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Steady-state Kalman filtering with an ${H}_{\infty }$ error bound. (English) Zbl 0684.93081

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