Institute of Institute of Technische Mathematical Modelling Flight System Dynamics Universität München 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 optimal control and related fields. Linz, Austria, November 21-25, 2016
2 Outline Value Viable function function Aircraft control Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
3 Differential Games • Subbotin A. I., Krasovskii N.N. Game-Theoretical Control Problems. Springer, New York (1988). Dynamics: , . Simplest payoff functional: Payoff function: Objectives of the players: 1 st player ( ) minimizes the payoff functional 2 nd player ( ) maximizes the payoff functional Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
4 Differential Games • Formalization Pure feedback strategies: , , . Counter-feedback strategies: , Bundles of all limiting functions as : Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
5 Differential Games • Value functions (lower and upper) Obviously: If the Isaacs saddle point condition holds: Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
6 Differential Games • Other payoff functionals (a) (already mentioned) (b) (c) Interpretation: (a) result at time (b) result by time (c) result by time subject to the state constraint : Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
7 Differential Games • Explanation of the functional (c) Set, e.g. . Consider the following target and state constraint sets: . such that : which implies that 1. for some 2. which implies that for all Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
8 Differential Games • 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 Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
9 Differential Games • Value function as viscosity solution Botkin, Hoffmann, Mayer, Turova. Analysis 31, 2011 A function is the value function in the case (c) iff: (i) (ii) If then If then (iii) Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
10 Differential Games • Grid algorithm Botkin, Hoffmann, Mayer, Turova. Analysis 31, 2011 (Souganidis, Barles) Botkin, Hoffmann, Turova: Applied Math. Modeling 35, 2011. Time and space discretizations Grid approx . Divided differences Operator defined on grid functions convergence rate ~ Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
11 Differential Games • 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. Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
12 Differential Games • 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 ( ) Upper v-stability Lower v-stability (lower u-stability, upper v-stability), (upper u-stability, lower v-stability) Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
13 Viability Kernels • 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 . Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
14 Viability Kernels • Properties (e.g. the case of upper u-stability) Theorem 2. Let be the minimal upper u- stable function. Then the following holds: On each finite interval the function is 1. the upper value function of the differential game with the payoff functional 2. is the maximal closed upper u-stable subset of . Denote 3. a) b) Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
15 Viability Kernels • Numerical algorithm (e.g. the case of upper u-stability) Lemma. as If then Moreover, as Thus, approximates if is large and are small. Criterion of the accuracy: Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
16 Viability Kernels • Control design (e.g. case of discriminating 1 st player) Let be sufficient large, and small. Feedback strategy of the first player Counter-feedback strategy of the second player Here, is a grid interpolation operator, is a shift parameter, which is larger than to regularize the control. Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
17 Research Flight Simulator at TUM-FSD 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 Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
18 Overview: Frames for Modeling the Aircraft Dynamics Propulsion System Rotated Kinematic- Kinematic- P System System K , K K , K K 𝜆, 𝜏 𝑦 K K 𝑧 𝑨 𝑨 𝑧 NED- , , Bodyfixed O B System 𝑨 𝑧 𝑦 System , A A A 𝑨 𝑧 , , A A A 𝑨 𝑧 𝑦 Aerodynamic- System 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). Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
2.19 North-East-Down (NED) Frame • Index: O • Role: Navigation Frame • N Orign: Referencepoint of the Aircraft x E • Translation: Moves with the Referencepoint y O O • ω EO Rotation: Rotates with the Transportrate, to fulfill the „NED condition “ Nordpol z O D K , K K , K K 𝑦 K K 𝑧 𝑨 𝑨 𝑧 , , O B 𝑨 𝑧 𝑦 , A A A 𝑨 𝑧 , , A A A 𝑨 𝑧 𝑦 Institute of Differential Games With State Constraints Institute of Institute of Flight System Dynamics Flight System Dynamics Mathematical Modelling and Viability Kernels
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