THE LINEAR QUADRATIC DYNAMIC GAME FOR DISCRETE-TIME DESCRIPTOR SYSTEMS

2003 ◽  
Vol 05 (04) ◽  
pp. 361-374
Author(s):  
HUA XU ◽  
HIROAKI MUKAIDANI
Keyword(s):  
State Space ◽  
Discrete Time ◽  
Dynamic Game ◽  
Descriptor Systems ◽  
Linear Feedback ◽  
Space System ◽  
Linear Quadratic ◽  
Reduced Order ◽  
State Space System ◽  
Zero Sum

The linear quadratic zero-sum dynamic game for discrete time descriptor systems is considered. A method, which involves solving a linear quadratic zero-sum dynamic game for a reduced-order discrete time state space system, is developed to find the linear feedback saddle-point solutions of the problem. Checkable conditions, which are described in terms of two dual algebraic Riccati equations and a Hamiltonian matrix, are given such that the linear quadratic zero-sum dynamic game for the reduced-order discrete time state space system is available. Sufficient conditions for the existence of the solutions are obtained. In contrast with the dynamic game in state space systems, the dynamic game in descriptor systems admits uncountably many linear feedback saddle-point solutions. All these solutions have the same existence conditions and achieve the same value of the dynamic game.

scholarly journals Stability Analysis of the Bat Algorithm Described as a Stochastic Discrete-Time State-Space System

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Janusz Piotr Paplinski
Keyword(s):  
Stability Analysis ◽  
State Space ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Bat Algorithm ◽  
Discrete Time System ◽  
Space System ◽  
State Matrix ◽  
State Space System

The main problem with the soft-computing algorithms is a determination of their parameters. The tuning rules are very general and need experiments during a trial and error method. The equations describing the bat algorithm have the form of difference equations, and the algorithm can be treated as a stochastic discrete-time system. The behaviour of this system depends on its dynamic and preservation stability conditions. The paper presents the stability analysis of the bat algorithm described as a stochastic discrete-time state-space system. The observability and controllability analyses were made in order to verify the correctness of the model describing the dynamic of BA. Sufficient conditions for stability are derived based on the Lyapunov stability theory. They indicate the recommended areas of the location of the parameters. The analysis of the position of eigenvalues of the state matrix shows how the different values of parameters affect the behaviour of the algorithm. They indicate the recommended area of the location of the parameters. Simulation results confirm the theory-based analysis.

State-Feedback Zero-Sum Dynamic Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Discrete Time ◽  
Information Structure ◽  
Dynamic Games ◽  
State Feedback ◽  
Linear Quadratic ◽  
Feedback Information ◽  
Time Dynamic ◽  
Finite State ◽  
Solution Methods ◽  
Zero Sum

This chapter focuses on the computation of the saddle-point equilibrium of a zero-sum discrete time dynamic game in a state-feedback policy. It begins by considering solution methods for two-player zero sum dynamic games in discrete time, assuming a finite horizon stage-additive cost that Player 1 wants to minimize and Player 2 wants to maximize, and taking into account a state feedback information structure. The discussion then turns to discrete time dynamic programming, the use of MATLAB to solve zero-sum games with finite state spaces and finite action spaces, and discrete time linear quadratic dynamic games. The chapter concludes with a practice exercise that requires computing the cost-to-go for each state of the tic-tac-toe game, and the corresponding solution.

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