Spline-type solution to parameter-dependent LMIs

Parameter-dependent quadratic forms (PDQF) play an important role in analysis and synthesis of linear parameter varying (LPV) systems. This paper proposes a method to find a PDQF that meets some criterion written in terms of LMI guaranteeing performances of LPV systems. Through approximation of PDQF with spline functions, we derive finite number of LMI from a given LMI-criterion on a PDQF, which is inherently a condition with infinitely many inequalities. This approximating solution to LMI on a PDQF is proved to have the following properties: (1) any solution to the derived LMI of finite number always produces a PDQF satisfying the original LMI-criterion, and (2) the finite LMI-condition always holds with sufficiently fine division of the parameter´s region if the original one is solvable. Thus the results of this paper enable to solve parameter-dependent LMI associated with PDQF without conservatism