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In this paper, we show that a minimal state space realization in Jordan canonical form for linear multivariable continuous‐time systems described by rational transfer function matrices could be obtained in a natural and basic way by using the concept of Nerode equivalence. While the minimal state space realization is known, the contribution of this note is to provide an alternative realization procedure which is directly introduced in a simple and self‐contained manner. Both scalar and multivariable cases in the continuous‐time setting are discussed. The presentation also provides a concrete construction of rational vector function with preassigned poles to interpolate any 2‐norm bounded analytical vector function in the open left‐half of the complex plane. The basic idea of Nerode equivalence is that the state can be identified with a corresponding equivalence class of inputs. For a linear finite dimensional time‐invariant continuous‐time system, the zero state is identified with the kernel of certain Hankel operator. This characterization via Nerode equivalence class sheds light on the construction of state for general nonlinear input–output systems. [ABSTRACT FROM AUTHOR]

Nanoscale zero-valent iron (nZVI) shows excellent reduction of Cr(Ⅵ), but the passivation on its outer surface can restrict its longevity and performance. To tackle this problem, this work introduced Shewanella oneidensis MR-1, a dissimilatory iron-reducing bacterium, into the chemical reduction system of aged nZVI/biochar (B) and Cr(Ⅵ). The potential synergistic effect of Cr(Ⅵ) reduction of aged nZVI/B and MR-1 was systematically investigated under varying conditions. The results indicated that aged nZVI/B and MR-1 exhibited a synergistic effect at a pH of 7, and the removal rate of Cr(Ⅵ) increased by 51.3%. Further research showed that the synergistic effect could be attenuated with the increase in the initial Cr(Ⅵ) concentration and enhanced with the increase in the MR-1 concentration. The XPS spectra confirmed that Cr(Ⅵ) was mainly removed through reduction. The dissimilatory iron-reducing ability of MR-1 played a key role in enhancing the Cr(Ⅵ) reduction. The reductive dissolution of the oxidation layers not only released reactive sites inside the nZVI, but also reduced Cr(Ⅵ) by producing ferrous ions. Moreover, B promoted the reduction by dispersing the nZVI and mediating the extracellular electron transfer. This study provides a new insight into solving the passivation problem of the long-term application of nZVI for Cr(Ⅵ) removal, which is considered a promising solution for synergistically improving the performance of nZVI in environmental remediation.