طراحی کنترل برای تغییر خطی سیستم های setups

In this paper we consider the control of a complex network of servers through which many types of jobs flow, where we assume that servers require a setup time when changing between types. Such networks can be used to model complex communication, traffic or manufacturing systems. Instead of starting with a given policy for controlling the network and then study the resulting dynamics, we start from a desired steady-state behavior and derive a policy which achieves this behavior. By means of an example we illustrate the way to derive a feedback controller from given desired steady-state behavior. Insights from this example can be used to deal with general networks, as illustrated by a more complex network example.

Consider a network of servers through which different types of jobs flow. One could think of a manufacturing system, i.e., a network of machines through which different types of products flow. An other example would be an urban road network of crossings with traffic lights through which cars flow. A third example would be a network of computers through which different streams of data flow. In this paper we take a fluid model (ODE) approach, where we assume that each server can only serve one type of job at a time, i.e., no processor sharing, and furthermore that when switching from one type of job to the other, a non-zero setup time is needed. This setup time might depend on the switch, that is, switching from Type 1 to Type 2 might take a different time than switching from Type 2 to Type 1 or from Type 3 to Type 2. Furthermore, we assume that each job type arrives to the network at a constant rate and that for each job type routes are specified a priori. It is allowed that a job visits the same server more than once (a so-called re-entrant system). The networks we consider might show some unexpected behavior. In Ref. [1] it was shown by simulation that even when each server has enough capacity to serve all jobs, these networks can be unstable in the sense that the total number of jobs in the network explodes as time evolves. Whether this happens depends on the policy used to control the flows through the network. In Ref. [2] it was shown analytically that using a clearing policy (serve the queue you are currently working on until it is empty, then switch to another queue) certain networks become unstable, even for deterministic systems with no setup times.

This paper deals with the problem of controlling a network consisting of a finite number of servers serving a finite number of job types where each job type requires a finite number of processing steps. We assume finite constant arrival rates and sufficient capacity at the servers. Furthermore, we assume finite strictly positive setup times when servers switch from one job type to the other. Most literature on this control problem starts from a policy and then analyzes the resulting closed-loop system. In this paper we start from a desired closed-loop behavior and then design a policy which achieves this behavior. The resulting policy is a feedback policy. The benefit of a feedback policy is that the closed-loop system is also robust against disturbances. Even though the policy has been derived for a deterministic system, the resulting policy can also be applied in case processing times and setup times are stochastic. The way to derive such a feedback policy has been illustrated extensively by means of an example. Based on the given desired periodic orbit, an “energy” of the system can be defined by considering the amount of work in the system. By controlling the network in such a way that this “energy” in the system is never increasing, the system stabilizes at a fixed energy level.