Parameter Estimation

The field of parameter estimation is an "inverse" method which requires calculation of a direct solution, usually several times, in order to fit measured information to a mathematical model.  For example, in the figure below, measurements were taken of the temperature on the back side of a heated plate.


A mathematical model was developed which described the heat transfer process.  The parameters within that model were found which minimized the sum of squares of the errors between the temperatures predicted by the mathematical model and the measured temperatures.


The expression used for finding this minimum of errors is where "T" represents the calculated temperatures and "Y" represents the measured temperatures.  The parameters of the mathematical model for the direct heat transfer solution are found such that "S" is minimized.  For a case of conductive heat transfer, the mathematical model for the temperature solution may be quite complex such as


where the eigen values are the solution to the following equation.


In this situation, the parameters cannot be solved for explicitly and the least-squares routine will have to iterate through non-linear regression.  The unknown parameters in this example are thermal diffusivity (designated by the greek letter alpha), thermal conductivity (designated by the letter k), and Biot number (designated by the symbol Bi).  The Biot number in this case is simply hL/k where h is the heat transfer coefficient and L is the thickness of the sample.

The iterations in a problem such as the one above are continued until convergence is obtained.  This is usually the point at which the changes in all parameters between iterations varies by less than 0.1%.  At this point, convergence is considered to have been obtained and the parameter estimates are reported by the computer program.  The above example involves one dimensional heat transfer through a heated flat plate, but this method can be applied to almost any geometry or heat transfer mechanism.

Although heat transfer is my main area of work, the field of parameter estimation applies to a wide range of areas which can be modeled and fitted to measurements from the field.  For example, a study undertaken for a co-generation power plant set out to determine the unit costs associated with steam delivered for heating and electrical power delivered.  The graph below shows a mathematical model of energy generation cost plotted against both heating steam and electrical energy delivered.


The axis "Sd" represents steam demand and the "E" axis represents electrical demand.  The "Sb" axis represents the total amount of boiler steam delivered in order to generate the coincident combination of low pressure steam and electrical energy.  The cost of boiler steam can then be traced directly in terms of dollars per pound.

The equation for the plane shown in this graph was generated by fitting the following equation to the monthly boiler log data.

  Sb = I + XsSd + XeE

In this equation, the "I" term represents the y-intercept on the graph, which corresponds to internal steam use and losses within the plant.  In essence, this is a "base" operating cost with no steam or electricity delivered.  The Xs and Xe terms are the parameters to be found where

  Xs=Unit cost of steam ($/lb)

  Xe=Unit cost of electrical energy ($/KWh)

These parameters are found by fitting the mathematical model shown above to the monthly boiler logs in order to minimize the sum of squares of the errors between the model and the measurements.  In other words, the following expression is minimized


where the "Y" values correspond to the measurements and the "T" values correspond to the prediction by the mathematical model.  In this way, several models can be fitted and the most appropriate model can be selected based on the comparison of the results generated from the competing models.  Once again, these models are normally based on heat transfer problems, but the example above is shown in order to highlight the diversity of problems which can be addressed using parameter estimation methods.