Proportional gain

The program can be used to generate time dependent displays of A, B and TR for input values of the effective proportional gain KC and set point temperature TSET, for initial conditions AO = BO = 0, T = TO at t = 0. 

Using the value of Sq = 330 kg/h, use program REFRIG2 to find the value of the proportional gain constant kp required to give stability. 

Feed temperature Latent heat of vaporization Average steam mass flow Proportional gain Set temperature of tank Time constant of thermocouple Time constant of thermowell constant of integral control 1,TFIN=30,NOCI=3 RESET GOTOl... 

For all commercial devices, the proportional gain is a positive quantity. Because we use negative feedback (see Fig. 5.2), the controller output moves in the reverse direction of the controlled variable.1 In the liquid level control example, if the inlet flow is disturbed such that h rises above hs, then e < 0, and that leads to p < ps, i.e., the controller output is decreased. In this case, we of course will have to select or purchase a valve such that a lowered signal means opening the valve (decreasing flow resistance). Mathematically, this valve has a negative steady state gain (-Kv)2... 

We may have introduced more confusion than texts that ignore this tiny detail. To reduce confusion, we will keep Kc a positive number. For problems in which the proportional gain is negative, we use the notation -Kc. We can think that the minus sign is likened to having flipped the action switch on the controller. 

Generally, the proportional gain is dimensionless (i.e.. p(t) and e(t) have the same units). Many controller manufacturers use the percent proportional band, which is defined as... 

A high proportional gain is equivalent to a narrow PB, and a low gain is wide PB. We can interpret PB as the range over which the error must change to drive the controller output over its full range.1... 

In some commercial devices, the proportional gain is defined as the ratio of the percent controller output to the percent controlled variable change [%/%]. In terms of the control system block diagram that we will go through in the next section, we just have to add gains to do the unit conversion. 

Because of the inherent underdamped behavior, we must be careful with the choice of the proportional gain. In fact, we usually lower the proportional gain (or detune the controller) when we add integral control. 

If simple proportional control works fine (in the sense of acceptable offset), we may try PD control. Similarly, we may try PID on top of PI control. The additional stabilizing action allows us to use a larger proportional gain and obtain a faster system response. 

We also find rather common that the proportional gain is multiplied into the bracket to give the integral and derivative gains ... 

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 ... 

While we have the analytical results, it is not obvious how choices of integral time constant and proportional gain may affect the closed-loop poles or the system damping ratio. (We may get a partial picture if we consider circumstances under which KcKp 1.) Again, we ll defer the analysis... 

In terms of the situation, if we use a PI controller on a slow multi-capacity process, the resulting system response will be even more sluggish. We should use PID control to increase the speed of the closed-loop response (being able to use a higher proportional gain) while maintaining stability and robustness. This comment applies to other cases such as temperature control as well. 

Consider the liquid flow rate controller in Fig R5.3. We want to keep the flow rate q constant no matter how the upstream pressure fluctuates. Consider if the upstream flow Q drops below the steady state value. How would you choose the regulating valve when you have (a) a positive and (b) a negative proportional gain ... 

All tuning relations provide different results. Generally, the Cohen and Coon relation has the largest proportional gain and the dynamic response tends to be the most underdamped. The Ciancone-Marlin relation provides the most conservative setting, and it uses a very small derivative time constant and a relatively large integral time constant. In a way, their correlation reflects a common industrial preference for PI controllers. 

While the calculations in the last example may appear as simple plug-and-chug, we should take a closer look at the tuning relations. The Cohen and Coon equations for the proportional gain taken from Table 6.1 are ... 

The choice of the proportional gain is affected by two quantities the product KcK, and the ratio of dead time to time constant, td/x. It may not be obvious why the product KCK is important now, but we shall see how it arises from direct synthesis in the next section and appreciate how it helps determine system stability in Chapter 8. 

B. Tuning relations based on closed-loop testing and the Ziegler-Nichols ultimate-gain (cycle) method with given ultimate proportional gain Kcu and ultimate period Tu. 

The proportional gain, integral time and derivative time constants are provided by the respective terms in the transfer function. If you have trouble spotting them, they are summarized in Table... 

The integral time constant is x = xb and the term multiplying the terms in the parentheses is the proportional gain Kc. In this problem, the system damping ratio Q is the only tuning parameter. 

Transient response criteria Analytical derivation Derive closed-loop damping ratio from a second order system characteristic polynomial. Relate the damping ratio to the proportional gain of the system. 

Ziegler-Nichols Continuous Cycling (empirical tuning with closed loop test) Increase proportional gain of only a proportional controller until system sustains oscillation. Measure ultimate gain and ultimate period. Apply empirical design relations. 

Root-locus With each chosen value of proportional gain, plot the closed-loop poles. Generate the loci with either hand-sketching or computer. 

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