Signal-flow graph: Difference between revisions

Revision as of 07:19, 11 December 2023

Revision as of 03:16, 27 August 2024

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=== Choosing the variables ===

{{quoteblockquote|In general, there are several ways of choosing the variables in a complex system. Corresponding to each choice, a [[system of equations]] can be written and each system of equations can be represented in a graph. This formulation of the equations becomes direct and automatic if one has at his disposal techniques which permit the drawing of a graph directly from the [[schematic diagram]] of the system under study. The structure of the graphs thus obtained is related in a simple manner to the [[topology]] of the [[schematic diagram]], and it becomes unnecessary to consider the [[equations]], even implicitly, to obtain the graph. In some cases, one has simply to imagine the flow graph in the schematic diagram and the desired answers can be obtained without even drawing the flow graph.|Robichaud<ref>{{harv|Robichaud|1962|p=ix}}</ref>}}

=== Non-uniqueness ===

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The rules presented below may be applied over and over until the signal flow graph is reduced to its "minimal residual form". Further reduction can require loop elimination or the use of a "reduction formula" with the goal to directly connect sink nodes representing the dependent variables to the source nodes representing the independent variables. By these means, any signal-flow graph can be simplified by successively removing internal nodes until only the input and output and index nodes remain.<ref>{{harv|Phang|2001|p=37}}</ref><ref>Examples of the signal-flow graph reduction can be found in {{harv|Robichaud|1962|p=186, Sec. 7-3 Algebraic reduction of signal flow graphs}}</ref> Robichaud described this process of systematic flow-graph reduction:

{{QuotationBlockquote|The reduction of a graph proceeds by the elimination of certain nodes to obtain a residual graph showing only the variables of interest. This elimination of nodes is called "'''node absorption'''". This method is close to the familiar process of successive eliminations of undesired variables in a system of equations. One can eliminate a variable by removing the corresponding node in the graph. If one reduces the graph sufficiently, it is possible to obtain the solution for any variable and this is the objective which will be kept in mind in this description of the different methods of reduction of the graph. In practice, however, the techniques of reduction will be used solely to transform the graph to a residual graph expressing some fundamental relationships. Complete solutions will be more easily obtained by application of [[Mason's gain formula|Mason's rule]].<ref name="Robichaud 1962 9–10, Sec. 1–5: Reduction of the flow graph">{{harv|Robichaud|1962|pp=9–10, Sec. 1–5: Reduction of the flow graph}}</ref>

The graph itself programs the reduction process. Indeed a simple inspection of the graph readily suggests the different steps of the reduction which are carried out by elementary transformations, by loop elimination, or by the use of a reduction formula.<ref name="Robichaud 1962 9–10, Sec. 1–5: Reduction of the flow graph"/>|Robichaud|Signal flow graphs and applications, 1962}}

For digitally reducing a flow graph using an algorithm, Robichaud extends the notion of a simple flow graph to a ''generalized'' flow graph:

{{QuotationBlockquote|Before describing the process of reduction...the correspondence between the graph and a system of linear equations ... must be generalized...''The generalized graphs will represent some operational relationships between groups of variables''...To each branch of the generalized graph is associated a matrix giving the relationships between the variables represented by the nodes at the extremities of that branch...<ref>{{harv|Robichaud|1962|pp=182, 183 Sec. 7-1, 7-2 of Chapter 7: Algebraic reduction of signal flow graphs using a digital computer}}</ref>

The elementary transformations [defined by Robichaud in his Figure 7.2, p. 184] and the loop reduction permit the elimination of any node ''j'' of the graph by the ''reduction formula'':[described in Robichaud's Equation 7-1]. With the reduction formula, it is always possible to reduce a graph of any order... [After reduction] the final graph will be a cascade graph in which the variables of the sink nodes are explicitly expressed as functions of the sources. This is the only method for reducing the generalized graph since Mason's rule is obviously inapplicable.<ref>{{harv|Robichaud|1962|p=185, Sec. 7-2: Generalization of flow graphs}}</ref>|Robichaud|Signal flow graphs and applications, 1962}}

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==== Applying Mason's gain formula ====

{{detailsfurther|Mason's gain formula}}

In the most general case, the values for all the x_k variables can be calculated by computing Mason's gain formula for the path from each y_j to each x_k and using superposition.

:<math>\begin{align} x_\mathrm{k} &= \sum_{\mathrm{j}=1}^{\mathrm{M}} ( G_\mathrm{kj} ) y_\mathrm{j} \end{align} </math>

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** Simulation on analog computers<ref>{{harv|Robichaud|1962|loc=chapter 5 Direct Simulation on Analog Computers Through Signal Flow Graphs}}</ref>

*[[Neuroscience]] and [[Combinatorics]]

** Study of Polychrony<ref>{{Cite journal | title = Polychronization: computation with spikes|last = Izhikevich |first = Eugene M |journal = Neural Computation | date = Feb 2006|doi= 10.1162/089976606775093882 | pmid = 16378515 | volume = 18 | issue = 2 |pages = 245–282 |s2cid = 14253998 }}</ref> <ref>{{Cite journal | title = Polychrony as Chinampas

|author = Dolores-Cuenca, E.& Arciniega-Nevárez, J.A.& Nguyen, A.& Zou, A.Y.& Van Popering, L.& Crock, N.& Erlebacher, G.& Mendoza-Cortes, J.L.

|journal = Algorithms

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* [https://web.archive.org/web/20040325002849/http://www.apl.jhu.edu/Classes/Notes/Penn/EE774/Chap_03r.pdf M. L. Edwards: ''S-parameters, signal flow graphs, and other matrix representations'' ] All Rights Reserved

* [https://tube.switch.ch/channels/d206c96c H Schmid: ''Signal-Flow Graphs in 12 Short Lessons'' ]

* {{wikibooks- inline|Control Systems/Signal Flow Diagrams}}

* {{commons category-inline|Signal flow graphs}}