Robust Control of Twin-Rotor MIMO Systems Under Unmodeled Dynamics: Comparative Experimental Validation of Hybrid BSMC and Online QBHO Strategies

Abstract

The control of Twin-Rotor Multi-Input Multi-Output (TRMS) systems presents a significant challenge due to high nonlinearity, strong aerodynamic cross-coupling, and the inevitable discrepancies between theoretical models and physical plants. This paper first exposes the instability of conventional Backstepping control under real hardware conditions, where unmodeled dynamics and parametric uncertainties drive the yaw subsystem into divergent oscillation, then proposes and experimentally validates two advanced architectures to overcome this limitation. The first is an online adaptive Backstepping gain-tuning scheme based on a novel Rate-Constrained Sequential Quantum Black Hole Optimization (RS-QBHO) algorithm. The second is a Hybrid Backstepping–Sliding Mode Control (BSMC) architecture that integrates structural disturbance rejection directly into the recursive design. Both schemes are formally verified via Lyapunov stability analysis and validated on a physical TRMS rig under identical hardware-in-the-loop conditions. Experimental results confirm that while the standard Backstepping controller failed in the yaw axis with an RMSE of 2.5624 rad, both proposed methods achieved stabilization. The QBHO-tuned controller yielded RMSE values of 0.0799 rad for pitch and 0.2305 rad for yaw, while the BSMC strategy proved superior, achieving 0.0682 rad and 0.1858 rad, respectively. These findings demonstrate that while meta-heuristic optimization effectively compensates for parametric mismatches, the passive disturbance rejection of the sliding mode term offers a more effective solution for mitigating unmodeled aerodynamic dynamics in MIMO flight platforms.

1. Introduction

The control of Twin-Rotor Multi-Input Multi-Output (TRMS) systems represents a classic benchmark in aerospace engineering, serving as a high-fidelity laboratory model for validating flight control algorithms intended for helicopters and Vertical Take-Off and Landing (VTOL) vehicles. Characterized by high nonlinearity, significant aerodynamic cross-coupling between the main and tail rotors, and intricate gyroscopic effects [1], TRMS remains a formidable challenge for traditional control strategies. Recent advancements in nonlinear control, such as observer-enhanced Backstepping schemes [2], have further highlighted the complexities involved in achieving robust performance for this platform. Consequently, linear methodologies, such as the Linear Quadratic Regulator (LQR) [3] and Model Predictive Control (MPC) [4], often struggle to maintain stability when the system operates beyond the localized linearized region or is subjected to significant model mismatch.

To address these nonlinearities, recursive control strategies such as Backstepping and Sliding Mode Control (SMC) have been widely adopted [5]. Nevertheless, these conventional implementations exhibit inherent vulnerabilities; specifically, standard Backstepping relies heavily on the precision of model-based cancellation, rendering it sensitive to unmodeled dynamics [6,7], while SMC is prone to the “chattering” phenomenon, which can induce severe mechanical stress in the TRMS rotors.

Recent literature has addressed these limitations via two distinct paradigms: hybridization and intelligent optimization. The hybridization paradigm integrates multiple control structures to exploit their complementary strengths. The fusion of Backstepping and Sliding Mode Control (BSMC) is particularly compelling for TRMS-class systems: whereas pure Backstepping achieves model-based cancellation but fails under parametric mismatch [7], the addition of a discontinuous sliding term passively rejects the residual unmodeled dynamics without requiring an explicit disturbance model. This structural advantage has been validated on TRMS platforms directly, including adaptive SMC designs for TRMS under rotor mass uncertainty [8], and has been extended to lateral aircraft control [9]. Beyond TRMS, BSMC architectures have demonstrated versatility across high-dynamic applications: deep-sea hydraulic manipulators [10], series-connected PMSMs [11], highly coupled UAVs [12], wind energy conversion systems [13], microgrids [14], electric vehicle regulation [15], and linear synchronous motors [16]. The consistent conclusion across this body of literature is that the sliding mode term provides a structural safety net that compensates for the fundamental limitation of model-based cancellation—a conclusion that directly motivates the Hybrid BSMC proposed in this work.

The alternative paradigm focuses on intelligent optimization, utilizing metaheuristic algorithms to tune control parameters without altering the underlying controller structure. This trend is evident in recent TRMS studies utilizing metaheuristic PID tuning [17], multi-objective genetic algorithms for stability [18], and fractional-order PID schemes [19,20], all of which improve nominal performance but remain sensitive to the unmodeled dynamics revealed under physical deployment. Active disturbance rejection via Particle Swarm Optimization (PSO) [21] and optimized Backstepping for valve-controlled motors [22] further highlight the potential of intelligent tuning. Emerging quantum-inspired methods, such as Quantum Black Hole Optimization (QBHO), offer superior convergence properties for high-dimensional tuning problems [23], as evidenced by recent comparative studies against conventional training strategies [24]. In the broader context of adaptive control for nonlinear multi-subsystem architectures, Zhao et al. [25] recently proposed an adaptive fuzzy decentralized controller for large-scale cyber–physical systems under denial-of-service attacks, employing adaptive Backstepping with Lyapunov stability guarantees that share structural foundations with the decentralized topology adopted in the present work further underscoring the versatility of recursive adaptive control frameworks in complex MIMO settings.

Despite these advancements, there remains a critical need for comparative experimental validation of these two philosophies, Robustness vs. Intelligence, under strict real-time constraints on a physical TRMS rig. The prior hybridization studies cited above either rely on simulation-only validation or target different platforms, leaving a gap in hardware-validated comparisons for the TRMS benchmark. This paper addresses this gap by implementing and comparing two advanced strategies on a physical TRMS rig: (1) an Online Adaptive Backstepping controller tuned by the RS-QBHO algorithm and (2) a Robust Hybrid Backstepping-SMC (BSMC).

The remainder of this paper is organized as follows: Section 2 details the TRMS mathematical model and the standard Backstepping baseline. Section 3 presents the robust Hybrid BSMC architecture. Section 4 introduces the Online RS-QBHO algorithm. Section 5 provides the comparative evaluation through numerical simulation and real-time HIL implementation, followed by concluding remarks in Section 6.

2. Materials and Methods

2.1. Plant Model

The experimental platform utilized in this study is the Twin Rotor Multi-Input Multi-Output System (TRMS), shown in Figure 1, a high-fidelity laboratory benchmark designed to emulate the flight dynamics of a helicopter. Structurally, the apparatus consists of a beam mounted on a pivotal tower, allowing for rotation in both the horizontal and vertical planes. The beam extremities are fitted with two propellers: the main rotor, which drives the pitch motion, and the tail rotor, which controls the yaw motion. A counterweight is attached to the beam to ensure static equilibrium.

The nonlinear dynamics of the system can be represented by the following state-space formulation:

$$\left\{\begin{array}{c}\dot{x}=f(x,u,t)\hfill \\ y=g(x,u,t)\hfill \end{array}\right.$$

(1)

The TRMS is described by a set of six first-order differential equations. The state vector comprises the angular positions and velocities, as well as the rotor torques: $x={[x_1,x_2,x_3,x_4,x_5,x_6]}^T={[\psi ,\dot{\psi },{\tau }_1,\varphi ,\dot{\varphi },{\tau }_2]}^T$. Note that for controller derivation, we organize the states such that $x_3$ is the main rotor torque and $x_6$ is the tail rotor torque. The control inputs $u={[U_1,U_2]}^T$ correspond to the DC voltages applied to the respective motors. The comprehensive nonlinear model is expressed as:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2={\displaystyle \frac{1}{I_1}}\left(a_1{x}_3^2+b_1{x}_3-M_{g}sin{x}_1-B_{1\psi }{x}_2-B_{2\psi }sgn\left(x_2\right)-K_{gy}{x}_{5}cos\left(x_1\right)(a_1{x}_3^2+b_1{x}_3)\right)\hfill \\ {\dot{x}}_3=-{\displaystyle \frac{T_{10}}{T_{11}}}{x}_3+{\displaystyle \frac{K_1}{T_{11}}}{U}_1\hfill \\ {\dot{x}}_4=x_5\hfill \\ {\dot{x}}_5={\displaystyle \frac{1}{I_2}}\left(a_2{x}_6^2+b_2{x}_6-B_{1\varphi }{x}_5-B_{2\varphi }sgn\left(x_5\right)-1.75K_c(a_1{x}_3^2+b_1{x}_3)\right)\hfill \\ {\dot{x}}_6=-{\displaystyle \frac{T_{20}}{T_{21}}}{x}_6+{\displaystyle \frac{K_2}{T_{21}}}{U}_2\hfill \end{array}\right.$$

(2)

To provide physical clarity, the state variables are interpreted as follows: $x_1=\psi$ is the pitch angle (rotation in the vertical plane); $x_2=\dot{\psi }$ is the pitch angular velocity; $x_3={\tau }_1$ is the main rotor aerodynamic torque; $x_4=\varphi$ is the yaw angle (rotation in the horizontal plane); $x_5=\dot{\varphi }$ is the yaw angular velocity; and $x_6={\tau }_2$ is the tail rotor aerodynamic torque. The control inputs $U_1$ and $U_2$ represent the DC motor voltages applied to the main and tail rotors, respectively.

The parameters in Equation (2) represent the physical properties of the system, including moments of inertia ($I_1,I_2$), aerodynamic force and torque constants ($a_1,b_1,a_2,b_2$), gravitational moments ($M_g$), friction coefficients ($B_{1\psi },B_{2\psi },B_{1\varphi },B_{2\varphi }$), gyroscopic cross-coupling gains ($K_{gy},K_c$), and motor dynamics parameters. The specific numerical values utilized in this study are detailed in

Table 1. Physical parameters of the TRMS platform.

Param.	Value	Param.	Value
$I_1$	$6.8\times {10}^{-2}$ kg·m2	$I_2$	$2.0\times {10}^{-2}$ kg·m2
$a_1$	$0.0135$	$a_2$	$0.02$
$b_1$	$0.0924$	$b_2$	$0.09$
$M_g$	$0.32$ N·m	$B_{1\psi }$	$6.0\times {10}^{-3}$
$B_{2\psi }$	$1.0\times {10}^{-3}$	$B_{1\varphi }$	$1.0\times {10}^{-1}$
$B_{2\varphi }$	$1.0\times {10}^{-2}$	$K_{gy}$	$0.05$ s/rad
$K_1$	$1.1$	$K_2$	$0.8$
$T_{11}$	$1.1$	$T_{21}$	$1.0$
$T_{10}$	$1.0$	$T_{20}$	$1.0$

, corresponding to the specifications provided in the manufacturer’s manual [26].

2.2. Nominal Backstepping Control (Baseline)

To establish a rigorous performance baseline for comparative analysis, a standard decentralized Backstepping controller is utilized. The synthesis follows a systematic Lyapunov-based recursive procedure, wherein the TRMS is decomposed into two lower-dimensional subsystems, the vertical (pitch) and horizontal (yaw) planes, with aerodynamic cross-coupling treated as exogenous bounded disturbances. The complete step-by-step derivation and stability proofs have been detailed and validated in our recent work [27], and are omitted here to maintain brevity and focus on the novel architectures proposed in this study. While this controller demonstrates mathematically guaranteed stability under nominal conditions, its real-time performance under physical model mismatch is critically examined in Section 5.

3. Robust Hybrid Control Law Implementation

This section details the synthesis of the Hybrid Backstepping–Sliding Mode Control (HBSMC). To guarantee asymptotic stability while counteracting unmodeled dynamics, the control law is derived using a step-by-step Lyapunov approach.

3.1. Pitch Axis Control Algorithm

The decentralized design model for the pitch subsystem, treating cross-coupling as external disturbances, is expressed as:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2={\displaystyle \frac{1}{I_1}}(a_1{x}_3^2+b_1{x}_3-M_{g}sin{x}_1-B_{1\psi }{x}_2-B_{2\psi }{x}_1)\hfill \\ {\dot{x}}_3=-{\displaystyle \frac{T_{10}}{T_{11}}}{x}_3+{\displaystyle \frac{K_1}{T_{11}}}{U}_1\hfill \end{array}\right.$$

(3)

Step 1: Tracking Error Stabilization

We define the primary tracking error $z_1$ for the pitch angle:

$$z_1=x_1-x_{1d}$$

(4)

Its time derivative is given by:

$${\dot{z}}_1={\dot{x}}_1-{\dot{x}}_{1d}=x_2-{\dot{x}}_{1d}$$

(5)

Treating $x_2$ as a virtual control input, we define a stabilizing function ${\alpha }_1$ and a second error variable $z_2$ representing the discrepancy between the actual state and the virtual command:

$$z_2=x_2-{\alpha }_1\Rightarrow x_2=z_2+{\alpha }_1$$

(6)

To ensure $z_1\to 0$, we select the first Lyapunov candidate function:

$$V_1=\frac{1}{2}{z}_1^2$$

(7)

The derivative with respect to time is:

$${\dot{V}}_1=z_1{\dot{z}}_1=z_1(x_2-{\dot{x}}_{1d})=z_1(z_2+{\alpha }_1-{\dot{x}}_{1d})$$

(8)

To render this subsystem asymptotically stable, the virtual control ${\alpha }_1$ is chosen as:

$${\alpha }_1=-c_1{z}_1+{\dot{x}}_{1d}$$

(9)

where $c_1>0$. Consequently, the derivative becomes:

$${\dot{V}}_1=-c_1{z}_1^2+z_1{z}_2$$

(10)

Clearly, if $z_2=0$, then ${\dot{V}}_1=-c_1{z}_1^2\le 0$, ensuring asymptotic convergence.

Step 2: Virtual Control for Velocity

We compute the derivative of the second error $z_2=x_2-{\alpha }_1$:

$${\dot{z}}_2={\displaystyle \frac{1}{I_1}}(a_1{x}_3^2+b_1{x}_3-M_{g}sin{x}_1-B_{1\psi }{x}_2-B_{2\psi }{x}_1)+c_1{x}_2-c_1{\dot{x}}_{1d}-{\ddot{x}}_{1d}$$

(11)

To isolate the thrust-related terms, we define an auxiliary variable $E={\displaystyle \frac{1}{I_1}}(a_1{x}_3^2+b_1{x}_3-B_{1\psi }{x}_2)$, and set a third error state $E=z_3+{\alpha }_2$, where the second virtual control ${\alpha }_2$ is designed as:

$${\alpha }_2=-c_2{z}_2-z_1+{\displaystyle \frac{1}{I_1}}(M_{g}sin{x}_1+B_{2\psi }{x}_1)-c_1{x}_2+c_1{\dot{x}}_{1d}+{\ddot{x}}_{1d}$$

(12)

with $c_2>0$. Augmenting the Lyapunov function to $V_2=V_1+{\displaystyle \frac{1}{2}}{z}_2^2$, we obtain:

$${\dot{V}}_2=-c_1{z}_1^2-c_2{z}_2^2+z_2{z}_3$$

(13)

Step 3: Sliding Surface and Final Control Law

To introduce structural robustness, we define the sliding surface $S_1=z_3=E-{\alpha }_2$. Its derivative is:

$${\dot{S}}_1={\dot{z}}_3=\dot{E}-{\dot{\alpha }}_2$$

(14)

Using the complete Lyapunov function $V_3=V_2+{\displaystyle \frac{1}{2}}{S}_1^2$, the time derivative is evaluated as:

$${\dot{V}}_3=-c_1{z}_1^2-c_2{z}_2^2+S_1{\dot{S}}_1=-c_1{z}_1^2-c_2{z}_2^2+z_2{z}_3+S_1(\dot{E}-{\dot{\alpha }}_2)$$

(15)

To enforce negative definiteness, we select the reaching law:

$${\dot{S}}_1=-k_{s1}sign\left(\epsilon S_1\right)-c_3{S}_1=\dot{E}-{\dot{\alpha }}_2$$

(16)

The required derivatives are analytically expanded as:

$${\dot{\alpha }}_2=-(c_1+c_2){\dot{x}}_2+\left({\displaystyle \frac{M_{g}cos{x}_1+B_{2\psi }}{I_1}}-c_1{c}_2-1\right)x_2+(c_1{c}_2+1){\dot{x}}_{1d}+(c_1+c_2){\ddot{x}}_{1d}+x_{1d}^{\left(3\right)}$$

(17)

$$\dot{E}={\displaystyle \frac{1}{I_1}}\left[(2a_1{x}_3+b_1){\dot{x}}_3-B_{1\psi }{\dot{x}}_2\right]$$

(18)

Substituting the motor dynamics ${\dot{x}}_3=-{\displaystyle \frac{T_{10}}{T_{11}}}{x}_3+{\displaystyle \frac{K_1}{T_{11}}}{U}_1$, the equation becomes:

$$\dot{E}={\displaystyle \frac{1}{I_1}}\left[(2a_1{x}_3+b_1)\left(-{\displaystyle \frac{T_{10}}{T_{11}}}{x}_3+{\displaystyle \frac{K_1}{T_{11}}}{U}_1\right)-B_{1\psi }{\dot{x}}_2\right]$$

(19)

By isolating the control input $U_1$, the final robust hybrid control law is synthesized as:

$$U_1=\left({\displaystyle \frac{I_1{T}_{11}}{K_1(2a_1{x}_3+b_1)}}\right)\left(-z_2-c_3{z}_3+{\displaystyle \frac{T_{10}}{T_{11}}}{x}_3+B_{1\psi }{\dot{x}}_2-{\dot{\alpha }}_2-k_{s1}sign\left(\epsilon S_1\right)\right)$$

(20)

Applying this control law reduces the Lyapunov derivative to:

$${\dot{V}}_3=-c_1{z}_1^2-c_2{z}_2^2-c_3{S}_1^2-S_1{k}_{s1}sign\left(\epsilon S_1\right)$$

(21)

Since $S_{1}sign\left(\epsilon S_1\right)\approx \left|S_1\right|$, this guarantees that:

$${\dot{V}}_3\le -c_1{z}_1^2-c_2{z}_2^2-c_3{S}_1^2<0$$

(22)

Thus, global asymptotic stability of the origin is mathematically ensured, while the inclusion of the signum function actively mitigates bounded modeling uncertainties.

3.2. Yaw Axis Control Algorithm

Following a symmetrical design philosophy, the decentralized model for the yaw subsystem is extracted by treating the main rotor cross-coupling as a bounded disturbance. The decoupled dynamics are given by:

$$\left\{\begin{array}{c}{\dot{x}}_4=x_5\hfill \\ {\dot{x}}_5={\displaystyle \frac{1}{I_2}}(a_2{x}_6^2+b_2{x}_6-B_{1\varphi }{x}_5-B_{2\varphi }{x}_4)\hfill \\ {\dot{x}}_6=-{\displaystyle \frac{T_{20}}{T_{21}}}{x}_6+{\displaystyle \frac{K_2}{T_{21}}}{U}_2\hfill \end{array}\right.$$

(23)

Step 1: Position Error and Virtual Control

Let $z_4=x_4-x_{d4}$ represent the tracking error for the yaw angle. Its time derivative is:

$${\dot{z}}_4={\dot{x}}_4-{\dot{x}}_{d4}=x_5-{\dot{x}}_{d4}$$

(24)

We define $x_5$ as the virtual command input for this step, leading to the stabilizing control law ${\alpha }_4$ and the second error variable $z_5$:

$$z_5=x_5-{\alpha }_4\Rightarrow x_5=z_5+{\alpha }_4$$

(25)

Considering the Lyapunov function $V_4={\displaystyle \frac{1}{2}}{z}_4^2$, its time derivative is:

$${\dot{V}}_4=z_4{\dot{z}}_4=z_4(z_5+{\alpha }_4-{\dot{x}}_{d4})$$

(26)

To ensure the asymptotic stability of the position error, the first virtual control ${\alpha }_4$ is chosen as:

$${\alpha }_4=-c_4{z}_4+{\dot{x}}_{d4}$$

(27)

where $c_4>0$. This yields ${\dot{V}}_4=-c_4{z}_4^2+z_4{z}_5$.

Step 2: Velocity Error and Auxiliary Dynamics

The derivative of the velocity error $z_5=x_5-{\alpha }_4$ is computed as:

$${\dot{z}}_5={\displaystyle \frac{1}{I_2}}(a_2{x}_6^2+b_2{x}_6-B_{1\varphi }{x}_5-B_{2\varphi }{x}_4)+c_4{x}_5-c_4{\dot{x}}_{d4}-{\ddot{x}}_{d4}$$

(28)

To isolate the tail rotor thrust terms, we introduce an auxiliary variable $F={\displaystyle \frac{1}{I_2}}(a_2{x}_6^2+b_2{x}_6-B_{1\varphi }{x}_5)$. We then define the third error state $z_6$ representing the difference between F and the second virtual control ${\alpha }_5$:

$$z_6=F-{\alpha }_5\Rightarrow F=z_6+{\alpha }_5$$

(29)

The virtual control ${\alpha }_5$ is synthesized to stabilize $z_5$ and cancel the remaining dynamic terms:

$${\alpha }_5=-c_5{z}_5-z_4+{\displaystyle \frac{B_{2\varphi }}{I_2}}{x}_4-c_4{x}_5+c_4{\dot{x}}_{d4}+{\ddot{x}}_{d4}$$

(30)

where $c_5>0$. Substituting this into the augmented Lyapunov function $V_5=V_4+{\displaystyle \frac{1}{2}}{z}_5^2$ yields:

$${\dot{V}}_5=-c_4{z}_4^2-c_5{z}_5^2+z_5{z}_6$$

(31)

Step 3: Sliding Surface and Final Yaw Control Law

To maximize structural robustness against the aerodynamic disturbances inherent in the horizontal plane, we define the sliding surface $S_2=z_6=F-{\alpha }_5$. The time derivative is:

$${\dot{S}}_2={\dot{z}}_6=\dot{F}-{\dot{\alpha }}_5$$

(32)

The final Lyapunov function for the yaw subsystem is $V_6=V_5+{\displaystyle \frac{1}{2}}{S}_2^2$, with its derivative given by:

$${\dot{V}}_6=-c_4{z}_4^2-c_5{z}_5^2+S_2(z_5+\dot{F}-{\dot{\alpha }}_5)$$

(33)

To ensure the derivative is negative definite, we impose the following reaching law condition:

$${\dot{S}}_2=-k_{s2}sign\left(\epsilon S_2\right)-c_6{S}_2=\dot{F}-{\dot{\alpha }}_5$$

(34)

The required analytical derivatives are computed as:

$${\dot{\alpha }}_5=-(c_4+c_5){\dot{x}}_5+\left({\displaystyle \frac{B_{2\varphi }}{I_2}}-c_4{c}_5-1\right)x_5+(c_4{c}_5+1){\dot{x}}_{d4}+(c_4+c_5){\ddot{x}}_{d4}+x_{d4}^{\left(3\right)}$$

(35)

$$\dot{F}={\displaystyle \frac{1}{I_2}}\left[(2a_2{x}_6+b_2)\left(-{\displaystyle \frac{T_{20}}{T_{21}}}{x}_6+{\displaystyle \frac{K_2}{T_{21}}}{U}_2\right)-B_{1\varphi }{\dot{x}}_5\right]$$

(36)

By substituting these derivatives into the stability condition and isolating the control input $U_2$, the final robust hybrid control law for the yaw axis is obtained:

$$U_2=\left({\displaystyle \frac{I_2{T}_{21}}{K_2(2a_2{x}_6+b_2)}}\right)\left[-z_5-c_6{S}_2+{\dot{\alpha }}_5+{\displaystyle \frac{B_{1\varphi }}{I_2}}{\dot{x}}_5-k_{s2}sign\left(\epsilon S_2\right)\right]+{\displaystyle \frac{T_{20}}{K_2}}{x}_6$$

(37)

Applying this control input forces the Lyapunov derivative to satisfy:

$${\dot{V}}_6=-c_4{z}_4^2-c_5{z}_5^2-c_6{S}_2^2-S_2{k}_{s2}sign\left(\epsilon S_2\right)\le -c_4{z}_4^2-c_5{z}_5^2-c_6{S}_2^2<0$$

(38)

This mathematically guarantees global asymptotic stability for the yaw subsystem. The inclusion of the signum function ensures instantaneous, passive rejection of the reactive torques generated by the main rotor, providing the necessary robustness for real-time deployment.

4. Online Adaptive Tuning via Rate-Constrained Sequential QBHO

While the Hybrid BSMC architecture provides structural robustness against unmodeled dynamics, its tracking performance is highly sensitive to the selection of the control gains. To address this sensitivity, we propose the Rate-Constrained Sequential QBHO (RS-QBHO), which introduces three key innovations beyond the standard QBHO framework [28]: (1) Sequential single-candidate evaluation, which reformulates the standard batch population evaluation (all N candidates per iteration) into a time-distributed process evaluating only one candidate per sample period, thereby resolving the real-time computational bottleneck; (2) a rate-limiting transition filter that smooths inter-candidate gain transitions to prevent mechanical stress in the physical actuators; and (3) a stability-augmented fitness function that penalizes aggressive parameter deviations, preventing plant destabilization during the online exploration phase. These innovations make the algorithm deployable within the strict $T_s=0.01$ s real-time loop of the TRMS rig.

Consistent with the decentralized control topology established in Section 3, we implement an intelligent adaptation layer comprising two parallel instances of the RS-QBHO algorithm.

4.1. Decentralized Optimization Topology

Optimizing the full gain vector $\mathrm{\Theta }\in {\mathbb{R}}^6$ simultaneously in real-time can lead to slow convergence due to the “curse of dimensionality.” To ensure rapid adaptation, the tuning problem is decomposed into two lower-dimensional subspaces:

• Pitch Optimizer: Tunes ${\mathrm{\Theta }}_{\psi }={[k_1,k_2,k_3]}^T$ by minimizing the pitch tracking error $e_{\psi }$.
• Yaw Optimizer: Tunes ${\mathrm{\Theta }}_{\varphi }={[k_4,k_5,k_6]}^T$ by minimizing the yaw tracking error $e_{\varphi }$.

Both optimizers operate independently but simultaneously, treating the aerodynamic cross-coupling effects as exogenous disturbances to be rejected.

4.2. Sequential Evaluation Logic

Standard meta-heuristics require evaluating an entire population in a single iteration, which is computationally prohibitive for real-time control loops ($T_s=0.01$ s). To resolve this, the proposed RS-QBHO reformulates the optimization as a sequential process distributed over time. Each optimizer manages a swarm of $N=30$ particles. The algorithm evaluates only one candidate per subsystem at a time, alternating between two operational modes.

The swarm size $N=30$ represents a balance between exploration diversity and computational feasibility for real-time deployment: it is large enough to sample the gain space representatively across the sinusoidal reference cycle while ensuring that all candidates can be evaluated within a practically relevant time horizon. The evaluation window $T_w=50$ samples (0.5 s) was empirically calibrated such that sufficient ITAE accumulation occurs within the expected settling dynamics of each TRMS subsystem, while remaining short enough to allow multiple candidate evaluations before aerodynamic operating conditions change significantly. This sequential architecture reduces the per-sample computational burden to a single ITAE accumulation, one rate-limited gain update, and a quantum attraction step executed only at window boundaries—making the algorithm fully compatible with the strict $T_s=0.01$ s control loop.

4.2.1. Exploration Mode (Test Phase)

For a deterministic window of $T_w$ samples, the controller applies the gains from the i-th candidate particle ${\mathrm{\Theta }}_i$. The system’s tracking performance is monitored using the subsystem-specific Integral of Time-weighted Absolute Error (ITAE):

$$J_{\mathrm{perf}}={\int }_{t_0}^{t_0+T_w}\tau ·\left|e_{\mathrm{sub}}\left(\tau \right)\right|\mathrm{d}\tau$$

(39)

where $\mathrm{sub}\in \{\psi ,\varphi \}$.

4.2.2. Exploitation Mode (Settle Phase)

Immediately following the test phase, the algorithm reverts to the current “Black Hole” (global best) gains ${\mathrm{\Theta }}_{BH}$. This phase is critical for experimental safety, as it allows the TRMS to stabilize and eliminates residual transient effects before the next candidate is evaluated.

4.3. Rate Limiting and Stability Augmentation

To prevent “Gain Jitter” and mechanical stress on the TRMS propellers, two safety mechanisms are integrated into the adaptive law.

1. Rate-Limiting Transition: The transition between the current gains and the target gains (${\mathrm{\Theta }}_{\mathrm{target}}$) is smoothed using a first-order lag filter:

$$\mathrm{\Theta }\left(k\right)=\mathrm{\Theta }(k-1)+\lambda ·({\mathrm{\Theta }}_{\mathrm{target}}-\mathrm{\Theta }(k-1))$$

(40)

where $\lambda \in (0,1]$ is the smoothing factor. The value $\lambda =0.1$ was selected through a sensitivity study: smaller values produce smoother transitions but slow adaptation, while larger values risk introducing gain jitter. This value was found to provide the best compromise between adaptation speed and actuator smoothness under the TRMS operating conditions.

2. Augmented Fitness Function: To encourage smooth adaptation, the fitness function $f_i$ incorporates a stability penalty based on the parameter deviation:

$$f_i=-\left(J_{\mathrm{perf}}+\beta ·{\parallel {\mathrm{\Theta }}_i-{\mathrm{\Theta }}_{\mathrm{prev}}\parallel }^2\right)$$

(41)

where $\beta$ is the stability weighting factor. The coefficient $\beta =0.01$ was chosen to impose a soft penalty on large parameter deviations without overly constraining the search space. A sensitivity analysis confirmed that values in the range $\beta \in [0.005,0.05]$ yield stable operation on the physical rig, with lower values prioritizing tracking accuracy and higher values prioritizing parameter smoothness. The selected value of $0.01$ provides the optimal trade-off under the identified TRMS operating envelope. This ensures that the optimizer seeks gains that provide high tracking accuracy without inducing aggressive parameter fluctuations.

4.4. Quantum Update Law

At the end of an evaluation cycle, the swarm evolves. If the tested particle outperforms the Black Hole for its respective axis, it captures the global best position. The remaining particles are then updated via the quantum attraction rule derived from Hatamlou [28]:

$$X_j(t+1)=X_j\left(t\right)+\rho ·(X_{BH}\left(t\right)-X_j\left(t\right))$$

(42)

where $\rho$ is a stochastic vector representing the quantum uncertainty. The complete sequence of the RS-QBHO tuning process is summarized in Algorithm 1, and the overall adaptive architecture is illustrated in Figure 2.

Algorithm 1 Online RS-QBHO Tuning (Executed for Pitch and Yaw in Parallel)

Require:  Subsystem error $e\left(k\right)$, previous gains $\mathrm{\Theta }(k-1)$, rate limit $\lambda$, stability weight $\beta$. Ensure:  Tuned gains $\mathrm{\Theta }\left(k\right)\in {\mathbb{R}}^3$.        Initialization (First Call Only): 1: Initialize $N=30$ particles and set $evalMode\leftarrow \mathrm{True}$ 2: Set $fitness\leftarrow -\infty$, $evalIdx\leftarrow 1$, $J_{acc}\leftarrow 0$        Main Execution (Per Sample k): 3: $evalCounter\leftarrow evalCounter+1$ 4: $J_{acc}\leftarrow J_{acc}+(k·T_s)·\left|e\left(k\right)\right|$                                                               ▹ Accumulate ITAE 5: if  $evalMode=\mathrm{True}$  then 6:     ${\mathrm{\Theta }}_{target}\leftarrow particles(evalIdx,:)$                                                             ▹ Test Candidate 7: else 8:     ${\mathrm{\Theta }}_{target}\leftarrow blackHole$                                                                               ▹ Settle at Best 9: end if 10: $\mathrm{\Theta }\left(k\right)\leftarrow \mathrm{\Theta }(k-1)+\lambda ·({\mathrm{\Theta }}_{target}-\mathrm{\Theta }(k-1))$                             ▹ Apply Equation (40) 11: if  $evalCounter=evalWindowSize$  then 12:     if $evalMode=\mathrm{True}$ then 13:         Compute fitness $f_i$ using Equation (41) 14:         Update $blackHole\leftarrow argmax\left(fitness\right)$ 15:         Evolve swarm using Quantum Update Equation (42) 16:         $evalIdx\leftarrow mod(evalIdx,N)+1$ 17:     end if 18:     $evalMode\leftarrow \mathrm{not}evalMode$, $evalCounter\leftarrow 0$, $J_{acc}\leftarrow 0$ 19: end if 20: Return  $\mathrm{\Theta }\left(k\right)$

5. Simulation Results and Comparative Analysis

To establish a theoretical baseline, we first validated the proposed control architectures through a series of numerical simulations in MATLAB/Simulink (version R2014a). This phase allows for a direct comparison of the controllers under nominal conditions, where the plant model is perfectly known and external disturbances are absent.

5.1. Numerical Configuration and Setup

We utilized the ode1 (Euler) fixed-step solver with a sample time of $T_s=1$ ms for all simulation cases. To ensure a fair comparative analysis, each controller was subjected to the same sinusoidal reference trajectories over a consistent observation window of $T_{\mathrm{sim}}=100$ s. These sine waves were chosen to excite the system’s nonlinear aerodynamic cross-coupling by forcing the rotors to continuously vary their angular velocities and directions. For physical realism, the control input voltages were bounded within the $\pm 2.5$ V saturation limit of the physical rig.

5.2. Performance Evaluation Metrics

The controllers were assessed using two primary performance indices: the Root Mean Square Error (RMSE) to quantify absolute tracking precision, and the Integral of Time-weighted Absolute Error (ITAE) to evaluate the convergence speed and transient efficiency. The quantitative metrics for the 100 s simulation window are presented in

Table 2. Quantitative performance metrics for a 100 s simulation window.

Controller	Pitch Subsystem		Yaw Subsystem
	RMSE (rad)	ITAE (rad·s)	RMSE (rad)	ITAE (rad·s)
Conventional Backstepping	0.0144	65.0287	0.0258	94.9400
Online RS-QBHO	0.0211	97.2441	0.0278	107.6615
Hybrid BSMC	0.0144	65.0287	0.0258	94.9400

Bold indicates the superior (lowest) performance metric achieved.

.

5.3. Comparative Analysis and Discussion

The tracking trajectories for the three strategies are depicted in Figure 3, Figure 4 and Figure 5. A comparative review of these results provides several key insights into the operational characteristics of the proposed laws.

5.3.1. Structural Convergence of the Hybrid Law

A noteworthy theoretical observation is that the Conventional Backstepping and Hybrid BSMC produced identical numerical results. This outcome is a deliberate consequence of the hybrid design and constitutes an important theoretical confirmation: the sliding mode switching term is specifically engineered to reject unmodeled dynamics $\mathrm{\Delta }$. Since the simulation environment uses the exact nominal model from which the controller was synthesized, the model uncertainty is identically zero and the switching term correctly remains inactive throughout. The Hybrid BSMC therefore degrades gracefully to a pure Backstepping law under ideal conditions, avoiding unnecessary actuator chattering. This theoretical convergence property is preserved intentionally in the simulation results to establish an unambiguous baseline: any performance difference observed in Section 6 between the Conventional Backstepping and Hybrid BSMC controllers is exclusively attributable to the sliding mode term’s activation by real unmodeled dynamics, and not to any intrinsic difference in their nominal tracking capability.

5.3.2. Performance Characteristics of Intelligent Tuning

The Online RS-QBHO strategy presents a distinct performance profile. In this idealized numerical environment, the meta-heuristic tuning yields a slightly higher cumulative error compared to the exact model-based cancellation of the Backstepping variants. Specifically, the pitch RMSE increased from $0.0144$ to $0.0211$ rad. This suggests that the stochastic search for optimal gains prioritizes a dynamic response that, while stable and effective, does not achieve the mathematical “perfection” of model cancellation when the model is $100\%$ accurate. However, this adaptive capability is expected to be the deciding factor in the real-time phase, where the model used for Backstepping design will no longer perfectly match the physical hardware.

6. Real-Time Experimental Validation and Discussion

The transition from a deterministic numerical environment to a physical laboratory setup represents the most rigorous validation for any flight control strategy. While simulation results established a theoretical baseline, the physical TRMS rig introduces non-linear friction, mechanical cable tension, and high-frequency aerodynamic turbulence that are omitted in idealized models. This section details the hardware-in-the-loop (HIL) configuration and provides a comparative critique of the performance trade-offs between the proposed intelligent and robust strategies.

6.1. Experimental Setup and HIL Configuration

The experimental platform utilized in this study is the Twin Rotor Multi-Input Multi-Output System (TRMS), shown in Figure 1, a high-fidelity laboratory benchmark. To bridge the gap between the discrete control law and the continuous plant dynamics, a deterministic HIL architecture was established, as illustrated in Figure 6.

The workstation, operating under the Simulink Desktop Real-Time kernel, executes the control algorithms at a sampling frequency of 100 Hz ($T_s=0.01$ s). Interfacing with the TRMS rig is facilitated by an Advantech PCI-1711 (Advantech Co., Ltd., Taipei, Taiwan) data acquisition board. This card serves as the critical communication bridge, utilizing its 12-bit DAC channels to modulate the motor voltages while simultaneously processing high-speed digital pulses from the quadrature optical encoders. With a sensing resolution of 2000 pulses per revolution, the HIL setup provides the high-fidelity state feedback necessary for the derivative-intensive Backstepping terms. To ensure the safety of the DC motors, all control commands were strictly saturated at $\pm 2.5$ V.

6.2. Empirical Evaluation of the Conventional Backstepping (Baseline)

The experimental sequence began with the deployment of the Conventional Backstepping controller derived in Section 2. While numerically stable in simulation, the physical results (Figure 7) revealed a significant and quantifiable sensitivity to model mismatch. Although the pitch axis maintained marginal stability, the yaw subsystem suffered from divergent oscillations, yielding a failure-state RMSE of $2.5624$ rad.

This instability is a direct consequence of the “model-reality gap.” The physical tail rotor is subjected to significant aerodynamic drag and friction that were not captured in the nominal design model. Because standard Backstepping relies on the exact cancellation of dynamics, these unmodeled terms were interpreted by the recursive law as state variations, causing the controller to over-compensate and drive the yaw subsystem into an unstable limit cycle. This observation underscores the necessity of the robust and adaptive layers proposed in this study.

6.3. Intelligent Adaptation via Online RS-QBHO

To address the baseline failure, the Online RS-QBHO strategy was implemented. As shown in Figure 8, the adaptive tuner successfully stabilized the plant. By sequentially evaluating candidate gain sets, the algorithm identified the optimal stiffness required to reject the physical cross-coupling.

MIMO Parameter Convergence Analysis

The real-time evolution of the 6-DOF control gains is presented in Figure 9. A rigorous analysis of these trajectories reveals a distinct two-stage adaptive process. During the first 10 s, the swarm undergoes an intensive exploration phase, characterized by high-frequency parameter fluctuations as the QBHO evaluates the initial population. Following this transient, the gains converge toward an optimal “uncertainty cloud.”

Notably, the proportional gains ($k_1$ and $k_4$) reach the predefined saturation limit of $5.0$, suggesting that the algorithm correctly identified the maximum allowable loop-gain to minimize tracking error within the safety bounds of the “event horizon.” The derivative and integral gains exhibit smooth, bounded variations, providing empirical proof that the rate-limiting filter (Equation (40)) effectively suppresses gain jitter. This sustained adaptation allows the controller to remain optimal even as the aerodynamic conditions vary throughout the sinusoidal cycle.

6.4. Performance Analysis of the Hybrid BSMC Strategy

The Hybrid BSMC strategy was subsequently implemented to evaluate structural robustness. As illustrated in Figure 10, this strategy provided the most consistent tracking performance. Unlike the QBHO, which must iteratively search for optimal gains, the Hybrid BSMC utilizes its sliding mode component to passively and instantaneously reject disturbances. This allows for superior tracking with significantly reduced phase lag and minimal overshoot compared to the adaptive tuner. The structural robustness is particularly evident in the yaw axis, where the sliding term successfully mitigated the reactive torques generated by the main rotor.

Regarding the chattering behavior inherent to sliding mode control, the $\epsilon$-modification of the signum function implemented as $sign\left(\epsilon S\right)$ with a small boundary layer parameter $\epsilon$ converts the ideal discontinuous relay action into a smooth, bounded approximation within a thin layer around the sliding surface. This modification confines switching activity to frequencies within the bandwidth of the motor drive system, as confirmed by the physical control signal profiles recorded throughout the 100 s experimental run. No high-frequency mechanical vibration was detected in the rotor shafts, providing empirical evidence that the boundary layer approach successfully suppresses actuator chattering under the operating conditions of the TRMS rig.

6.5. Comparative Synthesis and Quantitative Analysis

The final comparative performance is summarized in Figure 11 and

Table 3. Consolidated real-time performance metrics for the physical TRMS experiment.

Controller	Pitch Axis		Yaw Axis
	RMSE (rad)	ITAE (rad·s)	RMSE (rad)	ITAE (rad·s)
Conventional Backstepping	0.2571	1990.24	2.5624	21,281.99
Online RS-QBHO	0.0799	3382.29	0.2305	10,268.60
Hybrid BSMC	0.0682	2733.71	0.1858	8624.69

Bold indicates the superior (lowest) performance metric achieved.

. While both proposed strategies achieved stabilization, the Hybrid BSMC achieved lower values across all recorded performance metrics, reaching an RMSE of $0.0682$ rad for pitch and the lowest ITAE among all tested controllers.

It is important to interpret the RS-QBHO ITAE values in their correct context. The elevated cumulative error relative to the Hybrid BSMC is primarily attributable to the stochastic exploration transient during the first $\approx 10$ s of operation, during which the algorithm evaluates suboptimal candidate gain sets before converging to the global best (as visible in the gain trajectories of Figure 9). Once convergence is achieved, the steady-state tracking precision as quantified by the post-convergence RMSE is substantially improved relative to the baseline Backstepping controller. This transient exploration cost is an inherent and expected characteristic of online metaheuristic adaptation, and the RS-QBHO remains a viable alternative in scenarios where structural modifications to the control architecture are not permissible.

The synthesis of these results leads to a critical scientific insight: while intelligent meta-heuristic optimization (QBHO) is a powerful tool for compensating for parametric mismatches, the passive structural robustness of a Hybrid BSMC architecture is more efficacious for mitigating the high-dynamic, unmodeled aerodynamic fluctuations of the TRMS platform.

7. Conclusions

This study presented a comprehensive comparative evaluation of three nonlinear control strategies on the Twin-Rotor MIMO System (TRMS), bridging the gap between theoretical derivation and physical implementation. By deploying each controller under identical hardware-in-the-loop conditions, we exposed the inherent fragility of purely model-dependent control and validated two advanced layers, intelligent adaptation and structural robustness, as effective mechanisms for ensuring flight stability in the presence of unmodeled dynamics.

The experimental findings yield three principal scientific insights. First, the Conventional Backstepping controller, despite its mathematical guarantees under nominal conditions, failed on the physical rig due to the “simulation-to-reality” discrepancy: unmodeled aerodynamic drag and non-linear friction were misinterpreted by the recursive law, driving the yaw subsystem into divergent oscillation with a failure-state RMSE of 2.5624 rad. Second, the Online RS-QBHO strategy demonstrated the effectiveness of intelligent adaptation, successfully navigating the non-convex parameter space to achieve stabilization. The elevated ITAE relative to the Hybrid BSMC is attributable to the initial stochastic exploration phase and represents an expected, transient cost of online learning rather than a deficiency in steady-state performance. Third, the Hybrid BSMC architecture proved to be an effective and practically validated approach for high-dynamic aerodynamic MIMO systems: by integrating a sliding mode switching term into the Backstepping framework, the controller passively and instantaneously rejected unmodeled disturbances without requiring a learning phase, achieving the lowest RMSE values of 0.0682 rad and 0.1858 rad for the pitch and yaw subsystems, respectively.

In conclusion, this work demonstrates that while meta-heuristic optimization is an invaluable tool for parametric gain-scheduling, the direct, passive rejection capability of a hybrid robust architecture is more efficacious for mitigating rapid, unmodeled aerodynamic fluctuations of MIMO flight platforms. It should be noted that the experimental validation was conducted under fixed environmental conditions, and the robustness of both proposed architectures under time-varying parametric perturbations or structured external disturbances remains a subject for future investigation. Future research will focus on the integration of neural-network-based observers to provide active estimation of the unmodeled dynamics, potentially combining the strengths of intelligent estimation with the proven stability of structural sliding-mode robustness, while also extending the validation to more extreme, disturbance-rich flight-like scenarios.