Proportional controller Nyquist plot

Nyquist plots for a three-CSTR system with proportional controllers. 

An openloop unstable, second-order process has one positive pole at + 1/ti and one negative pole at — I/tj. If a proportional controller is used and if ti < show by using a root locus plot and then by using the Nyquist stability criterion that the system is always unstable. 

The first reactor in the 3-CSTR process has a conversion rate of 72.8%, and the reactant concentration in this first reactor is 2.18 kmol/m3. The reactor volume is low (14.3 m3), and the jacket heat transfer area is only 24.5 m2. The resulting jacket temperature (300 K) is almost down to the inlet cooling water temperature of 294 K. Linear analysis gives a Nyquist plot that never drops into the third quadrant, so the critical (—1,0) point cannot be encircled in a counterclockwise direction. This is required for closedloop stability because the openloop system is unstable and has a positive pole. Thus a proportional controller cannot stabilize this first reactor. 

We found in Chapter 9, using Laplace-domain root locus plots, that we could make this system closedloop stable by using a proportional controller with a gain Kc greater than MKp. Let us see if the Nyquist stability criterion leads us to the same conclusion. It certainly should if it is any good, because a table in Chinese must be a table in Russian ... 

Electrochemical Impedance Spectroscopy (EIS) as schematically shown in Eigure 3.10 requires an alternating current (AC) and the output is a Nyquist plot for charge-transfer or diffusion control process, which can be used to determine Rv, which in turn, is inversely proportional to the corrosion current density icorr-... 

The Faradaic resistance or polarization resistance Rp is inversely proportional to the corrosion rate. It is evident from the Nyquist plot that the solution resistance. Rn, measured at high frequency can be subtracted from the sum of Rp and Rn at low frequency to give the value of Rp corrected for ohmic interferences from solution resistance. For processes controlled by diffusion in the electrolyte (concentration polarization) or in a surface film or coating, an additional resistive element called the Warburg impedance, W, must be included in the circuit. The Warburg impedance appears at low frequencies on the Nyquist plot as a straight line superimposed at 45° (slope = 1) to both axes, as shown in Fig. 31.5. 

The concept of gain and phase margins derived from the Nyquist criterion provides a general relative stability criterion. Frequency response graphical tools such as Bode, Nyquist and Nichols plots can all be used in ensuring that a control system is stable. As in root locus plots, we can only vary one parameter at a time, and the common practice is to vary the proportional gain. 

For a first order function with deadtime, the proportional gain, integral and derivative time constants of an ideal PID controller. Can handle dead-time easily and rigorously. The Nyquist criterion allows the use of open-loop functions in Nyquist or Bode plots to analyze the closed-loop problem. The stability criteria have no use for simple first and second order systems with no positive open-loop zeros. 

Sketch Nyquist, Bode, and Nichols plots for the proportional-integral feedback controller... 

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