Proportional control constant

The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control. The natural period of the cycle—the proportional controller contributes no phase shift to alter it—is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band P , which produced the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced using on/off control to cycle the manipulated variable between two limits. The period of the cycle will be close to if the cycle is symmetrical the peak-to-peak amphtude of the controlled variable divided by the difference between the output limits A, is a measure of process gain at that period and is therefore related to for the proportional cycle ... 

The response of a controller to an error depends on its mode. In the proportional mode (P), the output signal is proportional to the detected error, e. Systems with proportional control often exhibit pronounced oscillations, and for sustained changes in load, the controlled variable attains a new equilibrium or steady-state position. The difference between this point and the set point is the offset. Proportional control always results in either an oscillatory behaviour or retains a constant offset error. 

The maintenance of constant liquid level in the reflux drum can be expressed by the following proportional control equation... 

When the reactor temperature (Tl) becomes greater than Tmax (=240 F), PERIOD = 2, and the program turns the cooling water on with flow rate Fw. This flow is controlled with a proportional controller using control constant Kc, whose set point (Pset) is varied according to the time ramp function with setting kR and whose output to the valve is Pc. This ramp is horizontal until time period Tihold has passed. Then the setpoint is decreased linearly. The temperature is sensed using a pressure transmitter with output Ptt. 

Example 5.2 Derive the closed-loop transfer function of a system with proportional control and a second order overdamped process. If the second order process has time constants 2 and 4 min and process gain 1.0 [units], what proportional gain would provide us with a system with damping ratio of 0.7 ... 

The system steady state gain is the same as that with proportional control in Example 5.1. We, of course, expect the same offset with PD control too. The system time constant depends on various parameters. Again, we defer this analysis to when we discuss root locus. 

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 (). 

A proportional controller is used to control a process which may be represented as two non-interacting first-order lags each having a time constant of 600 s (10 min). The only other lag in the closed loop is the measuring unit which can be approximated by a distance/velocity lag equal to 60 s (1 min). Show that, when the gain of a proportional controller is set such that the loop is on the limit of stability, the frequency of the oscillation is given by ... 

A control loop consists of a proportional controller, a first-order control valve of time constant rv and gain Kv and a first-order process of time constant T and gain Kx. Show that, when the system is critically damped, the controller gain is given by ... 

Assuming that Gi and H are constant and written as Ki and K2 respectively, that is the time constants of the final control element and measuring element are negligible in comparison with those of the process, and that the proportional controller has a gain Kc,... 

Choose a suitable temperature setpoint and simulate the reactor with control, first with proportional control only and then including integral control. Adjust the controller constants to obtain adequate control. 

The liquid flow rate from a vertical cylindrical tank, 10 feet in diameter, is flow-controlled. The liquid flow into the tank is manipulated to control liquid level in the tank. The control valve on the inQow stream has linear installed characteristics and can pass 1000 gpm when wide open. The level transmitter has a span of 6 feet of liquid. A proportional controller is uaed with a gain of 2. Liquid density is constant. 

A proportional controller merely multiplies the magnitude of at every frequency by a constant. On a Bode plot, this means a proportional controller raises the log modulus curve by 20 logiQ decibels but has no effect on the phase-angle curve. See Fig. 13.13n. 

Suppose the temperature control of a bioreactor using heat supply with a proportional controller. When a proportional controller is tuned at a set point of 30 °C, as long as the set point remains constant, the temperature will remain at 30 °C successfully. Then, if the set point is changed to 40 °C, the proportional controller increases the output (heat supply) proportional to the error (temperature difference). Consequently, a heat supply will continue until the temperature gets to 40 °C and would be off at 40 °C. However, the temperature of a bioreactor will not reach 40 °C because a heat loss from the bioreactor increases due to the temperature increase. Finally, the heat supply matches the heat loss, at this point, the temperature error will remain constant therefore, proportional controller will keep its output constant. Now the system keeps in a steady state, but the temperature of a bioreactor is below its set point. This residual error is called Offset. 

Power proportioning controllers are generally more sophisticated forms of the constant voltage units, employing solid state circuitry and rapid time-constant sensing circuits. An isolation transformer isolates the controller from line voltage fluctuations. If adequate insulation of the column from ambient air changes is provided, the controllers can be quite accurate. 

The level of liquid in a tank is controlled using a pneumatic proportional controller as shown in Fig. 7.6. The level sensor is able to measure over the range 1.85 to 2.2S m. It is found that, after adjustment, the controller output pressure changes by 4 kN/m2 for a 0.01 m variation in level with the desired value held constant. If a variation in output pressure of 80 kN/m2 moves the control valve from fully open to fully closed, determine the gain and the proportional band. 

Referring to Fig. 7.36, for a proportional controller Gc(j) = Kc (equation 7.62). Assume for simplicity that the time constants of G,(j) (the final control element) and H(j) (the measuring element) are negligible compared with that of G2(j) (the... 

Reactor volume VR can vary with time and is controlled by a proportional controller that manipulates the reactor effluent stream /%, Heat capacity cP is assumed constant (3137 J kg-1 K-1), as is the heat of vaporization AHv (23.24 x 106 J/kmol). The heat of reaction is removed by vaporizing the liquid. Note that there is no heat transfer in the reactor ... 

It is well known that there are closed loop locations which can not be reached by constant proportional control using less than full state feedback. The common approach in the case where proportional output feedback cannot yield a satisfactory design is to add an observer to the system. A similar but somewhat different approach is to use a dynamic controller. As an example, consider the control of a second order SISO plant by an ideal PID controller cascaded with a first order filter, which is... 

Proportional-only control should be used in nonreactive level loops for cascaded units in series. Even in reactor level control, proportional control should be considered to help filter flowrate disturbances to the downstream separation system. There is nothing necessarily sacred about holding reactor level constant. 

The critical product-quality and safety-constraint loops were tuned by using a relay -feedback test to determine ultimate gains and periods. The Tyreus-Luyben PI controller tuning constants were then implemented. Table 11.12 summarizes transmitter and valve spans and gives controller tuning constants for the important loops. Proportional control was used for all liquid levels and pressure loops. 

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