Variable aggregation

Variable aggregation, along with functional substitution, are the key components of the composition process in psymple.

Mathematical basis

Variable aggregation is the process of combining two ordinary differential equations linearly, as follows.

Definition

Let \( \frac{dx}{dt} = f(x,y,t,\underline{a}) \) and \( \frac{dy}{dt} = g(x,y,t,\underline{b}) \) be differential equations. Their aggregation under the identification \( (x,y) \longrightarrow z \) is the differential equation

\[ \frac{dz}{dt} = f(z,z,t,\underline{a}) + g(z,z,t,\underline{b}). \]

For example:

Example

The aggregation of the differential equations \( \frac{dx}{dt} = axt + by + c \) and \( \frac{dy}{dt} = dsin(y) \) under the identification \( (x,y) \longrightarrow z \) is given by

\[ \frac{dz}{dt} = (azt + bz + c) + (dsin(z)). \]

Discussion

In psymple, a key assumption is that complex dynamic systems are constructed by aggregating the differential equations of smaller components. For example, consider a tank of water with \( n \) pipes in adding water at rates \( \{r_1,\dots,r_n\} \) and \( m \) pipes out extracting water at rates \( \{s_1,\dots,s_m \} \). Then the rate of change of the volume \( V \) of water in the tank is given by

\[ \frac{dV}{dt} = \sum_{i=1}^n r_i - \sum_{i=1}^m s_i. \]

If this equation were implemented statically, then adapting the model to account for, for example, a leak whose flow rate is proportional to \( V \), would require writing a completely new differential equation. With the concept of variable aggregation, the contribution equation for \( V \) can instead be considered at each entry and exit point of the tank. For each entry pipe, this is

\[ \frac{dV}{dt} = r_i \]

and at each exit pipe this is

\[ \frac{dV}{dt} = -s_i. \]

The original tank model is then the aggregation of all of these component equations. If the leak were to be modelled too, by

\[ \frac{dV}{dt} = -\alpha V \]

then this can simply be added to the list of variable aggregations to perform.

Implementation detail

In psymple, variable aggregation is stored formally by connecting the ports of ported object instances by VariableAggregationWiring instances. A variable aggregation wire \( W_v(C, P_d, I) \) from child ports \( C = \{P_1,\dots,P_k \} \) to destination port \( P_d \) with identification variable \( I \) stores the information associated to variable aggregation.

Concretely, the child ports \( C \) each expose a differential assignment to be aggregated together, the identification variable \( I \) stores the identification to be made, and the destination port \( P_d \) exposes the aggregated assignment.

On model compilation, the variable aggregation wires in a system are interpreted and the variable aggregations are performed accordingly.