Proportional control

The simplest idea is that the compensation signal (actual controller output) is proportional to the error e(t):

p(t) = ps + Kce(t) = ps + Kc[hs – h(t)] (5-2) where Kc is the proportional gain of the controller. It is obvious that the value of Kc

determines the controller "sensitivity"—how much compensation to enact for a given change in error.

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

Now what if our valve has a positive gain? (Increased signal means opening the valve.) In this case, we need a negative proportional gain. Commercial devices provide such a “switch” on the controller box to invert the signal. Mathematically, we have changed the sign of the compensation term to: p = ps – Kce. 3

By the definition of a control problem, there should be no error at t = 0, i.e., es = 0, and the deviation variable of the error is simply the error itself:

e'(t) = e(t) – es = e(t)

1 _{You may come across the terms reverse and direct acting and they are extremely} confusing. Some authors consider the action between the controller output and the controlled variable, and thus a negative feedback loop with a positive Kc is considered reverse-acting. However, most commercial vendors consider the action between the error (controller input) and the controller output, and now, a controller with a positive Kc is direct-acting, exactly the opposite terminology. We’ll avoid using these confusing terms. The important point is to select the proper signs for all the steady state gains, and we'll get back to this issue in Section 5.4.

2 _{Take note that from the mass balance of the tank, the process gain associated with the outlet} flow rate is also negative. A simple-minded check is that in a negative feedback system, there can only be one net negative sign—at the feedback summing point. If one unit in the system has a negative steady state gain, we know something else must have a negative steady state gain too. 3 _{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.

Hence Eq. (5-2) is a relation between the deviation variables of the error and the controller output: p(t) – ps = Kc [e(t) – es] , or p'(t) = Kce'(t)

and the transfer function of a proportional controller is simply

Gc(s) = P(s)_E(s) = Kc (5-3)

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

PB = 100

Kc (5-4)

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

Before doing any formal analysis, we state a few qualitative features of each type of controller. This is one advantage of classical control. We can make fairly easy physical interpretation of the control algorithm. The analyses that come later will confirm these qualitative observations.

General qualitative features of proportional control

• We expect that a proportional controller will improve or accelerate the response of a process. The larger Kc is, the faster and more sensitive is the change in the compensation with respect to a given error. However, if Kc is too large, we expect the control compensation to overreact, leading to oscillatory response. In the worst case scenario, the system may become unstable. • There are physical limits to a control mechanism. A controller (like an amplifier) can deliver only so much voltage or current; a valve can deliver only so much fluid when fully opened. At these limits, the control system is saturated.2

• 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

• To tackle a problem, consider a simple proportional controller first. This may be all we need (lucky break!) if the offset is small enough (for us to bear with) and the response is adequately fast. Even if this is not the case, the analysis should help us plan the next step.

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.

2 _{Typical ranges of device outputs are 0–10 V, 0–1 V, 4–20 mA, and 3–15 psi.} 3 _{The exception is when a process contains integrating action, i.e., 1/s in the transfer} functions—a point that we will illustrate later.