- Email: [email protected]
Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method
G Model
ARTICLE IN PRESS
JAMM-2431; No. of Pages 7
Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method夽 N.M. Grevtsov a,∗ , S.A. Kumakshev b , A.M. Shmatkov b a b
Zhukovsky Central Aerohydrodynamic Institute, Zhukovsky, Russia Ishlinsky Institute for Problems in Mechanics, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 23 June 2017 Available online xxx Keywords: Control parameters Control functions
a b s t r a c t Using Bellman’s dynamic programming method, a fuel-consumption optimum flight trajectory of a typical mid-range aircraft is constructed for a prescribed range. The optimum trajectory was calculated for the full model of motion of the aircraft along a trajectory in a vertical plane. Optimization was realized simultaneously for the whole trajectory without decomposing the process into individual steps. The method made it possible to take into account all restrictions imposed both by technical peculiarities of the aircraft and also by other requirements – safety and comfort of the passengers. © 2018 Elsevier Ltd. All rights reserved.
In connection with the fact that transport aircraft, unlike fighter jets, are characterized by low manoeuvrability, individual segments of the flight trajectory – climb, change of level, descent, motion along a glide path – have been investigated relatively deeply. The problem of optimization of the trajectory as a whole to minimize fuel consumption is of practical interest. Aircraft trajectory control algorithms should ensure satisfaction of conditions of safety and economy, and at the same time require minimum computation time. Below, we consider the solution of the problem of minimum fuel consumption during flight from one point to another. Similar problems have been solved by decomposition of the trajectory into three parts: cruise climb, cruising segment, and descent with approach to landing.1–8 The problem of minimizing fuel consumption (or the generalized criterion consisting of a linear combination of fuel consumption and flight time) was solved on each segment separately. Of course, the mutual influence of adjoining segments was taken into account. To construct the entire flight trajectory, the solutions obtained on individual segments are joined together using some intermediate segments. In the optimization of the segments, an energy approach was applied to describe the motion of the aircraft, apparently first proposed in Ref. 9 for the optimization of trajectories of flying machines and then widely used to solve different problems of optimization of trajectories, in particular, to solve the problem of four-dimensional navigation.1–5 As a rule, when using this optimization approach, Pontryagin’s maximum principle is applied.10 The need arises to check the results obtained by the indicated method by optimizing the trajectory as a whole without subdividing the problem into individual tasks. An exact solution by methods proposed earlier1 in the presence of complicated and numerous phase constraints is exceedingly difficult. Bellman’s dynamic programming method makes it possible to take numerous and complicated constraints into account, both those imposed by technical peculiarities of a definite type of aircraft and those arising from other requirements, for example, safety and passenger comfort. Many recent studies on optimization of trajectories of non-manoeuvrable aircraft2–18 have been based on significantly simplified models of motion, not taking phase and other constraints into account. In this connection, a study including a significant number of elements inherent in real-world conditions would be of interest. The present paper is dedicated to that goal. 1. Statement of the problem The main role in ensuring economization is played by control of the aircraft in the vertical plane. We will assume that the flight trajectory is always found strictly in the vertical plane. The coordinates of the centre of mass of the aircraft x, y are assigned in a fixed right-handed
夽 Prikl. Mat. Mekh. Vol. 81, No.5, pp. 534–543, 2017. ∗ Corresponding author. E-mail addresses: [email protected] (N.M. Grevtsov), [email protected] (S.A. Kumakshev), [email protected] (A.M. Shmatkov). https://doi.org/10.1016/j.jappmathmech.2018.03.004 0021-8928/© 2018 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7 2
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
Cartesian coordinate system OXY (the Y axis is directed along the vertical and the X axis is horizontal). Then, the equations of motion of the aircraft have the form19
(1.1) Here, g is the acceleration due to gravity, V is the magnitude of the aircraft velocity, m is its mass, is the pitch angle, nx is the tangential load, ny is the load factor, Qt is the fuel consumption per second, which depends on the Mach number M, the flight altitude y, and the thrust P. We will neglect the normal component of the tractive force. Thus, for the load factor ny and the tangential load nx the following expressions are valid:
(1.2) where q is the dynamic pressure, (y) is the mass density of the atmosphere as a function of altitude, S is the wing area, taken in what follows to be equal to 168.3 m2 , Cx is the drag coefficient, which is a function of the lift coefficient Cy and the Mach number M. Eliminating time from the equations of motion by using the fourth of Eqs (1.1), we obtain
(1.3) The controlling parameters are the lift coefficient Cy and the magnitude of the tractive force P. Substituting the expression for ny in relations (1.2) into the second of Eqs (1.3), we find
(1.4) Substituting the expression for nx (1.2) into the first of Eqs (1.3), we have (1.5) It follows from system (1.3) that to describe the motion of the object it suffices to know the three functions V(x), (x), and y(x), on the basis of which it is possible to find the control functions (parameters) Cx (x) and P(x) from formulae (1.4) and (1.5), and also the aircraft mass m(x) by bringing to bear the last equation of system (1.3). The flight time T is not fixed. The initial and final positions of the aircraft are assigned as follows:
It is required to choose the control parameters such that the aircraft travels from the prescribed initial position to the prescribed final position with minimum fuel consumption. The optimization criterion has the form
Fig. 1.
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model
ARTICLE IN PRESS
JAMM-2431; No. of Pages 7
N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
3
Fig. 2.
2. Approximation and restrictions We denote by (y) the dependence of the density of the standard atmosphere on altitude, and by as (y) the dependence of the velocity of sound in a standard atmosphere on altitude.19 We represent the dependence of the drag coefficient on the Mach number and the lift coefficient in the form
where Cx0 (M) is a polynomial of third degree, Cxi (Cy ) is a quadratic function, and M = V/as (y). The fuel-consumption-rate function Qt can be approximated by a polynomial of second order in the altitude and thrust and of first order in the Mach number. The left-hand side of Fig. 1 plots the fuel consumption as a function of the thrust at an altitude of 10 km for different admissible velocities. The control variables at each instant of time should satisfy the restrictions (2.1) Cymin
Cymax
The lift coefficient Cy varies over the range extending from = −0.1 to = 1.2. The functions Pmin (y, V) and Pmax (y, V) have been approximated by second-order polynomials in the altitude and Mach number. The right-hand side of Fig. 1 shows four pairs of dependences of Pmin and Pmax in the range of admissible velocities for different values of the altitude indicated above the curves. The aircraft velocity satisfies the restrictions (2.2) indicated in Fig. 2 by dashed lines. Here Vmin and Vmax have been approximated by second-order polynomials in the altitude. The altitude of the aircraft also satisfies the restrictions (Fig. 2) (2.3) where Hmin = 100 m and Hmax = 14 km. 3. Description of the algorithm As is well known,20 the dynamic programming method is based on the principle of optimality: “whatever the initial state and solution at the initial time may be, subsequent solutions should prescribe optimal behaviour with respect to the state obtained as a result of the first solution”. Here we have applied the direct pass algorithm for the discrete case; therefore, in what follows we will understand the corresponding differentials as finite increments and use system (1.3) as a system of linear algebraic equations to calculate the values of the increments of the functions from increments of the argument dx. The method requires, for each admissible state of the system, to indicate the preceding state, the transition from which to the admissible state is optimal in the sense of a prescribed functional. In the problem under consideration, the state is defined by the range, altitude, and velocity; therefore, to apply the dynamic programming method it is necessary to save the values of these three functions. With the aim of economizing computer memory, we replace the values of the functions and their arguments by the values of the corresponding integer indices ky , k , kV , and kx . For example, instead of the altitude y we will consider the index ky in the formula y = ky y + ymin , where ymin is the minimum value of the altitude and y is the corresponding step. For each value of the range kx we form an array of cells. Each cell contains a pair of values: the angle index and the velocity index. The value of the altitude index here is equal to the cell number. We shall call a set of three specific values of these indices a node. We will call the array corresponding to the current value of kx the left layer, and the array corresponding to the value kx + 1 the right layer; they differ in their value of the x coordinate by the step dx = x. With each node of the left layer we will associate the value of the mass corresponding to the optimal trajectory ending at that node. We describe the transition from layer to layer within the framework of direct pass-through of the dynamic programming method. We fix an arbitrary admissible node in the right layer by choosing the indices of the coordinates, angle, and velocity. At the same time we eliminate transitions not corresponding to the conditions imposed on the admissible velocity (2.2) and altitude (2.3). We then take a node of the left layer from which it is possible to realize a transition to the fixed node without violating the imposed restrictions. In this way, on the basis of the indices corresponding to these two nodes, the values of the increments of the three functions Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7 4
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
Fig. 3.
Fig. 4.
Fig. 5.
describing the object according to Eqs (1.3) become known, and it is possible to calculate the control parameters Cy and P according to formulae (1.4) and (1.5). At this step, transitions are eliminated which do not satisfy conditions (2.1). We sort the nodes of the left layer according to the angle and velocity indices; the corresponding altitude index is uniquely determined on the basis of the altitude index of the fixed right node and the angle index of the current left node. We obtain the set of all possible transitions for each of which we shall determine the mass consumption dm. The total mass of the object as a result of a transition to a node of the right layer consists of the difference between the optimal mass obtained on the preceding step for the node of the left layer and the mass consumption dm for the given transition. We choose the transition for which the mass of the object at the fixed node of the right layer is the smallest and write the angle and velocity indices into the corresponding cell of the right layer, and also save the obtained value of the mass. Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
5
Fig. 6.
At the initial point of the trajectory, all indices of the object are known on the basis of the initial data. Thus, at the first step the left layer is filled. Next, we fill layer after layer all the way to the last one, corresponding to the range x(T) − x. Note that it is not necessary to save the value of the mass for all layers; in the calculations it suffices to have the values of the mass for the layers that are the left layer and the right layer at the given step. The terminal state is assigned by the node in this layer delivering the maximum value of the functional. We thereupon begin the reverse pass-through of the dynamic programming method and find the node in the pre-previous layer through which the optimal trajectory passes. Continuation of this procedure allows us to reconstruct the entire optimal trajectory. To improve the result, it is necessary to decrease the size of the increments of all of the functions describing the object. So as not to increase the number of possible values of the indices, it is possible to employ the fact that the improved trajectory lies in the vicinity of the found trajectory and to perform the calculations only within a narrow range of the corresponding values of the parameters. It should be borne in mind that the equations of motion and both of the controls are represented in the International System units, but the function Qt (M, y, P), needed to calculate the mass consumption, and also the restrictions on the thrust, Pmin and Pmax , are represented in the CGS system units for convenience in the use of the previously obtained experimental data. 4. Results of calculations Calculations were performed with the following parameters: 10,000 layers over the range with a step of 50 m, 504 possible values of the altitude with a step of 0.25 m in a width corridor of 126 m, 252 possible values of the velocity V with a step of 0.025 m/s in a range of width 6.3 m/s, 20 possible values of the angle in a range of width about 4.8◦ .
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
6
Fig. 7.
The optimum trajectory is represented in Fig. 3 (left axis), and the pitch angle is plotted by the curve in Fig. 4, where curves 1 and 3 show the boundaries of the range of this angle used in the calculations. With a fixed degree of accuracy, it is possible to divide the trajectory into three segments: climb, cruising segment, and descent. The points in Fig. 2 corresponding to the optimal values of the velocity V are joined by a solid line, and the arrows indicate the direction of increasing range x. It can be seen that in the first segment the velocity of motion, whose x dependence is plotted in Fig. 3 (right-hand axis), rapidly reaches its maximum possible values. Here, as is shown in Fig. 5, the thrust (curve 2) is also maximum up to a range of about 10 km. Note that it is precisely the restrictions on the thrust (curves 1 and 3 in Fig. 5) that can be used to delineate the second segment since the restrictions on the velocity cease to act at an altitude of about 5 km (Fig. 2). As can be seen from Fig. 3 (left axis), the cruising segment of the flight is not strictly horizontal. It is characterized by the absence of approaches of the phase and control variables to their limits. The descent segment can also be delineated with the help of the thrust graph (Fig. 5). It becomes minimal at a range of roughly 350 km. Descent terminates with a specific manoeuvre: before reaching the final point, the aircraft drops to the minimally admissible altitude of 100 m and abruptly reduces its speed, lifting up to the altitude y(T) = 600 m. The dependence of the optimal control Cy on x in the upper part of Fig. 6, like the corresponding dependence of Cx in the middle part of Fig. 6, is characterized by significant spikes arising due to errors in the calculations based on an approximation of the solution by broken lines. This also explains the form of the function ny (lower part of Fig. 6). The dependence of the current fuel consumption on range is described by the smooth curve (Fig. 7, left axis). The reason consists in the relatively moderate sensitivity of the fuel consumption to abrupt short-time changes in the control parameters. The final mass of the aircraft is approximately equal to 67,670 kg. The fuel consumption is approximately 2,330 kg. The same arguments also apply to the flight time. It is interesting that the corresponding graph (Fig. 7, right-hand axis) is close to a straight line. This is explained by the fact that significant changes in the velocity occur only in the very beginning and at the very end of the trajectory. The flight time is approximately equal to 2,471 s.
5. Conclusion The calculations performed here show that it is possible conditionally to divide the entire trajectory into three segments: 1) acceleration and climb, 2) flight with slope angle of the trajectory close to zero, and 3) braking and descent. In the first segment, acceleration and climb take place with the maximum admissible velocity and here the thrust takes its maximum values. The slope angle of the trajectory reaches 8.5◦ . In the second segment, the altitude and velocity reach their maximum values over the entire flight – about 10 km and 230 m/s, respectively. The thrust here is approximately 5.5 kgf. These values are close to those that deliver minimum fuel consumption per kilometre. In the third segment, the thrust is minimal. The slope (pitch) angle of the trajectory is approximately −4◦ . Our attention is drawn to the manoeuvre in the final segment: the aircraft at first descends to the minimum admissible altitude of 100 m, and then lifts up to the altitude of 600 m prescribed by the boundary condition. Our attention is also drawn to the dependence of the actual flight time on the traversed distance (Fig. 7, right-hand axis). Despite the wide range of the velocity values (Fig. 3, right-hand axis), over most of the trajectory it varies only weakly.
Acknowledgement This work was supported in part by the Russian Foundation for Basic Research (17-01-00538 and 17-08-00742).
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
7
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Rutowski ES. Energy approach to the general aircraft performance problem. J Aeronaut Sci 1954;21(3):187–95. Zagalsky NR, Irons RP, Schultz RL. Energy state approximation and minimum-fuel fixed-range trajectories. J Aircraft 1971;8(6):488–90. Andreevskii VV, Goroshchenko LB. Flight Control and Efficiency of the Aviation Complex. Moscow: Mashinostr; 1974. Ardema MD. Singular Perturbations in Flight Mechanics.. CA: NASA Techn Memor. Moffett Field; 1974. TM-X-62-380. Galise AJ. Extended energy management methods for flight performance optimization. AIAA J 1977;15(3):314–21. Erzberger H, Lee H. Constrained optimum trajectories with specified range. J Guid Control 1980;3(1):78–85. Kosminkov KYu. Optimization of the Flight Regime of a Subsonic Passenger Aircraft for Given Range and Flight Time. Moscow: TsAGI; 1984. Skripnichenko SYu. Optimization of the Flight Regimes According to Economic Criteria. Moscow: Mashinostr; 1988. Ostoslavskii IV, Lebedev AA. Calculation of the lift of a high-speed aircraft. Tekh Vozd Flota 1946;(9):21–7. Pontryagin LS, Boltyanskii VG, Gamkredlidze RV, Mishchenko EF. The Mathematical Theory of Optimal Processes. New York: Wiley; 1967. Bellman R. Dynamic Programming. Princeton, NJ: Princeton Univ Press; 1957. Bonami P, Olivares A, Soler M, Staffetti E. Multiphase mixed-integer optimal control approach to aircraft trajectory optimization. J Guid Control Dyn 2013;36(5):1267–77. Li X, Nair PB, Zhang Z, Gao L, Gao C. Aircraft robust trajectory optimization using nonintrusive polynomial chaos. J Aircraft 2014;51(5):1592–603. Mendozo MA, Botez Mr. Vertical navigation trajectory optimization algorithm for a commercial aircraft. In: AIAA 2014-3019, AIAA/3AF Aircraft Noise and Emissions Reduction Symp.. 2014., http://dx.doi.org/10.2514/6.2014-3019. Franco A, Rivas D. Optimization of multiphase aircraft trajectories using hybrid optimal control. J Guid Control Dyn 2015;38(3):452–67. Soler M, Olivares A, Staffetti E. Multiphase optimal control framework for commercial aircraft four dimensional flight planning problems. J Aircraft 2015;52(1):274–86. Rosenow J, Förster, Lindner M, Fricke H. Multi-objective trajectory optimization. Int Transport 2016;68:40–3. Gracia-Heras J, Soler M, Saez FJ. Collocation methods to minimum-fuel trajectory problems with required time of arrival in ATM. J Aerospace Inform Syst 2016;13:243–65. Bochkarev AF, Andreevskii VV, Belokonov VM, et al. Aeromechanics of Aircraft: Flight Dynamics. Moscow: Mashinostr; 1985. Roitenberg YaN. Automatic Control. Moscow: Nauka; 1971.
Translated by P.F.S.
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
ARTICLE IN PRESS
JAMM-2431; No. of Pages 7
Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method夽 N.M. Grevtsov a,∗ , S.A. Kumakshev b , A.M. Shmatkov b a b
Zhukovsky Central Aerohydrodynamic Institute, Zhukovsky, Russia Ishlinsky Institute for Problems in Mechanics, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 23 June 2017 Available online xxx Keywords: Control parameters Control functions
a b s t r a c t Using Bellman’s dynamic programming method, a fuel-consumption optimum flight trajectory of a typical mid-range aircraft is constructed for a prescribed range. The optimum trajectory was calculated for the full model of motion of the aircraft along a trajectory in a vertical plane. Optimization was realized simultaneously for the whole trajectory without decomposing the process into individual steps. The method made it possible to take into account all restrictions imposed both by technical peculiarities of the aircraft and also by other requirements – safety and comfort of the passengers. © 2018 Elsevier Ltd. All rights reserved.
In connection with the fact that transport aircraft, unlike fighter jets, are characterized by low manoeuvrability, individual segments of the flight trajectory – climb, change of level, descent, motion along a glide path – have been investigated relatively deeply. The problem of optimization of the trajectory as a whole to minimize fuel consumption is of practical interest. Aircraft trajectory control algorithms should ensure satisfaction of conditions of safety and economy, and at the same time require minimum computation time. Below, we consider the solution of the problem of minimum fuel consumption during flight from one point to another. Similar problems have been solved by decomposition of the trajectory into three parts: cruise climb, cruising segment, and descent with approach to landing.1–8 The problem of minimizing fuel consumption (or the generalized criterion consisting of a linear combination of fuel consumption and flight time) was solved on each segment separately. Of course, the mutual influence of adjoining segments was taken into account. To construct the entire flight trajectory, the solutions obtained on individual segments are joined together using some intermediate segments. In the optimization of the segments, an energy approach was applied to describe the motion of the aircraft, apparently first proposed in Ref. 9 for the optimization of trajectories of flying machines and then widely used to solve different problems of optimization of trajectories, in particular, to solve the problem of four-dimensional navigation.1–5 As a rule, when using this optimization approach, Pontryagin’s maximum principle is applied.10 The need arises to check the results obtained by the indicated method by optimizing the trajectory as a whole without subdividing the problem into individual tasks. An exact solution by methods proposed earlier1 in the presence of complicated and numerous phase constraints is exceedingly difficult. Bellman’s dynamic programming method makes it possible to take numerous and complicated constraints into account, both those imposed by technical peculiarities of a definite type of aircraft and those arising from other requirements, for example, safety and passenger comfort. Many recent studies on optimization of trajectories of non-manoeuvrable aircraft2–18 have been based on significantly simplified models of motion, not taking phase and other constraints into account. In this connection, a study including a significant number of elements inherent in real-world conditions would be of interest. The present paper is dedicated to that goal. 1. Statement of the problem The main role in ensuring economization is played by control of the aircraft in the vertical plane. We will assume that the flight trajectory is always found strictly in the vertical plane. The coordinates of the centre of mass of the aircraft x, y are assigned in a fixed right-handed
夽 Prikl. Mat. Mekh. Vol. 81, No.5, pp. 534–543, 2017. ∗ Corresponding author. E-mail addresses: [email protected] (N.M. Grevtsov), [email protected] (S.A. Kumakshev), [email protected] (A.M. Shmatkov). https://doi.org/10.1016/j.jappmathmech.2018.03.004 0021-8928/© 2018 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7 2
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
Cartesian coordinate system OXY (the Y axis is directed along the vertical and the X axis is horizontal). Then, the equations of motion of the aircraft have the form19
(1.1) Here, g is the acceleration due to gravity, V is the magnitude of the aircraft velocity, m is its mass, is the pitch angle, nx is the tangential load, ny is the load factor, Qt is the fuel consumption per second, which depends on the Mach number M, the flight altitude y, and the thrust P. We will neglect the normal component of the tractive force. Thus, for the load factor ny and the tangential load nx the following expressions are valid:
(1.2) where q is the dynamic pressure, (y) is the mass density of the atmosphere as a function of altitude, S is the wing area, taken in what follows to be equal to 168.3 m2 , Cx is the drag coefficient, which is a function of the lift coefficient Cy and the Mach number M. Eliminating time from the equations of motion by using the fourth of Eqs (1.1), we obtain
(1.3) The controlling parameters are the lift coefficient Cy and the magnitude of the tractive force P. Substituting the expression for ny in relations (1.2) into the second of Eqs (1.3), we find
(1.4) Substituting the expression for nx (1.2) into the first of Eqs (1.3), we have (1.5) It follows from system (1.3) that to describe the motion of the object it suffices to know the three functions V(x), (x), and y(x), on the basis of which it is possible to find the control functions (parameters) Cx (x) and P(x) from formulae (1.4) and (1.5), and also the aircraft mass m(x) by bringing to bear the last equation of system (1.3). The flight time T is not fixed. The initial and final positions of the aircraft are assigned as follows:
It is required to choose the control parameters such that the aircraft travels from the prescribed initial position to the prescribed final position with minimum fuel consumption. The optimization criterion has the form
Fig. 1.
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model
ARTICLE IN PRESS
JAMM-2431; No. of Pages 7
N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
3
Fig. 2.
2. Approximation and restrictions We denote by (y) the dependence of the density of the standard atmosphere on altitude, and by as (y) the dependence of the velocity of sound in a standard atmosphere on altitude.19 We represent the dependence of the drag coefficient on the Mach number and the lift coefficient in the form
where Cx0 (M) is a polynomial of third degree, Cxi (Cy ) is a quadratic function, and M = V/as (y). The fuel-consumption-rate function Qt can be approximated by a polynomial of second order in the altitude and thrust and of first order in the Mach number. The left-hand side of Fig. 1 plots the fuel consumption as a function of the thrust at an altitude of 10 km for different admissible velocities. The control variables at each instant of time should satisfy the restrictions (2.1) Cymin
Cymax
The lift coefficient Cy varies over the range extending from = −0.1 to = 1.2. The functions Pmin (y, V) and Pmax (y, V) have been approximated by second-order polynomials in the altitude and Mach number. The right-hand side of Fig. 1 shows four pairs of dependences of Pmin and Pmax in the range of admissible velocities for different values of the altitude indicated above the curves. The aircraft velocity satisfies the restrictions (2.2) indicated in Fig. 2 by dashed lines. Here Vmin and Vmax have been approximated by second-order polynomials in the altitude. The altitude of the aircraft also satisfies the restrictions (Fig. 2) (2.3) where Hmin = 100 m and Hmax = 14 km. 3. Description of the algorithm As is well known,20 the dynamic programming method is based on the principle of optimality: “whatever the initial state and solution at the initial time may be, subsequent solutions should prescribe optimal behaviour with respect to the state obtained as a result of the first solution”. Here we have applied the direct pass algorithm for the discrete case; therefore, in what follows we will understand the corresponding differentials as finite increments and use system (1.3) as a system of linear algebraic equations to calculate the values of the increments of the functions from increments of the argument dx. The method requires, for each admissible state of the system, to indicate the preceding state, the transition from which to the admissible state is optimal in the sense of a prescribed functional. In the problem under consideration, the state is defined by the range, altitude, and velocity; therefore, to apply the dynamic programming method it is necessary to save the values of these three functions. With the aim of economizing computer memory, we replace the values of the functions and their arguments by the values of the corresponding integer indices ky , k , kV , and kx . For example, instead of the altitude y we will consider the index ky in the formula y = ky y + ymin , where ymin is the minimum value of the altitude and y is the corresponding step. For each value of the range kx we form an array of cells. Each cell contains a pair of values: the angle index and the velocity index. The value of the altitude index here is equal to the cell number. We shall call a set of three specific values of these indices a node. We will call the array corresponding to the current value of kx the left layer, and the array corresponding to the value kx + 1 the right layer; they differ in their value of the x coordinate by the step dx = x. With each node of the left layer we will associate the value of the mass corresponding to the optimal trajectory ending at that node. We describe the transition from layer to layer within the framework of direct pass-through of the dynamic programming method. We fix an arbitrary admissible node in the right layer by choosing the indices of the coordinates, angle, and velocity. At the same time we eliminate transitions not corresponding to the conditions imposed on the admissible velocity (2.2) and altitude (2.3). We then take a node of the left layer from which it is possible to realize a transition to the fixed node without violating the imposed restrictions. In this way, on the basis of the indices corresponding to these two nodes, the values of the increments of the three functions Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7 4
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
Fig. 3.
Fig. 4.
Fig. 5.
describing the object according to Eqs (1.3) become known, and it is possible to calculate the control parameters Cy and P according to formulae (1.4) and (1.5). At this step, transitions are eliminated which do not satisfy conditions (2.1). We sort the nodes of the left layer according to the angle and velocity indices; the corresponding altitude index is uniquely determined on the basis of the altitude index of the fixed right node and the angle index of the current left node. We obtain the set of all possible transitions for each of which we shall determine the mass consumption dm. The total mass of the object as a result of a transition to a node of the right layer consists of the difference between the optimal mass obtained on the preceding step for the node of the left layer and the mass consumption dm for the given transition. We choose the transition for which the mass of the object at the fixed node of the right layer is the smallest and write the angle and velocity indices into the corresponding cell of the right layer, and also save the obtained value of the mass. Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
5
Fig. 6.
At the initial point of the trajectory, all indices of the object are known on the basis of the initial data. Thus, at the first step the left layer is filled. Next, we fill layer after layer all the way to the last one, corresponding to the range x(T) − x. Note that it is not necessary to save the value of the mass for all layers; in the calculations it suffices to have the values of the mass for the layers that are the left layer and the right layer at the given step. The terminal state is assigned by the node in this layer delivering the maximum value of the functional. We thereupon begin the reverse pass-through of the dynamic programming method and find the node in the pre-previous layer through which the optimal trajectory passes. Continuation of this procedure allows us to reconstruct the entire optimal trajectory. To improve the result, it is necessary to decrease the size of the increments of all of the functions describing the object. So as not to increase the number of possible values of the indices, it is possible to employ the fact that the improved trajectory lies in the vicinity of the found trajectory and to perform the calculations only within a narrow range of the corresponding values of the parameters. It should be borne in mind that the equations of motion and both of the controls are represented in the International System units, but the function Qt (M, y, P), needed to calculate the mass consumption, and also the restrictions on the thrust, Pmin and Pmax , are represented in the CGS system units for convenience in the use of the previously obtained experimental data. 4. Results of calculations Calculations were performed with the following parameters: 10,000 layers over the range with a step of 50 m, 504 possible values of the altitude with a step of 0.25 m in a width corridor of 126 m, 252 possible values of the velocity V with a step of 0.025 m/s in a range of width 6.3 m/s, 20 possible values of the angle in a range of width about 4.8◦ .
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
6
Fig. 7.
The optimum trajectory is represented in Fig. 3 (left axis), and the pitch angle is plotted by the curve in Fig. 4, where curves 1 and 3 show the boundaries of the range of this angle used in the calculations. With a fixed degree of accuracy, it is possible to divide the trajectory into three segments: climb, cruising segment, and descent. The points in Fig. 2 corresponding to the optimal values of the velocity V are joined by a solid line, and the arrows indicate the direction of increasing range x. It can be seen that in the first segment the velocity of motion, whose x dependence is plotted in Fig. 3 (right-hand axis), rapidly reaches its maximum possible values. Here, as is shown in Fig. 5, the thrust (curve 2) is also maximum up to a range of about 10 km. Note that it is precisely the restrictions on the thrust (curves 1 and 3 in Fig. 5) that can be used to delineate the second segment since the restrictions on the velocity cease to act at an altitude of about 5 km (Fig. 2). As can be seen from Fig. 3 (left axis), the cruising segment of the flight is not strictly horizontal. It is characterized by the absence of approaches of the phase and control variables to their limits. The descent segment can also be delineated with the help of the thrust graph (Fig. 5). It becomes minimal at a range of roughly 350 km. Descent terminates with a specific manoeuvre: before reaching the final point, the aircraft drops to the minimally admissible altitude of 100 m and abruptly reduces its speed, lifting up to the altitude y(T) = 600 m. The dependence of the optimal control Cy on x in the upper part of Fig. 6, like the corresponding dependence of Cx in the middle part of Fig. 6, is characterized by significant spikes arising due to errors in the calculations based on an approximation of the solution by broken lines. This also explains the form of the function ny (lower part of Fig. 6). The dependence of the current fuel consumption on range is described by the smooth curve (Fig. 7, left axis). The reason consists in the relatively moderate sensitivity of the fuel consumption to abrupt short-time changes in the control parameters. The final mass of the aircraft is approximately equal to 67,670 kg. The fuel consumption is approximately 2,330 kg. The same arguments also apply to the flight time. It is interesting that the corresponding graph (Fig. 7, right-hand axis) is close to a straight line. This is explained by the fact that significant changes in the velocity occur only in the very beginning and at the very end of the trajectory. The flight time is approximately equal to 2,471 s.
5. Conclusion The calculations performed here show that it is possible conditionally to divide the entire trajectory into three segments: 1) acceleration and climb, 2) flight with slope angle of the trajectory close to zero, and 3) braking and descent. In the first segment, acceleration and climb take place with the maximum admissible velocity and here the thrust takes its maximum values. The slope angle of the trajectory reaches 8.5◦ . In the second segment, the altitude and velocity reach their maximum values over the entire flight – about 10 km and 230 m/s, respectively. The thrust here is approximately 5.5 kgf. These values are close to those that deliver minimum fuel consumption per kilometre. In the third segment, the thrust is minimal. The slope (pitch) angle of the trajectory is approximately −4◦ . Our attention is drawn to the manoeuvre in the final segment: the aircraft at first descends to the minimum admissible altitude of 100 m, and then lifts up to the altitude of 600 m prescribed by the boundary condition. Our attention is also drawn to the dependence of the actual flight time on the traversed distance (Fig. 7, right-hand axis). Despite the wide range of the velocity values (Fig. 3, right-hand axis), over most of the trajectory it varies only weakly.
Acknowledgement This work was supported in part by the Russian Foundation for Basic Research (17-01-00538 and 17-08-00742).
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004
G Model JAMM-2431; No. of Pages 7
ARTICLE IN PRESS N.M. Grevtsov et al. / Journal of Applied Mathematics and Mechanics xxx (2018) xxx–xxx
7
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Rutowski ES. Energy approach to the general aircraft performance problem. J Aeronaut Sci 1954;21(3):187–95. Zagalsky NR, Irons RP, Schultz RL. Energy state approximation and minimum-fuel fixed-range trajectories. J Aircraft 1971;8(6):488–90. Andreevskii VV, Goroshchenko LB. Flight Control and Efficiency of the Aviation Complex. Moscow: Mashinostr; 1974. Ardema MD. Singular Perturbations in Flight Mechanics.. CA: NASA Techn Memor. Moffett Field; 1974. TM-X-62-380. Galise AJ. Extended energy management methods for flight performance optimization. AIAA J 1977;15(3):314–21. Erzberger H, Lee H. Constrained optimum trajectories with specified range. J Guid Control 1980;3(1):78–85. Kosminkov KYu. Optimization of the Flight Regime of a Subsonic Passenger Aircraft for Given Range and Flight Time. Moscow: TsAGI; 1984. Skripnichenko SYu. Optimization of the Flight Regimes According to Economic Criteria. Moscow: Mashinostr; 1988. Ostoslavskii IV, Lebedev AA. Calculation of the lift of a high-speed aircraft. Tekh Vozd Flota 1946;(9):21–7. Pontryagin LS, Boltyanskii VG, Gamkredlidze RV, Mishchenko EF. The Mathematical Theory of Optimal Processes. New York: Wiley; 1967. Bellman R. Dynamic Programming. Princeton, NJ: Princeton Univ Press; 1957. Bonami P, Olivares A, Soler M, Staffetti E. Multiphase mixed-integer optimal control approach to aircraft trajectory optimization. J Guid Control Dyn 2013;36(5):1267–77. Li X, Nair PB, Zhang Z, Gao L, Gao C. Aircraft robust trajectory optimization using nonintrusive polynomial chaos. J Aircraft 2014;51(5):1592–603. Mendozo MA, Botez Mr. Vertical navigation trajectory optimization algorithm for a commercial aircraft. In: AIAA 2014-3019, AIAA/3AF Aircraft Noise and Emissions Reduction Symp.. 2014., http://dx.doi.org/10.2514/6.2014-3019. Franco A, Rivas D. Optimization of multiphase aircraft trajectories using hybrid optimal control. J Guid Control Dyn 2015;38(3):452–67. Soler M, Olivares A, Staffetti E. Multiphase optimal control framework for commercial aircraft four dimensional flight planning problems. J Aircraft 2015;52(1):274–86. Rosenow J, Förster, Lindner M, Fricke H. Multi-objective trajectory optimization. Int Transport 2016;68:40–3. Gracia-Heras J, Soler M, Saez FJ. Collocation methods to minimum-fuel trajectory problems with required time of arrival in ATM. J Aerospace Inform Syst 2016;13:243–65. Bochkarev AF, Andreevskii VV, Belokonov VM, et al. Aeromechanics of Aircraft: Flight Dynamics. Moscow: Mashinostr; 1985. Roitenberg YaN. Automatic Control. Moscow: Nauka; 1971.
Translated by P.F.S.
Please cite this article in press as: Grevtsov NM, et al. Optimization of the flight trajectory of a non-manoeuvrable aircraft to minimize fuel consumption by the dynamic programming method. J Appl Math Mech (2018), https://doi.org/10.1016/j.jappmathmech.2018.03.004