Summary: An estimator design problem is considered which involves both ${L}_{2}$ (least squares) and ${H}_{\infty}$ (worst-case frequency-domain) aspects. Specifically, the goal of the problem is to minimize an ${L}_{2}$ state- estimation error criterion subject to a prespecified ${H}_{\infty}$ constraint on the state-estimation error. The ${H}_{\infty}$ estimation- error constraint is embedded within the optimization process by replacing the covariance Lyapunov equation by a Riccati equation whose solution leads to an upper bound on the ${L}_{2}$ state-estimation error.

The principal result is a sufficient condition for characterizing fixed- order (i.e., full- and reduced-order) estimators with bounded ${L}_{2}$ and ${H}_{\infty}$ estimation error. The sufficient condition involves a system of modified Riccati equations coupled by an oblique projection, i.e., idempotent matrix. When the ${H}_{\infty}$ constraint is absent, the sufficient condition specializes to the ${L}_{2}$ state-estimation result given by the first author and *D. C. Hyland* [IEEE Trans. Autom. Control AC-30, 583-585 (1985; Zbl 0555.93056)].

##### MSC:

93E11 | Filtering in stochastic control |

46J15 | Banach algebras of differentiable or analytic functions, ${H}^{p}$-spaces |

15A24 | Matrix equations and identities |