Open-loop time constant

Certain quantitative measures from linear control theory may help at various steps to assess relationships between the controlled and manipulated variables. These include steady-state process gains, open-loop time constants, singular value decomposition, condition numbers, eigenvalue analysis for stability, etc. These techniques are described in... 

The closed-loop time constant is smaller than the open-loop time constant. That is, proportional action makes the closed-loop process respond faster than the open-loop process. 

The open loop time constant in Equation l-2a is either the digital signal filter time in a Distributed Control System (DCS) for smoothing out pH oscillations and noise or the residence time for a well-mixed vessel for smoothing out concentration oscillations. Equation l-2a is useful for estimating the size of vessels needed to smooth out stick-slip and the size of a signal filter to keep short term pH fluctuations within the control band [Ref. 1.3]. For example, if the oscillation period of pH noise is 0.012 minutes, a digital filter of 0.04 minutes would reduce the noise amplitude by a factor of 20. 

The open loop time constant is the residence time of a back mixed volume ... 

Figure S-3c. Approxlmetion of (he Dead Time and Open Loop Time Constant on a Trend...

In 1953, Cohen and Coon [2] developed a set of controller tuning recommendations that correct for one deficiency in the Ziegler-Nichols open-loop rules. This deficiency is the sluggish closed-loop response given by the Ziegler-Nichols rules on the relatively rare occasion when process dead time is large relative to the dominant open-loop time constant. 

Figure 6.2. Illustration of fitting Eq. (6-2, solid curve) to open-loop step test data representative of self-regulating and multi-capacity processes (dotted curve). The time constant estimation shown here is based on the initial slope and a visual estimation of dead time. The Ziegler-Nichols tuning relation (Table 6.1) also uses the slope through the inflection point of the data (not shown). Alternative estimation methods are provided on our Web Support.

To find the new state feedback gain is a matter of applying Eq. (9-29) and the Ackermann s formula. The hard part is to make an intelligent decision on the choice of closed-loop poles. Following the lead of Example 4.7B, we use root locus plots to help us. With the understanding that we have two open-loop poles at -4 and -5, a reasonable choice of the integral time constant is 1/3 min. With the open-loop zero at -3, the reactor system is always stable, and the dominant closed-loop pole is real and the reactor system will not suffer from excessive oscillation. 

Finally, let s take a look at the probable root loci of a system with an ideal PID controller, which introduces one open-loop pole at the origin and two open-loop zeros. For illustration, we will not use the integral and derivative time constants explicitly, but only refer to the two zeros that the controller may introduce. We will also use zpk () to generate the transfer functions. 

Let s consider an overdamped process with two open-loop poles at -1 and -2 (time constants at 1 and 0.5 time units). A system with a proportional controller would have a root locus plot as follows. We stay with tf (), but you can always use zpk (). 

The major regions for placing the zero are the same, but the interpretation as to the choice of the integral time constant is very different. We now repeat adding the open-loop zeros ... 

You may want to try some sample calculations using a PID controller. One way of thinking we need to add a second open-loop zero. We can limit the number of cases if we assume that the value of the derivative time constant is usually smaller than the integral time constant. 

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. 

The longest time constant is slightly over 1.5 hours. The open loop system requires approximately four times this time constant (6 hours) to reach a new steady state after an upset, as Figure 3 clearly shows. With the added 83 minute time constant for the Q lag described above, this time becomes even longer, as seen in Figure 4. 

The dominant pole of this temperature control system is also determined by the thermal time constant of the microhotplate, which is approximately 20 ms. The open-loop gain of the differential analog architecture (Aql daa) is given by Eq. (5.8) ... 

INFICON s Auto Control Tune is based on measurements of the system response w/ith an open loop. The characteristic of the system response is calculated on the basis of a step change in the control signal. It is determined experimentally through two kinds of curve accordance at two points. This can be done either quickly w/ith a random rate or more precisely with a rate close to the desired setpoint. Since the process response depends on the position of the system (in our case the coating growth rate), it is best measured near the desired virork point. The process information measured in this vray (process amplification Kp, time constant T., and dead time L) are used to generate the most appropriate PID control parameters. 

Open-loop systems have inherently long residence times which may be detrimental if the retentate is susceptible to degradation by shear or microbiological contamination. A feed-bleed or closed-loop configuration is a one-stage continuous membrane system. At steady state, the upstream concentration is constant at Cj (Fig. 9). For concentration, a single-stage continuous system is the least efficient (maximum membrane area). 

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