Large-Slenderness-Ratio Supersonic Aircraft Lateral – Directional Coupling Dynamics Analysis and Control

Large-slenderness-ratio aircraft in which a large portion of the aircraft ’ s mass is concentrated along its centreline face the problem of severe lateral-directional coupling. Previous research has not fully considered the inertia product when conducting stability analyses of such aircraft. However, neglecting the inertia product may threaten the safety of the aircraft during high-speed ﬂ ight. This paper investigates the e ﬀ ect of the inertia product on the lateral – directional stability. Two methods, one based on eigenstructure assignment (EA) and the other based on an extended state observer (ESO), are developed for the lateral – directional decoupling of a research large-slenderness-ratio aircraft. Simulation results show that both methods can realize decoupling control. The EA-based control achieves better decoupling performance, whereas the ESO-based control has an advantage in terms of disturbance rejection. Finally, the ESO-based attitude controller is validated through a high subsonic ﬂ ight test. The decoupling and tracking performance of the attitude control demonstrate the e ﬀ ectiveness and practicality of the ESO-based controller.


Introduction
High-speed aircraft are faced with the problem of coupling dynamics, especially those configured with a large slenderness ratio (LSR). Historically, these coupling dynamics have resulted in various accidents [1]. In recent years, hypersonic vehicles have experienced rapid development, and an LSR is a common characteristic of these aircraft [2]. To ensure the flight safety of LSR aircraft, stability assessments and control design methods need to be developed.
With large portions of their mass concentrated along the centreline of the airplane, LSR vehicles with symmetric structure often exhibit undesirable stability and control characteristics due to coupling dynamics in the high subsonic, transonic, and supersonic flight regime [1]. The coupling dynamics can be catastrophic in the flight. In order to predict the departure characteristics arising from the coupling dynamics, significant research efforts have been conducted. In 1958, Moul and Paulson [3] proposed the concept of a lateral departure parameter (C nβdyn ) and lateral control departure parameter (LCDP) for high-performance aircraft. In 1966, Lutze et al. [4] presented a unified analytical approach based on the linearized equations of motion for calculating the deviation of departure-predicting parameters. Weissman [5] combined C nβdyn with LCDP and divided the plane into areas associated with different departure characteristics. This work gave rise to the "Weissman chart," which is widely used in assessing the departure characteristics of high-speed aircraft [6,7]. In 2007, Lee et al. proposed a criterion for estimating the optimum lateral static stability margin through experimental research and validated this criterion using flight tests [8]. Previous research has focused on the evaluation and prediction of the lateral-directional stability and departure characteristics of aircraft and has successfully explained the departure phenomenon arising from the coupling dynamics. For symmetric aircraft without the LSR, the inertia product is relatively small compared with the inertia moment around the body roll axis. Therefore, little attention has been paid to the inertia product in the coupling dynamics analysis, and the value of the inertia product is often ignored in the previous research. However, ignorance of the inertial product in the stability analysis and flight control law design will pose a potential risk to the LSR aircraft flight safety.
Many studies have examined the decoupling control due to the undesired coupling dynamics of aircraft. For example, an adaptive flight control law was developed to enforce performance across all flight conditions of an X-15 aircraft, although the severe disturbance introduced by mechanical nonlinearities and electrical saturation resulted in divergent airplane motions [9,10]. For the lateral-directional decoupling control of an F-18 with a high angle-of-attack [11], nonlinear dynamic inversion (NDI) in a multivariable system has achieved good decoupling performance. The disadvantage of NDI-based control is that accurate knowledge of the nonlinear dynamics is required [12], but this is very difficult to obtain. To reduce the dependence on the model information, incremental (INDI) was developed to realize three-axis decoupling control of a novel unmanned tiltrotor [13]. For the multiple-input-multiple-output (MIMO) design method based on a linearized model, eigenstructure assignment (EA) exhibits significant advantages in the decoupling control due to the capability of assigning the eigenvalues and eigenvectors simultaneously [14,15]. The EA-based controller is widely used in lateral-directional decoupling control, especially for tailless aircraft. The decoupled control of a flying-wing aircraft has been achieved under consideration of the necessary flying qualities [16,17], and EA-based decoupling control has been validated in a wind tunnel [18]. However, EA-based control requires gain scheduling for aircraft with large flight envelopes. In recent years, extended state observer-(ESO-) based controllers [19] have displayed great potential for decoupling control. The decoupling capability of ESO-based control results from treating the coupling dynamics and uncertainties as a disturbance to be estimated and then compensated. For the decoupling attitude control of a hypersonic glide vehicle, Chen et al. [20] used a nonlinear extended observer, which has low storage and computational requirements compared with the traditional gain scheduling. For outer-loop trajectory control, altitude and velocity decoupling control have been achieved with ESO in simulations [21]. Although plenty of simulations have validated the effectiveness of ESO-based control, few high-speed aircraft have been reported to use this control method in real flight tests; thus, the engineering practice of ESO-based decoupling control needs to be further researched and developed.
Faced with the problems of coupling characteristics including the inertial roll coupling, control coupling, Dutch-roll coupling, and so on [1], the lateral-directional dynamics of an LSR aircraft are analyzed considering the inertial product, and then, the decoupling control law is designed. First, the configuration of the research LSR aircraft is described, and the comparison between various couplingprone aircraft in history and the research LSR aircraft is given which shows the existence of severe coupling characteristics. In addition, a nonlinear model of the lateral-directional coupling dynamics is established. Second, the coupling dynamics are analyzed based on C nβdyn , LCDP, and the roll-yaw coupling parameter. Different from the previous work above which ignores the inertial product in analysis, the effect of the inertia product is emphasized, and a stability criterion related to the inertia product is firstly derived based on linear analysis. Besides, the control coupling with the inertial product is also dealt with. Third, EA-based MIMO and ESO-based single-input-single-output (SISO) control laws are designed for lateral-directional decoupling control with the hard cross-connection. Fourth, the linear analysis based on the robust control theory of the EA-based controller is performed, and the necessary flying qualities are shown to be satisfied. The Monte Carlo (MC) simulations are conducted and the root-mean-square (RMS) of the tracking error is used to compare the control performance between the ESO-based and EA-based controllers. To further evaluate the disturbance rejection characteristics of the two controllers, gusts of wind are introduced to the simulation. The advantages and disadvantages of the MIMO and SISO design methods are discussed for a better selection of these two methods. Finally, the ESO-based controller is validated in a real flight test. As a result, the proposed control structure can effectively realize decoupling control for high-speed LSR aircraft in engineering practice which provides some new ideas.
The remainder of this paper is organized as follows. The aircraft dynamics model and coupling dynamics analysis are presented in Section 2. The two decoupling control strategies based on EA and ESO are described in Section 3. The simulation results and comparisons with parameter perturbations and wind gusts are presented in Section 4. Section 5 discusses the results of the flight test, while Section 6 states the conclusions to this study.

Lateral-Directional Coupling
Dynamics Analysis 2.1. Configuration Description. In terms of aircraft design, the lateral static stability of an aircraft is mainly affected by the dihedral and sweep angle of the wing [22]. As depicted in Figure 1, the research LSR aircraft has a large sweep angle, which improves the flight qualities during supersonic flight. In addition, the research LSR aircraft uses ailerons, elevators, and the rudder for roll, pitch, and yaw control, respectively. To increase the moment of inertia around the x-axis, additional weights are attached to the wingtips of the aircraft. The aircraft parameters are summarized in Table 1.
Historically, various aircraft with the LSR configuration have suffered from coupling problems. Table 2 compares the mass properties between several coupling-prone aircraft [1] and the research LSR aircraft considered in this paper.
As shown in Figure 2, the angle of inclination of the principal axis is calculated as [1] In Table 2, the inclination angle of the research LSR aircraft is between that of the shuttle and the F-100A 2 International Journal of Aerospace Engineering aircraft. The former experienced control coupling and Dutch-roll coupling during the entry phase, while the latter had serious problems with inertia roll coupling. The value of ðI zz − I xx Þ/I yy among all these coupling-prone aircraft is around 1 because I xx is relatively small and I yy and I zz have very similar values. The value of I xz /I xx is 0.67 for the LSR aircraft, which is greater than the equivalent value of all other aircraft in Table 2 except for the X-3. In 1954, the X-3 aircraft showed a series of unexpected, extremely violent motions while testing rudder-fixed aileron rolls. Note that the LSR aircraft has the maximum value of I zz /I xx , which is about twice that of the X-3. The coupling dynamics of the research LSR aircraft must be taken into consideration because the poor mass properties may cause coupling problems.

Lateral-Directional
Cross-Coupled Dynamics. The nonlinearities induced by the pitch motion are ignored in the lateral-directional coupling dynamics because of the small angle-of-attack over the whole flight mission. Compared with longitudinal motion, the lateral-directional dynamics of the research LSR aircraft are more complicated due to the complex inertial and aerodynamic interactions. For flight safety, the instability and divergence of the lateral-directional modes should be analyzed in the high-subsonic and transonic flight regimes [23], where the center-of-pressure position changes dramatically under the unsteady flow field [24]. Based on computational fluid dynamics (CFD), an accurate aerodynamic model is built using a highdimensional linear interpolation table. In terms of the inertial parameters, the nonlinear lateral-directional dynamic equations are [25] _ where L, M, and N are the roll, pitch, and yaw moments induced by the aerodynamic forces, respectively; p, q, and r are the angular velocity for the body roll, pitch, and yaw axes; α and β are the angle-of-attack and the sideslip angle, respectively; ϕ, θ, and ψ are the roll, pitch, and yaw Euler angles, respectively; Y is the side force acting on the aircraft; and V is the true airspeed. The main feature of Equation (2) is that the roll and yaw dynamics are severely coupled with each other. The coupling dynamics are composed of aerodynamic coupling, control coupling, motion coupling, and inertial coupling [7]. Based on the small disturbance theory, the nonlinear model in Equation (2)    |fflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflffl ffl} where g is the gravitational acceleration; δ a and δ r are the aileron and rudder deflection angles, respectively; A lat is the system matrix; and B lat is the control input matrix. The parameters of the moments and forces induced by the sideslip angle, roll angular rate, and yaw angular rate are defined as  Figure 1: Plan view of the research LSR aircraft.

International Journal of Aerospace Engineering
where i ∈ fβ, p, r, δ a , δ r g. For level flight, the assumption that θ = α can be made.

Stability and Departure Analysis with Inertial Product.
In general, static stability and dynamic stability are considered in the stability and departure analysis [4]. The former requires only the knowledge of aerodynamic characteristics, while the latter requires the linear system defined in Equation (3) to be analyzed. All the dynamic stability criteria based on linear system analysis can be derived using the Routh-Hurwitz stability criterion, based on the coefficients of the characteristic polynomial. For simplicity, the effect of the aerodynamic forces and moments resulting from the angular rates and the acceleration of gravity are disregarded.
The simplified system matrix A s is given as : ð5Þ The characteristic equation of A s can be written as where λ is the eigenvalue associated with the system matrix.
According to the Routh-Hurwitz stability criterion, the stability requirements are The first term in Equation (7) naturally holds because the side force coefficient of the sideslip angle C sβ is negative. The second term can be expanded and simplified as where C lβ and C nβ are the rolling and yawing static stability parameters for the sideslip angle, respectively. Equation (8)  Figure 2: Angle of inclination of the principal axis. 4 International Journal of Aerospace Engineering defines the dynamic lateral-directional stability parameter C nβdyn . Note that the inertia product I xz has been ignored in previous research; however, neglecting this parameter may introduce a significant bias to the LSR aircraft. The dynamic lateral-directional stability parameter of the research LSR aircraft is calculated along the flight trajectory of the mission, and the center of gravity (CG) position is located at a reference point, as shown in Figure 3(a). There exists an approximate 20% bias in the results when the value of I xz is not introduced, and the results considering I xz are more conservative. For high-subsonic or transonic flights, the trim value of the angle of attack is quite small, especially for relatively lightweight vehicles such as the research LSR aircraft. The following inequality can be derived from Equation (8):    As depicted in Figure 3(b), C lβ < 0 and C nβ > 0 for all Mach numbers, confirming the static stability of rolling and yawing aerodynamic performance. Under the small angle-of-attack assumption, the necessary condition for the inertia product so that lateral-directional stability is satisfied can be concluded as To further evaluate the effect of the inertia product on the stability of lateral-directional modes, the research LSR aircraft model is trimmed at a steady level-flight condition with an altitude of H = 10,000 m and a speed of Ma = 0:8. Figure 4 depicts the movement of the eigenvalues at the root locus for the LSR vehicle, where the multiples of the inertia product are in the range 0.5-4.
In Figure 4, the frequency of the Dutch-roll mode decreases with increasing inertia product; in contrast, the frequency of the roll subsidence mode increases. The Dutch-roll mode tends to become unstable as the multiples of the inertia product increases and crosses the imaginary axis at a value of 2.3. Table 3 presents the effect of the wingtip additional weight (WAW) on the mass properties. The value of I xz /I xx is 2.0 without the WAW, which is close to the boundary of stability. Furthermore, the ratio of I xz /I xx is sensitive to the WAW: adding a weight of 2.5 kg results in a threefold increase. Considering the uncertainties of the inertia product, some allowance must be given when designing the aircraft weight distribution.
To evaluate the sensitivity of the Dutch-roll mode to the inertia product in different flight phases, the Dutch-roll mode with respect to changes in the inertia product and Mach number is shown in Figure 5. For the subsonic (Ma < 0:8) and supersonic (Ma > 1:2) flight phases, the Dutch-roll mode is less sensitive to the inertia product and remains stable over the range of multiples considered here. However, the Dutch-roll mode becomes unstable at multiples of around 2 in the transonic phase. The conclusion is that the inertia product should be the focus of attention when analyzing the stability of the Dutch-roll mode in the transonic flight phase.
Another widely used indicator for roll departure susceptibility is the LCDP [26], which can be derived using the roll angle and roll angle rate feedback to the aileron. The closed form of the system matrix A cl with roll angle feedback is where k is the feedback gain. Similar to the derivation process for Equation (6), the Routh-Hurwitz stability criterion in Equation (12) can be established; the detailed derivation is shown in Reference [4].
If we retain the inertia product, the inequality can be expanded as As shown in Figure 6(a), the LCDP calculated using Equation (13) is about 5% less than the value without the  International Journal of Aerospace Engineering inertia product. Weissman's chart [6] gives a criterion combining the LCDP and C nβdyn . The research LSR aircraft is in region A in the flight phases of the mission, which indicates no roll departure, as shown in Figure 6(b). Besides the LCDP and C nβdyn , the roll-yaw coupling is evaluated as a metric of flight quality. The amplitude ratio of the bank angle over the sideslip angle quantifies the roll reaction to a disturbance in the sideslip angle. An approximation of the roll-yaw coupling can be written as [27] The roll-yaw coupling prevents the flying qualities requirements from being satisfied when the flight speed is less than 1.1 Ma [28], having a maximum value of 27 considering the inertia product and 18 without the inertia product, as shown in Figure 7. Without considering the inertia product, Equation (14) can be simplified as In Equation (15), the roll-yaw coupling is proportional to the value of I zz /I xx , which reflects the slender configuration of the aircraft. Higher values of I zz /I xx indicate worse roll-yaw coupling of the aircraft. As depicted in Figure 7, the roll-yaw coupling value decreases as the Mach number increases. This is because C nβ and the absolute value of C lβ decrease, as shown in Figure 3(b). In addition, the negative effect of the inertia product on the roll-yaw coupling weakens as the Mach number increases, allowing the flight quality requirements to be met in the supersonic flight phase. Thus, to avoid roll departure in the subsonic flight phase, the sideslip angle should be suppressed.

Control Coupling Analysis.
During the mission flight phase, the ailerons and rudder are used to control the roll and yaw motions, respectively. However, the control coupling between the ailerons and rudder threatens the flight safety through the undesired moment generated by the deflection in each control channel. As shown in Figure 8, there exists a strong coupling effect between the ailerons and the rudder (C lδ a and C nδ a are the roll yaw control coefficients of the ailerons, respectively; C lδ r and C nδ r are the roll and yaw control coefficients of the rudder, respectively).
According to Equation (3), the control matrix for the roll and yaw angular acceleration is where b ij is the element in the i-th row and j-th column of the control matrix and m is the number of inputs. The parameter c i characterizes the degree of control coupling-larger values indicate more serious coupling. Consider the parameter c 1 as an example, which represents the control-coupling characteristics of the roll channel.
where L δ a = qS bC lδ a I xx , in which q is the dynamic pressure. Equation (19) can be simplified as As shown in Figure 9, the control-coupling characteristics of the roll channel get worse in the transonic flight phase, resulting in considerable control deficiencies. In contrast, the yaw control coupling effect decreases as the Mach number increases. The value of jC δ r /C nδ a j plays an important role in roll control coupling at small values of I xz /I zz . According to Figure 8(a), the absolute value of C lδ r dominates the roll control channel when the Mach number is greater than 0.9, which is undesirable for decoupled control. Therefore, it is difficult for the aircraft to maintain independent roll and yaw manipulation without the coordinated deflection of the control surfaces.

Attitude Decoupling Control
Motivated by the above analysis, a lateral-directional decoupling controller for the attitude control of the research LSR aircraft is now designed. The difficulties can be summarized as follows: First, a high roll-yaw coupling characteristic requires the suppression of the sideslip angle during the whole flight mission. Second, the sensitivity of the Dutchroll mode stability to the inertia product should be taken into consideration, especially in the transonic flight phase. Third, the crossed-coupling between the ailerons and the 3.1. MIMO Design Using EA. The EA control method is widely used in the control augmentation system (CAS) design to realize the roll-yaw decoupling control [16]. The output feedback EA is applied here; for the detailed mathematic theory, see reference [29]. First, the state-space model in Equation (3)    |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} where U 0 is the trim velocity. Note that the control-coupling parameters N δ a and L δ r are disregarded here and will be dealt with in Section 3.3. The modification of the control matrix in Equation (21) separates the control decoupling process from the EA design process, resulting in a more specific target in every design step. To realize tracking control for the lateral acceleration and bank angle, two integrator states are introduced into the augmented lateral-directional coupled model. The extended system matrix and control input matrix are where H = 1 0 0 0 Because two integrator states are introduced, the output matrix must be extended as where the states e 1 and e 2 are the integrators of the tracking error for the lateral acceleration and bank angle, respectively. The eigenvalues of the closed loop are the Dutch-roll mode, roll mode, spiral mode, and command tracking integrators. According to the specifications in reference [28], the desired eigenvalues can be chosen to meet the requirement of level 1 flight quality. In addition, the desired eigenvectors V d of the closed loop should be assigned, as follows: where v DR , v R , and v SP are the desired eigenvectors of the Dutch-roll mode, roll mode, and spiral mode, respectively; v e 1 and v e 2 are the desired eigenvectors of the integrators for the tracking error of the lateral acceleration and bank angle, respectively; and x represents the unrestricted state elements, whereas "0" and "1" denote constrained elements. As shown in Equation (24), the number of outputs is equal to the number of eigenvectors that must be assigned, and so the desired closed-loop eigenvalues can be determined [30]. Furthermore, the tracking speed of the lateral acceleration and bank angle commands depends on the eigenvalues of the integrator modes. Theoretically, the controller can achieve an arbitrary fast rate of error convergence if the poles of the integrators are assigned far away from the imaginary axis in the left-half plane. However, the eigenvalues of the integrators must be chosen according to the control power limit and the rate limit of the effectors. Following reference [16], the stabilization feedback gain K and the command tracking gain L can be obtained. The control diagram of the lateral-directional decoupling controller is shown in Figure 10, in which the parameters ϕ c and a yc are the bank angle and lateral acceleration command values, respectively.

SISO Design Using ESO.
Another control method based on the SISO design method is established to realize decoupled control for the lateral-directional dynamics of the research LSR aircraft. This controller divides the lateral-directional motion of the aircraft into fast-speed inner loops and low-speed outer loops and then uses the timescale-separation principle to realize attitude control. First, 9 International Journal of Aerospace Engineering the lateral acceleration and bank angle commands are given to the outer loops, which generate the yaw and roll angular rate commands for the inner loops. Second, the inner loops realize precise angular rate tracking commands using the rudder and ailerons.
For the angular rate control loops, the equations of the dynamics can be written as where f p and f r are the lumped disturbance of the roll and yaw angular rate control loops, including the internal coupling dynamics, model uncertainties, and environmental disturbances. In addition, the derivatives of f p and f r are assumed to be unknown but bounded [31], as in the following equation: where L and N are the roll and yaw moments imposed on the aircraft, respectively, which are composed of the moments induced by the aerodynamic effectors and wind disturbance. In the ESO-based controller, the inner-loop coupling dynamics are regarded as the disturbance that must be estimated and then compensated in real time. For simplicity, the original nonlinear ESO is replaced with a linear form that simplifies the adjustment of the control parameters [32]. However, the nonlinear ESO has some advantages in terms of low peaking levels and more accurate estimation [33]. A second-order nonlinear ESO is used in the angular rate control loop: where z 1 and z 2 are the estimations of the angular rate and lumped disturbance for each control channel, respectively; y is the measurement value of the angular rate; b 0 is the parameter related to the control efficiency; and β 1 and β 2 are the ESO parameters, which need to be tuned carefully.
The nonlinear function f al is defined as where 0 < α < 1 and δ > 0 are the observer parameters, and the sign function is defined as The control law of the angular rate control loop is where u 0 is the output of the proportional controller, given by where ω c ∈ ½p c , r c denotes the roll and yaw angular rate command values; ω ∈ ½p, r denotes the roll and yaw angular rate measured values; k ∈ ½k p , k r represents the gains of the proportional controller, where k p and k r are the closed-loop control bandwidths of the roll and yaw channels, respectively. According to Equation (31), the original system in Equation (26) can be converted to a pure integrator system under the assumption that the lumped disturbance is accurately estimated using ESO. As a result, the closed loop of  Figure 10: Lateral-directional EA control loop. 10 International Journal of Aerospace Engineering the angular rate control can be regarded as a first-order transfer function: For the stability analysis of the ESO-based controller, Guo et al. first gave the convergence analysis for the nonlinear systems with uncertainty [34]. The system in Equation (26) can be rewritten as where x 1 is the state variable; x 2 is the lumped disturbance, unknown but bounded and continuously differentiable is assumed; and w is the derivative of the lumped disturbance which satisfies jwj < w H according to Equation (27). Combing the second-order nonlinear ESO in Equation (28), the equations of estimation error can be constructed as According to reference [35], the estimation errors e 1 and e 2 are bounded when the observer gain β 1 and β 2 are adjusted properly. If the parameter α is equal to one, the function f al becomes a linear form, and the error bound becomes jwj/β 2 . However, the error bound becomes ðjwj/β 2 Þ 2 using the nonlinear form in Equation (29) which shows the efficiency of the nonlinear ESO, and the value of β 2 needs to be bigger than w H .
For the outer-loop bank angle control, NDI is used. The kinematic relationship between the bank angle and roll angular rate is given by Equation (2). The bank angle rate is replaced by a virtual control v ϕ , which is defined as where the parameter k ϕ is the closed-loop control bandwidth of the bank angle. Furthermore, the roll angular rate command value can be derived as According to Equation (3), the transfer function from the yaw angular rate to the sideslip angle can be obtained without considering gravity, inertial coupling, or damping effects, as shown in the following equation. where in which C Yβ < 0 is the side force coefficient induced by the sideslip angle. The relationship between the lateral acceleration and the sideslip angle can be written as According to Equation (33), the inner closed loop can be simplified as a first-order inertia element, and a proportional-integral (PI) compensator is used for tracking control of the lateral acceleration, as depicted in Figure 11.
With PI compensation, the closed transfer function for the lateral acceleration command to the measured output is a second-order nominal model that can be expressed as The damping ratio is selected as 0.8, and the controller gain k a is calculated as In addition, the yaw angular rate command is compensated using the maneuvering turn angular rate for coordinate flight, given by where Ω c is the maneuvering turn rate, V g is the ground velocity, and r co is the compensated yaw angular rate command value. The overall control structure of the lateraldirectional decoupling control based on ESO is shown in Figure 12.

Hard Cross-Connection.
According to the controlcoupling analysis in Section 2.4, there exists a strong coupling between the roll and yaw control effectors. To overcome the control coupling between the ailerons and rudder, a hard cross-connection method [16] is applied. Both the aileron-rudder interconnect (ARI) and the rudder-aileron interconnect (RAI) are used here [36], whereby the command deflection angle of one channel is fed directly into the other.
The control matrix of the roll and yaw angular acceleration is presented in Equation (16). The desired decoupled control matrix is 11 International Journal of Aerospace Engineering In Equation (44), the off-diagonal elements are zero, which means the roll and yaw channel are decoupled. The hard cross-connection matrix K hc ∈ R 2×2 is defined here, and the linear control allocation problem with equality constrained expressed as Clearly, the control matrix B is square and invertible [37]. Thus, the pseudoinverse method is applied to realize control decoupling as where B † = B T ðBB T Þ −1 is the pseudoinverse of the control matrix. The saturation of the control surfaces is ignored due to the requirement for the deflection angle to be quite small because of the high dynamic pressure over the whole flight regime. Additionally, under significant changes of the control matrix during flight, the hard cross-connection decoupling matrix K hc should be scheduled according to the Mach number.

Linear Analysis and Simulation of the EA-Based Control.
The research LSR aircraft is trimmed at a speed of Mach 0.8 and an altitude of 10,000 m. The effectors are modeled as a second-order dynamic system with position and rate limitations, as summarized in Table 4.

12
International Journal of Aerospace Engineering    Table 5. The real parts of the eigenvalues are all negative, implying that the lateral-directional modes are all stable. However, the damping ratio of the Dutch-roll mode is only 0.05, which does not satisfy the military specifications. The minimum damping ratio and natural frequency for level 1, category A, class IV flight vehicles [38] are 0.4 and 1 rad/s, respectively. Furthermore, the eigenvectors of the roll and yaw channels are coupled.
To realize decoupling control and satisfy the flight quality requirements, the desired eigenvalues are as listed in The MATLAB control toolbox [39] is used to compute the linear response and stability margins for the parameter uncertainties. The uncertainties of the stability derivatives are Y β = 10%, Y p = 10%, Y r = 10%, L β = 30%, L p = 50%, L r = 50%, N β = 30%, N p = 50%, and N r = 50%. The control uncertainties are 15% for all control derivatives. A step command value is supplied to the linear closed-loop system by the actuator model, and the responses are shown in Figure 13.
As shown in Figure 13, the requirements for precise tracking control and lateral-directional decoupling control are achieved under model uncertainties. Furthermore, the time constants of the lateral-acceleration and bank-angle tracking control are about 0.3 and 0.5, respectively. These time constants are consistent with the assigned eigenvalues. To further evaluate the frequency characteristics of the controller, the phase margin and gain margin at the breakpoints of the control input and model output are depicted in Figure 14. In Figure 14(a), the gain margins with the actuator model are all greater than 6 dB, while the phase margins are greater than 45 degrees in Figure 14(b), satisfying the specifications of 6 dB and 45 deg. In addition, the margins remain within the specifi-cations under parameter perturbations, demonstrating the robustness of the controller.

Nonlinear Simulation of the ESO-Based
Control. First, the control effectiveness parameter b 0 in Equation (28) is selected as where b p and b r are the parameters of the roll and yaw control channels, respectively. The control coefficients of the roll moment C lδ a and yaw moment C nδ r are scheduled according to the Mach number over the whole flight regime based on a reliable CFD calculation. The left parameters of the ESO and outer-loop controllers are listed in Table 7. According to reference [31], the gains of β 1 and β 2 in the ESO controller can be approximately as where ω o can be adjusted using the bandwidth method [32]. By trial and error, the bandwidths of the observers are chosen as 5 rad/s for both the roll and yaw control channels. The absolute stability convergence analysis is given in reference [40]. Second, to verify the tracking performance and robustness of the ESO-based controller, Monte Carlo simulations are conducted using the nonlinear model with the actuator model. The simulations are initialized using the trim condition in Section 4.1. For longitudinal control, the flying altitude and speed are maintained. The uncertainties of the inertial parameters and aerodynamic parameters are listed in Table 8.
Latin hypercube sampling [41] is used in the MC simulations to improve the degree of coverage with fewer sampling events. A doublet bank angle command is given to     Figure 15.
As shown in Figure 15(f), the aircraft enters the transonic flight regime due to airspeed uncertainties. The transonic flight regime leads to a significant change in aerodynamic performance which poses a great threat to the control quality. In Figures 15(e) and 15(a), the maximum value of jβ/ϕj is about 0.033, resulting in decoupled control of the roll and yaw channels. The maximum deflection angles of the ailerons and rudder are 2.8 and 2.1 degrees, respectively, as a result of high dynamic pressure, as shown in Figures 15(g) and 15(h). Precise tracking performance of the bank angle is achieved without overshoot, as depicted in Figure 15 Table 8 are applied to the nonlinear model for MC simulations using ESO-based and EA-based attitude control. To quantify the attitude tracking performance, the RMS of the tracking error is evaluated: where x(t) and x c (t) are the state response and command value, respectively, and N is the number of MC simulations. The RMS calculation results of the two controllers are shown in Figure 16. Figure 16(a) gives the RMS of the lateral acceleration control, where the EA-based control exhibits better performance than the ESO-based control. However, in Figure 16(b), the ESO-based control has a slight advantage in the RMS value for the bank angle tracking control. The conclusion is that the EA-based controller achieves better decoupling characteristics for the lateral-directional dynamics resulting from the MIMO design method. Furthermore, the SISO design method means that the ESO-based control-ler struggles to completely compensate for the residual coupling nonlinearities. Second, to further evaluate the disturbance rejection characteristics of the two controllers, discrete gusts of wind are taken into consideration. The discrete gusts have a "1cosine" shape given by [28] V wind = 0, where V m is the gust amplitude, d m is the gust length, x d is the distance travelled, and V wind is the resultant wind velocity. The gust length d m = 120 m and the gust amplitude V m = 20 m/s are chosen according to the specifications in [28].
To evaluate the control performance under the most extreme scenario, another step gust wind with an amplitude of 10 m/s is added to the 1-cosine wind gust model. The wind gust occurs after 40 s, as shown in Figure 17. The tracking performance of the bank angle is almost the same for the two controllers without the wind gusts, as shown in Figure 17(b). When wind gusts occur, the EA-based controller produces sharp oscillations in both the roll and yaw channels. In addition, the oscillations last 5 s. Under parameter perturbations, this oscillation may easily activate rollyaw coupling, whereby the aircraft loses control. Contrary to the EA-based controller, the ESO-based controller enables rapid convergence of the tracking error, with a convergence period of less than 1.5 s and little oscillation. Thus, the ESObased attitude controller has better disturbance rejection performance.

Flight Test Validation
Considering the complexity of tuning the parameters of the EA-based control and the risk associated with discontinuities while switching the control parameters due to gain scheduling, the SISO ESO-based control was chosen as the flight test control law. A large number of MC simulations were conducted before the flight to ensure flight safety. In addition, detailed emergency measures were proposed to avoid various flight risks. Hardware-in-the-loop (HIL) simulations were compared with software-in-the-loop (SIL) results, and the sample time of the autopilot was set to 5 ms. An extended Kalman filter was used to estimate the attitude angles and angular rates in the flight test, and the airflow angles were measured using a wind vane at the nose position, as shown in Figure 1. During the flight mission, the parameters of the ESO were fixed, except for the effectiveness-related parameter b 0 , which was scheduled according to the Mach number. Thus, the ESO-based attitude controller was used in the entire flight mission with a simple SISO design method and without the need to tune various control parameters. Figure 18 shows the threedimensional flight trajectory of the mission.
The attitude tracking performance of the ESO-based controller is shown in Figure 19. During the flight test, lateral-directional decoupling control was achieved. Table 9 compares the RMS values between the flight test and HIL simulations. The flight test values are consistently close to the HIL simulation. Therefore, the simulations provide reliable results because of the high-fidelity dynamics model and the realistic simulation environment.
As depicted in Figure 19(a), the bank angle was tracked without any steady errors or overshoot, illustrating the precise tracking performance of the roll angular rate. This is consistent with the simulation results shown in Section 4.3. In addition, the maximum value of the ailerons is about 2 degrees, which is close to the simulation results in Figure 15(g). The angle-of-attack is less than 4 degrees over the whole flight, satisfying the small angle-of-attack assumption in Equation (9), as shown in Figure 19(c). In Figure 19(b), the lateral acceleration is less than 1 m/s 2 during the flight, and the yaw angular rate exhibits good command tracking performance. According to Figures 19(b) and 19(c), the maximum value of jβ/ϕj is about 0.045, which is slightly higher than the simulation results shown in Section 4.2.

Conclusions and Further Work
For the lateral-directional coupling dynamics analysis and control of an LSR supersonic aircraft, this study has elicited the following research results: (1) The inertia product plays an important role in the coupling dynamics of the LSR aircraft and cannot be ignored in the design process. In addition, the stability of the Dutch-roll mode is most sensitive to the value of the inertia product during the transonic phase. Additional wingtip weights should be considered when there is a risk of instability (2) Roll-yaw coupling poses a great threat to flight safety in the case of LSR aircraft. As the angle-ofattack increases, the value of roll-yaw coupling increases sharply. Therefore, the attitude controller   (3) The EA-based controller exhibits a better decoupling performance than the ESO-based controller because the eigenvalues and eigenvectors are assigned simultaneously. However, the complexity of parameter scheduling and the relative weak disturbance rejection capability of EA-based control means that the ESO-based controller is more practical in engineering terms. A large number of simulations and assessments, including flight tests, have demonstrated the effectiveness of the ESO-based controller Further work will focus on utilizing the disturbance rejection capability of the ESO-based controller and the decoupling characteristics of the EA-based controller. One approach may be to use the MIMO design method for the ESO-based controller. Another method would involve combining the disturbance rejection module of ESO in the EA design process. These aspects require further research to achieve complete and robust decoupling control.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon reasonable request.