טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentOr Barak
SubjectIntegrated Guidance / Estimation in Linear Quadratic
Differential Games
DepartmentDepartment of Aerospace Engineering
Supervisor Professor Yoseph Ben-Asher


Abstract

Pursuit-evasion differential games have attracted considerable attention since the seminal works of Baron and Bryson & Ho. A class of differential games, with a couple of players driving linear state equations both affecting a quadratic cost function, is called Linear Quadratic Differential Game (LQDG).  In these games, the pursuer tries to minimize a quadratic cost function, whereas the evader tries to maximize the same cost function (zero-sum games). The cost function includes weights on the squared miss-distance, the control efforts of both players, as well as occasionally trajectory shaping terms. The main LQDG formulation leads to a derivation of popular guidance laws such as Proportional Navigation (PN), Optimal Rendezvous (OR), etc. A problem which is closely related to the LQDG problem is the one of Disturbance Attenuation (DA), where pursuer actions are considered to be control actions, whereas all external actions such as target maneuvers and measurement errors, are considered to be disturbances (Speyer & Jacobson). DA problems can either deal with perfect information patterns, where both pursuer and evader share perfect information regarding the full state-vector, or imperfect information formulation, where both players have access only to noisy measurements of a linear combination of the state-vector.

The present work revisits DA problems in the latter formulation where we introduce the equivalence between two main implementations of the DA control, one formulated by Speyer and Jacobson and the other by Green and Limebeer. Then, we perform detailed analyses for two simple pursuit-evasion cases which provide insight into the interplay between the control and the estimation parts of the pursuer strategy. We introduce and discuss a representative case study of a Simple Boat Guidance Problem (SBGP), with perfect and imperfect information patterns, with trajectory shaping element and without. We derive the optimal solution, and present some numerical results, for the SBGP using critical and non-critical values of the DA. The trade-off between the noise magnitude, DA value and trajectory shaping terms is shown. We then introduce another representative case study, concerning Missile Guidance Engagement (MGE). The qualitative and quantitative properties of the MGE solution, based on the critical and non-critical DA values, are studied by extensive numerical simulations. One main interesting result related to the MGE can be found in Speyer, 1976. A case study for MGE with jamming power was introduced. The Navigation Constant (connecting line-of-sight rate with the acceleration’s command) grows with the jamming magnitude (in fact, it grows when the cost imposed on the noise in the game is reduced). This result is somewhat counter-intuitive. For example, in another case study (Gutman) the Navigation Constant gets lower as the disturbances grow. The present work clarifies this result as well, where a comparison between the implementation of the critical DA and the non-critical DA values is done. It appears that working with the critical value yields more reasonable results and improves guidance performances. Lastly, the various factors that influence the choice of parameters for the optimal Trajectory Shaping matrix are introduced.