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Steady-state Kalman filtering with an $H\sb{\infty}$ error bound. (English) Zbl 0684.93081
Summary: An estimator design problem is considered which involves both $L\sb 2$ (least squares) and $H\sb{\infty}$ (worst-case frequency-domain) aspects. Specifically, the goal of the problem is to minimize an $L\sb 2$ state- estimation error criterion subject to a prespecified $H\sb{\infty}$ constraint on the state-estimation error. The $H\sb{\infty}$ 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 $L\sb 2$ state-estimation error. The principal result is a sufficient condition for characterizing fixed- order (i.e., full- and reduced-order) estimators with bounded $L\sb 2$ and $H\sb{\infty}$ estimation error. The sufficient condition involves a system of modified Riccati equations coupled by an oblique projection, i.e., idempotent matrix. When the $H\sb{\infty}$ constraint is absent, the sufficient condition specializes to the $L\sb 2$ state-estimation result given by the first author and {\it D. C. Hyland} [IEEE Trans. Autom. Control AC-30, 583-585 (1985; Zbl 0555.93056)].

93E11Filtering in stochastic control
46J15Banach algebras of differentiable or analytic functions, $H^p$-spaces
15A24Matrix equations and identities
Full Text: DOI
[1] Berstein, D. S.; Haddad, W. M.: LQG control with an H$\infty $performance bound: A Riccati equation approach. Proc. amer. Control conf., 796-802 (June 1988)
[2] Bernstein, D. S.; Hyland, D. C.: The optimal projection equations for reduced-order state estimation. IEEE trans. Automat. control 30, 583-585 (1985) · Zbl 0555.93056
[3] Brockett, R. W.: Finite dimensional linear systems. (1970) · Zbl 0216.27401
[4] Doyle, J. C.; Glover, K.; Khargonekar, P. P.; Francis, B. A.: State-space solutions to standard H2 and H$\infty $control problems. Proc. amer. Control. conf., 1691-1696 (June 1988)
[5] Khargonekar, P. P.; Petersen, I. R.; Rotea, M. A.: H\infty-optimal control with state-feedback. IEEE trans. Automat. control 33, 786-788 (1988) · Zbl 0655.93026
[6] Petersen, I. R.: Disturbance attenuation and H$\infty $optimization: A design method based on the algebraic Riccati equation. IEEE trans. Automat. control 32, 427-429 (1987) · Zbl 0626.93063
[7] Rao, C. R.; Mitra, S. K.: Generalized inverse of matrices and its applications. (1971) · Zbl 0236.15004
[8] Willems, J. C.: Least squares stationary optimal control and the algebraic Riccati equation. IEEE trans. Automat. control 16, 621-634 (1971)
[9] Wonham, W. M.: Linear multivariable control: A geometric approach. (1979) · Zbl 0424.93001