A State-Space based Interval Estimator for LTI Systems under $H_{1}$ and/or $H_{\infty}$ Constraints

This paper proposes a interval filter, for linear time-invariant systems. The goal is to provide upper and lower bounds of a linear combination of the system states, in a guaranteed manner, so that the influence of unknown inputs on the envelop size, is minimised in the
$H_{1}\text{and}/\text{or}H_{\infty}$
-norm sense. A key feature of the proposed interval filter is that it consists of a general state-space realization, so the filter can be of higher order than the system's model. This is also a major difference compared to current existing interval approaches that are based on observer structures. Using the Lyapunov theory, it is shown that the design of the filter state space matrices can be formulated as solving a nonlinear semi-definite programming (NL-SDP) problem. The feasibility of the proposed approach is demonstrated on an academic example.