This paper proposes a new cost criterion to enhance the precision of a zonotopic state estimator for discrete-time descriptor linear systems. Originally, the algorithm solves a minimum-trace problem involving zonotopes, whose evolution is given by an interval observer structure containing extra design matrices, called degrees of freedom. Although the minimization of trace yields explicit solutions, it does not necessarily imply minimization of volume, and thereby, the precision of the output zonotope cannot be improved effectively. The volume measure for zonotopes is computationally expensive and, when used as cost criterion, implies nonlinear optimization problems. Motivated by such issues, we here propose a minimum-radius criterion where the smallest box enclosing the output zonotope is minimized. The resulting optimization problem is nonlinear, but its convexity is exploited to yield an equivalent linear program. The effectiveness of our approach is illustrated over two numerical examples.

This paper presents two novel mixed-uncertainty state estimators for discrete-time descriptor linear systems, namely linear time-varying mixed-uncertainty filter (LTVMF) and linear time-invariant mixed-uncertainty filter (LTIMF). The former estimator is based on the minimum-variance approach, from which quadratic and explicit formulations are derived and addressed to LTI and LTV systems. Both formulations incorporate the knowledge of state linear constraints, such as equality (in the descriptor form) and inequality, to mitigate precision and accuracy issues related to initialization and evolution of the state estimates. The explicit version is developed to reduce the computational burden of quadratic solvers, which is based on a particularity of the state inequality constraints. The LTIMF algorithm is based on the mixed H 2 / H ∞ criterion motivated by performing low-cost computations. This speed benefit is originated from a reachability analysis involving constant design matrices. Both LTVMF and LTIMF algorithms solve state-estimation problems in which the uncertainties are combined to yield the so-called mixed-uncertainty vector, which is composed by set-bounded uncertainties, characterized by constrained zonotopes, and stochastic uncertainties, characterized by Gaussian random vectors. As mixed-uncertainty vectors imply biobjective optimization problems, we innovatively present multiobjective arguments to justify the choice of the solution on the Pareto-optimal front. According to these arguments, LTVMF is introduced with a cost normalization, which enables the combination of beyond minimum-variance approaches. Likewise, the mixed H 2 / H ∞ criterion of LTIMF is introduced with slack variables to improve the quality of the state estimates. In order to discuss the advantages and drawbacks, the state estimators are tested in two numerical examples.