ISA Transactions 55 (2015) 81–91
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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Information fusion based optimal control for large civil aircraft system Ziyang Zhen n, Ju Jiang, Xinhua Wang, Chen Gao College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, No.29, Yudao Street, Nanjing, China
art ic l e i nf o
a b s t r a c t
Article history: Received 9 July 2014 Received in revised form 4 September 2014 Accepted 20 September 2014 Available online 23 October 2014
Wind disturbance has a great influence on landing security of Large Civil Aircraft. Through simulation research and engineering experience, it can be found that PID control is not good enough to solve the problem of restraining the wind disturbance. This paper focuses on anti-wind attitude control for Large Civil Aircraft in landing phase. In order to improve the riding comfort and the flight security, an information fusion based optimal control strategy is presented to restrain the wind in landing phase for maintaining attitudes and airspeed. Data of Boeing707 is used to establish a nonlinear mode with total variables of Large Civil Aircraft, and then two linear models are obtained which are divided into longitudinal and lateral equations. Based on engineering experience, the longitudinal channel adopts PID control and Cn inner control to keep longitudinal attitude constant, and applies autothrottle system for keeping airspeed constant, while an information fusion based optimal regulator in the lateral control channel is designed to achieve lateral attitude holding. According to information fusion estimation, by fusing hard constraint information of system dynamic equations and the soft constraint information of performance index function, optimal estimation of the control sequence is derived. Based on this, an information fusion state regulator is deduced for discrete time linear system with disturbance. The simulation results of nonlinear model of aircraft indicate that the information fusion optimal control is better than traditional PID control, LQR control and LQR control with integral action, in anti-wind disturbance performance in the landing phase. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Large civil aircraft Automatic flight control Linear quadratic optimal control Information fusion Optimal estimation
1. Introduction The automatic flight control systems of Large Civil Aircrafts play important roles in reducing the pilot's workload and achieving accurate attitude/trajectory control. As one of the key technologies of automatic flight control system, design of control laws directly determine the flight performance of Large Civil Aircraft. Control laws should be designed to meet all the requirements of performance index, and regard security, amenity and economical efficiency of flight as the major tasks of automatic flight control systems design. There are several kinds of flight control methods presented for civil aircrafts. Nguyen (2009) uses robust optimal adaptive control to improve the tracking performance at a large adaptive gain, but the performance of restraining wind disturbance has not been tested [1]. Looye (2011) uses dynamic inversion and multi-objective optimization to design the attitude control system for civil aircraft, the controller is robustness to the parametric uncertainties, but performance index and flight quality have not been taken into account [2]. Gregory (2011) designs L1 adaptive control architecture to directly compensate for significant uncertain cross-coupling in nonlinear systems [3]. Shin (2007) uses a linear parameter-varying control synthesis method to design fault tolerant controllers for Boeing 747-100/200, and presents an application of robust gain-scheduled control methods [4]. References [5,6] study an adaptive control algorithm. Its advantages are the capability of improving the reliability and handling the aerodynamic parameter uncertainties. However, most references do not refer to concrete performance index, and do not test the control system's ability of anti-gust disturbance. Mohammad (2011) designs a closed-loop decision-making system to command the aircraft based on fuzzy logic controller for terrain following flight, which is important in approaching an airport with low or non-visibility for civil aircraft [7]. However, it only utilizes the system input–output data so that it is difficult to analyze and design the controllers, furthermore, theory basis of such intelligent methods is not perfect enough, and the computation costs of the intelligent methods generally become large.
n
Corresponding author. E-mail address: [email protected] (Z. Zhen).
http://dx.doi.org/10.1016/j.isatra.2014.09.017 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
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Development of Chinese Large Civil Aircraft is in the beginning stage, research on the automatic control law of the Large Civil Aircraft is not enough. Li (2010) uses LQG/LTR based method to design a longitudinal command augmentation system (CAS), this method is verified to enhance the stability, robustness and anti-jamming capacity, but the simulation is implemented based on linear system of civil aircraft [8]. Fan (2010) presents a modeling and control strategy for an advanced configured large civil aircraft with aeroservoelasticity via LQG method and control allocation [9]. LQ optimal control is a well-known fundamental theory in modern control theory. The LQ optimal control has received a great deal of attention from control theorists and engineers, since the resulting control law is linear with respect to the state and is therefore easy to compute [10,11]. There are many different approaches to the solution of LQ optimal control problems, such as the minimization method using Lagrange multipliers, dynamic programming, and Lyapunov function. Information fusion estimation (IFE) is the problem of how to best utilize useful information obtained from multiple sources or from a single source over a time period, for estimating an unknown parameter or process. The most important application area of IFE is track fusion or track-to-track fusion in target tracking system, which has been investigated for more than two decades [12]. Li (2003) establishes three estimation fusion architectures including centralized, distributed, and hybrid, moreover, he proposes an unified linear data model and two optimal fusion rules based on the best linear unbiased estimation (BLUE) and the weighted least squares (WLS) [13]. It is much more general and flexible than the previous approach of matching centralized and distributed fusion rules based on Kalman filters. Sun (2004) presents a new multi-sensor optimal information fusion criterion weighted by matrices in the linear minimum variance (LMV) sense which is equivalent to the maximum likelihood fusion criterion under the assumption of normal distribution [14]. Wang (2007) proposes a concept of the information weight which represents the contribution of the measurement on the estimated variable, and presents unified information fusion rules based on least square estimate criterion for linear and nonlinear measurements [15]. The information weight is essentially equal to the Fisher information matrix in case of Gaussian white noise. Information fusion is an ideological which universally exists in decision making problems. Both estimation problem and control problem belong to the decision making problem. In view of this, Zhen (2010) proposes a new optimal control method called information fusion control method for nonlinear systems [16,17]. It regards information of the desired trajectory, system dynamic equation and ideal control strategy as the measurement information of the control variable, and obtains an optimal estimate of the control variable by IFE, and then transfers the optimal control problem into the optimal estimation problem. In summary, the previous references in designing the automatic control system of Large Civil Aircraft are few, in which the control strategies for anti-wind are especially little studied. Furthermore, previous research works always use linear models of Large Civil Aircraft in simulation, which is not accurate enough for real aircraft. Aiming to solve these problems, first, a nonlinear model with total variables is established for system simulation and natural analysis. Second, according to the aerial engineering experience, in landing phase with wind disturbance, the pitch attitude and flight speed can be well controlled by the traditional engineering methods, while the inclined attitude is difficult to be hold. Therefore, PID control and Cn control are introduced to design the pitch attitude controller, an airspeed keeping autothrottle system is designed for flight speed control, and an optimal controller is designed for controlling inclined attitude. Third, in order to utilize the disturbance information to improve the control performance, this work originally propose an information fusion method to design the optimal regulator problem of discrete time linear system with disturbance. The paper is organized as following: models of Large Civil Aircraft are established for natural analysis in Section 2, information fusion estimation and information fusion optimal regulator are shown in Section 3. Automatic flight control system which is composed of longitudinal channel and lateral channel and their control laws are designed in Section 4, simulations are carried out for comparing different control methods in Section 5, and finally the conclusions are shown in Section 6.
2. Modeling and property analyzing of large civil aircraft According to the aerodynamic parameters of Boeing707 given in [18], we establish a nonlinear numerical model of Large Civil Aircraft, which is shown in Appendix A.1. In order to study the natural properties of Large Civil Aircraft and design the control laws, firstly linear models need to be obtained. The selected trim point of landing phase is shown in Table 1. By decoupling and linearization near the trim point, the linear longitudinal state equation of Large Civil Aircraft is expressed by x_ lon ¼ Alon xlon þ Blon ulon where xlon ¼ ½ ΔV follows. 2
0:0473
6 0:0030 6 Alon ¼ 6 4 0:0006
ð1Þ
Δα
Δq
Δθ is state vector, ulon ¼ Δδe T
5:9918
0
0:5121
1:0000
0:7086
0:2730
0
1
0
9:7866
3
0:0064 7 7 7; 0:0004 5 0
ΔδT 2
T
is control input vector. System matrices Alon and Blon are given as
0:1156
6 0:0299 6 Blon ¼ 6 6 0:7558 4 0
6:4151
3
0:0014 7 7 7 0:1419 7 5 0
The lateral state equation is expressed by x_ lat ¼ Alat xlat þ Blat ulat
ð2Þ
Table 1 Trim state in landing phase. State variable
Airspeed V0
Altitude H
Angle of attack α
Track bank angle μ
Unit Value
m/s 80
m 500
deg 0.97
deg 3
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where xlat ¼ ½ β p r ϕ is the state vector, ulat ¼ ½ δa δr is the control input vector. System matrices Alat and Blat are as follows. 2 3 2 3 0:0000 0:0320 0:0984 0:0088 0:9851 0:1224 6 1:8129 6 3:8071 2:2866 0:2901 7 6 7 6:9582 0:0000 7 6 7 7 Alat ¼ 6 7; Blat ¼ 6 6 0:7661 7 4 0:8256 0:4328 0:3319 0:0000 5 4 0:0000 5 0:0000 0:0000 0:0000 1:0000 0:0355 0:0000 Based on above nonlinear and linear models, we can easily analyze the natural properties of the Large Civil Aircraft. The Eigen values of Alat are as follows: 0.6547 71.6643i (representing Dutch-roll mode), 1.5174 (representing Roll-damping mode), 0.1098 (representing Spiral mode). We find that lateral Roll damping mode and Holland roll mode are stable, while the Spiral mode is unstable which makes the roll and yaw motions divergence. Fig. 1 shows the lateral states responses of zero-inputs or zero-states. It is found that the aircraft lateral states will divergence under small disturbances in initial states or small control inputs. Therefore, controllers should be designed to make the system stable and satisfy the control requirements.
3. Information fusion based optimal regulator for linear system under disturbance 3.1. Information fusion estimation model Consider a distributed system with a fusion center and M sensors connecting to the fusion center. Let zi A Rmi be the i-th sensor measurement of state x A Rn ; mi r n, then a unified linear measurement model of the data available to the fusion center is [13,15] z i ¼ H i x þ vi ;
i ¼ 1; 2; ⋯; M
ð3Þ
mi n
mi
where H i A R is measurement matrix and vi A R is measurement noise, E½vi ¼ 0; E½vi vj T ¼ Ri δij . Let z ¼ ½z1 ; z2 ; ⋯; zM T ; v ¼ ½v1 ; v2 ; ⋯; vM T ; H ¼ ½H 1 ; H 2 ; ⋯; H M T , then the corresponding batch form of (3) is z ¼ Hx þ v
ð4Þ
~ be the estimate of x and covariance of its associated error x~ ¼ x x^ at the fusion center, respectively. The problem is Let x^ and P ¼ covðxÞ how to obtain x^ and P using all available information Z ¼ fz1 ; z2 ; ⋯; zM g at the fusion center. According to the BLUE fusion rule without prior information [13], the optimal fusion estimate of x is x^ ¼ ðH T R 1 HÞ 1 H T R 1 z
ð5Þ
P ¼ ðH T R 1 HÞ 1
ð6Þ
It is identical to the Gauss–Markov estimation.
Fig. 1. Response of aircraft: (a) β0 ¼ 11 ; (b) ϕ0 ¼ 11; (c) Δδa ¼ 11; (d) Δδr ¼ 11
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According to least square estimate criterion, the optimal fusion estimation problem is equivalent to M
x^ ¼ arg min ∑ ðzi H i xÞT Ri 1 ðzi H i xÞ x
Supposing that all the measurements are mutually independent and !1 x^ ¼
ð7Þ
i¼1
M
T 1 Hi ∑M i ¼ 1 H i Ri
is nonsingular, we get [15]
M
∑ H Ti Ri 1 H i
∑ H Ti Ri 1 zi
i¼1
ð8Þ
i¼1
M
P 1 ¼ ∑ H Ti Ri 1 H i
ð9Þ
i¼1
Formulas (8), and (9) show a linear model of information fusion estimation, in which P 1 represents information weight of x^ on itself, H Ti Ri 1 Hi represents the information weight of zi on x, Ri 1 represents the information weight of zi on itself, denoted as I½x^ ¼ P 1 ; I½zi x^ ¼ H Ti Ri 1 H i ; I½zi ¼ Ri 1 , respectively. From the viewpoint of information fusion, Eq. (3) is called information equation or information model. 3.2. Information fusion based optimal regulator for disturbed linear system The traditional LQ optimal regulator is for the linear system without disturbance. Here we try to solve a LQ optimal regulator of a linear system with disturbance, based on IFE theory. And then, an information fusion optimal regulator is designed, which is based on the discrete time system model. Hence, the nonlinear model of the Large Civil Aircraft should be transformed to a linear discrete time model. Consider the following discrete linear system xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ þ EdðkÞ
ð10Þ
where d(k) is a disturbance vector. The state regulator is defined as solving the optimal control sequence by minimizing the following LQ performance index function. N 1
J ¼ xT ðNÞQ ðNÞxðNÞ þ ∑ ½xT ðkÞQ ðkÞxðkÞ þ uT ðkÞRðkÞuðkÞ
ð11Þ
k¼0
where Q ðkÞ ¼ Q T ðkÞ 4 0; RðkÞ ¼ RT ðkÞ 40, N is a terminal time. Theorem 3.1. The optimal control sequence of the state regulator problem (10) and (11) can be described as ^ uðkÞ ¼ ½KðkÞxðkÞ þvd ðkÞ;
k ¼ 0; 1; …; N 1
ð12Þ
while the minimum of the performance index function (11) is described as min J ¼ xT ð0ÞP 1 ð0Þxð0Þ 2½P 1 ð0Þx^ ð0ÞT xð0Þ þ hð0Þ
ð13Þ
where KðkÞ ¼ ½RðkÞ þ BT P 1 ðk þ 1ÞB 1 BT P 1 ðk þ 1ÞA
ð14Þ
vd ðkÞ ¼ ½RðkÞ þ BT P 1 ðk þ 1ÞB 1 BT P 1 ðk þ1Þ U½x^ ðk þ 1Þ EdðkÞ
ð15Þ
(
(
x^ ðkÞ ¼ PðkÞAT ½Pðk þ 1Þ þ BR 1 ðkÞBT 1 ½x^ ðk þ1Þ EdðkÞ x^ ðNÞ ¼ 0 hðkÞ ¼ hðk þ 1Þ þ d ðkÞET P 1 ðk þ 1Þ½ 2x^ ðk þ 1Þ þ EdðkÞ vd T ðkÞ½RðkÞ þ BT P 1 ðk þ 1ÞBvd ðkÞ T
hðNÞ ¼ 0 (
ð16Þ
P 1 ðjÞ ¼ Q ðjÞ þ AT ½Pðj þ 1Þ þBR 1 ðjÞBT 1 A P 1 ðNÞ ¼ Q ðNÞ
ð17Þ
ð18Þ
Remark 3.1. For above information fusion based state regulator, suppose that the weighted matrices of performance index function (11) are constant, that is Q ðkÞ ¼ Q ; RðkÞ ¼ R, and the disturbance is also constant, that is dðkÞ ¼ d, according to optimal control theory, it is obviously that the Riccati equation in (18) has a steady solution, thus (
x^ ðjÞ ¼ PAT ðP þ BR 1 BT Þ 1 ½x^ ðjþ 1Þ Ed P 1 ¼ Q þ AT ðP þ BR 1 BT Þ 1 A
ð19Þ
Considering the stability condition of the iterative Eq. (19), if spectral radius ρ satisfies ρ½PAT ðP þ BR 1 BT Þ 1 o 1, filtering equation in (19) tends to converge, that is lim x^ ðjÞ x^ ¼ ½P 1 AT ðP þ BR 1 BT Þ 1 1 UAT ðP þBR 1 BT Þ 1 Ed
j-1
ð20Þ
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Thus an approximate optimal estimation of control sequence of state regulator is described as ^ uðkÞ ¼ ðR þ BT P 1 BÞ 1 BT P 1 f½P 1 AT ðP þ BR 1 BT Þ 1 1 AT ðP þBR 1 BT Þ 1 þ IgEd ðR þ BT P 1 BÞ 1 BT P 1 AxðkÞ ¼ ½KxðkÞ þvd where ( K ¼ ðR þBT P 1 BÞ 1 BT P 1 A vd ¼ ðR þ BT P 1 BÞ 1 BT P 1 f½P 1 AT ðP þ BR 1 BT Þ 1 1 AT ðP þ BR 1 BT Þ 1 þ IgEd
ð21Þ
ð22Þ
It should be mentioned that K and vd are constant, therefore it can be calculated off-line in advance, which is very useful in engineering applications. 4. Design of flight control system and control laws It is usually considered 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 landing phase. The task of lateral control is mainly keeping balanced roll attitude. However, for the Large Civil Aircraft, it is actually that the state variables and the control variables of longitudinal and lateral channels are coupled, especially for the rudder channel and aileron channel. The traditional PID control is based on single-loop design method, which is difficult to obtain the high performance for this multi-variable coupled system. Therefore, a modern control method is proposed to solve this problem. The proposed control structure flowchart of landing phase is shown as Fig. 2. It consists of speed control loop, pitch attitude control loop and roll attitude control loop. The longitudinal control loops use PID control strategy and Cn control strategy. In order to inhibit the crosswind for the aircraft and keep the lateral balance, here an information fusion based optimal control strategy is adopted. The design process of each controller is as below. This work focuses on attitude control. At first, the root locus method has been used to design PID controller for longitudinal channel. Design of pitch attitude control loop will not only consider introducing the pitch rate to improve the damping property, but also introduce the normal overload nz to increase the longitudinal static stability. Hence, the pitch rate signal and the normal overload signal are feedback combined to the Cn signal. Based on the Cn inner loop, the control law of pitch attitude loop is designed. The flight trajectory not only related with the attitude but also related with the airspeed. If the natural aircraft has no power compensation in landing phase, the flight track angle will not well follow the attitude, which induces the trajectory uncontrollable. Therefore the control law of the autothrottle is designed based on PID. In lateral control loops, there is strong coupling between the aileron channel and rudder channel. Hence the single-variable design method is difficult to achieve the good performance. Therefore, an optimal control method is designed. Moreover, in order to improve the anti-wind performance, it is necessary for considering the wind disturbance in controller design, and the wind disturbance can be measured and feedback to the controller. This work mainly investigates the anti-crosswind performance in keeping incline attitude process.
Fig. 2. Attitude control structure flowchart of landing phase.
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In landing phase, the lateral attitude need to be kept under the disturbance of the crosswind, then an optimal performance index function is given by N1
J¼ ∑
k¼1
xTlat ðkÞQ xlat ðkÞ þ uTlat ðkÞRulat ðkÞ
ð23Þ
where QZ0 and R40, and the linear system Eq. (2) should be transformed into discrete time equation, and the crosswind disturbance is considered, so we get xlat ðk þ 1Þ ¼ Alat xlat ðkÞ þ Blat ulat ðkÞ þ Elat vw ðkÞ
ð24Þ
Based on the information fusion optimal regulator (12)–(18), the optimal control law for lateral channel is designed as 8 u^ lat ðkÞ ¼ ½K lat ðkÞxlat ðkÞ þ vd ðkÞ > > > > K ðkÞ ¼ ½R þ BT P 1 ðk þ 1ÞB 1 BT P 1 ðk þ 1ÞA > > lat lat lat lat lat > < vd ðkÞ ¼ ½R þ BTlat P 1 ðk þ 1ÞBlat 1 BTlat P 1 ðk þ 1Þ U ½x^ lat ðk þ 1Þ Elat vw ðkÞ > > T 1 T > > x^ lat ðkÞ ¼ PðkÞAlat ½Pðk þ1Þ þBlat R Blat 1 U ½x^ lat ðk þ1Þ Elat vw ðkÞ > > > : P 1 ðkÞ ¼ Q þ AT ½Pðk þ 1Þ þ B R 1 BT 1 A lat
lat
lat
ð25Þ
lat
where k ¼ 0; 1; ⋯; N 1. We can use steady solution of Riccati equation (18) of information weight, and then there will no online matrix inversion computation. Moreover, if crosswind disturbance is constant, then an approximate optimal control law is derived as 8 u^ lat ðkÞ ¼ ½K lat xlat ðkÞ þ vd > > > > < K lat ¼ ðR þ BTlat P 1 Blat Þ 1 BTlat P 1 Alat ð26Þ vd ¼ fðR þ BTlat P 1 Blat Þ 1 BTlat P 1 ½P 1 ATlat ðP þ Blat R 1 BTlat Þ 1 1 ATlat ðP þ Blat R 1 BTlat Þ 1 þ IgElat vw > > > > : P 1 ¼ Q þ AT ðP þ B R 1 BT Þ 1 A lat lat lat lat Here vd is constant signal related with the crosswind. Thus, there is even no online matrix calculation, so the calculation cost of approximate optimal control law is much smaller, which is more suitable for engineering applications.
5. Simulation study To verify the effect of the attitude controllers, the designed control system is implemented in the nonlinear model with total variables of the Large Civil Aircraft. The longitudinal channel includes automatic throttle control for airspeed keeping and pitch attitude keeping control. The lateral channel adopts the information fusion regulator (IFR). The comparison simulations among IFR, LQR with integral action (ILQR) [19], LQR and PID control methods for the lateral control channel are implemented. The weighted matrices in IFR, ILQR and LQR are determined same as Q ¼ diagð200; 1; 100; 50Þ; R ¼ diagð1; 1Þ.
Fig. 3. Lateral attitudes responses under a period of constant wind, (a) Roll angle responses, (b) Sideslip angle responses, (c) Roll rate responses, (d) Yaw rate responses, (e) Aileron responses, (f) Rudder responses.
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Fig. 4. Lateral attitudes responses under permanent constant wind, (a) Roll angle responses, (b) Sideslip angle responses, (c) Roll rate responses, (d) Yaw rate responses, (e) Aileron responses, (f) Rudder responses.
(1) Assume that a constant wind is appeared at t¼5 s with duration of 10 s. The constant wind is discomposed according to the body axis, that is uw ¼ 10 m/s, vw ¼10 m/s, ww ¼10 m/s. Fig. 3 shows the Large Civil Aircraft lateral control responses under the constant wind disturbance. The longitudinal attitudes responses are not available here, because the longitudinal control channels only use one control strategy. (2) Assume that a permanent constant wind is appeared at t ¼5 s. Fig. 4 shows the Large Civil Aircraft lateral control responses under the permanent wind disturbance conditions. Viewing from the simulation results, several points can be concluded: (i) Once the wind appears or disappears, the modern control methods such as IFR, ILQR and LQR can respond fast, to reject the wind disturbance influence, while the PID has slowest response. Moreover, IFR has the fastest response, which spends only 2–3 s for balance state recovery. It is because IFR directly utilizes the disturbance feedforward and compensation, while the other methods can also gradually reject the wind disturbance, but by indirect methods. (ii) IFR achieves smallest perturbation peaks and most smooth of roll angle, sideslip angle, roll angle rate and yaw angle rate, ILQR takes the second place, LQR is the next, and PID is the worst. The responses of PID control are most oscillating and have longest recovery time. (iii) IFR and LQR have the same dimension of matrices computation, while ILQR has larger dimension of matrices computation because it is based on augmented state equation. In all, information fusion optimal control method is characterized by best anti-wind ability and improves the ride quality and landing safety under the condition of wind disturbance. However, PID is independent of the system nonlinear and linear models, therefore, it is reliable in control engineering.
6. Conclusion This work establishes a quite complete nonlinear model of Boeing707, based on which the linear models are obtained and the natural properties are studied. For the attitude and velocity keeping control problem, we adopts the classical method to design the attitude control laws of automatic landing phase. The inner loop of pitch control uses Cn criterion to achieve better longitudinal attitude control effect. By applying autothrottle control system, the aircraft can keep airspeed and angle of attack constant in landing phase. An optimal state regulator for discrete time linear system with disturbance is presented by applying the IFE theory. The IFE method has explicit physical significance and is easily initialized since it is expressed in terms of information matrices and vectors. And then, an information fusion optimal regulator is designed in lateral channel to improve the anti-wind performance of the aircraft. The simulation results show that the lateral attitude control based on information fusion optimal control method obviously improves the anti-disturbance performance, responses fasted and smoothest, and makes the aircraft return to the equilibrium fastest after the wind disappearing, when comparing with the PID control, LQ control, and LQ control with integral action, so that enhances the riding comfort and safety of the Large Civil Aircraft.
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The information fusion optimal control is derived from information fusion theory, which exhibits the homogeneity of estimation and control problems. The proposed information fusion optimal control is a novel modern control method, the structure of which is composed of state feedback channel that is same with the traditional LQ control and disturbance/desired state feedforward channel that improves anti-disturbance or tacking performance. Therefore, it is stable for linear systems and is more effective than the traditional LQ control method. Moreover, the control matrices can be computed off-line in advance, which illustrates that the information fusion optimal control is completely suitable for engineering application in Large Civil Aircraft.
Acknowledgments This work is based on research supported by the NUAA Fundamental Research Fund under Grant no. NS2013029.
Appendix A.1 nonlinear dynamics and kinematics equations The structure of nonlinear model of Large Civil Aircraft is shown in Fig. A1. The variables definition is given in Table A1. Based on aircraft body axis, the line dynamics and angle dynamics equations are, respectively given by [18] 8 u_ ¼ vr wq g sin θ þ Fmx > > < F v_ ¼ ur þ wp þg cos θ sin ϕ þ my ðA:1Þ > > :w _ ¼ uq vp þ g cos θ cos ϕ þ F z m
Fig. A1. Nonlinear model of civil aircraft.
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Table A1 Variables definition. Variables
Notations
δe, δa, δr, δT CL, CD, CY Cl, Cm, Cn V, nz α, β μ, φ x,y,z θ,ϕ,ψ p,q,r Fx,Fy,Fz
Elevator, aileron, rudder and throttle opening angles Lift, drag and side forces coefficients Roll, pitch and yaw moments coefficients Airspeed, normal overload Angles of attack and sideslip Bank angle and azimuth angle of flight path Spatial location Pitch angle, roll angle and yaw angle Roll rate, pitch rate and yaw rate Axial, side and normal forces, Roll, pitch and yaw moments
L; M; N u, v, w, uw, vw, ww
Axial, side and normal velocities and wind velocities
8 _ > < p ¼ ðc1 r þc2 pÞq þ c3 L þ c4 N q_ ¼ c5 pr c6 ðp2 r 2 Þ þc7 M > :_ r ¼ ðc8 p c2 rÞq þc4 L þ c9 N
ðA:2Þ
Here c1–c9 are inertia coefficients. The line kinematics and angle kinematics equations are, respectively given by [18] 8_ x ¼ V cos μ cos φ > < g y_ g ¼ V cos μ sin φ > :_ h ¼ V sin μ
ðA:3Þ
8_ > < ϕ ¼ p þ ðr cos ϕ þ q sin ϕÞ tan θ ψ_ ¼ cos1 θðr cos ϕ þ q sin ϕÞ > :_ θ ¼ q cos ϕ r sin ϕ
ðA:4Þ
According to the relationship between airspeed in body axis and in velocity axis, we get [18] 8 > < u ¼ V cos α cos β v ¼ V sin β > : w ¼ V sin α cos β Thus we get pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ u2 þ v2 þ w2
ðA:5Þ
ðA:6Þ
w α ¼ arctan u
ðA:7Þ
v V
ðA:8Þ
β ¼ arctan
_ uu_ þ vv_ þ ww V_ ¼ V
ðA:9Þ
_ wu_ uw u2 þ w 2
ðA:10Þ
v_ V vV_ β_ ¼ 2 V cos β
ðA:11Þ
α_ ¼
The relationship among pitch angle, bank angle and attack angle is described as θ ¼ μþα
ðA:12Þ
Above equations constitute a nonlinear mathematical model with total variables of the Large Civil Aircraft. A.2 proof of theorem 3.1 Step 1: From performance index function (11), we get two information equations expressed by 8 Δ < z ðjÞ ¼ 0 ¼ xðjÞ þ mðjÞ x;1 : I½zx;1 ðjÞxðjÞ ¼ Q ðjÞ
ðA:13Þ
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Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Information fusion based optimal control for large civil aircraft system Ziyang Zhen n, Ju Jiang, Xinhua Wang, Chen Gao College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, No.29, Yudao Street, Nanjing, China
art ic l e i nf o
a b s t r a c t
Article history: Received 9 July 2014 Received in revised form 4 September 2014 Accepted 20 September 2014 Available online 23 October 2014
Wind disturbance has a great influence on landing security of Large Civil Aircraft. Through simulation research and engineering experience, it can be found that PID control is not good enough to solve the problem of restraining the wind disturbance. This paper focuses on anti-wind attitude control for Large Civil Aircraft in landing phase. In order to improve the riding comfort and the flight security, an information fusion based optimal control strategy is presented to restrain the wind in landing phase for maintaining attitudes and airspeed. Data of Boeing707 is used to establish a nonlinear mode with total variables of Large Civil Aircraft, and then two linear models are obtained which are divided into longitudinal and lateral equations. Based on engineering experience, the longitudinal channel adopts PID control and Cn inner control to keep longitudinal attitude constant, and applies autothrottle system for keeping airspeed constant, while an information fusion based optimal regulator in the lateral control channel is designed to achieve lateral attitude holding. According to information fusion estimation, by fusing hard constraint information of system dynamic equations and the soft constraint information of performance index function, optimal estimation of the control sequence is derived. Based on this, an information fusion state regulator is deduced for discrete time linear system with disturbance. The simulation results of nonlinear model of aircraft indicate that the information fusion optimal control is better than traditional PID control, LQR control and LQR control with integral action, in anti-wind disturbance performance in the landing phase. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Large civil aircraft Automatic flight control Linear quadratic optimal control Information fusion Optimal estimation
1. Introduction The automatic flight control systems of Large Civil Aircrafts play important roles in reducing the pilot's workload and achieving accurate attitude/trajectory control. As one of the key technologies of automatic flight control system, design of control laws directly determine the flight performance of Large Civil Aircraft. Control laws should be designed to meet all the requirements of performance index, and regard security, amenity and economical efficiency of flight as the major tasks of automatic flight control systems design. There are several kinds of flight control methods presented for civil aircrafts. Nguyen (2009) uses robust optimal adaptive control to improve the tracking performance at a large adaptive gain, but the performance of restraining wind disturbance has not been tested [1]. Looye (2011) uses dynamic inversion and multi-objective optimization to design the attitude control system for civil aircraft, the controller is robustness to the parametric uncertainties, but performance index and flight quality have not been taken into account [2]. Gregory (2011) designs L1 adaptive control architecture to directly compensate for significant uncertain cross-coupling in nonlinear systems [3]. Shin (2007) uses a linear parameter-varying control synthesis method to design fault tolerant controllers for Boeing 747-100/200, and presents an application of robust gain-scheduled control methods [4]. References [5,6] study an adaptive control algorithm. Its advantages are the capability of improving the reliability and handling the aerodynamic parameter uncertainties. However, most references do not refer to concrete performance index, and do not test the control system's ability of anti-gust disturbance. Mohammad (2011) designs a closed-loop decision-making system to command the aircraft based on fuzzy logic controller for terrain following flight, which is important in approaching an airport with low or non-visibility for civil aircraft [7]. However, it only utilizes the system input–output data so that it is difficult to analyze and design the controllers, furthermore, theory basis of such intelligent methods is not perfect enough, and the computation costs of the intelligent methods generally become large.
n
Corresponding author. E-mail address: [email protected] (Z. Zhen).
http://dx.doi.org/10.1016/j.isatra.2014.09.017 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
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Development of Chinese Large Civil Aircraft is in the beginning stage, research on the automatic control law of the Large Civil Aircraft is not enough. Li (2010) uses LQG/LTR based method to design a longitudinal command augmentation system (CAS), this method is verified to enhance the stability, robustness and anti-jamming capacity, but the simulation is implemented based on linear system of civil aircraft [8]. Fan (2010) presents a modeling and control strategy for an advanced configured large civil aircraft with aeroservoelasticity via LQG method and control allocation [9]. LQ optimal control is a well-known fundamental theory in modern control theory. The LQ optimal control has received a great deal of attention from control theorists and engineers, since the resulting control law is linear with respect to the state and is therefore easy to compute [10,11]. There are many different approaches to the solution of LQ optimal control problems, such as the minimization method using Lagrange multipliers, dynamic programming, and Lyapunov function. Information fusion estimation (IFE) is the problem of how to best utilize useful information obtained from multiple sources or from a single source over a time period, for estimating an unknown parameter or process. The most important application area of IFE is track fusion or track-to-track fusion in target tracking system, which has been investigated for more than two decades [12]. Li (2003) establishes three estimation fusion architectures including centralized, distributed, and hybrid, moreover, he proposes an unified linear data model and two optimal fusion rules based on the best linear unbiased estimation (BLUE) and the weighted least squares (WLS) [13]. It is much more general and flexible than the previous approach of matching centralized and distributed fusion rules based on Kalman filters. Sun (2004) presents a new multi-sensor optimal information fusion criterion weighted by matrices in the linear minimum variance (LMV) sense which is equivalent to the maximum likelihood fusion criterion under the assumption of normal distribution [14]. Wang (2007) proposes a concept of the information weight which represents the contribution of the measurement on the estimated variable, and presents unified information fusion rules based on least square estimate criterion for linear and nonlinear measurements [15]. The information weight is essentially equal to the Fisher information matrix in case of Gaussian white noise. Information fusion is an ideological which universally exists in decision making problems. Both estimation problem and control problem belong to the decision making problem. In view of this, Zhen (2010) proposes a new optimal control method called information fusion control method for nonlinear systems [16,17]. It regards information of the desired trajectory, system dynamic equation and ideal control strategy as the measurement information of the control variable, and obtains an optimal estimate of the control variable by IFE, and then transfers the optimal control problem into the optimal estimation problem. In summary, the previous references in designing the automatic control system of Large Civil Aircraft are few, in which the control strategies for anti-wind are especially little studied. Furthermore, previous research works always use linear models of Large Civil Aircraft in simulation, which is not accurate enough for real aircraft. Aiming to solve these problems, first, a nonlinear model with total variables is established for system simulation and natural analysis. Second, according to the aerial engineering experience, in landing phase with wind disturbance, the pitch attitude and flight speed can be well controlled by the traditional engineering methods, while the inclined attitude is difficult to be hold. Therefore, PID control and Cn control are introduced to design the pitch attitude controller, an airspeed keeping autothrottle system is designed for flight speed control, and an optimal controller is designed for controlling inclined attitude. Third, in order to utilize the disturbance information to improve the control performance, this work originally propose an information fusion method to design the optimal regulator problem of discrete time linear system with disturbance. The paper is organized as following: models of Large Civil Aircraft are established for natural analysis in Section 2, information fusion estimation and information fusion optimal regulator are shown in Section 3. Automatic flight control system which is composed of longitudinal channel and lateral channel and their control laws are designed in Section 4, simulations are carried out for comparing different control methods in Section 5, and finally the conclusions are shown in Section 6.
2. Modeling and property analyzing of large civil aircraft According to the aerodynamic parameters of Boeing707 given in [18], we establish a nonlinear numerical model of Large Civil Aircraft, which is shown in Appendix A.1. In order to study the natural properties of Large Civil Aircraft and design the control laws, firstly linear models need to be obtained. The selected trim point of landing phase is shown in Table 1. By decoupling and linearization near the trim point, the linear longitudinal state equation of Large Civil Aircraft is expressed by x_ lon ¼ Alon xlon þ Blon ulon where xlon ¼ ½ ΔV follows. 2
0:0473
6 0:0030 6 Alon ¼ 6 4 0:0006
ð1Þ
Δα
Δq
Δθ is state vector, ulon ¼ Δδe T
5:9918
0
0:5121
1:0000
0:7086
0:2730
0
1
0
9:7866
3
0:0064 7 7 7; 0:0004 5 0
ΔδT 2
T
is control input vector. System matrices Alon and Blon are given as
0:1156
6 0:0299 6 Blon ¼ 6 6 0:7558 4 0
6:4151
3
0:0014 7 7 7 0:1419 7 5 0
The lateral state equation is expressed by x_ lat ¼ Alat xlat þ Blat ulat
ð2Þ
Table 1 Trim state in landing phase. State variable
Airspeed V0
Altitude H
Angle of attack α
Track bank angle μ
Unit Value
m/s 80
m 500
deg 0.97
deg 3
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where xlat ¼ ½ β p r ϕ is the state vector, ulat ¼ ½ δa δr is the control input vector. System matrices Alat and Blat are as follows. 2 3 2 3 0:0000 0:0320 0:0984 0:0088 0:9851 0:1224 6 1:8129 6 3:8071 2:2866 0:2901 7 6 7 6:9582 0:0000 7 6 7 7 Alat ¼ 6 7; Blat ¼ 6 6 0:7661 7 4 0:8256 0:4328 0:3319 0:0000 5 4 0:0000 5 0:0000 0:0000 0:0000 1:0000 0:0355 0:0000 Based on above nonlinear and linear models, we can easily analyze the natural properties of the Large Civil Aircraft. The Eigen values of Alat are as follows: 0.6547 71.6643i (representing Dutch-roll mode), 1.5174 (representing Roll-damping mode), 0.1098 (representing Spiral mode). We find that lateral Roll damping mode and Holland roll mode are stable, while the Spiral mode is unstable which makes the roll and yaw motions divergence. Fig. 1 shows the lateral states responses of zero-inputs or zero-states. It is found that the aircraft lateral states will divergence under small disturbances in initial states or small control inputs. Therefore, controllers should be designed to make the system stable and satisfy the control requirements.
3. Information fusion based optimal regulator for linear system under disturbance 3.1. Information fusion estimation model Consider a distributed system with a fusion center and M sensors connecting to the fusion center. Let zi A Rmi be the i-th sensor measurement of state x A Rn ; mi r n, then a unified linear measurement model of the data available to the fusion center is [13,15] z i ¼ H i x þ vi ;
i ¼ 1; 2; ⋯; M
ð3Þ
mi n
mi
where H i A R is measurement matrix and vi A R is measurement noise, E½vi ¼ 0; E½vi vj T ¼ Ri δij . Let z ¼ ½z1 ; z2 ; ⋯; zM T ; v ¼ ½v1 ; v2 ; ⋯; vM T ; H ¼ ½H 1 ; H 2 ; ⋯; H M T , then the corresponding batch form of (3) is z ¼ Hx þ v
ð4Þ
~ be the estimate of x and covariance of its associated error x~ ¼ x x^ at the fusion center, respectively. The problem is Let x^ and P ¼ covðxÞ how to obtain x^ and P using all available information Z ¼ fz1 ; z2 ; ⋯; zM g at the fusion center. According to the BLUE fusion rule without prior information [13], the optimal fusion estimate of x is x^ ¼ ðH T R 1 HÞ 1 H T R 1 z
ð5Þ
P ¼ ðH T R 1 HÞ 1
ð6Þ
It is identical to the Gauss–Markov estimation.
Fig. 1. Response of aircraft: (a) β0 ¼ 11 ; (b) ϕ0 ¼ 11; (c) Δδa ¼ 11; (d) Δδr ¼ 11
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According to least square estimate criterion, the optimal fusion estimation problem is equivalent to M
x^ ¼ arg min ∑ ðzi H i xÞT Ri 1 ðzi H i xÞ x
Supposing that all the measurements are mutually independent and !1 x^ ¼
ð7Þ
i¼1
M
T 1 Hi ∑M i ¼ 1 H i Ri
is nonsingular, we get [15]
M
∑ H Ti Ri 1 H i
∑ H Ti Ri 1 zi
i¼1
ð8Þ
i¼1
M
P 1 ¼ ∑ H Ti Ri 1 H i
ð9Þ
i¼1
Formulas (8), and (9) show a linear model of information fusion estimation, in which P 1 represents information weight of x^ on itself, H Ti Ri 1 Hi represents the information weight of zi on x, Ri 1 represents the information weight of zi on itself, denoted as I½x^ ¼ P 1 ; I½zi x^ ¼ H Ti Ri 1 H i ; I½zi ¼ Ri 1 , respectively. From the viewpoint of information fusion, Eq. (3) is called information equation or information model. 3.2. Information fusion based optimal regulator for disturbed linear system The traditional LQ optimal regulator is for the linear system without disturbance. Here we try to solve a LQ optimal regulator of a linear system with disturbance, based on IFE theory. And then, an information fusion optimal regulator is designed, which is based on the discrete time system model. Hence, the nonlinear model of the Large Civil Aircraft should be transformed to a linear discrete time model. Consider the following discrete linear system xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ þ EdðkÞ
ð10Þ
where d(k) is a disturbance vector. The state regulator is defined as solving the optimal control sequence by minimizing the following LQ performance index function. N 1
J ¼ xT ðNÞQ ðNÞxðNÞ þ ∑ ½xT ðkÞQ ðkÞxðkÞ þ uT ðkÞRðkÞuðkÞ
ð11Þ
k¼0
where Q ðkÞ ¼ Q T ðkÞ 4 0; RðkÞ ¼ RT ðkÞ 40, N is a terminal time. Theorem 3.1. The optimal control sequence of the state regulator problem (10) and (11) can be described as ^ uðkÞ ¼ ½KðkÞxðkÞ þvd ðkÞ;
k ¼ 0; 1; …; N 1
ð12Þ
while the minimum of the performance index function (11) is described as min J ¼ xT ð0ÞP 1 ð0Þxð0Þ 2½P 1 ð0Þx^ ð0ÞT xð0Þ þ hð0Þ
ð13Þ
where KðkÞ ¼ ½RðkÞ þ BT P 1 ðk þ 1ÞB 1 BT P 1 ðk þ 1ÞA
ð14Þ
vd ðkÞ ¼ ½RðkÞ þ BT P 1 ðk þ 1ÞB 1 BT P 1 ðk þ1Þ U½x^ ðk þ 1Þ EdðkÞ
ð15Þ
(
(
x^ ðkÞ ¼ PðkÞAT ½Pðk þ 1Þ þ BR 1 ðkÞBT 1 ½x^ ðk þ1Þ EdðkÞ x^ ðNÞ ¼ 0 hðkÞ ¼ hðk þ 1Þ þ d ðkÞET P 1 ðk þ 1Þ½ 2x^ ðk þ 1Þ þ EdðkÞ vd T ðkÞ½RðkÞ þ BT P 1 ðk þ 1ÞBvd ðkÞ T
hðNÞ ¼ 0 (
ð16Þ
P 1 ðjÞ ¼ Q ðjÞ þ AT ½Pðj þ 1Þ þBR 1 ðjÞBT 1 A P 1 ðNÞ ¼ Q ðNÞ
ð17Þ
ð18Þ
Remark 3.1. For above information fusion based state regulator, suppose that the weighted matrices of performance index function (11) are constant, that is Q ðkÞ ¼ Q ; RðkÞ ¼ R, and the disturbance is also constant, that is dðkÞ ¼ d, according to optimal control theory, it is obviously that the Riccati equation in (18) has a steady solution, thus (
x^ ðjÞ ¼ PAT ðP þ BR 1 BT Þ 1 ½x^ ðjþ 1Þ Ed P 1 ¼ Q þ AT ðP þ BR 1 BT Þ 1 A
ð19Þ
Considering the stability condition of the iterative Eq. (19), if spectral radius ρ satisfies ρ½PAT ðP þ BR 1 BT Þ 1 o 1, filtering equation in (19) tends to converge, that is lim x^ ðjÞ x^ ¼ ½P 1 AT ðP þ BR 1 BT Þ 1 1 UAT ðP þBR 1 BT Þ 1 Ed
j-1
ð20Þ
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Thus an approximate optimal estimation of control sequence of state regulator is described as ^ uðkÞ ¼ ðR þ BT P 1 BÞ 1 BT P 1 f½P 1 AT ðP þ BR 1 BT Þ 1 1 AT ðP þBR 1 BT Þ 1 þ IgEd ðR þ BT P 1 BÞ 1 BT P 1 AxðkÞ ¼ ½KxðkÞ þvd where ( K ¼ ðR þBT P 1 BÞ 1 BT P 1 A vd ¼ ðR þ BT P 1 BÞ 1 BT P 1 f½P 1 AT ðP þ BR 1 BT Þ 1 1 AT ðP þ BR 1 BT Þ 1 þ IgEd
ð21Þ
ð22Þ
It should be mentioned that K and vd are constant, therefore it can be calculated off-line in advance, which is very useful in engineering applications. 4. Design of flight control system and control laws It is usually considered 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 landing phase. The task of lateral control is mainly keeping balanced roll attitude. However, for the Large Civil Aircraft, it is actually that the state variables and the control variables of longitudinal and lateral channels are coupled, especially for the rudder channel and aileron channel. The traditional PID control is based on single-loop design method, which is difficult to obtain the high performance for this multi-variable coupled system. Therefore, a modern control method is proposed to solve this problem. The proposed control structure flowchart of landing phase is shown as Fig. 2. It consists of speed control loop, pitch attitude control loop and roll attitude control loop. The longitudinal control loops use PID control strategy and Cn control strategy. In order to inhibit the crosswind for the aircraft and keep the lateral balance, here an information fusion based optimal control strategy is adopted. The design process of each controller is as below. This work focuses on attitude control. At first, the root locus method has been used to design PID controller for longitudinal channel. Design of pitch attitude control loop will not only consider introducing the pitch rate to improve the damping property, but also introduce the normal overload nz to increase the longitudinal static stability. Hence, the pitch rate signal and the normal overload signal are feedback combined to the Cn signal. Based on the Cn inner loop, the control law of pitch attitude loop is designed. The flight trajectory not only related with the attitude but also related with the airspeed. If the natural aircraft has no power compensation in landing phase, the flight track angle will not well follow the attitude, which induces the trajectory uncontrollable. Therefore the control law of the autothrottle is designed based on PID. In lateral control loops, there is strong coupling between the aileron channel and rudder channel. Hence the single-variable design method is difficult to achieve the good performance. Therefore, an optimal control method is designed. Moreover, in order to improve the anti-wind performance, it is necessary for considering the wind disturbance in controller design, and the wind disturbance can be measured and feedback to the controller. This work mainly investigates the anti-crosswind performance in keeping incline attitude process.
Fig. 2. Attitude control structure flowchart of landing phase.
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In landing phase, the lateral attitude need to be kept under the disturbance of the crosswind, then an optimal performance index function is given by N1
J¼ ∑
k¼1
xTlat ðkÞQ xlat ðkÞ þ uTlat ðkÞRulat ðkÞ
ð23Þ
where QZ0 and R40, and the linear system Eq. (2) should be transformed into discrete time equation, and the crosswind disturbance is considered, so we get xlat ðk þ 1Þ ¼ Alat xlat ðkÞ þ Blat ulat ðkÞ þ Elat vw ðkÞ
ð24Þ
Based on the information fusion optimal regulator (12)–(18), the optimal control law for lateral channel is designed as 8 u^ lat ðkÞ ¼ ½K lat ðkÞxlat ðkÞ þ vd ðkÞ > > > > K ðkÞ ¼ ½R þ BT P 1 ðk þ 1ÞB 1 BT P 1 ðk þ 1ÞA > > lat lat lat lat lat > < vd ðkÞ ¼ ½R þ BTlat P 1 ðk þ 1ÞBlat 1 BTlat P 1 ðk þ 1Þ U ½x^ lat ðk þ 1Þ Elat vw ðkÞ > > T 1 T > > x^ lat ðkÞ ¼ PðkÞAlat ½Pðk þ1Þ þBlat R Blat 1 U ½x^ lat ðk þ1Þ Elat vw ðkÞ > > > : P 1 ðkÞ ¼ Q þ AT ½Pðk þ 1Þ þ B R 1 BT 1 A lat
lat
lat
ð25Þ
lat
where k ¼ 0; 1; ⋯; N 1. We can use steady solution of Riccati equation (18) of information weight, and then there will no online matrix inversion computation. Moreover, if crosswind disturbance is constant, then an approximate optimal control law is derived as 8 u^ lat ðkÞ ¼ ½K lat xlat ðkÞ þ vd > > > > < K lat ¼ ðR þ BTlat P 1 Blat Þ 1 BTlat P 1 Alat ð26Þ vd ¼ fðR þ BTlat P 1 Blat Þ 1 BTlat P 1 ½P 1 ATlat ðP þ Blat R 1 BTlat Þ 1 1 ATlat ðP þ Blat R 1 BTlat Þ 1 þ IgElat vw > > > > : P 1 ¼ Q þ AT ðP þ B R 1 BT Þ 1 A lat lat lat lat Here vd is constant signal related with the crosswind. Thus, there is even no online matrix calculation, so the calculation cost of approximate optimal control law is much smaller, which is more suitable for engineering applications.
5. Simulation study To verify the effect of the attitude controllers, the designed control system is implemented in the nonlinear model with total variables of the Large Civil Aircraft. The longitudinal channel includes automatic throttle control for airspeed keeping and pitch attitude keeping control. The lateral channel adopts the information fusion regulator (IFR). The comparison simulations among IFR, LQR with integral action (ILQR) [19], LQR and PID control methods for the lateral control channel are implemented. The weighted matrices in IFR, ILQR and LQR are determined same as Q ¼ diagð200; 1; 100; 50Þ; R ¼ diagð1; 1Þ.
Fig. 3. Lateral attitudes responses under a period of constant wind, (a) Roll angle responses, (b) Sideslip angle responses, (c) Roll rate responses, (d) Yaw rate responses, (e) Aileron responses, (f) Rudder responses.
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Fig. 4. Lateral attitudes responses under permanent constant wind, (a) Roll angle responses, (b) Sideslip angle responses, (c) Roll rate responses, (d) Yaw rate responses, (e) Aileron responses, (f) Rudder responses.
(1) Assume that a constant wind is appeared at t¼5 s with duration of 10 s. The constant wind is discomposed according to the body axis, that is uw ¼ 10 m/s, vw ¼10 m/s, ww ¼10 m/s. Fig. 3 shows the Large Civil Aircraft lateral control responses under the constant wind disturbance. The longitudinal attitudes responses are not available here, because the longitudinal control channels only use one control strategy. (2) Assume that a permanent constant wind is appeared at t ¼5 s. Fig. 4 shows the Large Civil Aircraft lateral control responses under the permanent wind disturbance conditions. Viewing from the simulation results, several points can be concluded: (i) Once the wind appears or disappears, the modern control methods such as IFR, ILQR and LQR can respond fast, to reject the wind disturbance influence, while the PID has slowest response. Moreover, IFR has the fastest response, which spends only 2–3 s for balance state recovery. It is because IFR directly utilizes the disturbance feedforward and compensation, while the other methods can also gradually reject the wind disturbance, but by indirect methods. (ii) IFR achieves smallest perturbation peaks and most smooth of roll angle, sideslip angle, roll angle rate and yaw angle rate, ILQR takes the second place, LQR is the next, and PID is the worst. The responses of PID control are most oscillating and have longest recovery time. (iii) IFR and LQR have the same dimension of matrices computation, while ILQR has larger dimension of matrices computation because it is based on augmented state equation. In all, information fusion optimal control method is characterized by best anti-wind ability and improves the ride quality and landing safety under the condition of wind disturbance. However, PID is independent of the system nonlinear and linear models, therefore, it is reliable in control engineering.
6. Conclusion This work establishes a quite complete nonlinear model of Boeing707, based on which the linear models are obtained and the natural properties are studied. For the attitude and velocity keeping control problem, we adopts the classical method to design the attitude control laws of automatic landing phase. The inner loop of pitch control uses Cn criterion to achieve better longitudinal attitude control effect. By applying autothrottle control system, the aircraft can keep airspeed and angle of attack constant in landing phase. An optimal state regulator for discrete time linear system with disturbance is presented by applying the IFE theory. The IFE method has explicit physical significance and is easily initialized since it is expressed in terms of information matrices and vectors. And then, an information fusion optimal regulator is designed in lateral channel to improve the anti-wind performance of the aircraft. The simulation results show that the lateral attitude control based on information fusion optimal control method obviously improves the anti-disturbance performance, responses fasted and smoothest, and makes the aircraft return to the equilibrium fastest after the wind disappearing, when comparing with the PID control, LQ control, and LQ control with integral action, so that enhances the riding comfort and safety of the Large Civil Aircraft.
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The information fusion optimal control is derived from information fusion theory, which exhibits the homogeneity of estimation and control problems. The proposed information fusion optimal control is a novel modern control method, the structure of which is composed of state feedback channel that is same with the traditional LQ control and disturbance/desired state feedforward channel that improves anti-disturbance or tacking performance. Therefore, it is stable for linear systems and is more effective than the traditional LQ control method. Moreover, the control matrices can be computed off-line in advance, which illustrates that the information fusion optimal control is completely suitable for engineering application in Large Civil Aircraft.
Acknowledgments This work is based on research supported by the NUAA Fundamental Research Fund under Grant no. NS2013029.
Appendix A.1 nonlinear dynamics and kinematics equations The structure of nonlinear model of Large Civil Aircraft is shown in Fig. A1. The variables definition is given in Table A1. Based on aircraft body axis, the line dynamics and angle dynamics equations are, respectively given by [18] 8 u_ ¼ vr wq g sin θ þ Fmx > > < F v_ ¼ ur þ wp þg cos θ sin ϕ þ my ðA:1Þ > > :w _ ¼ uq vp þ g cos θ cos ϕ þ F z m
Fig. A1. Nonlinear model of civil aircraft.
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Table A1 Variables definition. Variables
Notations
δe, δa, δr, δT CL, CD, CY Cl, Cm, Cn V, nz α, β μ, φ x,y,z θ,ϕ,ψ p,q,r Fx,Fy,Fz
Elevator, aileron, rudder and throttle opening angles Lift, drag and side forces coefficients Roll, pitch and yaw moments coefficients Airspeed, normal overload Angles of attack and sideslip Bank angle and azimuth angle of flight path Spatial location Pitch angle, roll angle and yaw angle Roll rate, pitch rate and yaw rate Axial, side and normal forces, Roll, pitch and yaw moments
L; M; N u, v, w, uw, vw, ww
Axial, side and normal velocities and wind velocities
8 _ > < p ¼ ðc1 r þc2 pÞq þ c3 L þ c4 N q_ ¼ c5 pr c6 ðp2 r 2 Þ þc7 M > :_ r ¼ ðc8 p c2 rÞq þc4 L þ c9 N
ðA:2Þ
Here c1–c9 are inertia coefficients. The line kinematics and angle kinematics equations are, respectively given by [18] 8_ x ¼ V cos μ cos φ > < g y_ g ¼ V cos μ sin φ > :_ h ¼ V sin μ
ðA:3Þ
8_ > < ϕ ¼ p þ ðr cos ϕ þ q sin ϕÞ tan θ ψ_ ¼ cos1 θðr cos ϕ þ q sin ϕÞ > :_ θ ¼ q cos ϕ r sin ϕ
ðA:4Þ
According to the relationship between airspeed in body axis and in velocity axis, we get [18] 8 > < u ¼ V cos α cos β v ¼ V sin β > : w ¼ V sin α cos β Thus we get pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ u2 þ v2 þ w2
ðA:5Þ
ðA:6Þ
w α ¼ arctan u
ðA:7Þ
v V
ðA:8Þ
β ¼ arctan
_ uu_ þ vv_ þ ww V_ ¼ V
ðA:9Þ
_ wu_ uw u2 þ w 2
ðA:10Þ
v_ V vV_ β_ ¼ 2 V cos β
ðA:11Þ
α_ ¼
The relationship among pitch angle, bank angle and attack angle is described as θ ¼ μþα
ðA:12Þ
Above equations constitute a nonlinear mathematical model with total variables of the Large Civil Aircraft. A.2 proof of theorem 3.1 Step 1: From performance index function (11), we get two information equations expressed by 8 Δ < z ðjÞ ¼ 0 ¼ xðjÞ þ mðjÞ x;1 : I½zx;1 ðjÞxðjÞ ¼ Q ðjÞ
ðA:13Þ
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