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A multi-resolution theory for systems and control across scales

Periodic Reporting for period 3 - switchlet (A multi-resolution theory for systems and control across scales)

Reporting period: 2018-10-01 to 2020-03-31

Switchlets are a novel development in control theory. Analog to the concept of wavelets in signal processing, switchlet theory seeks to model systems that can amplify and modulate signals at particular scales. The project aims at demonstrating that such selective amplifiers are key to reliably signalling across scales.
Feedback theory is at the core of switchlet modelling. While control theory is fundamentally a theory of regulation, centered on the concept of negative feedback, the project aims at developing a theory of feedback systems that balance positive and negative feedback at different scales. This is thought to be the core mechanism of robust selective amplification. The project aims at understanding the prevalence of this mechanism in natural systems, in particular neural circuits, with the goal of inspiring novel developments in artificial systems with multiscale sensing and actuating capabilities. Such systems become widespread in an increasingly interconnected technology.
The central current effort of the project is to bridge the separate worlds of neurophysiology and control theory through the common modelling language of circuit theory. Core current questions of experimental neurophysiology include the neuromodulation of excitable circuits, and how those circuits can be at the same time robust to biological variability and controllable by tiny changes in the balance of ionic or synaptic conductances. Those questions translate into controllability and robustness questions of feedback nonlinear circuits. Key results so far have shown the role of specific feedback loops at the cellular level in enabling robustness and modulation at the population level. Dominance theory has been proposed to address the need for a control theory of mixed feedback amplifiers. The theory is differential, that is, studies nonlinear behaviors away from equilibrium through the analysis of linearized models along arbitrary trajectories. The conventional analysis of stability and dissipativity through linear matrix inequalities is generalised to dominance and p-dissipativity, which guarantee convergence of trajectories to low dimensional attractors such as bistable switches or limit cycle oscillations.
Multiscale modelling of complex dynamical systems in todays society is dominated by detailed computational models that integrate quantitative details at the micro-scale with the hope of faithfully capturing the macro-scale behaviour. The resulting models lack robustness, modulation, and tractability. Circuit theory aims at capturing the internal behaviour of a model only to the extent by which it affects its external variables, that is, the variables shared with other parts of the system. This interconnection viewpoint is central to control theory, which precisely focuses on robustness and modulation questions. Such a theory is mature for systems exhibiting linear phenomena but currently lacking for systems exhibiting nonlinear phenomena such as switches, clocks, and nonlinear resonances. The novel differential modelling and analysis methods developed in this project will provide a general system theory for such nonlinear behaviors.