Remarks about numerical conditioning of discrete-time Riccati equations

The numerical conditioning of discrete-time Riccati equations is discussed. The sensitivity properties of these equations are evaluated by using a suitable defined relative condition number. It is demonstrated that for standard problems the δ-domain approach is much better conditioned then the conventional shift operator approach as the sampling period tends to zero. It is also observed that for some singular cases the condition numbers of both the δ-domain and q-domain Riccati equations are equivalent and the corresponding solutions have poor numerical behaviour. However, even for singular problems the δ-domain approach can lead to satisfactory numerically robust solutions for some kind of structured plant model perturbations. In conclusion, it is claimed that for regular plants the δ-domain method is always no worse than the corresponding shift-domain method and should be recommended, especially for standard non-singular problems. An numerical example concerning the numerical robustness of estimation in the H_∞ is given to illustrate