Integrated algorithm for solving H2-optimal fault detection and isolation problems

The design problem of fault detection and isolation filters can be formulated as a model matching problem and solved using an H2-norm minimization approach. A systematic procedure is proposed to choose appropriate filter specifications which guarantee the existence of proper and stable solutions of the model matching problem. This selection is integral part of a numerically reliable computational method to design of H2optimal fault detection filters. The proposed design approach is completely general, being applicable to both continuousand discrete-time systems, and can easily handle even unstable and/or improper systems. I. THE FAULT DETECTION AND ISOLATION PROBLEM Consider additive fault models described by input-output representations of the form y(λ) = Gu(λ)u(λ)+Gd(λ)d(λ)+Gw(λ)w(λ)+Gf (λ)f(λ), (1) where y(λ), u(λ), d(λ), w(λ), and f(λ) are Laplaceor Z-transformed vectors of the p-dimensional system output vector y(t), mu-dimensional control input vector u(t), mddimensional disturbance vector d(t), mw-dimensional noise vector w(t) and mf -dimensional fault vector f(t), respectively, and where Gu(λ), Gd(λ), Gw(λ) and Gf (λ) are the transfer-function matrices (TFMs) from the control inputs to outputs, disturbance inputs to outputs, noise inputs to outputs, and fault inputs to outputs, respectively. According to the system type, the frequency variable λ is either s, the complex variable in the Laplace-transform in the case of a continuous-time system or z, the complex variable in the Ztransform in the case of a discrete-time system. For most of practical applications, the TFMs Gu(λ), Gd(λ), Gw(λ) and Gf (λ) are proper rational matrices. However, for complete generality of our problem setting, we will allow that these TFMs are general non-proper rational matrices for which we will not a priori assume any further properties (e.g., stability, full rank). A linear residual generator (or fault detection filter) processes the measurable system outputs y(t) and control inputs u(t) and generates the residual signals r(t) which serve for decision making on the presence or absence of faults. The input-output form of this filter is