Mason's rule

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Mason's Gain Formula is a method for finding the transfer function of a given control circuit/loop when you have the signal flow graph. It is used frequently in control theory, and was derived by Samuel Jefferson Mason. It can be determined by looking at a signal-flow graph, or a block diagram. Mason's Gain Formula provides a step by step method to obtain the transfer function from a block diagram or signal flow graph. An alternate method would be to find the transfer function algebraically by labelling each signal, writing down the equation for how that signal depends on other signals, and then solving the multiple equations for the output signal in terms of the input signal. Some people prefer a more structured approach, and Mason's Formula may be easier or more difficult depending on the graph in question.

[edit] Formula

The gain formula is as follows:

G = \frac{y_\mathrm{out}}{y_\mathrm{in}} = \sum_{k=1}^N \frac{G_k \Delta\ _k}{ \Delta\ }


{ \Delta\ } = 1 - \sum_{} L_i + \sum_{} L_iL_j- \sum_{} L_iL_jL_k + \cdots + (-1)^m \sum_{} ...

where:

  • Δ = the determinant of the graph.
  • yin = input-node variable.
  • yout = output-node variable
  • G = gain between yin and yout
  • N = total number of forward paths between yin and yout
  • Gk = gain of the kth forward path between yin and yout
  • Li = loop gain of each closed loop in the system
  • LiLj = product of the loop gains of any two non-touching loops (no common nodes)
  • LiLjLk = product of the loop gains of any three pairwise nontouching loops
  • Δk = the cofactor value of Δ for the kth forward path, with the loops touching the kth forward path removed. I.e. Remove those parts of the graph which form the loop, while retaining the parts needed for the forward path.

[edit] Usage

To use this technique,

  1. Make a list of all forward paths, and their gains, and label these Gk.
  2. Make a list of all the loops and their gains, and label these Li (for i loops). Make a list of all pairs of non-touching loops, and the products of their gains (LiLj). Make a list of all pairwise non-touching loops taken three at a time (LiLjLk), then four at a time, and so forth, until there are no more.
  3. Compute the determinant Δ and cofactors Δk.
  4. Apply the formula.

[edit] References

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