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We revisit the extremum seeking scheme whose local stability properties were analyzed in and propose its simplified version that still achieves extremum seeking. We show under slightly stronger conditions that this simplified scheme achieves extremum seeking from arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted. This non-local convergence result is proved by showing semi-global practical stability of the closed-loop system with respect to the design parameters. Moreover, we show at the same time that reducing the parameters typically slows down the convergence of the extremum seeking controller. Hence, the control designer faces a tradeoff between the size of the domain of attraction and the speed of convergence when tuning the extremum seeking controller. We present a simulation example to illustrate our results.

Abstract-In the literature on dynamical systems analysis and the control of systems with complex behavior, the topic of synchronization of the response of systems has received considerable attention. This concept is revisited in the light of the classical notion of observers from (non)linear control theory.

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FMRI has revealed the presence of correlated low-frequency cerebro-vascular oscillations within functional brain systems, which are thought to reflect an intrinsic feature of large-scale neural activity. The spatial correlations shown by these fluctuations has been their identifying feature, distinguishing them from fluctuations associated with other processes. Major analysis methods characterize these correlations, identifying networks and their interactions with various factors. However, other analysis approaches are required to fully characterize the regional signal dynamics contributing to these correlations between regions. In this study we show that analysis of the power spectral density (PSD) of regional signals can identify changes in oscillatory dynamics across conditions, and is able to characterize the nature and spatial extent of signal changes underlying changes in measures of connectivity. We analyzed spectral density changes in sessions consisting of both resting-state scans and scans recording 2 min blocks of continuous unilateral finger tapping and rest. We assessed the relationship of PSD and connectivity measures by additionally tracking correlations between selected motor regions. Spectral density gradually increased in gray and white matter during the experiment. Finger tapping produced widespread decreases in low-frequency spectral density. This change was symmetric across the cortex, and extended beyond both the lateralized task-related signal increases, and the established “resting-state” motor network. Correlations between motor regions also reduced with task performance. In conclusion, analysis of PSD is a sensitive method for detecting and characterizing BOLD signal oscillations that can enhance the analysis of network connectivity.