Characterization of ℋ−ℋ− index for linear time-varying systems

The increasing interest in ℋ−ℋ− index in fault detection has led to some recent development in time domain investigation. This paper at first presents a necessary and sufficient condition of the ℋ−ℋ− index in finite time horizon for linear time-varying systems. It is shown that the feasibility of the ℋ−ℋ− index, which is given when the ℋ−ℋ− index is greater than a predefined value, is equivalent to the existence of a certain backward differential Riccati equation under a constraint. The square system as a special case is discussed. The corresponding results are extended to systems with unknown initial condition with a modified definition of the ℋ−ℋ− index. The condition for weighted index to emphasize the signal effect at different time instant is also derived by transformation. The result is also compared with the famous bounded real lemma. The connection with the linear time-invariant (LTI) case is also discussed. Finally, examples are given to illustrate the main results.