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Design of automatic climbing controller for large civil aircraft
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Journal of the Franklin Institute 350 (2013) 2442–2454 www.elsevier.com/locate/jfranklin
Design of automatic climbing controller for large civil aircraft Huajun Gong, Ziyang Zhen*, Xin Li, Ju Jiang, Xinhua Wang College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Received 4 April 2012; received in revised form 28 September 2012; accepted 25 October 2012 Available online 23 November 2012
Abstract This paper focuses on automatic climbing control methods for large civil aircraft. The key technique is how to design automatic flight control laws that meet the requirements of flight performance indexes and have good characteristics of restraining various disturbances. The classical engineering methods are used to solve the above problem. Based on the aerodynamic data of Boeing707, a nonlinear model of large civil aircraft is established. Linear models which are divided into longitudinal and lateral equations are obtained by trim and linearization. The design of longitudinal control laws uses Cn criterion, three climbing schemes including pitch control mode, vertical velocity control mode and altitude control mode are designed and mutually compared. For lateral control problem, by the feedback angle of sideslip, the bank attitude control with good effect is achieved. The simulation results indicate the designed control laws can meet the requirements of performance indexes, and have satisfied characteristics of anti-gust disturbance. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Large civil aircraft; Flight control; Performance indexes; Classical control
1. Introduction The flight control designs for different kinds of aeronautics and astronautics aircrafts are becoming a hot topic. Ref. [1] develops an adaptive actuator fault tolerant control system for near-space-vehicle modeled by Takagi–Sugeno, which is based on online fault estimation. Ref. [2] presents a multiple-model-based direct adaptive control approach for *
Corresponding author. E-mail address: [email protected] (Z.Y. Zhen).
0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.10.011
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actuator fault accommodation in the flight control system, to increase robustness and provide stable adaptation of unknown faults. An attitude coordination controller for timevarying reference attitude tracking of a formation flying spacecraft subject to control input saturation is investigated in Ref. [3]. Ref. [4] presents a passive fault tolerant control scheme for a near-space hypersonic vehicle with both parameter uncertainty and actuator faults. Furthermore, Ref. [5] studies an active fault tolerant control for a reusable launch vehicle, which uses adaptive and sliding mode techniques. Recently, more and more attention has been attracted to the large civil aircraft, due to its military value and commercial value. The automatic flight control system for large civil aircraft has obvious advantages in improving security, reducing the pilot’s workload and achieving accurate attitude and trajectory control. As one of the key technologies of the automatic control system, design of control laws immediately concerns the flight performance of large civil aircraft. The designer must devote himself to make the control laws meet the requirements of performance indexes grimly, and regard security, amenity and economical efficiency of flight as major task of automatic flight control system designing. There are many kinds of control methods to design the flight control laws. Ref. [6] uses a robust optimal adaptive control to make the tracking performance can be achieved at a much larger adaptive gain than the standard direct model-reference adaptive control, but the performance of restraining wind disturbance hasn’t been tested. Ref. [7] uses dynamic inversion and multi-objective optimization to design the inner loop of an auto-land control for civil aircraft, the controller is robust to parametric uncertainties, but performance index and flight quality haven’t been taken into account. Ref. [8] designs L1 adaptive control architecture to directly compensate for significant uncertain cross-coupling in nonlinear systems. Ref. [9] uses a linear parameter-varying (LPV) control synthesis method to design fault tolerant controllers for Boeing 747-100/200, and presents an application of robust gain-scheduled control concepts. Ref. [10] uses adaptive control algorithms, its advantages are the capabilities to improve performance and reliability and handle aerodynamic parameter uncertainties. However, most references do not refer to concrete performance index, and does not test the control system’s ability of anti-gust disturbance. Development of Chinese large civil aircraft is in the beginning stage, research about automatic control system design of the large civil aircraft is not enough. Ref. [11] designs a pitching attitude control law. However, it is based on a linear model. Ref. [12] uses the LQG/LTR method to design a longitudinal command augmentation system (CAS), this method acts to enhance stability, robustness and anti-jamming capacity, but the simulation is also based on the linear system. This paper researches concrete performance indexes and uses classical engineering control method to design automatic climbing control laws. Simulations of nonlinear model described large civil aircraft with the designed automatic control system are carried out.
2. Mathematic model and performance indexes This paper uses aerodynamic data of Boeing707 to establish a mathematical model, and summarizes performance indexes for designing the automatic flight control laws of large civil aircraft.
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2.1. Mathematic model According to the method from Ref. [13] and data from Ref. [14], the nonlinear model is established. In order to design control laws, the linear model needs to be obtained. The selected trim point of climbing phase is as follows: airspeed is 130 m/s, flying altitude is 1500 m, angle of attack is 1.361. By decoupling and linearization near the trim point, the longitudinal and lateral equations can be obtained, which are respectively shown as x_ lon ¼ Alon xlon þ Blon ulon
ð1Þ
x_ lat ¼ Alat xlat þ Blat ulat
ð2Þ
in Eq. (1) is xlon ¼ ½ DV Da Dq Dy T , the control vector is ulon ¼ The statevector Dde DdT T , among these variables, the unit of airspeed increment DV is m/s, and the units of attack angle increment Da, pitch angle increment Dy, elevator angle increment Dde, throttle opening increment DdT are all rad, the unit of pitch rate increment Dq is rad/s. Matrices Alon and Blon are as follows: 2 3 0:0085 7:2787 0 9:7761 6 0:0011 0:7101 1:0000 0:0053 7 6 7 Alon ¼ 6 7 4 0:0004 1:4546 0:3656 0:0004 5 2
0 0:2636
6 0:0396 6 Blon ¼ 6 6 1:6239 4 0
0 3 6:4142 0:0012 7 7 7 0:1430 7 5 0
1
0
The state vector in Eq. (2) is xlat ¼ ½ b p r f T , control vector is ulat ¼ ½ da dr T , the units of sideslip angle b, roll angle f, aileron angle da and rudder angle dr are rad, the units of roll angle rate p and yaw angle rate r are rad/s. Matrices Alat and Blat are as follows: 2 3 0:1206 0:0172 0:9869 0:0751 6 5:3548 2:9093 3:5546 0:0000 7 6 7 Alat ¼ 6 7 4 1:5014 0:2846 0:4658 0:0000 5 2
0:0000 0:0000
6 2:1708 6 Blat ¼ 6 6 0:0000 4 0:0000
1:0000 0:0939 3 0:0450 0:6616 7 7 7 1:7474 7 5 0:0000
0:0000
2.2. Performance indexes As criterions of control law designing, performance indexes should be investigated firstly. Ref. [15] puts forward concrete performance indexes. At the same time, by referring to design experience of foreign civil aircraft, the performance indexes about automatic
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climbing can be summarized as follows. Roll angle must be within 7601 limits, and its control precision should not be lower than 71.01. Pitch angle must be within 7151 limits, and its control precision should not be lower than 70.51. Angle of sideslip should not exceed 71.01. In steady rectilinear flight, including climbing and declining, according to airspeed when airspeed hold mode is turned on, the departing should keep within 79 km/h, or 72% relative to the standard airspeed. To guarantee flying qualities, A320 adopts Cn criterion to design pitch control channel. Maneuverable rate of roll angle of A320 is 131/s, practical using roll angle is 331. The presented automatic climbing control system of large civil aircraft must meet the above performance indexes.
3. Designing of control laws In this study, the root locus method is used to design PID controllers. The root locus method is based on transfer function of the controlled system, but the obtained linear model is still multi-input and multi-output. So how to get transfer function that precisely describes relationship between input and output variables is very important. Based on analyzing natural characteristic, by neglecting the coupling between different channels, each channel’s transfer function is obtained. From system matrices A and B, it can be found that the throttle mainly controls the airspeed, the elevator mainly controls the pitch angle, the aileron mainly controls the roll angle, and the rudder mainly affects the Dutch Roll mode. The task of throttle channel control is making airspeed and angle of attack remain constant during the climbing phase. The task of elevator channel control is achieving three different modes of climbing. The task of lateral control is mainly keeping roll attitude unchanged. A control structure flowchart of climbing phase is shown as Fig. 1. The designing process of each channel is as below.
Fig. 1. Control structure flowchart of climbing phase.
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3.1. Throttle channel Auto-throttle control system can achieve accurate airspeed control, and plays an important role in improving the flight safety. This study uses auto-throttle control method to keep the airspeed constant. Feedback signal of airspeed error and airspeed reference signal are utilized to control throttle and keep airspeed constant. The control structure is shown as Fig. 1. To design PID controller, at first the transfer function must be obtained. The transfer function of throttle channel can be obtained from Eq. (1). Considering long period mode that is only thinking influence of throttle on airspeed and angle of attack. Let throttle value increment as input and airspeed increment as output, the transfer function of throttle channel can be represented as GV dT ¼
DV ðsÞ 6:4142s þ 4:5460 ¼ 2 DdT ðsÞ s þ 0:7186s þ 0:0140
ð3Þ
Considering the actual conditions, the transfer function of engine is shown as Eq. (4), and the transfer function of throttle server is shown as Eq. (5). GE ¼
DdT ðsÞ 1 ¼ DudT ðsÞ sþ1
ð4Þ
GS ¼
DudT ðsÞ 1 20 ¼ ¼ DuðsÞ 0:05s þ 1 s þ 20
ð5Þ
Using PID control structure, the control law is shown as Eq. (6). Since pole of GS is 20, compared with other poles, it is big enough to become a non-leading pole. Hence, when designing control law, it can be ignored. ki DdT ¼ W duT ðsÞðDV c DV Þ ¼ GE kp þ þ kd s ðDV c DV Þ ð6Þ s Use MATLAB/sisotool to design PID parameters. The designing method is the same with elevator channel which will be introduced below, the designed PI control gains are kp ¼ 0.12, ki ¼ 0.015, at this time, the damping ratio of the leading pole is 0.53. 3.2. Elevator channel The simulation results indicate that the airspeed control cannot achieve satisfied effect without pitch attitude control. Similarly, without airspeed control, the pitch attitude control also cannot achieve good results. Here we consider the short period mode of aircraft, that is, only take the influence of elevator on attack angle and pitch angle rate into account. Then the longitudinal short period mode can be represented as Eq. (7). That is, variations of airspeed and pitch angle are very small so that they can be ignored in such a short time. " # # " Da_ Da 0:5121 1 0:0299 ¼ þ Dde ð7Þ Dq_ 0:7086 0:2730 Dq 0:7558 Regard elevator value increment as input of the transfer function and pitch angle increment as output of the transfer function, then the transfer function of throttle channel
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can be represented as Eq. (8). G yde ¼
DyðsÞ 0:7558s0:3659 ¼ 2 Dde ðsÞ sðs þ 0:7851s þ 0:8484Þ
ð8Þ
Considering the actual conditions, the transfer function of elevator steering engine is shown as Eq. (9). The mathematical models of aileron steering engine and rudder steering engine can also be represented as Eq. (9). It’s an inertial first-order system whose time constant is 0.1 s. G de ¼ G da ¼ G dr ¼
10 s þ 10
ð9Þ
To guarantee flying qualities of the aircraft, this work adopts Cn criterion to design the pitch control channel. In the designing of longitudinal inner loop, feedback of pitch angle rate can improve damping characteristic of pitching, and feedback of normal overload nz can increase longitudinal static stability. Combine q with nz to get Cn signal that can be represented as C n ¼ Dnz þ
V co Dq g
ð10Þ
When the aircraft is at high speed, nz acts on dominant action, and when the aircraft is at low speed, pitch angle rate acts on dominant action. In flight engineering field, it is usually Vco ¼ 120 m/s. To design control law of Cn inner loop, Dnz signal needs be transformed to be a state signal. Eq. (11) gives the relationships of this conversion. From Eq. (11), it can be found that Dnz is positive proportional to Da. Dnz ¼
V0 Z a Da g
ð11Þ
Control structure flowchart of Cn inner loop is shown as Fig. 2. C* C*
2.0 IV
II
I
III
1.0
0
0.5
1.0
1.5
2.0
2.5
Fig. 2. Control structure of Cn inner loop.
t (s)
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According to a large amount of ground-based tests and results of flight testing, Cn envelope obtained is shown as Fig. 3 [15]. The horizontal ordinate is time t, while the vertical ordinate is ratio of Cnand its stead value. In Fig. 3, there are four zones. Zone I is the best response zone. Zone II is used for non-critical phase, such as cruising and in flight refueling. Zone III is used for the other phases. Zone IV is used for landing phase. Transfer function of Cn inner loop can be represented as GCn ¼
C n ðsÞ 20:5122s29:1326 ¼ 2 Dde ðsÞ sðs þ 1:0757s þ 1:7142Þ
ð12Þ
The objective of Cn inner loop designing is guaranteeing Cn signal in the envelope of Cn performance. The Cn controller uses proportional form, through the method of root locus, we get a suitable proportional gain kC n ¼ 0.2, and the root locus curve is shown as Fig. 4. Through checking with the linear model by simulation, Cn controller can make the Cn signal inside the envelope. At this time, damping ratio of the leading pole of Cn inner loop is 0.761. As the designed Cn controller has already met the requirements, so no matter designing the outer loop, or testing in the nonlinear system, it’s better to not change the Cn controller. The primary purpose of the climbing control is not to follow a trajectory, it actually does not exist a standard climbing trajectory. The goal of the climbing control is to control the vertical velocity. Therefore, three climbing strategies are presented, including climbing mode 1 based on giving pitch angle increment reference signal, climbing mode 2 based on giving vertical velocity reference signal and climbing mode 3 based on giving slope height reference signal. (1) Climbing mode 1—pitch control Since auto-throttle control is applied into the automatic climbing control system, the angle of track can follow variation of pitch angle. That is, we can control rate of climbing by controlling pitch angle. Under the condition of keeping the airspeed unchanged, the airspeed vector direction can be changed by regulating the pitch angle, thus the vertical velocity is changed so that the climbing control is realized.
Fig. 3. Envelope of Cn performance.
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Fig. 4. Root locus of Cn inner loop
Pitch control loop is the outer loop of Cn loop. The transfer function of the Cn closed loop is represented as Cn 2:637s2 þ 3:558s þ 1:2 ¼ 0:1624s4 þ 1:908s3 þ 9:9s2 þ 18:11s þ 8:261 Dde
ð13Þ
Change Eq. (13) into the form of Zero-Pole, it can be represented as Cn 16:239ðs þ 0:6746Þ2 ¼ Dde ðs þ 2:235Þðs þ 0:6746Þðs2 þ 8:84s þ 33:73Þ
ð14Þ
It can be found that there is a canceling of zero and pole. Thus the equivalent transfer function can be represented as Cn 16:239ðs þ 0:6746Þ ¼ ðs þ 2:235Þðs2 þ 8:84s þ 33:73Þ Dde
ð15Þ
Based on Eq. (15), we can design the pitch controller. Here we use PI control structure. According to the control parameters determination based on root locus method, the design process is same with the Cn control parameters. When kp ¼ 3.35 and ki ¼ 0.5, pitch control loop can meet the requirements, and damping ratio of the leading pole is 0.675.
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Apply the designed control law to the nonlinear system to test the control effect. By properly adjusting the control parameters, the final longitudinal control laws that use the climbing mode 1 are shown as 10 ½2:48Dq þ 0:2Dnz 4:8ðDyc DyÞ Dde ¼ s þ 10Z 0:8 DdT ¼
ðDyc DyÞdt
ð16Þ
20 1 ½0:18ðDV c DV Þ s þ 20 sZ þ 1 þ0:04
ðDV c DV Þdt
ð17Þ
Hear, Dyc is constant reference command, and DVc ¼ 0. (2) Climbing mode 2—vertical velocity control It is a direct climbing control strategy by controlling the vertical velocity. In the case of airspeed keeping constant, track angle Dg and pitch angle Dy have the relationship as Dg Z a ¼ ð18Þ Dy sZ a At the same time, the relationship of altitude rate Dh_ and track angle Dg is shown as Dh_ ¼ V0 ð19Þ g0 þ Dg It can be found that loop of vertical velocity control is outer loop of pitch control loop. The control structure flowchart can refer to Fig. 1. By using root locus method to initially design control law, and adjusting the control parameters by simulation with nonlinear model, the final longitudinal control laws which use the climbing mode 2 have the same form as Eqs. (16–17). The only difference is the reference signal form of the pitch attitude angle, which is designed as Z _ _ _ h_ c dt Dyc ¼ 0:02 hh c 0:004 h ð20Þ where h_ c is rate of altitude command. (3) Climbing mode 3—altitude control This mode utilizes the height controller to control the climbing speed by giving different slope signals. Altitude Dh and track angle Dg have relationship as Dh V0 ¼ ð21Þ g0 þ Dg s The final longitudinal control laws which use the climbing mode 3 also have the same forms as Eqs. (16–17), while the only difference is the reference signal form of the pitch
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attitude angle, which is designed as Dyc ¼ 0:02h_ þ 0:01ðhc hÞ þ 0:002
2451
Z ðhc hÞdt
ð22Þ
Among which, hc is altitude command, by giving the slope reference signal it can achieve the good climbing control results. 3.3. Aileron channel and rudder channel The task of lateral control is keeping roll attitude constant and reducing even eliminating sideslip angle. From Eq. (2), it can be found that roll attitude is mainly controlled by aileron. For all the different climbing modes, the lateral controllers are same in this work. Roll attitude controller uses PID structure, and feedback of roll angle rate r is equivalent to the differential part of PID control. Feedback of roll angle rate is used for improving roll damping. By using PID structure, roll angle can achieve zero steady error control. The control structure flowchart can refer to Fig. 1. Dutch Roll mode is mainly caused by the deflection of rudder. At the same time, from Eq. (2), it can be found that deflection of rudder has greater influence on sideslip angle than deflection of aileron. Therefore this work uses feedback of yaw rate to improve the damp of Dutch Roll, and imports feedback of sideslip angle to reduce or even eliminate the sideslip. By properly adjusting the control parameters in control laws based on the nonlinear system simulation test, the final lateral control laws are designed as Z 10 Dda ¼ 0:8Dp þ 4 DfDfc þ 0:3 DfDfc dt ð23Þ s þ 10 Ddr ¼
Z 10 3Dr6:5 DbDbc 5 DbDbc dt s þ 10
ð24Þ
Among which Dfc and Dbc are reference commands of roll angle and sideslip angle. Usually, Dbc ¼ 0. 4. Simulation and analysis Using the designed control law, the simulation of nonlinear model described large civil aircraft is implemented. The purpose of simulation is comparing the proposed different climbing modes. Correspondingly, there are three kinds of reference commands that can achieve the purpose of climbing control, including pitch command, vertical velocity command and altitude command. The initial state of the aircraft is set as trim state. For testing whether lateral control can satisfy performance indexes, from 10 to 20 s, a reference command Dfc ¼ 301of roll angle is joined into the automatic flight control system. From 50 to 60 s, a constant gust is joined into the automatic flight control system, and the speed components of the gust under the body frame of axes are respectively uwind ¼ vwind ¼ wwind ¼ 5 m/s.
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H. Gong et al. / Journal of the Franklin Institute 350 (2013) 2442–2454 Table 1 Different climbing mode simulation conditions. Number
Climbing mode
Command
Command1 Command2 Command3
Pitch control Vertical velocity control Altitude control
11Pitch increment 11.3 m/s step signal Slope signal
Fig. 5. Simulation results of three different climbing modes.
The simulation conditions are shown in Table 1. In order to get the comparison results, there reference commands with same vertical velocity are successively added into the automatic flight control system. Simulation results of the climbing control system of a large civil aircraft are shown as Fig. 5. In Fig. 5, command 1 represents response curves of climbing mode 1 through controlling the pitch angle, command 2 represents response curves of climbing mode 2 through controlling the vertical velocity, command 3 represents response curves of climbing mode 3 through controlling the altitude. Flight control system structure in simulation is shown as Fig. 1. For climbing mode 1, it should be mentioned that when the pitch control is connected, the vertical velocity control and altitude control must be cut off. Similarly, for climbing mode 2, when the vertical velocity control is connected, altitude control must be cut off. From Fig. 5, some conclusions can be summarized as below. 1) Viewing from the roll angle and sideslip response curves, the overshoot of roll angle is below 20%, and setting time is shorter than 5 s. When roll angle reference command is 301, sideslip angle is less than 0.51, which is less than 11 that satisfies the performance requirement. 2) Viewing from the airspeed control response curves, airspeed can keep constant when the aircraft is climbing. The control precision meets the requirements performance indexes. 3) Viewing from the attack angle and pitch angle response curves, they can also keep constant, and steady state of pitch control is approximately zero. As the automatic throttle control is applied into the control system, both the airspeed and angle of attack can keep constant, so the angle of track can follow the variation of pitch angle. It should be mentioned that mode 1 has the best anti wind disturbance performance, while mode 3 is worst. 4) Viewing from the height control response curves, mode 1 and mode 2 can accurately control altitude when the gust is affecting. However, mode 3 which controls altitude easily causes the greater oscillation, this happens both in roll controlling phase and gust disturbance phase. Although only mode 3 makes the aircraft track the fixed slope, it is not necessary for climbing problem.
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In all, for the climbing control problem of large civil aircraft, all the three climbing modes can achieve the desired results that the vertical speed is well controlled. The effect of mode 1 and mode 2 is almost the same. Mode 1 does not require controlling the altitude accurately and has a simpler control structure with fewest feedback signals, while mode 3 has the most complex control structure with most feedback signals. Therefore, climbing mode 1 is first choice, climbing mode 2 is the second choice, and climbing mode 3 is the third choice. The simulation results show that mode 1 can meet the all requirements’ performance indexes. 5. Conclusion This work establishes a nonlinear model of large civil aircraft based on the data of Boeing707, summarizes the control performance indexes for designing of automatic climbing control laws for large civil aircraft, and uses the classical engineering method to design the control laws of the aircraft in automatic climbing phase. The inner loop of pitch control uses Cn criterion to achieve better longitudinal attitude control effect. The rudder channel imports the feedback of sideslip angle to achieve the zero sideslip control. By developing the auto-throttle control system, the aircraft can keep airspeed and angle of attack constant. Three climbing modes are presented and mutually compared, the simulation results show that the mode 1 (Pitch Control) is the best choice. Using classical method to design control law can also meet the performance indexes requirements, at the same time, the automatic climbing control system has a better characteristic of anti-gust disturbance. Since there are few references that investigate the climbing control problem of large civil aircraft, it is not compared with other climbing control methods. This study is desired to have an important meaning to the development of Chinese large civil aircraft. Acknowledgment This work is supported by NUAA Research Funding (NP2011049, NP2011012, 56YAH10029, NP2012101, NP4003-56 1100), Aeronautical Science Foundation of China (No. 2010ZA52002), and Open Fund of Postgraduate Innovation Base of Nanjing University of Aeronautics and Astronautics (kfjj20110207)
References [1] B. Jiang, Z. Gao, P. Shi, Y. Xu, Adaptive fault-tolerant tracking control of near space vehicle using Takagi– Sugeno fuzzy models, IEEE Transactions on Fuzzy Systems 18 (5) (2010) 1000–1007. [2] Y. Guo, B. Jiang, P. Shi, Adaptive reconfiguration scheme for flight control systems, Proceedings of the Institution of Mechanical Engineers, Part I, Journal of Systems and Control Engineering 224 (6) (2010) 713–723. [3] Baoqun Zhang, Shenmin Song, Robust attitude coordination control of formation flying spacecraft under control input saturation, International Journal of Innovative Computing, Information and Control 7 (7(B)) (2011) 4223–4236.
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[4] Z. Gao, B. Jiang, P. Shi, J. Liu, Y. Xu, Passive fault tolerant control design for near space hypersonic vehicle dynamical system, Circuits Systems and Signal Processing 31 (2) (2012) 565–581. [5] Z. Gao, B. Jiang, P. Shi, M. Qian, J. Lin, Active fault tolerant control design for reusable launch vehicle using adaptive sliding mode approach, Journal of the Franklin Institute 349 (4) (2012) 1543–1560. [6] Nhan T. Nguyen, Robust optimal adaptive control method with large adaptive gain, in: Proceedings of the AIAA Infotech@Aerospace Conference, AIAA-2009-1802, 2009. [7] Gertjan Looye, Hans-Dieter Joos, Design of robust dynamic inversion control laws using multi-objective optimization for civil aircraft, in: Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit 6–9 August 2001, Montreal, Canada, AIAA-2001-4285. [8] Irene M.. Gregory, Enric Xargay, Chengyu Cao, Naira Hovakimyan, L1 Adaptive Control Law in Support of Large Flight Envelope Modeling Work, Nasa Report NF1676L-12060, Langley Research Center, April 13, 2011. [9] Jong-Yeob Shin Irene Gregory, Robust gain-scheduled fault tolerant control for a transport aircraft, in: Proceedings of the IEEE International Conference on Control Applications, 2007, pp. 1209–1214. [10] H. Bouadi, H. Wu F. Mora-Camino, Flight path tracking based-on direct adaptive sliding mode control, in: Proceedings of the 2011 IEEE Intelligent Vehicles Symposium (IV), 2011, pp. 25–30. [11] Wang Pengpeng, Liu Chao, Xu Zhou, Cui Degang, Autopilot system of CJ818 and its control law designing, Civil Aircraft Design and Research (2009) (in Chinese). [12] Li Yan Li Leyao, Wang Xinmin, Design of longitudinal command augmentation system for large civil aircraft relaxing static stability, Flight Dynamics 28 (4) (2010) (in Chinese). [13] Donald McLean, Automatic Flight Control Systems, Prentice Hall, New Jersey, USA, 1990. [14] Rudolf Brockhaus, Flight Control, Springer, Berlin, Germany, 1994. [15] Zhang Minglian, Flight Control System, Aerospace Industry Press, Beijing, China, 1994 (in Chinese) p. 172, pp. 289–290.
Journal of the Franklin Institute 350 (2013) 2442–2454 www.elsevier.com/locate/jfranklin
Design of automatic climbing controller for large civil aircraft Huajun Gong, Ziyang Zhen*, Xin Li, Ju Jiang, Xinhua Wang College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Received 4 April 2012; received in revised form 28 September 2012; accepted 25 October 2012 Available online 23 November 2012
Abstract This paper focuses on automatic climbing control methods for large civil aircraft. The key technique is how to design automatic flight control laws that meet the requirements of flight performance indexes and have good characteristics of restraining various disturbances. The classical engineering methods are used to solve the above problem. Based on the aerodynamic data of Boeing707, a nonlinear model of large civil aircraft is established. Linear models which are divided into longitudinal and lateral equations are obtained by trim and linearization. The design of longitudinal control laws uses Cn criterion, three climbing schemes including pitch control mode, vertical velocity control mode and altitude control mode are designed and mutually compared. For lateral control problem, by the feedback angle of sideslip, the bank attitude control with good effect is achieved. The simulation results indicate the designed control laws can meet the requirements of performance indexes, and have satisfied characteristics of anti-gust disturbance. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Large civil aircraft; Flight control; Performance indexes; Classical control
1. Introduction The flight control designs for different kinds of aeronautics and astronautics aircrafts are becoming a hot topic. Ref. [1] develops an adaptive actuator fault tolerant control system for near-space-vehicle modeled by Takagi–Sugeno, which is based on online fault estimation. Ref. [2] presents a multiple-model-based direct adaptive control approach for *
Corresponding author. E-mail address: [email protected] (Z.Y. Zhen).
0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.10.011
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actuator fault accommodation in the flight control system, to increase robustness and provide stable adaptation of unknown faults. An attitude coordination controller for timevarying reference attitude tracking of a formation flying spacecraft subject to control input saturation is investigated in Ref. [3]. Ref. [4] presents a passive fault tolerant control scheme for a near-space hypersonic vehicle with both parameter uncertainty and actuator faults. Furthermore, Ref. [5] studies an active fault tolerant control for a reusable launch vehicle, which uses adaptive and sliding mode techniques. Recently, more and more attention has been attracted to the large civil aircraft, due to its military value and commercial value. The automatic flight control system for large civil aircraft has obvious advantages in improving security, reducing the pilot’s workload and achieving accurate attitude and trajectory control. As one of the key technologies of the automatic control system, design of control laws immediately concerns the flight performance of large civil aircraft. The designer must devote himself to make the control laws meet the requirements of performance indexes grimly, and regard security, amenity and economical efficiency of flight as major task of automatic flight control system designing. There are many kinds of control methods to design the flight control laws. Ref. [6] uses a robust optimal adaptive control to make the tracking performance can be achieved at a much larger adaptive gain than the standard direct model-reference adaptive control, but the performance of restraining wind disturbance hasn’t been tested. Ref. [7] uses dynamic inversion and multi-objective optimization to design the inner loop of an auto-land control for civil aircraft, the controller is robust to parametric uncertainties, but performance index and flight quality haven’t been taken into account. Ref. [8] designs L1 adaptive control architecture to directly compensate for significant uncertain cross-coupling in nonlinear systems. Ref. [9] uses a linear parameter-varying (LPV) control synthesis method to design fault tolerant controllers for Boeing 747-100/200, and presents an application of robust gain-scheduled control concepts. Ref. [10] uses adaptive control algorithms, its advantages are the capabilities to improve performance and reliability and handle aerodynamic parameter uncertainties. However, most references do not refer to concrete performance index, and does not test the control system’s ability of anti-gust disturbance. Development of Chinese large civil aircraft is in the beginning stage, research about automatic control system design of the large civil aircraft is not enough. Ref. [11] designs a pitching attitude control law. However, it is based on a linear model. Ref. [12] uses the LQG/LTR method to design a longitudinal command augmentation system (CAS), this method acts to enhance stability, robustness and anti-jamming capacity, but the simulation is also based on the linear system. This paper researches concrete performance indexes and uses classical engineering control method to design automatic climbing control laws. Simulations of nonlinear model described large civil aircraft with the designed automatic control system are carried out.
2. Mathematic model and performance indexes This paper uses aerodynamic data of Boeing707 to establish a mathematical model, and summarizes performance indexes for designing the automatic flight control laws of large civil aircraft.
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2.1. Mathematic model According to the method from Ref. [13] and data from Ref. [14], the nonlinear model is established. In order to design control laws, the linear model needs to be obtained. The selected trim point of climbing phase is as follows: airspeed is 130 m/s, flying altitude is 1500 m, angle of attack is 1.361. By decoupling and linearization near the trim point, the longitudinal and lateral equations can be obtained, which are respectively shown as x_ lon ¼ Alon xlon þ Blon ulon
ð1Þ
x_ lat ¼ Alat xlat þ Blat ulat
ð2Þ
in Eq. (1) is xlon ¼ ½ DV Da Dq Dy T , the control vector is ulon ¼ The statevector Dde DdT T , among these variables, the unit of airspeed increment DV is m/s, and the units of attack angle increment Da, pitch angle increment Dy, elevator angle increment Dde, throttle opening increment DdT are all rad, the unit of pitch rate increment Dq is rad/s. Matrices Alon and Blon are as follows: 2 3 0:0085 7:2787 0 9:7761 6 0:0011 0:7101 1:0000 0:0053 7 6 7 Alon ¼ 6 7 4 0:0004 1:4546 0:3656 0:0004 5 2
0 0:2636
6 0:0396 6 Blon ¼ 6 6 1:6239 4 0
0 3 6:4142 0:0012 7 7 7 0:1430 7 5 0
1
0
The state vector in Eq. (2) is xlat ¼ ½ b p r f T , control vector is ulat ¼ ½ da dr T , the units of sideslip angle b, roll angle f, aileron angle da and rudder angle dr are rad, the units of roll angle rate p and yaw angle rate r are rad/s. Matrices Alat and Blat are as follows: 2 3 0:1206 0:0172 0:9869 0:0751 6 5:3548 2:9093 3:5546 0:0000 7 6 7 Alat ¼ 6 7 4 1:5014 0:2846 0:4658 0:0000 5 2
0:0000 0:0000
6 2:1708 6 Blat ¼ 6 6 0:0000 4 0:0000
1:0000 0:0939 3 0:0450 0:6616 7 7 7 1:7474 7 5 0:0000
0:0000
2.2. Performance indexes As criterions of control law designing, performance indexes should be investigated firstly. Ref. [15] puts forward concrete performance indexes. At the same time, by referring to design experience of foreign civil aircraft, the performance indexes about automatic
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climbing can be summarized as follows. Roll angle must be within 7601 limits, and its control precision should not be lower than 71.01. Pitch angle must be within 7151 limits, and its control precision should not be lower than 70.51. Angle of sideslip should not exceed 71.01. In steady rectilinear flight, including climbing and declining, according to airspeed when airspeed hold mode is turned on, the departing should keep within 79 km/h, or 72% relative to the standard airspeed. To guarantee flying qualities, A320 adopts Cn criterion to design pitch control channel. Maneuverable rate of roll angle of A320 is 131/s, practical using roll angle is 331. The presented automatic climbing control system of large civil aircraft must meet the above performance indexes.
3. Designing of control laws In this study, the root locus method is used to design PID controllers. The root locus method is based on transfer function of the controlled system, but the obtained linear model is still multi-input and multi-output. So how to get transfer function that precisely describes relationship between input and output variables is very important. Based on analyzing natural characteristic, by neglecting the coupling between different channels, each channel’s transfer function is obtained. From system matrices A and B, it can be found that the throttle mainly controls the airspeed, the elevator mainly controls the pitch angle, the aileron mainly controls the roll angle, and the rudder mainly affects the Dutch Roll mode. The task of throttle channel control is making airspeed and angle of attack remain constant during the climbing phase. The task of elevator channel control is achieving three different modes of climbing. The task of lateral control is mainly keeping roll attitude unchanged. A control structure flowchart of climbing phase is shown as Fig. 1. The designing process of each channel is as below.
Fig. 1. Control structure flowchart of climbing phase.
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3.1. Throttle channel Auto-throttle control system can achieve accurate airspeed control, and plays an important role in improving the flight safety. This study uses auto-throttle control method to keep the airspeed constant. Feedback signal of airspeed error and airspeed reference signal are utilized to control throttle and keep airspeed constant. The control structure is shown as Fig. 1. To design PID controller, at first the transfer function must be obtained. The transfer function of throttle channel can be obtained from Eq. (1). Considering long period mode that is only thinking influence of throttle on airspeed and angle of attack. Let throttle value increment as input and airspeed increment as output, the transfer function of throttle channel can be represented as GV dT ¼
DV ðsÞ 6:4142s þ 4:5460 ¼ 2 DdT ðsÞ s þ 0:7186s þ 0:0140
ð3Þ
Considering the actual conditions, the transfer function of engine is shown as Eq. (4), and the transfer function of throttle server is shown as Eq. (5). GE ¼
DdT ðsÞ 1 ¼ DudT ðsÞ sþ1
ð4Þ
GS ¼
DudT ðsÞ 1 20 ¼ ¼ DuðsÞ 0:05s þ 1 s þ 20
ð5Þ
Using PID control structure, the control law is shown as Eq. (6). Since pole of GS is 20, compared with other poles, it is big enough to become a non-leading pole. Hence, when designing control law, it can be ignored. ki DdT ¼ W duT ðsÞðDV c DV Þ ¼ GE kp þ þ kd s ðDV c DV Þ ð6Þ s Use MATLAB/sisotool to design PID parameters. The designing method is the same with elevator channel which will be introduced below, the designed PI control gains are kp ¼ 0.12, ki ¼ 0.015, at this time, the damping ratio of the leading pole is 0.53. 3.2. Elevator channel The simulation results indicate that the airspeed control cannot achieve satisfied effect without pitch attitude control. Similarly, without airspeed control, the pitch attitude control also cannot achieve good results. Here we consider the short period mode of aircraft, that is, only take the influence of elevator on attack angle and pitch angle rate into account. Then the longitudinal short period mode can be represented as Eq. (7). That is, variations of airspeed and pitch angle are very small so that they can be ignored in such a short time. " # # " Da_ Da 0:5121 1 0:0299 ¼ þ Dde ð7Þ Dq_ 0:7086 0:2730 Dq 0:7558 Regard elevator value increment as input of the transfer function and pitch angle increment as output of the transfer function, then the transfer function of throttle channel
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can be represented as Eq. (8). G yde ¼
DyðsÞ 0:7558s0:3659 ¼ 2 Dde ðsÞ sðs þ 0:7851s þ 0:8484Þ
ð8Þ
Considering the actual conditions, the transfer function of elevator steering engine is shown as Eq. (9). The mathematical models of aileron steering engine and rudder steering engine can also be represented as Eq. (9). It’s an inertial first-order system whose time constant is 0.1 s. G de ¼ G da ¼ G dr ¼
10 s þ 10
ð9Þ
To guarantee flying qualities of the aircraft, this work adopts Cn criterion to design the pitch control channel. In the designing of longitudinal inner loop, feedback of pitch angle rate can improve damping characteristic of pitching, and feedback of normal overload nz can increase longitudinal static stability. Combine q with nz to get Cn signal that can be represented as C n ¼ Dnz þ
V co Dq g
ð10Þ
When the aircraft is at high speed, nz acts on dominant action, and when the aircraft is at low speed, pitch angle rate acts on dominant action. In flight engineering field, it is usually Vco ¼ 120 m/s. To design control law of Cn inner loop, Dnz signal needs be transformed to be a state signal. Eq. (11) gives the relationships of this conversion. From Eq. (11), it can be found that Dnz is positive proportional to Da. Dnz ¼
V0 Z a Da g
ð11Þ
Control structure flowchart of Cn inner loop is shown as Fig. 2. C* C*
2.0 IV
II
I
III
1.0
0
0.5
1.0
1.5
2.0
2.5
Fig. 2. Control structure of Cn inner loop.
t (s)
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According to a large amount of ground-based tests and results of flight testing, Cn envelope obtained is shown as Fig. 3 [15]. The horizontal ordinate is time t, while the vertical ordinate is ratio of Cnand its stead value. In Fig. 3, there are four zones. Zone I is the best response zone. Zone II is used for non-critical phase, such as cruising and in flight refueling. Zone III is used for the other phases. Zone IV is used for landing phase. Transfer function of Cn inner loop can be represented as GCn ¼
C n ðsÞ 20:5122s29:1326 ¼ 2 Dde ðsÞ sðs þ 1:0757s þ 1:7142Þ
ð12Þ
The objective of Cn inner loop designing is guaranteeing Cn signal in the envelope of Cn performance. The Cn controller uses proportional form, through the method of root locus, we get a suitable proportional gain kC n ¼ 0.2, and the root locus curve is shown as Fig. 4. Through checking with the linear model by simulation, Cn controller can make the Cn signal inside the envelope. At this time, damping ratio of the leading pole of Cn inner loop is 0.761. As the designed Cn controller has already met the requirements, so no matter designing the outer loop, or testing in the nonlinear system, it’s better to not change the Cn controller. The primary purpose of the climbing control is not to follow a trajectory, it actually does not exist a standard climbing trajectory. The goal of the climbing control is to control the vertical velocity. Therefore, three climbing strategies are presented, including climbing mode 1 based on giving pitch angle increment reference signal, climbing mode 2 based on giving vertical velocity reference signal and climbing mode 3 based on giving slope height reference signal. (1) Climbing mode 1—pitch control Since auto-throttle control is applied into the automatic climbing control system, the angle of track can follow variation of pitch angle. That is, we can control rate of climbing by controlling pitch angle. Under the condition of keeping the airspeed unchanged, the airspeed vector direction can be changed by regulating the pitch angle, thus the vertical velocity is changed so that the climbing control is realized.
Fig. 3. Envelope of Cn performance.
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Fig. 4. Root locus of Cn inner loop
Pitch control loop is the outer loop of Cn loop. The transfer function of the Cn closed loop is represented as Cn 2:637s2 þ 3:558s þ 1:2 ¼ 0:1624s4 þ 1:908s3 þ 9:9s2 þ 18:11s þ 8:261 Dde
ð13Þ
Change Eq. (13) into the form of Zero-Pole, it can be represented as Cn 16:239ðs þ 0:6746Þ2 ¼ Dde ðs þ 2:235Þðs þ 0:6746Þðs2 þ 8:84s þ 33:73Þ
ð14Þ
It can be found that there is a canceling of zero and pole. Thus the equivalent transfer function can be represented as Cn 16:239ðs þ 0:6746Þ ¼ ðs þ 2:235Þðs2 þ 8:84s þ 33:73Þ Dde
ð15Þ
Based on Eq. (15), we can design the pitch controller. Here we use PI control structure. According to the control parameters determination based on root locus method, the design process is same with the Cn control parameters. When kp ¼ 3.35 and ki ¼ 0.5, pitch control loop can meet the requirements, and damping ratio of the leading pole is 0.675.
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Apply the designed control law to the nonlinear system to test the control effect. By properly adjusting the control parameters, the final longitudinal control laws that use the climbing mode 1 are shown as 10 ½2:48Dq þ 0:2Dnz 4:8ðDyc DyÞ Dde ¼ s þ 10Z 0:8 DdT ¼
ðDyc DyÞdt
ð16Þ
20 1 ½0:18ðDV c DV Þ s þ 20 sZ þ 1 þ0:04
ðDV c DV Þdt
ð17Þ
Hear, Dyc is constant reference command, and DVc ¼ 0. (2) Climbing mode 2—vertical velocity control It is a direct climbing control strategy by controlling the vertical velocity. In the case of airspeed keeping constant, track angle Dg and pitch angle Dy have the relationship as Dg Z a ¼ ð18Þ Dy sZ a At the same time, the relationship of altitude rate Dh_ and track angle Dg is shown as Dh_ ¼ V0 ð19Þ g0 þ Dg It can be found that loop of vertical velocity control is outer loop of pitch control loop. The control structure flowchart can refer to Fig. 1. By using root locus method to initially design control law, and adjusting the control parameters by simulation with nonlinear model, the final longitudinal control laws which use the climbing mode 2 have the same form as Eqs. (16–17). The only difference is the reference signal form of the pitch attitude angle, which is designed as Z _ _ _ h_ c dt Dyc ¼ 0:02 hh c 0:004 h ð20Þ where h_ c is rate of altitude command. (3) Climbing mode 3—altitude control This mode utilizes the height controller to control the climbing speed by giving different slope signals. Altitude Dh and track angle Dg have relationship as Dh V0 ¼ ð21Þ g0 þ Dg s The final longitudinal control laws which use the climbing mode 3 also have the same forms as Eqs. (16–17), while the only difference is the reference signal form of the pitch
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attitude angle, which is designed as Dyc ¼ 0:02h_ þ 0:01ðhc hÞ þ 0:002
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Z ðhc hÞdt
ð22Þ
Among which, hc is altitude command, by giving the slope reference signal it can achieve the good climbing control results. 3.3. Aileron channel and rudder channel The task of lateral control is keeping roll attitude constant and reducing even eliminating sideslip angle. From Eq. (2), it can be found that roll attitude is mainly controlled by aileron. For all the different climbing modes, the lateral controllers are same in this work. Roll attitude controller uses PID structure, and feedback of roll angle rate r is equivalent to the differential part of PID control. Feedback of roll angle rate is used for improving roll damping. By using PID structure, roll angle can achieve zero steady error control. The control structure flowchart can refer to Fig. 1. Dutch Roll mode is mainly caused by the deflection of rudder. At the same time, from Eq. (2), it can be found that deflection of rudder has greater influence on sideslip angle than deflection of aileron. Therefore this work uses feedback of yaw rate to improve the damp of Dutch Roll, and imports feedback of sideslip angle to reduce or even eliminate the sideslip. By properly adjusting the control parameters in control laws based on the nonlinear system simulation test, the final lateral control laws are designed as Z 10 Dda ¼ 0:8Dp þ 4 DfDfc þ 0:3 DfDfc dt ð23Þ s þ 10 Ddr ¼
Z 10 3Dr6:5 DbDbc 5 DbDbc dt s þ 10
ð24Þ
Among which Dfc and Dbc are reference commands of roll angle and sideslip angle. Usually, Dbc ¼ 0. 4. Simulation and analysis Using the designed control law, the simulation of nonlinear model described large civil aircraft is implemented. The purpose of simulation is comparing the proposed different climbing modes. Correspondingly, there are three kinds of reference commands that can achieve the purpose of climbing control, including pitch command, vertical velocity command and altitude command. The initial state of the aircraft is set as trim state. For testing whether lateral control can satisfy performance indexes, from 10 to 20 s, a reference command Dfc ¼ 301of roll angle is joined into the automatic flight control system. From 50 to 60 s, a constant gust is joined into the automatic flight control system, and the speed components of the gust under the body frame of axes are respectively uwind ¼ vwind ¼ wwind ¼ 5 m/s.
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H. Gong et al. / Journal of the Franklin Institute 350 (2013) 2442–2454 Table 1 Different climbing mode simulation conditions. Number
Climbing mode
Command
Command1 Command2 Command3
Pitch control Vertical velocity control Altitude control
11Pitch increment 11.3 m/s step signal Slope signal
Fig. 5. Simulation results of three different climbing modes.
The simulation conditions are shown in Table 1. In order to get the comparison results, there reference commands with same vertical velocity are successively added into the automatic flight control system. Simulation results of the climbing control system of a large civil aircraft are shown as Fig. 5. In Fig. 5, command 1 represents response curves of climbing mode 1 through controlling the pitch angle, command 2 represents response curves of climbing mode 2 through controlling the vertical velocity, command 3 represents response curves of climbing mode 3 through controlling the altitude. Flight control system structure in simulation is shown as Fig. 1. For climbing mode 1, it should be mentioned that when the pitch control is connected, the vertical velocity control and altitude control must be cut off. Similarly, for climbing mode 2, when the vertical velocity control is connected, altitude control must be cut off. From Fig. 5, some conclusions can be summarized as below. 1) Viewing from the roll angle and sideslip response curves, the overshoot of roll angle is below 20%, and setting time is shorter than 5 s. When roll angle reference command is 301, sideslip angle is less than 0.51, which is less than 11 that satisfies the performance requirement. 2) Viewing from the airspeed control response curves, airspeed can keep constant when the aircraft is climbing. The control precision meets the requirements performance indexes. 3) Viewing from the attack angle and pitch angle response curves, they can also keep constant, and steady state of pitch control is approximately zero. As the automatic throttle control is applied into the control system, both the airspeed and angle of attack can keep constant, so the angle of track can follow the variation of pitch angle. It should be mentioned that mode 1 has the best anti wind disturbance performance, while mode 3 is worst. 4) Viewing from the height control response curves, mode 1 and mode 2 can accurately control altitude when the gust is affecting. However, mode 3 which controls altitude easily causes the greater oscillation, this happens both in roll controlling phase and gust disturbance phase. Although only mode 3 makes the aircraft track the fixed slope, it is not necessary for climbing problem.
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In all, for the climbing control problem of large civil aircraft, all the three climbing modes can achieve the desired results that the vertical speed is well controlled. The effect of mode 1 and mode 2 is almost the same. Mode 1 does not require controlling the altitude accurately and has a simpler control structure with fewest feedback signals, while mode 3 has the most complex control structure with most feedback signals. Therefore, climbing mode 1 is first choice, climbing mode 2 is the second choice, and climbing mode 3 is the third choice. The simulation results show that mode 1 can meet the all requirements’ performance indexes. 5. Conclusion This work establishes a nonlinear model of large civil aircraft based on the data of Boeing707, summarizes the control performance indexes for designing of automatic climbing control laws for large civil aircraft, and uses the classical engineering method to design the control laws of the aircraft in automatic climbing phase. The inner loop of pitch control uses Cn criterion to achieve better longitudinal attitude control effect. The rudder channel imports the feedback of sideslip angle to achieve the zero sideslip control. By developing the auto-throttle control system, the aircraft can keep airspeed and angle of attack constant. Three climbing modes are presented and mutually compared, the simulation results show that the mode 1 (Pitch Control) is the best choice. Using classical method to design control law can also meet the performance indexes requirements, at the same time, the automatic climbing control system has a better characteristic of anti-gust disturbance. Since there are few references that investigate the climbing control problem of large civil aircraft, it is not compared with other climbing control methods. This study is desired to have an important meaning to the development of Chinese large civil aircraft. Acknowledgment This work is supported by NUAA Research Funding (NP2011049, NP2011012, 56YAH10029, NP2012101, NP4003-56 1100), Aeronautical Science Foundation of China (No. 2010ZA52002), and Open Fund of Postgraduate Innovation Base of Nanjing University of Aeronautics and Astronautics (kfjj20110207)
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