This paper is devoted to studying the issue of modeling, forecasting, and optimal feedback control of discrete-time first-order linear stochastic systems with prospective strong intervention (PSI). First, one type of PSI is formulated. Then, one class of induced mechanisms of PSI is characterized, and, with respect to the induced mechanisms, two multi-step forecasting methods for estimating the occurrence time and terminal time of PSI are discussed. Based on these fundamental results, the problem of optimal control with respect to an improved minimum variance performance index is discussed. Finally, the effectiveness and advantages of the proposed control strategy and approach are verified and demonstrated through numerical simulations.

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The Delta-modulated feedback control of a linear system introduces nonlinearity into the system through switchings between two input values. It has been found that Delta-modulation gives rise to periodic orbits. The existence of periodic points of all orders of Sigma-Delta modulation with "leaky" integration is completely characterized by some interesting groups of polynomials with "sign" coefficients. The results are naturally generalized to Sigma-Delta modulations with multiple delays, Delta-modulations in the "downlink", unbalanced Delta-modulations and systems with two-level quantized feedback. Further extensions relate to the existence of periodic points arising from Delta-modulated feedback control of a stable linear system in an arbitrary direction, for which some necessary and sufficient conditions are given.