General Matrix Pencil Techniques for Solving Discrete-Time Nonsymmetric Algebraic Riccati Equations

This technical note revisits the open-loop Stackelberg strategy for a two-player game. By introducing a new costate, which captures the future information of the control input, we present a necessary and sufficient condition for the existence and uniqueness of the two-player game. The optimal strategy is designed in terms of three decoupled and symmetric Riccati equations which improves the existing results greatly on computation.Index Terms-Leader-follower, necessary and sufficient condition, Riccati equation, Stackelberg strategy. I. INTRODUCTIONGame theory is of great significance in numerous fields such as economic, social science, political science and biology. Considerable attention has been devoted to the differential/difference game in [1], [3]-[5], [10], [21] and references therein. In general, the game is divided into zero-sum game [4] and nonzero-sum game [21] based on the difference of the performance criterions. According to the roles of the players, the strategies can be classified into Nash, minmax, noninferior, Stackelberg strategy, etc. [1] and [3] study the necessary conditions for the Nash game in terms of two coupled and nonsymmetric Riccati equations. The strategies of minmax and noninferior set are discussed in [21] by the value function approach. For the Stackelberg game, [18] presents an implicit necessary and sufficient condition for the unique existence of the solution based on the operator technique. It is shown that the positive definiteness of the weighting matrices in the performance is sufficient to ensure the uniqueness of the Stackelberg strategy. With the convex condition, [6] studies the properties of the solution to the corresponding algebraic Riccati equations. Reference [8] proposes a new sufficient existence condition for the open-loop equilibrium by adopting a Lyapunov-type approach. Both the finite-horizon and infinite-horizon linear-quadratic Stackelberg game including time preference rates are investigated in [11] via the matrix block technique. It should be noted that the aforementioned results are based on solving three coupled and nonsymmetric Riccati equations, under the stronger assumption that the weighting matrices of the cost functions are positive-definite.In this technical note, we revisit the Stackelberg strategy for a twoplayer game in discrete-time dynamic setting under a mild assumption. Our main contribution is the introduction of an appropriate costate which is defined as a vector capturing the effects of the future inputs.

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“…which implies that (7) holds at time s − 1 with ζ 1 s−1 and P 1 s satisfying (12) and (13). The proof is completed.…”

Section: A Optimization For the Followermentioning

confidence: 67%

“…. , N where P 1 k+1 , P 2 k+1 are the solutions to the equations (13) and (22), respectively. In this case, the unique open-loop Stackelberg strategy is given by…”

Section: Resultsmentioning

confidence: 99%

Necessary and Sufficient Condition for Two-Player Stackelberg Strategy

Zhang

Chai

2015

IEEE Trans. Automat. Contr.

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“…If the matrix A is not invertible, the result of theorem (6) can be extended in the framework of deflating subspaces of the pencil (L, G) (see [22]). Notice also that the controls u * i defined by (25) can be rewritten as…”

Section: Remarkmentioning

confidence: 99%

A gametheoretic approach for non-uniform pole shifting and pole homothety

Jungers

Abou-Kandil

Castelan³

et al. 2013

Automatica

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This paper investigates the problems of non-uniform pole shifting or pole homothety in multivariable linear systems by using a gametheoretic approach and more specifically a particular Nash strategy with an open-loop information structure for a game including a time preference rate in the quadratic criteria. Such a result is possible due to properties of non-symmetric Algebraic Riccati equations associated with an open-loop Nash strategy. Numerical examples allow to illustrate the efficiency of the proposed approach.

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Feedback strategies for discrete-time linear-quadratic two-player descriptor games

Jungers¹

2014

Linear Algebra and its Applications

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International audienceThis paper deals with discrete-time linear-quadratic descriptor two-player games. The two frameworks of Nash and Stackelberg strategies with feedback information structure are considered. The aim is to provide sufficient conditions ensuring, for any initial state, the existence and the uniqueness of a tri-trajectory, gathering the state and the controls of the two players, which reaches the equilibrium. The provided results are mainly based on a discrete-time matrix block formulation. The provided algorithms consist in an iterative backward in time procedure, issued from Dynamic Programming. Numerical examples illustrate this approach and the particular case of explicit systems is recovered for both kinds of strategies

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General Matrix Pencil Techniques for Solving Discrete-Time Nonsymmetric Algebraic Riccati Equations

Cited by 5 publications

References 41 publications

Necessary and Sufficient Condition for Two-Player Stackelberg Strategy

Necessary and Sufficient Condition for Two-Player Stackelberg Strategy

A gametheoretic approach for non-uniform pole shifting and pole homothety

Feedback strategies for discrete-time linear-quadratic two-player descriptor games

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