - K_{2, k} V_{g} (a cos (χ_{k}) + b sin (χ_{k})) \end{matrix}) \end{matrix}) \end{matrix}

Then, the optimized parameters $K_{i}$ can be obtained by solving the optimization problem shown as

K_{i} = min_{\begin{matrix} s . t . K_{i} \in P S_{l} \\ e q . (24) \end{matrix}} (ϕ (x_{N}) + + \sum_{i = 0}^{N - 1} L (x_{i}, K_{i}))

Similar to the indirect method of Lagrange Multipliers for solving the optimization problems using open-loop NMPC technology,³¹ the cost function J can be rewritten as

J = ϕ (x_{N}) + \sum_{i = 0}^{N - 1} L (x_{i}, K_{i}) + λ_{i + 1}^{T} (f (x_{i}, K_{i}) - x_{i + 1})

where ${λ_{i} : i = 0, \dots, N}$ is the Lagrange multiplier.

Here, we can get the Hamiltonian term $H_{i} (x_{i}, K_{i})$ as

H_{i} (x_{i}, K_{i}) = L (x_{i}, K_{i}) + λ_{i + 1}^{T} (f (x_{i}, K_{i}) - x_{i + 1})

In addition, the derivative of J can be derived as

\begin{matrix} d J = (\frac{\partial ϕ (x_{N})}{\partial x_{N}} - λ_{N}^{T}) d x_{N} \\ + \sum_{i = 1}^{N - 1} ((\frac{\partial H_{i} (x_{i}, K_{i})}{\partial x_{i}} - λ_{i}^{T}) d x_{i} + \frac{\partial H_{i} (x_{i}, K_{i})}{\partial K_{i}} d K_{i}) \\ + \frac{\partial H_{0} (x_{0}, K_{i, 0})}{\partial x_{0}} d x_{0} + \frac{\partial H_{0} (x_{0}, K_{i, 0})}{\partial K_{i, 0}} d K_{i, 0} . \end{matrix}

The Lagrange multiplier ${λ_{i} : i = 0, \dots, N}$ can be chosen as³¹

λ_{N} = {(\frac{\partial ϕ (x_{N})}{\partial x_{N}})}^{T}

\begin{matrix} λ_{i} = {(\frac{\partial H_{i} (x_{i}, K_{i})}{\partial x_{i}})}^{T} = {(\frac{\partial L (x_{i}, K_{i})}{\partial x_{i}})}^{T} + {(\frac{\partial f (x_{i}, K_{i})}{\partial x_{i}})}^{T} λ_{i + 1} \\ i = 0, \dots, N - 1 \end{matrix}

Then, $d J$ can be simplified as

d J = \sum_{i = 0}^{N - 1} \frac{\partial H_{i} (x_{i}, K_{i})}{\partial K_{i}} d K_{i} + λ_{0}^{T} d x_{0}

Thus, if x ₀ is constant, an N predictive control parameters $(K_{0}, K_{1}, \dots, K_{N - 1})$ can be obtained through optimization calculation. At time t_k , given the initial value of the parameters $(K_{0}, K_{1}, \dots, K_{N - 1})$ , the algorithm for the parameter optimization process is

while $| Δ J_{k} ≜ J_{k} - J_{k - 1} | > δ$ and $I t e r_n u m < I t e r_{max}$

for $j = 0, \dots, N$ -1

compute $u (K_{j})$ using (23) with the parameter sequence $(K_{0}, \dots, K_{N - 1})$ , and limit $u (K_{j})$ within $[- u_{max}, u_{max}]$ .

compute $x_{j + 1}$ using (25) $.$

end for

compute $λ_{0}$ using (30)

for $j = 0, \dots, N - 1$

compute $λ_{j}$ and $\frac{\partial H_{j} (x_{i}, K_{i})}{\partial K_{j}}$ using (31).

end for

if $Δ J_{k} \leq 0$

for $j = 0, \dots, N - 1$

update $K_{j}$ via $K_{j}^{k + 1} ≜ K_{j}^{k} + h \frac{\partial H_{j} (x_{i}, K_{i})}{\partial K_{j}^{k}}$

and limit $K_{j}^{k + 1}$ within the set $P S_{l}$ (19)

end for

else

reduce the step size h by $h = β \cdot h$

end if

update $I t e r_n u m$ by $I t e r_n u m + +$

end while

The parameters related to the above iterative process, including $I t e r_{max}$ , $δ$ , h, and $β < 1$ are all positive constants.

The stability analysis of systems with time-varying parameters

Lemma 1. ³⁰

Suppose we have a finite number of Lyapunov functions V_i , $i = 1, \dots, N_{t}$ corresponding to the continuous-time vector fields $\dot{x} = f_{i} (x)$ . Let s_k be the switching times of the system. If, whenever we switch in mode (or region) i, with corresponding Lyapunov function V_i , we have $V_{i} (x (s_{k})) < V_{i} (x (s_{j}))$ , where $s_{j} < s_{k}$ is the last time we switched out of mode (or region) i, then the system is stable in the sense of Lyapunov.

In this article, the parameters of each sampling interval remain unchanged, so during a complete flight time, we obtain the following time-varying nonlinear model as

{\begin{matrix} {\dot{e}}_{d, i} = V_{g} ∥ \nabla f ∥ sin (e_{χ, i}) \\ {\dot{e}}_{χ, i} = - K_{1, i} V_{g} ∥ \nabla f ∥ f_{s a t} (e_{d, i}) - K_{2, i} V_{g}^{2} ∥ \nabla f ∥ sin (e_{χ, i}) \end{matrix}

Here, i represents the model within the time range of $[(i - 1) Δ t, i Δ t)$ , where the corresponding control parameters are $(K_{1, i}, K_{2, i})$ . Equation (33) can be regarded as the continuous-time vector fields in Lemma 1. By choosing the Lyapunov functions V_i as $V_{i} = K_{1, i} \int_{0}^{e_{d, i}} f_{sat} (y) d y + (1 - cos (e_{χ, i}))$ , $i = 1, \dots, N_{t}$ , it can derives that the unique equipment point ${[0, 0]}^{T}$ for each mode of equation (33) is asymptotic stability. For a given limited flight path, the flight time is limited, so the number of parameter optimizations is limited. In this case, each mode can be regarded as unique and being switched only once. Then, the conditions $V_{i} (x (s_{k})) < V_{i} (x (s_{j}))$ in Lemma 1 can be satisfied. The above analysis derives that the equilibrium points are stable for the switched system in the sense of Lyapunov. In implementation, when the UAV is tracking close enough to the desired path in finite time, the PFC_NMPC is stopped, while the PFC with the final optimization parameters works, and the relative control parameters sequence is stored for the restarting of PFC_NMPC.

Circular path following controller with parameters optimized by NMPC

To facilitate the optimal calculation of the control parameters of the circular path following controller, the polar coordinates are applied to describe the kinematics of the vehicle. In Figure 6, the polar coordinates are established with the canter of the circle as the origin. The position and the velocities of the vehicle can be described as

{\begin{matrix} x = C_{x} + d cos (γ) \\ y = C_{y} + d sin (γ) \end{matrix}

{\begin{matrix} \dot{x} = V_{g} cos (χ_{c}) \\ \dot{y} = V_{g} sin (χ_{c}) \end{matrix}

Figure 6.

The polar coordinates for the UAV. UAV: unmanned aerial vehicle.

Where $(C_{x}, C_{y})$ is the center of the circle in in the Cartesian coordinate system, C_R is radius, and $γ$ is the angular position of the vehicle relative to the center of the circle.

Taking the derivative of the two sides of (34) with respect to time, and considering the dynamics (35), it can be derived that

{\begin{matrix} \dot{d} = V_{g} cos (χ_{c} - γ) \\ \dot{γ} = \frac{V_{g} sin (χ_{c} - γ)}{d} \end{matrix}

Thus, the model used for the design of a circular path following controller can be represented as

{\begin{matrix} \dot{d} = V_{g} cos (χ_{c} - γ) \\ \begin{matrix} \dot{γ} = V_{g} sin (χ_{c} - γ) / d \end{matrix} \\ {\dot{χ}}_{c} = u_{c, cmd} \end{matrix}

Furthermore, the guidance law (8) for circular path following can be expressed as

u_{c, cmd} = - K_{c 1} V_{g} f_{sat} (d - C_{R}) - K_{c 2} V_{g} \dot{d} + \dot{γ}

where the parameters $K_{c 1}$ and $K_{c 2}$ are positive, and the subscript c represents the circle.

The relationship between $K_{c i}$ and $u_{cmd}$ with constraint

For any given state $(d, γ, χ_{c})$ , the control parameters set $P S_{c}$ with the control input $u_{c,cmd}$ constraint as ${u_{c, cmd} | - u_{max} \leq u_{c,cmd} \leq u_{max}}$ is

P S_{c} = {(K_{c 1}, K_{c 2}) | \begin{matrix} \begin{matrix} | K_{c 1} f_{sat} (d - C_{R}) + K_{c 2} \dot{d} + \frac{V_{g} sin (χ_{c} - γ)}{d} | \\ \leq u_{max} \end{matrix} \\ 0 \leq K_{c 1, min} \leq K_{c 1} \leq K_{c 1, max} \\ 0 \leq K_{c 2, min} \leq K_{c 2} \leq K_{c 2, max} \end{matrix}}

The set $P S_{c}$ is different from $P S_{l}$ in (16), because sometimes the component $\frac{V_{g} sin (χ_{c} - γ)}{d}$ of the circle tracking controller will cause the set to be empty. In this case, the set $P S_{c}$ is replaced with the non empty point set ${(K_{c 1}, K_{c 2}) | K_{c 1} = K_{c 1, min}, K_{c 2} = K_{c 2, min}}$ arbitrarily, and the corresponding control input u is constrained by the saturation function $f_{sat} (x)$ with a new saturation value equals $u_{max}$ . Similar to the process of the straight-line control parameters optimization, if $(K_{c 1}, K_{c 2})$ is outside $P S_{c}$ , the current optimized parameters couple $(K_{c 1}, K_{c 2})$ without inequality constraint by using NMPC is projected to the bound of the set $P S_{c}$ . The processing is equivalent to solving the following problem

min_{K_{c 1}^{'}, K_{c 2}^{'} \in P S_{c}} (| | K_{c 1} - K_{c 1}^{'}, K_{c 2} - K_{c 2}^{'} | |)

The parameters optimization algorithm for circular path following

While using the NMPC technique to optimize the control parameters for circular path following, the discrete-time equality constraints can be derived from (37) as

\begin{matrix} x_{c, k + 1} = f (x_{c, k}, K_{c, k}) = \\ (\begin{matrix} d_{k} + Δ t V_{g} cos (χ_{c, k} - γ_{k}) \\ γ_{k} + Δ t V_{g} sin (χ_{c, k} - γ_{k}) / d_{k} \\ χ_{c, k} + Δ t (\begin{matrix} - K_{c 1, k} V_{g} f_{sat} (d_{k} - C_{R}) - \\ \begin{matrix} - K_{c 2, k} V_{g}^{2} cos (χ_{c, k} - γ_{k}) \\ + \frac{V_{g} sin (χ_{c, k} - γ_{k})}{d_{k}} \end{matrix} \end{matrix}) \end{matrix}) \end{matrix}

The cost function with equality constraints can be defined as

\begin{matrix} J_{c} = ϕ_{c} (x_{c, N}) + \sum_{i = 0}^{N - 1} L_{c} (x_{c, i}, K_{c, i}) \\ + λ_{c, i + 1}^{T} (f_{c} (x_{c, i}, K_{c 2, i}) - x_{c, i + 1}) \end{matrix}

For the circular path following, we have

\begin{matrix} \begin{matrix} ϕ_{c} (x_{c, N}) = S_{c, D} {(d_{N} - C_{R})}^{2} + S_{c, χ} {(χ_{c, N} - γ_{N} + π / 2)}^{2} \\ L_{c} (x_{c, i}, K_{c, i}) = Q_{c, D} {(d_{i} - C_{R})}^{2} + Q_{c, χ} {(χ_{c, i} - γ_{i} + π / 2)}^{2} \end{matrix} \\ + R_{c} u_{c, cmd, i}^{2} \end{matrix}

where $S_{c, D}$ , $S_{c, χ}$ , $Q_{c, D}$ , $Q_{c, χ}$ , and $R_{c}$ are weights, and

\begin{matrix} u_{c, cmd, i} = - K_{c 1, i} V_{g} f_{sat} (d_{i} - C_{R}) - \\ K_{c 2, i} V_{g}^{2} cos (χ_{c, i} - γ_{i}) + V_{g} \frac{V_{g} sin (χ_{c, i} - γ_{i})}{d_{i}} \end{matrix}

Then, the optimized parameters can be obtained by solving the optimization problem shown as

K_{c, i} = min_{s . t . K_{c, i} \in P S_{c}} J_{c}

The processing for calculating the parameters $K_{c, i}$ is similar to the NMPC for optimizing the control parameters for straight-line path following.

Simulation

The simulation research was carried out in the environment of Matlab/Simulink. An Aerosonde UAV model containing standard 6-DOF nonlinear dynamics is used for simulation purposes.²³ The wingspan of the UAV is 2.9 m. Figure 7 shows the overall framework of the simulation, which includes the Aerosonde UAV model, the stable inner-loop controller, and the proposed path following controller. During the simulation, the UAV adopts the bank-to-turn manoeuvre without sideslip. The commanded bank angle was tracked by a well-tunned inner-loop PD controller. With the command $\dot{χ} = u_{cmd}$ shown in (3), the bank angle command $ϕ_{cmd}$ can be derived as³²

ϕ_{cmd} = {tan}^{- 1} (\frac{V_{g} u_{cmd}}{g})

where g is the acceleration of gravity.

Figure 7.

The simulation in the Matlab/Simulink environment.

The parameters involved in the optimization processing, the wind, and the controller are listed in Table 1. The methods of using the fuzzy logic to adjust the parameter K ₂ in the control law (8) (PFC_FL),^28,30 VF,⁶ and PLOS²⁵ were also applied for comparison.

Table 1.

The parameters for simulation.

Parameter	Value	Parameter	Value
Initial V_g	25 m/s	S_D , Q_D	$2.0 \times 10^{- 7}$
Simulation period	50 ms	$S_{χ},$ $Q_{χ}$	$2.5 \times 10^{- 4}$
Wind speed	8 m/s	$S_{C, D}$ , $Q_{C, D}$	$2.0 \times 10^{- 7}$
Wind direction	180° South	$S_{C, χ}$ , $Q_{C, χ}$	$1.6 \times 10^{- 4}$
x ₀	25	$R, R_{C}$	$1.0 \times 10^{- 7}$
$u_{max}$	0.25 rad/s	N	120
Nominal values of K ₁, $K_{c 1}$	$2.0 \times 10^{- 4}$	$δ$	$1.0 \times 10^{- 3}$
Nominal values of K ₂, $K_{c 2}$	$8.0 \times 10^{- 4}$	h	$5.0 \times 10^{- 5}$
$K_{1,max}$ , $K_{2,max}$	$10$	h_c	$6.25 \times 10^{- 6}$
$K_{c 1, max}$ , $K_{c 2, max}$	$10$	$β$	0.7
α	$100$	$I t e r_{max}$	$20$

Square path following

The desired planar square path is depicted in Figure 8(a). In addition, Figure 8(b) presents a comparative analysis of path following results employing five distinct methods: PFC, PFC_FL, PFC_NMPC, VF, and PLOS. During the UAV’s navigation along the square path, the aircraft initiates route switching approximately 200 m in advance to effectively manage transitions.

Figure 8.

The square path following result using the five methods. (a) the position, (b) $f (x, y)$ , (c) K ₁, (d) K ₂, (e) $u_{cmd}$ , (f) $ϕ_{cmd}$ .

Table 2.

Performance indexes of square path following using the five methods.

Segment (which lap)	LOPLPF (m)					Convergence time (s)					Error overshoot (m)
PFC	PFC_FL	PFC_NMPC	VF	PLOS	PFC	PFC_FL	PFC_NMPC	VF	PLOS	PFC	PFC_FL	PFC_NMPC	VF	PLOS
AB (1st)	2263.04	2166.16	2548.19	2374.70	2676.72	21.10	24.75	12.40	20.05	7.45	<1.00	<1.00	<1.00	<1.00	1.643
BC (1st)	2162.26	2126.19	2456.15	2142.68	2440.52	29.05	29.95	13.10	27.80	16.50	8.80	6.82	<1.00	<1.00	<1.00
CD (1st)	2240.33	2154.45	2456.13	2147.67	2361.36	30.80	35.45	18.20	34.80	24.80	17.37	11.44	<1.00	<1.00	<1.00
DA (1st)	2311.88	2115.24	2392.27	2111.60	2366.10	17.95	24.10	16.45	24.50	18.30	1.32	<1.00	<1.00	<1.00	<1.00
AB (2nd)	2256.86	2205.87	2436.29	2202.68	2325.52	15.70	17.55	10.20	17.90	17.00	<1.00	<1.00	<1.00	<1.00	23.34
BC (2nd)	2161.95	2125.23	2446.12	2137.72	2434.04	29.00	31.10	13.60	30.40	16.60	8.43	6.54	<1.00	<1.00	<1.00

The response performance to linear paths can be reflected by the following three indexes: (1) the effective length of the predefined line path followed (LOPLPF), (2) the convergence time, and (3) the error overshoot. First, the effective flight time of the vehicle on a given straight line path is defined as the time from $f (x, y)$ converges to less than 2.9 m until the aircraft starts switching to another path. And then, the index LOPLPF is defined as the projected length of the actual flight path of the vehicle on the target straight line path within the effective path-following time. The convergence time is defined as the time from the beginning of the vehicle’s flight towards the desired path until $| f (x, y) |$ is less than the 2.9 m. The error overshoot is defined as the maximum of $| f (x, y) |$ after it crosses zero.

According to the definition of the straight-line equation, $f (x, y)$ is the lateral offset distance of the aircraft relative to the target line. The comparison of the lateral offset distance $f (x, y)$ was depicted in Figure 8(b). The elliptical regions II, III, IV, and VI in Figure 8(b) show that the method PFC will result the largest error overshoots, which are all larger than 1m. While those by using the method PFC_NMPC are all smaller than 1m.

The three performance indexes LOPLPF, convergence time, and error overshoot are listed in Table 2. Compared to the other two methods, the LOPLPF of each segment for the aircraft to follow the square path based on the method PFC_NMPC is the longest. During the flight of the first six segments, the total LOPLPF obtained by PFC, PFC_FL, PFC_NMPC, VF, and PLOS are 11234.37 m, 10767.91 m, 12289.03 m, 10979.33 m, and 12170.22 m, respectively. This indicates that the vehicle can obtain the longest effective flight path using the method PFC_NMPC. The convergence time listed in Table 2 also shows that the method PFC_NMPC makes the UAV converge the predefined path with the shortest time.

Figure 8(c) to (f) presents the comparison of the parameters K ₁, K ₂, the course rate command $u_{cmd}$ , and the bank angle command $ϕ_{cmd}$ during the flight, respectively. When the UAV turns near the waypoint, the NMPC technology makes the parameters K ₁ and K ₂ change obviously to adapt to the turning process. The resulting control parameters will make the corresponding guidance law and bank angle command adapt to the new segment flight. When using the NMPC technique to get the optimal control parameters K ₁ and K ₂, the specific process is to solve the current cost function based on the current measured values and predictive models. Thus, the optimized parameters K ₁ and K ₂ have the ability of predicting the following error e_d and $e_{χ}$ . The parameter K ₂ in PFC_FL is only tuned based on the fuzzy rules set by human experience. Thus, the method PFC_NMPC can make the UAV fly toward the new line path faster.

Circular path following

When designing the circular path following control simulation scenario, we designed the corresponding circular paths based on two concentric circles and set the aircraft to switch between the two circular paths at a certain position. The expressions for the two concentric circles are defined as $\sqrt{x^{2} + {(y - 400)}^{2}} - 300 = 0$ and $\sqrt{x^{2} + {(y - 400)}^{2}} - 250 = 0$ , respectively, which are marked with circle A and circle B shown in Figure 9(a). In this simulation, approximately 104 s later, the aircraft switched from circle A to circle B near the point (−300, 400).

Figure 9.

The circular path following result using the five methods: (a) the position, (b) $f (x, y)$ , (c) K ₁, (d) K ₂, (e) $u_{cmd}$ , and (f) $ϕ_{cmd}$ .

In a manner similar to the definition of LOPLPF for straight-line path following, we introduce the concept of the effective length of UAV following a predefined circular path (LOPCPF). LOPCPF is defined as the length of the actual flight path after the flight distance error $f (x, y)$ converges to less than 2.9 m. During the UAV’s traversal of circles A and B, three key indices are applied for evaluation: (1) LOPCPF, (2) convergence time, and (3) error overshoot. The corresponding values for these indices are summarized in Table 3. Over a 200-s flight duration, the LOPCPF values obtained using PFC, PFC_FL, PFC_NMPC, VF, and PLOS are 4128.00 m, 4153.00 m, 4323.60 m, 3703.33 m, and 0 m, respectively. Notably, when employing the PFC_NMPC method, the UAV exhibits the shortest convergence time to reach circle A, which is 5.30 s. In contrast, the other four methods all require over 10 s for convergence. Furthermore, when the UAV transitions from circle A to circle B, the PFC_NMPC method again demonstrates the shortest convergence time, recorded at 9.95 s. Importantly, during flight between the two concentric circles, the error overshoots observed with PFC_NMPC do not exceed the wingspan threshold of 2.9 m. Conversely, when using the PFL and PFC_FL methods, the corresponding error overshoots during circle A traversal both exceed 2.9 m.

Table 3.

Performance indexes of circular path following using the five methods.

Circle	LOPCPF (m)					Convergence time (s)					Error overshoot (m)
Circle	PFC	PFC_FL	PFC_NMPC	VF	PLOS	PFC	PFC_FL	PFC_NMPC	VF	PLOS	PFC	PFC_FL	PFC_NMPC	VF	PLOS
A	2132.79	2185.76	2297.49	2118.08	0	12.20	10.05	5.30	13.45	–	7.89	3.80	<1.0	<1.0	–
B	1995.21	1967.24	2026.11	1585.25	0	11.40	12.55	9.95	12.80	–	<1.0	<1.0	1.47	<1.0	–

The above findings highlight that, through the optimization of parameters K ₁ and K ₂ using NMPC, the PFC_NMPC method harnesses the predictive capabilities of the NMPC technique. Consequently, this approach continues to empower the UAV to achieve more efficient flight paths, thereby inheriting the advantages of NMPC’s predictive capabilities.

Conclusions and future works

This article puts forth an innovative path following control approach for fixed-wing unmanned aerial vehicles. The core novelty lies in harnessing NMPC to optimize adjustable parameters within a Lyapunov-stable guidance law. This addresses inherent limitations of fixed control schemes and enhances adaptability to varying flight conditions. Specifically, the contributions are threefold: (1) A smooth planar path following control law is formulated, grounded in Lyapunov stability arguments. This accounts for input constraints through a tailored saturation function. (2) Open-loop NMPC is uniquely employed to optimize proportional and derivative-like gains by transcribing the underlying continuous optimal control problem. This tuning empowers consistent tracking along straight and circular paths. (3) Stability of the resulting nonlinear switched system with time-varying parameters is ensured based on Lyapunov analysis for switched systems. Simulations showcase the proposed technique’s exceptional performance over alternatives, facilitating rapid path convergence and maximized trajectory consistency. The approach and results advance the state-of-the-art in real-time nonlinear path following control for unmanned aerial systems.

In summary, this study pioneers the optimization of adaptive guidance laws via NMPC for fixed-wing UAVs through a stability-assured, performance-centric integration. The domain-specific advancements hold promise in enhancing autonomous flight capabilities.

In future control design efforts, the dynamic response model of the roll angle during bank-to-turn maneuvers will be taken into account. The commanded bank angle will be derived from the course rate command using the backstepping technique. Furthermore, to address the presence of unmodeled components or other disturbances, consideration will be given to the application of fuzzy and neural approximation techniques.^33,34 These techniques have the potential to enhance the path following performance of unmanned aerial vehicles in real-world environments, making them more robust and adaptable.

Footnotes

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.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was partially supported by the National Natural Science Foundation of China (Grant No. 61503172), the Natural Science Foundation of Fujian Province (Grant No. 2020J05197), and the Xinluo District Industry University Research Technology Joint Innovation Project (Grant No. 2022XLXYZ002).

ORCID iD

Yang Chen

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Planar curved path following controller for a small fixed-wing unmanned aerial vehicle with constrained parameters optimized by nonlinear model predictive control

Abstract

Keywords

Introduction

Problem description

Assumption 1

Assumption 2

Assumption 3

Path following guidance law with fixed control parameters

Theorem 1

Proof

Path following controller with parameters optimized by NMPC (PFC_NMPC)

Straight-line path following controller with parameters optimized by NMPC

The relationship between Ki and u cmd with constraint

The parameters optimization algorithm for straight-line path following

The stability analysis of systems with time-varying parameters

Lemma 1. 30

Circular path following controller with parameters optimized by NMPC

The relationship between K c i and u cmd with constraint

The parameters optimization algorithm for circular path following

Simulation

Square path following

Circular path following

Conclusions and future works

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

The relationship between K_i and $u_{cmd}$ with constraint

Lemma 1. ³⁰

The relationship between $K_{c i}$ and $u_{cmd}$ with constraint