H-Infinity Artificial Bee Colony to Dampen Sideslip and Bankroll Disturbances of Boeing 747-100 Lateral Movement

Axial, directional and upright forces, along with pitch, yaw, and roll moments coupled with structural elastic forces, perform the overall dynamics of aircraft motions. The small perturbed equilibrium widely simplifies such dynamic complexities to decouple longitudinal and lateral motions. However, lateral-directional motion exposes rotations that happen about the x-axis and-z-axis. Their moments express coupled roll rate and yaw rate. The pilot workload may be neutralized in the existing reliable control mechanism. Stability augmentation systems in the inner loop control structure provide adequate damping characteristics and stability margins. The pioneer Sperry autopilots have become vital to the airline industry to hold an attitude using several high-speed processors [1]. They are also valuable for extremely long endurances to avoid pilot fatigue. The classical control approach was designed for ancient autopilot models [2, 3]. However, they provide the limited capability of disturbance rejections.

Robust control methods are widely approached in high-tech autopilot planes [4]. The position of the Qball-X4 quadrotor was controlled using the linear quadratic regulator (LQR) for desired attitudes [5]. The integral LQR showed firm robustness and highly interference elimination for six degrees of freedom control of a small-scale quadcopter [6]. The linear quadratic Gaussian (LQG) method had an excellent disturbance rejection of up to 3.5% plant noise and 1% measurement noise for the pitch angle in the longitudinal cruise aircraft [7]. The effectiveness of the integral LQR was approved in stabilizing the attitude and altitude of a star-shaped vehicle under 20% uncertainties [8]. Attitude microsatellite stabilization showed the efficiency of LQR and LQG controllers for large angles in terms of more accuracy compared with feedback quaternion and proportional-integral-derivative (PID) controllers [9]. The effectiveness of the controller gains found by the LQR method was also investigated under disturbances using the Kalman filter for a small un-crewed aerial vehicle in longitudinal flight where the reference speed was reached quickly without affecting altitude and pitch angle [10]. Darwish et al. [11] designed a compact aircraft autopilot system using an energy control approach based on the genetic algorithm (GA) for PID tuning. Their simulations exposed an exceptional solution compared with the predictive control law in the form of energy instead of altitude and velocity inputs. An LQR controller was successfully implemented in the real-time pitch axis helicopter stabilization. A good performance was found in a stable system and reference tracking as high as 55 degrees [12]. Satisfactory LQR performance was also found during all the yaw angles for a quadrotor tilt-wing uncrewed aerial vehicle [13]. The LQR method presented stable response effectiveness to poles assignment and fuzzy control approaches in investigating infrastructure collapse force due to earthquake disasters [14]. Aktas and Esen [15] suggested the LQR control design for the dynamic damping position of a competent, flexible cantilever. The LQR controller predicted optimal performance at the fixed-end actuation of the beam.

The H-infinity (H_∞) approach has been a challenging research area for two decades and is renowned as an efficient, robust design method [4]. The developed H_∞ may guarantee asymptotic stability, system performance, and tracking properties compared to LQR. H₂ control used a systematic optimization design for optimal autopilot pitch attitude control under icing conditions [16]. The dynamic inversion of H_∞ control simulation robustly met Category III Federal Aviation Association under wind shears and sensor errors for the Boeing 747 landing longitudinal plane [17]. The distributed H_∞ framework showed successful distance tracking performance and robust and string stabilities by a comparative simulation with no robust controllers for the rigid geometry of grouped autos [18]. A recursive H_∞ control solution was introduced to track all reference paths well and execute parallel parking for the mobile robot system [19]. A new H_∞ output dynamic feedback synthesis was offered based on linear matrix inequality and equality parameterizing [20].

The main design parameters in the optimal H_∞control algorithm are the selection of weighting matrices. However, traditional approaches are time-consuming and require a high experience to achieve a robust design in both the time and frequency domains. Genetic and particle swarm algorithms have been recently proposed in many artificial intelligent optimization procedures [21]. The handling qualities with zero steady-state error and 1.5% overshoot were obtained using linear matrix inequalities and multi-objective GA to weigh the sensitivities under a 5.72° step-change in aileron command for B747-200 lateral dynamics [17]. The bee swarms or artificial bee colony (ABC) procedure was first offered to optimize numeric benchmark functions [21]. A combined ABC and LQR optimal control was used for a nonlinear inverted pendulum and showed a reasonable optimizing efficiency in weighting matrices compared to the traditional methods [22]. Karaboga and Akay [23] extended the ABC procedure to be more robust, fast converged and higher flexible than element swarm optimization, GA and differential evolution scheme. The enhanced ABC method without random exploration showed a superb performance in the study of industrial discharge optimization when benchmarked with other metaheuristic methods [24]. Authors in [25] offered the new element swarm optimizer framework, which was well tested for several margin limits of the manufacturing discharge application. Al-awad [26] showed that the weighted GA optimization of the PID parameters controller for the rotational mechanical platform is more promising than the LQR and PID control methods.

Belletti et al. [27] used programmed synthesis of H_∞ and GA for the atmospheric flight of attitude launch vehicle control. A gain scheduling control system was realized over widespread conditions reducing the interferences and loads with achieved performance and stability. Hamza et al. [28] deployed mu-synthesis feedback linearization-based controller for low-frequency disturbance rejection problems in quadrotor flight. The proposed controller performed better than full-state feedback and mu-synthesis in providing robust performance, eliminating non-linearity and tracking trajectory under parametric uncertainty. Rachyd et al. [29] used the Monte Carlo algorithm to evaluate aircraft lateral downwind approaches based on the turn and flap scheduler of the pilot support system. The results indicated the possibility of stretching the approach path, and no constraints on the turn could be lagged or led towards the final as far as the scheduling timing and flow separation as concerned. The flap optimizer could handle the disturbance for various conditions. No wind conditions changed the aircraft performance was included in the model and simulation. Klyde et al. [30] evaluated a high-pitched bank turn flight at constant altitude case for 12 airlines Boing company. It is shown that the steep turn caused pilot spatial disorientation and control loss due to upset altitude variation. Authors in [31] used Kalman filters to estimate relative aircraft movements, particularly sideslip angle under a broad spatial environment. Their results showed there might be about two degrees of root square errors of sideslip estimations under aggressive flight conditions. Silva et al. [32] captured some effects of propeller slipstream problem on the Piper PA-30 aircraft vertical stabilization during the severe scenario of cross-wind or one engine failure.

A deep examination of the past literature [5-32] indicates that few researchers used optimizer algorithms for design gain matrices of their controller methods. Those authors [11, 26, 27] used GA to tune those matrices, and they all agreed that their results beat those who had missed out on optimization schemes [5-9]. Other authors [10, 31] preferred using the Kalman filter technique to estimate parameter uncertainty in their applications, giving good performance under various levels of disturbances. Rachyd et al. [29] employed the Monte Carlo algorithm to schedule the timing setting for disturbance handling. No obvious advantages are seen among those optimizers for specific applications. However, many of them used a glance of trial and error rules (TER) to weigh those matrices [5-9, 11-15, 18-20] even though stable systems and good reference tracking performance were claimed such tiresome procedure affects the controller robustness in many cases. Motivated by the ABC unique performance shown in [21-23], the ABC algorithm seems a much more suitable optimizer to the H_∞ control method in terms of robustly quadratic stable performance and fewer sensitivity influences on the controller parameters. No attention was paid to model the influence of the external environment, as lateral flights are less susceptible than longitudinal motion [1]. Such H_∞ and ABC combination was reasonably approached in the study of electric grid stability [33]. The lateral flight problem imposed multivariate states with cross-coupling between aileron and rudder channels which is more relevant to problem-solving with the synthesis of H_∞ control theory than PID, LQR, LQG, and predictive variable control implementations. Thus, the H_∞ and ABC platform is preferred to conduct the lateral flight control to dampen bankroll and sideslip perturbations due to the lateral coupling stick and rudder pedal inputs.

This paper investigates the lateral flight perturbations due to aileron (δ_a) and rudder (δ_r) coupling actuators of Boeing 747-100 (B747-100) Mach and altitude conditions covering CI [M =0.2, h=sea level]; CII [M=0.5, h=6096 m]; and CIII [M=0.9, h=12192 m]. The aircraft’s multivariable dynamics are linearized and modelled using a state-space system. The H_∞ stability augmentation design (H_∞SAD) is applied to manage moderate sideslip (β) and bankroll (ϕ) disturbances and to control the lateral states of lateral velocity (v), rolling rate (p) and yawing rate (r). The ABC scheme is synchronized to penalize the weight system matrices, which are expected to be large-scale coupling influences due to five states of (v, p, r, β and ϕ) and two commands of (δ_a and δ_r). Steady-state responses are realized for adequately accepted flying qualities in negligible overshoot and fast transient convergences based on one-degree step actuation of aileron and rudder coupling. In conclusion, the fine-tuning responses are attained based on the reference input full-state feedback autopilot implementation [34]. Those responses meet objectives for lateral velocity and bank attitude in those cases. Roll, spiral and Dutch roll modes have also been identified in those three cases. In particular, Dutch roll modes well meet good flying quality merits. Finally, 3D response surfaces of the flying qualities based on Dutch roll modes against flight cases (CI, CII and CIII) and free-disturbance bankroll or sideslip responses have met the minimum flying qualities merits (damping ratio × damped natural frequency=0.1 rad.sec^-1) [35, 36]. The minimum bankroll and sideslip flying qualities of 0.265 rad/sec and 0.137 rad/sec are found, respectively.

3.1 Uncontrolled lateral-directional characteristics

B747-100 lateral-directional uncontrolled flight was initially simulated at CI, CII, and CIII. This scenario mimics no pilot interference with the control mechanisms during stick-fixed flying. The plane was excited for unit step states, including 1° roll and yaw angles (the significant influence of altitude) and one m/sec resultant flight velocity (the primary influence of thrust). The linearized lateral-directional dynamics were modelled at those three trimmed flight conditions based on the A/C and stability derivatives [35] using Eq. (2). They can be summarized in Table 1. The B747-100 models of lateral-directional coupling variables were obtained with no aileron and rudder controls. Such itemizations gave ten TFs using Eq. (6) for lateral-directional variables at each Mach number and altitude case concerning the aileron and rudder inputs. The stability of lateral-directional motion can be evaluated by the eigenvalues of the A matrix or the characteristic equations given in Eq. (4). All those TFs had the same dominators. The coefficients of numerator TF aileron control off at three lateral-directional B747-100 flight cases are shown in Table 2.

Table 1. Uncontrolled B747-100 lateral characteristics

1. FCs (flight cases), 2. CI (M = 0.2, h = 0 m), 3. CII (M =0.5, h = 6096 m), 4. CIII (M = 0.9 and h = 12192 m)

Table 2. Numerator T.F. of aileron control off

LSs (lateral states)

The coefficients of numerator TF rudder control off at three lateral-directional B747-100 flight cases are shown in Table 3. The coefficients of denominator TF aileron and rudder control off at three lateral-directional B747-100 flight states are shown in Table 4. The characteristic equations are of the fourth order in the s Laplace variable. However, pole-zero cancellations at the origin may sometimes make the order less than a fourth. The eigenvalues show that lateral-directional dynamic motion consists of several oscillatory modes. Negative conjugate eigenvalues indicate the static stability of the airplane. However, a pair of eigenvalues was very close to the imaginary axis. One of them at the origin indicates that the plane may not sufficiently be dynamically stable to perform safe manoeuvring flight under the three conditions. The aircraft lateral-directional motion exhibited poor responses with high overshoot, long settling time and high oscillations. Aileron and rudder commands have to be controlled to enhance those responses. Prominent peaks in the oscillatory responses were found owing to the system zeroes’ effects on the underlying dynamics.

Table 3. Numerator TF of rudder control off

Table 4. Denominator TF of aileron/rudder control off

3.2 Controlled lateral-directional responses

The Q and H optimal matrices were reached with the ABC satisfying control requirements and good performance. In order to avoid the effects of sensitivity on the system control performance, the best solutions according to fitness were verified among 150 iterations of weighting attempts. Section 3.5 gives further discussions. The state vector had been utterly remodelled from the independent continuous measurement of w(t). The K gains of the H_∞ algorithm revised the controlled state matrix A_C as shown in Table 5. The stability of lateral-directional motion was re-evaluated by the eigenvalues of A_C using Eq. (14) and Eq. (15). The characteristic equations are of the fourth order in the “s” Laplace variable. The eigenvalues of the lateral-directional uncontrolled flight modes have now been moved farther away from the imaginary axis into the left s-plane. Negative real eigenvalues were found in CI compared with negative conjugates in CII and CIII. The plane tends to be dynamically stable to perform manoeuvring flights under those conditions. Convergent responses are obtained for the three cases with no harmful oscillations and overshoots. The resultant lateral-directional variables concerning the coupling control ($\Delta \delta=\Delta \delta_a \cup \Delta \delta_r$) were compared in the three cases.

Figure 2 shows the side velocity responses $\left(\frac{v}{\Delta \delta}\right)$ of the three cases. The settling time is almost the same as two sec for both CI and CII. CIII took a more settling time of seven sec to reach the steady-state lateral flight. The side velocity got higher with increasing the forward velocity, which is the reason for the difference in the amplitudes. The H_∞SAD produced well-converged flat responses with no overshoots. Figure 3 shows the bank “roll” angle responses ($\frac{\phi}{\Delta \delta}$) in the three cases. Both CI and CII almost had the same settling time of three secs. The steady-state is almost 4.3 sec for CIII. The steady-state amplitudes in CI, CII and CIII, are equal to -0.089°, -0.006° and -0.034°, respectively. The highest bankroll angle happened at a lower altitude and Mach number case, i.e., CI during the landing flight phase. However, the lowest bank angle occurred at M=0.5 and h=6096 m (CII). The roll subsidence convergence took a long time as the Mach number and altitude would get higher. Overall, the H_∞SAD attenuated all visible forms of bank angle perturbations for the three cases.

Table 5. Controlled B747-100 lateral characteristics

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Figure 2. Side velocities of lateral controlled flight

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Figure 3. Bank responses of lateral controlled flight

Figure 4 shows the roll rate responses ($\frac{p}{\Delta \delta}$) in the three cases. Large fluctuations found during the first seconds with a bit of overshoot, particularly in CIII. The overshot in CIII was a result of the roots of the spiral convergence mode, which are a little bit close to the imaginary axis. Quite similar roll rate responses are seen for CI and CII. The spiral model affects the time response of the three cases, which shows the lowest period of 0.281 sec in CI and the highest period in CIII of 1.176 sec. Overall the H_∞SAD well excluded the roll disturbances in less than five secs within zero amplitude steady states. Figure 5 shows the responses to the yaw rate ($\frac{r}{\Delta \delta}$) in the three cases. The settling times were almost the same for CI and CII. CIII took a more settling time of 5.8 sec to reach the steady state. Both CI and CIII had almost the same yaw rate perturbations of -0.181 rad/sec. The Dutch roll mode spent more time on high Mach number and altitude cases than the low one. Figure 6 shows the responses of the sideslip angle ($\frac{\beta}{\Delta \delta}$) in the three cases. CI converged at the settling time of 2.2 sec with amplitude of 0.0974°. CII took a settling time of 1.85 sec with amplitude of 0.0742°, and CIII converged at the settling time of 6.05 sec with the highest amplitude of 0.109°. Overall, the H_∞SAD attenuated perceptible sideslip perturbations for the three cases. Also, almost no opposite turn as the roll-yaw rotation rates incline toward being diminished due to the lateral coupling stick and rudder pedal rather than an un-modelled wind factor may typically hit the vertical tail.

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Figure 4. Roll rates of lateral controlled flight

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Figure 5. Yaw rates of lateral controlled flight

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Figure 6. Sideslip responses of lateral controlled flight

3.3 Controlled lateral-directional modes

Once stably convergent lateral-directional flight was achieved under coupling unit step aileron-rudder of 1°, the reference input full-state feedback autopilot designs [34] were searched based on lateral velocity and bank angle for no roll and yaw. The side velocity was almost 30% of the total velocity. From Figure 2, the side velocities converged to sitting values based on a factor of 27.5 for the three cases. The numerator TF coefficients based on combined aileron and rudder control at three lateral-directional B747-100 flight cases are shown in Table 6.

Table 6. Numerator TF aileron and rudder control

The denominator TF coefficients based on combined aileron and rudder control at three lateral-directional B747-100 flight states are shown in Table 7. The characteristic equations are of the fourth order in the s Laplace variable, where the eigenvalues show several lateral-directional flights of dynamic stability modes. The aircraft lateral-directional motion exhibited respectable responses with low overshoot, short settling time and minor oscillations. The flying quality is then checked for the three lateral-directional modes. Three well-known modes related to lateral-directional motion are already termed RS, SC and DR. These modes were recognized by the period and the damping ratio. Table 8 shows lateral-directional modes at the three flight conditions. The natural frequency, damping ratio and their product should generally exceed the minimum values of 0.4 rad/sec, 0.08 and 0.1 rad/sec, respectively, for the mode at the well-known Cooper-Harper scale [1, 35]. The DR modes are adequately damped for the three cases. However, the RS mode is relatively faster than the SC mode and is more pronounced at higher Mach number and altitude constraints.

Table 7. Denominator TF of aileron/rudder control

Table 8. B747-100 lateral-directional modes

LMs (lateral modes)

3.4 Flying quality assessments

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Figure 7. DR flying qualities of bankrolling control

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Figure 8. DR flying qualities of sideslipping control

The flying quality properties for the DR mode of B747-100 lateral controlled flight based on bankroll and sideslip responses are assessed in Figures 3 and 6. The federal aviation (FARs) or military specifications (MILSPECs) regulate permissible flying stability requirements [36]. These flying qualities are evaluated based on the level I mission flight phase [1, 35] for the product of damping ratio and undamped natural frequency (ξω_d) of the DR mode, namely, ξ×ω_d≥0.1 rad/sec [1, 36]. Moreover, ξ≥0.08 and ω_d≥0.4 rad/sec [35] were also widely verified. The DR lateral oscillation represents flat yaw and sideslip under no roll. The DR mode is similar to a short period of longitudinal flight [35]. Figure 7 shows DR flying qualities for B747-100 lateral-directional bank-rolling controlled flight. The 3D surface represents ξω_d the vertical axis, the bankroll ϕ alongside the cross-horizontal axis and the flight cases alongside the horizontal axis. The minimum flying qualities are based on ξω_d were overall met for the ϕ responses as high as 0.265 rad/sec. Figure 8 shows DR flying qualities for B747-100 lateral-directional sideslipping controlled flight. The 3D surface represents ξω_d alongside the vertical axis, sideslip β alongside the cross-horizontal axis and the flight cases alongside the horizontal axis. The minimum flying qualities based on ξω_d were overall met for the β responses as most values exceed 0.137 rad/sec.

The tolerable skidding turn has to fall within $1 \leq \frac{\left(\frac{\beta}{\phi}\right)}{\Delta \delta} \leq 15$ of the constricted turn radius in compliance with the airworthiness (authorities’) requirements [36]. Table 9 compare the B747-100 sideslip turn of three cases of steady-state responsiveness shown in Figures 3 and 6. All the lateral scenarios meet tolerable turn sideslip constraints, and the CI at sea level is more susceptible to sideslip turn effects due to the downward ground interaction. The sideslip response is also compared with flight testing from aircraft certification [36]. It shows satisfactory dampness in the sideslip incident by the tendency to raise the low wing with the aileron and rudder controls.

Table 9. B747-100 sideslip turn comparison

For benchmarking the sideslip dampening responsiveness, Figure 9 qualitatively compares the sideslip response obtained by the H_∞-ABC synthesis and typical DR manoeuvre flight-time histories of sideslip control effectiveness [36]. It indicated that the DR mode of the test aircraft was achieved by first commanding the aileron stick, and then the rudder pedals were pushed. The response settled in 5-10 sec for the DR flight test lasted 15-20 sec [36]. A skidding turn was manageable by keeping a constant heading through the aileron stick followed by the rudder pedal to maintain the heading angle constant with low side speed deviation (shown in Figure 2). It did not indicate relative roll oscillations concerning the sideslip practice. The flying qualities airworthiness requirement stated that β≤3° for level I flight [36]. No disrupted skid is recognized as the flight test showed an almost minimal steady levelled responsiveness. It looks like the promising implementation of the H_∞-ABC stability augmentation strategy to take place in the classical control system of tested aircraft. Flight tests based on steady heading sideslip manoeuvre [36] exhibited measured perturbations in sideslip-to-bankroll from 14.625 to 1.479 within the flight envelope ranging from CIII to CI cases studied here.

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Figure 9. DR sideslip responsiveness validation

3.5 Sensitivity assessments

The steady-state step responsiveness fulfilling $O_3=\lim _{t \rightarrow \infty}\left[\begin{array}{lllll}v & p & r & \beta & \phi\end{array}\right]^T=-A_C^{-1} B_C \delta$ must die out for sideslip and bankroll perturbations of the coupling control inputs to assure the dynamic stability of the aircraft. Therefore, ABC factors were set between 0.1 and 150, the inhabitant dimension was taken at 15, and the total cycle did not exceed 200. Employed bee matrices were initially chosen diagonal. The ABC maximum generations were fixed to 2000, and the search space was imposed as 13×10×2000; 5×5×2000; 4×5×2000 which copes with the design spaces matrices. The coefficient settings, colony dimension and total cycles of the ABC optimization procedure were implemented between 0.12 and 60, 15 and 100, respectively. The algorithm was exacted for 25 runtimes on each benchmark case. Beginning with the most achievable criteria and then gradually adding one-by-one constraints until the most control merits were achieved with a low level of sensitivity to parameter variations associated with the weighting process of design matrices. The finest design matrices from the ABC optimizer only proceeded for trade-offs amongst the overshot, settling epoch, the steady-state errors, the convergence and the computation effort. The accuracy of the H_∞ and ABC approach was qualitatively measured by root mean square error and absolute error metrics [21, 23]. The precision of the ABC optimizer in the estimations of the H_∞ weighting matrices regresses as high as 88% for the most lateral manoeuvre responses achieved.

The ABC parameters used for the lateral states are shown in Table 10. Those were supportive of minimizing the cost function (f_i, i=1:3) in Eq. (20) and thus obtaining the optimal weighting matrices. A demo is next given for the v state variable, which is relevant to all the other state variables in Table 10. A satisfactorily v response was reached using the ABC algorithm constraints (O₁<2% for desired overshoot, O₂<7% required settling time, and O₃<±1.7% the desired steady-state error). They had to be away from s=0, avoiding the ABC failure and H_∞ destabilization. Fitness residuals of F₁=0.6, F₂=0.4, and F₃=0.8 had been deployed to achieve reasonably optimized H_∞ design matrices to augment the v response of lateral flight. As shown in Figure 2, all constraints are satisfied for three flight conditions at CI, CII and CIII. The maximum settling time reached 6.45 sec for M=0.9 and h=12192 m (CIII), with negligible overshoot and steady-state error.

Table 11 quantitatively compares objective functions. The standard deviation, best, averaged and worst results are compared with TER [34]. The ABC algorithm converges much better than without systematic optimization of the TER. Moreover, the obtained flight responses had reasonable convergences based on the ABC limitations tabulated in Table 10.

Table 10. ABC sensitivity

Table 11. ABC cost functions’ accuracy

1. CF (cost function); 2. WS (worst solution); 3. BS (best solution); 4. AS (averaged solution); 4. SD (standard deviation); 5. TER (trial and error rules)

Thanks to the Mobility and Transport Research Centre (MTRC), Coventry University, the UK, for valuable supportive consultations. The University of Tripoli supports this research.