Differential games with state constraints and viability kernels N. - - PowerPoint PPT Presentation

Institute of Flight System Dynamics Technische Universität München Institute of Mathematical Modelling

Differential games with state constraints and viability kernels

• N. D. Botkin, J. Diepolder, V. L. Turova, M. Bittner, F

. Holzapfel

Center for Mathematics & Institute of Flight System Dynamics This work is supported by the DFG grants TU427/2-1 and HO4190/8-1. LRZ supercomputing support under grant pr74lu. Workshop 3. Numerical methods for Hamilton-Jacobi equations in

• ptimal control and related fields. Linz, Austria, November 21-25, 2016

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Outline

2

Value function Viable function

Aircraft control

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Differential Games

3

• Subbotin A. I., Krasovskii N.N. Game-Theoretical Control Problems.

Springer, New York (1988).

Dynamics: Simplest payoff functional: Payoff function: Objectives of the players:

1st player ( ) minimizes the payoff functional 2nd player ( ) maximizes the payoff functional

, .

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

• Formalization

4

Pure feedback strategies: Counter-feedback strategies: Bundles of all limiting functions as : , , . ,

Differential Games

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Differential Games

• Value functions (lower and upper)

5

Obviously: If the Isaacs saddle point condition holds:

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Differential Games

• Other payoff functionals

6

(a) (b) (c) Interpretation: (a) result at time (b) result by time

(c) result by time subject to the state constraint:

(already mentioned)

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Differential Games

7

• Explanation of the functional (c)

Set, e.g. . Consider the following target and state constraint sets: such that : 1. which implies that 2. which implies that for some for all .

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Differential Games

8

• Value function as viscosity solution

Lower and upper Hamiltonians: In the following, we omit lower and upper bars: upper or lover Hamiltonian and upper or lower value function

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Differential Games

9

• Value function as viscosity solution

(i) A function is the value function in the case (c) iff: (ii) If then (iii) If then

Botkin, Hoffmann, Mayer, Turova. Analysis 31, 2011

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Differential Games

10

• Grid algorithm

Botkin, Hoffmann, Mayer, Turova. Analysis 31, 2011 Botkin, Hoffmann, Turova: Applied Math. Modeling 35, 2011. Time and space discretizations Grid approx.

Divided differences Operator defined on grid functions

convergence rate ~ (Souganidis, Barles)

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Differential Games

11

• Upper u-stability of functions and sets

(1)

Upper u-stability of a function : Upper u-stability of a set : Proposition:

is u-stable iff all level sets are u-stable.

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Differential Games

12

• Lower u-stability of functions and sets

(2)

Lower u-stability of functions and sets is defined in a similar way: replacing by , and (1) by (2).

Upper and lower v-stabilities ( )

Lower v-stability Upper v-stability

(lower u-stability, upper v-stability), (upper u-stability, lower v-stability)

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Viability Kernels

13

• Assumptions, minimal u-stable functions.

Autonomous system:

• Lipschitz continuous, bounded function such that

are bounded and Let be the set of lower semi-continuous functions, , defined on and satisfying

Theorem 1. There exist unique minimal upper and

lower u- stable functions in . .

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Viability Kernels

14

• Properties (e.g. the case of upper u-stability)

Theorem 2. Let be the minimal upper u-

stable function. Then the following holds: 2.

upper u-stable subset of

. Denote

1.

On each finite interval the function is the upper value function of the differential game with the payoff functional

• 3. a)

b)

is the maximal closed

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Viability Kernels

15

• Numerical algorithm (e.g. the case of upper u-stability)

Lemma.

Thus, approximates if is large and are small. Criterion of the accuracy: If then as Moreover, as

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Viability Kernels

16

• Control design (e.g. case of discriminating 1st player)

Here, is a grid interpolation operator, is a shift parameter, which is larger than to regularize the control. Feedback strategy of the first player Counter-feedback strategy of the second player Let be sufficient large, and small.

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Research Flight Simulator at TUM-FSD

17

General Description:

• Generic cockpit layout due to focus on research; development started in 2000/2001
• Cooperation with Fairchild Dornier – who donated the shell
• One of the design philosophies was to achieve high flexibility and modularity
• Complete in-house development (besides the shell)
• Based on approximately 80-100 diploma theses

Computer Hardware (IT)

• High flexibility due to standard PC-Hardware (Windows XP)
• Server –Client architecture
• Three synchronized PCs for the visual system with CRT projectors

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Overview: Frames for Modeling the Aircraft Dynamics

The effects (e.g. forces) are described in the most convenient system and transformed into the system where they are needed for the dynamic modeling (i.e. translational and position propagation).

18

K

K B O

K



K K 

 , 

K K 

 ,

A

   , ,

A A A

   , ,

A A 

 , 

NED- System Bodyfixed System Kinematic- System Rotated Kinematic- System Aerodynamic- System

𝑦 𝑧 𝑨 𝑦 𝑦 𝑧 𝑧 𝑧 𝑧 𝑨 𝑨 𝑨 𝑨

P

𝜆, 𝜏

Propulsion System

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

2.19

• Index:

O

• Role:

Navigation Frame

• Orign:

Referencepoint of the Aircraft

• Translation:

Moves with the Referencepoint

• Rotation:

Rotates with the Transportrate, to fulfill the „NED condition“

N E D

O

x

O

y

O

z

Nordpol

 

EO

ω 

K

K B

O

K



K K 

 , 

K K 

 ,

A

   , ,

A A A

   , ,

A A 

 , 

𝑦 𝑧 𝑨 𝑦 𝑦 𝑧 𝑧 𝑧 𝑧 𝑨 𝑨 𝑨 𝑨

North-East-Down (NED) Frame

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yO xO zO yK xK zK

xKzK-plane xOyO-plane

2.20

• Index:

K

• Role:

Trajectory-System

• Orign:

Referencepoint of the Aircraft

• Translation: Moves with the Referencepoint
• Rotation:

Rotates with the Direction of the Kinematic Velocity of the Aircraft

K

K B

K



K K 

 , 

K K 

 ,

A

   , ,

A A A

   , ,

A A 

 , 

𝑦 𝑧 𝑨 𝑦 𝑦 𝑧 𝑧 𝑧 𝑧 𝑨 𝑨 𝑨 𝑨

O

Kinematic Frame

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

2.21

• Index:

B

• Role:

Description of Forces and Moments

• Orign:

Referencepoint of the Aircraft

• Translation:

Moves with the Referencepoint

• Rotation:

Rotates with the Aircraft xB zB yB

K

K B

K



K K 

 , 

K K 

 ,

A

   , ,

A A A

   , ,

A A 

 , 

𝑦 𝑧 𝑨 𝑦 𝑦 𝑧 𝑧 𝑧 𝑧 𝑨 𝑨 𝑨 𝑨

O

Body Fixed Frame

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

• Index:

A

• Role:

Description of Aerodynamic Quantities

• Orign:

Referencepoint of the Aircraft

• Translation:

Moves with the Referencepoint

• Rotation:

Rotates with the Direction of the Airspeed Relative to the Aircraft

2.22

xB yB zB xA yA zA

xBzB-plane xAyA-plane

Aerodynamic Frame

K

K

K B

K



K K 

 , 

K K 

 ,

A

   , ,

A A A

   , ,

A A 

 , 

𝑦 𝑧 𝑨 𝑦 𝑦 𝑧 𝑧 𝑧 𝑧 𝑨 𝑨 𝑨 𝑨

O

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Position Propagation (NED-Frame)

23

K

K B O

K



K K 

 , 

K K 

 ,

A

   , ,

A A A

   , ,

A A 

 , 

NED- System Bodyfixed System Kinematic- System Rotated Kinematic- System Aerodynamic- System

𝑦 𝑧 𝑨 𝑦 𝑦 𝑧 𝑧 𝑧 𝑧 𝑨 𝑨 𝑨 𝑨

P

𝜆, 𝜏

Propulsion System

𝑧 𝑨

𝑾𝐿

𝐻 𝑃 𝐹=

𝑊

𝐿 𝐻 ⋅

cos 𝜓 ⋅ cos(𝛿) sin 𝜓 ⋅ cos(𝛿) −sin(𝛿) 𝑾𝐿

𝐻 𝐿 𝐹=

𝑊

𝐿 𝐻

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Translational Propagation

• Newtons Law:
• Neglecting the influences from earth rotation and earth roundness:

 Now we need to determine the total forces in the kinematic frame in 𝑦𝐿, 𝑧𝐿, and 𝑨𝐿 direction: 𝑌𝐻 𝐿, 𝑍𝐻 𝐿, 𝑎𝐻 𝐿 which originate from gravitation, propulsion and aerodynamics.

24

 

 

II K G K K G T

m V F   



 

                 K

G E K G K EK K G K K G E K G K E K G K EK K G K K G EK K G K

Z V m Y V m X m V           1 cos 1 1      

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Computation of Forces (Gravitation, Propulsion, Aero)

25

K

K B O

K



K K 

 , 

K K 

 ,

A

   , ,

A A A

   , ,

A A 

 , 

NED- System Bodyfixed System Kinematic- System Rotated Kinematic- System Aerodynamic- System

𝑦 𝑧 𝑨 𝑦 𝑦 𝑧 𝑧 𝑧 𝑧 𝑨 𝑨 𝑨 𝑨

P

𝜆, 𝜏

Propulsion System

𝑧 𝑨

𝑮𝐻

𝐻 𝑃

𝑮𝑄

𝐻 𝑄

𝑮𝐵

𝐻 𝑩

𝑮T

G K =

𝑌𝐻 𝑍𝐻 𝑎𝐻 𝐿 = 𝑮𝐻

𝐻 𝐿 + 𝑮𝑄 𝐻 𝐿 + 𝑮𝐵 𝐻 𝐿

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Aerodynamic Force

26

• Aerodynamic Forces

𝑮𝐵

𝐻 𝐿

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Propulsion Force

27

• Propulsion and Gravity Forces

are approximated similar to ⋅ 𝑕 𝑃

𝑮𝑄

𝐻 𝐿

𝑮𝐻

𝐻 𝐿

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Overview: Aircraft Pointmass Equation of Motion

• Position Propagation (3 States)
• Translational Propagation (3 States):

28

ሶ 𝑦𝐿

𝐻

ሶ 𝑧𝐿

𝐻

ሶ 𝑨𝐿

𝐻 𝑃 𝐹

= 𝑊

𝐿 𝐻 ⋅

cos 𝜓 ⋅ cos(𝛿) sin 𝜓 ⋅ cos(𝛿) −sin(𝛿)

                 K

G E K G K EK K G K K G E K G K E K G K EK K G K K G EK K G K

Z V m Y V m X m V           1 cos 1 1      

• Aircraft Controls
• Disturbances (Wind):

𝛽𝐿 𝜈𝐿 𝜀𝑈 ≜ "𝑙𝑗𝑜𝑓𝑛𝑏𝑢𝑗𝑑 𝑏𝑜𝑕𝑚𝑓 𝑝𝑔 𝑏𝑢𝑢𝑏𝑑𝑙" "𝑙𝑗𝑜𝑓𝑛𝑏𝑢𝑗𝑑 𝑐𝑏𝑜𝑙 𝑏𝑜𝑕𝑚𝑓" "𝑢ℎ𝑠𝑣𝑡𝑢 𝑚𝑓𝑤𝑓𝑠 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜" 𝑣𝑋

𝐻 𝑃 𝐹

𝑤𝑋

𝐻 𝑃 𝐹

𝑥𝑋

𝐻 𝑃 𝐹

≜ "𝑋𝑗𝑜𝑒 𝑗𝑜 𝑦𝑃, 𝑧𝑃, 𝑨𝑃 −𝑒𝑗𝑠𝑓𝑑𝑢𝑗𝑝𝑜"

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Overview: Aircraft Pointmass Equation – Vertical/Lateral

• Position Propagation (3 States)
• Translational Propagation (3 States):
• Position Propagation (3 States)
• Translational Propagation (3 States):

29

         K

G E K G K EK K G K K G EK K G K

Z V m X m V       1 1   

• Aircraft Controls
• Disturbance (Wind):
• Aircraft Controls
• Disturbance (Wind):

𝛽𝐿 𝜀𝑈 ≜ "𝑙𝑗𝑜𝑓𝑛𝑏𝑢𝑗𝑑 𝑏𝑜𝑕𝑚𝑓 𝑝𝑔 𝑏𝑢𝑢𝑏𝑑𝑙" "𝑢ℎ𝑠𝑣𝑡𝑢 𝑚𝑓𝑤𝑓𝑠 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜" 𝑣𝑋

𝐻 𝑃 𝐹

𝑥𝑋

𝐻 𝑃 𝐹 ≜ "𝑋𝑗𝑜𝑒 𝑗𝑜 𝑦𝑃, 𝑨𝑃

−𝑒𝑗𝑠𝑓𝑑𝑢𝑗𝑝𝑜" ሶ 𝑦𝐿

𝐻

ሶ 𝑨𝐿

𝐻 𝑃 𝐹

= 𝑊

𝐿 𝐻 ⋅ cos 𝜓 ⋅ cos(𝛿)

sin(𝛿) 𝜈𝐿 𝜀𝑈 ≜ "𝑙𝑗𝑜𝑓𝑛𝑏𝑢𝑗𝑑 𝑐𝑏𝑜𝑙 𝑏𝑜𝑕𝑚𝑓" "𝑢ℎ𝑠𝑣𝑡𝑢 𝑚𝑓𝑤𝑓𝑠 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜" 𝑣𝑋

𝐻 𝑃 𝐹

𝑤𝑋

𝐻 𝑃 𝐹 ≜ "𝑋𝑗𝑜𝑒 𝑗𝑜 𝑦𝑃, 𝑧𝑃

−𝑒𝑗𝑠𝑓𝑑𝑢𝑗𝑝𝑜" ሶ 𝑦𝐿

𝐻

ሶ 𝑧𝐿

𝐻 𝑃 𝐹

= 𝑊

𝐿 𝐻 ⋅ cos 𝜓 ⋅ cos(𝛿)

sin 𝜓 ⋅ cos(𝛿) 𝑾𝒇𝒔𝒖𝒋𝒅𝒃𝒎 𝑸𝒎𝒃𝒐𝒇 (𝒚/𝒜) 𝑴𝒃𝒖𝒇𝒔𝒃𝒎 𝑸𝒎𝒃𝒐𝒇 (𝒚/𝒛)

           K

G E K G K E K G K EK K G K K G EK K G K

Y V m X m V         cos 1 1  

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Models

• Point-mass aircraft dynamics

30

5-d 3-d Controller providing constant altitude are components of the total force Lateral + vertical Lateral

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Viability Kernels

31

• Full model (5-dimensional, control is discriminated)

Grid parameters:

4 x 105

time steps Constraints on controls and disturbances: State constraints: Running time: 4 h on 100 compute nodes, 16 threads per node. , , .

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Viability Kernels

32

• Cross-section of the viability kernel

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Viability Kernels

33

• Lateral motion (3-dimensional, control is discriminated)

Grid parameters:

4 x 105 time steps

Constraints on controls and disturbances: State constraints: Running time: 30 min on 25 compute nodes, 16 threads per node. , , .

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Viability Kernels

34

• The viability kernel and an optimal trajectory

The control uses the optimal feedback strategy, the disturbance utilizes the optimal counter-feedback strategy.

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Viability Kernels

35

• Computer resources

Leibnitz Supercomputing Centre of the Bavarian Academy of Sciences and Humanities SuperMUC system

The strategies obtained will be tested in the flight simulator of the Institute of Flight System Dynamics.

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Outlook

• Computation of Viability Kernels for higher dimensional state spaces (5-7)
• The strategies obtained will be tested in the flight simulator of the

Institute of Flight System Dynamics:

– Integration in a suitable control structure (Envelope Protection) – Pilot training for severe wind conditions (Pilot vs. Wind) – Controller Testing

36

Differential Games With State Constraints and Viability Kernels Institute of Mathematical Modelling Institute of Flight System Dynamics Institute of Flight System Dynamics

Johannes Diepolder, johannes.diepolder@tum.de Matthias Bittner, matthias.bittner@tum.de Institute of Flight System Dynamics Technische Universität München Boltzmannstraße 15 D-85748 Garching bei München Deutschland / Germany Phone: +49 89 289-16080 Fax: +49 89 289-16058

Thank you for your attention!

Thank you for your attention

Institute of Mathematical Modelling Nikolai Botkin, Botkin@ma.tum.de Varvara Turova, Turova@ma.tum.de Institute for Mathematical Modelling (M6) Technische Universität München Boltzmannstraße 3 D-85747 Garching bei München Deutschland / Germany Phone: +49 89 289-16808 Fax: +49 89 289-16809 Institute of Flight System Dynamics