# Analysis¶

## Introduction¶

The equations generated by the processing module are unsuitable for the calculation process. The generated system of equation has many equations which are not useful to calculate the outputs of the system. Outputs are the values you are interested in. In the case of PyMbs these are usually sensors. An example for equations that are easy to simplify are variables that are zero/one or equal to another variable. The goal is to speed up the calculation by reducing the effort. The effort can be reduced by an analyses of the system of equations on the equation level and the system level. The resulting system of equations contains only the essential steps to calculate the outputs. Take for example this graph The corresponding system of equations is

(1)$\begin{split} a &= 1\\ b &= 0\\ c &= 5 \cdot k\\ d &= a \cdot c\\ e &= 2 \cdot a + 2\\ f &= b \cdot k\\ g &= d + e + f\end{split}$

with the output $$g$$. Seven steps needed to calculate $$g$$. In order to reduce the system all zeros are eliminated. The second step is to do the same for all variables equal one. The last step is to replace all variables used only once. The resulting system of equations reads as follows

(2)$\begin{split}c &= 5 \cdot k\\ g &= c + 4\end{split}$

Now, only two steps are required to calculate $$g$$.

Furthermore an abstract syntax tree of the system of equations is useful to calculate the dimensions from all variables and to sort the equations. Not all supported languages and corresponding simulation tools are able to derive the dimension of an variable from the equation. To sort the equations is necessary because not all supported simulation tools are able to do this. Note, that the implemented algorithm is not able to handle loops in the system of equations. For example the given system of equations

(3)$\begin{split}a &= 5\\ b &= c + 2a\\ c &= b + a \\ d &= b + c\end{split}$

contains a loop for the variables $$b$$ and $$c$$. The methods used in implementing the aforementioned tasks are described in the next section.

## Generating the Abstract Syntax Tree¶

The input from the Analysis Layer is a list of Expression objects. These objects have the following important information.

• The name of the variable called symbol, which represents the left hand side of the expression.
• The equation, which is the right hand side of the expression.

All information is stored in an node of the abstract syntax tree. A list of all required variables for each node can be generated using the symbolics module. With the list of required variables from each node the edges of the tree can be constructed. To store the edges, every node has two list. One for the parent nodes and one for the child nodes. A parent node is a node which is used to calculate the expression of the node. In addition a global list of all nodes exists to avoid to iterate through the tree when searching nodes.

## Reducing the System of Equation¶

The reduce algorithm has to find all nodes, which are

1. equal to zero or one or
2. have only one parent.

Because this is one of the main parts of the analysing module it should work efficient and fast.

The algorithm for the first point is divided in the following four steps. The first step is to find all nodes with no parents to extract all constants. This can be done simple by looking on the list of parents from each node. The second step is to check whether the node is replaceable. Not all nodes are replaceable, for example parameters or sensors should not disappear. To decide whether a node is replaceable or not categories were established. There are categories for state variables, parameter variables or sensor variables. The third step is to check whether the node is equal to one or zero. If this conditions is true the last step starts and replaces the node in the equations of the children and rearranges the abstract syntax tree by removing the corresponding edges. After this changes it is possible that a child node equation becomes equal to zero or one. Also this node should be replaced to obtain an equation system that is suitable for calculating the outputs efficiently. These four steps are executed recursively for each child node.

The algorithm for the second point looks similar to the algorithm for the first point. The difference is that this algorithm does not work recursively. The algorithm is executed for each node of the global list of nodes.

## Obtaining Sorted Equations¶

To sort the equations the nodes within the abstract syntax tree have to be marked with a level of depth. The level of depth is the longest path from a child to parent with no parents. This is done by a recursive algorithm that is executed for a list of nodes. The first list contains all nodes with no parent node. The level of depth is default set to zero for each node. This means all nodes have the same level of depth. The algorithm starts and tries to increase the level of depth for all children from a node of the list. If the level of depth of a child is not increased, the algorithm takes the next node from the list. The level of depth can be increased if the new value is greater than the old one. In addition this algorithm is used to find loops inside the system of equations. Therefore each node has a flag that indicates if the node has been checked. If the flag is active and it will be visited again a loop is detected and an error message is thrown. If the level of depth is set for all nodes the next step is to collect all nodes according to the desired category. Therefore a loop runs through the global list of nodes and collects the required nodes in a list. The next step is to collect all parents from the nodes inside the list. This is done by an recursive algorithm and a following loop. The recursive algorithm gets the list of required nodes. It steps through all parents of the nodes and sets the required flag. Afterwards the loop checks the global list of nodes and collects all nodes with active requried flag. With the list of all required nodes the third step starts to arrange the required nodes in a way that every expression of a node can be calculated. Therefore every variable inside the expression has to be calculated beforehand. This problem is solved by the level of depth. All nodes with equal level of depth are collected inside one list. Afterwards the lists are ordered by increasing level of depth and concatenated. The last step is to calculate the dimension of each expression and collect the expressions in a list according to the sequence.

## Calculating Value and Dimension of an Equation¶

For some layers there is a need to know the dimension of a variable. For example models designed in Modelica or C-code need the dimension for a variable. To get the dimension of a variable the expression need to be evaluated. To evaluate a expression all variables need to be known. This process is done recursively until a expression can be evaluated because it is a constant. The expression is evaluated using the symbolics module, which also returns the dimension of each variable.