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eHANDBOOK

Level Measurement

Spring 2019

www.controlglobal.com

eHandbook: Level Measurement, Spring 2019 2

TABLE OF CONTENTSWhen is reducing variability wrong? 4

Level may vary the most when process and product variation are minimized.

Solutions to prevent harmful feedforwards 8

How to correct issues in boiler, distillation column and neutralization control.

Understanding P, I and D 12

The simple mathematics can be clarified with mechanical analogies and an example of level control.

Is global warming like level control? 20

Comparing the processes sheds light on the problem of regulating Earth’s temperature.

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When is reducing variability wrong?Level may vary the most when product and process variation are minimized.by Greg McMillan

Having the blind wholesale goal of reducing variability can lead to doing the wrong thing that can reduce plant safety and performance. Here

we look at some common mistakes made

that users may not realize until they have

better concepts of what is really going on.

We seek to provide some insightful knowl-

edge here to keep you out of trouble.

Is a smoother data historian plot or a statis-

tical analysis showing less short-term vari-

ability good or bad? The answer is no for

the following situations, misleading users

and data analytics.

First of all, the most obvious case is surge

tank level control. Here we want to maxi-

mize the variation in level to minimize the

variation in manipulated flow, typically to

downstream users. This objective has a

positive name of absorption of variability.

What this is really indicative of is the prin-

ciple that control loops don’t make vari-

ability disappear, but transfer variability

from a controlled variable to a manipulated

variable. Process engineers often have a

problem with this concept because they

think of setting flows per a Process Flow

Diagram (PFD) and are reluctant to let a

controller freely move them per some al-

gorithm they don’t fully understand. This is

seen in predetermined sequential additions

of feeds or heating and cooling in a batch

operation rather allowing a concentration or

temperature controller do what is needed

via fed-batch control. No matter how smart

a process engineer is, not all of the situa-

tions, unknowns and disturbances can be

accounted for continuously. This is why fed-



batch control is called semi-continuous. I

have seen where process engineers, believe

or not, sequence air flows and reagent flows

to a batch bioreactor rather than going to

Dissolved Oxygen or pH control. We need

to teach chemical and biochemical engi-

neers process control fundamentals, includ-

ing the transfer of variability.

The variability of a controlled variable is

minimized by maximizing the transfer of

variability to the manipulated variable. Un-

necessary sharp movements of the ma-

nipulated variability can be prevented by

a setpoint rate of change limit on analog

output blocks for valve positioners or VFDs,

or directly on other secondary controllers

(e.g., flow or coolant temperature), and the

use of external-reset feedback (e.g., dy-

namic reset limit) with fast feedback of the

actual manipulated variable (e.g., position,

speed, flow or coolant temperature). There

is no need to re-tune the primary process

variable controller by the use of external-

reset feedback.

Data analytics programs need to use ma-

nipulated variables in addition to controlled

variables to indicate what is happening.

For tight control and infrequent setpoint

changes to a process controller, what is

really happening is seen in the manipulated

variable (e.g., analog output).

A frequent problem is data compression in

a data historian that conceals what is really

going on. Hopefully, this is only affecting

the trend displays and not the actual vari-

ables being used by a controller.

The next most common problem has been

extensively discussed by me, so at this point

you may want to move on to more pressing

needs. This problem is the excessive use of

signal filters that may even be more insidi-

ous because the controller doesn’t see a de-

veloping problem as quickly. A signal filter

that is less than the largest time constant in

the loop (hopefully in the process) creates

dead time. If the signal filter becomes the

largest time constant in the loop, the previ-

ously largest time constant creates dead

time. Since the controller tuning based on

largest time constant has no idea where

it is, the controller gain can be increased,

which, combined with the smoother trends,

We need to teach chemical and biochemical

engineers process control fundamentals,

including the transfer of variability.



can lead one to believe the large filter was

beneficial. The key here is a noticeable in-

crease in the oscillation period, particularly

if the reset time was not increased. Signal

filters become increasingly detrimental as

the process loses self-regulation. Integrat-

ing processes such as level, gas pressure

and batch temperature are particularly

sensitive. Extremely dangerous is the use of

a large filter on the temperature measure-

ment for a highly exothermic reaction. If the

PID gain window (ratio of maximum to mini-

mum PID gain) reduces due to measure-

ment lag to the point of not being able to

withstand nonlinearities (e.g., ratio less than

6), there is a significant safety risk.

A slow thermowell response, often due to

a sensor that is loose or not touching the

bottom of the thermowell, causes the same

problem as a signal filter. An electrode that

is old or coated can have a time constant

that is orders of magnitude larger (e.g., 300

sec) than a clean, new pH electrode. If the

velocity is slightly low (e.g., less than 5 fps),

pH electrodes become more likely to foul

and if the velocity is very low (e.g., less than

0.5 fps), the electrode time constant can

increase by one order of magnitude (e.g.,

30 sec) compared to an electrode seeing

recommended velocity. If the thermowell

or electrode is being hidden by a baffle, the

response is smoother but not representa-

tive of what is actually going on.

For gas pressure control, any measure-

ment filter, including that due to transmitter

damping, generally needs to be less than

0.2 sec, particularly if volume boosters on

valve positioner output(s) or a variable-

frequency drive is needed for a faster

response.

Practitioners experienced in doing Model

Predictive Control (MPC) want data com-

pression and signal filters to be completely

removed so that the noise can be seen and

a better identification of process dynamics,

especially dead time, is possible.

Virtual plants can show how fast the ac-

tual process variables should be changing,

revealing poor analyzer or sensor resolution

and response time, and excessive filtering.

In general, you want measurement lags to

total up to less than 10% of the total loop

dead time, or less than 5% of reset time.

However, you can’t get a good idea of the

loop dead time unless you remove the

filter and look for the time it takes to see a

change in the right direction, beyond noise,

after a controller setpoint or output change.

For more on the deception caused by a

measurement time constant, see the Con-

trol Talk Blog, “Measurement Attenuation

and Deception.”

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Solutions to prevent harmful feedforwardsHow to correct issues in boiler, distillation column and neutralization control.by Greg McMillan

Here we look at applications where feedforward can do more harm than good, and what to do to prevent this situation. This problem is

more common than one might think. In the

literature, we mostly hear how beneficial

feedforward can be for measured load

disturbances. Statements are made that

the limitation is the accuracy of the feed-

forward and that, consequently, an error of

2% can still result in a 50:1 improvement in

control. This optimistic view doesn’t take

into account process, load and valve dy-

namics. The feedforward correction needs

to arrive in the process at the same point

and the same time as the load disturbance.

This is traditionally achieved by passing

the feedforward (FF) through a deadtime

block and lead-lag block. The FF dead-

time is set equal to the load path deadtime

minus the correction path deadtime. The

FF lead time is set equal to the correction

path lag time. The FF lag time is set equal

to the load path lag time. If the FF arrives

too soon, we create inverse response,

and if the FF arrives too late, we create

a second disturbance. Setting up tuning

software to identify and compute the FF

dynamic can be challenging. Even more

problematic are the following feedforward

applications that do more harm than good

despite dynamic compensation.

1. Inverse response from the manipulated

flow causes excessive reaction in the

opposite direction of load. The inverse

response from a feedwater change can be

so large as to cause a boiler drum high or

low level trip, a situation that particularly

occurs for undersized drums and miss-



ing feedwater heaters due to misguided

attempts to save on capital costs. The

solution here is to use a traditional three-

element drum level control, but added to

the traditional feedforward is an uncon-

ventional feedforward with the opposite

sign that is decayed out over the period of

the inverse response. In other words, for a

step increase in steam flow, there would be

initially a step decrease in boiler feedwater

feedforward added to the three-element

drum level controller output that is trying

to increase feedwater flow. This prevents

shrink and a low level trip from bubbles

collapsing in the downcomers from an

increase in cold feedwater. For a step de-

crease in steam flow, there would be a step

increase in boiler feedwater feedforward

added to the three-element drum level

controller output that is trying to decrease

feedwater flow. This prevents swell and

a high level trip from bubbles forming in

the downcomers from a decrease in cold

feedwater. A severe problem of inverse re-

sponse can occur in furnace pressure con-

trol when the scale is a few inches of water

column and the incoming manipultaed

air is not sufficiently heated. The inverse

response from the ideal gas law can cause

a pressure trip. An increase in cold air flow

causes a decrease in gas temperature and,

consequently, a relatively large decrease

in gas pressure at the furnace pressure

sensor. A decrease in cold air flow causes

an increase in gas temperature and, con-

sequently, a relatively large increase in gas

pressure at the furnace pressure sensor.

2. Deadtime in the correction path is

greater than deadtime in the load path. The

result is a feedforward that arrives too late,

creating a second disturbance and worse

control than if there was no feedforward.

This occurs whenever the correction path

is longer than the load path. An example is

a distillation column control when the feed

load upset stream is closer to the tempera-

ture control tray than the corrective change

in reflux flow. The solution is to generate the

feedforward signal for ratio control based

on a setpoint change that is then delayed

before being used by the feed flow control-

ler. The delay is equal to the correction path

deadtime minus the load path deadtime. The

The result is a feedforward that arrives too late,

creating a second disturbance and worse control

than if there was no feedforward.



same problem can occur for a reagent injec-

tion delay that often occurs due to conven-

tionally-sized dip tubes and small reagent

flows. The same solution applies in terms of

using an influent flow controller setpoint for

feedforward ratio control of reagent, and

delaying the setpoint used by the influent

flow controller.

3. Feedforward correction makes re-

sponse from an unmeasured disturbance

worse. This occurs in unit operations

such as distillation columns and neutral-

izers where the unmeasured disturbance

from a feed composition change is made

worse by a feedforward correction based

on feed flow. Often, feed composition is

not measured and is large due to parallel

unit operations and a combination of flows

that become the feed flow. For pH, the

nonlinearity of titration curves increases

the sensitivity to feed composition. Even if

the influent pH is measured, the pH elec-

trode error or uncertainty of the titration

curve makes feedforward correction for

feed pH to do more harm than good for

setpoints on the steep part of the curve.

If the feed composition change requires

a decrease in manipulated flow and there

is a coincidental increase in feed flow that

corresponds to an increase in manipulated

flow or vice-versa, the feedforward does

more harm than good. The solution is to

compute the required rate of change of

manipulated flow from the unmeasured

disturbance, and add this to the computed

rate of change for the feedforward correc-

tion needed, paying attention to the signs

of the rate of change. If the required rate

of change of manipulated flow for the un-

measured disturbance is in the opposite di-

rection, the feedforward correction rate of

change in manipulated flow is decreased. If

it exceeds the computed feedforward cor-

rection rate of change in the manipulated

flow, the feedforward rate of change is

clamped at zero to prevent making con-

trol terribly worse. If the required rates of

change for the manipulated flow are in the

same direction, the magnitude of the feed-

forward rate of change is correspondingly

increased.

I am trying to see how all this applies in my

responses to known and unknown upsets to

my spouse.

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Understanding P, I and DThe simple mathematics can be clarified with mechanical analogies and an example of level control.by R. Russell Rhinehart

It is important to understand what the proportional, integral and derivative terms do within the PID controller. That understanding is essential to choose ap-

propriate action, troubleshoot controllers,

choose appropriate modifications, or set

up advanced controllers. Unfortunately, the

controller synthesis approach, in which PID

magically appears within Laplace Transform

analysis, does not provide that functional

understanding. Hopefully, this more intuitive

development of PID will be helpful.

Chosen as an example is a commonly un-

derstood process—level control in a tank of

liquid (Figure 1). The inflow is a wild variable,

or disturbance, that will upset the level, which

is an indication of liquid inventory. The slide

valve on the outflow will open or close to re-

lease more or less fluid to keep the level at

the desired setpoint. As notation, the con-

trolled variable, CV, is the liquid level in the

tank, h, and the manipulated variable, MV, is

the valve stem position, U. Recognize that

your region or community may use alternate

terminology. The nominal, initial steady state

values are h0 and U0.

The controller, also shown in Figure 1, is

a mechanical lever, a proportional-only

controller. If the liquid level rises somewhat,

then the float rises the same amount, which

raises the lever that opens the valve an ad-

ditional amount by the lever proportional-

ity. This releases fluid faster, which seeks

to counter the rising liquid inventory. If the

liquid level falls somewhat, the lever closes

the valve proportionally.

The liquid level may rise or fall for any

number of reasons. The inflow rate, Fin,

may change, the viscosity of the fluid may

change affecting its outflow speed, or the



downstream pressure or in-pipe flow restric-

tions may change, affecting Fout. The reason

for a rise or fall in level is immaterial for the

controller. This lever action moves the valve

in the appropriate direction, and in a manner

proportional to the level change.

PROPORTIONAL ACTIONThe lever arm length ratio, b/a, is the gain

of the controller. By changing the relative

lengths of the lever arms, perhaps by chang-

ing the position of the fulcrum, the controller

can be more or less aggressive. If the level, h,

starts at a base case of h0, which is also the

setpoint, hSP, then using simple relations, the

equation for the change in the valve stem po-

sition with respect to level is:

∆U = b a ∆h = - b a (hSP - h) = Kce (1)

Where Kc = - b a represents the controller gain

and e = (hSP - h) represents the actuating er-

ror, the deviation of the CV from its setpoint.

Since ∆U = U - U0, then the equation rep-

resenting the lever-type control is:

U = U0 + Kc e (2)

Figure 2 shows a block diagram of a con-

trolled process using generic symbols for

the MV, U; the CV, Y; and the disturbance, d.

Contrasting the illustration of the physical

process of Figure 1, this represents the path

of cause-and-effect information exchange.

The controller is shown acting on the actu-

ating error, e.

Note that although termed the process out-

put, Y, the level of the liquid does not come

out of the tank. The material liquid goes in

or comes out, and level is a measure of the

inventory response of the in-tank contents.

The block diagram reveals the information

LEVEL CONTROL EXAMPLEFigure 1: Here, inflow is a variable that will upset the level gauge float, which acts on the lever arm that controls the slide valve on the outflow. The controlled variable, CV, is the liquid level in the tank, h, and the manipulated variable, MV, is the valve stem position, U.

ho

Fin

Uo

b

a

Fout

ho

Fin

Uo

Uo

U

b

a

Fout

d

U Y

U

C+

-

+

+

Pe

e

YSP

Kc

Uo Fixed bias

Turnbuckle

Oppositethreads

+

+

+

+

e

ê

Kcess

+

+Kc

∫dt1τ1

1τ1s

τps

Û

LEVEL CONTROL BLOCK DIAGRAMFigure 2: A block diagram of the control scheme in Figure 1 shows the information transfer between controller and process, not material exchange. The block labeled C is the controller, which represents the lever, but could be a calculator that executes Equation (2): it multiplies Kc times e, then adds it to U0.

ho

Fin

Uo

b

a

Fout

ho

Fin

Uo

Uo

U

b

a

Fout

d

U Y

U

C+

-

+

+

Pe

e

YSP

Kc

Uo Fixed bias

Turnbuckle

Oppositethreads

+

+

+

+

e

ê

Kcess

+

+Kc

∫dt1τ1

1τ1s

τps

Û



transfer between controller and process, not

material exchange. The lines in Figure 2 are

not pipes. The block labeled C is the control-

ler, which represents the lever, but could be

a calculator that executes Equation (2). It is

simple arithmetic: the controller multiplies Kc

times e, then adds it to U0.

Equation (2) seems very much like what is

often presented as a proportional control-

ler, U = Kc e. However, if U = Kc e was the

relation, then if the CV were at the setpoint

(e = 0), the controller would set U = 0, and

close the valve, which would make the level

rise, and cause it to deviate from the set-

point. In Equation (2), the term U0 is the

controller bias—it’s the value of U for which

the initial MV position is required to hold

the CV at setpoint. As illustrated in Figure 1,

this is about 50%.

The user chooses the controller gain. Nor-

mally, the controller starts in manual mode

(MAN) with the user deciding the MV value,

then when the CV is at the setpoint, the

user switches the controller to automatic

(AUTO) mode. For bumpless transfer, the

bias is usually set by the controller as the

MV value when the controller is switched

from MAN to AUTO.

Figure 3 shows a block diagram of the arith-

metic operations in the P-only control logic

of Equation (2). The actuating error is multi-

plied by the controller gain, and then added

to the bias to determine the controller out-

put. These are simple arithmetic operations

(not calculus or Laplace-transformed magic).

Note that It does not matter whether the

controller in Figure 1 is an actual physical

float-and-lever device, or whether it’s a digi-

tal calculation of Equation (2) in a computer

that sends the valve stem position target to

an i/p device to move the valve stem. The

logic and action are identical.

STEADY STATE OFFSET AND INTEGRAL ACTIONP-only control is often good enough, but

its problem is steady state offset. Consider

what happens if Fin increases and holds at

a new value, when the level is initially at the

setpoint. Initially, Fout remains the same be-

cause h has not yet changed, and the valve

stem position is at the initial U0. Then, since

the new inflow is greater than the outflow,

level rises. As h rises, this increases U, which

increases Fout Eventually, Fout will match Fin

and the level will stop rising, but at this new

PROPORTIONAL ONLYFigure 3: A block diagram of the P-only control calculation of Equation (2) shows the actuating error is multiplied by the controller gain, and then added to the bias to deter-mine the controller output. These are simple arithmetic operations (not calculus or Laplace-transformed magic).

ho

Fin

Uo

b

a

Fout

ho

Fin

Uo

Uo

U

b

a

Fout

d

U Y

U

C+

-

+

+

Pe

e

YSP

Kc

Uo Fixed bias

Turnbuckle

Oppositethreads

+

+

+

+

e

ê

Kcess

+

+Kc

∫dt1τ1

1τ1s

τps

Û



steady state, h is not at the setpoint. It must

be above the setpoint for the valve to be

open enough to let the outflow be higher.

This h deviation is steady state offset.

It is immaterial whether the disturbance is

the inflow rate or some other aspect that af-

fects either the inflow or outflow, or wheth-

er the level falls or rises as a response to the

disturbance. If the disturbance persists, the

process will not settle at the setpoint.

A method to eliminate steady state offset

is to add integral action. But, that calculus

terminology has little physical meaning, so

to add understanding, consider the injection

of a turnbuckle to the valve stem (Figure 4).

In a turnbuckle, the ends of the rods are

threaded to fit into the threaded holes of

the buckle. The threads go in opposite

directions. So, if the buckle is turned in one

direction, the two sections of the valve stem

are pulled together, the stem is shortened,

and the valve opens. If turned in the other

direction, the stem is lengthened and the

valve closes. This permits opening and clos-

ing of the valve without a change in tank

level.

If you notice that the liquid level has risen,

you know that some disturbance is acting,

and the buckle needs to be turned to short-

en the stem, to open the valve a bit more

than the original, pre-disturbance position

for the tank level. If the level rises a little bit,

there is only a small upset, and the buckle

only needs to be turned a little. In contrast,

a large level deviation indicates a large

upset has occurred, which justifies a large

turnbuckle readjustment. And, of course, a

level rise or drop would direct turns in the

opposite direction.

So, let’s have an observer follow this rule:

At each sampling, observe the level devia-

tion from setpoint, and make an incremental

change in the turnbuckle angle that’s pro-

portional to the level deviation. In Equation

(3), ∝ represents the thread pitch (axial

distance per angle), β is the proportionality

rule to change angle due to level deviation

from setpoint (angle per h deviation), and

c=∝β is their product:

∆U = ∝ ∆θ = ∝ βe = ce (3)

After the most recent sampling, the ith ob-

servation, control action changes the valve

stem length from the previous length:

INTEGRAL IS LIKE A TURNBUCKLEFigure 4. Integral action eliminates steady state offset, like adding a turnbuckle to the valve stem. If the buckle is turned in one direction, the stem is shortened and the valve opens; if turned in the other direction, the stem is lengthened and the valve closes, without a change in the tank level.

ho

Fin

Uo

b

a

Fout

ho

Fin

Uo

Uo

U

b

a

Fout

d

U Y

U

C+

-

+

+

Pe

e

YSP

Kc

Uo Fixed bias

Turnbuckle

Oppositethreads

+

+

+

+

e

ê

Kcess

+

+Kc

∫dt1τ1

1τ1s

τps

Û



Ui = U i - 1 + cei (4)

The sequence of the past two adjustments is:

Ui = U i-2 + cei + ce i-1 (5)

Continuing to include past adjustments

generates an equation that indicates how all

of the adjustments have contributed to the

current valve stem length change from the

initial value, l0:

Ui = U0 + c∑Ni=1 ei (6)

In Equation (6), the number of items in the

sum is N = t ∆t , where t is the total time that

the controller has been in AUTO, and ∆t is the

controller sampling interval. Multiplying and

dividing the sum by ∆t reveals that the sum

of rectangles (e-height times ∆t-base) is just

the rectangle rule of integration, which can be

represented by the calculus notation.

U(t) = Ui = U0 + c ∆t ∑ ei ∆t = U0 +

c ∆t ∫

t o edt (7)

Note that, even though the integral symbol

is used in Equation (7), no calculus proce-

dure was used by the turnbuckle adjuster. If

this were to be implemented by a comput-

er, the Equation (4) adjustment in the valve

stem length is not calculus, but a simple

algebraic multiplication and addition. Fur-

ther, in the computer, the subscripts are not

needed. The assignment statement repre-

senting the Equation (4) action is U: = U +

ce. Don’t let the integral symbol misdirect

your understanding. There is no calculus to

the doing of control.

A common form of the controller calcula-

tion is to incrementally sum (integrate) the

scaled error, KC e, which means that the c ∆t

coefficient needs to be divided by KC. Since

the term tKCc only has dimensions of time, τi,

one can represent the PI controller function

as the block diagram of Figure 5. It shows

the integral value (really, it’s just the sum of

incremental changes) is added to the initial

bias to make the controller bias adjust at

each sampling. The block diagram nota-

tion indicates the function (inside the box)

and the function argument (the box input

value). But again, don’t be thinking calculus,

the integral operation is simply an arithme-

tic incremental accumulation.

In the standard form, KC is the controller

gain that multiplies both the P and I terms,

STANDARD PI CONTROLLERFigure 5. A block diagram of the standard form of the PI controller shows the integral value (the sum of incremental changes) is added to the initial bias to make the control-ler bias adjust at each sampling. The notation indicates the function (inside the box) and the function argument (the box input value).

ho

Fin

Uo

b

a

Fout

ho

Fin

Uo

Uo

U

b

a

Fout

d

U Y

U

C+

-

+

+

Pe

e

YSP

Kc

Uo Fixed bias

Turnbuckle

Oppositethreads

+

+

+

+

e

ê

Kcess

+

+Kc

∫dt1τ1

1τ1s

τps

Û



and the integral time, τi, divides the integral

(which is actually calculated by incremental

summation).

Recall, the purpose of the proportional ac-

tion was to immediately counter the effect

of a disturbance, but its problem is that it

leaves a steady-state offset. The integral

purpose was not to be the primary control

action, but to remove the offset left by the

P-action. Accordingly, tune the P first (KC)

to set the aggressiveness of the controller,

then adjust the I-action (τi) to remove the

residual offset at a desirable rate.

ANTICIPATED ERROR AND DERIVATIVE MODEIn the description of P-action, the controller

acts on the initial impact of the disturbance

on the CV. However, the initial reveal that a

disturbance has happened might be a small

CV deviation, but, if allowed to fully develop

over time, it might evolve to a large value.

The fully developed CV deviation, not the

initial indication, represents the magnitude

of the disturbance that’s causing the CV

to start deviating. With derivative action,

the controller will take proportional action

based on the anticipated error, not just on

the initial reveal of the CV deviation. The

question is how to forecast the anticipated

error, the fully developed result of a distur-

bance?

If the process is linear and has a first-order

response to a disturbance, then the model

of how e would respond to a change in a

disturbance, ∆d, is:

τd dedt

+ e = Kd (∆d) = eanticipated (8)

The value of eanticipated is the steady, fully de-

veloped, anticipated value.

Note that if the disturbance could be mea-

sured and the process gain to the distur-

bance were known, then eanticipated could be

calculated from Kd (∆d) = eanticipated. However,

those are often unmeasured and unknown.

Fortunately, Equation (8) reveals that one

can estimate the anticipated future error

based on the current actuating error and

its rate of change, the values of which are

already known by the controller.

eanticipated = τd dedt

+ e (9)

Equation (9) does not specify what the dis-

turbance is. The deviation could indicate a

confluence of several disturbances, includ-

ing the MV. Equation (9) is called a “lead,”

which should be familiar. It represents what

a ball thrower must do to have a running

target catch the ball. The ball must be

thrown to where the receiver will be when

the ball gets there. The PI controller with

the P-action based on eanticipated is:

U = KC eanticipated + KCτi

∫ edt + U0 (10)

When Equation (9) is substituted into Equa-

tion (10) and rearranged, the PI controller



with P-action on the anticipated error is the

classical PID relation:

U = KC e + KCτi

∫ edt + KC τd dedt

+ U0 (11)

Although Equation (11) looks like calculus

with its ∫ edt and dedt representation, the

integral is actually just the incrementally

updated sum, and the derivative will be cal-

culated from a numerical approach (enew - eold)∆t, which, again, is simple arithmetic subtrac-

tion and division.

PD-action is equivalent to P-action on

the anticipated error. Whether D action

is used or not, one still needs incremental

adjustment to the bias, I-action, to remove

steady-state offset.

If the process measurement is noisy, the

numerical derivative amplifies the noise im-

pact. And, if the process is relatively quick

to respond, there’s no need to use the antic-

ipated error concept. So, only use D-action

on noiseless and slow-to-evolve processes.

In block diagram notation, the PID con-

troller (a PI controller with P based on the

anticipated error, and the incremental ad-

justment to remove steady state offset) is

represented in Figure 6. The figure uses the

Laplace transformed notation, where s in-

dicates the operation to take the derivative

of the input to the function block, and 1/s

indicates to integrate the input. But, again,

regardless of the symbols or calculus words

to describe the functions all are simple

arithmetic operations.

PID IN SUMMARYProportional control is the simple concept

of taking immediate proportional action on

the actuating error, but P-only control,

U = KC e + b0, with a fixed bias, leaves

steady state offset. The user chooses the

value for KC to set controller aggressive-

ness.

Integral action incrementally adjusts the

bias to remove steady state offset. Note, al-

though called “integral,” there’s no calculus

in the action in the incremental accumula-

tion. The user chooses the value for τi to set

the speed at which offset is removed.

Derivative action forecasts what the actuat-

ing error will be, as a result of past influenc-

es, if they’re left uncontrolled. It’s the lead

commonly used in hitting a moving target.

PD-action is equivalent to P-action on the

anticipated error, and leaves steady state

offset. Note, although called “derivative,”

STANDARD PID CONTROLLERFigure 6. This block diagram of a PID control-ler uses the Laplace transformed notation, where s indicates the operation to take the derivative of the input to the function block, and 1/s indicates to integrate the input.

ho

Fin

Uo

b

a

Fout

ho

Fin

Uo

Uo

U

b

a

Fout

d

U Y

U

C+

-

+

+

Pe

e

YSP

Kc

Uo Fixed bias

Turnbuckle

Oppositethreads

+

+

+

+

e

ê

Kcess

+

+Kc

∫dt1τ1

1τ1s

τps

Û



there is no calculus in the action of numeri-

cally estimating the CV rate of change. The

user chooses the value for τd to lead the aim

of the controller.

We communicate the PID procedure with

calculus, or Laplace or Z-transforms, or oth-

er advanced mathematical symbols. With a

bit of sarcasm, it seems the reason to use

fancy mathematics is to make people think

it’s difficult, so they need to hire an expert.

But, the reality is that the PID calculations

are simple arithmetic procedures. By con-

trast, an expert’s focus on the mathematics

distracts those intellects from the impor-

tant aspects of control, such as structuring

ratio, cascade and override, or choosing

appropriate modifications, anti-windup and

initialization procedures. The real experts

are not necessarily the mathematicians of

control theory. They are the ones who can

implement control.

There are many modifications to the PID

equation. Reset feedback, for example, is

an alternate method to incrementally adjust

the bias, which prevents integral windup,

and is especially useful in override and con-

straint strategies. Another common modifi-

cation is the rate-before-reset or interacting

controller, which can be created if incre-

mental changes to the bias are also based

on the anticipated error. For a discussion

of such modifications, see Rhinehart, R. R.,

H. L. Wade, and F. G. Shinskey in the Instru-

ment Engineers’ Handbook, Vol II, Process

Control and Analysis, 4th Edition, B. Liptak,

Editor, Section 2.3, “Control Modes – PID

Variations,” pp. 124-129, Taylor and Francis,

CRC Press, Boca Raton, Fla., 2005.

For a procedure to tune the controller,

see “Criteria and procedure for control-

ler tuning” by R.R. Rhinehart (Control, Jan

’17, p. 54-55, www.controlglobal.com/

articles/2017/criteria-and-procedure-for-

controller-tuning).

R. Russell Rhinehart, engineering coach, R3eda, North

Carolina State University, can be reached at russ@

r3eda.com.

The real experts are not necessarily

the mathematicians of control theory.

They are the ones who can implement control.



Is global warming like level control?Comparing the processes sheds light on the problem of regulating Earth’s temperature.by Béla Lipták

QIn a previous column, you wrote that understanding the control of global warming is very similar to understanding level control. Could you

explain this in more detail? In what respect

are the two processes similar?

Z. Friedmann

[email protected]

A: You hit upon a large subject, one that will

take up a full chapter in my upcoming book

to be published by ISA. I will try to give you

a brief answer.

Figure 1 shows the rise of the surface tem-

perature of the oceans during the indus-

trial age. Now, if we look at the protection

against climate change as a “control loop,”

the measurement of that loop is that tem-

perature, which has increased by only about

1.5 °C during the past century, and today

we’re just beginning to see its consequenc-

es (melting ice, wildfires, hurricanes). This

rate of rise is still slow (2.0-3.0 °C per cen-

tury) but is accelerating. My estimate is it

TEMPERATURE OVER TIMEFigure 1: The average surface temperature of the world’s oceans, using the baseline of 1971 to 2000 average. The shaded band shows the range of uncertainty in the data based on the number of measurements collected and the precision of the measurements used. Source: EPA

Tem

pera

ture

ano

mal

y (°F

)

2.0

1.5

1.0

0.5

0

-0.5

-1.0

-1.5

-2.01880 1900 1920 1940 1960 1980 2000 2020

Year



will reach a rise of about 5.0 °C by 2075. In

my view, we’ll never stop at 2.0°C (recom-

mended by the Paris Agreement of 2015)

and particularly not at 1.5 °C (recommended

by the U.N. Intergovernmental Panel on Cli-

mate Change – IPCC in 2018) because this

high-inertia process is totally out of control.

Some believe that because the numbers de-

scribing the temperature rise are small, the

problem we face is also small. This is not the

case. Let me compare them with our own

body temperature, which is accurately con-

trolled by our brain. The body temperature

of a healthy, resting adult human being is

98.6 °F (37.0 °C). Our “thermostat” (called

the hypothalamus, a portion of the brain)

controls body temperature. The span of our

thermostat is 36.4–37.1 °C (97.5–98.8 °F)

or about 0.7 °C. This thermostat turns on

shivering at 97.5 °F and initiates sweating at

98.8 °F. I mention this only to illustrate that

certain processes must be controlled within

small limits because small temperature

changes can have large effects.

According to all scientific data (www.nasa.

gov/topics/earth/features/co2-temperature.

html), the thermostat of global temperature

control is CO2 concentration in the atmo-

sphere. If it rises, the global temperature rises

(because the thermal insulation of the planet

increases), and when it dops, the planet cools.

During the past 1 million years, nature

“controlled” this concentration by keep-

ing the inflow of CO2 into the atmosphere

(generated by animal life and man) rough-

ly equal the outflow (intake of plants and

dissipation by the oceans), and therefore

the atmospheric concentration of CO2

stayed roughly constant, never exceeding

280 ppm even during ice ages, chang-

ing sun spot numbers, volcanic activity or

meteor impacts.

If we look at the atmosphere as a tank and

CO2 concentration as the level in that tank,

then we could say that this level stayed rea-

sonably constant for a million years because

it never exceeded 280 ppm (the planet didn’t

need to start “sweating”) as nature took care

of it. Since the beginning of the industrial

revolution, humans gradually took over this

control from nature, CO2 concentration in the

atmosphere increased from 280 to more than

410 ppm, and it’s predicted by most models

that it will reach or even exceed 500 ppm by

the end of this century.

If a control engineer was to bring this

process under control by returning CO2

concentration to the stable, pre-industrial

state, the task would be to balance the in

and outflows of this tank, and on top of

that, remove roughly half of the CO2 that

accumulated during the industrial age. If a

conventional level controller was installed

on this tank, it would see an error of 410 –

280 = 130 ppm and a past error accumula-

tion of some 400-500 Gt (vvvgigatons) of

carbon. It would immediately close the inlet



valve and open the outlet valve.

Unfortunately, in this process, these valves

are stuck. The outflow from the tank (the

CO2 intake of the plants and dissipation

by the oceans) can’t be increased. In fact,

it has probably decreased during the past

century because of deforestation, acidifica-

tion of the oceans, and building of dams/

reservoirs, holding 8,000 km3 of water,

which also emit carbon to the atmosphere.

In short, this outlet valve is almost com-

pletely stuck and we have no technology

to open it further except reforestation,

which is unlikely due to overpopulation

(during the industrial age, population in-

creased from 1.0 to 9.0 billion).

As shown in Figure 2, every year we send 9

Gt of carbon into the atmosphere. 3 Gt of

that is taken up by the photosynthesis of

CARBON IMBALANCEFigure 2: This figure shows the fast carbon cycle (left - on land, right - in the oceans) in billions of tons of carbon per year. Yellow numbers are natural fluxes, red are human contributions. White numbers refer to stored carbon. Source: NASA



plants, 2 Gt is dissipated by the oceans, and

4 Gt remains in the atmosphere for the next

20 to 200 years. So, the inflow exceeds the

outflow by 4 Gt/yr.

We do have some means to reduce this

inflow, such as using more bicycles, public

transport, converting to electric cars, insu-

lating our homes, using smart thermostats

and appliances, eliminating animal prod-

ucts from our diet (which cuts greenhouse

emissions by more than 10%), introduc-

ing carbon taxes (not cap-and-trade, but

taxes), and eventually, fully converting the

energy economy from fossil/nuclear fuels

to carbon-free ones. The speed of conver-

sion is a function of both the marketplace

and government support. Where both are

present, the conversion is faster (in Cali-

fornia today, green electricity is 30% of

the total), while where only the market-

place is driving the conversion, it is much

slower (15% in the U.S. overall).

It will probably take a generation or two

to overcome the resistance of the fossil

industry. Leaving some $35 trillion worth

of fossil fuel in the ground justifies some

resistance, and it will take time for society

as a whole to realize that we must con-

vert to solar energy and use hydrogen as

the means of storing, transporting and

distributing this energy to areas where

insolation is insufficient. (For more solar

storage technology, see my book, Post

Oil Energy Economy, or tune in the video:

http://techchannel.att.com/play-video.

cfm/2011/8/25/Science-&-Technology-

Author-Series-Bela-G-Liptak:-Post-Oil-

Energy-Technology.)

What even the best of our leaders seem

to not understand is that, even after we’ve

balanced the in and outflows, we will not

have returned the planet to pre-industrial

conditions because even after this tre-

mendous technological and political trans-

formation effort, we didn’t even start to

remove the already accumulated 400-500

Gt of carbon from the atmosphere, which

can stay there for 20-200 years. And, as

long as that accumulation remains, the

CO2 concentration does not drop and the

planet will keep warming.

This is obvious to a process control engi-

neer, but not to our well-intentioned leaders

who wrote the Paris Agreement in 2015 or

the smarter U.N. experts who participated

in the IPCC meeting in 2018. It is for this

reason that we who understand process

control have the responsibility to explain

that the present uncontrolled rate of carbon

accumulation will reach approximately 500

ppm by the end of this century, at which

point the tropical regions of the planet are

likely to become unlivable, and the resulting

biblical-scale migration could destroy hu-

man civilization.

Béla Lipták

[email protected]

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