Optimal Nemo-Controller in Longitudinal Autolanding of a Commercial Jet Transport Hojat had?,Mehrdad Pakmehr'. Nasser Sadat?
the last three decades, optimality-based autolanding designs have been considered to the most effective way by many authors. However, it is known that the straight fow'ard solution to the optimal control problem leads to Two Point Boundary Value Problem (TPBVP)(Riccati equation), which is usually too complex in solution, backward in the time, and real-time onboard implementation, or the final time, as a boundary condition, may also not be known precisely. To avoid these problems, first, a suboptimal solution by assuming im
Abstraa-
In
has been considered and its inapplicability has been discussed. Then an optimal controller for landing phase of a typical commercial aircraft has been designed. Finally, seven neural networks were being trained to learn the costates of the system to estimate the costates in similar scenarios without using the final time value, which usually is needed in solving the optimal control problems I n d a Terms- Optimal Control, Longitudinal Autolanding, Final Time Avoidance, Neuro-controller, Suboptimal Solution
1. INTRODUCTION
M
ANY research activities have heen done on designing an automatic landing controller for different classes of aircrafts especially jet transports; hereby some of them have been mentioned briefly. In [l], a bidimensional model for 6 and w of airplane has been used to design an optimal control for landing phase of airplane to obviate the complexity of stochastic differential equation that gives t h e i . In [Z], an extended Kalman filter based on non-linear longitudinal aircraft equations of motion is developed for estimation of horizontal and vertical atmospheric wind inputs and finally an optimal feedback control law based on non-linear inverse dynamics has been designed. In [3], the optimal control problem for landing phase of Boeing-747 has been set up to minimize the trajectory deviations and control efforts. Atmospheric
' Graduate Student, Mechanical Engineering Department, Tarbiyat Modarres University, Nasr Bridge, All-e-Ahmad High way, Tehran, IranIl.adi_hojjat~yahoo.com ' B.Sc. Graduate, Aerospace Engineering Department, Sharif University of Technology-
[email protected] Associate Professor, Electrical Engineering Department, Sharif University o f Technology-
[email protected]
0-7803-7729-X/03/$17.00 0 2 0 0 3 IEEE.
disturbances (wg , OJ also has been included. In [4], the landing phase of aircraft has been expressed as an example for a new technique in optimal control. This approach named as adaptivecritic-based neural network and the authors have extended this technique in [5]. In [ 5 ] , adaptive critic neural networks have been used to design a controller for a benchmark problem in aircraft autolanding. It will he shown, in this research, that the performance of the proposed controller, especially its followed sink rate, is better than the ones presented in aforementioned researches. In this paper, a new approach based on Multilayer Perceptron (MLP) neural networks have been proposed to solve the TPBVP associated with the optimal control of aircraft landing which its mathematical model is assumed to be linear quadratic. Optimal control of Linear Quadratic Tracking (LQT) system results in a TPBVP that one set of its boundary condition needs the final time of the process and in some cases the final time of the process doesn't exist. In other words, first it should he solved backward in time to obtain its costates, and then the optimal control law can be obtained. Finally, the applicability of the proposed approach is demonstrated through an application example. In some problems such as aircraft landing, it is not possible to obtain the final time precisely before landing initiation phase. This is because of sensor errors, sudden changes in atmospheric conditions near the runway and pilot error. Hence, we should try to design an architecture to be independent of time and in any case follows the optimal condition. Neural networks, due to their learning ability, can fulfill this job and due to their approximating ability can extend the flight envelope of aircraft in presence of sudden changes in atmospheric conditions and flight parameters. It seems they can be good candidates to solve the landing problem associated with the optimal control of a tracking problem. There is a classic method, called suboptimal solution, to solve the Riccati equation when the final time of the process doesn't exist. In suboptimal solutions, designers usually suppose that!, + -, and with this assumption the differential term in Riccati equation is omitted. Therefore Differential Riccati Equation leads to the Algebraic Riccati Equation, which is simpler to solve and doesn't need the final time of the process. The occurrence, acceptability and the existence of solution in suboptimal solution depends on some conditions, which must be satisfied by the system. These conditions are widely discussed later: one of these conditions mentions that; to have an acceptable solution for suboptimal method, one must have t , - f +- or N-K + w , in other words, for the preliminary 492
time stages of the process we have an acceptable solution. For t near tt the accuracy of the solution is not acceptable while in our problem, aircraft landing, the accuracy of control signal is so important at final time stages of the flare maneuver. On the other hand, in our case tr is not large enough to suppose that + w ,because approximately t = 45sec.
where the final time of the process is not defined. In the proceeding section, the new approach based on neural networks for obtaining the solution of the costate equation, has been proposed.
In this paper, first, the suboptimal solution for our case is
111. AIRCRAFT AUTOLANDING AND EQUATIONS OF
discussed. Then an optimal controller has been designed for aircraft landing and some neural networks have been trained to learn the output costates of the system. Finally, a new scenario, which its final time isn't known in advance, is considered to show the abilities of the proposed neurocontroller based approach. The results show that the neurocontroller lands the aircraft as well as the optimal controller with the benefit that it doesn't need the landing final time value to fulfill its operation.
Two major modes of landing phase are studied here; glideslope-hold and intercept, and flare and touch down [7]. In present case, glide mode begins at 500 ft altitude and finishes at 45 ft altitude, which is the start point of the flare mode. These landing phases are shown in Figure I . The flare mode continues until a smooth touch down is achieved. During glide-slope, autolanding guides the aircraft along a straight line with a constant slope (with a constant glide angle; "/.
I,
1-
11. OPTIMAL CoNTROL OF LQT SYSTEMS
In optimal control of LQT systems the output of a linear quadratic system has to track a reference signal so that the cost function, which is usually taken to be quadratic function, is minimized. Now to formulate the problem, let's use the following discrete state space model for LQT system [6]: (1) X K + , = 4Xk + 4%
(4 ?, = c , x * + D , u * Where the 4, C, and 9; are the constant matrices of system, & is the state vector, 4 is control matrix and yk is output which must follows the reference signal denoted by ri, The performance index is formulated as follows: 1
J=-(y 2
N
-rN)TF(?N
-rN)
Where, F, Q and R are the weighting matrices with appropriate dimensions and definiteness. Using the calculus of variations [6].we find the optimal control as (4): uK = -R- I B T PKtl (4) Where P is the matrix of costates and it is obtained by solving the Riccati Equation (equations 5 through 11) backward in time: P, -V, (5) and Skand V, are obtained as follows:
=s,x,
'K = ~ ~ ~ ~ + , B t R ) - i ~ ( ~ K + I ~ x
K , =(B'S,,,B+
R)-'B'S,+,A
SE =A'S,,(A-BK,)
t C'QC V, =(A-BK,)'V,+,+CTQr,
(6)
Autopilot also attempts to prevent any changes in aircraft vertical and horizontal speeds, and hold them constant. In other words, during glide mode the sink rate is constant. As flare mode starts, the autopilot starts to nose up the aircraft by changing the glide angle to prepare the aircraft for a smooth touchdown. The trajectory of aircrafi during this mode is estimated by an exponential function (Fig. 1). Throughout this mode, the sink rate is reduced to the desired value of -1.5 ftisec, to land the aircraft safely and smoothly on the runway and to provide the passengers comfort. The elevator and throttle are the two controls during these two modes. Since studying the aircraft dynamics in its sixdegree of freedom is time-consuming and complex, usually 3-DOF equations of motion in longitudinal mode are being used to analyze the aircraft behaviour. To describe the parameters of motion, perturbed parameters are used with the aid of "Small Perturbation Theory". Also to linearize the aircraft aerodynamics, the equations of motion are expanded by the longitudinal stability derivatives. The equations of motion are linearized at an altitude of 500 A and in the trimmed speed of 235 Wsec. The dynamics of aircraft with the above conditions is also represented by:
w = z,u
+ z,w + (Z,q
x 180
- (-)U0)q
8
g(-jsin(y,je I80
q=M"u+M~wtMUqtM,6,tM,6,
(8) (9)
il=usino-wcoso
(10)
(11) Equations (IO) and (11) are the boundary conditions for equations (7), (8) and (9). These equations need the final time (N) for the process to be solved, which is hard in some cases = +C'Fr,
493
+
+z,6, + Zd,
0=q L = U cos0 t wsin 0
(7)
where S, =C'FC
MOTIONS
The initial conditions are assumed to he: ~ ~ h(O)=SOO ft, x(O)=h(O)/tam yo
t(0) = U, The variables of interest are defined as follows: U : perturbed longitudinal velocity (fVsec),
~
~
e, x and h. Adding
variables namely w, q.
: perturbed pitch rate (degisec),
important variables to the above states, we will have seven state variables, defined as: (26) X =[ w ,q ,e,x ,.ii,h ,h ] The output vector of the system is also considered to he: Y T = [ h , h l (27) The performance index is now considered as (23):
: perturbed pitch angle (deg),
: horizontal position ofaircnft (I?),
I) and
I; as two
: perturbed lateral velocity (ftisec),
'
:altitude (ft), : flare initiation altitude, : glide initiation altitude, : elevator angle setting (deg),'
~
S,
: throttle setting (deg),
J =
U. : normal speed (235 Wsec), g :gravity (32.2 ft/sec*), Y o : flight path angle (-3 deg).
IV.
U (.x(f)) ,.o
where U(x (t)) is called the utilityand obtained as follows:
U(x(t))=q[h(t)-h,(t)Y +a2[l;(t)-hc(t)lZ +4[6,12
LONGITUDINAL AUTOLANDING CONTROLLER DESIGN
The autolanding controller should land the aircraft automatically in an acceptable range of a g h t conditions such as sink rate and speed, to achieve 'a good performance in landing. AS previously mentioned, in this paper, an optimal controller an5also an optimal neuro-controller are designed as autolanding sydems. ~.
-
I
(29)
in which a,, a2 and a? are the respective. elements of utility and performance index matrices which are named as design parameters. In other words, if these design parameters hecome greater, the terms which are multiplied in them hecome more important So the controller behaves in such a way that the more important parameter among h,hand& has more weight in equation(29) and respectively, it has more impact on the performance. For our case these values are chosen as follows: a,= 0.01, aP1.0, ai= 0.009 This formulation demands that V,, he introduced as a state in .the following equation:
A. Suboptimal Solution . 'As previously mentioned,h somecases,.due to the lack of the i =VrRS (30) final time of the process, design& tend to solve the optimal which triggers the uncontrollahility in the system. To prevent .problem of LQT with the aid of suboptimal method. In other words, they suppose -that f, + and under this condition . this, we suppose V, as a control input which its multiplier is considered to he I and also a high weight associated with this S 3 0 or Sk=Ski,. Consequently the Differential .Riccati imaginary control, in matrix R, is considered. With this .Equation leads to Algebraic Riccati Equation those results in a formulation all required matrixes namely ,A, B, C, D, F, Q and R solution for S which is independent of time. With regards to are known, .and so the optimal control theory mentioned m . the above conditions, equations (7) through (9) are expressed - . section 11; can be applied now. Note that the reference.signal is -asfoliows: . presented with the following definition: K p _ ! B r S B t R. .) T ' B r S A ~ - ~ .; . (21) ,. .. I
'
'-
~
~
~
S =ArS(AiBK;)+dTQC
.. :..
.
I~
. I
'KL e-(BrfB+R)-'Br
- I.
..- .
-
_ -
V,
,
~
-
.
1
-(22)
.
.
-.
= (.%BK_)-r(VKCrQtK)
(23) (24Y
U K =-KJK+K:vK+,. (25) To- solve- the above equations; first, the Algebraic 'Riccati Equation (22) h a s ~ t ohe solved [XI. By Substituting S into. equations (21) and (23), K-aniKL havebeen obtained. To.
we can-solve equation (9) off-. obtain the auxiliary part, k+], .line rand backward in time by a hypothetical ,t to .calculate _.-V(0); the initial-condition of equation (24); Then we can~usd V(0) to solve equation (24) on-line &d also forward in time. Suboptimal method was described above, to solve the optimal problem of DLQT independent of time. It is important to note .that these assumptions are true and the suboptimal solution exists only if the system satisfies the related conditions which are mentioned in the following paragraph.
C. Optimal Neuro-Controller
After obtaining the optimal control 4aw by solving equations defined in section 1I;backward n time; we focus on this problem to solve it directly in time -from t = ~ Oto t = N (orm), With respect to the optimal control law given by equation (4); we get: Uop,jm;= - R-IBPK+, (32)
To compute directly, we .need an architecture, which as ,it is -shown schematically in inputs X, and outputs Figure 2. Now, since we have seven state.variables, an MLP neural netwoik has been considered to estimate each of their costates correspondingly. To train the networks, the optimal control pxameters, which are obtained in previous.sections, have been used. Distance (x), which is scaled hy its maximum absolute value, has been used as an input for the networks, so the scaled costates are obtained in output of these networks (scaled values are used to train the networks more rapidly and B.. Optimal Controller -'-I n landing phase of the flight, thrust is used to counter the efficiently). All of these networks except network- 3 have the .same " variations in' distributed forward velocity U. Hence.the effect of structures. They I&e 3 hidden layers eacti having 7, 10 and 5 U is neglected and in equations of motion; U is set to be~zero. .. neurons respectively (N,,,,,n,5,,). The activation function- used Consequently, .the remainder states of model are five state j
-- -
--.
~i
.
.~
.. 494
-
for the first layers of these networks is Tangent Hyperbolic and for hidden layers is Logarithm Sigmoid. Also, in output layers linear functions are used. Network 3 which estimates the third costate, bas 4 hidden layers constructed by 6,9,6 and 3 neurons ( N I , ~ , , , , , )The . reason for the relative largeness of the third network is likely having high nonlinearity in the third costate (PJ in comparison with the others. Now to train the networks, the classical error back propagafion method (Levenberg-Marquardt back propagafion) is used. This method updates the weight and bias values according to Levenherg-Marquardt optimization
isn't known and in cases that, these conditions do not exist, selecting the neurooptimal method to solve the landing problem of a commercial aircraft is properlydone. 2) Neurooptimal controller can estimate the costates without knowing the final hme value of the process and without solving the Riccati equations backward in time. 3) The optimal neuro-controller behaves well in different cases and adapts itselfproperly in new conditions, Inother words, these types of controllers can develop the flight envelope of commercial jet transports with less expenses and flight tests, and with more reliability and preciseness; because the optimal neuro-controller can guide the aircraft properly with in the conditions that has not been trained and tested. 4) The optimal neurocontroller can land the aircraft smoothly on runway and it is easy to be implemented. In the current research, only longitudinal motion of aircraft is considered. This can be easily extended to include the lateral motion.
191. V. SIMULATIONS RESULTS To train the networks and simulate the model three cases have been studied. First, the optimal control problem, as mentioned previously, is simulated to obtain the costates and optimal control law. The initial condition for these cases is taken as in section 3 (ha8=500 ft). Then, the seven costate-networks have been trained, simulated and compared with optimal controller and finally, to test our networks, a new case with hog=475ft has been presented. The results are shown in Figures 3 through 16. In Figure 3, the optimal and neurocontroller followed trajectories were compared. The Figure shows that the neurocontroller lands the aircraft smoothly on runway and follows the command trajectory well. In Figure 4, the followed sink rate for optimal and neuro-controllers have been shown. There is only one disturbance in the curve, which is due to the initiation of flare mode. In Figures 5, 6, 7 and 8 the state variables w,q,8 and x of optimaband neurocontrollers have
ACKNOWLEDGMENT The authors would like to thank Dr. Malaek of Sharif University of Technology for his useful remarks in flight dynamics.
REFERENCES [I] Lehebvre Mario, "A Bidimensional Optimal Landing Problem", Journal of Automatica, vol. 34, no. 5, PP 655-651, Elsevier
been compared. Only one oscillation in w,q a n d e is seen, in Figure 8, and this is because of the initiation of flare mode and = 235 f p s , a s we expected these noises. In Figures Y through 16, we study the important case 3; the initial condition for this case is presented with (33): x, = [0,0,0,-Y072,235,475,0] (33) In Figure 9,,the followed trajectory is depicted and in Figure 10, the followed sink rate is shown. It is shown that neurocontroller lands the aircraft smoothly on the runway and the
x
final value of h i s reduced to desired value (about -1.5 fps) at main gear touch down point. In Figures 1 I through 14, the other state variables are shown. As we know, variations in all parameters are in an acceptable range. To better show the performance of neuro-controller, we compare it with optimal controller in Figure 15 and 16. For hog=475ft,inFigure 15, the followed trajectory and in Figure 16, the followed sink rate are shown for both neuro and optimal controllers. It is shown that the ncurocontroller acts as good as optimal controller in this case, even it performs its job better than its reference case (hD8 = 500 ft), which was trained in that condition (in comparison with Figures 3 and 4).
VI.
..
Inc., 1995. [71 Roskam J., Airplane Flight Dynamics and Automatic Flight Control. Roskam Publishing Incorp.. 1979 (Pat I & 11). 18) Optimization Toolbox (For Use with Matlab), User's Guide, Version 2, Mathworks Inc., 2000. [9] Neural Networks Toolbox (For Use with Matlab), User's Guide, Version 4, Mathworks Inc., 2000.
CONCLUSION
The results of this research are as follows: I) Since suboptimal solution requires special conditions to besatisfied in cases in which the,final time of the process .~ ..
Science Ltd, 1998. 121 Malgond Sandeep S., Stengel Roben F:, "Optimal Nonlinear Estimation for Aircrafl Control in Wind Shear", Joumal of Autamatica, voI 32, no. 1, PP 3-13, Elsevier Science Ltd.1996. [3] Fazeladeh S. A., Pourtakdoust S. H., "Development of an Optimal Approach Control System", Proceedings of fourth Annual ISME & . Second lntemaiional Mechanical Engineering Conference, ISME, S h i m University, Shiraz, Iran, May 14-17, 1996. [4] Balakrishnan S. N., Biega Victor, 'Adaptive-Critic-BasedNeural Networks for Aircraft Optimal Control", Joumal of Guidance, Control, and Dynamics, vol. 19, no. 4, July-August 1996. [5] Saini Gaurav, Balakrishnan S. N., "Adaptive Critic Based NeuroController for Autolanding of Aircraft", Proceedings of the American Control Conference, Albuquerque, New Mexico, June 1991. [6] Lewis, F. L., Syrmos V. L., Optimal Control. John Wiley and Sons
. .
.. -
.
.
. .
-495.
~
'
.
~
_-:
'
Fig. 6 : Pitch rate time history for both controllers
Fig. 2: Structure of required Neural Network architecture r
