Optimization methods can be used to fit equations to data. Parameter estimation involves the calculation of unknown parameters that minimize the squared error between data and the proposed mathematical model. The step response of an overdamped second-order dynamic process can be described using the equation
\frac{y(t)}{K}=\left(1-\frac{\tau_{1} e^{-t / \tau_{1}}-\tau_{2} e^{-t / r _{2}}}{\tau_{1}-\tau_{2}}\right)
where \tau_{1} and \tau_{2} are process time constants and K is the process gain. The following normalized data have been obtained from a unit step test (K=1.0) :
| time, t | 0 | 1 | 2 | 3 | 4 | 5 |
| y_{i} | 0.0 | 0.0583 | 0.217 | 0.360 | 0.488 | 0.600 |
| t | 6 | 7 | 8 | 9 | 10 | 11 |
| y_{i} | 0.692 | 0.772 | 0.833 | 0.888 | 0.925 | 0.942 |
Use Excel with a starting point (1,0) to find values of \tau_{1} and \tau_{2} that minimize the sum of squares of the errors.
