Process Identification

Although the fundamental model described earlier is a reasonably accurate description of the system dynamics, many of the parameters are not available a priori, which requires the estimation of several model parameters. The tank areas A_i can be measured directly from the apparatus. Using tank drainage data, the cross sectional outlet areas a_i can also be determined. The steady-state operating point of  and  were used for subsequent results. The system valves were set such that the operating point exhibits inverse response (). Time constants, T_i, for the linear system model were on the order of 40 seconds. The students designed a suitable test input sequence to generate data for the estimation of the remaining parameters. In this case, they elected to identify the parameters of the original nonlinear model, requiring the solution of a nonlinear optimization problem. The problem was formulated to minimize the 2-norm of the difference between the nonlinear model and actual measurements, searching over four parameters. Using dynamic data from the experiment, the optimization routine found the optimal pump gains k_i and gamma values  as depicted in Table 4. A similar routine was employed to model the characteristics of the disturbance introduced by the submersible pumps, k_d1 and k_d2. A critical step in any identification procedure is the validation of the model against novel data. The students were successful in validating the model that resulted from the previous optimization problem. They were able to capture the known inverse response in the system, and they also were able to compare the nonlinear model response to a linear approximation, which was subsequently used for analysis. 
1999-07-20