From STP to game-based control

Daizhan Cheng, Hongsheng Qi, Zequn Liu
2017 Science China Information Sciences  
This paper provides a comprehensive survey on semi-tensor product (STP) of matrices and its applications to different disciplines. First of all, the STP and its basic properties are introduced. Meanwhile, its inside physical meaning is explained. Second, its application to conventional dynamic systems is presented. As an example, the region of attraction of stable equilibriums is discussed. Third, its application to logical systems is presented. Particularly, the algebraic state space
more » ... tion of logical systems and the important role it plays in analysis and control of logical systems are emphasized. Fourth, its application to finite games is discussed. The most interesting problems include potential game, evolutionary game, and game theoretic control. Finally, the mathematical essence of STP is briefly introduced. (7) Given a (C ω ) manifold M , the set of C ω functions is C ω (M ); the set of vector fields is V ω (M ); the set of co-vector fields is V * ω (M ). Geometric inside of STP After unsuccessful submissions for several years, the first paper on STP of matrices was published in 2001 [1]. Since then, the criticism, questioning or doubts on STP had been lasted for years. The most frequently asked question is: is it reasonable to generalize the conventional matrix product to two arbitrary matrices? To answer this question we first give a motivation to reveal the insight structure of STP. Consider a linear function f ∈ L(R n , R), where L(R n , R) is the set of linear functions from R n to R. Then there exists a row vector a = (a 1 , . . . , a n ) such that
doi:10.1007/s11432-017-9265-2 fatcat:vckb3ifiavh4rjriq4vn4w4num