Fault detection filter design for linear systems

Abstract

This dissertation considers residual generation for robust fault detection of linear systems with control inputs, unknown disturbances and possible faults. First, multi-objective fault detection problems such as $\mathscr{H_-}/ \mathscr{H_\infty}$, $\mathscr{H}_2/\mathscr{H_\infty}$ and $\mathscr{H_\infty}/\mathscr{H_\infty}$ have been formulated for linear continuous time-varying systems (LCTVS) in time domain for finite horizon and infinite horizon case, respectively. It is shown that under mild assumptions, the optimal solution is an observer determined by solving a standard differential Riccati equation (DRE). The solution is also extended to the case when the initial state for the system is unknown. Second, the parallel problems are also solved for linear discrete time-varying systems in time domain. The solution is again an observer whose gain is determined by solving a standard recursive difference Riccati equation (DDRE). The solution is also extended to the case when the initial state for the system is unknown. Third, for the general case in which $G_d$ (the transfer matrix from disturbance to output) may be a tall or square transfer matrix, and $D_d$ may not have full column rank for linear discrete time invariant systems (LDTIS), the common $\mathscr{H_-}/ \mathscr{H_\infty}$, $\mathscr{H}_2/\mathscr{H_\infty}$ and $\mathscr{H_\infty}/\mathscr{H_\infty}$ frameworks are not applicable. Based on several novel definitions of norms over a certain subspace, we propose a new problem formulation with both disturbance decoupling and fault sensitivity optimization. It is shown that the solution is an observer determined by a generalized Riccati equation (or Riccati system, alternatively). To be more specific, with this filter, some faults in certain subspace can be completely decoupled from the residual signal, while the others are optimized in terms of fault sensitivity. Furthermore, the completely non-decoupling and decoupling conditions are given. Disturbance rejection based on the solution is discussed. A direct procedure for deriving the fault detection filter in transfer matrix form is also proposed. Finally, some potential further research problems are outlined.