Nonlinear Control of Networked Dynamical Systems

Abstract

We develop a principled mathematical framework for controlling nonlinear, networked dynamical systems. Our method integrates dimensionality reduction, bifurcation theory and emerging model discovery tools to find low-dimensional subspaces where feed-forward control can be used to manipulate a system to a desired outcome. The method leverages the fact that many high-dimensional networked systems have many fixed points, allowing for the computation of control signals that will move the system between any pair of fixed points. The sparse identification of nonlinear dynamics (SINDy) algorithm is used to fit a nonlinear dynamical system to the evolution on the dominant, low-rank subspace. This then allows us to use bifurcation theory to find collections of constant control signals that will produce the desired objective path for a prescribed outcome. Specifically, we can destabilize a given fixed point while making the target fixed point an attractor. The discovered control signals can be easily projected back to the original high-dimensional state and control space. We illustrate our nonlinear control procedure on established bistable, low-dimensional biological systems, showing how control signals are found that generate switches between the fixed points. We then demonstrate our control procedure for high-dimensional systems on random high-dimensional networks and Hopfield memory networks.

Nonlinear Control in the Nematode C. elegans

“…This model is minimally parameterized and changes in several parameters can reproduce changes in behavioral distributions akin to that of known neuro-modulators, thus producing a unifying framework for analyzing various changes in distributions of behavior at multiple timescales. In addition, the framework for building this model can be extended to other complex systems with more behavioral states which are defined by fixed points as discussed in Morrison and Kutz (2020).…”

“…Our objective is to create the lowest dimension model with the fewest number of parameters that is able to represent three features of the C. elegans neural activity: (1) the intrinsic stability of the neural activity underlying the forward and reversal behaviors, (2) the variability in transition trajectories, and (3) the destabilization of the stable states under the influence of feed-forward control signals, that is, control of the network's state. A nonlinear control model that is higher-dimensional, or that has more parameters, can be found using the methods outlined in Morrison and Kutz (2020). We fit the model parameters to the low-dimensional C. elegans activity by minimizing the error between the dominant PCA trajectory v 1 (t) and the trajectory of the corresponding model variable x(t) (see section 4).…”

Fine-grained classification based on multi-scale pyramid convolution networks

“…Visual attention is widely used in various deep learning tasks [18][19][20]. It can be employed to discover the subtle inter-class differences in fine-grained image categorization.…”

Optimizing hyperparameters of deep reinforcement learning for autonomous driving based on whale optimization algorithm

“…The annual social benefits of AD systems, if widely adopted, are expected to hit approximately $800 billion by 2050, thanks to traffic reduction, reduced road fatalities, lower energy usage, and improved efficiency due to the reallocation of driving time [ 29 ]. In this context, respectful efforts have been put into the research to provide a reliable and safe experience to the future of AD for both connected vehicles [ 30 , 31 ] and ego vehicles [ 32 , 33 ]. On the other hand, the DRL technique can be seen as a promising technique to be applied in the field of AD.…”