In this paper, we discuss various topological and metrical aspects of the set of stabilizing static feedback gains for multiple-input-multiple-output (MIMO) linear-time-invariant (LTI) systems, in both continuous and discrete-time. Recently, connectivity properties of this set (for continuous time) have been reported in the literature, along with a discussion on how this connectivity is affected by restricting the feedback gain to linear subspaces. We show that analogous to the continuous-time case, one can construct instances where the set of stabilizing feedback gains for discrete time LTI systems has exponentially many connected components.

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The potential impacts of data integrity attacks on multi-settlement electricity markets have been recently investigated and have sent a strong message to power grids independent system operators (ISOs) that adversaries could launch profitable cyber attacks by casting an incorrect image of transmission lines congestion pattern. However, these cautionary messages may be underestimated due to the adversaries unrealistic requirements (e.g. having access to real-time measurements) to launch a successful stealthy and profitable attack. This study examines the potential of the aforementioned risk by demonstrating how a malicious power market participant could disturb the electricity market operation, using a pre-designed false data injection attack along with bogus electricity trades in both day-ahead and real-time markets. The proposed attack design is robust against market uncertainties and the adversary can guarantee the success of the attack in advance. Hence, the existence of such cyber attacks against electricity markets can make the adversaries more aggressive. The numerical results on the IEEE 14-bus test system confirm the vulnerability of multi-settlement electricity markets to such financial cyber attacks. The results obtained from investigating such an attack design can be employed by ISOs in order to provide appropriate countermeasures.

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