Subsystem Interface¶

Hierarchical Connections¶

Because subsystems are typically composed exclusively of components or other subsystems, any physical interaction with that subsystem is usually redirected to some nested component or subsystem. We’ve seen this pattern many times in this chapter where connectors at the subsystem boundary really act as “proxies” for internal connectors. A simple example of this can be seen in the GearWithBacklash model:

  connect(flange_a, inertia_a.flange_a)

Note that this connect statement connects flange_a, a connector instance that belongs to the subsystem, with inertia_a.flange_a, a connector instance belonging to the subcomponent inertia_a. This is a common pattern in subsystem models and it can be easily recognized because one of the connectors named in the connect statement includes a “.” and the other one does not. All the “internal” connections that wire together internal components directly to other internal components will have a “.” for both of the named connectors in the connect statement, e.g.,

  connect(idealGear.flange_b, inertia_b.flange_a)

Equation Generation for Hierarchical Connections¶

In this book, the point has been made repeatedly (e.g., in our discussion of Acausal Connections) that the sign convention for flow variables is such that a positive value for the flow variable represents a flow into the component or subsystem. At the same time, we also pointed out that Connections generate equations that sum all the corresponding flow variables in the connection set to zero.

But, when dealing with hierarchical subsystem definitions there is a modification to this rule. For a subsystem, the sign convention for flow variables remains intact. So, a positive value for the flow variable on a connector still represents a flow of a conserved quantity into that component. And it is still the case that a conservation equation will be generated for each flow variable in a connection set. However, in that conservation equation, the sign for flow variables on connectors belonging to internal components will have the opposite sign as flow variables on connectors belonging to the subsystem itself.

To understand the implications of this, let us consider the following two connect statements from the GearWithBacklash model:

  connect(flange_a, inertia_a.flange_a)
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  connect(idealGear.flange_b, inertia_b.flange_a)
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From these equations, we get the following two connection sets:

• Connection Set #1: flange_a, inertia_a.flange_a
• Connection Set #2: idealGear.flange_b, inertia_b.flange_a

In each of these connection sets, there is a flow variable, tau. Using the rules for Connections described previously, we might expect the following two equations to be generated for the flow variables on these connectors:

flange_a.tau + inertia_a.flange_a.tau = 0;
idealGear.flange_b.tau + inertia_b.flange_a = 0;

However, in our previous discussion on Connections, all the components were at the same hierarchical level (i.e., as idealGear.flange_b and inertia_b.flange_a are). But with subsystem models, this isn’t always the case. And, as described a moment ago, for cases where the connections are at different levels, we need to introduce a sign difference for contributions at different levels. Taking that into account, the actual equations that will be generated will be:

-flange_a.tau + inertia_a.flange_a.tau = 0;
idealGear.flange_b.tau + inertia_b.flange_a = 0;

Note the minus sign in front of the flange_a.tau. Remember that flange_a is meant to be acting as a proxy for inertia_a.flange_a. If that is the case, then by changing the sign of flange_a.tau, the first equation above can be transformed into:

inertia_a.flange_a.tau = flange_a.tau;

In other words, by introducing the sign change on flange_a.tau, any conserved quantity the flows in through flange_a also flows into inertia_a.flange_a, which is exactly what we expect from this kind of “proxy” relationship.