TY - GEN

T1 - The Inverse Problem of Linear-Quadratic Differential Games

T2 - 58th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2022

AU - Huang, Yunhan

AU - Zhang, Tao

AU - Zhu, Quanyan

N1 - Publisher Copyright:
© 2022 IEEE.

PY - 2022

Y1 - 2022

N2 - This paper aims to formulate and study the inverse problem of non-cooperative linear quadratic games: Given a profile of control strategies, find cost parameters for which this profile of control strategies is Nash. We formulate the problem as a leader-followers problem, where a leader aims to implant a desired profile of control strategies among selfish players. In this paper, we leverage frequency-domain techniques to develop a necessary and sufficient condition on the existence of cost parameters for a given profile of stabilizing control strategies to be Nash under a given linear system. The necessary and sufficient condition includes the circle criterion for each player and a rank condition related to the transfer function of each player. The condition provides an analytical method to check the existence of such cost parameters, while previous studies need to solve a convex feasibility problem numerically to answer the same question. We develop an identity in frequency-domain representation to characterize the cost parameters, which we refer to as the Kalman equation. The Kalman equation reduces redundancy in the time-domain analysis that involves solving a convex feasibility problem. Using the Kalman equation, we also show the leader can enforce the same Nash profile by applying penalties on the shared state instead of penalizing the player for other players' actions to avoid the impression of unfairness.

AB - This paper aims to formulate and study the inverse problem of non-cooperative linear quadratic games: Given a profile of control strategies, find cost parameters for which this profile of control strategies is Nash. We formulate the problem as a leader-followers problem, where a leader aims to implant a desired profile of control strategies among selfish players. In this paper, we leverage frequency-domain techniques to develop a necessary and sufficient condition on the existence of cost parameters for a given profile of stabilizing control strategies to be Nash under a given linear system. The necessary and sufficient condition includes the circle criterion for each player and a rank condition related to the transfer function of each player. The condition provides an analytical method to check the existence of such cost parameters, while previous studies need to solve a convex feasibility problem numerically to answer the same question. We develop an identity in frequency-domain representation to characterize the cost parameters, which we refer to as the Kalman equation. The Kalman equation reduces redundancy in the time-domain analysis that involves solving a convex feasibility problem. Using the Kalman equation, we also show the leader can enforce the same Nash profile by applying penalties on the shared state instead of penalizing the player for other players' actions to avoid the impression of unfairness.

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U2 - 10.1109/Allerton49937.2022.9929408

DO - 10.1109/Allerton49937.2022.9929408

M3 - Conference contribution

AN - SCOPUS:85142658497

T3 - 2022 58th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2022

BT - 2022 58th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2022

PB - Institute of Electrical and Electronics Engineers Inc.

Y2 - 27 September 2022 through 30 September 2022

ER -