Data-driven system modeling and optimal control for nonlinear dynamical systems

Das, Apurba Kumar
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With the increasing complexity of modern industry processes, robotics, transportation, aerospace,

power grids, an exact model of the physical systems are extremely hard to obtain whereas abundant

of time-series data can be captured from these systems. This makes it a important and

demanding research area to investigate feasibility of using data to learn behaviours of systems and

design controllers where the end goal generally evolves around stabilization. Transfer operators

i.e. Perron-Frobenius and Koopman operators play an undeniable role in advanced research of

nonlinear dynamical system stabilization. These operators have been a alternate direction of how

we generally approach dynamical systems, providing linear representations for even strongly nonlinear

dynamics. There is tremendous benet of acquiring a linear model of a system using these

models but, there remains a challenge of innite dimension for such models. To deal with it, we can

approximate a nite dimensional matrix of these operators e.g. Koopman matrix using Extended

Dynamic Mode Decomposition (EDMD) or Naturally Structured Dynamic Mode Decomposition

(NSDMD). Using duality property of Koopman and P-F operators we can derive formulation for

P-F matrix from Koopman matrix. Once we have a linear approximation of the system, Lyapunov

measure approach can be used along with a linear programming based computational framework for

stability analysis and design of almost everywhere stabilizing controller. In this work, we propose

a complete structure to stabilize a system that does not have an explicit model and only requires

black box input output time-series data. On a separate work, we show a set-oriented approach can

be used to control and stabilize systems with known dynamics model however having stochastic

parameters. Essentially, this work proposes two approaches to stabilize a nonlinear system using

both of known system model with inherent uncertainty and stabilize a black box system entirely

using input-output data.

Control design, Dynamical System, Nonlinear System, Optimization, Stabilization, Statistics