LMI solutions to the mixed ℋ−/ℋ∞fault detection observer design for linear parameter‐varying systems

This paper addresses the mixed ℋ︁−/ℋ︁∞fault detection observer design issue for a class of linear parameter‐varying (LPV) systems. Analogous to the definition of the quadratic ℋ︁∞performance for LPV systems and the ℋ︁−index for linear time invariant (LTI) systems, the quadratic ℋ︁−index and the affine quadratic ℋ︁−index for LPV systems are defined in terms of linear matrix inequalities (LMIs). The first algorithm for designing the mixed ℋ︁−/H∞observer is proposed, which aims at minimizing the quadratic ℋ︁∞performance and maximizing the quadratic ℋ︁−index of the observer error dynamic systems. To reduce the conservativeness of this algorithm, the affine quadratic ℋ︁∞performance and the affine ℋ︁−index for LPV systems are utilized. The robustness conditions and affine ℋ︁−index conditions for the underlying observer optimization issue are formulated as parameter‐dependent LMIs. The Gridding technique and multi‐convexity concept are applied, respectively, for reducing the parameter‐dependent LMIs to finite LMI constraints. Correspondingly, two iterative algorithms are proposed. Furthermore, the threshold design and the estimation of the worst undetectable fault size are investigated. An example is studied to demonstrate the effectiveness of the proposed algorithms.