Adaptive Pinning Control using Network Controlability Gramian

Note: this implementation has been incorporated into a larger project here.

This project implements adaptive pinning control on a network of dynamic agents. Energy efficient pins are automatically selected using the Network Controlability Gramian. We consider the network as a Graph, $G=${ $V$, $E$ } such that:

• Vertices ( $V$ ) are the agents (nodes)
• Edges ( $E$ ) are a set of links in the form of ordered pairs
• Edges are defined by the Cartesian Product $V \times V$, $E=$ {( $a$ , $b$ ) such that $a \in V$ and $b \in V$}
• $G$ is simple, or ( $a$ , $a$) $\notin E~\forall~a \in V$
• $G$ is undirected, or ( $a$ , $b$ ) $\in E <=>$ ( $b$ , $a$) $\in E$
• Nodes $i$ and $j$ are neighbours if they share an edge
• Adjacency matrix $A=$ { $a_{ij}$ } with dimensions $N \times N$ describes the connections between nodes where $a_{ij}$ is $1$ if $i$ and $j$ are neighbours or $0$ otherwise
• Component set of the graph is a set with no neighbours outside itself

We define a pin for each component such that it consumes the minimal amount of energy to control its respective component. Here, we achieve this by computing the Network Controlability Gramian:

$W_j=$ $\sum\limits_{h=0}^{H-1}$ $A_d^hb_jb_j^T\left(A_d^T\right)^h$

where:

• $A_d = AD^{-1}$
• $D$ is the diagonal augmented in-degree matrix of $A$, composed by the sum of the columns of $A$ on its diagonal (or a $1$ when this sum is $0$)
• $b_j$ is a column vector with all terms equal to $0$ except at the pin location $j$
• $H$ is the control horizon

The trace of $W_j$ is inversely proportional to the average energy required to control the graph component when $j$ is selected as the pin. Therefore, we compute the trace of $W_j$ for each candidate pin (within a given component set) and select the one with the lowest value in order to minimize the control energy.

The graph is built using Olfati-Saber flocking (Reference 4, below) such that the agents are considered to be connected when they are within radius $r$ of eachother. These connections are what define the component sets of the graph. The pins share a common target, which draws the separate components together. As the components meet, they combine to form a larger component. New pins are computed in real-time, until all agents form part of a single component representing the entire graph. This convergence is guaranteed, so long as the components are observable, controlable, stable, and share the same target.

Plots

Below are plots with adaptive pins, selected to conserve energy using the Network Controlability Gramian:

References

1. Kléber M. Cabral, Sidney N. Givigi, and Peter T. Jardine, Autonomous assembly of structures using pinning control and formation algorithms in 2020 IEEE International Systems Conference (SysCon), 07 Dec 2020

2. Erfan Nozari, Fabio Pasqualetti, and Jorge Cortes,Heterogeneity of Central Nodes Explains the Benefits of Time-Varying Control in Complex Dynamical Networks in arXiv, 26 Dec 2018

3. Fabio Pasqualetti, Sandro Zampieri, and Francesco Bullo, Controllability Metrics, Limitations and Algorithms for Complex Networks in IEEE Transactions on Control of Network Systems, 11 Mar 2014

4. Reza Olfati-Saber, "Flocking for Multi-Agent Dynamic Systems: Algorithms and Theory" in IEEE Transactions on Automatic Control, Vol. 51 (3), 3 Mar 2006.

5. The code is opensource but, if you reference this work in your own reserach, please cite me. I have provided an example bibtex citation below:

@techreport{Jardine-2022, title={Adaptive Pinning Control using Network Controlability Gramian}, author={Jardine, P.T.}, year={2022}, institution={Royal Military College of Canada, Kingston, Ontario}, type={GitHub Code Repository}, }

6. Alternatively, you can cite any of my related papers, which are listed in Google Scholar.