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Feedback representations of critical controls for well‐posed linear systems
Author(s) -
Staffans Olof J.
Publication year - 1998
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/(sici)1099-1239(19981215)8:14<1189::aid-rnc385>3.0.co;2-4
Subject(s) - mathematics , operator (biology) , factorization , state (computer science) , riccati equation , linear system , subspace topology , pure mathematics , function (biology) , mathematical analysis , algorithm , partial differential equation , biochemistry , chemistry , repressor , evolutionary biology , biology , transcription factor , gene
This is the first part in a three part study of the suboptimal full information H ∞ problem for a well‐posed linear system with input space U , state space H , and output space Y . We define a cost function Q ( x 0 , u )=∫ R +〈 y ( s ), Jy ( s )〉 Y d s , where y ∈ L 2 loc ( R + ; Y ) is the output of the system with initial state x 0 ∈ H and control u ∈ L 2 loc ( R + ; U ), and J is a self‐adjoint operator on Y . The cost function Q is quadratic in x 0 and u , and we suppose (in the stable case) that the second derivative of Q ( x 0 , u ) with respect to u is non‐singular. This implies that, for each x 0 ∈ H , there is a unique critical control u crit such that the derivative of Q ( x 0 , u ) with respect to u vanishes at u = u crit . We show that u crit can be written in feedback form whenever the input/output map of the system has a coprime factorization with a ( J, S )‐inner numerator; here S is a particular self‐adjoint operator on U . A number of properties of this feedback representation are established, such as the equivalence of the ( J, S )‐losslessness of the factorization and the positivity of the Riccati operator on the reachable subspace. © 1998 John Wiley & Sons, Ltd.