Scheduling Schemes using Connective Stability

Abstract

Large systems are often constructed using small subsystems which are connected. These interconnections can lead to complex behavior; for example, the entire system may become unstable even if each of the individual subsystems are stable by themselves. The unstable systems can be stabilized with the use of a shared feedback controller. The effects of one subsystem on the state of other subsystems (coupling) can be reduced if each subsystem has access to the state information of the subsystems that are affecting its state. However, this solution requires communication between the controller and the subsystems and between subsystems. If there are limited communication resources, management of this resource is also required. Hence there is a need for a scheduling policy that specifies which subsystem should use the communication resource at any given time.

We start our formulation by first investigating systems that contain only stable subsystems. If the connected system is unstable due to coupling, the system cannot be scheduled. Therefore, we first proceed to extend previous work on stability of connected systems in order to formulate computationally efficient schedulability checks for these systems. We provide sufficient and necessary conditions for certain topologies and results for scalar systems that are dependent on the number of subsystems.

Then we proceed to formulate a centralized scheduling policy based on results of connective stability. Here we constrain ourselves to first studying systems with only a single communication resource that restricts only one subsystem to transmit its state in a given time slot. We study the best input a subsystem may apply once it has knowledge of the state of another subsystem that is affecting its state. We also provide evidence from simulations to support the performance increase in using the proposed algorithm.

Finally, we extend these results to formulate a decentralized scheduling policy that supports multiple communication resources. We also analyze a possible way of improving the scheduling policy using similarity transformations and show that such a methodology does not guarantee performance improvement and in-fact may lead to worse performance.