Controlling Linear Networks with Minimally Novel Inputs (original) (raw)

Input Novelty as a Control Metric for Time Varying Linear Systems

This paper introduces a framework for quantitative characterization of the controllability of time-varying linear systems (or networks) in terms of input novelty. The motivation for such an approach comes from the study of biophysical sensory networks in the brain, wherein responsiveness to both energy and salience (or novelty) are presumably critical for mediating behavior and function. Here, we use an inner product to define the angular separation of the current input with respect to the past input history. Then, by constraining input energy, we define a non-convex optimal control problem to obtain the minimally novel input that effects a given state transfer. We provide analytical conditions for existence and uniqueness in continuous-time, as well as an explicit closed-form expression for the solution. In discrete time, we show that a relaxed convex optimization formulation provides the global optimal solution of the original non-convex problem. Finally, we show how the minimum novelty control can be used as a metric to study control properties of large scale recurrent neuronal networks and other complex linear systems. In particular, we highlight unique aspects of a system's controllability that are captured through the novelty-based metric. The result suggests that a multifaceted approach, combining energy-based analysis with other specific metrics, may be useful for obtaining more complete controllability characterizations.

On controllability of neuronal networks with constraints on the average of control gains

IEEE transactions on cybernetics, 2014

Control gains play an important role in the control of a natural or a technical system since they reflect how much resource is required to optimize a certain control objective. This paper is concerned with the controllability of neuronal networks with constraints on the average value of the control gains injected in driver nodes, which are in accordance with engineering and biological backgrounds. In order to deal with the constraints on control gains, the controllability problem is transformed into a constrained optimization problem (COP). The introduction of the constraints on the control gains unavoidably leads to substantial difficulty in finding feasible as well as refining solutions. As such, a modified dynamic hybrid framework (MDyHF) is developed to solve this COP, based on an adaptive differential evolution and the concept of Pareto dominance. By comparing with statistical methods and several recently reported constrained optimization evolutionary algorithms (COEAs), we sho...

Analyzing Controllability of Neural Networks

2019

In recent years, due to the relation between cognitive control and mathematical concept of control dynamical systems, there has been growing interest in the descriptive analysis of complex networks with linear dynamics, permeating many aspects from everyday life, obtaining considerable advances in the description of their structural and dynamical properties. Nevertheless, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Concretely, for complex systems is of interest to study the exact controllability, this measure is defined as the minimum set of controls that are needed to steer the whole system toward any desired state. In this paper, a revision of controllability concepts is presented and provides conditions for exact controllability for the multiagent systems

A geometrical approach to control and controllability of nonlinear dynamical networks

Nature communications, 2016

In spite of the recent interest and advances in linear controllability of complex networks, controlling nonlinear network dynamics remains an outstanding problem. Here we develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability. The control objective is to apply parameter perturbation to drive the system from one attractor to another, assuming that the former is undesired and the latter is desired. To make our framework practically meaningful, we consider restricted parameter perturbation by imposing two constraints: it must be experimentally realizable and applied only temporarily. We introduce the concept of attractor network, which allows us to formulate a quantifiable controllability framework for nonlinear dynamical networks: a network is more controllable if the attractor network is more strongly connected. We test our control framework using examples from various models of experimental gene regulatory networks and demon...

Network Design for Controllability Metrics

IEEE Transactions on Control of Network Systems, 2020

In this paper, we consider the problem of tuning the edge weights of a networked system described by linear time-invariant dynamics. We assume that the topology of the underlying network is fixed and that the set of feasible edge weights is a given polytope. In this setting, we first consider a feasibility problem consisting of tuning the edge weights such that certain controllability properties are satisfied. The particular controllability properties under consideration are (i) a lower bound on the smallest eigenvalue of the controllability Gramian, and (ii) an upper bound on the trace of the Gramian inverse. In both cases, the edge-tuning problem can be stated as a feasibility problem involving bilinear matrix equalities, which we approach using a sequence of convex relaxations. Furthermore, we also address a design problem consisting of finding edge weights able to satisfy the aforementioned controllability constraints while seeking to minimize a cost function of the edge weights, which we assume to be convex. Finally, we verify our results with numerical simulations over many random network realizations, as well as with an IEEE 14-bus power system topology. Index Terms Networked dynamics, network design, controllability Gramian, bilinear matrix equality, convex optimization. I. INTRODUCTION Many technological, biological, chemical, and social systems can be modeled as large ensembles of dynamical units connected via an intricate pattern of interactions [1]. From an engineering perspective, we are interested in efficiently steering the dynamics of these complex systems via external actuation. In this direction, control theory provides us with the notion of controllability to decide whether a given system can be steered towards an arbitrary state [2]. Furthermore, the so-called controllability Gramian of a system, which implicitly depends on the system's dynamics and the configuration of its actuators, can be used to quantify the energy required to steer the system, assuming the system is controllable [2]. Leveraging these notions, several papers have recently focused on the problem of optimally allocating actuators throughout the network under several performance metrics [3]-[12].

Control and controllability of nonlinear dynamical networks: a geometrical approach

ArXiv, 2015

In spite of the recent interest and advances in linear controllability of complex networks, controlling nonlinear network dynamics remains to be an outstanding problem. We develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability (multiple coexisting final states or attractors), which are representative of, e.g., gene regulatory networks (GRNs). The control objective is to apply parameter perturbation to drive the system from one attractor to another, assuming that the former is undesired and the latter is desired. To make our framework practically useful, we consider RESTRICTED parameter perturbation by imposing the following two constraints: (a) it must be experimentally realizable and (b) it is applied only temporarily. We introduce the concept of ATTRACTOR NETWORK, in which the nodes are the distinct attractors of the system, and there is a directional link from one attractor to another if the system can be driven from the ...

Some Control Theoretic Issues in Neural Networks

1970

We have observed that many neural network models can be written as a bilinear system with a speci c form of nonlinear state-to-input feedback. This framework includes the ART architecture among others. There are two signi cant results which follow from this observation. First, the parameters of the model determine the controllability of the system. A system is controllable if there exists some input which transfers any initial state to any desired nal state in a nite time. If for a given set of these parameters the system is not controllable, then there are regions of the state space which the system can never enter in a nite time for any input. Because of this restriction the learning ability of the system may be severely limited. Second, the multiplicative equation is linear in all of the parameters, and all of the adjustable weights. This means that a provably convergent learning algorithm can be devised for all of these quantities. This does not however circumvent the learning limitation since the learning algorithm is not guaranteed to converge in a nite time. In the paper, we will study these issues as they apply to the ART architecture.

Trajectory control of convergent networks

1998

We present a class of feedback control functions which increase the convergence rates of nonlinear dynamical systems. A simple sign function is used to obtain convergence in finite time. We describe a trajectory learning procedure which preserves the convergence property of the system. Based on the proposed feedback, we developed a new neural network model which converges in finite time.

Appropriate time to apply control input to complex dynamical systems

Scientific Reports

Controlling a network structure has many potential applications many fields. In order to have an effective network control, not only finding good driver nodes is important, but also finding the optimal time to apply the external control signals to network nodes has a critical role. If applied in an appropriate time, one might be to control a network with a smaller control signals, and thus less energy. In this manuscript, we show that there is a relationship between the strength of the internal fluxes and the effectiveness of the external control signal. To be more effective, external control signals should be applied when the strength of the internal states is the smallest. We validate this claim on synthetic networks as well as a number of real networks. Our results may have important implications in systems medicine, in order to find the most appropriate time to inject drugs as a signal to control diseases.

Minimum-cost control of complex networks

New Journal of Physics, 2015

Finding the solution for driving a complex network at the minimum energy cost with a given number of controllers, known as the minimum-cost control problem, is critically important but remains largely open. We propose a projected gradient method to tackle this problem, which works efficiently in both synthetic and real-life networks. The study is then extended to the case where each controller can only be connected to a single network node to have the lowest connection complexity. We obtain the interesting insight that such connections basically avoid high-degree nodes of the network, which is in resonance with recent observations on controllability of complex networks. Our results provide the first technical path to enabling minimum-cost control of complex networks, and contribute new insights to locating the key nodes from a minimum-cost control perspective.