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The Hinfinity method is used to augment the lateral stability of Boeing 747100 flight at Mach numbers and altitudes of (0.2, sealevel), (0.5, 6096 m), and (0.9, 12192 m). The aim is to attenuate the lateraldirectional states’ perturbations coupling with the aileron and rudder. The method is synthesized with the artificial bee colony algorithm to ensure a robust quadratic performance under moderate sideslip and bankroll disturbances based on the mixed Hinfinity sensitivity criteria. Such an optimizer effectively weighs the design gain matrices for at least a degree of freedom higher than without it. Stable eigenvalues and steadystate responses are reached for the step states. The controller appropriately tracks reference side velocity, roll rate and yaw rate and effectively compensates for sideslip and bankroll disturbances. Despite the transient peaks for the roll and yaw rates, level convergences are obtained for the other states. Dutch roll mode meets flying qualities of airworthiness requirements, whereas roll and spiral modes slightly diverge nearby the landing conditions. The Hinfinity and artificial bee colony synthesis were well performed for bankroll and sideslip references of small to moderate perturbations. A highfidelity optimizer may be considered for a severe level of disturbance and transitory behaviour.
artificial bee colony, Boeing 747100 lateral movement, Hinfinity, sideslip and bankroll disturbances
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, lateraldirectional motion exposes rotations that happen about the xaxis andzaxis. 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 highspeed 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 hightech autopilot planes [4]. The position of the QballX4 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 smallscale 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 starshaped 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 proportionalintegralderivative (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 uncrewed 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 realtime 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 tiltwing 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 fixedend actuation of the beam.
The Hinfinity (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. H_{2} 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 timeconsuming 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 steadystate error and 1.5% overshoot were obtained using linear matrix inequalities and multiobjective GA to weigh the sensitivities under a 5.72° stepchange in aileron command for B747200 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. Alawad [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 musynthesis feedback linearizationbased controller for lowfrequency disturbance rejection problems in quadrotor flight. The proposed controller performed better than fullstate feedback and musynthesis in providing robust performance, eliminating nonlinearity 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 highpitched 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 PA30 aircraft vertical stabilization during the severe scenario of crosswind or one engine failure.
A deep examination of the past literature [532] 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 [59]. 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 [59, 1115, 1820] 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 [2123], 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 crosscoupling between aileron and rudder channels which is more relevant to problemsolving 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 747100 (B747100) 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 statespace 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 largescale coupling influences due to five states of (v, p, r, β and ϕ) and two commands of (δ_{a} and δ_{r}). Steadystate responses are realized for adequately accepted flying qualities in negligible overshoot and fast transient convergences based on onedegree step actuation of aileron and rudder coupling. In conclusion, the finetuning responses are attained based on the reference input fullstate 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 freedisturbance 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.
2.1 Lateraldirectional model
B747100 lateraldirectional sideslipping, rolling, yawing, and banking coupling with aileron and rudder are modelled as in [1, 35]. Based on a small perturbation statespace model [1], the linearized lateraldirectional motions can be shown below:
$\begin{align} & \left[ \begin{matrix} \dot{\beta }\cong {{\dot{v}}}/{V}\; \\ {\dot{p}} \\ {\dot{r}} \\ {\dot{\phi }} \\\end{matrix} \right]= \\ & \left[ \begin{matrix} \tfrac{{{Y}_{v}}}{mV} & \tfrac{{{Y}_{p}}}{mV} & \left( \tfrac{{{Y}_{r}}}{mV}\frac{{{u}_{0}}}{V} \right) & \frac{g}{V} \\ \left( \tfrac{{{L}_{v}}}{{{{{I}'}}_{x}}}+{{{{I}'}}_{zx}}{{N}_{v}} \right) & \left( \tfrac{{{L}_{p}}}{{{{{I}'}}_{x}}}+{{{{I}'}}_{zx}}{{N}_{p}} \right) & \left( \tfrac{{{L}_{r}}}{{{{{I}'}}_{x}}}+{{{{I}'}}_{zx}}{{N}_{r}} \right) & 0 \\ \left( \tfrac{{{N}_{v}}}{{{{{I}'}}_{z}}}+{{{{I}'}}_{zx}}{{L}_{v}} \right) & \left( \tfrac{{{N}_{p}}}{{{{{I}'}}_{z}}}+{{{{I}'}}_{zx}}{{L}_{p}} \right) & \left( \tfrac{{{N}_{r}}}{{{{{I}'}}_{z}}}+{{{{I}'}}_{zx}}{{L}_{r}} \right) & 0 \\ 0 & 1 & 0 & 0 \\\end{matrix} \right]\left[ \begin{matrix} \beta \\ p \\ r \\ \phi \\\end{matrix} \right]+ \\ & \left[ \begin{matrix} \tfrac{\Delta {{Y}_{c}}}{m} \\ \tfrac{\Delta {{L}_{c}}}{{{{{I}'}}_{x}}}+{{{{I}'}}_{zx}}\Delta {{N}_{c}} \\ \tfrac{\Delta {{N}_{c}}}{{{{{I}'}}_{z}}}+{{{{I}'}}_{zx}}\Delta {{L}_{c}} \\ 0 \\\end{matrix} \right] \\\end{align}$ (1)
where, Y_{v}, Y_{p} and Y_{r} are derivatives of vertical forces concerning side velocity, roll rate and yaw rate, respectively. The subscript “c” designates derivatives concerning δ_{a} and δ_{r}. Y_{v}, Y_{p} and Y_{r} are derivatives of roll moment concerning side velocity, roll rate and yaw rate, respectively. N_{v}, N_{p} and N_{r} are derivatives of yaw moment concerning side velocity, roll rate and yaw rate, respectively. $I_x^{\prime}, I_z^{\prime}$ and $I_{z x}^{\prime}$ are the reformed moments and moment inertia products when the xz is a plane of symmetry. u_{0} is a steadystate velocity, m is aircraft mass ranging from 288, 660255, 740 kg and g is gravity acceleration (9.81 m/sec). V is the aircraft velocity {67.4 m/sec (CI), 157.9 m/sec (CII) and 265.5 m/sec (CIII)}.
The lateraldirectional states and response equations can also be shown as follows:
$\left[ \begin{matrix} {\dot{v}} \\ {\dot{p}} \\ {\dot{r}} \\ {\dot{\phi }} \\\end{matrix} \right]=A\left[ \begin{matrix} v \\ p \\ r \\ \phi \\\end{matrix} \right]+B\left[ \begin{matrix} {{\delta }_{a}} \\ {{\delta }_{r}} \\\end{matrix} \right]$ (2)
$\left[ \begin{matrix} v \\ p \\ r \\ \phi \\\end{matrix} \right]=C\left[ \begin{matrix} v \\ p \\ r \\ \phi \\\end{matrix} \right]+D\left[ \begin{matrix} {{\delta }_{a}} \\ {{\delta }_{r}} \\\end{matrix} \right]$ (3)
where, A and B are system and control matrices of the size of 4×4 and 4×2, respectively. C and D are the output observation matrix and the state transition matrix, respectively. Since the states were taken as system outputs, the C and D would be 4×4 unity and 4×2 nullity matrices, respectively.
$\begin{align} & A= \\ & \left[ \begin{matrix} \tfrac{{{Y}_{v}}}{m} & \tfrac{{{Y}_{p}}}{m} & \left( \tfrac{{{Y}_{r}}}{m}{{u}_{0}} \right) & g\cos {{\theta }_{0}} \\ \left( \tfrac{{{L}_{v}}}{{{{{I}'}}_{x}}}+{{{{I}'}}_{zx}}{{N}_{v}} \right) & \left( \tfrac{{{L}_{p}}}{{{{{I}'}}_{x}}}+{{{{I}'}}_{zx}}{{N}_{p}} \right) & \left( \tfrac{{{L}_{r}}}{{{{{I}'}}_{x}}}+{{{{I}'}}_{zx}}{{N}_{r}} \right) & 0 \\ \left( \tfrac{{{N}_{v}}}{{{{{I}'}}_{z}}}+{{{{I}'}}_{zx}}{{L}_{v}} \right) & \left( \tfrac{{{N}_{p}}}{{{{{I}'}}_{z}}}+{{{{I}'}}_{zx}}{{L}_{p}} \right) & \left( \tfrac{{{N}_{r}}}{{{{{I}'}}_{z}}}+{{{{I}'}}_{zx}}{{L}_{r}} \right) & 0 \\ 0 & 1 & \tan {{\theta }_{0}} & 0 \\\end{matrix} \right] \\\end{align}$ (4)
$B=\left[ \begin{matrix} \begin{matrix} \tfrac{\Delta {{Y}_{{{\delta }_{a}}}}}{m} \\ \tfrac{\Delta {{L}_{{{\delta }_{a}}}}}{{{{{I}'}}_{x}}}+{{{{I}'}}_{zx}}\Delta {{N}_{{{\delta }_{a}}}} \\ \tfrac{\Delta {{N}_{{{\delta }_{a}}}}}{{{{{I}'}}_{z}}}+{{{{I}'}}_{zx}}\Delta {{L}_{{{\delta }_{a}}}} \\ 0 \\\end{matrix} & \begin{matrix} \tfrac{\Delta {{Y}_{{{\delta }_{r}}}}}{m} \\ \tfrac{\Delta {{L}_{{{\delta }_{r}}}}}{{{{{I}'}}_{x}}}+{{{{I}'}}_{zx}}\Delta {{N}_{{{\delta }_{r}}}} \\ \tfrac{\Delta {{N}_{\delta r}}}{{{{{I}'}}_{z}}}+{{{{I}'}}_{zx}}\Delta {{L}_{{{\delta }_{r}}}} \\ 0 \\\end{matrix} \\\end{matrix} \right]$ (5)
where, θ_{0} is the reference trim angle. The lateraldirectional flight behaviour has to be obtained by dynamic analysis and extensive simulations. Timedomain step response is most important to finetune the design and evaluate whether design criteria met targets. The swiftness and tracking accuracy can be found in terms of the step response of a system. Also, the crosscoupled aileron and rudder control significantly produce yawing and rolling moments. Such lateral controls may not be individually helpful in managing the steadystate conditions [1]. The sideslip response, the rectilinear flight path and the turn response, which is the vertical angular velocity vector, represent a lateral steadystate joint application of the aileron and rudder. The H_{∞} tunings of entire state feedback controller gains are implemented to obtain the optimum responses of crosscoupling lateraldirectional variables and to reject such unwanted banking and side slipping disturbances.
The transfer functions (TF) of side velocity, roll rate, yaw rate, bank and sideslip angles to aileron and rudder control inputs were derived as,
$\frac{vpr\phi \beta }{\Delta {{\delta }_{a}}\Delta {{\delta }_{r}}}=\frac{{{a}_{3}}{{s}^{3}}+{{a}_{2}}{{s}^{2}}+{{a}_{1}}s+{{a}_{0}}}{{{b}_{4}}{{s}^{4}}+{{b}_{3}}{{s}^{3}}+{{b}_{2}}{{s}^{2}}+{{b}_{1}}s+{{b}_{0}}}$ (6)
The polynomial characteristic equation of lateral motion can usually be factorized into the following formula:
$\begin{align} & {{b}_{4}}{{s}^{4}}+{{b}_{3}}{{s}^{3}}+{{b}_{2}}{{s}^{2}}+{{b}_{1}}s+{{b}_{0}}= \\ & \lambda \left( \lambda +e \right)\left( \lambda +f \right)\left( \lambda +2\xi {{\omega }_{d}}\lambda +{{\omega }_{d}}^{2} \right) \\\end{align}$ (7)
(a) The term (λ+e) represents spiral convergence mode (SC) with a very sluggish motion during the wings level or ‘roll off’ in a divergent spiral.
(b) The term (λ+f) represents rolling subsidence mode (RS) for proper quicker than the SC mode.
(c) The term $\left(\lambda+2 \xi \omega_d \lambda+\omega_d^2\right)$ represents Dutch roll oscillatory mode (DR) with a small damping ratio.
Also, the simple directional mode at λ=0 represents an aircraft’s heading that has been changed without restoring to an equilibrium. An aircraft may use corrective control to bypass perturbed heading and obtain neutral stability.
2.2 H_{∞} control algorithm
The H_{∞ }sufficient condition considers linear matrix inequality and equality for robust disturbance attenuation. H_{∞} defines the space of all stable linear systems of the ultimate energy gain. The response matrix is searched for singularity over the entire frequency domain [37]. The timeinvariant dynamic output control effort is shown below.
$\left[ \begin{matrix} {{{\dot{q}}}_{1}} \\ {{{\dot{q}}}_{2}} \\ {{{\dot{q}}}_{3}} \\ {{{\dot{q}}}_{4}} \\\end{matrix} \right]=J\left[ \begin{matrix} {{q}_{1}} \\ {{q}_{2}} \\ {{q}_{3}} \\ {{q}_{4}} \\\end{matrix} \right]+L\left[ \begin{matrix} v \\ p \\ r \\ \phi \\\end{matrix} \right]$ (8)
$u(t)=M\left[ \begin{matrix} {{q}_{1}} \\ {{q}_{2}} \\ {{q}_{3}} \\ {{q}_{4}} \\\end{matrix} \right]+N\left[ \begin{matrix} v \\ p \\ r \\ \phi \\\end{matrix} \right]$ (9)
where, $q=\left[\begin{array}{llll}q_1 & q_2 & q_3 & q_4\end{array}\right]^T \in \Re^4$ indicates the controller variable vector. Because A/C control systems are strictly proper, i.e., D = 0, the commands are decoupled from the responses, then the weighting design matrix may be written as:
$K=\left[ \begin{matrix} {\overset{\scriptscriptstyle\smile}{J}} & {\overset{\scriptscriptstyle\smile}{L}} \\ {\overset{\scriptscriptstyle\smile}{M}} & {\overset{\scriptscriptstyle\smile}{N}} \\\end{matrix} \right]$ (10)
where, $K\in {{\Re }^{6\times 8}}$ has the prescribed dynamic wrt the real matrices $\overset{\scriptscriptstyle\smile}{J}\in {{\Re }^{4\times 4}}$, $\overset{\scriptscriptstyle\smile}{L}\in {{\Re }^{4\times 4}}$, $\overset{\scriptscriptstyle\smile}{M}\in {{\Re }^{2\times 4}}$, and $\overset{\scriptscriptstyle\smile}{N}\in {{\Re }^{2\times 4}}$ to be designed where the objective was to design the control law matrix parameters not exceeding a specified limit defined as the guaranteed quadratic performance and optimized in the sense of the TF matrix H_{∞} norm concerning the unknown disturbance. By considering in control only the measured variable output vector y(t) and the impact of the disturbance on y(t) expressed in terms of the quad H_{∞} norm of the transfer function matrix. Accordingly, the control law design was based on the mixed sensitivity approach and mutually enough to produce quadratic performance [37]. Thus, it has to attenuate dynamic disturbances ε→0 and reduce control energy $\sigma \simeq 1$.
$\left\\left(I+\left[\begin{array}{ll}A & B \\ C & 0\end{array}\right] K\right)^{1}\right\_{\infty} \rightarrow \varepsilon$ (11a)
$\left\K\left(I+\left[\begin{array}{ll}A & B \\ C & 0\end{array}\right] K\right)^{1}\right\_{\infty} \rightarrow \sigma$ (11b)
Using Schurz’s complement property then, the inequality matrix implies [37]:
$\left[ \begin{matrix} AQ+Q{{A}^{T}}+BYC+{{C}^{T}}{{Y}^{T}}{{B}^{T}} & {{S}^{T}} & CQ \\ {{S}^{T}} & \gamma {{I}_{S}} & 0 \\ CQ & 0 & {{I}_{CQ}} \\\end{matrix} \right]<0$ (12)
where, $K \in \Re^{2 \times 4}$ signifies an unknown real matrix, Q=Q^{T}>0 and γ>0. Analyzing the matrix element in the upper left corner of Eq. (12) when CQ=HC. Thus, the following expression is reached:
$BKCQ=BKH{{H}^{1}}CQ$ (13)
The control gain matrix may then be written as:
$K=Y{{H}^{1}}$ (14)
The controlled system matrix A_{C} is noted as:
${{A}_{C}}=\left( A+BKC \right)$ (15)
The static output controller is stable with the quadratic performance for a positive scalar $\gamma \in \Re$ if there may be a positive definite symmetric matrix $Q \in \Re^{4 \times 4}$, a systematic matrix $H \in \Re^{4 \times 4}$ and an output matrix $Y \in \Re^{2 \times 4}$. The control policy gives the output action:
$u(t)=\left[ \begin{matrix} {{\delta }_{a}} \\ {{\delta }_{r}} \\\end{matrix} \right]=K\left[ \begin{matrix} v \\ p \\ r \\ \phi \\\end{matrix} \right]+Ww\left( t \right)$ (16)
where, $w(t) \in \Re^4$ is the preferred response signal vector and $W \in \mathfrak{\Re}^{4 \times 4}$ is the signal gain matrix. The static decoupling matrix $W$ is then considered below:
$W={{\left( C{{A}_{C}}^{1}B \right)}^{1}}$ (17)
The W matrix is the inverse of the closedloop gain matrix. Such a former procedure can reasonably track the command value w(t) for slowly enough variations, i.e. y(t), to follow w(t).
2.3 Artificial bee colony
The ABC swarm intelligent optimizer procedure is broadly approached to penalize the design matrices in many largescale applications nowadays. The ABC stages comprise initializing, employing bees, arranging onlooker bees and scouting bees. The quasiABC flowchart is shown in Figure 1. The attained populations are first reset and evaluated; the algorithm is then iterated for the first cycle and follows the diagram, and uses the provided equations to achieve the calculations. The algorithm will be terminated when the design objective is met. Furthermore, the iterations will be kept till the maximum cycles are reached or no adequate evaluation is found.
The produced solutions Λ_{ij} nearby the employed bees can be evaluated by:
${{\Omega }_{ij}}{{\Lambda }_{ij}}={{\Omega }_{ij}}+{{\mu }_{ij}}\left( {{\Omega }_{ij}}{{\Omega }_{kj}} \right)$ (18)
where, i, j and k are i, j and kareas evaluating parameters, and μ is an arbitrary array around minus or plus one.
An evaluation of the likelihood P_{i} can be computed by:
${{P}_{i}}=\frac{{1}/{\left( 1+{{f}_{0}} \right)}\;}{\sum\nolimits_{i=1}^{N}{{1}/{\left( 1+{{f}_{i}} \right)}\;}}\,\,\,\,:\,\,\,{{f}_{i}}\ge 0$ (19)
where, N is several feasible solutions.
Figure 1. ABC algorithm illustration
The unique objective function f_{i} maybe constructed by multiobjective optimization for certain objective functions [21].
${{f}_{i}}=\sum\nolimits_{i=1}^{n}{{{F}_{i}}{{O}_{i}}}$ (20)
where, O_{i} is the design objective requirements holding desired overshoot, settling criteria, and steadystate errors. F_{i} is the control limit residual.
The accessible abandoned evaluation has to be replaced with fresh ones Ω_{i} for the lookout horizon.
${{\Omega }_{ij}}=\min {{\Omega }_{j}}+rand(0,1)\left( \max {{\Omega }_{j}}\min {{\Omega }_{j}} \right)$ (21)
3.1 Uncontrolled lateraldirectional characteristics
B747100 lateraldirectional uncontrolled flight was initially simulated at CI, CII, and CIII. This scenario mimics no pilot interference with the control mechanisms during stickfixed 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 lateraldirectional 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 B747100 models of lateraldirectional coupling variables were obtained with no aileron and rudder controls. Such itemizations gave ten TFs using Eq. (6) for lateraldirectional variables at each Mach number and altitude case concerning the aileron and rudder inputs. The stability of lateraldirectional 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 lateraldirectional B747100 flight cases are shown in Table 2.
Table 1. Uncontrolled B747100 lateral characteristics
FCs 
A 
B 
CI 
$\left[\begin{array}{cccc}0.09 & 0 & 67.36 & 9.81 \\ 0.02 & 0.97 & 0.331 & 0 \\ 0.003 & 0.16 & 0.219 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$ 
$\left[\begin{array}{cc}0 & 0.99 \\ 0.227 & 0.07 \\ 0.025 & 0.15 \\ 0 & 0\end{array}\right]$ 
CII 
$\left[\begin{array}{cccc}0.08 & 0 & 157.9 & 9.81 \\ 0.0010.65 & 0.378 & 0 \\ 0.003 & 0.07 & 0.142 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$ 
$\left[\begin{array}{cc}0 & 2.068 \\ 0.128 & 0.154 \\ 0.017 & 0.39 \\ 0 & 0\end{array}\right]$ 
CIII 
$\left[\begin{array}{cccc}0.061 & 0 & 265.5 & 9.81 \\ 0.005 & 0.46 & 0.282 & 0 \\ 0.004 & 0.02 & 0.142 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$ 
$\left[\begin{array}{cc}0 & 1.233 \\ 0.19 & 0.106 \\ 0.01 & 0.44 \\ 0 & 0\end{array}\right]$ 
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
FCs 
LSs 
a_{0} 
a_{1} 
a_{2} 
a_{3} 
CI 
v 
0.568 
3.027 
1.701 
0 
p 
0 
0.084 
0.078 
0.227 

r 
0.0107 
0.001 
0.01 
0.025 

ϕ 
0.079 
3.026 
0.227 
0 

β 
0.568 
3.026 
1.701 
0 

CII 
v 
0.241 
0.847 
2.69 
0 
p 
0 
0.062 
0.035 
0.128 

r 
0.004 
0 
0.004 
0.017 

ϕ 
0.062 
0.035 
0.128 
0 

β 
0.241 
0.847 
2.69 
0 

CIII 
v 
0.24 
3.595 
1.872 
0 
p 
0 
0.174 
0.036 
0.186 

r 
0.007 
0.001 
0.007 
0.008 

ϕ 
0.174 
0.036 
0.186 
0 

β 
0.24 
3.595 
1.872 
0 
LSs (lateral states)
The coefficients of numerator TF rudder control off at three lateraldirectional B747100 flight cases are shown in Table 3. The coefficients of denominator TF aileron and rudder control off at three lateraldirectional B747100 flight states are shown in Table 4. The characteristic equations are of the fourth order in the s Laplace variable. However, polezero cancellations at the origin may sometimes make the order less than a fourth. The eigenvalues show that lateraldirectional 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 lateraldirectional 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
FCs 
LSs 
a_{0} 
a_{1} 
a_{2} 
a_{3} 
CI 
v 
0.349 
11.53 
11.38 
0.997 
p 
0 
0.197 
0.049 
0.066 

r 
0.028 
0.008 
0.169 
0.151 

ϕ 
0.197 
0.049 
0.066 
0 

β 
0.349 
11.53 
11.38 
0.997 

CII 
v 
1.239 
43.63 
63.49 
2.068 
p 
0 
0.038 
0.115 
0.154 

r 
0.003 
0.0173 
0.291 
0.392 

ϕ 
0.038 
0.115 
0.153 
0 

β 
1.24 
43.63 
63.49 
2.068 

CIII 
v 
1.074 
55.58 
118 
1.231 
p 
0 
0.492 
0.109 
0.106 

r 
0.018 
0.010 
0.227 
0.442 

ϕ 
0.492 
0.109 
0.106 
0 

β 
1.074 
55.58 
118 
1.231 
Table 4. Denominator TF of aileron/rudder control off
FCs 
LSs 
b_{0} 
b_{1} 
b_{2} 
b_{3} 
b_{4} 
CI 
v p r ϕ 
0.03 
0.60 
0.55 
1.3 
1 
β 
2.29 
40.6 
36.8 
86 
68 

CII 
v p r ϕ 
0.01 
0.32 
0.64 
0.9 
1 
β 
1.6 
49.9 
101 
138 
158 

CIII 
v p r ϕ 
0 
0.53 
1.08 
0.66 
1 
β 
0.8 
139.9 
288 
176 
266 
3.2 Controlled lateraldirectional 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 lateraldirectional motion was reevaluated 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 lateraldirectional uncontrolled flight modes have now been moved farther away from the imaginary axis into the left splane. 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 lateraldirectional 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 steadystate 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 wellconverged 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 steadystate is almost 4.3 sec for CIII. The steadystate 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 B747100 lateral characteristics
FCs 
A_{C} 
CI 
$\left[\begin{array}{cccc}1.725 & 6.609 & 20.62 & 12.098 \\ 0.052 & 9.567 & 0.840 & 23.55 \\ 0.259 & 0.066 & 7.603 & 0.862 \\ 0 & 1 & 0 & 0\end{array}\right]$ 
CII 
$\left[\begin{array}{cccc}3.562 & 9.909 & 54.952 & 41.815 \\ 0.225 & 7.485 & 5.722 & 21.52 \\ 0.667 & 0.996 & 19.94 & 7.419 \\ 0 & 1 & 0 & 0\end{array}\right]$ 
CIII 
$\left[\begin{array}{cccc}0.417 & 0.145 & 141.599 & 8.323 \\ 0.035 & 3.460 & 10.532 & 6.025 \\ 0.131 & 0.043 & 44.567 & 0.765 \\ 0 & 1 & 0 & 0\end{array}\right]$ 
Figure 2. Side velocities of lateral controlled flight
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 rollyaw rotation rates incline toward being diminished due to the lateral coupling stick and rudder pedal rather than an unmodelled wind factor may typically hit the vertical tail.
Figure 4. Roll rates of lateral controlled flight
Figure 5. Yaw rates of lateral controlled flight
Figure 6. Sideslip responses of lateral controlled flight
3.3 Controlled lateraldirectional modes
Once stably convergent lateraldirectional flight was achieved under coupling unit step aileronrudder of 1°, the reference input fullstate 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 lateraldirectional B747100 flight cases are shown in Table 6.
Table 6. Numerator TF aileron and rudder control
FCs 
LSs 
a_{0} 
a_{1} 
a_{2} 
a_{3} 
CI 
v 
14,300 
7,381 
1,207 
61.04 
p 
0 
35.67 
24.63 
3.842 

r 
7.129 
209.7 
88.16 
9.29 

ϕ 
35.67 
24.63 
3.842 
0 

β 
14,300 
7,381 
1,207 
61.04 

CII 
v 
94,480 
38,720 
6,475 
177 
p 
0 
309.2 
131.2 
15.82 

r 
20.96 
602.2 
225.1 
33.17 

ϕ 
309.2 
131.2 
15.82 
0 

β 
94,480 
38,720 
6,475 
177 

CIII 
v 
29,560 
17,320 
5,090 
51.32 
p 
0 
95.54 
143.9 
7.578 

r 
2.118 
109.2 
64.77 
18.53 

ϕ 
95.54 
143.9 
7.578 
0 

β 
29,560 
17,320 
5,090 
51.32 
The denominator TF coefficients based on combined aileron and rudder control at three lateraldirectional B747100 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 lateraldirectional flights of dynamic stability modes. The aircraft lateraldirectional motion exhibited respectable responses with low overshoot, short settling time and minor oscillations. The flying quality is then checked for the three lateraldirectional modes. Three wellknown modes related to lateraldirectional motion are already termed RS, SC and DR. These modes were recognized by the period and the damping ratio. Table 8 shows lateraldirectional 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 wellknown CooperHarper 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
FCs 
LSs 
b_{0} 
b_{1} 
b_{2} 
b_{3} 
b_{4} 
CI 
v p r ϕ 
410.4 
354.58 
115.22 
17.342 
1 
β 
27,646 
23,884 
7,761 
1,168 
67.36 

CII 
v p r ϕ 
1,181 
840.2 
231.7 
28.55 
1 
β 
186,500 
132,700 
36,580 
4,507 
157.9 

CIII 
v p r ϕ 
219 
391.9 
198.5 
48.45 
1 
β 
58,130 
104,000 
52,690 
12,860 
265.5 
Table 8. B747100 lateraldirectional modes
FCs 
LMs 
λ 
ξ 
Frequency (rad/sec) 
Period (sec) 
CI 
RS 
6.58 
1.00 
6.58 
0.151 
SC 
2.86 
1.00 
2.86 
0.349 

DR 
3.95±2.49i 
0.846 
4.67 
0.214 

CII 
RS 
18.1 
1.00 
18.1 
0.055 
SC 
3.55 
1.00 
3.55 
0.281 

DR 
3.44±2.56i 
0.802 
4.29 
0.233 

CIII 
RS 
44.14 
1.00 
44.1 
0.022 
SC 
0.851 
1.00 
0.850 
1.176 

DR 
1.72±1.69i 
0.714 
2.42 
0.413 
LMs (lateral modes)
3.4 Flying quality assessments
Figure 7. DR flying qualities of bankrolling control
Figure 8. DR flying qualities of sideslipping control
The flying quality properties for the DR mode of B747100 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 B747100 lateraldirectional bankrolling controlled flight. The 3D surface represents ξω_{d} the vertical axis, the bankroll ϕ alongside the crosshorizontal 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 B747100 lateraldirectional sideslipping controlled flight. The 3D surface represents ξω_{d} alongside the vertical axis, sideslip β alongside the crosshorizontal 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 B747100 sideslip turn of three cases of steadystate 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. B747100 sideslip turn comparison
FCs 
β(º) 
ϕ (º) 
((β/ϕ))/Δδ 
CI 
0.0974 
0.0862 
1.1309 
CII 
0.0737 
0.0063 
11.692 
CIII 
0.1162 
0.0326 
3.565 
For benchmarking the sideslip dampening responsiveness, Figure 9 qualitatively compares the sideslip response obtained by the H_{∞}ABC synthesis and typical DR manoeuvre flighttime 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 510 sec for the DR flight test lasted 1520 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 sidesliptobankroll from 14.625 to 1.479 within the flight envelope ranging from CIII to CI cases studied here.
Figure 9. DR sideslip responsiveness validation
3.5 Sensitivity assessments
The steadystate 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 onebyone 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 tradeoffs amongst the overshot, settling epoch, the steadystate 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 (O_{1}<2% for desired overshoot, O_{2}<7% required settling time, and O_{3}<±1.7% the desired steadystate error). They had to be away from s=0, avoiding the ABC failure and H_{∞ }destabilization. Fitness residuals of F_{1}=0.6, F_{2}=0.4, and F_{3}=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 steadystate 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
LSs 
Constraints 
Fitness 

O_{1} 
O_{2} 
O_{3} 
F_{1} 
F_{2} 
F_{3} 

v 
2 
7 
1.7 
0.6 
0.4 
0.8 
p 
1.8 
5 
1.4 
0.5 
0.7 
0.6 
r 
1.6 
4 
1.8 
0.6 
0.7 
0.9 
ϕ 
1.1 
4 
1.3 
0.8 
0.7 
0.6 
β 
1.1 
6 
1.5 
0.7 
0.6 
0.7 
Table 11. ABC cost functions’ accuracy
CF 
Optimizer 
WS 
BS 
AS 
SD 
f_{1} 
TER [33] 
1.18E02 
1.24E3 
1.73E02 
1.43E02 
ABC 
1.67E03 
1.35E6 
1.87E05 
1.31E05 

f_{2} 
TER [33] 
3.76E02 
4.47E3 
1.07E2 
2.48E2 
ABC 
4.48E04 
3.86E7 
3.60E6 
7.72E5 

f_{3} 
TER [33] 
1.93E02 
2.57E3 
4.62E3 
2.48E2 
ABC 
1.25E04 
8.97E6 
3.76E5 
3.91E4 
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)
The lateral flight modes significantly vary as the stability derivatives do with Mach number and altitude. Those modes normally behave at lower altitudes and have asymmetrical dampness at higher altitudes and Mach number. When substantial aeroelasticity and compressibility effects are added, more irregular lateral modes could be experienced. Additional freedom is admitted in guaranteeing the output feedback control quadratic performance concerning roll and yaw disturbances during the lateraldirectional B747100 flight. The H_{∞}SAD control design tasks, which are solvable numerical problems, sufficiently opt for the Lyapunov stability of the closedloop aircraft design. It minimizes the closedloop lateral flight responses of bankroll and sideslip perturbations. An ABC optimizer gives robust performance and stabilization of those lateral characteristics like the wellknown loopshaping approach [4, 37]. There is no guarantee that such synthesis behaves well for the proper plant encountered in practice as did here for the nominal plant design. Many other flight disruptions may encounter causing sideslip and bankroll perturbations such as wind, faulty stabilizer and engine inoperative not being included here. An adequate response may be essential as the H_{∞} and ABC synthesis is flexible enough to handle addon flight conditions. The ABC algorithm collaborates well to ensure the H_{∞}SAD robust performance and consequently augment the stability and robustness. The Dutch roll modes reasonably meet (damping ratio × damping ratio × undamped frequency) flying qualities for the flight cases studied. The effectiveness of the controllers is pronounced for a safe recovery to routine flight from perturbed lateral conditions, particularly sideslip and roll bank disturbance attenuations. However, the flying quality of roll convergence and spiral modes may be further enhanced. Overall, 3D response surfaces, which signify tradeoff patterns of the DRmode flying quality versus the bankroll or sideslip freedisturbance responses and the flight conditions, indicate the minimum merits of flying qualities (ξ×ω_{d} ≥ 0.1 rad/sec) for wide flight missions met [35].
As might be noticed, peak transitional behaviours were produced for the yaw rate and roll rate responses, further identifications of whether the nature of the H_{∞} method or the ABC algorithm or even the complete synthesis is behind such biased responses. Fortunately, these temporary behaviours quickly died out, and nominal roll and yaw rates were produced for three flight cases. Furthermore, stable, robust performance could also be analyzed using the gain scheduling or μcontrol approaches through the whole lateral flight envelope. The YK parametrization might also be used to meet addon criteria of unmodelled aircraft dynamics.
Thanks to the Mobility and Transport Research Centre (MTRC), Coventry University, the UK, for valuable supportive consultations. The University of Tripoli supports this research.
a_{0} 
transfer function numerator coefficient of s^{0 }term 
a_{1} 
transfer function numerator coefficient of s term 
a_{2} 
transfer function numerator coefficient of s^{2} term 
a_{3} 
transfer function numerator coefficient of s^{3} term 
A 
aircraft system matrix 
ABC 
Artificial bee colony 
A_{C} 
controlled aircraft matrix 
A/C 
aircraft 
b_{0} 
transfer function denominator coefficient of s^{0} term 
b_{1} 
transfer function denominator coefficient of s term 
b_{2} 
transfer function denominator coefficient of s^{2} term 
b_{3} 
transfer function denominator coefficient of s^{3} term 
b_{4} 
transfer function denominator coefficient of s^{4} term 
B 
aircraft stability derivative matrix 
C 
output observation matrix 
D 
state transition matrix 
DR 
Dutch roll oscillatory mode 
e 
the eigenvalue of spiral convergence mode 
f 
eigenvalues of rolling subsidence mode 
f_{0} 
baseline objective function 
f_{i} 
unique objective function 
F_{i} 
control limit residual for design variables 
g 
gravity acceleration, m.sec^{2} 
GA 
genetic algorithm 
h 
altitude, m 
H 
systematic matrix 
H_{∞} 
Hinfinity 
H_{∞}SAD 
Hinfinity stability augmentation design 
I_{CQ} 
inequality matrix of CQ dimension 
I_{S} 
inequality matrix of S dimension 
${{{I}'}_{x}}$ 
the reformed moment around the xaxis, kg.m^{2} 
${{{I}'}_{z}}$ 
the reformed moment around the zaxis, kg.m^{2} 
${{{I}'}_{zx}}$ 
xz symmetry plane moment inertia product, kg.m^{2} 
$\overset{\scriptscriptstyle\smile}{J}$ 
first design matrix of 4 by 4 size 
$K$ 
real matrix dynamics 
$\overset{\scriptscriptstyle\smile}{L}$ 
second design matrix of 4 by 4 size 
${{L}_{p}}$ 
roll moment derivative with respect to roll rate, kg.m^{2}/sec/rad 
${{L}_{r}}$ 
roll moment derivative with respect to yaw rate, kg.m^{2}/sec/rad 
${{L}_{v}}$ 
roll moment derivative with respect to side velocity, kg.m^{2}/sec/rad 
LQR 
linear quadratic regulator 
LQG 
linear quadratic Gaussian 
$m$ 
aircraft mass, kg 
M 
Mach number, dimensionless 
$\overset{\scriptscriptstyle\smile}{M}$ 
third design matrix of 2 by 4 size 
$n$ 
the upper limit of search space 
$\overset{\scriptscriptstyle\smile}{N}$ 
forth design matrix of 2 by 4 size 
N 
likely solutions number 
${{N}_{p}}$ 
yaw moment derivative with respect to roll rate, kg.m^{2}/sec/rad 
${{N}_{r}}$ 
yaw moment derivative with respect to yaw rate, kg.m^{2}/sec/rad 
${{N}_{v}}$ 
yaw moment derivative with respect to side velocity, kg.m^{2}/sec/rad 
${{O}_{i}}$ 
design objectives 
$p$ 
rolling rate, rad.sec^{1} 
${{P}_{i}}$ 
likelihood function 
PID 
proportionalintegralderivative 
$q$ 
controller variable vector 
${{q}_{1}}$ 
first controller variable 
${{q}_{2}}$ 
second controller variable 
${{q}_{3}}$ 
third controller variable 
${{q}_{4}}$ 
fourth controller variable 
$Q$ 
positive definite symmetric matrix 
$r$ 
yawing rate, rad.sec^{1} 
RS 
rolling subsidence mode 
s 
Laplace variable 
SAD 
stability augmentation design 
SC 
spiral convergence mode 
TER 
trail and error rule 
TF 
transfer function 
${{u}_{0}}$ 
steadystate velocity, m.sec^{1} 
$u\left( t \right)$ 
control law 
$v$ 
side velocity, m.sec^{1} 
$V$ 
aircraft velocity, m.sec^{1} 
$w\left( t \right)$ 
preferred response signal vector 
$W$ 
static decoupling signal gain matrix 
$Y$ 
output matrix 
${{Y}_{p}}$ 
normal force derivative with respect to roll rate, kg.m/sec/rad 
${{Y}_{r}}$ 
normal force derivative with respect to yaw rate, kg.m/sec/rad 
${{Y}_{v}}$ 
normal force derivative with respect to side velocity, kg.m/sec/rad 
Greek symbols 

$\beta $ 
sideslip angle, rad, deg. 
$\gamma $ 
positive scalar 
${{\delta }_{a}}$ 
aileron control action 
${{\delta }_{r}}$ 
rudder control action 
$\Delta $ 
perturbation 
$\varepsilon $ 
dynamic disturbance parameter 
${{\theta }_{0}}$ 
reference trim angle, rad, deg. 
$\lambda $ 
eigenvalue, dimensionless 
${{\Lambda }_{ij}}$ 
solutions nearby the employed bees 
$\mu $ 
arbitrary array 
$\xi $ 
damping coefficient, dimensionless 
$\sigma $ 
control energy parameter 
$\phi $ 
bankroll angle, rad, deg. 
${{\omega }_{d}}$ 
undamped natural frequency, rad.sec^{1} 
${{\Omega }_{j}}$ 
fresh abandoned evaluation of j element 
${{\Omega }_{ij}}$ 
abandoned evaluation of $ij$ neighbour 
${{\Omega }_{kj}}$ 
abandoned evaluation of $kj$ neighbour 
Subscripts 

0 
steadystate 
$i$ 
iarea evaluating parameter 
$j$ 
jarea evaluating parameter 
$k$ 
karea evaluating parameter 
$c$ 
derivatives with respect to ${{\delta }_{a}}$ and ${{\delta }_{r}}$ 
$p$ 
rolling rate 
$r$ 
yawing rate 
$v$ 
side velocity 
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