Asymptotic Stackelberg Optimal Control Design for an Uncertain Euler Lagrange System M. Johnson, T. Hiramatsu, N. Fitz-Coy, and W. E. Dixon Abstract— Game theory methods have advanced various dis- ciplines from social science, notably economics and biology, and engineering. Game theory establishes an optimal strategy for multiple players in either a cooperative or noncooperative manner where the objective is to reach an equilibrium state among the players. A Stackelberg game strategy involves a leader and a follower that follow a hierarchy relationship where the leader enforces its strategy on the follower. In this paper, a general framework is developed for feedback control of an Euler Lagrange system using an open-loop Stackelberg differential game. A Robust Integral Sign of the Error (RISE) controller is used to cancel uncertain nonlinearities in the system and a Stackelberg optimal controller is used for stabilization in the presence of uncertainty. A Lyapunov analysis is provided to examine the stability of the developed controller. I. I NTRODUCTION 1 Noncooperative game theory has been applied to a variety of control problems [1]–[14]. While zero sum differential games have been heavily exploited in nonlinear ∞ control theory, non-zero sum differential games have had limited application in feedback control. In particular, the Stackelberg differential game which is based on a hierarchal relationship that exists between the players of the game, has been utilized in a decentralized control system [5], hierarchal control prob- lems [3], [4], [13], and nonclassical control problems [6]. This paper is focused on the leader and a follower two player Stackelberg game. The leader is able to enforce a strategy on the follower and the leader knows how the follower will rationally react; however, the follower does not know the leader’s rational reaction. The strategy space for the game is the set of all available information for the players to make their decisions. The players are committed to following a predeter- mined strategy based on their knowledge of the initial state, the system model and the cost functional to be minimized. In this open-loop game formulation each player has access to state measurements and can adapt the respective strategy as a function of the system’s evolution. Previous research of the open-loop Stackelberg game in control theory is mainly focused on deriving an analytic solution and providing gain constraints; however these results are limited to linear systems and dont demonstrate stabilization properties of the control law. The main contribution of this work is the development of an optimal open-loop Stackelberg-based feedback control law, in conjunction with a robust feedback control law, that 1 This project was supported by the National Aeronautics and Space Administration through the University of Central Florida’s Space Grant Consortium. M. Johnson, T. Hiramatsu, N. Fitz-Coy, and W. E. Dixon are with the Dept. of Mechanical and Aerospace Engineering, University of Florida, Gainesville, Florida 32611,{marc1518,takashi,nfc,wdixon}@ufl.edu. is shown to stabilize a nonlinear system with exogenous disturbances. In addition, the control law is shown to account for unknown state varying and bounded disturbances. To formulate the Stackelberg game for a nonlinear Euler Lagrange system, an implicit learning robust controller is used to cancel nonlinearities in the system. The control development is based on the continuous Robust Integral of the Sign of the Error (RISE) [15]–[17] technique. The RISE architecture is adopted since this method can accommodate for 2 disturbances and yield an asymptotic result. This paper investigates the development of the RISE controller in conjunction with an optimal Stackelberg feedback controller for an Euler Lagrange system with additive disturbances. The formulation of the Stackelberg game derives a structured uncertainty as one of the players; however, this paper also considers an additional bounded unstructured uncertainty that cannot be accounted for by the Stackelberg feedback. An optimal feedback controller based on a open-loop Stackelberg game is shown to minimize a cost functional in the presence of structured uncertainty, while a Lyapunov-based asymptotic tracking result is obtained through the amalgam of the RISE and Stackelberg feedback methods. II. DYNAMIC MODEL AND PROPERTIES The class of nonlinear dynamic systems considered in this paper is assumed to be modeled by the following Euler- Lagrange [18] formulation: ()¨ + ( ˙ ) ˙ + ()+ ( ˙ )+ + = () (1) In (1), () ∈ R × denotes the generalized inertia matrix, ( ˙ ) ∈ R × denotes the generalized centripetal-Coriolis matrix, () ∈ R denotes the generalized gravity vector, ( ˙ ) ∈ R denotes the generalized friction vector, ∈ R denotes a general uncertain disturbance with a structure de- rived from the formulation of the Stackelberg game, () ∈ R is a general bounded unstructured disturbance, () ∈ R represents the input control vector, and (), ˙ (), ¨ () ∈ R denote the generalized position, velocity, and acceleration vectors, respectively. The subsequent development is based on the assumption that () and ˙ () are measurable, and (), ( ˙ ), (), ( ˙ ), and are unknown. Moreover, the following properties and assumptions will be exploited in the subsequent development. Assumption 1: The inertia matrix () is symmetric, posi- tive definite, and satisfies the following inequality ∀() ∈ R : 1 kk 2 ≤ () ≤ ¯ () k k 2 (2)