(42c) Distributed Extremum Seeking Control over Unknown Network

When it comes to solving large-scale real-time optimization problems involving multi-agents, control approaches employed could either be centralized or distributed (decentralized).
In a centralized approach, the individual agents that make up a system are controlled by a single decision maker. This decision maker monitors, receives and processes the information from all agents, it also sends processed results back to the agents. The main advantage of this control approach is that the decision maker has full knowledge of the state of the system at any given time and can make useful decision(s) that help meet global objective(s). As the number of agents increases, it becomes difficult for the decision maker to perform its task due to increase in computational complexity. This phenomenon could result in the transmission of inaccurate information, loss of information and increase in computation time. In addition, centralized approaches are complex and most importantly lack system robustness as failure of the decision maker could mean failure of the entire system.

Most of the challenges highlighted above can be effectively avoided if one adopts a distributed approach. In a distributed environment, multiple decision makers can be utilized. Each subsystem or agent is controlled by a local decision maker. With the help of the decision maker, each agent's task can be limited to the solution of a simpler local problem. Each local problem can be solved with or without the cooperation of neighbouring agents. In a cooperative environment, agents work with other agents through local communication to find the best solution that meets global objective(s) of the system. Some of the advantages of distributed control include but are not limited to effectiveness, flexibility, scalability and adaptiveness. The greatest advantage of distributed control is system robustness. This implies that the failure of a decision maker does not necessarily mean overall system failure as the system can absorb the effect of a failure and quickly recover through the help of other decision makers.

In this paper, the solution of large-scale real-time optimization problems in the absence of precise knowledge of network connectivity in a distributed environment is considered. The knowledge of the communication network is assumed to be unknown and the agents are faced with the task of ensuring that the system's unknown overall cost (i.e., the sum of the local cost of all the agents) is minimized. Each agent has access to the measurement of two unknown cost functions referred to as the local cost and the local disagreement cost respectively. The minimization of the system's unknown overall cost to its unknown optimum depends on the minimization of the local disagreement cost of all the agents. For the local disagreement cost of the agents to be minimized, dynamic consensus estimation is required. This ensures that the agents reach agreement on their inputs (the decision variables). To help tackle this challenging problem, a distributed proportional-integral extremum seeking control technique is proposed, one that solves both problems simultaneously. A two-time scale approach is employed where consensus is achieved at a faster time-scale than the minimization of the overall cost. Included are two simulation examples, they show the effectiveness and robustness of this technique.