Observability and Controllability of Nonlinear Networks: The Role of Symmetry

Abstract

Observability and controllability are essential concepts to the design of predictive observer models and feedback controllers of networked systems. For example, noncontrollable mathematical models of real systems have subspaces that influence model behavior, but cannot be controlled by an input. Such subspaces can be difficult to determine in complex nonlinear networks. Since almost all of the present theory was developed for linear networks without symmetries, here we present a numerical and group representational framework, to quantify the observability and controllability of nonlinear networks with explicit symmetries that shows the connection between symmetries and nonlinear measures of observability and controllability. We numerically observe and theoretically predict that not all symmetries have the same effect on network observation and control. Our analysis shows that the presence of symmetry in a network may decrease observability and controllability, although networks containing only rotational symmetries remain controllable and observable. These results alter our view of the nature of observability and controllability in complex networks, change our understanding of structural controllability, and affect the design of mathematical models to observe and control such networks.

Physical controllability of complex networks

“…Although the mathematical controllability theories410 offer theoretically justified frameworks to guide us to apply external inputs on a minimum set of driver nodes, when we implement control to steer a system to a desired state, the energy consumption is likely to be too large to be affordable. For nonlinear dynamical networks, we continue to lack a general controllability framework and an understanding of required control energy, although progress has been made22414243444546, in spite of the fact that for specific types of systems, e.g., gene regulatory networks, controllability can be defined in terms of the coexisting attractors (final destinations) of the system45. Unlike linear networked systems, controllability of a nonlinear network depends on both the network structure and the system dynamics.…”

“…The past few years have witnessed great progress toward understanding the linear controllability of complex networks12345678910111213141516171819202122232425262728. Given a linear and time-invariant dynamical system, the traditional approach to assessing its controllability is through the Kalman rank condition29.…”

Neural mass model-based tracking of anesthetic brain states

“…This can be understood in part by the fact that the standard parameters correspond to a region in parameter space involving an alpha rhythm limit cycle (Jansen & Rit, 1995) and state observability is generally higher when limit cycle dynamics are present as compared say to a stable fixed point (Whalen et al, 2015). These results highlight the importance of considering monotonicity, observability and error analysis when investigating other neural mass model inversion procedures.…”

“…Given that the JR model is a nonlinear system, a method for nonlinear observability (Hermann & Krener, 1977) has been employed that follows the approach of Whalen et al (2015). The reader is referred to this 265 paper for mathematical details, however, it is briefly noted that the observability index is taken to be the average over many simulated samples and state trajectories of the absolute ratio of the minimum and maximum singular values of the inner product of the observability matrix evaluated at each sample using the Jacobian of the Lie derivative map of the JR model (Whalen et al, 2015). This definition produces an observability index with the range 0 ≤ δ ≤ 1 where values closer to 1 indicate full observability of the states.…”

Effects of symmetry on the structural controllability of neural networks: A perspective

“…These two theorems together paint a more complete picture of controllability than either alone as shown in [8], where both are used in concert to explain and understand why certain neural networks were not controllable from particular inputs. Including symmetry constraints makes structural controllability a more general concept, as it does not depend on the explicit non-zero entries of the system pair ( A, B ) (necessary, but not sufficient), while a network that has the NCS property possesses specific sets of the non-zero entries in ( A, B ) that define the symmetry contained by the system.…”