Scalability in multi-agent systems: analysis and control

Network systems have considerably evolved in both their size and complexity of topology. For these systems, the presence of delays and disturbances cannot be neglected when designing the control protocol. The overarching goal of this thesis is to design control protocols for networks consisting of possibly heterogeneous nonlinearly coupled agents guaranteeing that: (i) the desired network behaviour is achieved; (ii) certain classes of disturbances are rejected; (iii) disturbances that are not rejected are not amplified through the rest of the network. These are important requirements for a wide range of applications. For example, the vehicle platooning, power grid control and robotic formation control etc. We explore the fulfilment of these requirements via the scalability property, formalised as $\mathcal{L}_\infty$-scalable Input-to-State Stability and $\mathcal{L}_\infty$-scalable Input-Output Stability.

First, leveraging contraction theory, a sufficient condition for scalability of networks affected by homogeneous time-varying delays is derived. The results can serve as guidelines for e.g. the design of control protocols for a group of unicycle robots, the selection of weights and activation functions for recurrent neural networks with constant inputs, both of which satisfy the desired scalability property. Next, for the same type of network systems, we consider the problem of rejecting polynomial disturbances. We show that integral actions delivered via multiplex architecture, which we term as multiplex integral control, are able to reject polynomial disturbances. A sufficient condition guaranteeing scalability is derived which guide the design of such control protocol. Experiments on hardware unicycle robots via the test bed Robotarium confirm the effectiveness of the results. Then, we remove the hypothesis of homogeneous delays and give a sufficient condition on Input-to-State Stability for networks affected by heterogeneous time-varying delays. We validate our results via the application of voltage control for a multi-terminal high-voltage DC grid. Finally, we consider an application of designing intervention strategies for the epidemic control in a network model of Italy. A stability condition is derived which can be used to guarantee that the population of the infected will decrease. This condition can also be used to enforce a scalability condition which guarantees that the break out of the disease in one region will not lead to explosive growth in the infected population across the rest of the country. We then include satisfaction of such a condition as a constraint in a Model Predictive Control problem so as to mitigate (or suppress) the spread of the epidemic while minimizing the economic impact of the interventions. We conclude that our results can be used in the control design for a wide variety of network systems, from a group of robots, power grids to epidemic in networked regions. The results open a few interesting directions, namely: (i) deriving conditions for scalability of networks affected by heterogeneous time-varying delays; (ii) exploring the conservativeness of the conditions and finding the sufficient conditions on the loss of scalability; (iii) designing scalable recurrent neural networks dealing with sequence of inputs; (iv) extending the results to time scale dynamics.