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[https://gregstanleyandassociates.com/CES-1981a-ObservabilityRedundancy.pdf Stanley G.M. and Mah, R.S.H., "Observability and Redundancy in Process Data Estimation, Chem. Engng. Sci. 36, 259 (1981)]</ref> for these cases with set constraints such as algebraic equations and inequalities. Next, we illustrate some special cases:
Topological redundancy is intimately linked with the [[degrees of freedom (physics and chemistry)|degrees of freedom]] (<math>dof\,\!</math>) of a mathematical system,<ref name="vdi">VDI-Gesellschaft Energie und Umwelt, "Guidelines - VDI 2048 Blatt 1 - “Control and quality improvement of process data and their uncertainties by means of correction calculation for operation and acceptance tests”; VDI 2048 Part 1; September 2017", ''[http://www.vdi.de/401.0.html Association of German Engineers] {{Webarchive|url=https://web.archive.org/web/20100325223512/http://www.vdi.de/401.0.html |date=2010-03-25 }}'', 2017.</ref> i.e. the minimum number of pieces of information (i.e. measurements) that are required in order to calculate all of the system variables. For instance, in the example above the flow conservation requires that <math>a=b+c\,</math>. One needs to know the value of two of the 3 variables in order to calculate the third one. The degrees of freedom for the model in that case is equal to 2. At least 2 measurements are needed to estimate all the variables, and 3 would be needed for redundancy.
When speaking about topological redundancy we have to distinguish between measured and unmeasured variables. In the following let us denote by <math>x\,\!</math> the unmeasured variables and <math>y\,\!</math> the measured variables. Then the system of the process constraints becomes <math>F(x,y)=0\,\!</math>, which is a nonlinear system in <math>y\,\!</math> and <math>x\,\!</math>.
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