Robust L∞-induced filtering and deconvolution of a wide class of linear discrete-time stochastic systems

• The problems of filtering and deconvolution are addressed for discrete-time linear systems with deterministic and stochastic uncertainties.
• A new lemma is derived which provides the sufficient conditions in terms of LMIs for the guarantee of an upper bound on the induced L∞L∞ norm of this class of systems.
• Two induced L∞L∞ estimator design methods have been proposed based on quadratic and parameter-dependent stability ideas.

The problems of stationary robust L∞L∞-induced filtering and deconvolution are addressed for discrete-time linear systems with deterministic and stochastic uncertainties in the state–space model. Stochastic uncertainties are in the form of state- and input-dependent multiplicative white noises which appear in both state and the measurement equations. The deterministic part of the system matrices and covariance matrices of the stochastic parameters is unknown and resides in a given polytopic-type domain. For this system a new lemma is derived which characterizes the induced L∞L∞ norm disturbance attenuation performance by linear matrix inequalities (LMIs). According to this lemma, the problem of estimator design is solved for stochastic uncertain systems based on the notion of quadratic stability. To further reduce the overdesign in the quadratic framework, this paper also proposes a parameter-dependent design procedure, which is much less conservative than the quadratic approach. The proposed estimators guarantee the mean-square exponential stability of estimation error dynamics and satisfy the prescribed induced L∞L∞ performance index. Two examples are used to demonstrate the proposed methods and their effectiveness.