Sampled data feedback control systems

We developed the mathematical tool of z transformation in the last chapter. Now we are ready to apply it to analyze the dynamics of sampled-data systems. Our primary task is to design sampled-data feedback controllers for these systems. We will explore the very important impact of sampling period 7 on these designs. 

The available data from emulsion polymerization systems have been obtained almost exclusively through manual, off-line analysis of monomer conversion, emulsifier concentration, particle size, molecular weight, etc. For batch systems this results in a large expenditure of time in order to sample with sufficient frequency to accurately observe the system kinetics. In continuous systems a large number of samples are required to observe interesting system dynamics such as multiple steady states or limit cycles. In addition, feedback control of any process variable other than temperature or pressure is impossible without specialized on-line sensors. This note describes the initial stages of development of two such sensors, (one for the monitoring of reactor conversion and the other for the continuous measurement of surface tension), and their implementation as part of a computer data acquisition system for the emulsion polymerization of methyl methacrylate. 

Data processed by off-line systems have the disadvantage that they provide information retrospectively about an analytical process that has been completed. Any indication of deteriorating performance, as shown for example by control samples, therefore cannot be used to initiate corrective action. On-line systems, in which data are continuously processed as they emerge from the measuring modules of the instruments, provide an opportunity for feedback control of the analytical process, as well as for reducing the overall time lapse between starting the analysis and reporting the results. 

There is a need in both the laboratory and process reactors to minimize the time delay between sample extraction and analysis results if the data are going to be used effectively to determine an end point or be used for feedback control. For these reasons, there has long been a desire for process analyzers to be able to collect data directly from the process stream itself without the need for a sample-handling system. Conceptually, one could simply mount the analyzer to the process stream to accomplish this goal. In reality however, safety, maintenance, environmental, and other practical concerns make this undesirable. 

To carry out a measurement, when the test sample was in place the variable pneumatic resistor was adjusted until the pressure drop over the powder plug equaled that over the variable resistor. In such a situation the two have the same pneumatic resistance (the electrical analog is the adjusting of a slide wire until the potential drop over the unknown resistor is the same of that of the known resistor.) In such a situation the dial or other display system of the variable pneumatic resistor can be calibrated directly in fineness units and the ultimate position of an indicator, or data from the system, could be used directly as a feedback control to adjust the operating parameters of the mill feed and the powder production unit. 

The controller receives the on-line composition measurement of the product outlets (extract and raffinate) as feedback data from the plant. These measurements are filtered through a periodic Kalman filter and used together with the simplified SMB model results to estimate the state of the system and to remove the possible moidel errors. The formulation of RMPC is based on the assumption that possible errors or disturbances are likely to repeat and will have a periodic effect on the output, which is the most likely correlation between disturbances and output in a SMB unit. The estimated future concentration profile in the SMB is used to optimize the future behaviour of the plant over a predefined prediction horizon. The controller implements the calculated optimal plant input by changing the external flow rates in order to control the internal flow rates, which are the manipulated variables. Time lags, e.g. between online concentration measurements and optimizer or between optimizer and SMB plant, are insignificant relative to the process dynamics and sampling time for the planned scheme. 

The SPM feedback mechanism, if applicable, has important implications for resolution. When Dpj is controlled by feedback, instability of the probe must be reduced to ensure proper tip position is measnred. When probe position is not precisely maintained, image convolution can occnr. Qnantilication of data can also be difficult, when Dps is not highly controlled. Inherent differences among various feedback modes for a given system affect image resolution, thus care must be taken to select the mechanism most suitable for the chosen sample. 

In a typical experiment, the test sample and a suitable reference material are contained in two separate, identical ampoules kept at constant temperature in separate, identically constructed wells of the calorimeter. Ideally, the reference material is identical or very similar to the test sample in mass, heat capacity and thermal conductivity, but, unlike the test sample, it is thermally inert (i.e. the reference material will not undergo changes that result in heat production or absorption under the conditions of the experiment). One example is a small quantity of ordinary glass beads in air at room temperature used as reference for the same amoimt of a hydrated ceramic material which is expected to lose water under the same conditions. Consequently, most of the noise arising from temperature fluctuations is removed when the reference data are subtracted. A feedback temperature control system between the wells (a) serves to ensure that the temperature difference between the weUs is zero and (b) provides an output that measures any difference in electric power requirement of one well relative to the other, needed to keep the temperature of both weUs the same. This power difference, as a function of time, is the output from the calorimeter, which is recorded continuously or intermittently over the duration of the test. 

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