Optimal design and experimental validation of a two-loop fractional-order controller for a linearized 2-DOF helicopter system

Abstract

This paper addresses the problem of reference tracking in nonlinear multivariable systems by proposing an optimal two-loop controller design. The methodology begins with input-output linearization and decoupling via state feedback, transforming the original two-input two-output (TITO) nonlinear system into decoupled second-order single-input single-output (SISO) subsystems, which correspond to standard double-integrator systems. For each subsystem, a dual-loop control structure is implemented: the inner loop employs a classical proportional–derivative (PD) controller to stabilize the fast dynamics, while the outer loop uses a fractional-order integral (FI) compensator to enhance steady-state performance and robustness. Controller parameters are optimally tuned using a novel analytical approach based on Bode’s ideal transfer function as an open-loop reference model, in combination with the generic particle swarm optimization (GPSO) algorithm. The proposed method minimizes the mean square tracking error while satisfying input constraints. Comparative evaluations with other optimally tuned fractional-order controllers, including fractional-order PI (FPI), fractional-order PID (FOPID), and fractional-order Tilt–Integral–Derivative (FTID), indicate that the proposed optimal PD + $I^{α}$ controller achieves substantially lower overshoot and tracking error, while offering markedly higher phase and gain margins. Although the rise and settling times are longer, this trade-off prioritizes robustness and smooth transient response over aggressive dynamics, which is particularly desirable for safety-critical systems. Simulation and experimental validations conducted on a 2-degree-of-freedom (2-DOF) helicopter system confirm precise trajectory tracking, effective disturbance rejection, and enhanced robustness against model uncertainties. These results demonstrate that the proposed approach provides an improved compromise between transient performance and robustness, making it particularly suitable for safety-critical multivariable nonlinear systems.

Keywords

particle swarm optimization fractional-order systems feedback linearization nonlinear systems

Introduction

Context and literature review

In recent years, studies have shown that controller design techniques leveraging fractional controllers combined with optimization methods yield superior results compared to traditional approaches, as evidenced in Yang et al. (2023) and Jekan and Subramani (2021).

A wide variety of optimization algorithms are employed in the control domain, including both direct and indirect methods (Wang et al., 2021) as well as metaheuristics (Borne and Gharbi, 2019). Among these, metaheuristic approaches are favored by researchers due to their ability to generate high-quality solutions within reasonable computational times. Metaheuristics can be divided into two main categories: local search techniques and global search techniques. Local search techniques include methods such as tabu search and simulated annealing, while global search techniques encompass algorithms like genetic algorithms (GA), introduced by John Holland in 1975 (Al Tobi et al., 2022), ant colony optimization, proposed by Marco Dorigo in 1992, and particle swarm optimization (PSO), introduced by Kennedy and Eberhart in 1995 (Borne and Gharbi, 2019; Vanneschi and Silva, 2023). PSO is a powerful metaheuristic algorithm inspired by social behaviors found in nature, such as the flocking of birds and the schooling of fish (Liu, 2023). Drawing on concepts from artificial intelligence, PSO is an effective optimization technique capable of addressing complex, nonlinear, and multidimensional problems (Karunanithi et al., 2023). The algorithm operates by initializing a swarm of candidate solutions, known as particles, that iteratively adjust their positions based on both their own experiences and the experiences of their neighboring particles. The ultimate goal is for the particles to converge toward an optimal solution. Chang (2022) suggests the use of an enhanced PSO algorithm for the design of PID controller parameters. In addition, Salamat and Tonello (2019) proposes an adaptive nonlinear PID controller based on multi-objective PSO, which demonstrates PSO’s versatility in handling dynamic and complex control systems. Moreover, PSO has been extended to the design of fractional-order PID (FOPID) controllers, as presented in Safarzadeh and Noori-Kalkhoran (2021), highlighting its broad applicability in the optimization of advanced control strategies. Several studies have explored the design of optimal fixed-structure controllers, such as FOPID, using heuristic optimization algorithms across various engineering applications. For instance, in Mousavi and Alfi (2015), a novel fractional particle swarm optimization–based memetic algorithm (FPSOMA) is proposed for trajectory control by optimizing the parameters of an FOPID controller. Similarly, Shahri et al. (2019) introduce an enhanced PSO variant, known as the augmented Lagrangian particle swarm optimization with fractional-order velocity, which is employed to tune an FOPID controller in an $H_{\infty}$ control framework. These works demonstrate the growing interest and effectiveness of metaheuristic approaches in the optimal tuning of fractional-order fixed-structure controllers. Constrained particle swarm optimization (CPSO) extends PSO to effectively handle constrained optimization problems. Various methods have been proposed to enhance PSO performance in such environments. For instance, Chai et al. (2021) addresses constrained trajectory planning using biased PSO, while (Yang et al., 2021) introduces a two-level balance strategy for constrained multi/many-objective optimization. In recent years, a wide variety of metaheuristic optimization algorithms have been developed to address complex engineering and control problems. Algorithms such as GAs, differential evolution (DE), ant colony optimization (ACO), and PSO have gained popularity due to their ability to solve nonlinear, constrained, and multimodal problems without requiring gradient information. Several recent studies have explored and compared these methods in various applications. For instance, the study in Shokri-Ghaleh et al. (2020) introduces the unequal limit cuckoo optimization algorithm (ULCOA) and demonstrates its effectiveness in the optimal calibration of triaxial accelerometers under real-world uncertainties. The results highlight the robustness and superior convergence properties of ULCOA compared to traditional techniques. Similarly, the comprehensive review in Pahnehkolaei et al. (2017) analyzes the performance of different metaheuristics, including PSO, DE, and ACO, over a set of constrained engineering benchmarks. This work shows that PSO-based algorithms offer a good compromise between solution quality, robustness, and ease of implementation. In addition, Eskandar et al. (2012) applies PSO and other evolutionary strategies to structural control problems, confirming their suitability for optimizing damping and vibration control systems with high nonlinearity and coupling. In this work, the controller design involves the tuning of a two-loop structure composed of a PD controller in the inner loop and a fractional-order integral controller in the outer loop. This leads to a nonlinear and constrained optimization problem in a continuous parameter space. We adopt the generic particle swarm optimization (GPSO) algorithm due to the following reasons:

Suitability for constrained and continuous optimization.

GPSO provides a good balance between exploring different solutions and improving the best ones, which helps to find the optimal controller settings.

Algorithmic simplicity.

Proven success in control applications: PSO and its variants have consistently demonstrated high performance in tuning both classical and fractional-order controllers for complex multivariable systems, as supported by the aforementioned literature. Therefore, based on comparative insights from the literature and the specific nature of our control problem, GPSO is deemed an appropriate and effective optimization strategy for the controller tuning task addressed in this paper.

Recent developments in the PSO literature have introduced advanced variants designed to improve convergence speed, solution diversity, and adaptability in complex dynamic environments. For instance, the study in Pahnehkolaei et al. (2021) presents a novel PSO variant known as Complex-Order Particle Swarm Optimization (CPSO), which incorporates complex-order derivatives and conjugate-order differentials into the update rules of velocity and position. This approach allows particles to model memory effects and learning dynamics more precisely, leading to improved exploration–exploitation trade-offs. A comprehensive sensitivity analysis conducted in the study highlights the impact of fractional parameters on convergence behavior and solution quality. In another recent work, the Enhanced Leader Particle Swarm Optimization (ELPSO) algorithm is introduced in Rezaee Jordehi (2020) for the optimal scheduling of household appliances in Home Energy Management Systems (HEMS). The optimization task is formulated as a constrained multi-objective problem, aiming to minimize electricity cost and peak demand while maintaining user comfort. ELPSO employs a dynamic leader-based mechanism to guide the swarm toward optimal solutions under diverse demand response scenarios. Experimental results across 10 test cases show that ELPSO significantly outperforms traditional PSO and several other state-of-the-art algorithms in terms of convergence efficiency and solution robustness. These recent contributions exemplify the ongoing evolution of PSO and confirm its continued relevance and adaptability to modern engineering challenges. By situating our work within this broader context, we aim to provide readers with a clearer understanding of PSO’s versatility and justify our adoption of the Generic PSO (GPSO) variant as a practical and effective choice for solving the constrained and nonlinear controller tuning problem addressed in this study.

Fractional calculus is a major advancement in mathematical analysis, extending traditional calculus to include derivatives and integrals of fractional orders. This approach has been applied across various fields, including robotics (Singh and Bingi, 2024), image processing (Zhang et al., 2024), and biochemical modeling (Da Silva et al., 2024). In control systems, FOPID controllers offer significant advantages over traditional PID controllers, enhancing system performance through additional tuning parameters that allow finer adjustments to system dynamics (Duarte-Mermoud et al., 2024). Fractional-order (FO) controllers provide greater flexibility and robustness than classical PID controllers due to their additional parameters, though this also makes their design more challenging. They exhibit enhanced performance in the presence of parametric uncertainties in both the system and the controller (Mousavi and Alfi, 2015), and numerous studies have demonstrated the superiority of FOPID controllers in setpoint tracking and disturbance rejection compared to conventional PID controllers (Betala and Nangrani, 2023).

A two-loop control structure provides superior stability, faster response to disturbances, and more precise control compared to a single-loop system. It consists of an inner loop and an outer loop and is widely used in industrial and engineering applications. In Hung et al. (2019), a double-loop control structure is employed in which an outer voltage loop ensures accurate voltage regulation under sag/swell conditions, while an inner current loop provides fast dynamic response and robustness using resonant controllers. Similarly, Mu et al. (2021) proposes a double-loop sliding mode control strategy for trajectory tracking of remotely operated vehicles (ROVs) under strong ocean current disturbances. An outer position control loop generates reference velocities for an inner velocity control loop, while an ocean current observer is integrated to enhance tracking accuracy and robustness. In Lu et al. (2012), a two-loop control architecture is adopted, where the inner loop employs a state-feedback control law with pole placement to ensure stability and strong disturbance rejection. The outer loop integrates an online trajectory replanning strategy to generate constraint-satisfying reference signals, enabling fast and accurate tip-position tracking. This method is implemented on a linear-motor-driven flexible beam system.

A fractional-order two-loop control structure (also known as a dual-loop or cascade control structure) is an advanced strategy that employs fractional-order calculus in its controllers to enhance system performance, stability, and robustness compared to traditional integer-order systems. This structure typically consists of two loops. The inner loop is often an FO proportional–derivative (FOPD) or a simple integer-order PD controller, designed primarily to stabilize the plant and improve disturbance rejection. The outer loop is typically an FO proportional–integral (FOPI) or a fractional-order internal model controller (FOIMC), responsible for controlling the overall system and achieving the desired set-point tracking performance. In Kumar and Ajmeri (2023), the inner loop uses a PD controller tuned for maximum sensitivity to stabilize the plant and improve disturbance rejection, while the outer loop employs a fractional-order IMC to ensure overall performance and robustness. This work is further extended to unstable systems with time delays in Kumar et al. (2025). In addition, a Smith-predictor-based two-loop fractional-order control law for integrating-type chemical processes is presented in Sarkar et al. (2025).

Double-integrator systems appear in many control applications, such as satellites, inverted pendulums, and actuated mechanisms. Classical PID or output-feedback controllers ensure stability, but fractional-order controllers improve robustness, disturbance rejection, and tracking performance. Typically, a dual-loop structure is used: the inner loop stabilizes fast dynamics, while the outer loop achieves accurate reference tracking. Recent studies show that fractional-order dual-loop controllers outperform integer-order controllers in both simulations and experiments. In Saleh et al. (2006), control strategies for double-integrator systems with input saturation nonlinearity are reviewed, covering both classical controllers (PID and conventional compensators) and intelligent controllers, including fuzzy logic and genetic-algorithm-tuned PID controllers. In Meena et al. (2023), a dual-loop structure is adopted where the inner loop employs a fractional-order Tilt–Integral–Derivative (FTID) controller designed via an IMC-based Smith predictor, while the outer loop shapes the closed-loop response through a fractional-order IMC filter to enhance tracking and robustness. A fractional-order $P I^{λ} - D$ controller is proposed in Shankaran et al. (2022) for second-order integrating plants to improve stability and robustness, with parameters determined using explicit formulas and validated via simulations and quadrotor altitude control experiments. In Mehedi et al. (2019), a 2-DOF fractional-order control is implemented for the ball-and-beam system, a typical double-integrator example. The outer loop regulates the ball position via a fractional integral controller, while the inner loop stabilizes the beam angle using an FPD controller to enhance disturbance rejection and robustness. The approach is validated through simulations and experiments, showing superior performance compared to integer-order controllers. However, despite the extensive use of dual-loop and fractional-order control strategies, most reported approaches either employ fully fractional controllers in both loops, rely on heuristic/optimization-based tuning, or target specific application-driven models rather than a generic double-integrator structure. From the above review, it can be observed that a systematic analytical design of a dual-loop controller combining a classical PD inner loop with a fractional-order integrator outer loop for double-integrator-type systems has not yet been explicitly addressed.

Most physical systems exhibit nonlinear behavior, and many industrial processes fall into the category of multiple-input multiple-output (MIMO) systems. Analyzing and controlling nonlinear MIMO systems remains a challenging problem, particularly in fractional control, where limited research has been conducted. In Sivananaithaperumal and Baskar (2014), an automatic tuning method for MIMO FOPID controllers using the covariance matrix adaptation evolutionary strategy was proposed. Similarly, Moradi (2014) applied a GA to design FOPID controllers for a multi-variable process. A novel frequency-domain design for an FOPID controller was introduced in Abdulwahhab and Abbas (2017) to control a nonlinear MIMO system, specifically the twin-rotor aerodynamic system. In addition, Rojas-Moreno (2016) presented a method for controlling both stable and unstable nonlinear MIMO square plants using MIMO FOPID controllers.

In MIMO systems, feedback linearization compensates for nonlinearities while decoupling and linearizing input-output behavior into independent single-input single-output (SISO) systems. The theory is well-documented in Isidori (1985); Sastry (2013), with most research focusing on improving tracking performance. For instance, Manring et al. (2018) applies feedback linearization to enhance the tracking of a hydraulic actuator, while (Moradi et al., 2009) uses pole placement for position control of a single-rod hydraulic actuator. More recently, Boussalem et al. (2019, 2024, 2025) proposed feedback linearization combined with fractional-order integral and PI controllers and applied the approach to a hydraulic system. However, research on tracking control using feedback linearization with fractional-order controllers remains limited.

Objectives

The objective of this paper is to develop a novel, analytically derived dual-loop fractional-order control methodology for nonlinear two-input two-output (TITO) systems that can be transformed into an equivalent double-integrator form. The proposed approach focuses on establishing a structured and physically interpretable analytical framework for cascade control architectures, rather than introducing a new optimization algorithm. Specifically, the inner loop employs a classical proportional–derivative (PD) controller, analytically designed to stabilize the fast dynamics of the double integrator and shape the transient response. The outer loop incorporates a fractional-order integral ( $I^{α}$ ) controller, whose parameters are analytically derived to ensure accurate reference tracking, enhance robustness, and achieve an ideal open-loop behavior based on Bode’s ideal transfer function (BITF), thereby guaranteeing iso-damping properties. Unlike many existing approaches that rely primarily on heuristic or purely optimization-based tuning of fractional-order controllers, the proposed methodology provides explicit analytical expressions for both the PD gains and the fractional-order integral parameters, offering clear insight into the role of each control component. Input–output feedback linearization is used solely as a preprocessing step to obtain a decoupled double-integrator representation, while the GPSO (Ebbesen et al., 2012) algorithm is employed only as an auxiliary refinement tool to slightly improve the analytically obtained parameters, without altering the controller structure. The effectiveness and practical relevance of the proposed analytical design framework are demonstrated through numerical simulations and real-time experimental validation on a 2-DOF helicopter system, including robustness tests under parameter variations and external disturbances. A 2-DOF helicopter is a nonlinear, MIMO system that is unstable and highly coupled. It is often regarded as an unmanned aerial vehicle (UAV) with various civil and military applications. Due to its complexity, it serves as a testbed for validating advanced control strategies. Controlling this system involves addressing challenges such as external disturbances, nonlinearity, and reference tracking. Recent research explores several approaches, including a robust PID controller with a low-pass filter (Mohammed and Mary, 2025), an adaptive neural network controller (Zhu et al., 2024), and a higher-order sliding mode controller combined with a linear quadratic regulator (Boukadida et al., 2019).

The Quanser Aero2 experimental testbed is a nonlinear system used to validate theoretical control approaches. In integer-order control, various methods have been explored through simulation and experimental validation. For instance, a state-feedback and observer-based control strategy is presented in Fellag and Belhocine (2024). A novel active fault-tolerance controller is presented in Ma et al. (2025). However, the implementation of fractional-order controllers on the 2-DOF helicopter testbed remains limited. Notable works include an optimized fractional sliding mode controller (Labdai et al., 2020) and a model reference fractional-order control approach using a type-2 fuzzy neural network.

Contributions

Our main contributions are as follows:

Proposing a novel analytically designed PD + $I^{α}$ fractional-order dual-loop controller for decoupled nonlinear TITO systems.

Ensuring closed-loop robustness by adopting Bode’s ideal transfer function (BITF) as the open-loop reference model.

Validating the proposed approach through numerical simulations and experimental implementation on a decoupled TITO representation of a 2-DOF helicopter system.

Demonstrating the effectiveness of the proposed method through comprehensive comparative analyses against a non-optimal tuning strategy and other fractional-order control schemes.

Paper organization

The paper is organized as follows:

Section “Input-output feedback linearization for MIMO systems” introduces input-output feedback linearization for MIMO systems.

Section “Optimal fractional-order controller Design for Double Integrating Systems” presents the design of the optimal PD + $I^{α}$ fractional-order controller for double-integrator systems. This section is structured into three main parts: (1) the analytical design of the PD + $I^{α}$ fractional-order controller, (2) the constrained PSO algorithm, and (3) the synthesis of the optimal PD + $I^{α}$ fractional-order controller.

Section “Application of feedback linearization and the optimal fractional-order controller on a 2-DOF helicopter” details the modeling of the 2-DOF helicopter, followed by feedback linearization and the application of the optimal PD + $I^{α}$ Fractional-Order Controller.

Section “Simulation and experimental results” presents the simulation and experimental results. First, simulation results are reported for both optimal and non-optimal tuning approaches, followed by a comparative performance analysis. Then, a simulation-based robustness analysis is carried out for the optimal PD + $I^{α}$ fractional-order controller. Finally, the proposed controller is compared in simulation with three fractional-order controllers, namely, the fractional-order PI (FPI), FOPID, and Fractional Tilt–Integral–Derivative (FTID) controllers. Experimental results are provided exclusively for the proposed optimal PD + $I^{α}$ fractional-order controller.

Section “Conclusion” concludes the paper.

Input-output feedback linearization for MIMO systems

Consider a nonlinear multivariable (multi inputs, multi outputs) system having the same number of inputs as outputs (square multivariable system) described by the following state space model

\begin{matrix} {\begin{matrix} ẋ (t) = f (x (t)) + \sum_{i = 1}^{m} g_{i} (x (t)) u_{i} (t) \\ y_{i} (t) = h_{i} (x (t)), i = \bar{1, m} \end{matrix} \end{matrix}

(1)

where $x (t) \in Ω \subset ℝ^{n}$ denotes the state vector and Ω is an open set of $ℝ^{n}$ containing the operating point $x^{0}$ , $u (t) \in ℝ^{m}$ is the input control vector with components $u_{i} (t)$ and $y (t) \in ℝ^{m}$ is the output vector with components $y_{i} (t)$ , m is the number of inputs and outputs. The vector fields $f (x)$ and $g_{i} (x)$ are assumed to be smooth on Ω. $h_{i} (x)$ , $i = \bar{1, m}$ are scalar non linear functions. Feedback linearization is a fundamental strategy in nonlinear control design, rooted in Lie algebra. It aims to transform the input-output dynamics of a nonlinear system into a linear form through coordinate transformations and state feedback, (Isidori, 1985). Alternatively, the input-output feedback linearization of MIMO systems is achieved by differentiating the output variables $y_{i} (t)$ repeatedly until the control inputs appear (Isidori, 1985).

To begin, the time derivative of the i-th output yields to

\begin{matrix} ẏ_{i} (t) = L_{f} h_{i} (x) + \sum_{j = 1}^{m} L_{g_{j}} h_{i} (x) u_{j} (t) \end{matrix}

(2)

where $L_{f} h_{i} (x) = \frac{\partial h_{i} (x)}{∂x} f (x)$ and $L_{g} h_{i} (x) = \frac{\partial h_{i} (x)}{∂x} g (x)$ denote the Lie derivative of the scalar nonlinear function $h_{i} (x)$ with respect to the vector fields $f (x)$ and $g (x)$ , respectively. We define $r_{i}$ as the smallest integer for which at least one control input appears in the $r_{i}$ -th time derivative of $y_{i} (t)$ , that is,

\begin{matrix} y_{i}^{(r_{i})} (t) = L_{f}^{(r_{i})} h_{i} (x) + \sum_{j = 1}^{m} L_{g_{j}} L_{f}^{(r_{i} - 1)} h_{i} (x) u_{j} (t), \end{matrix}

(3)

with the fact that one of the terms $L_{g_{j}} L_{f}^{(r_{i} - 1)} h_{i} (x)$ , $j = \bar{1, m}$ is non null, i.e., $L_{g_{j}} L_{f}^{(r_{i} - 1)} h_{i} (x) \neq 0$ for at least on j, $j = \bar{1, m}$ .

Definition 1. (Sastry, 2013) The multivariable nonlinear system (1) has a vector relative degree $(r_{1}, r_{2}, \dots, r_{m})$ at point $x^{0}$ of Ω if

$L_{g_{j}} L_{f}^{k} h_{i} (x) = 0$ for all $1 \leq j \leq m$ , $k \leq r_{i} - 1$ , $1 \leq i \leq m$ and for all $x \in Ω$ in a neighborhood of $x^{0}$

The $m \times m$ matrix

\begin{matrix} A (x) = [\begin{matrix} L_{g_{1}} L_{f}^{r_{1} - 1} h_{1} (x) & \dots & L_{g_{m}} L_{f}^{r_{1} - 1} h_{1} (x) \\ L_{g_{1}} L_{f}^{r_{2} - 1} h_{2} (x) & \dots & L_{g_{m}} L_{f}^{r_{2} - 1} h_{2} (x) \\ ⋮ & ⋮ & ⋮ \\ L_{g_{1}} L_{f}^{r_{m} - 1} h_{m} (x) & \dots & L_{g_{m}} L_{f}^{r_{m} - 1} h_{m} (x) \end{matrix}] \end{matrix}

(4)

is non singular at $x^{0}$ .

The system (1) is said to have relative degree $(r_{i}, \dots, r_{m})$ at $x_{0}$ , and the sum $r = r_{i} + \dots + r_{m}$ is called the total relative degree of the system at $x^{0}$ . If $r = n$ , then the system can be completely linearized without the presence of internal zeros dynamic. However, if $r < n$ then the input-output linearization can be achieved with the presence of zeros dynamic. In this case, the zeros dynamics should be stable.

Rewriting all relations (3) in vector form, we obtain

\begin{matrix} [\begin{matrix} y_{1}^{(r_{1})} (t) \\ y_{2}^{(r_{2})} (t) \\ ⋮ \\ y_{m}^{(r_{m})} (t) \end{matrix}] = [\begin{matrix} L_{f}^{r_{1}} h_{1} \\ L_{f}^{r_{2}} h_{2} \\ ⋮ \\ L_{f}^{r_{m}} h_{m} \end{matrix}] + A (x) [\begin{matrix} u_{1} (t) \\ ⋮ \\ u_{m} (t) \end{matrix}] \end{matrix}

(5)

If a system has a well-defined vector relative degree, then $A (x)$ is a nonsingular matrix for $x \in Ω$ , a neighborhood of $x_{0}$ . This implies that $A^{- 1} (x)$ has a bounded norm on Ω. Consequently, the input-output decoupling state feedback control law

\begin{matrix} [\begin{matrix} u_{1} (t) \\ ⋮ \\ u_{m} (t) \end{matrix}] = A^{- 1} [\begin{matrix} - L_{f}^{r_{1}} h_{1} + υ_{1} \\ ⋮ \\ - L_{f}^{r_{m}} h_{m} + υ_{m} \end{matrix}] \end{matrix}

(6)

yields to m decoupled linear monovariable systems described by the following equation (Slotine and Li, 1991)

\begin{matrix} y_{i}^{(r_{i})} (t) = v_{i} (t) \end{matrix}

(7)

where $v_{i} (t)$ , $i = \bar{1, m}$ denotes auxiliary control inputs. In the sequel, we focus on the control of nonlinear TITO systems of order $n = 4$ , assuming relative degrees $r_{1} = r_{2} = 2$ . The control methodology begins with input-output feedback linearization, which transforms the original nonlinear multivariable system into two decoupled second-order linear SISO subsystems. This transformation process is schematically represented in Figure 1, which shows the overall TITO system structure with state feedback linearization. The resulting decoupled linear subsystems, each corresponding to a double-integrator model, are depicted in Figure 2. These subsystems form the basis for the proposed two-loop control strategy, where a PD controller is used in the inner loop for stabilization, and a fractional-order integral ( $I^{α}$ ) compensator is employed in the outer loop to enhance robustness and steady-state performance. The design and implementation details of this control architecture are presented in the following section.

Figure 1.

TITO system with state feedback linearization.

Figure 2.

Decoupled linear monovariable subsystems (i=1,2) after state feedback linearization of the TITO system.

Optimal fractional-order controller design for double integrating systems

Fractional-order controller design for linearized TITO system

After applying feedback linearization to the nonlinear TITO system, the resulting dynamics are decoupled into two independent linear subsystems, each characterized by a double-integrator structure, as shown in Figure 2. To effectively control these canonical forms, we propose a fractional-order $PD + I^{α}$ controller, specifically designed to exploit the structure of the linearized model.

As depicted in Figure 3, the controller is organized into a two-loop hierarchical architecture. The inner loop employs a classical proportional-derivative (PD) compensator, primarily aimed at stabilizing the double-integrator system and improving the transient performance, particularly in terms of rise time and damping. The outer loop introduces a fractional-order integrator of order α, where $0 < α < 1$ , designed to approximate Bode’s ideal open-loop transfer function over a defined frequency range around the gain crossover frequency.

Figure 3.

PD + $I^{α}$ control scheme.

This structure enhances the system’s robustness to parameter variations, improves reference tracking accuracy, and contributes to superior steady-state performance. Moreover, the fractional integrator’s inherent memory and non-local properties improve the system’s resilience to external disturbances and modeling uncertainties. By combining the fast dynamic response of the PD controller with the flexibility and robustness offered by the fractional-order integrator, the proposed control scheme achieves a well-balanced compromise between transient behavior and steady-state precision, making it particularly suitable for controlling nonlinear multivariable systems. This compensator is designed to approximate Bode’s ideal open-loop transfer function over a defined frequency range around the gain crossover frequency. As a result, it improves robustness with respect to parameter variations, enhances reference tracking, and contributes to improved steady-state performance. Furthermore, the fractional-order integrator increases the system’s resilience to disturbances and modeling uncertainties.

The controller design methodology is based on frequency-domain specifications and the ideal transfer function of Bode as an open-loop reference model given by

\begin{matrix} M_{B}^{i} (s) = (\frac{Ψ_{c}^{i}}{s})^{δ^{i}}, 1 < δ^{i} < 2 \end{matrix}

(8)

where $Ψ_{c}^{i}$ denotes the gain crossover frequency and $δ^{i}$ the fractional order.

To explain our contribution, let $G^{i} (s)$ be the double-integrator system to be controlled given by (9)

\begin{matrix} G^{i} (s) = \frac{y_{i}}{v_{i}} = \frac{1}{s^{2}} \end{matrix}

(9)

and let $v_{i} (t)$ be the PD + $I^{α}$ fractional-order controller given by (10)

\begin{matrix} v_{i} (t) = - k_{p_{0}}^{i} y_{i} (t) - k_{p_{1}}^{i} \frac{d}{dt} y_{i} (t) + k_{f}^{i} I_{α^{i}} (e_{i} (t)), 0 < α^{i} < 1 \end{matrix}

(10)

where $k_{p_{0}}^{i}, k_{p_{1}}^{i}, k_{f}^{i} \in R^{1 \times 1}$ represent the controller gains to be determined. $y_{i}$ and $\frac{d}{dt} y_{i} (t)$ represent the output and its derivative, respectively. $I_{α^{i}} (e_{i} (t))$ denotes the fractional integration operator of order $α^{i}$ and $e_{i} (t) = {y_{ref}}_{i} - y_{i}$ is the error between the desired setpoint ${y_{ref}}_{i}$ and the output $y_{i} (t)$ . The fractional-order integrator $s^{- α^{i}}$ was implemented using Oustaloup’s recursive approximation via the CRONE toolbox, over a specified frequency range.

The proposed controller is a two-loop control structure composed of a PD controller in the inner loop and a fractional-order integrator $(I^{α})$ in the outer loop, which can be seen as a $PD + I^{α}$ configuration. The proposed two-loop control structure is specifically chosen to handle the instability and performance requirements of the system. The original nonlinear system, after feedback linearization, is transformed into decoupled double-integrator subsystems, which are inherently unstable, with a transfer function of the form $\frac{1}{s^{2}}$ .

The primary role of the inner loop is to stabilize the unstable double-integrator dynamics. This is achieved using a classical proportional-derivative (PD) controller. The resulting closed-loop transfer function of the inner loop becomes: $\frac{1}{s^{2} + k_{p_{1}} s + k_{p_{0}}}$ . By selecting $k_{p_{1}} > 0$ and $k_{p_{0}} > 0$ , the inner loop is guaranteed to be stable with a well-damped response. Hence, the PD controller effectively stabilizes the fast dynamics of the system.

Once stability is ensured by the inner loop, the outer loop employs a fractional-order integral controller $I^{α}$ , $0 < α < 1$ to shape the overall system behavior and improve robustness. The objective is to achieve the ideal open-loop Bode transfer function, which is known to provide desirable robustness margins (in both gain and phase). Designing the open-loop system to match this ideal reference ensures robustness to model uncertainties and external disturbances.

Remark 1. The stability of the overall system is further confirmed by the use of BITF as the open-loop reference model, which inherently guarantees desired phase and gain margins. This separation between stabilization (via the inner PD loop) and performance/robustness (via the outer fractional-order integrator) is a key advantage of the proposed two-loop control structure.

When this control structure is applied to the double-integrator system, it provides a fractional integrator and a second-order transfer function in the open-loop. To ensure a constant phase margin, the second-order part is designed with a damping ratio greater than 1, leading to two real poles. This ensures a constant phase of $- π ∕ 2$ within the cutoff frequency range $[\frac{1}{T_{1}^{i}} \frac{1}{T_{2}^{i}}]$ . The gain crossover frequency is selected within this interval to match the phase behavior of Bode’s ideal model. In this paper, we focus on the practical design and validation of the proposed controller via simulation and experimental results. While a formal proof of stability is beyond the scope of this study, the controller is designed based on BITF, which ensures sufficient stability margins by design. This choice ensures stable behavior in the closed loop.

In this paper, the design results of the PD + $I^{α}$ fractional-order controller for double-integrator systems, derived using input-output feedback linearization, are formalized in Theorem 1.

Theorem 1. Consider a nonlinear MIMO system that has been decoupled into multiple double-integrator subsystems through input-output feedback linearization, as described in (7), with the corresponding control law defined in (10). The controller parameters $k_{p_{0}}^{i}, k_{p_{1}}^{i}, k_{f}^{i}$ , $α^{i}$ and $k^{i}$ are designed to achieve Bode’s ideal open-loop transfer function given by (8), with a specified phase margin ϕ . To ensure a desired gain crossover frequency $ξ_{c}^{i} \leq {(0.091)}^{\frac{1}{α^{i} + 2}}$ , these parameters are determined by the following expressions

\begin{matrix} {\begin{matrix} k_{f}^{i} = 1 \\ k_{p_{0}}^{i} = {Ψ_{c}^{i}}^{2} \\ k^{i} = lo g_{10} \frac{1}{1.1 {Ψ_{c}^{i}}^{2 + α_{i}}}, for Ψ_{c}^{i} < {(\frac{1}{11})}^{\frac{1}{2 + α^{i}}} \\ k_{p_{1}}^{i} = \frac{Ψ_{c}^{i} (1 + 1 0^{2 k^{i}})}{1 0^{k^{i}}} \\ α^{i} = \frac{- 2 Φ^{i} + π}{π} \end{matrix} \end{matrix}

(11)

Proof. Equation (9), can be written as

\begin{matrix} s^{2} y_{i} = v_{i} \end{matrix}

(12)

Using (12) and the Laplace transformation of (10) under zero initial conditions, the resulting closed-loop transfer function is obtained as

\begin{matrix} F^{i} (s) = \frac{y_{i} (s)}{{y_{ref}}_{i} (s)} = \frac{k_{f}^{i}}{s^{α} (s^{2} + k_{p_{1}}^{i} s + k_{p_{0}}^{i}) + k_{f}^{i}} \end{matrix}

(13)

Let $L^{i} (s)$ denote the open-loop transfer function from (13); it is expressed as

\begin{matrix} L^{i} (s) = \frac{k_{f}^{i}}{s^{α^{i}} (s^{2} + k_{p_{1}}^{i} s + k_{p_{0}}^{i})} \end{matrix}

(14)

We observe that the open-loop transfer function consists of two components: the first is a fractional integrator with the transfer function $\frac{k_{f}^{i}}{s^{α^{i}}}$ and the second is a second-order transfer function $\frac{1}{s^{2} + k_{p_{1}}^{i} s + k_{p_{0}}^{i}}$ .

To achieve a behavior similar to that of Bode’ s ideal transfer function (8), the damping coefficient of the second-order system must be greater than or equal to 1. Under this condition, the poles of the second-order transfer function are real and negative, ensuring system stability. Consequently, the second-order transfer function can be factorized into two first-order systems.

Thus, $L^{i} (s)$ consists of three transfer functions and can be expressed in the following general form

\begin{matrix} L^{i} (s) = \frac{b^{i}}{{(s)}^{α^{i}} (1 + T_{1}^{i} s) (1 + T_{2}^{i} s)} \end{matrix}

(15)

Let $L_{1}^{i} (s) = \frac{1}{s^{α^{i}}}$ be the fractional integrator transfer function with a cutoff frequency of $Ψ_{L_{1}^{i}}^{i} = 1$ . Let $L_{2}^{i} (s) = \frac{b^{i}}{1 + T_{1}^{i} s}$ be the first-order transfer function with a cutoff frequency of $Ψ_{L_{2}^{i}}^{i} = \frac{1}{T_{1}^{i}}$ . Let $L_{3}^{i} (s) = \frac{1}{1 + T_{2}^{i} s}$ be the second first-order transfer function with a cutoff frequency of $Ψ_{L_{3}^{i}}^{i} = \frac{1}{T_{2}^{i}}$ . Figure 4 illustrates the asymptotic phase diagram of the open-loop transfer function $L^{i} (s)$ along with its individual components: $L_{1}^{i}, L_{2}^{i}$ and $L_{3}^{i}$ .

Figure 4.

Asymptotic phase diagram of the open-loop transfer function and its various terms.

The Bode diagram in Figure 4 features an interval between $[Ψ_{L_{2}^{i}}^{i} Ψ_{L_{3}^{i}}^{i}]$ where the phase remains constant. Within this region, the phase margin is given by

\begin{matrix} Φ^{i} = π - \frac{π}{2} - \frac{α^{i} π}{2} = π - \frac{π}{2} (1 + α^{i}) \end{matrix}

(16)

The phase margin described above is identical to that of BITF. Therefore, $L^{i} (s)$ can be considered equivalent to the BITF within the interval $[Ψ_{L_{2}^{i}}^{i} Ψ_{L_{3}^{i}}^{i}]$ .

After some mathematical manipulation, an alternative form of (15) is

\begin{matrix} L^{i} (s) = \frac{b^{i} ∕ T_{1}^{i} T_{2}^{i}}{s^{α^{i}} (s^{2} + \frac{T_{1}^{i} + T_{2}^{i}}{T_{1}^{i} T_{2}^{i}} s + \frac{1}{T_{1}^{i} T_{2}^{i}})} \end{matrix}

(17)

A term-by-term identification between transfer functions (14) and (17) leads to

\begin{matrix} {\begin{matrix} k_{p_{0}}^{i} = \frac{1}{T_{1}^{i} T_{2}^{i}} \\ k_{p_{1}}^{i} = \frac{T_{1}^{i} + T_{2}^{i}}{T_{1}^{i} T_{2}^{i}} \\ k_{f}^{i} = \frac{b^{i}}{T_{1}^{i} T_{2}^{i}} \end{matrix} \end{matrix}

(18)

The controller parameters depend on the variables $b^{i}$ , $T_{1}^{i}$ , and $T_{2}^{i}$ , which must be determined analytically. Our primary objective is to impose BITF (8) in the open loop. To design the fractional controller parameters $k_{p_{0}}^{i}$ , $k_{p_{1}}^{i}$ , $k_{f}^{i}$ , and $α^{i}$ , it is necessary to specify the desired phase margin $Φ^{i}$ and the gain crossover frequency $Ψ_{c}^{i}$ for each subsystem i.

From (16), the non-integer order $α^{i}$ of the proposed controller is given by

\begin{matrix} α^{i} = \frac{- 2 Φ^{i} + π}{π} \end{matrix}

(19)

The gain crossover frequency $Ψ_{c}^{i}$ must be selected within the interval $[Ψ_{L_{2}^{i}}^{i} Ψ_{L_{3}^{i}}^{i}]$ i.e. $Ψ_{T_{2}^{i}}^{i} < Ψ_{c}^{i} < Ψ_{T_{3}^{i}}^{i}$ , where the phase remains constant. To satisfy the condition $Ψ_{c}^{i} > Ψ_{L_{2}^{i}}^{i}$ , we introduce a positive real number $k^{i}$ which defines the distance in decades between $Ψ_{T_{2}^{i}}^{i}$ and $Ψ_{c}^{i}$ . Hence, we have

\begin{matrix} \frac{Ψ_{c}^{i}}{Ψ_{L_{2}^{i}}^{i}} = \frac{Ψ_{c}^{i}}{\frac{1}{T_{1}^{i}}} = 1 0^{k^{i}} \end{matrix}

(20)

Consequently, the time constant $T_{1}^{i}$ is defined as

\begin{matrix} T_{1}^{i} = \frac{1 0^{k^{i}}}{Ψ_{c}^{i}} \end{matrix}

(21)

To satisfy the condition $Ψ_{c}^{i} < Ψ_{L_{3}^{i}}^{i}$ , we use the same positive real number $k^{i}$ to define the distance, in decades, between $Ψ_{c}^{i}$ and $Ψ_{L_{3}^{i}}$ . Hence, we have

\begin{matrix} \frac{Ψ_{L_{3}^{i}}^{i}}{Ψ_{c}^{i}} = \frac{\frac{1}{T_{2}^{i}}}{Ψ_{c}^{i}} = 1 0^{k^{i}} \end{matrix}

(22)

Consequently, the time constant $T_{2}^{i}$ is defined as

\begin{matrix} T_{2}^{i} = \frac{1}{Ψ_{c}^{i} 1 0^{k^{i}}} \end{matrix}

(23)

To determine the value of $k^{i}$ , we impose the following condition

\begin{matrix} \frac{b^{i}}{T_{1}^{i} T_{2}^{i}} = 1 \end{matrix}

(24)

Equation (24) yields

\begin{matrix} b^{i} = T_{1}^{i} T_{2}^{i} \end{matrix}

(25)

To achieve the desired phase margin at the specified cutoff frequency $Ψ_{c}^{i}$ , the magnitude of the open-loop transfer function $L^{i} (s)$ must be equal to 1. Therefore, $| L^{i} (j Ψ_{c}^{i}) | = 1$ .

The magnitude of (15) is given by

\begin{matrix} | (L (j Ψ_{c}^{i})) | = \frac{b^{i}}{{Ψ_{c}^{i}}^{α^{i}} \sqrt{(1 + {Ψ_{c}^{i}}^{2} {T_{1}^{i}}^{2}) (1 + {Ψ_{c}^{i}}^{2} {T_{2}^{i}}^{2})}} = 1 \end{matrix}

(26)

Thus, $b^{i}$ is given by

\begin{matrix} b^{i} = {Ψ_{c}^{i}}^{α^{i}} \sqrt{(1 + {Ψ_{c}^{i}}^{2} {T_{1}^{i}}^{2}) (1 + {Ψ_{c}^{i}}^{2} {T_{2}^{i}}^{2})} \end{matrix}

(27)

Using (25) and (27), we derive

\begin{matrix} {Ψ_{c}^{i}}^{α^{i}} \sqrt{(1 + {Ψ_{c}^{i}}^{2} {T_{1}^{i}}^{2}) (1 + {Ψ_{c}^{i}}^{2} {T_{2}^{i}}^{2})} = T_{1}^{i} T_{2}^{i} \end{matrix}

(28)

By substituting (21) and (23) into (28), we determine $k^{i}$ as

\begin{matrix} k^{i} = lo g_{10} (\frac{1}{1.1 {Ψ_{c}^{i}}^{α^{i} + 2}}) \end{matrix}

(29)

To have $k^{i} \geq 1$ , we need to verify the following condition

\begin{matrix} \frac{1}{1.1 {Ψ_{c}^{i}}^{α^{i} + 2}} \geq 10 \end{matrix}

(30)

Equation (30) enables us to determine the desired gain crossover frequency $Ψ_{c}^{i}$ , which is given by

\begin{matrix} Ψ_{c}^{i} \leq {(0.091)}^{\frac{1}{α^{i} + 2}} \end{matrix}

(31)

Taking into account (19) and (29), and substituting (21), (23), and (24) into (18), the parameters of the ${PD + I}^{α}$ controller are given by

\begin{matrix} {\begin{matrix} α^{i} = \frac{- 2 * Φ^{i} + π}{π} \\ k^{i} = lo g_{10} (\frac{1}{1.1 {Ψ_{c}^{i}}^{α^{i} + 2}}) \\ k_{p_{0}}^{i} = {Ψ_{c}^{i}}^{2} \\ k_{p_{1}}^{i} = \frac{Ψ_{c}^{i} (1 + 1 0^{2 k^{i}})}{1 0^{k^{i}}} \\ k_{f}^{i} = 1 \end{matrix} \end{matrix}

(32)

Stability considerations

The design of the proposed fractional-order controller is based on shaping the open-loop system to match BITF, which serves as a reference model. Specifically, a desired phase margin and a gain crossover frequency are imposed during the design process to construct the ideal open-loop profile. This approach ensures that the resulting open-loop system exhibits favorable frequency-domain characteristics, such as robustness and stability. By enforcing these criteria, the closed-loop system inherits a sufficient stability margin without the need for an explicit theoretical stability proof. This strategy, commonly used in robust and fractional-order control design, provides a practical and systematic framework for achieving reliable control performance even in the presence of modeling uncertainties and external disturbances.

A GPSO

In this work, we employed a GPSO algorithm, implemented as a MATLAB function developed by Ebbesen et al. (2012), to achieve the optimal synthesis of the PD + $I^{α}$ fractional-order controller parameters described above. This algorithm is capable of solving problems of the following form

\begin{matrix} \min_{x} : g (x) \end{matrix}

(33)

under conditions

\begin{matrix} {\begin{matrix} Hx \leq b \\ H_{eq} x = b_{eq} \\ C (x) \leq 0 \\ C_{eq} (x) = 0 \\ L_{b} \leq x \leq U_{b} \end{matrix} \end{matrix}

(34)

where $g (x)$ is the fitness function, H and $H_{eq}$ are matrices, and b, $L_{b}$ , $b_{eq}$ , and $U_{b}$ are vectors. The functions $C (x)$ , $C_{eq} (x)$ , and $g (x)$ can be either linear or nonlinear.

To ensure convergence to the optimal solution, the PSO algorithm relies on adjusting the velocity and position vectors using

\begin{matrix} {\begin{matrix} {ve}_{i}^{k + 1} = ϕ^{k} {ve}_{i}^{k} + μ_{1} [r_{1, i} (P a_{i} - {x_{i}}^{k})] + μ_{2} [r_{2, i} (Gl - {x_{i}}^{k})] \\ {x_{i}}^{k + 1} = {x_{i}}^{k} + {ve}_{i}^{k + 1} \end{matrix} \end{matrix}

(35)

where ${v e_{i}}^{k}$ and ${x_{i}}^{k}$ represent the current velocity and position of the i-th generation, respectively. Here, $i = 1, . . ., N$ , where N is the total number of particles. $P a_{i}$ denotes the best position found by each particle, while $Gl$ represents the global best position discovered among all particles up to the current generation. The terms $r_{1}$ and $r_{2}$ are uniformly distributed random values in the range $[0, 1]$ , and $μ_{1}$ and $μ_{2}$ are acceleration constants. The function ϕ represents the particle inertia, which generates a certain momentum for the particles.

To ensure the stability of the swarm, the algorithm must satisfy condition (36)

\begin{matrix} {\begin{matrix} μ_{1} + μ_{2} < 4 \\ \frac{μ_{1} + μ_{2}}{2} - 1 < ϕ < 1 \end{matrix} \end{matrix}

(36)

Optimal controller design

In section “Optimal fractional-order controller Design for Double Integrating Systems,” we introduced a method for designing a PD + $I^{α}$ fractional-order controller based on BITF and frequency-domain specifications. The controller parameters in (32) consist of five elements. The fractional integral gain parameter, $k_{f}^{i}$ is a constant, while the fractional-order $α^{i}$ is a function of the fixed phase margin $Φ_{i}$ . The remaining three parameters $k^{i}$ , $k_{p_{0}}^{i}$ and $k_{p_{1}}^{i}$ are functions of the gain crossover frequency $Ψ_{c}^{i}$ which is constrained by condition (31). For each subsystem i, we designed an optimal fractional-order controller by enforcing a fixed phase margin $Φ^{i}$ , and defining a range for the desired gain crossover frequency $Ψ_{c}^{i}$ . A PSO algorithm was employed to determine the optimal $Ψ_{c}^{i}$ that minimizes the total mean square error of all tracking errors while satisfying control constraints. The objective function is given by

\begin{matrix} g = \frac{1}{N} \sum_{k = 1}^{N} {E^{i}}^{2} \end{matrix}

(37)

where i the number of subsystems.

Application of feedback linearization and the optimal fractional-order controller on a 2-DOF helicopter

The experimental testbed for the 2-DOF helicopter, known as the Quanser Aero2 (Figure 5), features two identical rotors mounted on a base with an integrated data acquisition board. The front rotor, oriented horizontally, controls motion around the pitch axis, while the tail rotor governs yaw-axis movement and counteracts the yawing torque generated by the main rotor. Both rotors are powered by DC motors and equipped with 3D-printed, eight-bladed counter-rotating propellers designed to achieve high dynamic coupling. Encoders measure the pitch angle of the aero-body and the angular positions of the DC motors. In addition, an integrated inertial measurement unit (IMU) is mounted on the Aero2’s core board, enabling real-time measurement of angular position and velocity across all three principal axes of the aero-body. A detailed description of the Quanser Aero2 components can be found in Quanser (2022).

Figure 5.

2-DOF helicopter testbed.

The total forces acting on the free-body Quanser Aero2 system, with a mass $m_{H}$ , are illustrated in Figure 6. The reference frame $B_{r}$ is fixed to the right of the system, with body-fixed axes x, y, and z. Thrust forces $f_{p}$ and $f_{y}$ generated by the propellers, act at distances $h_{p}$ and $h_{y}$ . from the pivot point, respectively. The gravitational force $f_{g}$ acts vertically downward, while $h_{c}$ represents the distance between the system’s center of mass and the pivot of rotation. The horizontal propeller produces torque around the y axis, resulting in pitch motion, while the vertical propeller generates torque around the z axis, causing yaw motion supplied to the DC motors, with the corresponding output responses being the pitch angle θ and the yaw angle ξ.

Figure 6.

Free-body’s applied forces.

The dynamic model of the Quanser Aero2 system, derived using the Lagrangian formulation (Lambert and Reyhanoglu, 2018), is presented in (38)

\begin{matrix} {\begin{matrix} (I_{p} + m_{H} h_{c}^{2}) \overset{··}{θ} = k_{pp} v_{p} + k_{py} v_{y} - D_{p} \overset{\cdot}{θ} - m_{H} g h_{c} \cos (θ) - m h_{c}^{2} {\overset{\cdot}{ξ}}^{2} \sin θ \cos θ \\ (I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})) \overset{··}{ξ} = k_{yp} v_{p} + k_{yy} v_{y} - D_{y} \overset{\cdot}{ξ} + 2 m_{H} h_{c}^{2} \overset{\cdot}{ξ} \overset{\cdot}{θ} \sin (θ) \cos (θ) \end{matrix} \end{matrix}

(38)

All numerical values and variable definitions of the model are provided below: Pitch inertia: $I_{p} = 0.0215$ kg.m², Yaw inertia: $I_{y} = 0.037$ kg.m², Center of mass displacement: $h_{c} = 0.002$ m, Mass : $m_{H} = 1.075$ kg, Pitch viscous friction coefficient: $D_{p} = 0.0071$ N/v, Yaw viscous friction coefficient: $D_{y} = 0.022$ N/v, Pitch torque thrust gain from pitch propeller: $k_{pp} = 0.0011$ Nm/V, Pitch torque thrust gain from yaw propeller: $k_{py} = 0.0007$ Nm/V, Yaw torque thrust gain from pitch propeller: $k_{yp} = - 0.0027$ Nm/V, Yaw torque thrust gain from yaw propeller: $k_{yy} = 0.0022$ Nm/V, voltage applied to the pitch rotor: $v_{p}$ , voltage applied to the yaw rotor: $v_{y}$ .

Input-output feedback linearization for the 2-DOF helicopter system

Consider a TITO nonlinear system represented by a 2-DOF helicopter. Let $x = {[\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \end{matrix}]}^{T} = {[\begin{matrix} θ & \overset{\cdot}{θ} & ξ & \overset{\cdot}{ξ} \end{matrix}]}^{T}$ be the state vector, $u = {[\begin{matrix} v_{p} & v_{y} \end{matrix}]}^{T}$ the control vector, $h = {[\begin{matrix} h_{1} & h_{2} \end{matrix}]}^{T} = {[\begin{matrix} θ & ξ \end{matrix}]}^{T}$ the output vector. The state-space model of the 2-DOF helicopter is expressed as follows

\begin{matrix} {\begin{matrix} \overset{\cdot}{x_{1}} = x_{2} \\ \overset{\cdot}{x_{2}} = \frac{k_{pp}}{I_{p} + m_{H} h_{c}^{2}} v_{p} + \frac{k_{py}}{I_{p} + m_{H} h_{c}^{2}} v_{y} - \frac{D_{p}}{I_{p} + m_{H} h_{c}^{2}} x_{2} \\ - \frac{m_{H} g h_{c}}{I_{p} + m_{H} h_{c}^{2}} \cos (x_{1}) - \frac{m_{H} h_{c}^{2}}{I_{p} + m h_{c}^{2}} {x_{4}}^{2} \sin (x_{1}) \cos (x_{1}) \\ \overset{\cdot}{x_{3}} = x_{4} \\ \overset{\cdot}{x_{4}} = \frac{k_{yp}}{I_{y} + m_{H} h_{c}^{2} co s^{2} (x_{1})} v_{p} + \frac{k_{yy}}{I_{y} + m_{H} h_{c}^{2} co s^{2} (x_{1})} v_{y} - \frac{D_{y}}{I_{y} + m_{H} h_{c}^{2} co s^{2} (x_{1})} x_{4} \\ + \frac{2 m_{H} h_{c}^{2} x_{4} x_{2}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} {x_{4}}^{2} \sin (x_{1}) \cos (x_{1}) \\ y_{1} = x_{1} \\ y_{2} = x_{3} \end{matrix} \end{matrix}

(39)

The 2-DOF helicopter is an MIMO system represented by the nonlinear model (1), where

\begin{matrix} {\begin{matrix} f (x) = [\begin{matrix} f_{1} (x) \\ f_{2} (x) \\ f_{3} (x) \\ f_{4} (x) \end{matrix}] = [\begin{matrix} x_{2} \\ - \frac{D_{p}}{I_{p} + m_{H} h_{c}^{2}} x_{2} - \frac{m_{H} h_{c}^{2} {x_{4}}^{2} \sin (x_{1}) \cos (x_{1})}{I_{p} + m_{H} h_{c}^{2}} - \frac{m_{H} g h_{c} \cos (x_{1})}{I_{p} + m_{H} h_{c}^{2}} \\ x_{4} \\ - \frac{D_{y}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} x_{4} + \frac{2 m_{H} h_{c}^{2} x_{4} x_{2}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} \sin (x_{1}) \cos (x_{1}) \end{matrix}] \\ g (x) = (\begin{matrix} 0 & 0 \\ \frac{k_{pp}}{I_{p} + m_{H} h_{c}^{2}} & \frac{k_{py}}{I_{p} + m_{H} h_{c}^{2}} \\ 0 & 0 \\ \frac{k_{yp}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} & \frac{k_{yy}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} \end{matrix}) \end{matrix} \end{matrix}

(40)

The first and second derivatives of the first output $y_{1} = h_{1} = x_{1}$ can be expressed as follows

\begin{matrix} {\begin{matrix} \overset{\cdot}{y_{1}} = L_{f} h_{1} (x) + [L_{g_{1}} L_{f}^{0} h_{1} (x) L_{g_{2}} L_{f}^{0} h_{1} (x)] [v_{p} v_{y}] \\ ÿ_{1} = L_{f}^{2} h_{1} (x) + [L_{g_{1}} L_{f}^{1} h_{1} (x) L_{g_{2}} L_{f}^{1} h_{1} (x)] {[v_{p} v_{y}]}^{t} \end{matrix} \end{matrix}

(41)

The formula computation derived in (41) provides

\begin{matrix} {\begin{matrix} \overset{\cdot}{y_{1}} = x_{2} \\ ÿ_{1} = \frac{- m_{H} g h_{c} \cos (x_{1})}{I_{p} + m_{H} h_{c}^{2}} - \frac{D_{p} x_{2}}{I_{p} + m_{H} h_{c}^{2}} - \frac{m_{H} h_{c}^{2} {x_{4}}^{2} \sin (x_{1}) \cos (x_{1})}{I_{p} + m_{H} h_{c}^{2}} \\ + \frac{k_{pp}}{I_{p} + m_{H} h_{c}^{2}} v_{p} + \frac{k_{py}}{I_{p} + m_{H} h_{c}^{2}} v_{y} \end{matrix} \end{matrix}

(42)

The first and second derivatives of the second output $y_{2} = h_{2} = x_{3}$ can be expressed as follows

\begin{matrix} {\begin{matrix} \overset{\cdot}{y_{2}} = L_{f} h_{2} (x) + [L_{g_{1}} L_{f}^{0} h_{2} (x) L_{g_{2}} L_{f}^{0} h_{2} (x)] [v_{p} v_{y}] \\ ÿ_{2} = L_{f}^{2} h_{2} (x) + [L_{g_{1}} L_{f}^{1} h_{2} (x) L_{g_{2}} L_{f}^{1} h_{2} (x)] {[v_{p} v_{y}]}^{t} \end{matrix} \end{matrix}

(43)

The formula computation derived in (43) provides

\begin{matrix} {\begin{matrix} \overset{\cdot}{y_{2}} = x_{4} \\ ÿ_{2} = - \frac{D_{y} x_{4}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} + \frac{2 m_{H} h_{c}^{2} x_{4} x_{2} \sin (x_{1}) \cos (x_{1})}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} \\ + \frac{k_{yp} v_{p}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} + {\frac{k_{yy} v_{y}}{I_{y} + m_{H} h_{c}}}^{2} \cos^{2} (x_{1}) \end{matrix} \end{matrix}

(44)

The input emerges in the second derivative of each output $y_{1} = h_{1} = x_{1}$ and $y_{2} = h_{2} = x_{3}$ . The decoupling matrix is derived from (4) as follows

\begin{matrix} \begin{matrix} A (x) = [\begin{matrix} L_{g_{1}} L_{f}^{1} h_{1} L_{g_{2}} L_{f}^{1} h_{1} \\ L_{g_{1}} L_{f}^{1} h_{2} L_{g_{2}} L_{f}^{1} h_{2} \end{matrix}] = \\ [\begin{matrix} \frac{k_{pp}}{I_{p} + m_{H} h_{c}^{2}} \frac{k_{py}}{I_{p} + m_{H} h_{c}^{2}} \\ \frac{k_{yp}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} \frac{k_{yy}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} \end{matrix}] \end{matrix} \end{matrix}

(45)

The determinant of $A (x)$ evaluated at $x = x_{0} = (0 0 0 0)$ , is expressed as

\begin{matrix} \det (A) ∣_{x_{0}} = \frac{k_{pp} k_{yy} - k_{py} k_{yp}}{(I_{p} + m_{H} h_{c}^{2}) (I_{y} + m_{H} h_{c}^{2} co s^{2} (x_{1}))} ∣_{x_{0}} \neq 0 \end{matrix}

(46)

This indicates that $A (x)$ is nonsingular. Consequently, $r_{1} = 2$ and $r_{2} = 2$ represent the relative degrees of the outputs $h_{1} = x_{1}$ and $h_{2} = x_{3}$ , respectively. Therefore, $r_{1} + r_{2}$ equals the system’s order. This establishes the system as an exact feedback linearization, implying that there are no zero dynamics. Using (6), the decoupling control law is expressed as

\begin{matrix} {[\begin{matrix} v_{p} \\ v_{y} \end{matrix}]}_{m \times 1} = {[\begin{matrix} \frac{(I_{p} + m_{H} h_{c}^{2}) k_{yy}}{k_{pp} k_{yy} - k_{py} k_{yp}} - \frac{(I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})) k_{py}}{k_{pp} k_{yy} - k_{py} k_{yp}} \\ - \frac{(I_{p} + m_{H} h_{c}^{2}) k_{yp}}{k_{pp} k_{yy} - k_{py} k_{yp}} \frac{(I_{y} + m_{H} h_{c}^{2} co s^{2} (x_{1})) k_{pp}}{k_{pp} k_{yy} - k_{py} k_{yp}} \end{matrix}]}_{m_{H} \times m_{H}} \\ . [\begin{matrix} \frac{m_{H} g h_{c} \cos (x_{1})}{I_{p} + m_{H} h_{c}^{2}} + \frac{D_{p}}{I_{p} + m_{H} h_{c}^{2}} x_{2} + \frac{m_{H} h_{c}^{2} {x_{4}}^{2} \sin (x 1) \cos (x 1)}{I_{p} + m_{H} h_{c}^{2}} + v_{1} \\ \frac{D_{y} x_{4}}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} - \frac{2 m_{H} h_{c}^{2} x_{4} x_{2} \sin (x_{1}) \cos (x_{1})}{I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})} + v_{2} \end{matrix}] \end{matrix}

(47)

Finally, the control law is defined as

\begin{matrix} {\begin{matrix} v_{p} = \frac{m_{H} g h_{c} \cos (x_{1}) k_{yy}}{k_{pp} k_{yy} - k_{py} k_{yp}} + \frac{D_{p}}{k_{pp} k_{yy} - k_{py} k_{yp}} x_{2} + \frac{m_{H} h_{c}^{2} {x_{4}}^{2} \sin (x 1) \cos (x 1) k_{yy}}{k_{pp} k_{yy} - k_{py} k_{yp}} \\ + \frac{(I_{p} + m_{H} h_{c}^{2}) k_{yy}}{k_{pp} k_{yy} - k_{py} k_{yp}} v_{1} - \frac{D_{y} x_{4} k_{py}}{k_{pp} k_{yy} - k_{py} k_{yp}} + \frac{2 m_{H} h_{c}^{2} x_{4} x_{2} \sin (x 1) \cos (x 1) k_{py}}{k_{pp} k_{yy} - k_{py} k_{yp}} \\ - \frac{(I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})) k_{py}}{k_{pp} k_{yy} - k_{py} k_{yp}} v_{2} . \\ v_{y} = - \frac{m_{H} g h_{c} \cos (x_{1}) k_{yp}}{k_{pp} k_{yy} - k_{py} k_{yp}} - \frac{D_{p} k_{yp}}{k_{pp} k_{yy} - k_{py} k_{yp}} x_{2} - \frac{m_{H} h_{c}^{2} {x_{4}}^{2} \sin (x 1) \cos (x 1) k_{yp}}{k_{pp} k_{yy} - k_{py} k_{yp}} \\ - \frac{(I_{p} + m_{H} h_{c}^{2}) k_{yp}}{k_{pp} k_{yy} - k_{py} k_{yp}} v_{1} + \frac{D_{y} x_{4} k_{pp}}{k_{pp} k_{yy} - k_{py} k_{yp}} - \frac{2 m_{H} h_{c}^{2} x_{4} x_{2} \sin (x 1) \cos (x 1) k_{pp}}{k_{pp} k_{yy} - k_{py} k_{yp}} \\ + \frac{(I_{y} + m_{H} h_{c}^{2} \cos^{2} (x_{1})) k_{pp}}{k_{pp} k_{yy} - k_{py} k_{yp}} v_{2} . \end{matrix} \end{matrix}

(48)

results in two equations of the simple form $y_{i}^{r_{i}} = v_{i}$ , specifically, $y_{1}^{r_{1}} = y_{1}^{(2)} = v_{1}$ , and $y_{2}^{r_{(2)}} = y_{2}^{(2)} = v_{2}$ .

Optimal fractional-order controller for 2-DOF helicopter system control

As discussed above, the fractional controller parameters in (32) consist of five elements. The fractional integral gain parameter, $k_{f}^{i}$ , is a constant, while the fractional-order $α^{i}$ is a function of the fixed phase margin Φ. The remaining three parameters, $k^{i}$ , $k_{p_{0}}^{i}$ and $k_{p_{1}}^{i}$ are functions of the gain crossover frequency $Ψ_{c}^{i}$ , which is a positive value selected within the range $] 0, {(0.091)}^{\frac{1}{α^{i} + 2}} [$ . The PSO algorithm selects $Ψ_{c}^{i}$ , enabling the automatic computation of all dependent variables using equations (21), (23), (29), and (32). The optimal controller is designed by minimizing the total mean square error between the desired and actual pitch and yaw outputs, given by

\begin{matrix} g = \frac{1}{N} \sum_{k = 1}^{N} {E_{θ}}^{2} + {E_{ξ}}^{2} \end{matrix}

(49)

where $E_{θ} = θ_{ref} - θ$ and $E_{ξ} = ξ_{ref} - ξ$ under control constraint $- 24 < v_{p}, v_{y} < 24$ . The block diagram of the FL technique incorporating the optimal PD + $I^{α}$ fractional-order controller for the 2-DOF helicopter system is shown in Figure 7

Figure 7.

Block diagram of feedback linearization with optimal PD + $I^{α}$ fractional-order controller for the 2-DOF helicopter system.

Remark 2. The performance and convergence of GPSO are influenced by several control parameters, including the inertia weight, cognitive acceleration coefficient, social acceleration coefficient, random coefficients and the population size. In this study, these parameters were selected based on standard practices reported in the literature and refined through preliminary experiments to ensure a balance between exploration and exploitation capabilities. To assess the repeatability of the optimization process, the GPSO algorithm was executed multiple times (8 independent runs). The results showed a high degree of consistency across runs, with negligible variations in the obtained controller parameters and system performance. This confirms the robustness and stability of the tuning process under the adopted GPSO configuration. While this work focuses primarily on the analytical development of the proposed two-loop fractional-order controller, GPSO is utilized solely to optimize its parameters and improve the closed-loop performance, we acknowledge that a formal analytical study of the algorithm’s convergence such as that presented in Pahnehkolaei et al. (2022) offer valuable theoretical insights. We consider this as a promising avenue for future research.

Simulation and experimental results

To evaluate the performance of the proposed control strategy, extensive simulation and experimental studies are carried out on the 2-DOF helicopter system. The simulations are performed using a square reference input with an amplitude of $\pm 0.52$ rad ( $\pm 3 0^{\circ}$ ) and a frequency of 0.01 Hz.

First, simulation results are reported for both optimal and non-optimal tuning approaches of the proposed PD + $I^{α}$ fractional-order controller, followed by a comparative performance analysis based on standard time-domain indices. Subsequently, a simulation-based robustness analysis is conducted for the optimally tuned PD + $I^{α}$ controller to assess its ability to handle internal parametric uncertainties.

Furthermore, in simulation, the proposed controller is compared with three fractional-order control strategies, namely, the FPI, FOPID and FTID controllers, in order to highlight its relative advantages in terms of tracking accuracy and dynamic performance.

Finally, experimental results are presented exclusively for the proposed optimal PD + $I^{α}$ fractional-order controller to validate its practical feasibility and real-time performance. In addition to nominal experimental evaluation, robustness tests are carried out under external disturbances, including the addition of a 30 g mass to the helicopter and the application of a wind disturbance generated by a fan.

The following subsections detail these simulation and experimental investigations, aiming to assess the stability, robustness, and tracking performance of the proposed control scheme under both nominal and perturbed operating conditions.

Remark 3. The fractional-order differential operators were approximated using the Oustaloup recursive method and the CRONE toolbox, which replace $s^{α}$ with a rational transfer function over a specified frequency band. This allows the system to be simulated using standard numerical solvers without requiring the direct integration of fractional-order derivatives in the time domain.

Simulation results

To track the desired output vector $(θ_{ref}, ξ_{ref})$ with respect to the actual system outputs $(θ, ξ)$ , two PD + $I^{α}$ fractional-order controllers integrated with feedback linearization are investigated. The first controller corresponds to the proposed optimal design obtained using an analytical approach combined with the PSO algorithm, while the second represents a non-optimal PD + $I^{α}$ controller derived from the analytical formulation given in (32).

For a fair comparison, both controllers are designed to satisfy a desired phase margin of $7 5^{\circ}$ . In the non-optimal case, the controller performance is evaluated for different values of the design constant $ξ_{c}$ , namely, 0.09, 0.18, and 0.25, in order to analyze its influence on the closed-loop response. For the proposed optimal controller, the parameters are initialized as $μ_{1} = 0.5$ , $μ_{2} = 1$ , $N = 18$ , and $k = 18$ prior to the optimization process. The resulting controller parameters for both the optimal and non-optimal tuning approaches are summarized in Tables 1 and 2, respectively.

Table 1.

Controller parameters for the first subsystem pitch angle.

Controller parameter	$k_{f}^{1}$	$α^{1}$	$k_{p_{0}}^{1}$	$k^{1}$	$k_{p_{1}}^{1}$
OPD_FIC ( ${ξ_{c}}_{best} = 0.2308$ )	1	0.1667	0.0533	1.3383	5.0398
PD_FIC( $ξ_{c} = 0.09$ )	1	0.1667	0.0081	2.2244	15.0894
PD_FIC( $ξ_{c} = 0.18$ )	1	0.1667	0.0324	1.5722	6.7262
PD_FIC( $ξ_{c} = 0.25$ )	1	0.1667	0.0625	1.2631	4.5952

Table 2.

Controller parameters for the second subsystem yaw angle.

Controller parameter	$k_{f}^{2}$	$α^{2}$	$k_{p_{0}}^{2}$	$k^{2}$	$k_{p_{1}}^{2}$
OPD_FIC ( ${ξ_{c}}_{best} = 0.2308$ )	1	0.1667	0.0533	1.3383	5.0398
PD_FIC( $ξ_{c} = 0.09$ )	1	0.1667	0.0081	2.2244	15.0894
PD_FIC( $ξ_{c} = 0.18$ )	1	0.1667	0.0324	1.5722	6.7262
PD_FIC( $ξ_{c} = 0.25$ )	1	0.1667	0.0625	1.2631	4.5952

The system responses, $θ, ξ$ , to the previously defined square reference inputs are presented in Figures 8 and 9.

Figure 8.

Evolution of the output θ for square reference for optimal and non optimal fractional order PD + $I^{α}$ controllers combined with feedback linearization.

Figure 9.

Evolution of the output ξ for square reference for optimal and non optimal fractional order PD + $I^{α}$ controllers combined with feedback linearization.

It can be observed from Figures 8 and 9 that, for both controllers, the first output θ accurately tracks the setpoint $θ_ref$ while the second output ξ follows the setpoint $ξ_ref$ very quickly with zero steady-state error. The system achieves excellent tracking of pitch and yaw angles.

All responses exhibit nearly constant overshoot due to the fixed desired phase margin, which results from the robustness property of the ideal Bode function. For the fractional PD + $I^{α}$ controller, the system dynamics vary depending on the choice of $ξ_{c}$ . The optimal controllers provide superior performance, whereas the non-optimal controllers also perform well, provided that an appropriate $ξ_{c}$ is selected. A key advantage of PSO is its ability to automate parameter tuning, eliminating the need for manual adjustments and saving time in selecting the optimal $Ψ_{c}$ .

To provide a clearer comparison of control performance, Table 3 presents the main indicators—namely, tracking error (TE), overshoot (OS), rise time (RT), and settling time (ST) for the pitch subsystem, while Table 4 reports the same indicators for the yaw subsystem. These results highlight the performance improvements achieved by the proposed optimal fractional-order PD + $I^{α}$ controller compared to non-optimal control structures in both loops.

Table 3.

Comparison of control performance indicators for the pitch subsystem using the proposed optimal fractional-order PD + $I^{α}$ controller versus non-optimal control structures.

Controller	RT (s)	ST (s)	OS (%)	TE
OPD_FIC ( ${ξ_{c}}_{best} = 0.2308$ )	5.70	118.987	8.48	0.3473
PD_FIC ( $ξ_{c} = 0.09$ )	19.775	126.913	1.68	0.3756
PD_FIC ( $ξ_{c} = 0.18$ )	8.27	126.573	8.15	0.3589
PD_FIC ( $ξ_{c} = 0.25$ )	6.243	120.80	8.5	0.3508

Table 4.

Comparison of control performance indicators for the yaw subsystem using the proposed optimal fractional-order PD + $I^{α}$ controller versus non-optimal control structures.

Controller	RT (s)	ST (s)	OS (%)	TE
OPD_FIC ( ${ξ_{c}}_{best} = 0.2308$ )	5.7	118.987	8.48	0.347
PD_FIC ( $ξ_{c} = 0.09$ )	19.775	126.913	1.68	0.3756
PD_FIC ( $ξ_{c} = 0.18$ )	8.27	126.573	8.15	0.3589
PD_FIC ( $ξ_{c} = 0.25$ )	6.243	120.8	8.5	0.3508

As shown in Tables 3 and 4, the proposed optimal fractional-order PD+ $I^{α}$ controller clearly outperforms the non-optimal control structures in both the pitch and yaw subsystems. In the pitch loop, it achieves the best overall trade-off, combining the shortest rise time (5.70 s), a moderate overshoot (8.48%), and the lowest tracking error (0.3473). Similarly, in the yaw loop, it delivers a fast response (rise time of 5.70 s), moderate overshoot (8.48%), and superior tracking accuracy with the lowest tracking error (0.3473).

The PSO algorithm automatically determines the optimal gain crossover frequency $Ψ_{c}^{i}$ , from which all remaining controller parameters are computed using (21), (23), (29), and (32). The optimization objective is to minimize the total mean square error between the desired and actual pitch and yaw responses, while simultaneously enforcing robustness constraints.

In this study, a desired phase margin of $7 5^{\circ}$ is imposed for both the pitch and yaw subsystems. The PSO algorithm yields an optimal crossover frequency of $Ψ_{c}^{i} = 0.2308$ rad/s. The corresponding open-loop Bode diagram of the double-integrator plant controlled by the proposed optimal fractional-order PD+ $I^{α}$ controller is depicted in Figure 10, confirming that the specified phase margin is accurately achieved at the gain crossover frequency. In addition, the obtained gain margin is sufficiently large ( $GM = 49.7$ ), indicating strong robustness and further confirming the stability of the closed-loop system.

Figure 10.

Open-loop bode plots of the double-integrator pitch and yaw subsystems with the proposed controllers.

Simulation-based robustness analysis

To assess the robustness of the proposed controller with respect to internal model uncertainties, a parametric variation was introduced in simulation by increasing the system mass by $50 %$ . Figures 11 and 12, which correspond to the pitch and yaw responses, respectively, illustrate that the closed-loop system remained stable and maintained accurate setpoint tracking. These results confirm that the controller ensures robust performance even under significant parameter variations.

Figure 11.

Simulated pitch tracking response under a $50 %$ mass variation using feedback linearization and optimal fractional-order PD + $I^{α}$ control.

Figure 12.

Simulated yaw tracking response under a $50 %$ mass variation using feedback linearization and optimal fractional-order PD + $I^{α}$ control.

Comparison of the proposed controller with other optimal fractional-order controllers

To provide a comprehensive and fair evaluation, the proposed optimal fractional-order PD+ $I^{α}$ controller is compared with three optimal fractional-order controllers, namely, the FPI, FOPID, and FTID controllers. All controllers were optimized using the same PSO framework, identical performance indices, and identical operating conditions for both pitch (θ) and yaw (ξ) subsystems.

The mathematical form of the FOPID controller for each subsystem is given by Tejado et al. (2019)

\begin{matrix} C_{FOPID, i} (s) = K_{p, i} + K_{i, i} s^{- λ_{i}} + \frac{K_{d, i} s^{ν_{i}}}{1 + K_{d, i} s^{ν_{i}}}, i \in {θ, ξ} \end{matrix}

(50)

where $K_{p}$ , $K_{i}$ , and $K_{d}$ denote the proportional, integral, and derivative gains, respectively, and $λ \in (0, 1]$ and $ν \in (0, 1]$ correspond to the fractional orders of the integral and derivative actions. The fractional derivative term is realized using a filtered implementation of the form $\frac{1}{1 + K_{d} s^{ν}}$ , which ensures physical realizability and effectively attenuates high-frequency measurement noise.

The mathematical representation of the FPI controller for each subsystem is expressed as Terfia et al. (2023)

\begin{matrix} C_{FPI, i} (s) = K_{p, i} + K_{i, i} s^{- λ_{i}}, i \in {θ, ξ} \end{matrix}

(51)

where $K_{p}$ and $K_{i}$ are the proportional and integral gains, respectively, while λ is the fractional order of the integral action.

The mathematical form of the FTID controller for each subsystem is given by Daraz et al. (2022)

\begin{matrix} C_{FTID, i} (s) = K_{t, i} s^{- 1 ∕ N_{t_{i}}} + K_{i, i} s^{- λ_{i}} + \frac{N_{f_{i}} K_{d, i} s^{ν_{i}}}{N_{f_{i}} + s^{ν_{i}}}, i \in {θ, ξ} \end{matrix}

(52)

where $K_{t, i}$ is the tilt gain, with fractional-order $- 1 ∕ N_{t_{i}}$ , $K_{i, i}$ is the integral gain with fractional-order $λ_{i}$ , $K_{d, i}$ is the derivative gain with fractional-order $ν_{i}$ , $N_{f_{i}} ∕ (N_{f_{i}} + s^{ν_{i}})$ represents the filtered derivative term to limit high-frequency noise. $N_{f_{i}}$ is the coefficient of the proposed controller’s derivative filter, which can be set in the range [1, 500].

The PSO search space for the FOPID controller was defined as $K_{p, θ} \in [1, 4]$ , $K_{i, θ} \in [0.1, 5]$ , $K_{d, θ} \in [0.5, 5]$ , $λ_{θ} \in [0.6, 1.7]$ , $ν_{θ} \in [0.1, 1.4], K_{p, ξ} \in [1, 4]$ , $K_{i, ξ} \in [0.1, 5]$ , $K_{d, ξ} \in [0.5, 5]$ , $λ_{ξ} \in [0.6, 1.7]$ , $ν_{ξ} \in [0.1, 1.4]$ . The optimal FOPID parameters obtained are $[K_{p, θ}, K_{i, θ}, K_{d, θ}, λ_{θ}, ν_{θ}, K_{p, ξ}, K_{i, ξ}, K_{d, ξ}, λ_{ξ}, ν_{ξ}] = [1.4161, 0.1, 0.5574, 0.6, 1.4, 1, 0.1, 0.6911, 0.6, 1.4] .$

For the FPI controller, the PSO search space was selected as $K_{p, θ} \in [0.1, 10]$ , $K_{i, θ} \in [0.1, 5]$ , $λ_{θ} \in [1.2, 1.8]$ , $K_{p, ξ} \in [0.1, 10]$ , $K_{i, ξ} \in [0.1, 5]$ , $λ_{ξ} \in [1.2, 1.8]$ . The optimal FPI parameters are $[K_{p, θ}, K_{i, θ}, λ_{θ}, K_{p, ξ}, K_{i, ξ}, λ_{ξ}]$ = $[6.2309, 0.1027, 1.2040, 10, 0.1127, 1.6866]$ .

The PSO search space for the FTID controller was defined as $K_{t, θ} \in [0.01, 5]$ , $N_{t, θ} \in [2, 3]$ , $K_{i, θ} \in [0, 5]$ , $K_{d, θ} \in [0, 5]$ , $λ_{θ} \in [0.01, 0.99]$ , $ν_{θ} \in [0.01, 0.99]$ , $K_{t, ξ} \in [0.01, 5]$ , $N_{t, ξ} \in [2, 3]$ , $K_{i, ξ} \in [0, 5]$ , $K_{d, ξ} \in [0, 5]$ , $λ_{ξ} \in [0.01, 0.99]$ , and $ν_{ξ} \in [0.01, 0.99]$ . For each subsystem, the filter coefficient is fixed to $N_{f_{θ}} = N_{f_{ξ}} = 2$ . The optimal FTID parameters are $[K_{t, θ}, N_{t_{θ}}, K_{i, θ}, K_{d, θ}, λ_{θ}, ν_{θ}, K_{t, ξ}, N_{t_{ξ}}, K_{i, ξ}, K_{d, ξ}, λ_{ξ}, ν_{ξ}]$ = $[1.37, 2.992, 1.33, 4.9697, 0.0179, 0.99, 0.592, 2, 1.1572, 5, 0.01, 0.92]$ .

The pitch and yaw responses, along with the corresponding control signals, obtained with the proposed optimal fractional-order PD+ $I^{α}$ controller and the three other optimal fractional controllers are illustrated in Figures 13 and 14, respectively.

Figure 13.

Pitch and control responses under OPD_FIC, FPI, FOPID, and FTID controllers with feedback linearization.

Figure 14.

Yaw and control responses under OPD_FIC, FPI, FOPID, and FTID controllers with feedback linearization.

Tables 5 and 6 show that the proposed OPD+ $I^{α}$ controller significantly outperforms the conventional optimal fractional-order controllers (FPI, FOPID, FTID) for both pitch and yaw subsystems. While the other controllers exhibit very fast rise times, this comes at the cost of high overshoot, elevated tracking error, and limited stability margins. In contrast, the OPD+ $I^{α}$ controller achieves a balanced trade-off: drastically reduced overshoot (below 9%), the lowest tracking error, and substantially higher phase and gain margins, indicating superior robustness and steady-state accuracy. Although its settling time is longer, this reflects a deliberate prioritization of robustness and smooth transient response over aggressive dynamics—a highly desirable trade-off for safety-critical systems such as the 2-DOF helicopter.

Table 5.

Control performance comparison of the pitch subsystem with the proposed OPD+ $I^{α}$ and other fractional-order controllers.

Controller	RT (s)	OS (%)	ST (s)	TE	PM (^∘)	GM
FPI (optimal)	0.43	90.01	99.87	0.8045	-0.3	0
FOPID (optimal)	0.72	67.97	57.12	1.0471	24.91	0.1
FTID (optimal)	0.62	12.53	56.04	1.04	22.95	0.015
OPD+ $I^{α}$ (proposed)	5.70	8.48	118.99	0.3473	72.8	49.7

Bold values indicate the best performance achieved by the proposed controller among the compared controllers.

Table 6.

Control performance comparison of the yaw subsystem with the proposed OPD+ $I^{α}$ and other fractional-order controllers.

Controller	RT (s)	OS (%)	ST (s)	TE	PM (^∘)	GM
FPI (optimal)	0.43	90.01	99.4	0.8045	-0.11	0
FOPID (optimal)	0.72	67.97	60.62	1.0471	26.7	0.17
FTID (optimal)	0.62	12.53	56.04	1.04	26.81	0.01
OPD+ $I^{α}$ (proposed)	5.7	8.48	118.99	0.3473	72.8	49.7

Bold values indicate the best performance achieved by the proposed controller among the compared controllers.

Experimental results

For both subsystems, the experimental results were obtained using the same square reference inputs and initial conditions as in the simulations. The optimal values $Ψ_{c}^{1} = Ψ_{c}^{2} = 0.2308$ determined using PSO were applied to each subsystem. The controller parameters used in the experimental tests were scaled from their simulation values by a factor of 2.12 for the pitch subsystem and 3.78 for the yaw subsystem. To mitigate noise amplification in the derivative action, a second-order low-pass filter with a cutoff frequency of $ω_{0} = 17$ rad/s, and a damping ratio of $ε = 0.8$ was applied to both pitch and yaw output derivatives.

The effectiveness of the proposed controllers in achieving accurate reference tracking is demonstrated in Figures 15 and 16. These results confirm the controllers’ ability to precisely track the desired references.

Figure 15.

Experimental results for square reference using feedback linearization combined with an optimal fractional PD + $I^{α}$ controller, applied to the first subsystem (pitch).

Figure 16.

Experimental results for square reference, using FL combined with an optimal fractional PD + $I^{α}$ controller, applied to the second subsystem (yaw).

Discussion on control signal chattering

The experimental results shown in Figures 15 and 16 exhibit noticeable chattering in the control signals. This phenomenon is primarily due to the sensitivity of the feedback linearization approach, which, while effective in decoupling and linearizing the nonlinear MIMO system, can amplify modeling errors and measurement noise. Consequently, the control signals may exhibit high-frequency oscillations, particularly during rapid reference tracking.

Additional practical factors contribute to this effect, including actuator limitations, sensor quantization, and unmodeled nonlinear dynamics such as friction and valve behavior. These imperfections, not accounted for in the nominal model, collectively intensify the chattering observed in practice.

Although the proposed control scheme performs well in terms of tracking accuracy and stability, the chattering remains a relevant issue that could be addressed in future work. Potential improvements include the use of anti-windup mechanisms and robust or adaptive extensions of the feedback linearization law.

Experimental robustness tests

To further validate the robustness of the proposed control strategy under real-world disturbances, two experimental tests were performed.

The first test involved adding a 30g mass to the helicopter arm to simulate a structural perturbation. As shown in Figures 17 and 18, the system maintained closed-loop stability and exhibited consistent tracking performance in both pitch and yaw axes, confirming robustness against external mechanical disturbances.

Figure 17.

Experimental pitch response under 30 g mass disturbance using feedback linearization and optimal fractional-order PD + $I^{α}$ control.

Figure 18.

Experimental yaw response under 30 g mass disturbance using feedback linearization and optimal fractional-order PD + $I^{α}$ control.

The second test subjected the helicopter to an airflow disturbance generated by a fan operating at approximately 2.7 m/s. As shown in Figures 19 and 20, the system remained stable and accurately followed the reference trajectory in both pitch and yaw directions, even under this aerodynamic disturbance.

Figure 19.

Experimental pitch response under 2.7 m/s airflow disturbance using feedback linearization and optimal fractional-order PD + $I^{α}$ control.

Figure 20.

Experimental yaw response under 2.7 m/s airflow disturbance using feedback linearization and optimal fractional-order PD + $I^{α}$ control.

These experimental results indicate that the proposed fractional-order control strategy is capable of rejecting moderate external disturbances and maintaining stable operation under realistic conditions. These robustness tests confirm that the proposed controller exhibits adequate disturbance rejection capabilities while maintaining high tracking precision. The use of exact feedback linearization effectively decouples and linearizes the system dynamics under nominal conditions. However, future work may explore the integration of robust feedback linearization to improve disturbance rejection and ensure accurate linearization even in the presence of significant external perturbations, while also addressing the residual chattering effects observed in the control signals.

Remark 4. From the experimental tests, we observed that the system begins to lose performance beyond specific disturbance thresholds. In the case of added mass, the system remains stable and maintains acceptable tracking up to 30 g; however, when the added mass exceeds 40 g, the tracking accuracy significantly degrades. Similarly, under wind disturbances, the system performs well up to a speed of 2.7 m/s, but for speeds above 3 m/s, the system remains stable yet fails to follow the reference properly. These observations indicate practical robustness limits of the controller under external disturbances.

Limitations and future work

A comparative analysis with other fractional-order heuristic optimization algorithms, such as the Fractional calculus-based firefly algorithm applied to parameter estimation of chaotic systems presented in Mousavi and Alfi (2018), is planned as part of future work to further evaluate and enhance the performance of the proposed control strategy.

Conclusion

This paper presents two effective approaches for reference tracking in TITO nonlinear systems through the design of both optimal and non-optimal PD plus fractional-order integral (PD + $I^{α}$ ) controllers. By integrating feedback linearization, PSO, and BITF as an open-loop reference model, the proposed methods enhance tracking accuracy and system stability.

The application to a 2-DOF helicopter process demonstrates the practical viability of these controllers, showcasing their ability to achieve precise reference tracking. Compared to conventional optimal fractional-order controllers (FPI, FOPID, FTID), the proposed OPD+ $I^{α}$ controller provides substantially lower overshoot and tracking error, while offering markedly higher phase and gain margins. Although its settling time is longer, this reflects a deliberate trade-off favoring robustness and smooth transient response over aggressive dynamics, which is particularly desirable in safety-critical applications.

Furthermore, extensive robustness tests—carried out under both external disturbances (such as added mass and wind effects) and internal model uncertainties (parametric variations) confirm the reliability and resilience of the proposed control strategy. These results highlight the potential for broader implementation in complex control systems, making the proposed approach a valuable solution for nonlinear MIMO systems.

Footnotes

ORCID iDs

Chahira Boussalem

Fouad Yacef

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

Data availability statement

All data generated or analyzed during this study are included in this published article.

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