Central suboptimal H ∞ filter design for linear time-varying systems with state and measurement delays

Abstract

This article presents the central finite-dimensional H ∞ filters for linear systems with state and measurement delay that are suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the results previously obtained for linear time delay systems, this article reduces the original H ∞ filtering problem to H 2 (optimal mean-square) filtering problem using the technique proposed in Doyle, Glover, Khargonekar, and Francis (1989 ‘State-space Solutions to Standard H 2 and H ∞ Control Problems’, IEEE Transactions on Automatic Control, 34, 831–847). Application of the reduction technique becomes possible, since the optimal closed-form filtering equations solving the H 2 (mean-square) filtering problem have been obtained for linear systems with state and measurement delays. This article first presents the central suboptimal H ∞ filter for linear systems with state and measurement delays, based on the optimal H 2 filter from Basin, Alcorta-Garcia, and Rodriguez-Gonzalez (2005, ‘Optimal Filtering for Linear Systems with State and Observation Delays’, International Journal of Robust and Nonlinear Control, 15, 859–871), which consists, in the general case, of an infinite set of differential equations. Then, the finite-dimensional central suboptimal H ∞ filter is designed in case of linear systems with commensurable state and measurement delays, which contains a finite number of equations for any fixed filtering horizon; however, this number still grows unboundedly as time goes to infinity. To overcome that difficulty, the alternative central suboptimal H ∞ filter is designed for linear systems with state and measurement delays, which is based on the alternative optimal H 2 filter from Basin, Perez, and Martinez-Zuniga (2006, ‘Alternative Optimal Filter for Linear State Delay Systmes’, International Journal of Adaptive Control and Signal Processing, 20, 509–517). In all cases, the standard H ∞ filtering conditions of stabilisability, detectability and noise orthonormality are assumed. Finally, to relax the standard conditions, this article presents the generalised versions of the designed H ∞ filters in the absence of the noise orthonormality. The proposed H ∞ filtering algorithms provide direct methods to calculate the minimum achievable values of the threshold γ, based on the existence properties for a bounded solution of the gain matrix equation. Numerical simulations are conducted to verify the performance of the designed central suboptimal filters for linear systems with state and measurement delays against the central suboptimal H ∞ filter available for linear systems without delays. The simulation results show a definite advantage in the values of the noise-output transfer function H ∞ norms in favour of the designed filters.