- Email: [email protected]
Time-varying linear control for tiltrotor aircraft
CJA 998 20 February 2018 Chinese Journal of Aeronautics, (2018), xxx(xx): xxx–xxx
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Time-varying linear control for tiltrotor aircraft
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Jing ZHANG a, Liguo SUN a,*, Xiangju QU a, Liuping WANG b
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School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China School of Electrical and Computer Engineering, RMIT University, Melbourne, VIC 3001, Australia
Received 12 January 2017; revised 26 May 2017; accepted 5 September 2017
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KEYWORDS
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Constrained optimal control; Inertia/control couplings; Tiltrotor aircraft; Time-varying control; Transition mode
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Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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Abstract Tiltrotor aircraft have three flight modes: helicopter mode, airplane mode, and transition mode. A tiltrotor has characteristics of highly nonlinear, time-varying flight dynamics and inertial/control couplings in its transition mode. It can transit from the helicopter mode to the airplane mode by tilting its nacelles, and an effective controller is crucial to accomplish tilting transition missions. Longitudinal dynamic characteristics of the tiltrotor are described by a nonlinear Lagrangeform model, which takes into account inertial/control couplings and aerodynamic interferences. Reference commands for airspeed velocity and attitude in the transition mode are calculated dynamically by visiting a command library which is founded in advance by analyzing the flight envelope of the tiltrotor. A Time-Varying Linear (TVL) model is obtained using a Taylorexpansion based online linearization technique from the nonlinear model. Subsequently, based on an optimal control concept, an online optimization based control method with input constraints considered is proposed. To validate the proposed control method, three typical tilting transition missions are simulated using the nonlinear model of XV-15 tiltrotor aircraft. Simulation results show that the controller can be used to control the tiltrotor throughout its operating envelop which includes a transition flight, and can also deal with vertical gust disturbances.
1. Introduction A tiltrotor aircraft, which combines the characteristics of helicopters and fixed-wing aircraft, consists of an aircraft body (including the fuselage and wings), engine nacelles, and rotors. * Corresponding author. E-mail addresses: [email protected], [email protected] (L. SUN). Peer review under responsibility of Editorial Committee of CJA.
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Unlike a traditional aircraft, a tiltrotor aircraft is a complex multibody system which can change its configuration by tilting the nacelles, and it owns three different flight modes, namely helicopter mode, airplane mode, and transition mode. The transition mode, which means the conversion between the other two flight modes, is a special mode of tiltrotors. A tiltrotor aircraft with two side-by-side rotors such as the XV-15 tiltrotor is studied in this paper. A tiltrotor owns the advantages of both helicopters and fixed-wing aircraft. Firstly, a tiltrotor can fly freely in different directions like a helicopter, and can also hover in the air. Secondly, a tiltrotor has a faster cruising speed than that of a helicopter. Thirdly, a tiltrotor can enhance civil or military
https://doi.org/10.1016/j.cja.2018.01.025 1000-9361 Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: ZHANG J et al. Time-varying linear control for tiltrotor aircraft, Chin J Aeronaut (2018), https://doi.org/10.1016/j.cja.2018.01.025
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transportation capability because it has three flight modes. However, a tiltrotor aircraft has some disadvantages such as low safety and low reliability. Firstly, the flight dynamics of a tiltrotor is much more complex than that of a traditional aircraft since the tiltrotor demonstrates different flight dynamic characteristics in different flight modes, and there exist inertial couplings between the aircraft body and nacelles. Secondly, a tiltrotor has low reliability because there are control couplings between its control action paths. Thirdly, the pilot workload of a tiltrotor would be higher especially in the transition period, because it has the characteristics of complex flight dynamics, inertial couplings, and control couplings. This paper studies the flight control for tiltrotor aircraft to deal with the complex flight dynamics and inertial couplings. To obtain a reasonable and reliable dynamic model is crucial for aircraft controller design. A large amount of literature considers a tiltrotor as a single rigid body, and describes its dynamic characteristics through three-Degree-Of-Freedom (DOF) longitudinal modeling or six-DOF modeling, which is similar to that for traditional aircraft.1–3 However, a tiltrotor is a kind of morphing aircraft which tilts its nacelles in its special transition mode, so a traditional flight dynamic model cannot reveal the inertial couplings and dynamic characteristics in the transition mode. Certain literature regards a tiltrotor as a multi-rigid-body system in view of the inertial couplings. For example, Li et al. considered a tiltrotor as multiple entities, and developed a twelve-DOF dynamic model for tiltrotors based on multibody dynamics.4 The multibody model can clearly characterize the inertial couplings and complex flight dynamics of a tiltrotor, but its expression is too complicated for controller design. Later on, based on the Lagrange’s equation, Zhang et al. built a multibody longitudinal nonlinear model of a tiltrotor, and the derived model is in a more concise form and more suitable for the longitudinal controller design of the tiltrotor.5 It is worth noting that modeling of multibody systems has experienced a remarkable development in space flight.6,7 Furthermore, founding an aeroelastic model is beneficial to analyzing the stability of the tiltrotor.8 The literature about tiltrotor aircraft control can be classified into two categories: linear control and nonlinear control. Firstly, linear controllers were designed on the basis of a linear dynamic model and the information of certain flight conditions, and references are organized chronologically. A Model Predictive Control (MPC) method was adopted to design a flight controller for the hovering mode and three other typical flight scenarios of a tiltrotor.9 An attitude controller for a tiltrotor in the helicopter hovering mode was designed using MPC.10 An optimal control approach was proposed to deal with gusts effects on a tiltrotor in helicopter and airplane modes.11 Minimum energy controllers were designed based on the helicopter mode and the airplane mode, respectively.12 Based on optimal preview control, an attitude controller was developed for a particular flight state in the transition mode of a tiltrotor.13 In consideration of model errors, a controller was designed for the airplane mode of a tiltrotor based on a sliding mode method.14 It should be noted that the above mentioned linear controllers were mainly designed for one or several particular flight states of a tiltrotor, and they need gain scheduling to achieve control of the whole conversion process of the tiltrotor. Secondly, nonlinear control approaches have the advantage that the nacelle angle does not need to be assumed fixed when designing tiltrotor flight controllers. In
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J. ZHANG et al. Ref. 15, a robust nonlinear controller was designed for a tiltrotor in the hover mode. Considering system uncertainties and disturbances, an adaptive control method was developed based on Neural Networks (NNs),16 but it requires using different NN weights for different flight states, e.g., different nacelle angles. Based on the work,16 an online NN modelling method was introduced to develop a fully adaptive flight control method, which achieves not only longitudinal control but also lateral control of a tiltrotor. It should be mentioned that this method needs to update the entire full-state nonlinear model of a tiltrotor for different flight states.17 Considering the control difficulty brought by the tiltrotor mode switching process and the tiltrotor configuration change during the transition mode, Ref. 18 presented a nonlinear control method and specifically studied the transition process control. Nevertheless, the tiltrotor controller designed in Ref. 18 fails to consider input constraints and control couplings. A tiltrotor aircraft, especially in the transition mode, is a time-varying and strongly nonlinear system, and there exist couplings between different control action paths. As an alternative to linear controllers with a gain scheduling mechanism and nonlinear controllers mentioned above, it is a wise choice to convert a time-varying highly nonlinear system into a TimeVarying Linear (TVL) system, and then design a controller based on the TVL model. There exist a large number of approaches in literature considering founding a TVL model and TVL control.19–22 The Linear Parameter Varying (LPV) control method, which approximates a nonlinear system as a TVL system, is a classical control method which has been widely applied to aircraft with a variable structure.23–27 However, a single LPV controller designed for a single flight state cannot accomplish the whole transition control of tiltrotors operating in the transition mode. Moreover, the switching LPV control method based on the Lyapunov function is computationally expensive because it needs to solve a large number of linear matrix inequalities.28 The objective of this paper is to propose a control approach which is competent for achieving effective control for tiltrotor aircraft in the transition mode. More specifically, this paper contributes to online model linearization, dynamic generation strategy of reference commands, and real-time optimization based optimal control of a tiltrotor. Firstly, a nonlinear and time-varying Lagrange-form tiltrotor model is presented, which takes the control couplings of tiltrotor aircraft into account. Secondly, how to derive a TVL model from the time-varying nonlinear model is addressed. Note that the TVL model updates its parameters at each sampling instant according to the flight conditions. Thirdly, a trim-condition based approach is developed to calculate the reference commands for airspeed velocity and attitude. To update the reference commands in real-time, a library for the flight envelope of the tiltrotor and the static characteristics curves of the pitch angle is founded. Fourthly, inspired by the MPC concept, an online optimization based optimal control approach is proposed for tiltrotor aircraft, especially when considering transition-mode flight missions. Moreover, the proposed control approach allows for conducting input constraints, and the control inputs are the solution of a constrained online optimization problem. The structure of this paper is organized as follows. Section 2 presents the longitudinal dynamic model of a tiltrotor which is used for designing a controller. In Section 3, how to calculate
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the reference commands of the tiltrotor is introduced. The detailed controller design is described in Section 4. In Section 5, three cases of tilting transition missions are simulated and discussed. Finally, conclusions are given in Section 6.
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2. Longitudinal nonlinear model of a tiltrotor
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A nonlinear modeling method for tiltrotor aircraft was proposed in Ref. 5, and the formulation of the nonlinear modeling method is briefly given in this section for completeness concern. Fig. 1 shows the longitudinal diagram of a tiltrotor aircraft. As depicted in Fig. 1, the longitudinal diagram of the tiltrotor aircraft is described in the OgXgYg coordinate system. The OgXgYg is an inertial coordinate system which is fixed on the earth ground, and the axis OgYg is parallel to the local plumb line and points upward. The point O means the position of the nacelle shaft while point A indicates the position of the rotor hub. Points G1, G2, and G represent the mass centers of the aircraft body, the nacelle-rotor system, and the whole aircraft, respectively. Moreover, h describes the pitch angle, and hN represents the nacelle angle, while a indicates the Angle of Attack (AoA). In addition, V is the airspeed velocity of the tiltrotor. L, D, and M are the lift, drag, and pitching moments of the aircraft frame (including the aircraft body and the nacelles), respectively, and they are transited from the aerodynamic center to point G. What’s more, MN means the torque to tilt the nacelle-rotor system, and LT represents the aerodynamic pull of the rotors, while a1s is the angle between the rotor disc and the hub disc. According to the Lagrange’s equations, the position coordinates of point O (x, y), h, and hN are selected as the generalized coordinates. Let hAF ¼ h dAF , hTM ¼ h þ hN dTM , and hd ¼ h þ hN . Then, the expressions of generalized forces are described as follows: 8 Q1 ¼ L sinða hÞ D cosða hÞ þ LT cosða1s þ hd Þ > > > > Q > < 2 ¼ L cosða hÞ þ D sinða hÞ þ LT sinða1s þ hd Þ mg Q3 ¼ Q4 MN þ M þ m1 l1 ðL sin a D cos a þ mg cos hAF Þ=m > > > Q ¼ m2 l2 ½L cosða þ hN Þ þ D sinða þ hN Þ mg cos hTM =m > > : 4 þMN þ l3 LT sin a1s ð1Þ
Fig. 1
3 where m, m1, and m2 are the masses of the whole tiltrotor, the aircraft body, and the nacelle-rotor system, respectively, while l1, l2, and l3 denote the lengths of OA, OG1, and OG2, respectively. Similar to the case in Ref. 4, the nacelles are designed as uniform rotation in this paper. Taking into account that the nacelles should switch between rotating and stopping, the scheduling rules of hN are designed as follows: 8 b > < a 0 6 t t0 < a p € hN ðtÞ ¼ 0 ba 6 t t0 < 2b > : p p a 2b 6 t t0 < 2b þ ba Z t € ð2Þ hN ðsÞds h_ N ðtÞ ¼ t0
p hN ðtÞ ¼ þ 2
Z
t
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h_ N ðsÞds
t0
a < 0; b < 0
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where a and b are both constants, and they equal to –p/90 and –p/30 respectively in this paper. Substituting Eq. (2) into Eq. (1), MN in Eq. (1) can be eliminated. _ Consequently, the longitudinal Let Vx ¼ x_ and Vy ¼ y. dynamic model of the tiltrotor aircraft can be described as the following by adding one kinematic equation: ( _ y ; q_ T ¼ D2 _ x ; V D1 V ð3Þ h_ ¼ q
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where q denotes the pitch angular velocity, and 3 2 m 0 a13 7 6 D1 ¼ 4 0 m a23 5 a31 a32 a33
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8 > < a13 ¼ a31 ¼ m1 l1 cos h m2 l2 sin hd a23 ¼ a32 ¼ m1 l1 sin h þ m2 l2 cos hd > : a33 ¼ I1 þ I2 þ m1 l21 þ m2 l22
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hN sin hd Q1 þ m1 l1 q2 sin h þ m2 l2 ½ðh_ N þ qÞ cos hd þ € 2
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6 7 2 7 D2 ¼ 6 hN cos hd 5 4 Q2 m1 l1 q2 cos h þ m2 l2 ½ðh_ N þ qÞ sin hd € Q3 ðI2 þ m2 l22 Þ€ hN
Longitudinal diagram of a tiltrotor aircraft.
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where I1 and I2 are the moments of inertia of the aircraft body and the nacelle-rotor system, respectively. According to the aerodynamic theory of blades, LT and a1s can be calculated as follows: LT ¼ LT ðX; h0 ; B1 ; V; a; hN Þ ð4Þ a1s ¼ a1s ðh0 ; B1 ; V; a; hN Þ where X, h0, and B1 represent the rotational speed, collective pitch, and longitudinal cyclic pitch of the rotors, respectively. Aerodynamic coefficients of L, D, and M were obtained from wind tunnel test data of Bell Helicopter Company and NASA’s Langley Laboratory.29,30 Furthermore, the wings are disturbed by rotor wake in the helicopter and transition modes.1,31 As a result, the overall aerodynamic forces and moments without considering the contribution of the rotors can be written into the following form: 8 > < L ¼ LðV; a; hN ; Vi ; kÞ ð5Þ D ¼ DðV; a; hN ; Vi ; kÞ > : M ¼ MðV; a; hN ; Vi ; k; de Þ where k and Vi represent the influence coefficient and induced speed of the rotors, respectively, while de means the elevator angle. It is worth noting that there exists a control coupling between B1 and de. They are both controlled by the displacement of the longitudinal control bar dlong. Therefore, four flight states (Vx, Vy, q, h) and two control inputs (h0, dlong) used for describing the longitudinal dynamic characteristics of the tiltrotor aircraft are all acquired. Furthermore, let X ¼ ½Vx ; Vy ; q; hT mean the state variables, and U ¼ ½h0 ; dlong T mean the control variables. As a result, Eq. (3) can be rewritten into the following general form: d Xi ¼ fi ðX; UÞ; dt
i ¼ 1; 2; 3; 4
ð6Þ
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3. Generate reference flight state from nacelle angle
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The reference flight state is the desired motion of the tiltrotor in the transition mode. For the tiltrotor transiting from the helicopter mode to the airplane mode, the desired motion is the one that can keep the altitude unchanged and drive the airspeed velocity to increase monotonically during the whole transition process within the safe flight envelope. Note that before determining the reference flight state concerning airspeed velocity commands and attitude commands, the scheduling curve for the nacelle angle is assumed to be chosen according to Eq. (2). In this section, the reference flight state is generated by analyzing the flight envelope. An offline library of the reference commands is founded for real-time visiting during the transition flight.
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3.1. Flight envelope and tilting rule determination
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The flight envelope shows the safe airspeed velocity ranges of the tiltrotor aircraft at different nacelle angles. Taking the XV-15 tiltrotor aircraft as an example, its flight envelope is shown in Fig. 2. Moreover, the horizontal and vertical axes mean the airspeed velocity and nacelle angle, respectively.
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Fig. 2
Flight envelope and 3 typical transition curves.
There are so many possible tilting transition curves in the flight envelop if the tiltrotor aircraft transits from one certain flight condition in the helicopter mode to one certain flight condition in the airplane mode. For example, Fig. 2 demonstrates 3 possible tilting transition curves when the airspeed velocities of the tiltrotor in the helicopter and airplane modes are equal to 30 m=s and 90 m=s, respectively, and their numbers are also annotated in Fig. 2.
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3.2. Reference for velocity
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Velocity V can be expressed as a function of hN according to each tilting transition curve demonstrated in Fig. 2. Combining Fig. 2 with Eq. (2), the reference of V can be defined as follows:
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VRef ¼ VðhN Þ ¼ VðhN ðtÞÞ
ð7Þ
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Taking the No. 1 tilting transition curve as an example, specific expressions of Eq. (7) are written as follows: 8 VT þ p2 ðV0 VT Þ p2 þ a2 t2 > > > > > 0 6 t t0 < ba > > > > > < VT þ 2 ðV0 VT Þ 2 b2 þ bt 2a p p VRef ðtÞ ¼ ð8Þ b p > > 6 t t < 0 > a 2b > > 2 > > p > VT pa ðV0 VT Þ t þ 2b ba > > : p p 2b 6 t t0 < 2b þ ba
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where V0 and VT represent the airspeed velocities of the tiltrotor in the helicopter and airplane modes, respectively, while t0 is the time instant when the nacelles are beginning to tilt. Furthermore, V can be divided into the horizontal component Vx and the vertical component Vy. In order to keep the altitude, the reference value of Vy should always be zero. As a result, we have Vx;Ref ðtÞ ¼ VRef ðtÞ ð9Þ Vy;Ref ðtÞ 0
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3.3. References for pitch angle and angular velocity
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The reference of h is generated based on its static characteristics curves in this paper. First of all, we trim the tiltrotor with different nacelle angles for a straight-level flight, and compute static characteristic curves of h. When the aircraft is in a steady forward flight, h is equal to the AoA. Moreover, Vx, Vy, and hN are known variables while h, MN, h0, and dlong are unknown variables, and the four generalized forces in Eq. (1) are all zero at the equilibrium state. Then these four unknown variables
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5 d Xi ¼ fi ðX1 ; X2 ; X3 ; X4 ; U1 ; U2 Þ i ¼ 1; 2; 3; 4 dt
ð13Þ
Moreover, Eq. (13) can be rewritten based on the smalldisturbance theory as follows32: d ðX þ DXi Þ ¼ fi ðX þ DX; U þ DUÞ dt i d X ¼ fi ðX ; U Þ dt i i ¼ 1; 2; 3; 4 Fig. 3
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can be derived. As illustrated in Fig. 3, there are five static characteristic curves with different nacelle angles. As described in Fig. 3, one and only one trim pitch angle can be determined by one certain airspeed velocity and one certain nacelle angle. Combining Eqs. (7), (9), and Fig. 3, the reference of h can be expressed as the following equation: hRef ¼ hðVRef ; hN Þ ¼ hðVðhN ðtÞÞ; hN ðtÞÞ ¼ hðhN ðtÞÞ
ð10Þ
It can be seen that hRef is dependent on hN. Meanwhile, a large amount of computation is required if the static characteristics curves are computed with all nacelle angles, because calculation of the static characteristics curves is complex. Therefore, five trim pitch angles can be obtained according to the five static characteristics curves, and five time instants can be calculated based on Eq. (2). Then an approximate hRef can be obtained according to the curve fitting results, and it is written as ~ hRef ðtÞ in this paper. Furthermore, the reference of q can be expressed as follows: qRef ðtÞ ¼
d~ hRef ðtÞ dt
ð11Þ
It should be noted that the beginning of the transition mode is the end of the helicopter mode, and the end of the transition mode is the beginning of the airplane mode. Therefore, the values of qRef ðtÞ at these two time instants should both be zero, and the highest degree of function h~Ref ðtÞ should be six because of five known function values and two known derivative values. Consequently, three curves of the approximate reference pitch angles are illustrated in Fig. 3. In summary, the reference flight state can be determined by each tilting transition curve in the flight envelope, and the steps are demonstrated as follows. Step 1: select a tilting transition curve in the flight envelope. Step 2: calculate VRef according to the transition curve and Eq. (2). Step 3: based on Eqs. (2), (7), and the static characteristics curves of h, compute hRef and qRef . Finally, the reference flight state is obtained. 8 Vx;Ref ðtÞ ¼ VRef ðhN ðtÞÞ > > > > < Vy;Ref ðtÞ 0 ð12Þ > qRef ðtÞ ¼ dtd ~hRef ðtÞ > > > : hRef ðtÞ ¼ h~Ref ðtÞ
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4. Controller design
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It is seen from Eq. (6) that, at any instant, the dynamic models of the tiltrotor aircraft can be written as follows:
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Taking ð@f1 =@X1 Þ as an example, it describes the value of the partial derivative @f1 =@X1 at a certain sampling instant, and it can be solved as follows:
@f1 f ðX þ ½dX1 ; 0; 0; 0T ; U Þ 1 2dX1 @X1 f1 ðX ½dX1 ; 0; 0; 0T ; U Þ 2dX1
ð16Þ
where dX1 is a small quantity. Similarly, the other three equations of Eq. (13) can also be rewritten as the form of Eq. (15). In advanced control design for nonlinear time-varying systems, a nonlinear model is often linearized at each sampling point while the operating point of the controlled plant is changing. In Trajectory Linearization Control (TLC), Zhu et al. have been using Taylor expansion type real-time linearization and the corresponding linear models to update controller gains. TLC has been applied in the field of space shuttle control (e.g., application to X33, Marshall space flight center).34 Besides, Pu et al. applied TLC to the control of a flexible hypersonic vehicle.35 Let T represent the sampling interval, and t ¼ KT means the current sampling instant. As a result, we have 9t 2 ½kT; ðk þ 1ÞT d s:t: DXðtÞ ¼ AðtÞDXðtÞ þ BðtÞDUðtÞ dt
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where X and U represent the basic values of X and U at a certain sampling instant, respectively, while DX and DU mean the incremental values, respectively. The prerequisite for Taylor expansion is that the system dynamics have continuouslydifferentiable characteristics. In this paper, the tiltrotor flight dynamics have time-varying and continuously-differentiable characteristics in the transition mode; therefore, Eq. (14) can be transited into a linear model using Taylor expansion type real-time linearization as follows:33
4 2 X X d @fi @fi DXi ¼ DXj þ DUk ð15Þ dt @Xj @Uk j¼1;j–i k¼1
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ð14Þ
Static characteristic curves of pitch angle.
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ð17Þ 404
where the order of matrix A is 4 4, the order of matrix B is 4 2, and we have
@fi @fi Aði; jÞ ¼ ; Bði; jÞ ¼ @Xj @Uj Obviously, the values of A and B depend on the time, so Eq. (17) is a time-varying linear model. Furthermore, the discrete form of Eq. (17) is
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DXððk þ 1ÞTÞ ¼ ðTAðkTÞ þ I4 ÞDXðkTÞ þ TBðkTÞDUðkTÞ ð18Þ
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where I4 denotes the four-order unit matrix. It is known that DXðkTÞ ¼ XðkTÞ X ðkTÞ
Accordingly, the following formulation can be derived: DXððk þ 1ÞTÞ ¼ XðkTÞ X ðkTÞ þ TBðkTÞDUðkTÞ
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ð20Þ
This paper defines a function to calculate the basic values of state variables, and the expression of the function is as follows:
X ðtÞ ¼ FðXðkTÞ; U ðkTÞ; t kTÞ
ð21Þ
JðkÞ
In Eq. (21), U ðkTÞ are the basic control variables at the current sampling instant, X ðtÞ are the basic state variables after a period of time, and F means that X ðtÞ is calculated according to the nonlinear dynamic models which are represented as Eq. (3). What’s more, this function lets the initial values of the state and control variables equal to XðkTÞ and U ðkTÞ respectively, and then computes new state variables with the control inputs remaining unchanged. Therefore, X ðtÞis equal to the new state variables. Obviously, we can get
ð26Þ
where HðkÞ ¼ T
BTðkÞ QBðkÞ
þR
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KðkÞ ¼ HðkÞ Uðk1Þ þ TBTðkÞ QðRefðkþ1Þ FðkÞ Þ
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LðkÞ ¼ ðRefðkþ1Þ FðkÞ þ TBðkÞ Uðk1Þ ÞT QðRefðkþ1Þ FðkÞ þ TBðkÞ Uðk1Þ Þ þ UTðk1Þ RUðk1Þ
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In addition, the amplitudes and rates of the control inputs are also limited to certain ranges as _ min 6 UðkÞ Uðk1Þ 6 TU _ max TU Umin 6 UðkÞ 6 Umax
ð27Þ
Moreover, Eq. (27) can be rewritten as follows:
where variables with subscript (k) represent their values at the current sampling instant. It can be seen from Eq. (23) that the basic control variables at the current sampling instant are the real control variables at the last sampling instant. The basic state variables at the next sampling instant are calculated according to the real state variables, the basic control variables at the current sampling instant, and the sampling time interval. Moreover, Xðkþ1Þ are the state variables at the next sampling instant. As a result, Eq. (23) can be simplified as follows:
MUðkÞ 6 cðkÞ
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M ¼ ½I2 ; I2 ; I2 ; I2 T
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_ max þ Uðk1Þ ; TU _ min Uðk1Þ T cðkÞ ¼ Umax ; Umin ; TU
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ð28Þ
where I2 means the two-order unit matrix. To minimize JðkÞ subject to inequality constraints, let us consider the so-called Lagrange expression which takes into account the constraints as J1;ðkÞ ¼ JðkÞ þ
kTðkÞ ðMUðkÞ
cðkÞ Þ
ð29Þ
where kðkÞ contains the Lagrange multipliers. According to the Karush-Kuhn-Tucker condition, the following expression is obtained36,37: 8 @J 1;ðkÞ > > < @UðkÞ ¼ 0 ð30Þ kðkÞ P 0 > > : kT ðMU c Þ ¼ 0 ðkÞ
where Refðkþ1Þ ¼ ½Vx;Ref ; Vy;Ref ; qRef ; hRef Tðkþ1Þ , which represents the reference values of state variables at the next sampling instant. Q and R are the output weighting matrix and control input weighting matrix with appropriate dimensions, and their diagonal elements are all not zero. Furthermore, Eq. (25) can be reformulated as follows:
505 506 507 508 510 511 512 513 514
This problem is equivalent to
517 518
ð31Þ
The minimization over UðkÞ is unconstrained and attained by
ð25Þ
504
516
max minJ1;ðkÞ
ð24Þ
503
ðkÞ
ðkÞ
kðkÞ P0 UðkÞ
Eq. (24) reveals the nature of aircraft dynamics. In detail, the first term on the right indicates the change of the state caused by itself with the control unchanged, and the second term signifies the change of the state caused by a change of the control. The objective function JðkÞ is used to compute the optimal solution for UðkÞ and to make Xðkþ1Þ close to the references as much as possible, and it also should depress dramatic changes of the control variables as 1 1 JðkÞ ¼ kRefðkþ1Þ Xðkþ1Þ k2Q þ kUðkÞ Uðk1Þ k2R 2 2
490 491
498
ð22Þ
Consequently, the discrete model can be further written as follows: 8 UðkÞ ¼ Uðk1Þ > > > > > > < DUðkÞ ¼ UðkÞ UðkÞ Xðkþ1Þ ¼ FðXðkÞ ; UðkÞ ; TÞ ð23Þ > > > DXðkþ1Þ ¼ TBðkÞ DUðkÞ > > > : Xðkþ1Þ ¼ Xðkþ1Þ þ DXðkþ1Þ
Xðkþ1Þ ¼ FðkÞ þ TBðkÞ ðUðkÞ Uðk1Þ Þ FðkÞ ¼ FðXðkÞ ; Uðk1Þ ; TÞ
489
494 495
DXððk þ 1ÞTÞ ¼ TBðkTÞDUðkTÞ
478 479 480
2
X ðkTÞ ¼ FðXðkTÞ; U ðkTÞ; 0Þ ¼ XðkTÞ 440
ð19Þ
1 1 ¼ UTðkÞ HðkÞ UðkÞ UTðkÞ KðkÞ þ LðkÞ 2 2
UðkÞ ¼
520 521 522 523
H1 ðkÞ ðKðkÞ
M kðkÞ Þ
ð32Þ
T
It is known that R is reversible by its definition. Let ~ ¼ Q=b and b–0. Then it can be proven that HðkÞ will always Q be reversible by adjusting the values of Q as follows: HðkÞ ¼ ðbT
2
~ ðkÞ R1 BTðkÞ QB
þ I2 ÞR
ð33Þ
The determinant of HðkÞ can be computed by Eq. (33) with ~ ðkÞ R1 being li ði ¼ 1; 2Þ as follows: the eigenvalues of BTðkÞ QB ~ ðkÞ R detðHðkÞ Þ ¼ detðbT2 BTðkÞ QB ¼ detðRÞ
2 Y
1
525 526 527 528 529 531 532 533 534
þ I2 Þ detðRÞ
ðbT2 li þ 1Þ
ð34Þ
i¼1
Obviously, there always exists a b which satisfies the inequality that bT2 li –1ði ¼ 1; 2Þ, which ensures that inversion of HðkÞ exists. Substituting Eq. (32) into Eq. (31), we get
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Fig. 4 Control diagram of closed-loop tiltrotor aircraft system (the control law part and the controlled plant are located separately in the dashed box).
540
min
542
kðkÞ P0
543 544
where
1 T 1 k SðkÞ kðkÞ þ kTðkÞ TðkÞ þ VðkÞ 2 ðkÞ 2
ð35Þ
Parameter h0 dlong
T SðkÞ ¼ MH1 ðkÞ M
TðkÞ ¼ cðkÞ MH1 ðkÞ KðkÞ 546 547 548 549 551 552 553 554 555 557
VðkÞ ¼ KTðkÞ H1 ðkÞ KðkÞ LðkÞ Finally, we use Hildreth’s quadratic programming procedure to minimize the objective function as 1 1 J2;ðkÞ ¼ kTðkÞ SðkÞ kðkÞ þ kTðkÞ TðkÞ þ VðkÞ 2 2
ð36Þ
subject to kðkÞ P 0.38 Substituting the calculated kðkÞ into Eq. (32), the optimal control inputs can be obtained in the following form: ½h0 ; dlong TðkÞ ¼ UðkÞ
ð37Þ
559
Furthermore, the system structure of the tiltrotor aircraft with the controller is depicted in Fig. 4.
560
5. Simulation results and discussions
561
To demonstrate the performance of the controller, three scenarios are taken as examples, and the references of the state variables are calculated respectively. These three scenarios are one-by-one related to the three tilting transition curves in Fig. 2. It is worth noting that all these simulations are performed using the nonlinear model, and all the results are also obtained from the nonlinear model as described in Section 2. Some inherent parameters of the tiltrotor aircraft are shown in Table 1. Moreover, limitations of the control inputs are presented in Table 2. The specific expressions of the nacelle angle are described as follows:
558
562 563 564 565 566 567 568 569 570 571 572
Table 1
Table 2
Inherent parameters of the tilt rotor aircraft.
Parameter
Value
Parameter
Value
m1 (kg) m2 (kg) m (kg) I1 ðkg m2 Þ I2 ðkg m2 Þ
3844.4 2052.4 5896.8 17887.4 1916.32
l1 ðmÞ l2 ðmÞ l3 ðmÞ dAF ð Þ dTM ð Þ
0.717 0.574 1.42 14.28 19.57
8 € h > > > N > > € > < hN € hN > > > € > hN > > :€ hN
Limitations of control inputs. Amplitude limit
½30; 90 ð Þ ½0; 100%
¼ 0; h_ N ¼ 0; hN ¼ 90 ¼ 2; h_ N ¼ 2t; hN ¼ 90 ðt 10Þ2 ¼ 0; h_ N ¼ 6; hN ¼ 81 6ðt 13Þ ¼ 2; h_ N ¼ 56 þ 2t; hN ¼ 90 ¼ 0; h_ N ¼ 0; hN ¼ 0
Rate limit ½30; 30 ð Þ s1 ½75; 75% s1
573
0 s 6 t < 10 s 10 s 6 t < 13 s 13 s 6 t < 25 s 25 s 6 t < 28 s 28 s 6 t < 40 s ð38Þ 575
As can be seen from Eq. (38), the tiltrotor aircraft flies in the helicopter mode at the beginning, then transfers from the helicopter mode to the transition mode, and finally switches into the airplane mode with nacelle tilting completed. The entire flight process takes 40 s. In addition, there are three vertical gust disturbances during 4–6 s, 18–20 s, and 34–36 s of the process. Moreover, the model of the gust is a sine-squared one and its maximum speed is 1 m=s. As shown in Eq. (25), not only the references of the state variables, but also the values of Q and R are necessary. What’s more, the four parameters of Q are the weights of vx, vy, q, and h, respectively, while the two parameters of R are the weights of h0 and dlong, respectively. This paper selects different weight matrices and compares simulation results, and their values are finally determined as in Table 3. In addition, an i7-4790 CPU and the MATLAB 2016 software are used to calculate simulation results, and the sampling time interval for the controller is chosen as 100 ms while that for the nonlinear models is chosen as 10 ms. It is noted that the computational cost of solving the optimal control is approximately 2.6 ms. As a result, simulation results of Scenario 1 are shown in Fig. 5. The hN, Vx, Vy, q, h, h0, dlong, MN, a represent the nacelle angle, the horizontal velocity, the vertical velocity, the pitch angular velocity, the pitch angle, the collective pitch, the longitudinal control bar displacement, the nacelle tilt torque, and the AOA of the tiltrotor, respectively. References of the horizontal velocity, the vertical velocity, the pitch angular velocity and the pitch angle are calculated according to Eqs. (12) and (38).
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Values of matrix Q and Matrix R.
Transition curve
Q
R
Scenario 1 Scenario 2 Scenario 3
diagð1; 20; 300; 300Þ diagð1; 30; 300; 300Þ diagð1; 20; 400; 400Þ
diagð5; 5Þ diagð5; 5Þ diagð5; 5Þ
In Fig. 5, the three parts separated by two dotted lines represent the helicopter mode, the transition mode, and the airplane mode, respectively. As can be seen from these figures, the true states of the tiltrotor can track the reference states well and have good performances in dealing with disturbances caused by vertical gusts. Furthermore, simulation results of
Fig. 5
Scenario 2 are depicted in Fig. 6. Simulation results for Scenario 3 are shown in Fig. 7. It can be seen from Figs. 5–7 that the tracking performances for Vy, q, and h are better than that for Vx. This is because the rotor pull is mainly used for providing the aircraft lift when the tiltrotor has a large nacelle angle, and it is difficult to provide a sufficient acceleration for Vx in the early stage of the transition mode. This is also the reason why the tracking performance for Vx shown in Fig. 6 is the best of these three simulations. As can be seen from Figs. 5–7, in the early stage of the transition mode, the tracking performance for Vy in Scenario 2 is the best of those in the three scenarios, while the tracking performance for Vy in Scenario 3 is the worst. The reason is as follows: the reference airspeed velocity in Scenario 2
Simulation results for Scenario 1.
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Fig. 6
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9
Simulation results for Scenario 2.
increases slowly and smoothly over time in the early stage of the transition mode, but the reference airspeed velocity in Scenario 3 has a sudden change in the early stage of the transition mode. Meanwhile, in the late stage of the transition mode, the tracking performance for Vy in Scenario 3 is the best of those in the three scenarios, and the tracking performance for Vy in Scenario 2 is the worst. The reason is as follows: the reference airspeed velocity in Scenario 3 increased slowly and smoothly over time in the late stage of the transition mode, but the reference airspeed velocity in Scenario 2 appears a sudden change at the late stage of the transition mode, which can be seen in the first and second subfigures of Fig. 7 and Fig. 6, respectively. Considering the impacts of the gust disturbances on a flight, the simulation results show that the tiltrotor is more sensitive to the gust disturbance when the airspeed velocity is higher. The reason is that these disturbances lead to changes of the AoA, and even a small change of the AoA can still cause relatively larger aerodynamic changes when the airspeed velocity is high. In order to demonstrate the advantage of the controller with input constraints taken into account, this paper takes Scenario 1 as an example to compare the control method which considers the control input constraints to the one which does not consider the constraints. It is worth noting that the controller without considering the control input constraints is
derived based on Eq. (25). Responses of h are illustrated in Fig. 8, and the maximum speed of the gust is 14 m=s. Reference command of h for Fig. 8 is the same as that in Fig. 5. As shown in Fig. 8, the controller with input constraints taken into account has a better performance than that of the controller which does not consider input constraints. The latter controller results in obvious oscillations in all three flight modes, and the performance for the airplane mode is particularly unsatisfying in the case of gust disturbance. To summarize, the quadratic programming based optimal controller proposed in this paper is shown to be able to accomplish the tilting transition missions of the tiltrotor aircraft. Except for the transition mode, the controller can also result in satisfying command tracking performance when the tiltrotor operates in the helicopter or airplane mode. Furthermore, the controller can effectively deal with unexpected gust disturbances.
651
6. Conclusions
667
A tiltrotor aircraft is a nonlinear and multibody system with a high degree of freedom, and has strong time-varying characteristics when considering the nacelle tilting procedure. A quadratic programming based optimal control method is developed
668
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Fig. 7
Fig. 8
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Simulation results for Scenario 3.
Responses of h comparison between two controllers.
based on a tiltrotor Time-Varying Linear (TVL) model, and the proposed controller is thoroughly validated in this paper. (1) The TVL model is derived from a nonlinear Lagrangeform model through online linearization. The TVL model can effectively describe the dynamic characteristics of the tiltrotor in a finite time domain. Based on the TVL model and the regular optimal control strategy, an online optimization based optimal control approach which takes input constraints into account is proposed,
and a controller is presented. Moreover, the controller updates the parameters of the TVL model at each sampling instant according to flight conditions. Therefore, the controller can deal with the strong timevarying characteristics and control input constraints of the tiltrotor aircraft. (2) The proposed control approach is validated using data from an XV-15 tiltrotor. Three typical scenarios are simulated, and abrupt vertical gusts are introduced to show the interference immunity. The results show that the designed controller can achieve desired tracking performance for the tiltrotor throughout its operating envelop which includes a transition flight. In addition, the controller has sufficient capability to handle input constraints and accommodate gust disturbances. (3) In future research, attention will be paid to how to dynamically generate more reasonable reference commands for airspeed velocity and attitude by considering the tiltrotor maneuverability and energy consumption.
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Acknowledgement
701
This work was supported by the National Natural Science Foundation of China (No. 11502008).
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References
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No. of Pages 11
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Time-varying linear control for tiltrotor aircraft
4
Jing ZHANG a, Liguo SUN a,*, Xiangju QU a, Liuping WANG b
5 6
7
a b
School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China School of Electrical and Computer Engineering, RMIT University, Melbourne, VIC 3001, Australia
Received 12 January 2017; revised 26 May 2017; accepted 5 September 2017
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KEYWORDS
12
Constrained optimal control; Inertia/control couplings; Tiltrotor aircraft; Time-varying control; Transition mode
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Abstract Tiltrotor aircraft have three flight modes: helicopter mode, airplane mode, and transition mode. A tiltrotor has characteristics of highly nonlinear, time-varying flight dynamics and inertial/control couplings in its transition mode. It can transit from the helicopter mode to the airplane mode by tilting its nacelles, and an effective controller is crucial to accomplish tilting transition missions. Longitudinal dynamic characteristics of the tiltrotor are described by a nonlinear Lagrangeform model, which takes into account inertial/control couplings and aerodynamic interferences. Reference commands for airspeed velocity and attitude in the transition mode are calculated dynamically by visiting a command library which is founded in advance by analyzing the flight envelope of the tiltrotor. A Time-Varying Linear (TVL) model is obtained using a Taylorexpansion based online linearization technique from the nonlinear model. Subsequently, based on an optimal control concept, an online optimization based control method with input constraints considered is proposed. To validate the proposed control method, three typical tilting transition missions are simulated using the nonlinear model of XV-15 tiltrotor aircraft. Simulation results show that the controller can be used to control the tiltrotor throughout its operating envelop which includes a transition flight, and can also deal with vertical gust disturbances.
1. Introduction A tiltrotor aircraft, which combines the characteristics of helicopters and fixed-wing aircraft, consists of an aircraft body (including the fuselage and wings), engine nacelles, and rotors. * Corresponding author. E-mail addresses: [email protected], [email protected] (L. SUN). Peer review under responsibility of Editorial Committee of CJA.
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Unlike a traditional aircraft, a tiltrotor aircraft is a complex multibody system which can change its configuration by tilting the nacelles, and it owns three different flight modes, namely helicopter mode, airplane mode, and transition mode. The transition mode, which means the conversion between the other two flight modes, is a special mode of tiltrotors. A tiltrotor aircraft with two side-by-side rotors such as the XV-15 tiltrotor is studied in this paper. A tiltrotor owns the advantages of both helicopters and fixed-wing aircraft. Firstly, a tiltrotor can fly freely in different directions like a helicopter, and can also hover in the air. Secondly, a tiltrotor has a faster cruising speed than that of a helicopter. Thirdly, a tiltrotor can enhance civil or military
https://doi.org/10.1016/j.cja.2018.01.025 1000-9361 Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: ZHANG J et al. Time-varying linear control for tiltrotor aircraft, Chin J Aeronaut (2018), https://doi.org/10.1016/j.cja.2018.01.025
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transportation capability because it has three flight modes. However, a tiltrotor aircraft has some disadvantages such as low safety and low reliability. Firstly, the flight dynamics of a tiltrotor is much more complex than that of a traditional aircraft since the tiltrotor demonstrates different flight dynamic characteristics in different flight modes, and there exist inertial couplings between the aircraft body and nacelles. Secondly, a tiltrotor has low reliability because there are control couplings between its control action paths. Thirdly, the pilot workload of a tiltrotor would be higher especially in the transition period, because it has the characteristics of complex flight dynamics, inertial couplings, and control couplings. This paper studies the flight control for tiltrotor aircraft to deal with the complex flight dynamics and inertial couplings. To obtain a reasonable and reliable dynamic model is crucial for aircraft controller design. A large amount of literature considers a tiltrotor as a single rigid body, and describes its dynamic characteristics through three-Degree-Of-Freedom (DOF) longitudinal modeling or six-DOF modeling, which is similar to that for traditional aircraft.1–3 However, a tiltrotor is a kind of morphing aircraft which tilts its nacelles in its special transition mode, so a traditional flight dynamic model cannot reveal the inertial couplings and dynamic characteristics in the transition mode. Certain literature regards a tiltrotor as a multi-rigid-body system in view of the inertial couplings. For example, Li et al. considered a tiltrotor as multiple entities, and developed a twelve-DOF dynamic model for tiltrotors based on multibody dynamics.4 The multibody model can clearly characterize the inertial couplings and complex flight dynamics of a tiltrotor, but its expression is too complicated for controller design. Later on, based on the Lagrange’s equation, Zhang et al. built a multibody longitudinal nonlinear model of a tiltrotor, and the derived model is in a more concise form and more suitable for the longitudinal controller design of the tiltrotor.5 It is worth noting that modeling of multibody systems has experienced a remarkable development in space flight.6,7 Furthermore, founding an aeroelastic model is beneficial to analyzing the stability of the tiltrotor.8 The literature about tiltrotor aircraft control can be classified into two categories: linear control and nonlinear control. Firstly, linear controllers were designed on the basis of a linear dynamic model and the information of certain flight conditions, and references are organized chronologically. A Model Predictive Control (MPC) method was adopted to design a flight controller for the hovering mode and three other typical flight scenarios of a tiltrotor.9 An attitude controller for a tiltrotor in the helicopter hovering mode was designed using MPC.10 An optimal control approach was proposed to deal with gusts effects on a tiltrotor in helicopter and airplane modes.11 Minimum energy controllers were designed based on the helicopter mode and the airplane mode, respectively.12 Based on optimal preview control, an attitude controller was developed for a particular flight state in the transition mode of a tiltrotor.13 In consideration of model errors, a controller was designed for the airplane mode of a tiltrotor based on a sliding mode method.14 It should be noted that the above mentioned linear controllers were mainly designed for one or several particular flight states of a tiltrotor, and they need gain scheduling to achieve control of the whole conversion process of the tiltrotor. Secondly, nonlinear control approaches have the advantage that the nacelle angle does not need to be assumed fixed when designing tiltrotor flight controllers. In
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J. ZHANG et al. Ref. 15, a robust nonlinear controller was designed for a tiltrotor in the hover mode. Considering system uncertainties and disturbances, an adaptive control method was developed based on Neural Networks (NNs),16 but it requires using different NN weights for different flight states, e.g., different nacelle angles. Based on the work,16 an online NN modelling method was introduced to develop a fully adaptive flight control method, which achieves not only longitudinal control but also lateral control of a tiltrotor. It should be mentioned that this method needs to update the entire full-state nonlinear model of a tiltrotor for different flight states.17 Considering the control difficulty brought by the tiltrotor mode switching process and the tiltrotor configuration change during the transition mode, Ref. 18 presented a nonlinear control method and specifically studied the transition process control. Nevertheless, the tiltrotor controller designed in Ref. 18 fails to consider input constraints and control couplings. A tiltrotor aircraft, especially in the transition mode, is a time-varying and strongly nonlinear system, and there exist couplings between different control action paths. As an alternative to linear controllers with a gain scheduling mechanism and nonlinear controllers mentioned above, it is a wise choice to convert a time-varying highly nonlinear system into a TimeVarying Linear (TVL) system, and then design a controller based on the TVL model. There exist a large number of approaches in literature considering founding a TVL model and TVL control.19–22 The Linear Parameter Varying (LPV) control method, which approximates a nonlinear system as a TVL system, is a classical control method which has been widely applied to aircraft with a variable structure.23–27 However, a single LPV controller designed for a single flight state cannot accomplish the whole transition control of tiltrotors operating in the transition mode. Moreover, the switching LPV control method based on the Lyapunov function is computationally expensive because it needs to solve a large number of linear matrix inequalities.28 The objective of this paper is to propose a control approach which is competent for achieving effective control for tiltrotor aircraft in the transition mode. More specifically, this paper contributes to online model linearization, dynamic generation strategy of reference commands, and real-time optimization based optimal control of a tiltrotor. Firstly, a nonlinear and time-varying Lagrange-form tiltrotor model is presented, which takes the control couplings of tiltrotor aircraft into account. Secondly, how to derive a TVL model from the time-varying nonlinear model is addressed. Note that the TVL model updates its parameters at each sampling instant according to the flight conditions. Thirdly, a trim-condition based approach is developed to calculate the reference commands for airspeed velocity and attitude. To update the reference commands in real-time, a library for the flight envelope of the tiltrotor and the static characteristics curves of the pitch angle is founded. Fourthly, inspired by the MPC concept, an online optimization based optimal control approach is proposed for tiltrotor aircraft, especially when considering transition-mode flight missions. Moreover, the proposed control approach allows for conducting input constraints, and the control inputs are the solution of a constrained online optimization problem. The structure of this paper is organized as follows. Section 2 presents the longitudinal dynamic model of a tiltrotor which is used for designing a controller. In Section 3, how to calculate
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the reference commands of the tiltrotor is introduced. The detailed controller design is described in Section 4. In Section 5, three cases of tilting transition missions are simulated and discussed. Finally, conclusions are given in Section 6.
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2. Longitudinal nonlinear model of a tiltrotor
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A nonlinear modeling method for tiltrotor aircraft was proposed in Ref. 5, and the formulation of the nonlinear modeling method is briefly given in this section for completeness concern. Fig. 1 shows the longitudinal diagram of a tiltrotor aircraft. As depicted in Fig. 1, the longitudinal diagram of the tiltrotor aircraft is described in the OgXgYg coordinate system. The OgXgYg is an inertial coordinate system which is fixed on the earth ground, and the axis OgYg is parallel to the local plumb line and points upward. The point O means the position of the nacelle shaft while point A indicates the position of the rotor hub. Points G1, G2, and G represent the mass centers of the aircraft body, the nacelle-rotor system, and the whole aircraft, respectively. Moreover, h describes the pitch angle, and hN represents the nacelle angle, while a indicates the Angle of Attack (AoA). In addition, V is the airspeed velocity of the tiltrotor. L, D, and M are the lift, drag, and pitching moments of the aircraft frame (including the aircraft body and the nacelles), respectively, and they are transited from the aerodynamic center to point G. What’s more, MN means the torque to tilt the nacelle-rotor system, and LT represents the aerodynamic pull of the rotors, while a1s is the angle between the rotor disc and the hub disc. According to the Lagrange’s equations, the position coordinates of point O (x, y), h, and hN are selected as the generalized coordinates. Let hAF ¼ h dAF , hTM ¼ h þ hN dTM , and hd ¼ h þ hN . Then, the expressions of generalized forces are described as follows: 8 Q1 ¼ L sinða hÞ D cosða hÞ þ LT cosða1s þ hd Þ > > > > Q > < 2 ¼ L cosða hÞ þ D sinða hÞ þ LT sinða1s þ hd Þ mg Q3 ¼ Q4 MN þ M þ m1 l1 ðL sin a D cos a þ mg cos hAF Þ=m > > > Q ¼ m2 l2 ½L cosða þ hN Þ þ D sinða þ hN Þ mg cos hTM =m > > : 4 þMN þ l3 LT sin a1s ð1Þ
Fig. 1
3 where m, m1, and m2 are the masses of the whole tiltrotor, the aircraft body, and the nacelle-rotor system, respectively, while l1, l2, and l3 denote the lengths of OA, OG1, and OG2, respectively. Similar to the case in Ref. 4, the nacelles are designed as uniform rotation in this paper. Taking into account that the nacelles should switch between rotating and stopping, the scheduling rules of hN are designed as follows: 8 b > < a 0 6 t t0 < a p € hN ðtÞ ¼ 0 ba 6 t t0 < 2b > : p p a 2b 6 t t0 < 2b þ ba Z t € ð2Þ hN ðsÞds h_ N ðtÞ ¼ t0
p hN ðtÞ ¼ þ 2
Z
t
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h_ N ðsÞds
t0
a < 0; b < 0
204
where a and b are both constants, and they equal to –p/90 and –p/30 respectively in this paper. Substituting Eq. (2) into Eq. (1), MN in Eq. (1) can be eliminated. _ Consequently, the longitudinal Let Vx ¼ x_ and Vy ¼ y. dynamic model of the tiltrotor aircraft can be described as the following by adding one kinematic equation: ( _ y ; q_ T ¼ D2 _ x ; V D1 V ð3Þ h_ ¼ q
205
where q denotes the pitch angular velocity, and 3 2 m 0 a13 7 6 D1 ¼ 4 0 m a23 5 a31 a32 a33
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8 > < a13 ¼ a31 ¼ m1 l1 cos h m2 l2 sin hd a23 ¼ a32 ¼ m1 l1 sin h þ m2 l2 cos hd > : a33 ¼ I1 þ I2 þ m1 l21 þ m2 l22
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2
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213
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hN sin hd Q1 þ m1 l1 q2 sin h þ m2 l2 ½ðh_ N þ qÞ cos hd þ € 2
3
6 7 2 7 D2 ¼ 6 hN cos hd 5 4 Q2 m1 l1 q2 cos h þ m2 l2 ½ðh_ N þ qÞ sin hd € Q3 ðI2 þ m2 l22 Þ€ hN
Longitudinal diagram of a tiltrotor aircraft.
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J. ZHANG et al.
where I1 and I2 are the moments of inertia of the aircraft body and the nacelle-rotor system, respectively. According to the aerodynamic theory of blades, LT and a1s can be calculated as follows: LT ¼ LT ðX; h0 ; B1 ; V; a; hN Þ ð4Þ a1s ¼ a1s ðh0 ; B1 ; V; a; hN Þ where X, h0, and B1 represent the rotational speed, collective pitch, and longitudinal cyclic pitch of the rotors, respectively. Aerodynamic coefficients of L, D, and M were obtained from wind tunnel test data of Bell Helicopter Company and NASA’s Langley Laboratory.29,30 Furthermore, the wings are disturbed by rotor wake in the helicopter and transition modes.1,31 As a result, the overall aerodynamic forces and moments without considering the contribution of the rotors can be written into the following form: 8 > < L ¼ LðV; a; hN ; Vi ; kÞ ð5Þ D ¼ DðV; a; hN ; Vi ; kÞ > : M ¼ MðV; a; hN ; Vi ; k; de Þ where k and Vi represent the influence coefficient and induced speed of the rotors, respectively, while de means the elevator angle. It is worth noting that there exists a control coupling between B1 and de. They are both controlled by the displacement of the longitudinal control bar dlong. Therefore, four flight states (Vx, Vy, q, h) and two control inputs (h0, dlong) used for describing the longitudinal dynamic characteristics of the tiltrotor aircraft are all acquired. Furthermore, let X ¼ ½Vx ; Vy ; q; hT mean the state variables, and U ¼ ½h0 ; dlong T mean the control variables. As a result, Eq. (3) can be rewritten into the following general form: d Xi ¼ fi ðX; UÞ; dt
i ¼ 1; 2; 3; 4
ð6Þ
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3. Generate reference flight state from nacelle angle
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The reference flight state is the desired motion of the tiltrotor in the transition mode. For the tiltrotor transiting from the helicopter mode to the airplane mode, the desired motion is the one that can keep the altitude unchanged and drive the airspeed velocity to increase monotonically during the whole transition process within the safe flight envelope. Note that before determining the reference flight state concerning airspeed velocity commands and attitude commands, the scheduling curve for the nacelle angle is assumed to be chosen according to Eq. (2). In this section, the reference flight state is generated by analyzing the flight envelope. An offline library of the reference commands is founded for real-time visiting during the transition flight.
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3.1. Flight envelope and tilting rule determination
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The flight envelope shows the safe airspeed velocity ranges of the tiltrotor aircraft at different nacelle angles. Taking the XV-15 tiltrotor aircraft as an example, its flight envelope is shown in Fig. 2. Moreover, the horizontal and vertical axes mean the airspeed velocity and nacelle angle, respectively.
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Fig. 2
Flight envelope and 3 typical transition curves.
There are so many possible tilting transition curves in the flight envelop if the tiltrotor aircraft transits from one certain flight condition in the helicopter mode to one certain flight condition in the airplane mode. For example, Fig. 2 demonstrates 3 possible tilting transition curves when the airspeed velocities of the tiltrotor in the helicopter and airplane modes are equal to 30 m=s and 90 m=s, respectively, and their numbers are also annotated in Fig. 2.
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3.2. Reference for velocity
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Velocity V can be expressed as a function of hN according to each tilting transition curve demonstrated in Fig. 2. Combining Fig. 2 with Eq. (2), the reference of V can be defined as follows:
288
VRef ¼ VðhN Þ ¼ VðhN ðtÞÞ
ð7Þ
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Taking the No. 1 tilting transition curve as an example, specific expressions of Eq. (7) are written as follows: 8 VT þ p2 ðV0 VT Þ p2 þ a2 t2 > > > > > 0 6 t t0 < ba > > > > > < VT þ 2 ðV0 VT Þ 2 b2 þ bt 2a p p VRef ðtÞ ¼ ð8Þ b p > > 6 t t < 0 > a 2b > > 2 > > p > VT pa ðV0 VT Þ t þ 2b ba > > : p p 2b 6 t t0 < 2b þ ba
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where V0 and VT represent the airspeed velocities of the tiltrotor in the helicopter and airplane modes, respectively, while t0 is the time instant when the nacelles are beginning to tilt. Furthermore, V can be divided into the horizontal component Vx and the vertical component Vy. In order to keep the altitude, the reference value of Vy should always be zero. As a result, we have Vx;Ref ðtÞ ¼ VRef ðtÞ ð9Þ Vy;Ref ðtÞ 0
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3.3. References for pitch angle and angular velocity
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The reference of h is generated based on its static characteristics curves in this paper. First of all, we trim the tiltrotor with different nacelle angles for a straight-level flight, and compute static characteristic curves of h. When the aircraft is in a steady forward flight, h is equal to the AoA. Moreover, Vx, Vy, and hN are known variables while h, MN, h0, and dlong are unknown variables, and the four generalized forces in Eq. (1) are all zero at the equilibrium state. Then these four unknown variables
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5 d Xi ¼ fi ðX1 ; X2 ; X3 ; X4 ; U1 ; U2 Þ i ¼ 1; 2; 3; 4 dt
ð13Þ
Moreover, Eq. (13) can be rewritten based on the smalldisturbance theory as follows32: d ðX þ DXi Þ ¼ fi ðX þ DX; U þ DUÞ dt i d X ¼ fi ðX ; U Þ dt i i ¼ 1; 2; 3; 4 Fig. 3
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358
can be derived. As illustrated in Fig. 3, there are five static characteristic curves with different nacelle angles. As described in Fig. 3, one and only one trim pitch angle can be determined by one certain airspeed velocity and one certain nacelle angle. Combining Eqs. (7), (9), and Fig. 3, the reference of h can be expressed as the following equation: hRef ¼ hðVRef ; hN Þ ¼ hðVðhN ðtÞÞ; hN ðtÞÞ ¼ hðhN ðtÞÞ
ð10Þ
It can be seen that hRef is dependent on hN. Meanwhile, a large amount of computation is required if the static characteristics curves are computed with all nacelle angles, because calculation of the static characteristics curves is complex. Therefore, five trim pitch angles can be obtained according to the five static characteristics curves, and five time instants can be calculated based on Eq. (2). Then an approximate hRef can be obtained according to the curve fitting results, and it is written as ~ hRef ðtÞ in this paper. Furthermore, the reference of q can be expressed as follows: qRef ðtÞ ¼
d~ hRef ðtÞ dt
ð11Þ
It should be noted that the beginning of the transition mode is the end of the helicopter mode, and the end of the transition mode is the beginning of the airplane mode. Therefore, the values of qRef ðtÞ at these two time instants should both be zero, and the highest degree of function h~Ref ðtÞ should be six because of five known function values and two known derivative values. Consequently, three curves of the approximate reference pitch angles are illustrated in Fig. 3. In summary, the reference flight state can be determined by each tilting transition curve in the flight envelope, and the steps are demonstrated as follows. Step 1: select a tilting transition curve in the flight envelope. Step 2: calculate VRef according to the transition curve and Eq. (2). Step 3: based on Eqs. (2), (7), and the static characteristics curves of h, compute hRef and qRef . Finally, the reference flight state is obtained. 8 Vx;Ref ðtÞ ¼ VRef ðhN ðtÞÞ > > > > < Vy;Ref ðtÞ 0 ð12Þ > qRef ðtÞ ¼ dtd ~hRef ðtÞ > > > : hRef ðtÞ ¼ h~Ref ðtÞ
359
4. Controller design
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It is seen from Eq. (6) that, at any instant, the dynamic models of the tiltrotor aircraft can be written as follows:
361 362
Taking ð@f1 =@X1 Þ as an example, it describes the value of the partial derivative @f1 =@X1 at a certain sampling instant, and it can be solved as follows:
@f1 f ðX þ ½dX1 ; 0; 0; 0T ; U Þ 1 2dX1 @X1 f1 ðX ½dX1 ; 0; 0; 0T ; U Þ 2dX1
ð16Þ
where dX1 is a small quantity. Similarly, the other three equations of Eq. (13) can also be rewritten as the form of Eq. (15). In advanced control design for nonlinear time-varying systems, a nonlinear model is often linearized at each sampling point while the operating point of the controlled plant is changing. In Trajectory Linearization Control (TLC), Zhu et al. have been using Taylor expansion type real-time linearization and the corresponding linear models to update controller gains. TLC has been applied in the field of space shuttle control (e.g., application to X33, Marshall space flight center).34 Besides, Pu et al. applied TLC to the control of a flexible hypersonic vehicle.35 Let T represent the sampling interval, and t ¼ KT means the current sampling instant. As a result, we have 9t 2 ½kT; ðk þ 1ÞT d s:t: DXðtÞ ¼ AðtÞDXðtÞ þ BðtÞDUðtÞ dt
366 367
369
where X and U represent the basic values of X and U at a certain sampling instant, respectively, while DX and DU mean the incremental values, respectively. The prerequisite for Taylor expansion is that the system dynamics have continuouslydifferentiable characteristics. In this paper, the tiltrotor flight dynamics have time-varying and continuously-differentiable characteristics in the transition mode; therefore, Eq. (14) can be transited into a linear model using Taylor expansion type real-time linearization as follows:33
4 2 X X d @fi @fi DXi ¼ DXj þ DUk ð15Þ dt @Xj @Uk j¼1;j–i k¼1
365
ð14Þ
Static characteristic curves of pitch angle.
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370 371 372 373 374 375 376 377 378 379
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387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402
ð17Þ 404
where the order of matrix A is 4 4, the order of matrix B is 4 2, and we have
@fi @fi Aði; jÞ ¼ ; Bði; jÞ ¼ @Xj @Uj Obviously, the values of A and B depend on the time, so Eq. (17) is a time-varying linear model. Furthermore, the discrete form of Eq. (17) is
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DXððk þ 1ÞTÞ ¼ ðTAðkTÞ þ I4 ÞDXðkTÞ þ TBðkTÞDUðkTÞ ð18Þ
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J. ZHANG et al. 476
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where I4 denotes the four-order unit matrix. It is known that DXðkTÞ ¼ XðkTÞ X ðkTÞ
Accordingly, the following formulation can be derived: DXððk þ 1ÞTÞ ¼ XðkTÞ X ðkTÞ þ TBðkTÞDUðkTÞ
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457 458 459 460 461 462 463 464 465 466 467 469 470 471 472 473 474 475
ð20Þ
This paper defines a function to calculate the basic values of state variables, and the expression of the function is as follows:
X ðtÞ ¼ FðXðkTÞ; U ðkTÞ; t kTÞ
ð21Þ
JðkÞ
In Eq. (21), U ðkTÞ are the basic control variables at the current sampling instant, X ðtÞ are the basic state variables after a period of time, and F means that X ðtÞ is calculated according to the nonlinear dynamic models which are represented as Eq. (3). What’s more, this function lets the initial values of the state and control variables equal to XðkTÞ and U ðkTÞ respectively, and then computes new state variables with the control inputs remaining unchanged. Therefore, X ðtÞis equal to the new state variables. Obviously, we can get
ð26Þ
where HðkÞ ¼ T
BTðkÞ QBðkÞ
þR
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KðkÞ ¼ HðkÞ Uðk1Þ þ TBTðkÞ QðRefðkþ1Þ FðkÞ Þ
485 486
LðkÞ ¼ ðRefðkþ1Þ FðkÞ þ TBðkÞ Uðk1Þ ÞT QðRefðkþ1Þ FðkÞ þ TBðkÞ Uðk1Þ Þ þ UTðk1Þ RUðk1Þ
488
In addition, the amplitudes and rates of the control inputs are also limited to certain ranges as _ min 6 UðkÞ Uðk1Þ 6 TU _ max TU Umin 6 UðkÞ 6 Umax
ð27Þ
Moreover, Eq. (27) can be rewritten as follows:
where variables with subscript (k) represent their values at the current sampling instant. It can be seen from Eq. (23) that the basic control variables at the current sampling instant are the real control variables at the last sampling instant. The basic state variables at the next sampling instant are calculated according to the real state variables, the basic control variables at the current sampling instant, and the sampling time interval. Moreover, Xðkþ1Þ are the state variables at the next sampling instant. As a result, Eq. (23) can be simplified as follows:
MUðkÞ 6 cðkÞ
493
497
M ¼ ½I2 ; I2 ; I2 ; I2 T
500
_ max þ Uðk1Þ ; TU _ min Uðk1Þ T cðkÞ ¼ Umax ; Umin ; TU
501
ð28Þ
where I2 means the two-order unit matrix. To minimize JðkÞ subject to inequality constraints, let us consider the so-called Lagrange expression which takes into account the constraints as J1;ðkÞ ¼ JðkÞ þ
kTðkÞ ðMUðkÞ
cðkÞ Þ
ð29Þ
where kðkÞ contains the Lagrange multipliers. According to the Karush-Kuhn-Tucker condition, the following expression is obtained36,37: 8 @J 1;ðkÞ > > < @UðkÞ ¼ 0 ð30Þ kðkÞ P 0 > > : kT ðMU c Þ ¼ 0 ðkÞ
where Refðkþ1Þ ¼ ½Vx;Ref ; Vy;Ref ; qRef ; hRef Tðkþ1Þ , which represents the reference values of state variables at the next sampling instant. Q and R are the output weighting matrix and control input weighting matrix with appropriate dimensions, and their diagonal elements are all not zero. Furthermore, Eq. (25) can be reformulated as follows:
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This problem is equivalent to
517 518
ð31Þ
The minimization over UðkÞ is unconstrained and attained by
ð25Þ
504
516
max minJ1;ðkÞ
ð24Þ
503
ðkÞ
ðkÞ
kðkÞ P0 UðkÞ
Eq. (24) reveals the nature of aircraft dynamics. In detail, the first term on the right indicates the change of the state caused by itself with the control unchanged, and the second term signifies the change of the state caused by a change of the control. The objective function JðkÞ is used to compute the optimal solution for UðkÞ and to make Xðkþ1Þ close to the references as much as possible, and it also should depress dramatic changes of the control variables as 1 1 JðkÞ ¼ kRefðkþ1Þ Xðkþ1Þ k2Q þ kUðkÞ Uðk1Þ k2R 2 2
490 491
498
ð22Þ
Consequently, the discrete model can be further written as follows: 8 UðkÞ ¼ Uðk1Þ > > > > > > < DUðkÞ ¼ UðkÞ UðkÞ Xðkþ1Þ ¼ FðXðkÞ ; UðkÞ ; TÞ ð23Þ > > > DXðkþ1Þ ¼ TBðkÞ DUðkÞ > > > : Xðkþ1Þ ¼ Xðkþ1Þ þ DXðkþ1Þ
Xðkþ1Þ ¼ FðkÞ þ TBðkÞ ðUðkÞ Uðk1Þ Þ FðkÞ ¼ FðXðkÞ ; Uðk1Þ ; TÞ
489
494 495
DXððk þ 1ÞTÞ ¼ TBðkTÞDUðkTÞ
478 479 480
2
X ðkTÞ ¼ FðXðkTÞ; U ðkTÞ; 0Þ ¼ XðkTÞ 440
ð19Þ
1 1 ¼ UTðkÞ HðkÞ UðkÞ UTðkÞ KðkÞ þ LðkÞ 2 2
UðkÞ ¼
520 521 522 523
H1 ðkÞ ðKðkÞ
M kðkÞ Þ
ð32Þ
T
It is known that R is reversible by its definition. Let ~ ¼ Q=b and b–0. Then it can be proven that HðkÞ will always Q be reversible by adjusting the values of Q as follows: HðkÞ ¼ ðbT
2
~ ðkÞ R1 BTðkÞ QB
þ I2 ÞR
ð33Þ
The determinant of HðkÞ can be computed by Eq. (33) with ~ ðkÞ R1 being li ði ¼ 1; 2Þ as follows: the eigenvalues of BTðkÞ QB ~ ðkÞ R detðHðkÞ Þ ¼ detðbT2 BTðkÞ QB ¼ detðRÞ
2 Y
1
525 526 527 528 529 531 532 533 534
þ I2 Þ detðRÞ
ðbT2 li þ 1Þ
ð34Þ
i¼1
Obviously, there always exists a b which satisfies the inequality that bT2 li –1ði ¼ 1; 2Þ, which ensures that inversion of HðkÞ exists. Substituting Eq. (32) into Eq. (31), we get
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Time-varying linear control for tiltrotor aircraft
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Fig. 4 Control diagram of closed-loop tiltrotor aircraft system (the control law part and the controlled plant are located separately in the dashed box).
540
min
542
kðkÞ P0
543 544
where
1 T 1 k SðkÞ kðkÞ þ kTðkÞ TðkÞ þ VðkÞ 2 ðkÞ 2
ð35Þ
Parameter h0 dlong
T SðkÞ ¼ MH1 ðkÞ M
TðkÞ ¼ cðkÞ MH1 ðkÞ KðkÞ 546 547 548 549 551 552 553 554 555 557
VðkÞ ¼ KTðkÞ H1 ðkÞ KðkÞ LðkÞ Finally, we use Hildreth’s quadratic programming procedure to minimize the objective function as 1 1 J2;ðkÞ ¼ kTðkÞ SðkÞ kðkÞ þ kTðkÞ TðkÞ þ VðkÞ 2 2
ð36Þ
subject to kðkÞ P 0.38 Substituting the calculated kðkÞ into Eq. (32), the optimal control inputs can be obtained in the following form: ½h0 ; dlong TðkÞ ¼ UðkÞ
ð37Þ
559
Furthermore, the system structure of the tiltrotor aircraft with the controller is depicted in Fig. 4.
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5. Simulation results and discussions
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To demonstrate the performance of the controller, three scenarios are taken as examples, and the references of the state variables are calculated respectively. These three scenarios are one-by-one related to the three tilting transition curves in Fig. 2. It is worth noting that all these simulations are performed using the nonlinear model, and all the results are also obtained from the nonlinear model as described in Section 2. Some inherent parameters of the tiltrotor aircraft are shown in Table 1. Moreover, limitations of the control inputs are presented in Table 2. The specific expressions of the nacelle angle are described as follows:
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Table 1
Table 2
Inherent parameters of the tilt rotor aircraft.
Parameter
Value
Parameter
Value
m1 (kg) m2 (kg) m (kg) I1 ðkg m2 Þ I2 ðkg m2 Þ
3844.4 2052.4 5896.8 17887.4 1916.32
l1 ðmÞ l2 ðmÞ l3 ðmÞ dAF ð Þ dTM ð Þ
0.717 0.574 1.42 14.28 19.57
8 € h > > > N > > € > < hN € hN > > > € > hN > > :€ hN
Limitations of control inputs. Amplitude limit
½30; 90 ð Þ ½0; 100%
¼ 0; h_ N ¼ 0; hN ¼ 90 ¼ 2; h_ N ¼ 2t; hN ¼ 90 ðt 10Þ2 ¼ 0; h_ N ¼ 6; hN ¼ 81 6ðt 13Þ ¼ 2; h_ N ¼ 56 þ 2t; hN ¼ 90 ¼ 0; h_ N ¼ 0; hN ¼ 0
Rate limit ½30; 30 ð Þ s1 ½75; 75% s1
573
0 s 6 t < 10 s 10 s 6 t < 13 s 13 s 6 t < 25 s 25 s 6 t < 28 s 28 s 6 t < 40 s ð38Þ 575
As can be seen from Eq. (38), the tiltrotor aircraft flies in the helicopter mode at the beginning, then transfers from the helicopter mode to the transition mode, and finally switches into the airplane mode with nacelle tilting completed. The entire flight process takes 40 s. In addition, there are three vertical gust disturbances during 4–6 s, 18–20 s, and 34–36 s of the process. Moreover, the model of the gust is a sine-squared one and its maximum speed is 1 m=s. As shown in Eq. (25), not only the references of the state variables, but also the values of Q and R are necessary. What’s more, the four parameters of Q are the weights of vx, vy, q, and h, respectively, while the two parameters of R are the weights of h0 and dlong, respectively. This paper selects different weight matrices and compares simulation results, and their values are finally determined as in Table 3. In addition, an i7-4790 CPU and the MATLAB 2016 software are used to calculate simulation results, and the sampling time interval for the controller is chosen as 100 ms while that for the nonlinear models is chosen as 10 ms. It is noted that the computational cost of solving the optimal control is approximately 2.6 ms. As a result, simulation results of Scenario 1 are shown in Fig. 5. The hN, Vx, Vy, q, h, h0, dlong, MN, a represent the nacelle angle, the horizontal velocity, the vertical velocity, the pitch angular velocity, the pitch angle, the collective pitch, the longitudinal control bar displacement, the nacelle tilt torque, and the AOA of the tiltrotor, respectively. References of the horizontal velocity, the vertical velocity, the pitch angular velocity and the pitch angle are calculated according to Eqs. (12) and (38).
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Values of matrix Q and Matrix R.
Transition curve
Q
R
Scenario 1 Scenario 2 Scenario 3
diagð1; 20; 300; 300Þ diagð1; 30; 300; 300Þ diagð1; 20; 400; 400Þ
diagð5; 5Þ diagð5; 5Þ diagð5; 5Þ
In Fig. 5, the three parts separated by two dotted lines represent the helicopter mode, the transition mode, and the airplane mode, respectively. As can be seen from these figures, the true states of the tiltrotor can track the reference states well and have good performances in dealing with disturbances caused by vertical gusts. Furthermore, simulation results of
Fig. 5
Scenario 2 are depicted in Fig. 6. Simulation results for Scenario 3 are shown in Fig. 7. It can be seen from Figs. 5–7 that the tracking performances for Vy, q, and h are better than that for Vx. This is because the rotor pull is mainly used for providing the aircraft lift when the tiltrotor has a large nacelle angle, and it is difficult to provide a sufficient acceleration for Vx in the early stage of the transition mode. This is also the reason why the tracking performance for Vx shown in Fig. 6 is the best of these three simulations. As can be seen from Figs. 5–7, in the early stage of the transition mode, the tracking performance for Vy in Scenario 2 is the best of those in the three scenarios, while the tracking performance for Vy in Scenario 3 is the worst. The reason is as follows: the reference airspeed velocity in Scenario 2
Simulation results for Scenario 1.
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Fig. 6
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9
Simulation results for Scenario 2.
increases slowly and smoothly over time in the early stage of the transition mode, but the reference airspeed velocity in Scenario 3 has a sudden change in the early stage of the transition mode. Meanwhile, in the late stage of the transition mode, the tracking performance for Vy in Scenario 3 is the best of those in the three scenarios, and the tracking performance for Vy in Scenario 2 is the worst. The reason is as follows: the reference airspeed velocity in Scenario 3 increased slowly and smoothly over time in the late stage of the transition mode, but the reference airspeed velocity in Scenario 2 appears a sudden change at the late stage of the transition mode, which can be seen in the first and second subfigures of Fig. 7 and Fig. 6, respectively. Considering the impacts of the gust disturbances on a flight, the simulation results show that the tiltrotor is more sensitive to the gust disturbance when the airspeed velocity is higher. The reason is that these disturbances lead to changes of the AoA, and even a small change of the AoA can still cause relatively larger aerodynamic changes when the airspeed velocity is high. In order to demonstrate the advantage of the controller with input constraints taken into account, this paper takes Scenario 1 as an example to compare the control method which considers the control input constraints to the one which does not consider the constraints. It is worth noting that the controller without considering the control input constraints is
derived based on Eq. (25). Responses of h are illustrated in Fig. 8, and the maximum speed of the gust is 14 m=s. Reference command of h for Fig. 8 is the same as that in Fig. 5. As shown in Fig. 8, the controller with input constraints taken into account has a better performance than that of the controller which does not consider input constraints. The latter controller results in obvious oscillations in all three flight modes, and the performance for the airplane mode is particularly unsatisfying in the case of gust disturbance. To summarize, the quadratic programming based optimal controller proposed in this paper is shown to be able to accomplish the tilting transition missions of the tiltrotor aircraft. Except for the transition mode, the controller can also result in satisfying command tracking performance when the tiltrotor operates in the helicopter or airplane mode. Furthermore, the controller can effectively deal with unexpected gust disturbances.
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6. Conclusions
667
A tiltrotor aircraft is a nonlinear and multibody system with a high degree of freedom, and has strong time-varying characteristics when considering the nacelle tilting procedure. A quadratic programming based optimal control method is developed
668
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Fig. 7
Fig. 8
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Simulation results for Scenario 3.
Responses of h comparison between two controllers.
based on a tiltrotor Time-Varying Linear (TVL) model, and the proposed controller is thoroughly validated in this paper. (1) The TVL model is derived from a nonlinear Lagrangeform model through online linearization. The TVL model can effectively describe the dynamic characteristics of the tiltrotor in a finite time domain. Based on the TVL model and the regular optimal control strategy, an online optimization based optimal control approach which takes input constraints into account is proposed,
and a controller is presented. Moreover, the controller updates the parameters of the TVL model at each sampling instant according to flight conditions. Therefore, the controller can deal with the strong timevarying characteristics and control input constraints of the tiltrotor aircraft. (2) The proposed control approach is validated using data from an XV-15 tiltrotor. Three typical scenarios are simulated, and abrupt vertical gusts are introduced to show the interference immunity. The results show that the designed controller can achieve desired tracking performance for the tiltrotor throughout its operating envelop which includes a transition flight. In addition, the controller has sufficient capability to handle input constraints and accommodate gust disturbances. (3) In future research, attention will be paid to how to dynamically generate more reasonable reference commands for airspeed velocity and attitude by considering the tiltrotor maneuverability and energy consumption.
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Acknowledgement
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This work was supported by the National Natural Science Foundation of China (No. 11502008).
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