Full column rank preservers that preserve semipositivity of matrices

Left invertibility preservers on Mm,n(ℝ), m ≥ n, that preserve either semipositivity of matrices or the subset of minimally semipositive matrices are studied. We prove that such maps cannot be degenerate. We also highlight the structure of nonsingular subspaces of dimension 2 in M2(ℝ).

On the observer matching condition and unknown input observer design based on the system left-invertibility concept

This paper considers the problems of unknown input observer (UIO) designs when the so-called observer matching condition (OMC) is not satisfied. Firstly, the system left-invertibility (SLI) concept is investigated in detail and some new criteria are given. Secondly, based on the SLI concept, an augmented output vector formed partly by the original measurable output and partly by the auxiliary output is obtained. By this constructed output vector, the new system is allowed to be transformed into an equivalent linear dynamic system which contains no unknown input, and thus a simple Luenberger observer can be constructed to simultaneously estimate the original states and unknown inputs. Compared with the existing results, the present UIO design can be taken under the single strong detectability condition and the OMC is not needed any more. Finally, the simulations of a single-link flexible joint robotic model and a linearized vertical take-off and landing helicopter model are given to show the effectiveness of our method and its superiority over existing ones.

An analysis of the Rüschendorf transform - with a view towards Sklar’s Theorem

We revisit Sklar’s Theorem and give another proof, primarily based on the use of right quantile functions. To this end we slightly generalise the distributional transform approach of Rüschendorf and facilitate some new results including a rigorous characterisation of an almost surely existing “left-invertibility” of distribution functions.