Parametrization of optimal fault detection filters

This paper is concerned with the fault detection filter design problem for linear time-invariant systems subject to disturbance and possible faults. To make a tradeoff between fault sensitivity and disturbance sensitivity, frameworks such as ℋ−/ℋ∞ℋ−/ℋ∞, ℋ2/ℋ∞ℋ2/ℋ∞, and ℋ∞/ℋ∞ℋ∞/ℋ∞ are considered. It is shown that for ℋ−/ℋ∞ℋ−/ℋ∞ and ℋ∞/ℋ∞ℋ∞/ℋ∞ frameworks, all optimal fault detection filters can be characterized by an envelope besieged by the special solutions serving as upper and lower bounds. Furthermore, the explicit parametrization is given in terms of a free contractive system/parameter. This free parameter provides freedom to emphasize fault and disturbance sensitivities at different frequency ranges, which can be used to construct new optimal fault detection filter. The results are also extended to the optimization over finite frequency range, for which the upper and/or lower bound of the optimal solution set may not be rational. On the other hand, it is shown that the solution set of ℋ2/ℋ∞ℋ2/ℋ∞ problem is unique up to square inner system. Finally, examples are given to illustrate the results.