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Aircraft Guiding in Windshear through Differential Game-Based Overload Control ⁎
11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019
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IFAC PapersOnLine 52-16 (2019) 706–711
Aircraft Guiding in Windshear through Aircraft Guiding in Windshear through Aircraft Guiding in Windshear through ⋆⋆ Differential Game-Based Overload Control Aircraft Guiding in Windshear through Differential Game-Based Overload Control Differential Game-Based Overload Control ⋆⋆ Differential Game-Based Overload Control Nikolai D. Botkin ∗∗ Alexey E. Golubev ∗∗ ∗∗
Nikolai D. Botkin ∗ Alexey E. ∗∗ ∗∗∗Golubev Nikolai D.Varvara Botkin L.Alexey TurovaE. ∗∗∗Golubev Varvara ∗L. Turova ∗∗∗ Nikolai D.Varvara Botkin L.Alexey Golubev ∗∗ TurovaE. Varvara L. Turova ∗∗∗ ∗ ∗ Technical University of Munich, Boltzmannstraße 3, 85748 Garching, ∗ Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Technical University of Munich, 3, 85748 Garching, Germany (e-mail: [email protected]). Germany (e-mail: Boltzmannstraße [email protected]). ∗∗∗ Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Germany (e-mail: [email protected]). Bauman Moscow State Technical University, 2-nd ∗∗ Moscow State (e-mail: Technical University, 2-nd Baumanskaya Baumanskaya 5, 5, ∗∗ Bauman Germany [email protected]). Bauman Moscow State Technical University, 2-nd Baumanskaya 5, 105005 Moscow, Russia (e-mail: [email protected]) ∗∗ ∗∗∗ 105005 Moscow, Russia (e-mail: [email protected]) Bauman Moscow State Technical University, 2-nd Baumanskaya 5, Moscow, Russia (e-mail: [email protected]) Technical University of Boltzmannstraße 3, ∗∗∗ 105005 Technical University of Munich, Munich, Boltzmannstraße 3, 85748 85748 ∗∗∗ 105005 Moscow, Russia (e-mail:[email protected]). [email protected]) Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Germany (e-mail: Garching, Germany ∗∗∗ Technical University of (e-mail: Munich,[email protected]). Boltzmannstraße 3, 85748 Garching, Germany (e-mail: [email protected]). Garching, Germany (e-mail: [email protected]). Abstract: The The problem problem of of aircraft aircraft take-off take-off under under windshear windshear conditions conditions is is considered considered as as aa Abstract: Abstract: The problem of aircraft take-off under windshear conditions is considered as a differential game. A simplified five-dimensional nonlinear model of aircraft dynamics, assuming differential game. A simplified five-dimensional nonlinear model of aircraft dynamics, assuming Abstract: The problem of aircraft take-off under windshear conditions is considered as a differential game. A simplified five-dimensional nonlinear model of aircraft dynamics, assuming the flight in a vertical plane, is used. The overloads, differences between the projections the flight in a vertical plane,five-dimensional is used. The overloads, differences between the projections differential game. A drag, simplified nonlinear model of aircraft dynamics, assuming the flight in aand vertical is used.lift, Theforces overloads, differences the projections of the the thrust and respectively lift, forces are considered considered asbetween control variables. The of thrust drag,plane, respectively are as control variables. The the flight in a vertical plane, is used. The overloads, differences between the projections of the thrust and drag, respectively lift, forces are considered as control variables. The corresponding technically relevant control, the aerodynamic angle of attack, can always be corresponding technically control, the aerodynamic angle as of control attack, variables. can alwaysThe be of the thrust and drag, relevant respectively lift, forces are considered corresponding technically relevant control, the aerodynamic angle of attack, can always be retrieved. Stable numerical algorithms for solving Hamilton-Jacobi-Bellman-Isaacs equations retrieved. Stable numericalrelevant algorithms for solving Hamilton-Jacobi-Bellman-Isaacs equations corresponding technically the aerodynamic of attack, be retrieved. Stable numerical algorithms forconstraints solving Hamilton-Jacobi-Bellman-Isaacs equations arising from from differential games withcontrol, state constraints are used usedangle to compute compute the can valuealways function arising differential games with state are to the value function retrieved. Stable numerical algorithms forconstraints solving Hamilton-Jacobi-Bellman-Isaacs equations arising from differential games with state are used to compute the value function and design an appropriate feedback control. and design appropriate feedback arising froman games with control. state constraints are used to compute the value function and design andifferential appropriate feedback control. © 2019, IFAC Automatic Control) Hosting by Elsevier Ltd. All rights reserved. and design an(International appropriateFederation feedback of control. Keywords: Aircraft Aircraft control, control, Wind Wind microburst, microburst, Differential Differential games, games, Dynamic Dynamic programming, programming, Keywords: Keywords: Aircraft control, Wind microburst, Differential games, Dynamic programming, Hamilton-Jacobi equations, Grid methods, Multiprocessor systems, Feedback controls. Hamilton-Jacobi equations, Grid methods, Multiprocessor systems, Feedback controls. Keywords: Aircraft control, Wind microburst, Differential games, Dynamic programming, Hamilton-Jacobi equations, Grid methods, Multiprocessor systems, Feedback controls. Hamilton-Jacobi equations, Grid methods, Multiprocessor systems, Feedback controls. 1. Leitmann 1. INTRODUCTION INTRODUCTION Leitmann and and Pandey Pandey (1990) (1990) demonstrates demonstrates application application of of 1. INTRODUCTION Leitmann and Pandey (1990) demonstrates application of robust control theory to design of aa feedback control robust control theory to design of feedback control keepkeep1. INTRODUCTION Leitmann and Pandey (1990) demonstrates application of robust control theory to design of a feedback control keeping the relative path inclination, whereas papers Leitmann Very often, severe wind conditions arise from downbursts the control relativetheory path inclination, whereas papers Leitmann Very often, severe wind conditions arise from downbursts ing robust to design of a feedback control keeping relative(1991) path inclination, whereas papersand Leitmann andthe Pandey and Pandey, Ryan Very severe wind arise from downbursts of A is descending column of Pandey (1991) and Leitmann, Leitmann, Pandey, and Ryan of air. air.often, A downburst downburst is a aconditions descending column of air air that that and ing the relative path inclination, whereas papers Leitmann Very often, severe wind conditions arise from downbursts and Pandey (1991) and Leitmann, Pandey, and Ryan (1993), address the design of feedback controls keeping of air. A downburst is a descending column of air that hits the ground and then spreads horizontally. This phe(1993), address the design of feedback controls keeping hits the ground and then spreads horizontally. This pheand Pandey (1991) and Leitmann, Pandey, and Ryan of air. A downburst is a descending column of air that (1993), address the design of feedback controls keeping the climb rate. In papers Botkin and Turova (2012) and hits the ground and then spreads horizontally. This phenomenon is dangerous for aircrafts the climb rate. Inthe papers Botkin and Turova (2012) and nomenon is especially especially dangerous for aircrafts during during the the (1993), address design ofdifferential feedback controls keeping hits the phase ground and then spreads horizontally. This phethe climb rate. In papers Botkin and Turova (2012) and Botkin and Turova (2013), game approach nomenon is especially dangerous for aircrafts during the take-off because wind gusts occur at relatively low Botkin and Turova (2013), differential game approach take-off phase because dangerous wind gustsfor occur at relatively low climb rate. In papers Botkin and Turova (2012) and nomenon is especially aircrafts during can the the Botkin and Turova (2013), differential game approach (see e.g. Krasovskii and Subbotin (1988)) is used take-off phase because wind gusts occur at relatively low altitudes. An appropriate control theoretical problem (see e.g.and Krasovskii and Subbotin (1988)) isapproach used to to altitudes.phase An appropriate control theoretical problem can Botkin Turova (2013), differential game take-off because wind gusts occur at relatively low (see e.g. Krasovskii and Subbotin (1988)) is used to aa feedback control effective against downbursts. The altitudes. An appropriate control theoretical problem can design be formulated as differential game to guarantee the rejecdesign feedback control effective against downbursts. The be formulated as differential game to guarantee the rejec(see e.g. Krasovskii and Subbotin (1988)) is used to altitudes. An appropriate control theoretical problem can design a feedback control effective against downbursts. The main element of this approach is the value function that be formulated as differential game to guarantee the rejection wind main element of this approach is against the value function that tion of of unpredictable unpredictable wind disturbances. disturbances. a feedback control effective downbursts. The be as differential game to guarantee the rejec- design main element of this approach is the value function that coincides with a viscosity solution (see. e.g. Crandall and tionformulated of unpredictable wind disturbances. coincides with of a viscosity solution (see.value e.g. function Crandall that and The problem of control under element thisSubbotin approach is the tion unpredictable wind disturbances. coincides with aand viscosity solution (see. of e.g.an Crandall and Lions (1995) (1995)) appropriate The of problem of aircraft aircraft control under severe severe wind wind condicondi- main Lions (1995) and Subbotin (1995)) of an appropriate The problem of aircraft control under severe wind conditions has been considered by many authors. Papers Miele, coincides with aand viscosity solution (see. e.g. Crandall and Lions (1995) Subbotin (1995)) of an appropriate Hamilton-Jacobi equation, and can be computed using tions has been considered by many authors. Papers Miele, and(1995)) can beofcomputed using The problem ofconsidered aircraft control under severe wind(1989), conditions has been by many authors. Papers Miele, Hamilton-Jacobi Wang, and Melvin (1986), Chen and Pandey Lions (1995)described andequation, Subbotin an appropriate Hamilton-Jacobi equation, and can be computed using grid schemes in Botkin, Hoffmann, and Turova Wang, and Melvin (1986), Chen and Pandey (1989), in Botkin, Hoffmann, and Turova tions has been considered by many authors. Papers(2012), Miele, grid schemes described Wang, and Melvin and Leitmann and Pandey (1991), Botkin and Turova equation, and can bethecomputed using grid schemes described in(2011a). Botkin, Hoffmann, and (2011) and Botkin et al. Both cases of known Leitmann and Pandey(1986), (1991),Chen Botkin andPandey Turova (1989), (2012), Hamilton-Jacobi (2011) and Botkin et al. (2011a). Both the cases of Turova known Wang, and Melvin (1986), Chen and Pandey (1989), Leitmann and Pandey (1991), Botkin and Turova (2012), Botkin and Turova (2013) study the take-off phase. In grid schemes described in Botkin, Hoffmann, and Turova (2011) and Botkin et al. (2011a). Both the cases of known unknown wind field considered. Botkin and Turova (2013) study theand take-off phase. In and unknown windetvelocity velocity field are are considered. Leitmann and Pandeyand (1991), Botkin Turova (2012), Botkin Miele, and Turova (2013) study the take-off phase. In and papers Wang, Melvin (1986) and Miele, Wang, (2011) and Botkin al. (2011a). Both the cases of known and unknown wind velocity field are considered. papers Miele, Wang, and Melvin (1986) and Miele, Wang, Botkin and Turova (2013) study the field take-off phase. In The current paper addresses the problem of papers Miele, Wang, and Melvin (1986) and Miele, Wang, and Melvin (1986a), the wind velocity is assumed to and windaddresses velocity field are considered. The unknown current paper the problem of aircraft aircraft taketakeand Melvin (1986a), the wind velocity field is assumed to papers Miele, Wang, and Melvin (1986) and Miele, Wang, The current paper addresses the problem of aircraft takeoff considered in Botkin and Turova (2012) and Botkin and Melvin (1986a), the wind velocity field is assumed to be known so that open-loop controls are used. This yields off considered in Botkin and Turova (2012) and Botkin be known so that open-loop controls are used. This yields The current paper addresses the problem of aircraft takeand Melvin (1986a), the wind velocity field is assumed to off considered in Botkin and Turova (2012) and Botkin and Turova (2013) in the framework of differential game be known so for thatrelatively open-loop controls used. This yields good results severe wind disturbances. NevTurova (2013) in theand framework of differential game good results for relatively severe windare disturbances. Nev- and off considered in Botkin Turova (2012) and Botkin be known so that open-loop controls are used. This yields and Turova (2013) in the framework of differential game theory. The difference consists in the extension of the good results for relatively severe wind disturbances. Nevertheless, the assumption of known wind velocity is The (2013) difference consists in the extension ofgame the ertheless, thefor assumption of known wind velocity field field is theory. and Turova in the framework ofapplication differentialof good results relatively severe wind disturbances. Nevtheory. The difference consists in the extension of the aircraft dynamics to five equations and mulertheless, the assumption of known wind velocity field is unrealistic so that feedback principles of control design are aircraft dynamics to five equations and application of mulunrealistic so that feedback principles of control design are theory. Thecomputer difference consists inand theapplication extension ofmulthe ertheless, the assumption ofprinciples known wind velocity field is aircraft dynamics to five equations of tiprocessor platforms. Moreover, the utilization unrealistic so that feedback of control design are more appropriate. Papers Chen and Pandey (1989), Leittiprocessor computer platforms. Moreover, the utilization more appropriate. Papers Chen and Pandey (1989), Leitaircraft dynamics to five equations and application of mulunrealistic so that feedback principles of control design are tiprocessor computer platforms. Moreover, the utilization of overloads (see e.g. Tkachev and Liu (2015), Velishchanmore and Pandey (1989), Leit- of overloads (see e.g. Tkachev and Liu (2015), Velishchanmann and (1990), Leitmann and (1991), mann appropriate. and Pandey Pandey Papers (1990),Chen Leitmann and Pandey Pandey (1991), tiprocessor computer platforms. Moreover, the utilization more appropriate. Papers Chen and Pandey (1989), Leitof overloads e.g. Tkachev and Liu (2015), skiy (2016), Belinskaya (2016), Kanatnikov et al. mann and Pandey (1990), Leitmann and Pandey (1991), Leitmann, Pandey, and Ryan (1993) present different apskiy (2016), (see Belinskaya (2016), Kanatnikov etVelishchanal. (2018), (2018), Leitmann, Pandey, and Ryan (1993) present different apof overloads (see e.g. Tkachev and Liu (2015), Velishchanmann andtoPandey (1990), Leitmann Pandey (1991), (2016),and Belinskaya (2016), Kanatnikov et parameters al. (2018), Belinskaya Chetverikov (2018)) as control Leitmann, Pandey, and Ryan (1993) and present different ap- skiy proaches design of feedback controls. In paper Chen and Belinskaya and Chetverikov (2018)) as control parameters proaches to design of feedback controls. In paper Chen and skiy (2016), Belinskaya (2016), Kanatnikov et al. (2018), Leitmann, and Ryanrobust (1993)control present different ap- Belinskaya and Chetverikov (2018)) as control parameters allows us significantly reduce the of operations proaches toPandey, designa feedback controls. In paper Chen and Pandey feedback is allows us to toand significantly reduce the number number ofparameters operations Pandey (1989), (1989), a of feedback robust control is constructed constructed Belinskaya Chetverikov (2018)) as control proaches to design of feedback controls. In paper Chen and allows us to significantly reduce the number of operations when computing the minimums over control parameter Pandey (1989), a feedback robust control is constructed using Lyapunov function. Moreover, paper the minimums over control parameter using an an appropriate appropriate Lyapunov function. Moreover, paper when computing usbefore, to significantly reduce the number of operations Pandey (1989), a feedback robust control is constructed when computing the minimums over control parameter sets. As numerical methods described in Botkin, using an appropriate Lyapunov function. Moreover, paper allows sets. As before, numerical methods described in Botkin, when computing the minimums over control parameter using appropriate Lyapunov function. Moreover, paper sets. As before, numerical methods described in Botkin, ⋆ This an Hoffmann, and Turova (2011) and Botkin et al. (2011a) is supported by the DFG grant TU427/2-2. Computer ⋆ This work Hoffmann, and Turova (2011) and Botkin et al. (2011a) work is supported by the DFG grant TU427/2-2. Computer sets. As before, numerical methods described in (2011a) Botkin, ⋆ Hoffmann, and Turova (2011) and Botkin et al. are applied, and a performance index of Chebyshev type resources for is this project have been provided by the Gauss Centre This work supported by the DFG grant TU427/2-2. Computer are applied,and andTurova a performance index of Chebyshev type resources for this project have been provided by the Gauss Centre ⋆ Hoffmann, (2011) and Botkin et al. (2011a) are applied, and a performance index of Chebyshev type This work is supported by the DFG grant TU427/2-2. Computer is used. The quality of optimal trajectories is comparable for Supercomputing/Leibniz Supercomputing Centre under grant: resources for this project have been provided by the Gauss Centre is used. The quality of optimal trajectories is comparable for Supercomputing/Leibniz Supercomputing Centre under grant: are applied, and a performance index of Chebyshev type resources forsecond this project been provided the Gauss Centre pr74lu. The authorhave is Supercomputing supported by theby Russian Foundation is used. The quality of optimal trajectories is comparable for Supercomputing/Leibniz Centre under grant: pr74lu. The second author is supported by the Russian Foundation is used. The quality of optimal trajectories is comparable for Supercomputing/Leibniz Supercomputing Centre grant: of Basic Research 17-07-00653). pr74lu. The second(projects author is19-07-00817 supported and by the Russianunder Foundation
of Basic Research (projects 19-07-00817 and 17-07-00653). pr74lu. second(projects author is19-07-00817 supported and by the Russian Foundation of BasicThe Research 17-07-00653). of Basic Research (projects 19-07-00817Federation and 17-07-00653). 2405-8963 © 2019, IFAC (International of Automatic Control) Copyright © 2019 IFAC 1260Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 1260Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 1260 10.1016/j.ifacol.2019.12.045 Copyright © 2019 IFAC 1260
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with that obtained in Botkin and Turova Botkin and Turova (2013).
h˙ = V sin γ + Wh , � 1� T (V ) cos(α + δ) − D(V, α) − g sin γ V˙ = m ˙ h sin γ, ˙ x cos γ − W −W � 1 1 � T (V ) sin(α + δ) + L(V, α) − g cos γ γ˙ = mV V � 1� ˙ ˙ h cos γ , Wx sin γ − W + V ˙ x = −κ (Wx − v1 ), W
(2012) and
2. MODEL EQUATIONS A simplified point-mass aircraft model describing flight in a vertical plane, see papers Miele, Wang, and Melvin (1986) and Miele, Wang, and Melvin (1986a), is used. The dynamics is described by four ODEs comprising the horizontal distance x, the altitude h, the aircraft relative velocity V , and the relative path inclination γ: x˙ = V cos γ + Wx , h˙ = V sin γ + Wh , � 1� T (V ) cos(α + δ) − D(V, α) − g sin γ V˙ = m ˙ h sin γ, ˙ x cos γ − W −W � 1 1 � T (V ) sin(α + δ) + L(V, α) − g cos γ γ˙ = mV V � 1� ˙ ˙ h cos γ . Wx sin γ − W + V
(1)
CD = B0 + B1 α + B2 α2 ,
D(V, α) = L(V, α) =
� C0 + C1 α, CL = C0 + C1 α + C2 (α − α∗∗ )2 ,
(2)
˙ h = −κ (Wh − v2 ). W Here, v1 and v2 are the so-called command disturbances that may have instantaneous jumps. The variables Wx and Wh smoothly track v1 and v2 , respectively, and the parameter κ (κ = 0.2 is set) determines a time lag. The following constraints are imposed (cf. Turova (1992)): α ∈ [0, 16] deg, |v1 | ≤ 60 ft/s |v2 | ≤ 20 ft/s. (3) Thus, model (2) and constraints (3) form a differential game, where the aim of the control is to ensure a safe climb trajectory, and the objective of wind is opposite. 2.2 Control Change
Moreover, α is the aerodynamic angle of attack (control); δ the thrust inclination; g the acceleration of gravity; m the aircraft mass; T, D, and L are the thrust, drag, and lift forces, respectively; Wx and Wh are the horizontal and vertical components of the wind velocity, respectively. The following relations hold: T (V ) = A0 + A1 V + A2 V 2 ,
707
1 CD ρSV 2 , 2
Introduce the following variables that can be interpreted as scaled overloads: � 1 � T (V ) cos(α + δ) − D(V, α) , ux = mg (4) � 1 � uh = T (V ) sin(α + δ) + L(V, α) . mg These variables can be considered as new control parameters, which yields the following, equivalent with (2) and (3), differential game:
1 CL ρSV 2 , 2
h˙ = V sin γ + Wh , � ˙ x cos γ − W ˙ h sin γ, V˙ = ux − sin γ)g − W
α ∈ [0, α∗∗ ], α ∈ [α∗∗ , α∗ ],
� 1� 1� ˙ ˙ h cos γ , uh − cos γ)g + Wx sin γ − W V V
where α∗∗ and α∗ are given constants. The aerodynamic angle of attack α, considered as the control variable, is constrained as follows: 0 ≤ α ≤ α∗ .
γ˙ =
Note that the time derivatives of Wx and Wh appearing in (1) can be computed using the first two equations of (1) if Wx and Wh would be known functions of x and h. Unfortunately, this is not the case if differential game approach is used. The next subsection shows how model (1) can be extended to overcome this difficulty.
˙ h = −κ (Wh − v2 ), W
˙ x = −κ (Wx − v1 ), W
where the control variables are constrained now as follows: �
2.1 Model Extension Model (1) is being extended in the same way as in Turova (1992). Additionally, the first equation of (1) is unusable because the other equations do not involve x, and this variable should not be controlled as the process is considered after the initial climb phase (h = 50 ft). Therefore, we arrive at the following system:
(5)
ux uh
�
∈ P (V ) (6) � 1 � T (V ) cos(α + δ) − D(V, α) mg := , α ∈ [0, 16] . � 1 � T (V ) sin(α + δ) + L(V, α) mg
The constraints on the disturbance variables v1 and v2 are the same as in (3), that is
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� � v1 ∈ Q := [−60, 60] × [−20, 20]. v2
(7)
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uh
Note that the constraint P (V ) depends now on the state variable V . For example, Fig. 1 shows the set P (V ) at V = 220 ft/s.
1 0.9
uh
0.8
1
0.7
0.9
0.6
0.8
0.5
0.7
0.4
0.6
0.3
0.5
0.2
0.4
0.1
s
0 0.11
0.3
0.12
0.13
0.14
0.15
0.16
0.17
0.18
ux
0.19
0.2 0.1 0 0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
ux
Fig. 3. Triangle approximation (triangle line) of the set coP (V ) at V = 220 ft/s
0.19
Fig. 1. The control constraint set P (V ) at V = 220 ft/s
3. PROBLEM FORMULATION AND SOLUTION GRID METHOD
2.3 Processing of the Sets P (V ) Note that the grid method for solving the differential game (5), (6), and (7) involves the operation min max, where
The objective of the control u = (ux , uh )T in the differential game (5), (6), and (7) is to minimize the performance index
u∈P (V ) v∈Q
u := (ux , uh )T and v := (v1 , v2 )T . Since the variables u and v appear linearly in (5), differential game theory claims that the solution (value function) of the game under consideration remains the same if the above mentioned operation min max is being computed over the extremal points of the convex hulls of P (V ) and Q. The set Q has four of such points, and it is necessary to approximate the convex hull of the sets P (V ) (cf. Fig. 2) by polygons having few vertices, much better by triangles. This can be done in the manner shown in Fig. 3, where the middle point s is chosen to minimize the Hausdorff distance between coP (V ) and the triangle (cf. Fig. 3). Such an operation should be performed for each grid value of the variable V , and the results should be stored in a file. Thus, we can consider P (V ) and Q as three and four-point sets, respectively, which dramatically improve the performance.
J = −h(tf )
(8)
under the condition of keeping the state constraint −h(t) ≤ 0,
t0 ≤ t ≤ tf ,
(9)
where t0 and tf are the start and finish times of the process, respectively. The aims of the disturbance v = (v1 , v2 )T is opposite. The both players (control and disturbance) use feedback strategies. It would be convenient to rewrite the game (5), (6), and (7) in a general form: x˙ = f (x, u, v), x ∈ R5 , u ∈ P (x), v ∈ Q, (10)
J = σ0 (x(tf )),
uh 1
σ1 (x(t)) ≤ 0, t0 ≤ t ≤ tf .
0.9
Here, x and f denote now the state vector and the righthand side of the system (5), respectively. The set P (x) is defined as P (x) = P (V ). The function σ0 is defined as σ0 (x) = −h, and the function σ1 coincides with σ0 .
0.8 0.7 0.6 0.5
Note that the Isaacs saddle point condition
0.4 0.3
max min �p, f (x, u, v)� = min max�p, f (x, u, v)� v∈Q u∈P (x)
0.2
0 0.11
u∈P (x) v∈Q
def
(11)
= H(x, p)
0.1 0.12
0.13
0.14
0.15
0.16
0.17
0.18
ux
0.19
Fig. 2. The set coP (V ) (convex hull) at V = 220 ft/s
holds for any p ∈ R5 , (t, x) ∈ [0, tf ] × R5 . Therefore, the differential game (10) has the value function (t, x) → V(t, x) (see Krasovskii and Subbotin (1988)).
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3.1 Solution Scheme It is shown in Crandall and Lions (1995) that the value function is a viscosity solution of the Hamilton-Jacobi equation ∂V/∂t+H (x, ∂V/∂x) = 0.
(12)
It should be emphasized that viscosity solutions are generally not differentiable and are defined using upper and lower smooth test functions. Moreover, the Hamiltonian H is not globally convex or concave in the variable p, and, therefore, the computation of viscosity solutions to (12) is a difficult problem. As it is shown in Souganidis (1985), viscosity solutions can be approximately computed using monotone finite difference schemes. The monotonicity can easily be achieved if the Hamiltonian is monotone in components of the vector p, which is generally not true. The works Botkin, Hoffmann, and Turova (2011) and Botkin et al. (2011a), propose monotone finite difference upwind schemes for Hamilton-Jacobi equation (12) strengthened by an additional condition arising from the last inequality of (10). This subsection describes the application of the methods developed in Botkin, Hoffmann, and Turova (2011) and Botkin et al. (2011a) to the differential game (10).
Here, F Lξ [V ℓ ] is computed at the grid nodes only and then compared with the grid function σ1ξ . Thus, V ℓ−1 is a grid function. Theorem 1. The time-space grid function computed with (15) converges point wise to the value function V of differential game (10) as τ and |ξ| tend to zero under the condition τ /|ξ| ≤ C, where C is any positive constant. The proof of Theorem 1 is given in Botkin et al. (2011a). It should be noted that the proof is complicated technically and occupies several pages. The method consists in the construction of two special test functions (upper and lower) comprising signed deviation of the approximate solution from the exact one. The substitution of these test functions into the inequalities defining the viscosity solution allows us to estimate the signed deviation. 3.2 Control Design Let x be the current state of the system at a time instant tℓ . The control u(tℓ , x) is computed as follows: u(tℓ , x) = arg min max Lξ [V ℓ ] x + τ f (x, u, v) . u u∈P (x) v∈Q
Then, the feasible control α(tℓ , x), attack angle, is retrieved from u(tℓ , x).
Let τ and ξ := (ξ1 , ..., ξ5 ) be time and space discretization step sizes, and |ξ| = max{ξ1 , ..., ξ5 }. Let F be the following operator defined on continuous functions: F (V)(x) = min max V x + τ f (x, u, v) . u∈P (x) v∈Q
(13)
Set Λ = [tf /τ ], tℓ = ℓτ, ℓ = 0, ..., Λ, and introduce the following notation: V ℓ (xi1 , ..., xi5 ) = V(i1 ξ1 , ..., i5 ξ5 ), σ0ξ (xi1 , ..., xi5 ) = σ0 (i1 ξ1 , ..., i5 ξ5 ), σ1ξ (xi1 , ..., xi5 ) = σ1 (i1 ξ1 , ..., i5 ξ5 ). Let Lξ be an interpolation operator that maps grid functions to continuous functions and satisfies the estimate ˜ − φ� ≤ C|h| �D φ� �Lξ [φ] 2
2
(14)
for any smooth function φ. Here, φ˜ is the restriction of φ to the grid, � · � point-wise maximum norm, D2 φ the Hessian matrix of φ, and C a constant independent on φ and ξ.
709
4. SIMULATION RESULTS Parameter values used in the model are taken from Chen and Pandey (1989). They correspond to Boeing-727. In five dimensional state space (h, V, γ, Wx , Wh ), the grid size was equal to 200 × 200 × 100 × 20 × 20. The computation was performed on the SuperMUC multi processor system at the Leibniz Supercomputing Centre of the Bavarian Academy of Sciences and Humanities. The computation involved 30 compute nodes with 16 cores per node, which is regarded as “test task” for the SuperMUC system. The runtime was about 2 hours. Some details related to the parallelization technique can be found in paper Botkin et al. (2018). The feedback control was computed according to section 3.2. The disturbance was generated by a wind microburst described in Chen and Pandey (1989) (Fig. 4 shows the wind velocity field).
Notice that the estimate (14) is typical for interpolation operators (see e.g. M¨ oßner and Reif (2009)). Roughly speaking, interpolation operators reconstruct both the value and the gradient of interpolated functions, and, therefore, the expected error is of order |ξ|2 . For example, a multilinear interpolation operator from Botkin, Hoffmann, and Turova (2011) can be used.
h(ft)
Consider the following grid scheme: V ℓ−1 = max F Lξ [V ℓ ] , σ1ξ , V Λ = σ0ξ , ℓ = Λ, Λ−1, ..., 1. (15)
x(ft) Fig. 4. Wind streamlines generated by the microburst
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index and state constraints, which improves the resulting control but increases the dimension of the state vector. Another difference consists in using the overloads as new control parameters, which enables to significantly reduce the cardinality of the control constraint sets.
600
500
400
300
16 14
200
12 100 10 0
8 0
5
10
15
20
25
30
35
40 6
Fig. 5. Altitude h (ft) versus time (s)
4 2
340 0 330
0
5
10
15
20
25
30
35
40
320
Fig. 8. Control, attack angle α (deg), versus time (s)
310 300 290 280
60
270 260
40 250 20
240 230 0
5
10
15
20
25
30
35
40
0
Fig. 6. Velocity V (ft/s) versus time (s)
-20 -40
12 -60 10
0
8
5
10
15
20
25
30
35
40
Fig. 9. Wind component Wx (ft/s) versus time (s)
6 4 1 2 0.5 0
0
-2
-0.5 0
5
10
15
20
25
30
35
40 -1
Fig. 7. Path inclination γ (deg) versus time (s) -1.5
Figures 5–9 show the simulation results. It should be noted that these results are comparable with that reported in papers Chen and Pandey (1989), Botkin and Turova (2012), Botkin and Turova (2013). Remember that the novelty of the current paper, compared with Botkin and Turova (2012) and Botkin and Turova (2013), is that the altitude h is directly involved in the performance
-2 -2.5 -3 0
5
10
15
20
25
30
35
Fig. 10. Wind component Wh (ft/s) versus time (s)
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REFERENCES Belinskaya, Yu. S. (2016). Solution to point-to-point steering problem for constrained flat system by changing the flat output. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 6, 122–134. Belinskaya, Yu. S. and Chetverikov, V. N. (2018). Covering method for trajectory generation and orbital decomposition of systems. Differential Equations, 54(4), 497–508. Botkin, N. D., Hoffmann, K.-H., and Turova, V. L. (2011). Stable numerical schemes for solving Hamilton–Jacobi– Bellman–Isaacs equations. SIAM J. Sci. Comput., 33(2), 992–1007. Botkin, N. D., Hoffmann, K.-H., Mayer, N., and Turova, V. L. (2011a). Approximation schemes for solving disturbed control problems with non-terminal time and state constraints. Analysis, 31, 355–379. Botkin, N. D. and Turova, V. L. (2012). Application of dynamic programming approach to aircraft take-off in a windshear. In T. E. Simos, G. Psihoyios, Ch. Tsitouras, A. Zacharias (eds.), ICNAAM-2012. AIP Conference Proceedings, volume 1479, 1226–1229. AIP, Melville. Botkin, N. D. and Turova, V. L. (2013). Dynamic programming approach to aircraft control in a windshear. In V. Kˇrivan, G. Zaccour (eds.), Advances in Dynamic Games: Theory, Applications, and Numerical Methods. Annals of the International Society of Dynamic Games, volume 13, 53–69. Botkin, N. D., Turova, V. L., Diepolder, J., and Holzapfel, F. (2018). Computation of Viability Kernels on Grid Computers for Aircraft Control in Windshear. Advances in Science, Technology and Engineering Systems Journal (ASTES), 3(1), 502–510. Chen, Y. H. and Pandey, S. (1989). Robust control strategy for take-off performance in a windshear. Optim. Contr. Appl. Met., 10 (1), 65–79. Crandall, M. G. and Lions, P. L. (1983). Viscosity solutions of Hamilton-Jacobi equations. T. Am. Math. Soc., 277, 1–47. Kanatnikov, A. N., Liu, W., and Tkachev, S. B. (2018). Path coordinates in a 3D path following problem. Mathematical Models and Computer Simulations, 10(3), 265– 275. Krasovskii, N. N. and Subbotin, A. I. (1988). GameTheoretical Control Problems. Springer, New York. Leitmann, G. and Pandey, S. (1990). Aircraft control under conditions of windshear. In C. T. Leondes (ed.) Control and Dynamic Systems, volume 34, part 1, 1–79. Academic Press, New York. Leitmann, G. and Pandey, S. (1991). Aircraft control for flight in an uncertain environment: takeoff in windshear. J. Optimiz. Theory App., 70(1), 25–55. Leitmann, G., Pandey, S., and Ryan, E. (1993). Adaptive control of aircraft in windshear. Int. J. Robust Nonlin., 3, 133–153. Miele, A., Wang, T., and Melvin, W. W. (1986). Optimal take-off trajectories in the presence of windshear. J. Optimiz. Theory App., 49, 1–45. Miele, A., Wang, T., and Melvin, W. W. (1986). Guidance strategies for near-optimum take-off performance in windshear. J. Optimiz. Theory App., 50(1), 1–47.
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Aircraft Guiding in Windshear through Aircraft Guiding in Windshear through Aircraft Guiding in Windshear through ⋆⋆ Differential Game-Based Overload Control Aircraft Guiding in Windshear through Differential Game-Based Overload Control Differential Game-Based Overload Control ⋆⋆ Differential Game-Based Overload Control Nikolai D. Botkin ∗∗ Alexey E. Golubev ∗∗ ∗∗
Nikolai D. Botkin ∗ Alexey E. ∗∗ ∗∗∗Golubev Nikolai D.Varvara Botkin L.Alexey TurovaE. ∗∗∗Golubev Varvara ∗L. Turova ∗∗∗ Nikolai D.Varvara Botkin L.Alexey Golubev ∗∗ TurovaE. Varvara L. Turova ∗∗∗ ∗ ∗ Technical University of Munich, Boltzmannstraße 3, 85748 Garching, ∗ Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Technical University of Munich, 3, 85748 Garching, Germany (e-mail: [email protected]). Germany (e-mail: Boltzmannstraße [email protected]). ∗∗∗ Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Germany (e-mail: [email protected]). Bauman Moscow State Technical University, 2-nd ∗∗ Moscow State (e-mail: Technical University, 2-nd Baumanskaya Baumanskaya 5, 5, ∗∗ Bauman Germany [email protected]). Bauman Moscow State Technical University, 2-nd Baumanskaya 5, 105005 Moscow, Russia (e-mail: [email protected]) ∗∗ ∗∗∗ 105005 Moscow, Russia (e-mail: [email protected]) Bauman Moscow State Technical University, 2-nd Baumanskaya 5, Moscow, Russia (e-mail: [email protected]) Technical University of Boltzmannstraße 3, ∗∗∗ 105005 Technical University of Munich, Munich, Boltzmannstraße 3, 85748 85748 ∗∗∗ 105005 Moscow, Russia (e-mail:[email protected]). [email protected]) Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Germany (e-mail: Garching, Germany ∗∗∗ Technical University of (e-mail: Munich,[email protected]). Boltzmannstraße 3, 85748 Garching, Germany (e-mail: [email protected]). Garching, Germany (e-mail: [email protected]). Abstract: The The problem problem of of aircraft aircraft take-off take-off under under windshear windshear conditions conditions is is considered considered as as aa Abstract: Abstract: The problem of aircraft take-off under windshear conditions is considered as a differential game. A simplified five-dimensional nonlinear model of aircraft dynamics, assuming differential game. A simplified five-dimensional nonlinear model of aircraft dynamics, assuming Abstract: The problem of aircraft take-off under windshear conditions is considered as a differential game. A simplified five-dimensional nonlinear model of aircraft dynamics, assuming the flight in a vertical plane, is used. The overloads, differences between the projections the flight in a vertical plane,five-dimensional is used. The overloads, differences between the projections differential game. A drag, simplified nonlinear model of aircraft dynamics, assuming the flight in aand vertical is used.lift, Theforces overloads, differences the projections of the the thrust and respectively lift, forces are considered considered asbetween control variables. The of thrust drag,plane, respectively are as control variables. The the flight in a vertical plane, is used. The overloads, differences between the projections of the thrust and drag, respectively lift, forces are considered as control variables. The corresponding technically relevant control, the aerodynamic angle of attack, can always be corresponding technically control, the aerodynamic angle as of control attack, variables. can alwaysThe be of the thrust and drag, relevant respectively lift, forces are considered corresponding technically relevant control, the aerodynamic angle of attack, can always be retrieved. Stable numerical algorithms for solving Hamilton-Jacobi-Bellman-Isaacs equations retrieved. Stable numericalrelevant algorithms for solving Hamilton-Jacobi-Bellman-Isaacs equations corresponding technically the aerodynamic of attack, be retrieved. Stable numerical algorithms forconstraints solving Hamilton-Jacobi-Bellman-Isaacs equations arising from from differential games withcontrol, state constraints are used usedangle to compute compute the can valuealways function arising differential games with state are to the value function retrieved. Stable numerical algorithms forconstraints solving Hamilton-Jacobi-Bellman-Isaacs equations arising from differential games with state are used to compute the value function and design an appropriate feedback control. and design appropriate feedback arising froman games with control. state constraints are used to compute the value function and design andifferential appropriate feedback control. © 2019, IFAC Automatic Control) Hosting by Elsevier Ltd. All rights reserved. and design an(International appropriateFederation feedback of control. Keywords: Aircraft Aircraft control, control, Wind Wind microburst, microburst, Differential Differential games, games, Dynamic Dynamic programming, programming, Keywords: Keywords: Aircraft control, Wind microburst, Differential games, Dynamic programming, Hamilton-Jacobi equations, Grid methods, Multiprocessor systems, Feedback controls. Hamilton-Jacobi equations, Grid methods, Multiprocessor systems, Feedback controls. Keywords: Aircraft control, Wind microburst, Differential games, Dynamic programming, Hamilton-Jacobi equations, Grid methods, Multiprocessor systems, Feedback controls. Hamilton-Jacobi equations, Grid methods, Multiprocessor systems, Feedback controls. 1. Leitmann 1. INTRODUCTION INTRODUCTION Leitmann and and Pandey Pandey (1990) (1990) demonstrates demonstrates application application of of 1. INTRODUCTION Leitmann and Pandey (1990) demonstrates application of robust control theory to design of aa feedback control robust control theory to design of feedback control keepkeep1. INTRODUCTION Leitmann and Pandey (1990) demonstrates application of robust control theory to design of a feedback control keeping the relative path inclination, whereas papers Leitmann Very often, severe wind conditions arise from downbursts the control relativetheory path inclination, whereas papers Leitmann Very often, severe wind conditions arise from downbursts ing robust to design of a feedback control keeping relative(1991) path inclination, whereas papersand Leitmann andthe Pandey and Pandey, Ryan Very severe wind arise from downbursts of A is descending column of Pandey (1991) and Leitmann, Leitmann, Pandey, and Ryan of air. air.often, A downburst downburst is a aconditions descending column of air air that that and ing the relative path inclination, whereas papers Leitmann Very often, severe wind conditions arise from downbursts and Pandey (1991) and Leitmann, Pandey, and Ryan (1993), address the design of feedback controls keeping of air. A downburst is a descending column of air that hits the ground and then spreads horizontally. This phe(1993), address the design of feedback controls keeping hits the ground and then spreads horizontally. This pheand Pandey (1991) and Leitmann, Pandey, and Ryan of air. A downburst is a descending column of air that (1993), address the design of feedback controls keeping the climb rate. In papers Botkin and Turova (2012) and hits the ground and then spreads horizontally. This phenomenon is dangerous for aircrafts the climb rate. Inthe papers Botkin and Turova (2012) and nomenon is especially especially dangerous for aircrafts during during the the (1993), address design ofdifferential feedback controls keeping hits the phase ground and then spreads horizontally. This phethe climb rate. In papers Botkin and Turova (2012) and Botkin and Turova (2013), game approach nomenon is especially dangerous for aircrafts during the take-off because wind gusts occur at relatively low Botkin and Turova (2013), differential game approach take-off phase because dangerous wind gustsfor occur at relatively low climb rate. In papers Botkin and Turova (2012) and nomenon is especially aircrafts during can the the Botkin and Turova (2013), differential game approach (see e.g. Krasovskii and Subbotin (1988)) is used take-off phase because wind gusts occur at relatively low altitudes. An appropriate control theoretical problem (see e.g.and Krasovskii and Subbotin (1988)) isapproach used to to altitudes.phase An appropriate control theoretical problem can Botkin Turova (2013), differential game take-off because wind gusts occur at relatively low (see e.g. Krasovskii and Subbotin (1988)) is used to aa feedback control effective against downbursts. The altitudes. An appropriate control theoretical problem can design be formulated as differential game to guarantee the rejecdesign feedback control effective against downbursts. The be formulated as differential game to guarantee the rejec(see e.g. Krasovskii and Subbotin (1988)) is used to altitudes. An appropriate control theoretical problem can design a feedback control effective against downbursts. The main element of this approach is the value function that be formulated as differential game to guarantee the rejection wind main element of this approach is against the value function that tion of of unpredictable unpredictable wind disturbances. disturbances. a feedback control effective downbursts. The be as differential game to guarantee the rejec- design main element of this approach is the value function that coincides with a viscosity solution (see. e.g. Crandall and tionformulated of unpredictable wind disturbances. coincides with of a viscosity solution (see.value e.g. function Crandall that and The problem of control under element thisSubbotin approach is the tion unpredictable wind disturbances. coincides with aand viscosity solution (see. of e.g.an Crandall and Lions (1995) (1995)) appropriate The of problem of aircraft aircraft control under severe severe wind wind condicondi- main Lions (1995) and Subbotin (1995)) of an appropriate The problem of aircraft control under severe wind conditions has been considered by many authors. Papers Miele, coincides with aand viscosity solution (see. e.g. Crandall and Lions (1995) Subbotin (1995)) of an appropriate Hamilton-Jacobi equation, and can be computed using tions has been considered by many authors. Papers Miele, and(1995)) can beofcomputed using The problem ofconsidered aircraft control under severe wind(1989), conditions has been by many authors. Papers Miele, Hamilton-Jacobi Wang, and Melvin (1986), Chen and Pandey Lions (1995)described andequation, Subbotin an appropriate Hamilton-Jacobi equation, and can be computed using grid schemes in Botkin, Hoffmann, and Turova Wang, and Melvin (1986), Chen and Pandey (1989), in Botkin, Hoffmann, and Turova tions has been considered by many authors. Papers(2012), Miele, grid schemes described Wang, and Melvin and Leitmann and Pandey (1991), Botkin and Turova equation, and can bethecomputed using grid schemes described in(2011a). Botkin, Hoffmann, and (2011) and Botkin et al. Both cases of known Leitmann and Pandey(1986), (1991),Chen Botkin andPandey Turova (1989), (2012), Hamilton-Jacobi (2011) and Botkin et al. (2011a). Both the cases of Turova known Wang, and Melvin (1986), Chen and Pandey (1989), Leitmann and Pandey (1991), Botkin and Turova (2012), Botkin and Turova (2013) study the take-off phase. In grid schemes described in Botkin, Hoffmann, and Turova (2011) and Botkin et al. (2011a). Both the cases of known unknown wind field considered. Botkin and Turova (2013) study theand take-off phase. In and unknown windetvelocity velocity field are are considered. Leitmann and Pandeyand (1991), Botkin Turova (2012), Botkin Miele, and Turova (2013) study the take-off phase. In and papers Wang, Melvin (1986) and Miele, Wang, (2011) and Botkin al. (2011a). Both the cases of known and unknown wind velocity field are considered. papers Miele, Wang, and Melvin (1986) and Miele, Wang, Botkin and Turova (2013) study the field take-off phase. In The current paper addresses the problem of papers Miele, Wang, and Melvin (1986) and Miele, Wang, and Melvin (1986a), the wind velocity is assumed to and windaddresses velocity field are considered. The unknown current paper the problem of aircraft aircraft taketakeand Melvin (1986a), the wind velocity field is assumed to papers Miele, Wang, and Melvin (1986) and Miele, Wang, The current paper addresses the problem of aircraft takeoff considered in Botkin and Turova (2012) and Botkin and Melvin (1986a), the wind velocity field is assumed to be known so that open-loop controls are used. This yields off considered in Botkin and Turova (2012) and Botkin be known so that open-loop controls are used. This yields The current paper addresses the problem of aircraft takeand Melvin (1986a), the wind velocity field is assumed to off considered in Botkin and Turova (2012) and Botkin and Turova (2013) in the framework of differential game be known so for thatrelatively open-loop controls used. This yields good results severe wind disturbances. NevTurova (2013) in theand framework of differential game good results for relatively severe windare disturbances. Nev- and off considered in Botkin Turova (2012) and Botkin be known so that open-loop controls are used. This yields and Turova (2013) in the framework of differential game theory. The difference consists in the extension of the good results for relatively severe wind disturbances. Nevertheless, the assumption of known wind velocity is The (2013) difference consists in the extension ofgame the ertheless, thefor assumption of known wind velocity field field is theory. and Turova in the framework ofapplication differentialof good results relatively severe wind disturbances. Nevtheory. The difference consists in the extension of the aircraft dynamics to five equations and mulertheless, the assumption of known wind velocity field is unrealistic so that feedback principles of control design are aircraft dynamics to five equations and application of mulunrealistic so that feedback principles of control design are theory. Thecomputer difference consists inand theapplication extension ofmulthe ertheless, the assumption ofprinciples known wind velocity field is aircraft dynamics to five equations of tiprocessor platforms. Moreover, the utilization unrealistic so that feedback of control design are more appropriate. Papers Chen and Pandey (1989), Leittiprocessor computer platforms. Moreover, the utilization more appropriate. Papers Chen and Pandey (1989), Leitaircraft dynamics to five equations and application of mulunrealistic so that feedback principles of control design are tiprocessor computer platforms. Moreover, the utilization of overloads (see e.g. Tkachev and Liu (2015), Velishchanmore and Pandey (1989), Leit- of overloads (see e.g. Tkachev and Liu (2015), Velishchanmann and (1990), Leitmann and (1991), mann appropriate. and Pandey Pandey Papers (1990),Chen Leitmann and Pandey Pandey (1991), tiprocessor computer platforms. Moreover, the utilization more appropriate. Papers Chen and Pandey (1989), Leitof overloads e.g. Tkachev and Liu (2015), skiy (2016), Belinskaya (2016), Kanatnikov et al. mann and Pandey (1990), Leitmann and Pandey (1991), Leitmann, Pandey, and Ryan (1993) present different apskiy (2016), (see Belinskaya (2016), Kanatnikov etVelishchanal. (2018), (2018), Leitmann, Pandey, and Ryan (1993) present different apof overloads (see e.g. Tkachev and Liu (2015), Velishchanmann andtoPandey (1990), Leitmann Pandey (1991), (2016),and Belinskaya (2016), Kanatnikov et parameters al. (2018), Belinskaya Chetverikov (2018)) as control Leitmann, Pandey, and Ryan (1993) and present different ap- skiy proaches design of feedback controls. In paper Chen and Belinskaya and Chetverikov (2018)) as control parameters proaches to design of feedback controls. In paper Chen and skiy (2016), Belinskaya (2016), Kanatnikov et al. (2018), Leitmann, and Ryanrobust (1993)control present different ap- Belinskaya and Chetverikov (2018)) as control parameters allows us significantly reduce the of operations proaches toPandey, designa feedback controls. In paper Chen and Pandey feedback is allows us to toand significantly reduce the number number ofparameters operations Pandey (1989), (1989), a of feedback robust control is constructed constructed Belinskaya Chetverikov (2018)) as control proaches to design of feedback controls. In paper Chen and allows us to significantly reduce the number of operations when computing the minimums over control parameter Pandey (1989), a feedback robust control is constructed using Lyapunov function. Moreover, paper the minimums over control parameter using an an appropriate appropriate Lyapunov function. Moreover, paper when computing usbefore, to significantly reduce the number of operations Pandey (1989), a feedback robust control is constructed when computing the minimums over control parameter sets. As numerical methods described in Botkin, using an appropriate Lyapunov function. Moreover, paper allows sets. As before, numerical methods described in Botkin, when computing the minimums over control parameter using appropriate Lyapunov function. Moreover, paper sets. As before, numerical methods described in Botkin, ⋆ This an Hoffmann, and Turova (2011) and Botkin et al. (2011a) is supported by the DFG grant TU427/2-2. Computer ⋆ This work Hoffmann, and Turova (2011) and Botkin et al. (2011a) work is supported by the DFG grant TU427/2-2. Computer sets. As before, numerical methods described in (2011a) Botkin, ⋆ Hoffmann, and Turova (2011) and Botkin et al. are applied, and a performance index of Chebyshev type resources for is this project have been provided by the Gauss Centre This work supported by the DFG grant TU427/2-2. Computer are applied,and andTurova a performance index of Chebyshev type resources for this project have been provided by the Gauss Centre ⋆ Hoffmann, (2011) and Botkin et al. (2011a) are applied, and a performance index of Chebyshev type This work is supported by the DFG grant TU427/2-2. Computer is used. The quality of optimal trajectories is comparable for Supercomputing/Leibniz Supercomputing Centre under grant: resources for this project have been provided by the Gauss Centre is used. The quality of optimal trajectories is comparable for Supercomputing/Leibniz Supercomputing Centre under grant: are applied, and a performance index of Chebyshev type resources forsecond this project been provided the Gauss Centre pr74lu. The authorhave is Supercomputing supported by theby Russian Foundation is used. The quality of optimal trajectories is comparable for Supercomputing/Leibniz Centre under grant: pr74lu. The second author is supported by the Russian Foundation is used. The quality of optimal trajectories is comparable for Supercomputing/Leibniz Supercomputing Centre grant: of Basic Research 17-07-00653). pr74lu. The second(projects author is19-07-00817 supported and by the Russianunder Foundation
of Basic Research (projects 19-07-00817 and 17-07-00653). pr74lu. second(projects author is19-07-00817 supported and by the Russian Foundation of BasicThe Research 17-07-00653). of Basic Research (projects 19-07-00817Federation and 17-07-00653). 2405-8963 © 2019, IFAC (International of Automatic Control) Copyright © 2019 IFAC 1260Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 1260Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 1260 10.1016/j.ifacol.2019.12.045 Copyright © 2019 IFAC 1260
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with that obtained in Botkin and Turova Botkin and Turova (2013).
h˙ = V sin γ + Wh , � 1� T (V ) cos(α + δ) − D(V, α) − g sin γ V˙ = m ˙ h sin γ, ˙ x cos γ − W −W � 1 1 � T (V ) sin(α + δ) + L(V, α) − g cos γ γ˙ = mV V � 1� ˙ ˙ h cos γ , Wx sin γ − W + V ˙ x = −κ (Wx − v1 ), W
(2012) and
2. MODEL EQUATIONS A simplified point-mass aircraft model describing flight in a vertical plane, see papers Miele, Wang, and Melvin (1986) and Miele, Wang, and Melvin (1986a), is used. The dynamics is described by four ODEs comprising the horizontal distance x, the altitude h, the aircraft relative velocity V , and the relative path inclination γ: x˙ = V cos γ + Wx , h˙ = V sin γ + Wh , � 1� T (V ) cos(α + δ) − D(V, α) − g sin γ V˙ = m ˙ h sin γ, ˙ x cos γ − W −W � 1 1 � T (V ) sin(α + δ) + L(V, α) − g cos γ γ˙ = mV V � 1� ˙ ˙ h cos γ . Wx sin γ − W + V
(1)
CD = B0 + B1 α + B2 α2 ,
D(V, α) = L(V, α) =
� C0 + C1 α, CL = C0 + C1 α + C2 (α − α∗∗ )2 ,
(2)
˙ h = −κ (Wh − v2 ). W Here, v1 and v2 are the so-called command disturbances that may have instantaneous jumps. The variables Wx and Wh smoothly track v1 and v2 , respectively, and the parameter κ (κ = 0.2 is set) determines a time lag. The following constraints are imposed (cf. Turova (1992)): α ∈ [0, 16] deg, |v1 | ≤ 60 ft/s |v2 | ≤ 20 ft/s. (3) Thus, model (2) and constraints (3) form a differential game, where the aim of the control is to ensure a safe climb trajectory, and the objective of wind is opposite. 2.2 Control Change
Moreover, α is the aerodynamic angle of attack (control); δ the thrust inclination; g the acceleration of gravity; m the aircraft mass; T, D, and L are the thrust, drag, and lift forces, respectively; Wx and Wh are the horizontal and vertical components of the wind velocity, respectively. The following relations hold: T (V ) = A0 + A1 V + A2 V 2 ,
707
1 CD ρSV 2 , 2
Introduce the following variables that can be interpreted as scaled overloads: � 1 � T (V ) cos(α + δ) − D(V, α) , ux = mg (4) � 1 � uh = T (V ) sin(α + δ) + L(V, α) . mg These variables can be considered as new control parameters, which yields the following, equivalent with (2) and (3), differential game:
1 CL ρSV 2 , 2
h˙ = V sin γ + Wh , � ˙ x cos γ − W ˙ h sin γ, V˙ = ux − sin γ)g − W
α ∈ [0, α∗∗ ], α ∈ [α∗∗ , α∗ ],
� 1� 1� ˙ ˙ h cos γ , uh − cos γ)g + Wx sin γ − W V V
where α∗∗ and α∗ are given constants. The aerodynamic angle of attack α, considered as the control variable, is constrained as follows: 0 ≤ α ≤ α∗ .
γ˙ =
Note that the time derivatives of Wx and Wh appearing in (1) can be computed using the first two equations of (1) if Wx and Wh would be known functions of x and h. Unfortunately, this is not the case if differential game approach is used. The next subsection shows how model (1) can be extended to overcome this difficulty.
˙ h = −κ (Wh − v2 ), W
˙ x = −κ (Wx − v1 ), W
where the control variables are constrained now as follows: �
2.1 Model Extension Model (1) is being extended in the same way as in Turova (1992). Additionally, the first equation of (1) is unusable because the other equations do not involve x, and this variable should not be controlled as the process is considered after the initial climb phase (h = 50 ft). Therefore, we arrive at the following system:
(5)
ux uh
�
∈ P (V ) (6) � 1 � T (V ) cos(α + δ) − D(V, α) mg := , α ∈ [0, 16] . � 1 � T (V ) sin(α + δ) + L(V, α) mg
The constraints on the disturbance variables v1 and v2 are the same as in (3), that is
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� � v1 ∈ Q := [−60, 60] × [−20, 20]. v2
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uh
Note that the constraint P (V ) depends now on the state variable V . For example, Fig. 1 shows the set P (V ) at V = 220 ft/s.
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Fig. 3. Triangle approximation (triangle line) of the set coP (V ) at V = 220 ft/s
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Fig. 1. The control constraint set P (V ) at V = 220 ft/s
3. PROBLEM FORMULATION AND SOLUTION GRID METHOD
2.3 Processing of the Sets P (V ) Note that the grid method for solving the differential game (5), (6), and (7) involves the operation min max, where
The objective of the control u = (ux , uh )T in the differential game (5), (6), and (7) is to minimize the performance index
u∈P (V ) v∈Q
u := (ux , uh )T and v := (v1 , v2 )T . Since the variables u and v appear linearly in (5), differential game theory claims that the solution (value function) of the game under consideration remains the same if the above mentioned operation min max is being computed over the extremal points of the convex hulls of P (V ) and Q. The set Q has four of such points, and it is necessary to approximate the convex hull of the sets P (V ) (cf. Fig. 2) by polygons having few vertices, much better by triangles. This can be done in the manner shown in Fig. 3, where the middle point s is chosen to minimize the Hausdorff distance between coP (V ) and the triangle (cf. Fig. 3). Such an operation should be performed for each grid value of the variable V , and the results should be stored in a file. Thus, we can consider P (V ) and Q as three and four-point sets, respectively, which dramatically improve the performance.
J = −h(tf )
(8)
under the condition of keeping the state constraint −h(t) ≤ 0,
t0 ≤ t ≤ tf ,
(9)
where t0 and tf are the start and finish times of the process, respectively. The aims of the disturbance v = (v1 , v2 )T is opposite. The both players (control and disturbance) use feedback strategies. It would be convenient to rewrite the game (5), (6), and (7) in a general form: x˙ = f (x, u, v), x ∈ R5 , u ∈ P (x), v ∈ Q, (10)
J = σ0 (x(tf )),
uh 1
σ1 (x(t)) ≤ 0, t0 ≤ t ≤ tf .
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Here, x and f denote now the state vector and the righthand side of the system (5), respectively. The set P (x) is defined as P (x) = P (V ). The function σ0 is defined as σ0 (x) = −h, and the function σ1 coincides with σ0 .
0.8 0.7 0.6 0.5
Note that the Isaacs saddle point condition
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max min �p, f (x, u, v)� = min max�p, f (x, u, v)� v∈Q u∈P (x)
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u∈P (x) v∈Q
def
(11)
= H(x, p)
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Fig. 2. The set coP (V ) (convex hull) at V = 220 ft/s
holds for any p ∈ R5 , (t, x) ∈ [0, tf ] × R5 . Therefore, the differential game (10) has the value function (t, x) → V(t, x) (see Krasovskii and Subbotin (1988)).
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3.1 Solution Scheme It is shown in Crandall and Lions (1995) that the value function is a viscosity solution of the Hamilton-Jacobi equation ∂V/∂t+H (x, ∂V/∂x) = 0.
(12)
It should be emphasized that viscosity solutions are generally not differentiable and are defined using upper and lower smooth test functions. Moreover, the Hamiltonian H is not globally convex or concave in the variable p, and, therefore, the computation of viscosity solutions to (12) is a difficult problem. As it is shown in Souganidis (1985), viscosity solutions can be approximately computed using monotone finite difference schemes. The monotonicity can easily be achieved if the Hamiltonian is monotone in components of the vector p, which is generally not true. The works Botkin, Hoffmann, and Turova (2011) and Botkin et al. (2011a), propose monotone finite difference upwind schemes for Hamilton-Jacobi equation (12) strengthened by an additional condition arising from the last inequality of (10). This subsection describes the application of the methods developed in Botkin, Hoffmann, and Turova (2011) and Botkin et al. (2011a) to the differential game (10).
Here, F Lξ [V ℓ ] is computed at the grid nodes only and then compared with the grid function σ1ξ . Thus, V ℓ−1 is a grid function. Theorem 1. The time-space grid function computed with (15) converges point wise to the value function V of differential game (10) as τ and |ξ| tend to zero under the condition τ /|ξ| ≤ C, where C is any positive constant. The proof of Theorem 1 is given in Botkin et al. (2011a). It should be noted that the proof is complicated technically and occupies several pages. The method consists in the construction of two special test functions (upper and lower) comprising signed deviation of the approximate solution from the exact one. The substitution of these test functions into the inequalities defining the viscosity solution allows us to estimate the signed deviation. 3.2 Control Design Let x be the current state of the system at a time instant tℓ . The control u(tℓ , x) is computed as follows: u(tℓ , x) = arg min max Lξ [V ℓ ] x + τ f (x, u, v) . u u∈P (x) v∈Q
Then, the feasible control α(tℓ , x), attack angle, is retrieved from u(tℓ , x).
Let τ and ξ := (ξ1 , ..., ξ5 ) be time and space discretization step sizes, and |ξ| = max{ξ1 , ..., ξ5 }. Let F be the following operator defined on continuous functions: F (V)(x) = min max V x + τ f (x, u, v) . u∈P (x) v∈Q
(13)
Set Λ = [tf /τ ], tℓ = ℓτ, ℓ = 0, ..., Λ, and introduce the following notation: V ℓ (xi1 , ..., xi5 ) = V(i1 ξ1 , ..., i5 ξ5 ), σ0ξ (xi1 , ..., xi5 ) = σ0 (i1 ξ1 , ..., i5 ξ5 ), σ1ξ (xi1 , ..., xi5 ) = σ1 (i1 ξ1 , ..., i5 ξ5 ). Let Lξ be an interpolation operator that maps grid functions to continuous functions and satisfies the estimate ˜ − φ� ≤ C|h| �D φ� �Lξ [φ] 2
2
(14)
for any smooth function φ. Here, φ˜ is the restriction of φ to the grid, � · � point-wise maximum norm, D2 φ the Hessian matrix of φ, and C a constant independent on φ and ξ.
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4. SIMULATION RESULTS Parameter values used in the model are taken from Chen and Pandey (1989). They correspond to Boeing-727. In five dimensional state space (h, V, γ, Wx , Wh ), the grid size was equal to 200 × 200 × 100 × 20 × 20. The computation was performed on the SuperMUC multi processor system at the Leibniz Supercomputing Centre of the Bavarian Academy of Sciences and Humanities. The computation involved 30 compute nodes with 16 cores per node, which is regarded as “test task” for the SuperMUC system. The runtime was about 2 hours. Some details related to the parallelization technique can be found in paper Botkin et al. (2018). The feedback control was computed according to section 3.2. The disturbance was generated by a wind microburst described in Chen and Pandey (1989) (Fig. 4 shows the wind velocity field).
Notice that the estimate (14) is typical for interpolation operators (see e.g. M¨ oßner and Reif (2009)). Roughly speaking, interpolation operators reconstruct both the value and the gradient of interpolated functions, and, therefore, the expected error is of order |ξ|2 . For example, a multilinear interpolation operator from Botkin, Hoffmann, and Turova (2011) can be used.
h(ft)
Consider the following grid scheme: V ℓ−1 = max F Lξ [V ℓ ] , σ1ξ , V Λ = σ0ξ , ℓ = Λ, Λ−1, ..., 1. (15)
x(ft) Fig. 4. Wind streamlines generated by the microburst
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index and state constraints, which improves the resulting control but increases the dimension of the state vector. Another difference consists in using the overloads as new control parameters, which enables to significantly reduce the cardinality of the control constraint sets.
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Fig. 5. Altitude h (ft) versus time (s)
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Fig. 8. Control, attack angle α (deg), versus time (s)
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Fig. 6. Velocity V (ft/s) versus time (s)
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Fig. 9. Wind component Wx (ft/s) versus time (s)
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Fig. 7. Path inclination γ (deg) versus time (s) -1.5
Figures 5–9 show the simulation results. It should be noted that these results are comparable with that reported in papers Chen and Pandey (1989), Botkin and Turova (2012), Botkin and Turova (2013). Remember that the novelty of the current paper, compared with Botkin and Turova (2012) and Botkin and Turova (2013), is that the altitude h is directly involved in the performance
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Fig. 10. Wind component Wh (ft/s) versus time (s)
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