Frequency Analysis of Sampled-Data Systems Applied to a Lime Slaking Process
Single-rate and multirate sampled-data (SD) systems as well as linear time periodic (LTP) systems with discrete jumps are analyzed in the frequency domain. The focus of the analysis is both on rejection of disturbances and on robust stability properties. Due to the periodic behavior of LTP systems, a distinction is made between the performance frequency gain (PFG) and the robust frequency gain (RFG). The maximum power gain is computed for two different sets of input signals. In the analysis of the PFG a single sinusoidal signal v0ejwt is considered, while for the analysis of the RFG v0(t)ejwt is used as input, where v0(t) is a periodically time-varying complex vector. The RFG is related to current frequency response approaches for SD systems.
In a parallel work, a semi-physical model is developed for an industrial dry lime slaking process. The model is based on mass and heat balances for two non-stationary ideal stirred tank reactors in series. It is shown how the total reaction rate is derived from the conversion rate for lime particles. The identification of the conversion rate is based on a number of batch experiments, where the temperature and water evaporation flow from the batch is continuously monitored. Experiments have also been performed on a pilot plant slaker. It is verified that the slaker can be modeled as two ideal tank reactors in series. The evaporation model is based on experimental data from the pilot plant slaker. The slaker model is simulated and compared to a step response experiment.
An H¥-controller is designed for the lime slaker model using both continuous and discrete measurements. The design includes both H¥-synthesis and frequency analysis of the closed-loop system. An algorithm for the periodic solution of the Riccati differential equation with jumps, associated with the filtering update, is presented. A problem in LTP design is that the closed-loop system may never reach steady state when it is subjected to step disturbances. Instead the closed-loop system reaches a periodic steady state where the output oscillates. It is shown how weighting filters in the control design can be selected to avoid the oscillation problem.