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Sliding mode control design for oblique wing aircraft in wing skewing process
Chinese Journal of Aeronautics, (2019), 32(2): 263–271
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Sliding mode control design for oblique wing aircraft in wing skewing process Ting YUE a, Zijian XU b, Lixin WANG a,*, Tong WANG b a b
School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China China Ship Development and Design Center, Wuhan 430064, China
Received 2 April 2018; revised 22 May 2018; accepted 7 September 2018 Available online 20 November 2018
KEYWORDS Decoupling; Dynamic response; Oblique wing aircraft; Sliding mode control; Wing skewing process
Abstract When the wing of Oblique Wing Aircraft (OWA) is skewed, the center of gravity, inertia and aerodynamic characteristics of the aircraft all significantly change, causing an undesirable flight dynamic response, affecting the flying qualities, and even endangering the flight safety. In this study, the dynamic response of an OWA in the wing skewing process is simulated, showing that the threeaxis movements of the OWA are highly coupled and present nonlinear characteristics during the wing skewing. As the roll control efficiency of the aileron decreases due to the shortened control arm in an oblique configuration, the all-moving horizontal tail is used for additional roll and the control allocation is performed based on minimum control energy. Given the properties of pitchroll-yaw coupling and control input and state coupling, and the difficulty of establishing an accurate aerodynamic model in the wing skewing process due to unsteady aerodynamic force, a multi-loop sliding mode controller is formulated by the time-scale separation method. The closed-loop simulation results show that the asymmetric aerodynamics can be balanced and that the velocity and altitude of the aircraft maintain stable, which means that a smooth transition is obtained during the OWA’s wing skewing. Ó 2018 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Oblique Wing Aircraft (OWA) can rotate the wing at different flight velocities to form various wing sweep configurations. * Corresponding author. E-mail addresses: [email protected], [email protected] (L. WANG). Peer review under responsibility of Editorial Committee of CJA.
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Representing a simplified minimum drag solution in supersonic flow, the oblique elliptic wing can achieve efficient flight under subsonic cruise/loiter conditions while providing excellent supersonic dash/cruise capability.1,2 Therefore, OWA can adapt to multi-mission flight with extended flight envelope, and demonstrate higher combat effectiveness than conventional fixed-wing aircraft. A significant amount of early theoretical work was conducted by Jones starting from the 1950s.3 In the past decades, a number of aircrafts were flown to prove the feasibility of flight control of OWA, including small gliders, small remotely piloted aircraft and the manned AD-1 aircraft flown by NASA.4–8 Numerous conceptual design studies and research
https://doi.org/10.1016/j.cja.2018.11.002 1000-9361 Ó 2018 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
264 papers have addressed oblique wing-body-tail or oblique flying wing designs, and a number of wind tunnel tests and flight tests have been performed on both oblique wing and oblique flying wing designs.9–17 Recently, there has been renewed interest in developing OWA due to its aerodynamic and structural advantages. However, a variety of technical difficulties associated with the oblique wing configuration have prevented its application to operational aircraft. A critical technical challenge of OWA arises from its nonlinear and strongly coupled aerodynamic characteristics, and the inertial couplings further complicate the flight dynamics of such aircraft. These complicated characteristics are shown during the wing skewing of OWA, when a relative movement occurs between the wing and the fuselage. The aircraft in various configurations can be described as a time-varying dynamic system. Wing skewing largely changes the aircraft’s configuration parameters, including the moment of inertia, the location of the center of gravity, etc. It not only causes changes of lift, drag and pitching moment, but also generates side force, and rolling and yawing moments. This results in the aircraft’s highly coupled threeaxis movement and presents strong nonlinearity. In addition, wing skewing causes longitudinal and lateral couplings of aileron control, and lateral control inefficiency of the aileron at a large angle of the oblique wing. As a result, OWA will have an undesirable dynamic response in the presence of wing skews, and the flying qualities and flight safety will deteriorate. It is thus necessary to investigate the flight dynamic characteristics of OWA during wing skewing and design the flight control law to ensure a smooth transition between different configurations. The wing skewing process of OWA is similar to that of variable-sweep wing aircraft and morphing aircraft. Several studies have been carried out on the flight dynamic modelling and control of variable-sweep wing aircraft and morphing aircraft in configuration variations.18–23 However, unlike the aircraft designed in a symmetric configuration, OWA is characterized by the asymmetry configuration with strong three-axis aerodynamic and inertial couplings, resulting in a significant variation in three-axis motion parameters during wing skewing. Currently, there have been a number of studies mainly focusing on the analysis of flight dynamic characteristics24–26 and flight control design of OWA in static configurations (wing-fixed configurations at different oblique angles).27–32 However, the existing literature does not consider the influence of the wing skewing process on aircraft’s dynamic characteristics; additionally, there is lack of published literature about dynamic characteristics of and flight control design for the process of wing skewing. Due to the inertial and pitch-yaw-roll aerodynamic couplings in the wing skewing process, OWA is a non-linear time-varying system. This causes a difficulty in modeling precisely the aerodynamics of wing skewing which has uncertain dynamic behaviors. In addition, due to the inertial and aerodynamic cross-couplings, it is difficult for conventional control methods such as proportional-Integral-Derivative (PID) control to obtain acceptable control performance. A feasible approach for flight control design is sliding mode control theory, which can guarantee the dynamic qualities of the sliding mode motion and approaching motions by designing the sliding mode function and approaching rate, respectively.33,34
T. YUE et al. In this study, OWA is regarded as a rigid body system composed of the fuselage and rotating wing, and a time-varying and nonlinear dynamic model is built in order to examine the flight dynamic characteristics of OWA in the wing skewing process. The dynamic responses are numerically simulated and analyzed. According to the highly pitch-roll-yaw coupling, control input and state coupling, and nonlinear properties of OWA, a multi-loop sliding mode controller is designed based on the time-scale separation method. The all-moving horizontal tail is used for roll control due to the insufficient control effect of the aileron, and the control allocation of the control surfaces is performed based on minimum control energy. The simulation results show that the controller can ensure the good dynamic characteristics of OWA during wing skewing. The work introduced in this paper can provide a theoretical reference for flight control design for OWA. 2. Dynamic modeling and analysis 2.1. Layout and aerodynamic characteristics of OWA The OWA investigated in this study is presented in Fig. 1. The oblique wing is designed to rotate from 0° to 60° with the right wing forward. When the wing is skewed, the configuration parameters of the OWA change significantly, which mainly include the center of gravity position and the moment of inertia. Since the rotation axis of the wing is in front of the wing gravity center and parallel to the z-axis in the body axes, the aircraft’s gravity center shifts forward to the right when the wing is skewed, while the gravity center position along the z-axis remains unchanged. Meanwhile, the inertia moment and product of inertia also change significantly. As the symmetry of the conventional layout is destroyed, the cross inertial products Ixy and Iyz appear. The inertia coupling is strong between the longitudinal and lateral axes, which obviously affects the flight dynamic characteristics of the OWA.32 The asymmetric layout formed by the skewed wing also has a great influence on aerodynamic characteristics, which are stated as follows.32 The drag gradually decreases with the increase of the skew angle K. Since the effective wing span reduces after the wing is oblique, the lift decreases. In addition, the asymmetric wing layout generates side force, which is rather large and cannot be ignored. The oblique wing generates not only additional nose-down pitching moment, but also rolling moment and yawing moment. The rolling moment sinks the front wing and the yawing moment makes the aircraft rotate to the right.
Fig. 1
Layout of the oblique wing aircraft.
Sliding mode control design for oblique wing aircraft in wing skewing process After the wing is oblique, the ailerons are arranged back and forth. When the ailerons are deflected, the difference in lift will generate additional pitching moment.32 This should be considered in designing flight control law for the wing skewing process. In addition, the control arm of the aileron is shortened after the wing is oblique, which gradually decreases the roll control efficiency of the aileron with the increase of the wing skew angle. This may result in insufficient roll control ability of the ailerons. Therefore, it is necessary to use the allmoving horizontal tail for additional roll control of the OWA in the large skew angled configuration.27 2.2. Dynamic modeling During the oblique wing skewing, the OWA’s gravity center keeps changing. In this paper, the origin of the body axes is located at the center of gravity of the OWA at 0° skew angle. According to the momentum theorem and the theorem of moment of momentum for any moving point,18,35 we get the nonlinear dynamic equations of the OWA in the wing skewing process in a scalar form as below: 8 > _ z rS _ y 2rS_ y X ¼ mðu_ þ qw rvÞ þ qS > > > > 2 > þqðpSy qSx Þ r Sx þ S€x > > > < Y ¼ mðv_ þ ru pwÞ þ rS _ x pS _ z þ 2rS_ x ð1Þ > þr qSz rSy p pSy qSx þ S€y > > > > > _ y qS _ x þ prSx Z ¼ mðw_ þ pv quÞ þ pS > > > : _ _ 2 pSy qSx þ qrSy 8 L ¼ Ix p_ Ixy q_ Ixz r_ þ I_x p I_xy q I_xz r > > > > > þqðIxz p Iyz q þ Iz rÞ r Ixy p þ Iy q þ Iyz r > > > > > > þSy w_ þ Sy ðpv quÞ I1xz x_ 1z I_1xz x1z þ I1yz x21z > > > > > þ m1 S1z S€1y þ q S1x S_ 1y S1y S_ 1x rS1z S_ 1x > > > > > > M ¼ Ixy p_ þ Iy q_ Iyz r_ I_xy p þ I_y q I_yz r > > > < þr Ix p Ixy q Ixz r p Ixz p Iyz q þ Iz r > Sx w_ Sx ðpv quÞ I1yz x_ 1z I_1z x1z I1yz x21z > > > > > > þ m1 S1z S€1x rS1z S_ 1y pðS1x S_ 1y S1y S_ 1x Þ > > > > > N ¼ Ixz p_ Iyz q_ þ Iz r_ I_xz p I_yz q þ I_z r > > > > > þp Ixy p þ Iy q þ Iyz r q Ix p Ixy q Ixz r > > > > > > þSx v_ Sy u_ þ Sx ðru pwÞ Sy ðqw rvÞ þ I1z x_ 1z > > : þI_1z x1z þ m1 S1x S€1y S1y S€1x þ pS1z S_ 1x þ qS1z S_ 1y
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dynamic characteristics of the aircraft is small.18,21,22 Analogously, when the angular velocity of the OWA’s wing skew is comparatively small, the quasi-steady assumption can be used to simplify the calculation of aerodynamic force during the wing skewing process. It is considered that the aerodynamic force of the whole aircraft depends on the static configuration and the flight status of the aircraft at that time. In this study, we just investigate the case in which the angular velocity of the OWA’s wing skew is comparatively slow, and in which the aerodynamic force is simplified using the quasi-steady assumption. The aerodynamic uncertainty caused by simplification is solved by robust sliding mode control design. Therefore, here we simplify that the aerodynamic model during the wing skewing is the same with the static configuration.32 When the OWA is in an asymmetric configuration, the pitching motion will generate roll and yaw motions which will then generate pitching motion. Moreover, the deflection of the aileron will generate pitching moment. The aerodynamic crosscouplings are dependent on the wing skew angle and the angle of attack. As a result, the nonlinear equation of motion can be expressed as x_ ¼ fðx; uðxÞ; tÞ
ð3Þ
where f is a nonlinear time-varying function about x and u. The changes of the state variables affect the efficiency of the control input, and the control input can lead to changes of the state variables, i.e., coupling exists between x and u and cannot be overlooked. This is the main difference of the six degrees of freedom nonlinear model with that of conventional aircraft. 2.3. Analysis of dynamic characteristics
ð2Þ where [X, Y, Z] is the total force on the aircraft, [L, M, N] the total moment relative to the origin of the body axes, m the mass of the aircraft, [u, v, w] the flight velocity, [p, q, r] the angular velocity in the body axes relative to the inertial axes, [Sx, Sy, Sz] the static moment of aircraft, [S1x, S1y, S1z] the static moments of the wing, m the mass of the wing, I the rotational inertia, and [x1x, x1y, x1z] the projections of the angular velocity of the wing in the body axes. It can be seen from Eqs. (1) and (2) that, unlike the dynamic equations of conventional fixed-wing aircraft, there are additional terms caused by the center of gravity position shift and wing skew rate. In the variable-sweep process of swept-wing aircraft and the wing-folding process of morphing aircraft, when the changing rate of configurations is slow, the influence of unsteady aero-
The flight dynamics of the OWA in the wing skewing process without control is mathematically simulated. The aircraft’s CG position of the straight wing configuration is selected as the origin of the body axes. The wing skewing process and returning to straight wing process are simulated separately. The initial condition is a straight and level flight at H = 4000 m and Ma = 0.4. The dynamic responses are presented in Figs. 2 and 3, where c, /, and V are climb angle, roll angle and velocity respectively. It can be seen from Fig. 2 that the initial responses of the aircraft are mainly the changes of the three-axis angular velocities and attitudes when the wing is skewed. This is mainly caused by the asymmetric aerodynamic layout, resulting in a nose down pitching moment and positive rolling and yawing moments. The rolling moment sinks the front wing and the yawing moment makes the aircraft rotate to the right. As the skew angle continues to increase to 30°, the asymmetric rolling moment still exists. Then the roll angle increases rapidly and the lift direction keeps tilting. Meanwhile, the lift coefficient decreases with the increase of the skewing angle, so that the lift cannot balance the gravity of the aircraft. Therefore, the aircraft dives with acceleration and cannot reach a new steady state. For the process of returning to the straight wing from the oblique layout, the variations of the moments and the threeaxis angular velocities are opposite to those in the wing skewing process. As the initial roll inertia Ix in the returning process is relatively small, the roll angle varies faster. Then the lift
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Fig. 2
Dynamic responses in the wing skewing process.
direction rapidly changes, unable to balance the gravity. Like the wing skewing process, the aircraft dives with acceleration during the wing returning process. During the changes of configurations, the attitude, altitude and speed of the OWA are all obviously changed. Therefore, a flight control law should be designed to ensure the flight safety and flying qualities of the aircraft in the dynamic process.
Fig. 3 wing.
Dynamic responses during the returning to the straight
istics. Therefore, the flight control law using sliding mode control method is designed in this study. The control objective is to keep the altitude and velocity of the OWA constant during the wing skewing, so that a smooth transition of the configuration changes can be achieved. 3.1. Flight control system design based on sliding mode control
3. Flight control law design based on sliding mode control method In the wing skewing process, the three-axis movements of the OWA are highly coupled and strongly nonlinear, and it is thus difficult to establish an accurate aerodynamic model. This leads to a certain error between the simulation results and the real flight response, meaning that there exists uncertainty in the motion model of the aircraft. In addition, the range of the motion parameters is large, which requires the flight control system to be more robust. Since the sliding mode control method can ensure the dynamic quality through the design of the sliding mode function and the approach rate, the designed controller can adapt to the parameter changes of the OWA during the wing skewing. The sliding mode controller has strong robustness and can obtain precise control effect, more suitable for the OWA with a large range of parameter variations, and with time-varying and nonlinear character-
If a nonlinear system x_ ¼ fðx; u; tÞ can be transformed to an affine system x_ ¼ fðx; tÞ þ gðx; tÞu, the sliding mode control can be applied conveniently. However, the OWA’s dynamic equations of motion during the wing skewing exhibit inertia and aerodynamic cross couplings, and cannot be directly transformed to the affine form. According to the characteristics of the aircraft motions, we propose that the OWA’s variables of movement are divided into fast and slow variables by using the time-scale separation method. Then the complicated nonlinear equations can be simplified to an inner-loop and an outer-loop affine systems, and the sliding mode control design can be easily applied to the design of the inner-loop and outerloop sliding mode controllers. The sliding mode flight control system introduced in this paper mainly includes five modules: the modules of control command generation, outer-loop control law, inner-loop control law, control surface allocation, and automatic throttle.
Sliding mode control design for oblique wing aircraft in wing skewing process
Fig. 4
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Flight control structure.
The inner and the outer loops are separated by time-scale separation method for the aircraft motions. The control structure is shown in Fig. 4. The function of each module is introduced as follows: (1) The control command generation module generates control commands to the outer-loop control law module and the automatic throttle module based on the states of the aircraft. In order to ensure that the aircraft’s altitude and speed remain constant without sideslip, the roll angle is used to generate force to balance the side force generated by the asymmetric configuration. The output control commands are formulated as 8 DV ¼ Vc V > > > < a ¼ K ðH HÞ þ K _ H_ c DH c H ð4Þ > b ¼ 0 > c > : lc ¼ Ky_ y_ where DV, ac, bc and lc are the output command of speed, angle of attack, side slip angle and roll angle in the speed axis, respectively. Hc and Vc are the target altitude and speed, respectively. KDH , KH_ and Ky_ are the gains corresponding to the altitude error DH, the altitude varying rate H_ and the _ tuned to be 2 rad/m, 0.5 rad s/m and 6 rad s/ yaw rate y, m according to the control performance, respectively. (2) The outer-loop control model generates the input command ½pc ; qc ; rc T for the inner-loop model based on the control reference½ac ; bc ; lc T . According to the time-scale separation method for the state variables, the dynamic responses of the fast variables of the inner-loop are considered to have reached steady states and are neglected in the outer-loop design. Thus, the inner and outer-loop control laws can be designed separately. The differential equation can be expressed as 2 3 2 3 2 32 3 fl l_ p cosa=cosb 0 sina=cosb 6 7 6f 7 6 76 7 _ a ¼ ð5Þ þ cosatanb 1 sinatanb 4 5 4 a5 4 54 q 5 _b fb r sina 0 cosa where fl , fa , fb are continuous functions related to state variables. Eq. (5) can be transformed to a nonlinear multipleinput-multiple-output system as x_ s ðtÞ ¼ fs ðxÞ þ gs ðxÞus ðtÞ ð6Þ ys ðtÞ ¼ xs T where xs = [a, b, l]T ,fs ðxÞ ¼ fl ; fa ; fb , gs ðxÞ is the 3 3 T matrix in Eq. (5), us ðtÞ= [p, q, r] , ys ðtÞ is the output. For a system such as Eq. (6), according to the sliding mode control theory,36 the sliding mode surface can be designed as
3 Rt 3 2 ða ac Þ þ c1s 0 ða ac Þ Ss1 7 Rt 6 7 6 7 Ss ¼ 4 Ss2 5 ¼ 6 4 ðb bc Þ þ c2s 0 ðb bc Þ 5 Rt Ss3 ðl lc Þ þ c3s ðl lc Þ 2
ð7Þ
0
where the values of the sliding surface control parameters cis (i = 1,2,3) should make S_ s Hurwitz stable. The system can reach the sliding mode surface within a limited time and the error tends to be zero when the control law takes the following form37,38: u ¼ E1 ðw þ RRÞ
ð8Þ
where E = gs, w = fs + Usxs (x_ c + Usxc), Us = diag{c1s, c2s, c3s}, xc ¼ ½ac ; bc ; lc T , R = diag{xi/2ai }, R¼ n oT 2ai 1 sgnðSsi Þ . The bandwidth control parameters are jSsi j xi > 0, ai 2 ð0:5; 1:0Þ, i = 1,2,3. In order to suppress the vibration of the system, the signfunction sgn of the control law is replaced by the following saturation function 8 Ss > e > <1 satðSs Þ ¼ Ss =e jSs j 6 e ð9Þ > : 1 Ss < e where e is a small positive number. According to the control effect, the parameters are tuned as below: Rs = diag{2,2,2}, Us = diag{0.1,0.1,0.1}, and e = 0.1. (3) The inner loop calculates the required three-axis control moments to track the command ½pc ; qc ; rc T . The rotational dynamics Eq. (2) can be rewritten as 2 3 p_ 6 7 ð10Þ 4 q_ 5 ¼ ff ðxÞ þ I1 Mc r_ Similar to the nonlinear MIMO system of the outer loop, Mc can also be obtained by using the control law of Eq. (8). Here E = I1, w = ff + Ufxc (x_ c + Ufxc), Uf = diag{c1f,c2f, c3f} = diag{0.1,0.1,0.1}, xc = [pc,qc,rc]T, Rf = diag{4,4,4}, n oT andRf ¼ jSsi j2ai 1 sgnðSsi Þ (i = 1,2,3). The sign function is also replaced by a saturation function where a1 = a2 = a3 = 2/3, and e = 0.05. (4) The control allocation module calculates the required deflections of the control surfaces based on the demand of three-axis control moment Mc. Since the roll control efficiency of the OWA’s aileron is low in the large skew angled configuration, the all-moving horizontal tails are used to assist roll control.27 The control surface allocation can be expressed as
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gpda 6g Mc ¼ Gu ¼ 4 qda grda
2
gpdr gqdr grdr
gpdeL gqdeL grdeL
3
3 da gpdeR 6 7 6 dr 7 gqdeR 7 56 7 4 deL 5 grdeR deR
Mc ¼ Gu ¼ GDD1 u ¼ GD ^ u ð11Þ
where G is the matrix of the control coefficient, and u ¼ ½da ; dr ; deL ; deR ,da ,dr ,deL ,deR are deflections of the aileron, rudder, left elevator and right elevator, gij (i = p, q, r and j = da ,dr ,deL ,deR ) are the three-axis moment control efficiencies. In order to avoid the full or extremely small deflection of a certain control surface, we use D ¼ diag½da max ; dr max ; deL max ; deR max to weigh the allocation. Then Eq. (11) can be rewritten as
Fig. 5
ð12Þ
^ = [da/da max, dr/dr max, deL/deL max, deR/ where GD ¼ GD, and u deR max]T. The control input u can be obtained by a pseudoinverse of Eq. (12), an expenditure of minimum control energy,39 and can be expressed as u ¼ DðGD Þþ Mc
ð13Þ
(5) Since the drag and flight velocity of the OWA both change significantly in the wing skewing process, the throttle is used to control the speed of the aircraft. Designed by the traditional PID control method,40 the automatic throttle control law is expressed as
Closed-loop system responses (H = 4 km and Ma = 0.4).
Sliding mode control design for oblique wing aircraft in wing skewing process 269 Z the speed and altitude of the aircraft basically remain dT ¼ k1 DV þ k2 DVdt þ k3 V_ ð14Þ unchanged. We can see that the three-axis control surfaces deflect collaboratively to balance the asymmetric aerodynamic where k1, k2, and k3 are the gains of PID, and the parameters moments and that the throttle is decreased to balance the 1 2 are tuned as k1 = 6 s/m, k2 = 5 m and k3 = 1.5 s /m. reduced drag. The all-moving horizontal tail participates in roll control and involves in the control allocation based on 3.2. Closed-loop simulation and analysis minimal control energy, and the deflections of the control surfaces and the throttle are smooth. The sliding mode control law is verified by closed-loop simulaThe closed-loop simulation results for the process of wing tions for the wing skewing process and the process of wing returning to the straight position are shown in Fig. 5(b). It returning to the straight position. The initial flight conditions can be seen that the attitude of the aircraft changes exactly are H = 4 km and Ma = 0.4. The dynamic responses of the in the opposite direction of the wing skewing process. The lift parameters are shown in Fig. 5. coefficient increases, the lateral force decreases to zero, and the The closed-loop simulation results for the wing skewing angle of attack decreases. Moreover, the roll angle for balancprocess are shown in Fig. 5(a). It can be seen that since the lift ing the side force returns to zero, while the sideslip angle coefficient of the aircraft gradually decreases with the increase remains zero. After returning to the symmetrical configuration of the skewing angle, the angle of attack gradually increases to of the straight wing, the asymmetric moment disappears and achieve the balance of the longitudinal force. The sideslip angle the drag increases. As a result, the lateral control surfaces (rudis maintained as 0°, which is consistent with the target comder, ailerons, and differential horizontal tail) decrease to zero mand. The aircraft rolls to the left and the component of gravand the throttle increases accordingly. During the whole proity is used to balance the asymmetric side force; the aircraft cess, the flight velocity and altitude of the aircraft basically eventually maintains a certain roll angle. In the entire process, remain unchanged. It can be seen that the sliding mode flight
Fig. 6
Closed-loop system responses (H = 5 km and Ma = 0.5).
270 control law can fully guarantee the OWA flight at the required altitude and speed during the processes of wing skewing and returning to the straight position. To validate its robustness in the presence of the oblique wing, the sliding mode flight controller is applied to control the OWA in another flight state. Fig. 6 presents the closedloop dynamic responses of the OWA during both the wing skewing and the returning to the straight wing at H = 5 km and Ma = 0.5; the responses are controlled by the sliding mode flight controller designed for the state of H = 4 km and Ma = 0.4. We can see that the flight controller can still ensure that the altitude and speed in the wing skewing process are almost unchanged. This means that the sliding mode controller designed in this study has good robustness. 4. Conclusions (1) The three-axis motions of the OWA are significantly coupled and highly nonlinear in the wing skewing process. The initial dynamic responses of the aircraft in the processes of wing skewing and wing returning to the straight position are mainly the changes of threeaxis angular velocities and attitudes. With the rapid increase of the roll angle, the lift direction of the aircraft continuously changes, thus being unable to balance the gravity of the aircraft. The aircraft continues to dive with acceleration in the two processes mentioned above, and cannot reach a new balance state. (2) The moment arm of the aileron is shortened when the wing is skewed, which leads to an insufficient rolling control effectiveness. Therefore, the all-moving horizontal tail is used for assistant roll control. According to the three-axis coupling characteristics of aileron control, the control surfaces are allocated based on minimum control energy, achieving an accurate control effect. (3) According to the characteristics of pitch-roll-yaw coupling, control input and state coupling, and difficulty to accurately establish the aerodynamic model, a multi-loop sliding mode flight controller is designed by the time-scale separation method for the OWA to ensure the smooth transition during the wing skewing. The simulation results show that the robust sliding mode flight controller can smoothly adjust the attitude of the OWA to balance the asymmetric aerodynamic forces generated in the processes of wing skewing and returning. In addition, the speed and altitude of the aircraft can be kept constant.
Acknowledgement This work was supported by the National Natural Science Foundation of China (No. 11402010). References 1. Hirschberg MJ, Hart DM, Beautner TJ. A summary of a halfcentury of oblique wing research. Reston: AIAA; 2007. Report No.: AIAA-2007-0150. 2. Nelms JWP. Applications of oblique wing technology — An overview. Reston: AIAA; 1976. Report No.: AIAA-1976-0943.
T. YUE et al. 3. Jones RT. Aerodynamic design for supersonic speeds. Proceedings of the 1st international congress in the aeronautical sciences (ICAS), advances in aeronautical sciences.1959. 4. Boeing Commercial Airplane Company. Oblique wing transonic transport configuration development. Washington, D.C.: NASA; 1977. Report No.: NASA CR-151928. 5. Bailey RO, Putnam PA. Oblique wing, remotely piloted research aircraft. National association for remotely piloted vehicles’ 1975 meeting. 1975. 6. Curry RE, Sim AG. Unique flight characteristics of the AD-1 oblique-wing research airplane. J Aircraft 1983;20(6):564–8. 7. McMurty TC. AD-1 Oblique wing aircraft program. Reston: AIAA; 1981. Report No.: AIAA-1981-2354. 8. Painter WD. AD-1 oblique wing research aircraft pilot evaluation program. Reston: AIAA; 1983. Report No.: AIAA-1983-2509. 9. Kennelly RA, Carmichael RL, Smith SC. Transonic wind tunnel test of a 14% thick oblique wing. Washington, D.C.: NASA; 1990. Report No.: NASA-TM-102230. 10. Van der Velden AJM. The aerodynamic design of the oblique flying wing supersonic transport. Washington, D.C.: NASA; 1990. Report No.: NASA CR 177552. 11. Morris S, Tigner B. Flight tests of an oblique flying wing small scale demonstrator. Reston: AIAA; 1995. Report No.: AIAA1995-3327. 12. Li P, Sobieczky H, Seebass R. Manual aerodynamic optimization of an oblique wing supersonic transport. J Aircraft 1999;36 (6):907–13. 13. Wintzer M, Sturdza P, Kroo I. Conceptual design of conventional and oblique wing configurations for small supersonic aircraft. Reston: AIAA; 2006. Report No.: AIAA-2006-0930. 14. Dillsaver MJ, Franke ME. Wind tunnel study of oblique wing missile model. Reston: AIAA; 2008. Report No.: AIAA-20080318. 15. Tang L, Liu DD, Chen PC. Extension of projectile range using oblique-wing concept. J Aircraft 2007;44(3):774–9. 16. Center NA. Oblique flying wings: an introduction and white paper. Palo Alto: Desktop Aeronautics Inc.; 2005. 17. El-Salamony M. Oblique flying wing: Past and future. 3rd International Workshop ERBA. 2016. 18. An JG, Yan M, Zhou WB. Aircraft dynamic response to variable wing sweep geometry. J Aircraft 1988;25(3):216–21. 19. Seigler TM, Neal DA, Bae JS. Modeling and flight control of large-scale morphing aircraft. J Aircraft 2007;44(4):1077–87. 20. Ameri N, Lowenberg MH, Friswell MI. Modeling the dynamic response of a morphing wing with active winglets. Reston: AIAA; 2007. Report No.: AIAA-2007-6500. 21. Yue T, Wang L, Ai J. Gain self-scheduled H1 control for morphing aircraft in the wing transition process based on an LPV model. Chin J Aeronaut 2013;26(4):909–17. 22. Yue T, Wang L, Ai J. Longitudinal linear parameter varying modeling and simulation of morphing aircraft. J Aircraft 2013;50 (6):1673–81. 23. He Z, Yin M, Lu YP. Tensor product model-based control of morphing aircraft in transition process. P I Mech Eng G-J Aer 2016;230(2):378–91. 24. Weisshaar TA. Integrated structure/control concepts for oblique wing roll control and trim. J Aircraft 1994;31(1):117–24. 25. Raghavan B, Patil M. Aeroelasticity and flight dynamics of oblique flying wings. Reston: AIAA; 2007. Report No.: AIAA2007-2240. 26. Qiu L, Gao Z, Liu Y. Research on oblique wing air craft’s flight dynamics model. Fight Dynamics 2015; 33(2):106-10 [Chinese]. 27. Alag GS, Kempel RW, Pahle JW, Bresina JJ, Bartoli F. Modelfollowing control for an oblique-wing aircraft. Astrodynamics conference, fluid dynamics and co-located conferences, 1986. 28. Clark RN, Letron XJY. Oblique wing aircraft flight control system. J Guid Control Dyn 1989;12(2):201–8.
Sliding mode control design for oblique wing aircraft in wing skewing process 29. Jeffery J. The design and control of an oblique winged remote piloted vehicle. Reston: AIAA; 1995. Report No.: AIAA-19950744. 30. Pang J, Mei R, Chen M. Modeling and control for near-Space vehicles with oblique wing. Proceedings of the 10th world congress on intelligent control and automation. Piscataway: IEEE Press; 2012. p. 1773-8. 31. Matusˇ u˚ R, Prokop R. Robust stabilization of oblique wing aircraft model using PID controller. IFAC-PapersOnLine 2015;48 (14):265–70. 32. Wang LX, Xu Z, Yue T. Dynamic characteristics analysis and flight control design for oblique wing aircraft. Chin J Aeronaut 2016;29(6):1664–72. 33. Utkin VI. Sliding mode control design principles and applications to electric drives. IEEE Trans Ind Electron 1993;40(1):23–36. 34. Sabanovic A, Fridman LM, Spurgeon SK. Variable structure systems: from principles to implementation. London: The Institution of Electrical Engineers; 2004.
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35. Liu Y, Hong J, Yang H. Multibody system dynamics. Shanghai: Higher Education Press; 1989 [Chinese]. 36. Chen Z, Wang Z, Zhang J. Sliding mode variable control theory and application. Beijing: Electronic Industry Press; 2012. 37. Sun H, Li S, Sun C. Finite time integral sliding mode control of hypersonic vehicles. Nonlinear Dyn 2013;73(1–2):229–44. 38. Yang J, Li S, Yu X. Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Trans Ind Electron 2013;60(1):160–9. 39. Reigelsperger WC, Banda SS. Nonlinear simulation of a modified F-16 with full-enveloped control laws. Control Eng Pract 1998;6 (17):309–20. 40. Gong H, Zhen Z, Li X. Design of automatic climbing controller for large civil aircraft. J Franklin Inst 2013;350(9):2442–54.
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Sliding mode control design for oblique wing aircraft in wing skewing process Ting YUE a, Zijian XU b, Lixin WANG a,*, Tong WANG b a b
School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China China Ship Development and Design Center, Wuhan 430064, China
Received 2 April 2018; revised 22 May 2018; accepted 7 September 2018 Available online 20 November 2018
KEYWORDS Decoupling; Dynamic response; Oblique wing aircraft; Sliding mode control; Wing skewing process
Abstract When the wing of Oblique Wing Aircraft (OWA) is skewed, the center of gravity, inertia and aerodynamic characteristics of the aircraft all significantly change, causing an undesirable flight dynamic response, affecting the flying qualities, and even endangering the flight safety. In this study, the dynamic response of an OWA in the wing skewing process is simulated, showing that the threeaxis movements of the OWA are highly coupled and present nonlinear characteristics during the wing skewing. As the roll control efficiency of the aileron decreases due to the shortened control arm in an oblique configuration, the all-moving horizontal tail is used for additional roll and the control allocation is performed based on minimum control energy. Given the properties of pitchroll-yaw coupling and control input and state coupling, and the difficulty of establishing an accurate aerodynamic model in the wing skewing process due to unsteady aerodynamic force, a multi-loop sliding mode controller is formulated by the time-scale separation method. The closed-loop simulation results show that the asymmetric aerodynamics can be balanced and that the velocity and altitude of the aircraft maintain stable, which means that a smooth transition is obtained during the OWA’s wing skewing. Ó 2018 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Oblique Wing Aircraft (OWA) can rotate the wing at different flight velocities to form various wing sweep configurations. * Corresponding author. E-mail addresses: [email protected], [email protected] (L. WANG). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
Representing a simplified minimum drag solution in supersonic flow, the oblique elliptic wing can achieve efficient flight under subsonic cruise/loiter conditions while providing excellent supersonic dash/cruise capability.1,2 Therefore, OWA can adapt to multi-mission flight with extended flight envelope, and demonstrate higher combat effectiveness than conventional fixed-wing aircraft. A significant amount of early theoretical work was conducted by Jones starting from the 1950s.3 In the past decades, a number of aircrafts were flown to prove the feasibility of flight control of OWA, including small gliders, small remotely piloted aircraft and the manned AD-1 aircraft flown by NASA.4–8 Numerous conceptual design studies and research
https://doi.org/10.1016/j.cja.2018.11.002 1000-9361 Ó 2018 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
264 papers have addressed oblique wing-body-tail or oblique flying wing designs, and a number of wind tunnel tests and flight tests have been performed on both oblique wing and oblique flying wing designs.9–17 Recently, there has been renewed interest in developing OWA due to its aerodynamic and structural advantages. However, a variety of technical difficulties associated with the oblique wing configuration have prevented its application to operational aircraft. A critical technical challenge of OWA arises from its nonlinear and strongly coupled aerodynamic characteristics, and the inertial couplings further complicate the flight dynamics of such aircraft. These complicated characteristics are shown during the wing skewing of OWA, when a relative movement occurs between the wing and the fuselage. The aircraft in various configurations can be described as a time-varying dynamic system. Wing skewing largely changes the aircraft’s configuration parameters, including the moment of inertia, the location of the center of gravity, etc. It not only causes changes of lift, drag and pitching moment, but also generates side force, and rolling and yawing moments. This results in the aircraft’s highly coupled threeaxis movement and presents strong nonlinearity. In addition, wing skewing causes longitudinal and lateral couplings of aileron control, and lateral control inefficiency of the aileron at a large angle of the oblique wing. As a result, OWA will have an undesirable dynamic response in the presence of wing skews, and the flying qualities and flight safety will deteriorate. It is thus necessary to investigate the flight dynamic characteristics of OWA during wing skewing and design the flight control law to ensure a smooth transition between different configurations. The wing skewing process of OWA is similar to that of variable-sweep wing aircraft and morphing aircraft. Several studies have been carried out on the flight dynamic modelling and control of variable-sweep wing aircraft and morphing aircraft in configuration variations.18–23 However, unlike the aircraft designed in a symmetric configuration, OWA is characterized by the asymmetry configuration with strong three-axis aerodynamic and inertial couplings, resulting in a significant variation in three-axis motion parameters during wing skewing. Currently, there have been a number of studies mainly focusing on the analysis of flight dynamic characteristics24–26 and flight control design of OWA in static configurations (wing-fixed configurations at different oblique angles).27–32 However, the existing literature does not consider the influence of the wing skewing process on aircraft’s dynamic characteristics; additionally, there is lack of published literature about dynamic characteristics of and flight control design for the process of wing skewing. Due to the inertial and pitch-yaw-roll aerodynamic couplings in the wing skewing process, OWA is a non-linear time-varying system. This causes a difficulty in modeling precisely the aerodynamics of wing skewing which has uncertain dynamic behaviors. In addition, due to the inertial and aerodynamic cross-couplings, it is difficult for conventional control methods such as proportional-Integral-Derivative (PID) control to obtain acceptable control performance. A feasible approach for flight control design is sliding mode control theory, which can guarantee the dynamic qualities of the sliding mode motion and approaching motions by designing the sliding mode function and approaching rate, respectively.33,34
T. YUE et al. In this study, OWA is regarded as a rigid body system composed of the fuselage and rotating wing, and a time-varying and nonlinear dynamic model is built in order to examine the flight dynamic characteristics of OWA in the wing skewing process. The dynamic responses are numerically simulated and analyzed. According to the highly pitch-roll-yaw coupling, control input and state coupling, and nonlinear properties of OWA, a multi-loop sliding mode controller is designed based on the time-scale separation method. The all-moving horizontal tail is used for roll control due to the insufficient control effect of the aileron, and the control allocation of the control surfaces is performed based on minimum control energy. The simulation results show that the controller can ensure the good dynamic characteristics of OWA during wing skewing. The work introduced in this paper can provide a theoretical reference for flight control design for OWA. 2. Dynamic modeling and analysis 2.1. Layout and aerodynamic characteristics of OWA The OWA investigated in this study is presented in Fig. 1. The oblique wing is designed to rotate from 0° to 60° with the right wing forward. When the wing is skewed, the configuration parameters of the OWA change significantly, which mainly include the center of gravity position and the moment of inertia. Since the rotation axis of the wing is in front of the wing gravity center and parallel to the z-axis in the body axes, the aircraft’s gravity center shifts forward to the right when the wing is skewed, while the gravity center position along the z-axis remains unchanged. Meanwhile, the inertia moment and product of inertia also change significantly. As the symmetry of the conventional layout is destroyed, the cross inertial products Ixy and Iyz appear. The inertia coupling is strong between the longitudinal and lateral axes, which obviously affects the flight dynamic characteristics of the OWA.32 The asymmetric layout formed by the skewed wing also has a great influence on aerodynamic characteristics, which are stated as follows.32 The drag gradually decreases with the increase of the skew angle K. Since the effective wing span reduces after the wing is oblique, the lift decreases. In addition, the asymmetric wing layout generates side force, which is rather large and cannot be ignored. The oblique wing generates not only additional nose-down pitching moment, but also rolling moment and yawing moment. The rolling moment sinks the front wing and the yawing moment makes the aircraft rotate to the right.
Fig. 1
Layout of the oblique wing aircraft.
Sliding mode control design for oblique wing aircraft in wing skewing process After the wing is oblique, the ailerons are arranged back and forth. When the ailerons are deflected, the difference in lift will generate additional pitching moment.32 This should be considered in designing flight control law for the wing skewing process. In addition, the control arm of the aileron is shortened after the wing is oblique, which gradually decreases the roll control efficiency of the aileron with the increase of the wing skew angle. This may result in insufficient roll control ability of the ailerons. Therefore, it is necessary to use the allmoving horizontal tail for additional roll control of the OWA in the large skew angled configuration.27 2.2. Dynamic modeling During the oblique wing skewing, the OWA’s gravity center keeps changing. In this paper, the origin of the body axes is located at the center of gravity of the OWA at 0° skew angle. According to the momentum theorem and the theorem of moment of momentum for any moving point,18,35 we get the nonlinear dynamic equations of the OWA in the wing skewing process in a scalar form as below: 8 > _ z rS _ y 2rS_ y X ¼ mðu_ þ qw rvÞ þ qS > > > > 2 > þqðpSy qSx Þ r Sx þ S€x > > > < Y ¼ mðv_ þ ru pwÞ þ rS _ x pS _ z þ 2rS_ x ð1Þ > þr qSz rSy p pSy qSx þ S€y > > > > > _ y qS _ x þ prSx Z ¼ mðw_ þ pv quÞ þ pS > > > : _ _ 2 pSy qSx þ qrSy 8 L ¼ Ix p_ Ixy q_ Ixz r_ þ I_x p I_xy q I_xz r > > > > > þqðIxz p Iyz q þ Iz rÞ r Ixy p þ Iy q þ Iyz r > > > > > > þSy w_ þ Sy ðpv quÞ I1xz x_ 1z I_1xz x1z þ I1yz x21z > > > > > þ m1 S1z S€1y þ q S1x S_ 1y S1y S_ 1x rS1z S_ 1x > > > > > > M ¼ Ixy p_ þ Iy q_ Iyz r_ I_xy p þ I_y q I_yz r > > > < þr Ix p Ixy q Ixz r p Ixz p Iyz q þ Iz r > Sx w_ Sx ðpv quÞ I1yz x_ 1z I_1z x1z I1yz x21z > > > > > > þ m1 S1z S€1x rS1z S_ 1y pðS1x S_ 1y S1y S_ 1x Þ > > > > > N ¼ Ixz p_ Iyz q_ þ Iz r_ I_xz p I_yz q þ I_z r > > > > > þp Ixy p þ Iy q þ Iyz r q Ix p Ixy q Ixz r > > > > > > þSx v_ Sy u_ þ Sx ðru pwÞ Sy ðqw rvÞ þ I1z x_ 1z > > : þI_1z x1z þ m1 S1x S€1y S1y S€1x þ pS1z S_ 1x þ qS1z S_ 1y
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dynamic characteristics of the aircraft is small.18,21,22 Analogously, when the angular velocity of the OWA’s wing skew is comparatively small, the quasi-steady assumption can be used to simplify the calculation of aerodynamic force during the wing skewing process. It is considered that the aerodynamic force of the whole aircraft depends on the static configuration and the flight status of the aircraft at that time. In this study, we just investigate the case in which the angular velocity of the OWA’s wing skew is comparatively slow, and in which the aerodynamic force is simplified using the quasi-steady assumption. The aerodynamic uncertainty caused by simplification is solved by robust sliding mode control design. Therefore, here we simplify that the aerodynamic model during the wing skewing is the same with the static configuration.32 When the OWA is in an asymmetric configuration, the pitching motion will generate roll and yaw motions which will then generate pitching motion. Moreover, the deflection of the aileron will generate pitching moment. The aerodynamic crosscouplings are dependent on the wing skew angle and the angle of attack. As a result, the nonlinear equation of motion can be expressed as x_ ¼ fðx; uðxÞ; tÞ
ð3Þ
where f is a nonlinear time-varying function about x and u. The changes of the state variables affect the efficiency of the control input, and the control input can lead to changes of the state variables, i.e., coupling exists between x and u and cannot be overlooked. This is the main difference of the six degrees of freedom nonlinear model with that of conventional aircraft. 2.3. Analysis of dynamic characteristics
ð2Þ where [X, Y, Z] is the total force on the aircraft, [L, M, N] the total moment relative to the origin of the body axes, m the mass of the aircraft, [u, v, w] the flight velocity, [p, q, r] the angular velocity in the body axes relative to the inertial axes, [Sx, Sy, Sz] the static moment of aircraft, [S1x, S1y, S1z] the static moments of the wing, m the mass of the wing, I the rotational inertia, and [x1x, x1y, x1z] the projections of the angular velocity of the wing in the body axes. It can be seen from Eqs. (1) and (2) that, unlike the dynamic equations of conventional fixed-wing aircraft, there are additional terms caused by the center of gravity position shift and wing skew rate. In the variable-sweep process of swept-wing aircraft and the wing-folding process of morphing aircraft, when the changing rate of configurations is slow, the influence of unsteady aero-
The flight dynamics of the OWA in the wing skewing process without control is mathematically simulated. The aircraft’s CG position of the straight wing configuration is selected as the origin of the body axes. The wing skewing process and returning to straight wing process are simulated separately. The initial condition is a straight and level flight at H = 4000 m and Ma = 0.4. The dynamic responses are presented in Figs. 2 and 3, where c, /, and V are climb angle, roll angle and velocity respectively. It can be seen from Fig. 2 that the initial responses of the aircraft are mainly the changes of the three-axis angular velocities and attitudes when the wing is skewed. This is mainly caused by the asymmetric aerodynamic layout, resulting in a nose down pitching moment and positive rolling and yawing moments. The rolling moment sinks the front wing and the yawing moment makes the aircraft rotate to the right. As the skew angle continues to increase to 30°, the asymmetric rolling moment still exists. Then the roll angle increases rapidly and the lift direction keeps tilting. Meanwhile, the lift coefficient decreases with the increase of the skewing angle, so that the lift cannot balance the gravity of the aircraft. Therefore, the aircraft dives with acceleration and cannot reach a new steady state. For the process of returning to the straight wing from the oblique layout, the variations of the moments and the threeaxis angular velocities are opposite to those in the wing skewing process. As the initial roll inertia Ix in the returning process is relatively small, the roll angle varies faster. Then the lift
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Fig. 2
Dynamic responses in the wing skewing process.
direction rapidly changes, unable to balance the gravity. Like the wing skewing process, the aircraft dives with acceleration during the wing returning process. During the changes of configurations, the attitude, altitude and speed of the OWA are all obviously changed. Therefore, a flight control law should be designed to ensure the flight safety and flying qualities of the aircraft in the dynamic process.
Fig. 3 wing.
Dynamic responses during the returning to the straight
istics. Therefore, the flight control law using sliding mode control method is designed in this study. The control objective is to keep the altitude and velocity of the OWA constant during the wing skewing, so that a smooth transition of the configuration changes can be achieved. 3.1. Flight control system design based on sliding mode control
3. Flight control law design based on sliding mode control method In the wing skewing process, the three-axis movements of the OWA are highly coupled and strongly nonlinear, and it is thus difficult to establish an accurate aerodynamic model. This leads to a certain error between the simulation results and the real flight response, meaning that there exists uncertainty in the motion model of the aircraft. In addition, the range of the motion parameters is large, which requires the flight control system to be more robust. Since the sliding mode control method can ensure the dynamic quality through the design of the sliding mode function and the approach rate, the designed controller can adapt to the parameter changes of the OWA during the wing skewing. The sliding mode controller has strong robustness and can obtain precise control effect, more suitable for the OWA with a large range of parameter variations, and with time-varying and nonlinear character-
If a nonlinear system x_ ¼ fðx; u; tÞ can be transformed to an affine system x_ ¼ fðx; tÞ þ gðx; tÞu, the sliding mode control can be applied conveniently. However, the OWA’s dynamic equations of motion during the wing skewing exhibit inertia and aerodynamic cross couplings, and cannot be directly transformed to the affine form. According to the characteristics of the aircraft motions, we propose that the OWA’s variables of movement are divided into fast and slow variables by using the time-scale separation method. Then the complicated nonlinear equations can be simplified to an inner-loop and an outer-loop affine systems, and the sliding mode control design can be easily applied to the design of the inner-loop and outerloop sliding mode controllers. The sliding mode flight control system introduced in this paper mainly includes five modules: the modules of control command generation, outer-loop control law, inner-loop control law, control surface allocation, and automatic throttle.
Sliding mode control design for oblique wing aircraft in wing skewing process
Fig. 4
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Flight control structure.
The inner and the outer loops are separated by time-scale separation method for the aircraft motions. The control structure is shown in Fig. 4. The function of each module is introduced as follows: (1) The control command generation module generates control commands to the outer-loop control law module and the automatic throttle module based on the states of the aircraft. In order to ensure that the aircraft’s altitude and speed remain constant without sideslip, the roll angle is used to generate force to balance the side force generated by the asymmetric configuration. The output control commands are formulated as 8 DV ¼ Vc V > > > < a ¼ K ðH HÞ þ K _ H_ c DH c H ð4Þ > b ¼ 0 > c > : lc ¼ Ky_ y_ where DV, ac, bc and lc are the output command of speed, angle of attack, side slip angle and roll angle in the speed axis, respectively. Hc and Vc are the target altitude and speed, respectively. KDH , KH_ and Ky_ are the gains corresponding to the altitude error DH, the altitude varying rate H_ and the _ tuned to be 2 rad/m, 0.5 rad s/m and 6 rad s/ yaw rate y, m according to the control performance, respectively. (2) The outer-loop control model generates the input command ½pc ; qc ; rc T for the inner-loop model based on the control reference½ac ; bc ; lc T . According to the time-scale separation method for the state variables, the dynamic responses of the fast variables of the inner-loop are considered to have reached steady states and are neglected in the outer-loop design. Thus, the inner and outer-loop control laws can be designed separately. The differential equation can be expressed as 2 3 2 3 2 32 3 fl l_ p cosa=cosb 0 sina=cosb 6 7 6f 7 6 76 7 _ a ¼ ð5Þ þ cosatanb 1 sinatanb 4 5 4 a5 4 54 q 5 _b fb r sina 0 cosa where fl , fa , fb are continuous functions related to state variables. Eq. (5) can be transformed to a nonlinear multipleinput-multiple-output system as x_ s ðtÞ ¼ fs ðxÞ þ gs ðxÞus ðtÞ ð6Þ ys ðtÞ ¼ xs T where xs = [a, b, l]T ,fs ðxÞ ¼ fl ; fa ; fb , gs ðxÞ is the 3 3 T matrix in Eq. (5), us ðtÞ= [p, q, r] , ys ðtÞ is the output. For a system such as Eq. (6), according to the sliding mode control theory,36 the sliding mode surface can be designed as
3 Rt 3 2 ða ac Þ þ c1s 0 ða ac Þ Ss1 7 Rt 6 7 6 7 Ss ¼ 4 Ss2 5 ¼ 6 4 ðb bc Þ þ c2s 0 ðb bc Þ 5 Rt Ss3 ðl lc Þ þ c3s ðl lc Þ 2
ð7Þ
0
where the values of the sliding surface control parameters cis (i = 1,2,3) should make S_ s Hurwitz stable. The system can reach the sliding mode surface within a limited time and the error tends to be zero when the control law takes the following form37,38: u ¼ E1 ðw þ RRÞ
ð8Þ
where E = gs, w = fs + Usxs (x_ c + Usxc), Us = diag{c1s, c2s, c3s}, xc ¼ ½ac ; bc ; lc T , R = diag{xi/2ai }, R¼ n oT 2ai 1 sgnðSsi Þ . The bandwidth control parameters are jSsi j xi > 0, ai 2 ð0:5; 1:0Þ, i = 1,2,3. In order to suppress the vibration of the system, the signfunction sgn of the control law is replaced by the following saturation function 8 Ss > e > <1 satðSs Þ ¼ Ss =e jSs j 6 e ð9Þ > : 1 Ss < e where e is a small positive number. According to the control effect, the parameters are tuned as below: Rs = diag{2,2,2}, Us = diag{0.1,0.1,0.1}, and e = 0.1. (3) The inner loop calculates the required three-axis control moments to track the command ½pc ; qc ; rc T . The rotational dynamics Eq. (2) can be rewritten as 2 3 p_ 6 7 ð10Þ 4 q_ 5 ¼ ff ðxÞ þ I1 Mc r_ Similar to the nonlinear MIMO system of the outer loop, Mc can also be obtained by using the control law of Eq. (8). Here E = I1, w = ff + Ufxc (x_ c + Ufxc), Uf = diag{c1f,c2f, c3f} = diag{0.1,0.1,0.1}, xc = [pc,qc,rc]T, Rf = diag{4,4,4}, n oT andRf ¼ jSsi j2ai 1 sgnðSsi Þ (i = 1,2,3). The sign function is also replaced by a saturation function where a1 = a2 = a3 = 2/3, and e = 0.05. (4) The control allocation module calculates the required deflections of the control surfaces based on the demand of three-axis control moment Mc. Since the roll control efficiency of the OWA’s aileron is low in the large skew angled configuration, the all-moving horizontal tails are used to assist roll control.27 The control surface allocation can be expressed as
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gpda 6g Mc ¼ Gu ¼ 4 qda grda
2
gpdr gqdr grdr
gpdeL gqdeL grdeL
3
3 da gpdeR 6 7 6 dr 7 gqdeR 7 56 7 4 deL 5 grdeR deR
Mc ¼ Gu ¼ GDD1 u ¼ GD ^ u ð11Þ
where G is the matrix of the control coefficient, and u ¼ ½da ; dr ; deL ; deR ,da ,dr ,deL ,deR are deflections of the aileron, rudder, left elevator and right elevator, gij (i = p, q, r and j = da ,dr ,deL ,deR ) are the three-axis moment control efficiencies. In order to avoid the full or extremely small deflection of a certain control surface, we use D ¼ diag½da max ; dr max ; deL max ; deR max to weigh the allocation. Then Eq. (11) can be rewritten as
Fig. 5
ð12Þ
^ = [da/da max, dr/dr max, deL/deL max, deR/ where GD ¼ GD, and u deR max]T. The control input u can be obtained by a pseudoinverse of Eq. (12), an expenditure of minimum control energy,39 and can be expressed as u ¼ DðGD Þþ Mc
ð13Þ
(5) Since the drag and flight velocity of the OWA both change significantly in the wing skewing process, the throttle is used to control the speed of the aircraft. Designed by the traditional PID control method,40 the automatic throttle control law is expressed as
Closed-loop system responses (H = 4 km and Ma = 0.4).
Sliding mode control design for oblique wing aircraft in wing skewing process 269 Z the speed and altitude of the aircraft basically remain dT ¼ k1 DV þ k2 DVdt þ k3 V_ ð14Þ unchanged. We can see that the three-axis control surfaces deflect collaboratively to balance the asymmetric aerodynamic where k1, k2, and k3 are the gains of PID, and the parameters moments and that the throttle is decreased to balance the 1 2 are tuned as k1 = 6 s/m, k2 = 5 m and k3 = 1.5 s /m. reduced drag. The all-moving horizontal tail participates in roll control and involves in the control allocation based on 3.2. Closed-loop simulation and analysis minimal control energy, and the deflections of the control surfaces and the throttle are smooth. The sliding mode control law is verified by closed-loop simulaThe closed-loop simulation results for the process of wing tions for the wing skewing process and the process of wing returning to the straight position are shown in Fig. 5(b). It returning to the straight position. The initial flight conditions can be seen that the attitude of the aircraft changes exactly are H = 4 km and Ma = 0.4. The dynamic responses of the in the opposite direction of the wing skewing process. The lift parameters are shown in Fig. 5. coefficient increases, the lateral force decreases to zero, and the The closed-loop simulation results for the wing skewing angle of attack decreases. Moreover, the roll angle for balancprocess are shown in Fig. 5(a). It can be seen that since the lift ing the side force returns to zero, while the sideslip angle coefficient of the aircraft gradually decreases with the increase remains zero. After returning to the symmetrical configuration of the skewing angle, the angle of attack gradually increases to of the straight wing, the asymmetric moment disappears and achieve the balance of the longitudinal force. The sideslip angle the drag increases. As a result, the lateral control surfaces (rudis maintained as 0°, which is consistent with the target comder, ailerons, and differential horizontal tail) decrease to zero mand. The aircraft rolls to the left and the component of gravand the throttle increases accordingly. During the whole proity is used to balance the asymmetric side force; the aircraft cess, the flight velocity and altitude of the aircraft basically eventually maintains a certain roll angle. In the entire process, remain unchanged. It can be seen that the sliding mode flight
Fig. 6
Closed-loop system responses (H = 5 km and Ma = 0.5).
270 control law can fully guarantee the OWA flight at the required altitude and speed during the processes of wing skewing and returning to the straight position. To validate its robustness in the presence of the oblique wing, the sliding mode flight controller is applied to control the OWA in another flight state. Fig. 6 presents the closedloop dynamic responses of the OWA during both the wing skewing and the returning to the straight wing at H = 5 km and Ma = 0.5; the responses are controlled by the sliding mode flight controller designed for the state of H = 4 km and Ma = 0.4. We can see that the flight controller can still ensure that the altitude and speed in the wing skewing process are almost unchanged. This means that the sliding mode controller designed in this study has good robustness. 4. Conclusions (1) The three-axis motions of the OWA are significantly coupled and highly nonlinear in the wing skewing process. The initial dynamic responses of the aircraft in the processes of wing skewing and wing returning to the straight position are mainly the changes of threeaxis angular velocities and attitudes. With the rapid increase of the roll angle, the lift direction of the aircraft continuously changes, thus being unable to balance the gravity of the aircraft. The aircraft continues to dive with acceleration in the two processes mentioned above, and cannot reach a new balance state. (2) The moment arm of the aileron is shortened when the wing is skewed, which leads to an insufficient rolling control effectiveness. Therefore, the all-moving horizontal tail is used for assistant roll control. According to the three-axis coupling characteristics of aileron control, the control surfaces are allocated based on minimum control energy, achieving an accurate control effect. (3) According to the characteristics of pitch-roll-yaw coupling, control input and state coupling, and difficulty to accurately establish the aerodynamic model, a multi-loop sliding mode flight controller is designed by the time-scale separation method for the OWA to ensure the smooth transition during the wing skewing. The simulation results show that the robust sliding mode flight controller can smoothly adjust the attitude of the OWA to balance the asymmetric aerodynamic forces generated in the processes of wing skewing and returning. In addition, the speed and altitude of the aircraft can be kept constant.
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