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SUMMARYThis paper presents an integral sliding mode technique robustifying the optimal controller for linear stochastic systems with input and observation delays, which is based on integral sliding mode compensation of disturbances. The general principles of the integral sliding mode compensator design are modified to yield the basic control algorithm oriented to time-delay systems, which is then applied to robustify the optimal controller. As a result, two integral sliding mode control compensators are designed to suppress disturbances in state and observation equations, respectively, from the initial time moment. Moreover, it is shown that if certain matching conditions hold, the designed compensator in the state equation can simultaneously suppress observation disturbances, as well as the designed compensator in the observation equation can simultaneously suppress state disturbances. The obtained robust control algorithm is verified by simulations in the illustrative example, where the compensator in the observation equation provides simultaneous suppression of state and observation disturbances.

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In the last few years, the efforts to elucidate the mechanisms underlying the segmentation clock in various vertebrate species have multiplied. Early evidence suggested that oscillations are caused by one of the genes under the Notch signalling pathway (like those of the her or Hes families). Recently, Aulehla et al. [Wnt3a plays a major role in the segmentation clock controlling somitogenesis. Dev. Cell 4, 395-406] discovered that Axin2 (a gene under the Wnt3a signalling pathway) also oscillates in the presomitic mesoderm (PSM) of mice embryos and proposed some mechanisms through which the Notch and Wnt3a pathways may interact. They further suggested that a decreasing concentration of Wnt3a along the PSM may be the gradient the segmentation clock interacts with to form somites. These results were reviewed by Rida et al. [A notch feeling of somite segmentation and beyond. Dev. Biol. 265, 2-22], who introduced a complex clockwork comprising genes Hes1, Lfng (under the Notch pathway), and Axin2, as well as their multiple interactions. In the present work we develop a mathematical model based on the Rida et al. review and use it to tackle some of the questions raided by the Aulehla et al. paper: can the Axin2 feedback loop constitute a clock? Could a decreasing Wnt3a signaling constitute the wavefront, where phase is recorded and the spatial pattern laid down? What is the master oscillator?