An Equivalence Between a Control Network and a Switched Hybrid System Linda Bushnell? , Octavian Beldiman ?? and Gregory Walsh ???
Abstract. A simple model for ideal control networks is proposed in this
paper. A model for hybrid systems due to Witsenhausen is extended by adding both a discrete output and input. This extended model is used for modeling an ideal network of interactive hybrid systems. An equivalence is established between the network model and the Witsenhausen model. This equivalence allows for simulating complicated systems, and extending di erent properties of Witsenhausen type systems to control network systems. A simple HVAC application is modeled using the above equivalence.
1 Introduction Recently, the area of control networks has attracted increased interest. Control networks are seen as a possible way to analyze and design complex dynamical systems that either are scattered over a large area, or have real-time requirements that make the data transmission process a critical one. Several architecture standards have been developed in industry. Some of the most representative are the BACnet (building automation and control network) standard and the CAN (controller area network) standard. The BACnet has been designed to provide a standard communication and environmental control network for commercial and government buildings and campus environments. The primary application for this standard is HVAC control. CAN has been designed primarily for automotive applications. For instance, Brauninger et. al. [15] use the CAN standard to accommodate the growing need for data communications in trucks and busses. Several other standards are also uner et. al. [14] use the control network standard FOSU (Ford-Ohio available. Ozg State University) to control automotive suspension. Modeling and analysis of these networks have just started to develop. Although some introductory papers have been published ([2],[3]) very few papers discuss modeling or analysis issues. In [4] Walsh studies the race condition behavior for networks of hybrid systems. In [5], Tindell et. al. give bounds on the ??
Department of Electrical and Computer Engineering, Box 90291, Duke University, Durham, NC 27708-0291,
[email protected] and
[email protected] ??? Department of Mechanical Engineering, University of Maryland, College Park, MD 20742,
[email protected] ? Dr. Bushnell is also with the U.S. Army Research Oce, P.O. Box 12211, RTP, NC 27709-2211. This research was supported in part by the Army Research Oce grant number DAAH04-93-D-0002.
message response times in a CAN network. More recently, Wong and Brockett [10] study the e ect of the communication bandwidth constraints upon these systems. The subject of control networks is strongly connected to the modeling and analysis of hybrid systems. A vast amount of literature can be found in that direction, in both the eld of control and computer science ([6], [7], [8], [9], [11], [12]). The Witsenhausen model is an older and simpler model [1], and seems to be a good starting point for modeling control networks. Our proposed model allows for distinguishing between the low-level, continuous dynamics and the high-level, discrete switching commands in the network. The high-level strategy is implemented with regard to the way the systems respond to events. We will see that this is actually established by choosing the discrete input transition sets and network priorities assignments. A slightly modi ed version of the Witsenhausen model is presented in section 2. The modi cation consists of adding a discrete output to announce a transition of the discrete state. Then extensions of this model are presented in section 3. A network of these systems is proven to be equivalent to a Witsenhausen model in section 4. Each of the sections contains an example for the models presented. In section 5 we present a simple HVAC application and we used the equivalence in section 4 to build a simulator for it.
2 The Witsenhausen Model Without loss of generality we consider only autonomous models. Time-varying vector elds may be made autonomous by adding time as a new state. The Witsenhausen model for hybrid systems is developed as follows:
{ state: (m; x) 2 M Tmax o=0 0
v=0 u
