Constrained proportional integral control of dynamical distribution networks with state constraints

Abstract

Abstract-This paper studies a basic model of a dynamical distribution network, where the network topology is given by a directed graph with storage variables corresponding to the vertices and flow inputs corresponding to the edges. We aim at regulating the system to consensus, while the storage variables remain greater or equal than a given lower bound. The problem is solved by using a distributed PI controller structure with constraints which vary in time. It is shown how the constraints can be obtained by solving an optimization problem.

Optimal steady state regulation of distribution networks with input and flow constraints

“…and due to (42) it is easy to see that (44) coincides with (32). Lastly it can be checked that ( 31)-( 33) satisfies (30) identically, which concludes this proof.…”

“…The incremental states as in (29), where x, x e and x p are the solution to (1) in closed loop with ( 27) and ( 28), and xe as any solution to (15), xp as in (14) and xp , xe , x as defined as in (30), satisfy…”

“…It is shown that if the graph has uni-directional flow due to the saturation, a sufficient condition for output agreement is that the associated directed graph is strongly connected. The same authors recently provided a result in which proportionalintegral (PI) controllers are able to handle state constraints [30].…”

Structure and stability of the equilibrium set in potential-driven flow networks

“…In this generality, flow networks arise in many different application fields: let us mention for example distribution networks [11,14,27,43], supply chains [17], traffic networks [10,21], power systems [9,18], inventory systems [4,8], etc. We do not at all attempt to be exhaustive, and references are only a small sample of a huge amount of literature.…”

“…in the presence of external supplies or demands, the right-hand side of (4) takes the form -Au + s for a (possibly time-dependent) vector s = (s\,..., s n ) of supply/demands (cf. for instance [8,10,14,43]). When supplies and demands are balanced we have Y17=i Si = ® an d> using basic properties of the incidence matrix, this makes it possible to recast the vector field above as -A(u + s) for a certain vector s, driving the system to the form (4).…”