Proportional Controller P

Explain the reason why a proportional control (P action) results in the off-set between the set-point and the measured process variable at a steady state. 

The proportional controller (P) produces its output proportional to the error... 

The response of a system to a disturbance is shown schematically in Fig. 7.44. The behaviour without control is represented by the solid line. Proportional control (P) reduces the maximum deviation of the controlled variable but gives rise to long-lived oscillations. At the new steady state there is a finite deviation of the controlled variable, known as the offset. An offset is characteristic of proportional control which requires an error in order to generate a control signal, equation 7.7.2. If the error (offset) is reduced to zero there is no control action. 

The proportional controller (P controller) assigns to each value of the deviation a specific value of the manipulated variable. For our tank example, the P controller mode takes the form of Eq. (81), where q, are the new output and the steady-state values of the flow rate respectively, Kc is the controller gain, and h p, h are the current level input height set point and measured value respectively. 

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. 

In industrial practice it is common to combine all three modes. The action is proportional to the error (P) and its change (D) and it continues if residual error is present (I). This combination gives the best control using conventional feedback equipment. It retains the specific advantages of all three modes proportional correction (P), offset elimination (1) and stabilising, quick-acting character, especially suitable to overcome lag presence (D). 

We expect a system with only a proportional controller to have a steady state error (or an offset). A formal analysis will be introduced in the next section. This is one simplistic way to see why. Let s say we change the system to a new set point. The proportional controller output, p = ps + Kce, is required to shift away from the previous bias ps and move the system to a new steady state. For p to be different from ps, the error must have a finite non-zero value.3... 

On the plus side, the integration of the error allows us to detect and eliminate very small errors. To make a simple explanation of why integral control can eliminate offsets, refer back to our intuitive explanation of offset with only a proportional controller. If we desire e = 0 at steady state, and to shift controller output p away from the previous bias ps, we must have a nonzero term. Here, it is provided by the integral in Eq. (5-5). That is, as time progresses, the integral term takes on a final nonzero value, thus permitting the steady state error to stay at zero. 

Example 7.4A. This time, let s revisit Example 7.4 (p. 7-8), which is a system with dead time. We would like to know how to start designing a PI controller. The closed-loop characteristic equation with a proportional controller is (again assuming the time unit is in min)... 

B. PROPORTIONAL CONTROLLER. The output of a proportional controller changes only if the error signal changes. Since a load change requires a new control-valve position, the controller must end up with a new error signal. This means that a proportional controller usually gives a steadystate error or offset. This is an inherent limitation of P controllers and why integral action is usually added. 

The Bode plot of is given in Fig. 13.20 for D = 0.5. The ultimate gain is 3.9 (11.6 dB), and the ultimate frequency is 3.7 radians per minute. The ZN controller settings for P and PI controllers and the corresponding phase and gain margins and log moduli are shown in Table 13.2 for several values of deadtime D. Also shown are the values for a proportional controller that give +2-dB maximum closedloop log modulus. 

Notice in Fig. 13.20 that the curve for the P controller does not approach 0 dB at low frequencies. This shows that there is a steadystate offset with a proportional controller. The curve for the PI controller does go to 0 dB at low frequencies because the integrator drives the closedloop servo transfer function to unity (i.e., no offset). 

Also shown in Fig. 16.1 is the W plot when only proportional controllers are used. Note that the curves with P controllers start on the positive real axis. However, with PI controllers the curves start on the negative real axis. This is due to the two integrators, one in each controller, which give 180 degrees of phase angle lag at low frequencies. As shown in Eq. (16.3), the product of the and B2 controllers appears in the closedloop characteristic equation. 

The influence of data im and total photon count N = nm is exerted by adding the appropriate constraints to In P in proportions controlled by Lagrange multipliers Xm and p. The most likely solution hm is then found by maximizing... 

Proportional-lntegral-Derivative Control The most common algorithm for control action in the feedback loop of processing industries is the PID control, which is a combination of proportional action (P), integral action (1), and differential action (D). 

However, the ideal control algorithm would have no overshoot, no offset, and a quick response characteristic. For this purpose, a proportional action (P), an integral action (I), and a differential action (D) were combined as a PID controller as follows. 

By inserting a restrictor in the line to the proportional bellows, any change in P will not be transmitted immediately to the feedback system. Thus, initially, the arrangement shown in Fig. 7.117 will behave as a narrow-band proportional controller changing to wide-band action as the pressure in the feedback bellows Pd approaches P. The rate at which PD - P depends upon the resistance to flow RDr through the derivative restrictor. This mechanism thus simulates derivative action in that it is most sensitive when the error is changing the most rapidly (Section 7.2.3). The derivative time rd is varied by adjusting Rdr-... 

The response of the proportional (P) mode (as the name suggests) is proportional to the error. The proportional setting Kc, is called the proportional gain, frequently expressed in terms of percent proportional band, PB, which is inversely related to the proportional gain (Kc = 100/PB). "Wide bands" (high PB values) result in less sensitivity (lower gains), and "narrow bands" (low PB percentages) result in more "sensitive" controller response. The output of a proportional controller (m) is... 

Schwartz, P., Rosenthal, N., and Wehr, T., Serotonin 1A receptors, melatonin, and the proportional control thermostat in patients with winter depression, Archives of General Psychiatry, Vol. 55, No. 10, 1998, pp. 897-903. 

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