Tài liệu Process Dynamics, Operations, and Control pptx

Using knowledge of the controlled variable to motivate changes to the manipulated variable is a fundamental control structure, known as feedback control.. We might moderate these swings

1.0 context and direction

Process control is an application area of chemical engineering - an

identifiable specialty for the ChE It combines chemical process

knowledge (how physics, chemistry, and biology work in operating

equipment) and an understanding of dynamic systems, a topic important to

many fields of engineering Thus study of process control allows

chemical engineers to span their own field, as well as form a useful

acquaintance with allied fields Practitioners of process control find their

skills useful in design, operation, and troubleshooting - major categories of

chemical engineering practice

Process control, like any coherent topic, is an integrated body of

knowledge - it hangs together on a multidimensional framework, and

practitioners draw from many parts of the framework in doing their work

Yet in learning, we must receive information in sequence - following a

path through multidimensional space It is like entering a large building

with unlighted rooms, holding a dim flashlight and clutching a vague map

that omits some of the stairways and passages How best to learn one’s

way around?

In these lessons we will attempt to move through a significant portion of

the structure - say, half a textbook - in about two weeks Then we will

repeat the journey several times, each time inspecting the rooms more

thoroughly By this means we hope to gain, from the start, a sense of

doing an entire process control job, as well as approach each new topic in

the context of a familiar path

1.1 the job we will do, over and over

We encounter a process, learn how it behaves, specify how we wish to

control it, choose appropriate equipment, and then explore the behavior

under control to see if we have improved things

1.2 introducing a simple process

A large tank must be filled with liquid from a supply line One operator

stands at ground level to operate the feed valve Another stands on the

tank, gauging its level with a dipstick When the tank is near full, the stick

operator will instruct the other to start closing the valve Overfilling can

cause spills, but underfilling will cause later process problems

To learn how the process works, we write an overall material balance on

the tank

i

FV

The tank volume V can be expressed in terms of the liquid level h The

inlet volumetric flow rate Fi may vary with time due to supply pressure

fluctuations and valve manipulations by the operator The liquid density

depends on the temperature, but will usually not vary significantly with

time during the course of filling Thus (1.2-1) becomes

)t(Fdt

1)0(

h

1.3 planning a control scheme

Clearly the liquid level h is important, and we will call it the controlled

variable Our control objective is to bring h quickly to its target value hr

and not exceed it (To be realistic, we would specify allowable limits ± δh

on hr.) We will call the volumetric flow Fi the manipulated variable,

because we adjust it to achieve our objective for the controlled variable

The existing control scheme is to measure the controlled variable via

dipstick, decide when the controlled variable is near target, and instruct

the valve operator to change the manipulated variable The scheme suffers

from

• delay in measurement Overfilling can occur if the stick operator

cannot complete the measurement in time

• performance variations Both stick and valve operators may vary in

attentiveness and speed of execution

• resources required There are better uses for operating personnel

• unsafe conditions There is too much potential for chemical exposure

A new scheme is proposed: put a timer on the valve Calculate the time

required for filling from (1.2-3) Close the valve when time has expired

The timing scheme would no longer require an operator to be on the tank

top, and with a motor-driven valve actuator the entire operation could be

directed from a control room These are indeed improvements However,

the timing scheme abandons a crucial virtue of the existing scheme: by

measuring the controlled variable, the operators can react to unexpected

disturbances, such as changes in the filling rate Using knowledge of the

controlled variable to motivate changes to the manipulated variable is a

fundamental control structure, known as feedback control The proposed

timing scheme has no feedback mechanism, and thus cannot accommodate

changes to h(0) and Fi(t) in (1.2-3)

An alternative is to build on the feedback already inherent in the

two-operator scheme, but to improve its operation We propose an automatic

controller that behaves according to the following controller algorithm:

near i max

r near i max

Algorithm (1.3-1) is an idealization of what the operators are already

doing: filling occurs at maximum flow until the level reaches a value hnear

Beyond this point, the flow decreases linearly, reaching zero when h

reaches the target hr The setting of hnear may be adjusted to tune the

control performance

1.4 choosing equipment

We need a sensor to replace the dipstick, a valve actuator to replace the

valve operator, and a controller mechanism to replace the stick operator

We imagine a buoyant object floating on the liquid surface The float is

linked to a lever that drives the valve stem When the liquid level is low,

the float rests above it on a structure so that the valve is fully open

1.5 process behavior under automatic control

Typically these things work quite well We predict its performance by

combining our process model (1.2-2) with the controller algorithm (1.3-1),

which eliminates the manipulated variable between the equations We

take the simple case in which Fmax does not vary during filling due to

pressure fluctuations, etc For h less than hnear,

tA

F)0(

h

h

known)

0(hF

Equation (1.5-1) can be used to calculate tnear, the time at which h reaches

hnear For h greater than hnear,

Content removed due to copyright restrictions

(To see a cut-away diagram of a toilet, go to

http://www.toiletology.com/lg-views.shtml#cutaway2x)

where the parameter tfill is the time required for the level to reach hr at

flow Fmax, starting from an empty tank

r fill

The plot shows the filling profile from h(0) = 0.10hr with several values of

hnear/hr Certainly the filling goes faster if the flow can go instantaneously

from Fmax to zero at hr; however this will not be practical, so that hnear will

1.6 defining ‘system’

In Section 1.2, we introduced a process - a tank with feed piping - whose

inventory varied in time We thought of the process as a collection of

equipment and other material, marked off by a boundary in space,

communicating with its environment by energy and material streams

'Process' is a good notion, important to chemical engineers Another

useful notion is that of 'system' A system is some collection of equipment

and operations, usually with a boundary, communicating with its

environment by a set of input and output signals By these definitions, a

process is a type of system, but system is more abstract and general For

example, the system boundary is often tenuous: suppose that our system

comprises the equipment in the plant and the controller in the central

control room, with radio communication between the two A physical

boundary would be in two pieces, at least; perhaps we should regard this

system boundary as partly physical (around the chemical process) and

partly conceptual (around the controller)

Furthermore, the inputs and outputs of a system need not be material and

energy streams, as they are for a process System inputs are "things that

cause" or “stimuli”; outputs are "things that are affected" or “responses”

systeminputs

(causes)

outputs(responses)

To approach the problem of controlling our filling process in Section 1.3,

we thought of it in system terms: the primary output was the liquid level h

not a stream, certainly, but an important response variable of the system

and inlet stream Fi was an input And peculiar as it first seems, if the

tank had an outlet flow Fo, it would also be an input signal, because it

influences the liquid level, just as does Fi

The point of all this is to look at a single schematic and know how to view

it as a process, and as a system View it as a process (Fo as an outlet

stream) to write the material balance and make fluid mechanics

calculations View it as a system (Fo as an input) to analyze the dynamic

behavior implied by that material balance and make control calculations

System dynamics is an engineering science useful to mechanical,

electrical, and chemical engineers, as well as others This is because

transient behavior, for all the variety of systems in nature and technology,

can be described by a very few elements To do our job well, we must

understand more about system dynamics how systems behave in time

That is, we must be able to describe how important output variables react

to arbitrary disturbances

1.7 systems within systems

We call something a system and identify its inputs and outputs as a first

step toward understanding, predicting, and influencing its behavior In

some cases it may help to determine some of the structure within the

system boundaries; that is, if we identify some component systems Each

of these, of course, would have inputs and outputs, too

system

2

Considering the relationship of these component systems, we recognize

the existence of intermediate variables within a system Neither inputs

nor outputs of the main system, they connect the component systems

Intermediate variables may be useful in understanding and influencing

overall system behavior

1.8 the system of single-loop feedback control

When we add a controller to a process, we create a single time-varying

system; however, it is useful to keep process and controller conceptually

distinct as component systems This is because a repertoire of relatively

few control schemes (relationships between process and controller)

suffices for myriad process applications Using the terms we defined in

Section 1.3, we represent a control scheme called single-loop feedback

control in this fashion:

outputs

controlled variable

outputs

controlled variable system

Figure 1.8-1 The single-loop feedback control system and its

subsystems

We will see this structure repeatedly Inside the block called "process" is

the physical process, whatever it might be, and the block is the boundary

we would draw if we were doing an overall material or energy balance

HOWEVER, we remember that the inputs and outputs are NOT

necessarily the same as the material and energy streams that cross the

process boundary From among the outputs, we may select a controlled

variable (often a pressure, temperature, flow rate, liquid level, or

composition) and provide a suitable sensor to measure it From the inputs,

we choose a manipulated variable (often a flow rate) and install an

appropriate final control element (often a valve) The measurement is fed

to the controller, which decides how to adjust the manipulated variable to

keep the controlled variable at the desired condition: the set point The

other inputs are potential disturbances that affect the controlled variable,

and so require action by the controller

1.9 conclusion

Think of a chemical process as a dynamic system that responds in

particular ways to its inputs We attach other dynamic systems (sensor,

controller, etc.) to that process in a single-loop feedback structure and

arrive at a new dynamic system that responds in different ways to the

inputs If we do our job well, it responds in better ways, so to justify all

the trouble

2.0 context and direction

Imagine a system that varies in time; we might plot its output vs time A

plot might imply an equation, and the equation is usually an ODE

(ordinary differential equation) Therefore, we will review the math of the

first-order ODE while emphasizing how it can represent a dynamic

system We examine how the system is affected by its initial condition

and by disturbances, where the disturbances may be non-smooth, multiple,

or delayed

2.1 first-order, linear, variable-coefficient ODE

The dependent variable y(t) depends on its first derivative and forcing

function x(t) When the independent variable t is t0, y is y0

0

0) yt(y)t(Kx)t(ydt

In writing (2.1-1) we have arranged a coefficient of +1 for y Therefore

a(t) must have dimensions of independent variable t, and K has

dimensions of y/x We solve (2.1-1) by defining the integrating factor p(t)

∫

=

)(exp)

(

t a

dt t

Notice that p(t) is dimensionless, as is the quotient under the integral The

solution

∫+

t

0 0

0

dt)t(a

)t(x)t(p)t(p

K)

t(p

)t(y)t(p)

t

(

comprises contributions from the initial condition y(t0) and the forcing

function Kx(t) These are known as the homogeneous (as if the right-hand

side were zero) and particular (depends on the right-hand side) solutions

In the language of dynamic systems, we can think of y(t) as the response

of the system to input disturbances Kx(t) and y(t0)

2.2 first-order ODE, special case for process control applications

The independent variable t will represent time For many process control

applications, a(t) in (2.1-1) will be a positive constant; we call it the time

constant τ

0

0) yt(y)t(Kx)t(ydt

The integrating factor (2.1-2) is

=τ

Ke

y)

t 0

0

0

∫ τ τ

− τ

−

−

τ+

The initial condition affects the system response from the beginning, but

its effect decays to zero according to the magnitude of the time constant -

larger time constants represent slower decay If not further disturbed by

some x(t), the first order system reaches equilibrium at zero

However, most practical systems are disturbed K is a property of the

system, called the gain By its magnitude and sign, the gain influences

how strongly y responds to x The form of the response depends on the

nature of the disturbance

Example: suppose x is a unit step function at time t1 Before we proceed

formally, let us think intuitively From (2.2-3) we expect the response y to

decay toward zero from IC y0 At time t1, the system will respond to being

hit with a step disturbance After a long time, there will be no memory of

the initial condition, and the system will respond only to the disturbance

input Because this is constant after the step, we guess that the response

will also become constant

Now the math: from (2.2-3)

=

−τ

−

−

τ τ

− τ

−

−

∫

1 0

0 0

t t 1

t t 0

t

t

1 t

t t

t 0

e1)tt(KUe

y

dt)tt(Uee

Ke

y)

t

(

y

(2.2-4)

Figure 2.2-1 shows the solution Notice that the particular solution makes

no contribution before time t1 The initial condition decays, and with no

disturbance would continue to zero At t1, however, the system responds

to the step disturbance, approaching constant value K as time becomes

large This immediate response, followed by asymptotic approach to the

new steady state, is characteristic of first-order systems Because the

response does not track the step input faithfully, the response is said to lag

behind the input; the first-order system is sometimes called a first-order

lag

2.3 piecewise integration of non-smooth disturbances

The solution (2.2-3) is applied over succeeding time intervals, each

featuring an initial condition (from the preceding interval) and disturbance

<

<

τ+

∫

τ τ

− τ

−

−

τ τ

− τ

−

−

.etc

tttdt)t(xee

Ke

)t(y

tttdt)t(xee

Ke

)t(y)

t 1

1 0

t

t

t t t

t 0

1 1

0 0

(2.3-1)

Example: suppose

2t11t2

1t00

x

0)0(yxy

dt

dy

(2.3-2)

In this problem, variables t, x, and y should be presumed to have

appropriate, if unstated, units; in these units, both gain and time constant

are of magnitude 1 From (2.3-1),

e2

2t1e

2t2

1t00

1

With a zero initial condition and no disturbance, the system remains at

equilibrium until the ramp disturbance begins at t = 1 Then the output

immediately rises in response, lagging behind the linear ramp At t = 2,

the disturbance ceases, and the output decays back toward equilibrium

2.4 multiple disturbances and superimposition

Systems can have more than one input Consider a first-order system with

two disturbance functions

0 0 2

2 1

1x (t) K x (t) y(t ) yK

)t(ydt

Applying (2.2-3) and distributing the integral across the disturbances, we

find that the effects of the disturbances on y are additive

dt)t(xee

Kdt)t(xee

Ke

y)

t

t

t t 1 t

t 0

0 0

−

−

τ

+τ

+

This additive behavior is a happy characteristic of linear systems Thus

another way to view problem (2.4-1) is to decompose it into component

problems That is, define

2 1

H y y

y

and write (2.4-1) in three equations We put the initial condition with no

disturbances, and each disturbance with a zero initial condition

0)t(y)t(xK)t(ydt

dy

0)t(y)t(xK)t(ydt

dy

y)t(y0

)t(ydt

dy

0 2 2

2 2

2

0 1 1

1 1

1

0 0 H H

H

=

=+

τ

=

=+

τ

=

=+

τ

(2.4-4)

Equations and initial conditions (2.4-4) can be summed to recover the

original problem specification (2.4-1) The solutions are

dt)t(xee

K)

t

(

y

dt)t(xee

K)

t

(

y

ey)

t

(

y

2 t

t

t t 2 2

1 t

t

t t 1 1

t t 0 H

0 0 0

∫

∫

τ τ

−

τ τ

and of course these solutions can be added to recover original solution

(2.4-2) Thus we can view the problem of multiple disturbances as a

system responding to the sum of the disturbances, or as the sum of

responses from several identical systems, each responding to a single

disturbance

Example: consider

2)0(y)3t(U)1t(U4

3y4

1dt

Equation (2.4-7) shows us that the time constant is 1, and that the system

responds to the first disturbance with a gain of 3, and to the second with a

gain of -4 The solution is

t 3U(t 1)1 e 4U(t 3)1 ee

2

In Figure 2.4-1, the individual solution components are plotted as solid

traces; their sum, which is the system response, is a dashed trace Notice

how the first-order lag responds to each new disturbance as it occurs

Figure 2.4-1 first-order response to multiple disturbances

Writing the step functions explicitly in solution (2.4-8) emphasizes that

particular disturbances do not influence the solution until the time of their

occurrence For example, if they were omitted, some deceptively correct

but inappropriate rearrangement would lead to errors

) 3 t ) 1 t t

) 3 t )

1 t t

) 3 t )

1 t t

e4e

3e21

e44e

33e

2

e14e

13e

=

−

−

−+

=

(do not do this!) (2.4-9)

This notation at least implies that two of the exponential functions have

delayed onsets However, further correct-but-inappropriate rearrangement

makes things even worse

( 1 3) t

t 3 t 1 t

) 3 t ) 1 t t

ee4e321

ee4ee3e21

e4e

3e21

−

=

+

−+

−

=

+

−+

−

=

(do not do this!) (2.4-10)

The incorrect solutions are plotted with (2.4-8) in Figure 2.4-2 Equation

(2.4-9) has become discontinuous - the response takes non-physical leaps

at the onset of each new disturbance Equation (2.4-10) has lost all

dependence on the disturbances and decays from a non-physical initial

condition Even with the mistakes, both incorrect solutions lead to the

correct long-term condition

Figure 2.4-2 comparison of correct and incorrect solutions

2.5 delayed response to disturbances

Consider a system that reacts to a disturbance, but only after some

intervening time interval θ has passed That is

0

0) yt(y)t(Kx)t(ydt

Equation (2.5-1) shows the dependence of y, at any time t, on the value of

x at earlier time t - θ The solution is written directly from (2.2-3)

dt)t(xee

Ke

y)

t 0

0

0

θ

−τ

+

We must integrate the disturbance considering the time delay Take as an

example a disturbance x(t) occurring at time t1 The plot shows the

disturbance, as well as the disturbance as the system experiences it, which

begins at time t1 + θ We could express this disturbance-as-experienced as

some new function x1(t), occurring at time t1 + θ

=

=θ

θ

− τ θ + ξ τ

τx(t )dt e x (t)dt e x( )d

e

0 0

1 t

t

t t

t

t

(2.5-4) Therefore, solution (2.5-2) becomes

ξξτ

+

θ

− τ ξ τ τ

− τ

−

−

d)(xeee

Ke

y)

t

Example: consider a step disturbance at time t = 2 that affects the system

3 time units later

)2t(U)

t

(

x

0)0(y)3t(xy

(2.5-6)

Using (2.5-5)

[ ]

3 t 2 3 t 3 t

2 3 t 3 t 2 3

t

2

2

3 0

3 t

3 0

3 t

e1)5t(U

ee

)5t(U

eeee)23t(U

e)2(Uee

d)2(Ued)2(Ueee

d)2(Ueee

−

ξ ξ

ξ

−ξ

=

ξ

−ξ

Figure 2.5-1 shows that a typical first-order lag step response occurs 3

time units after being disturbed at t = 2

Figure 2.5-1 step response of first order system with dead time

The time delay in responding to a disturbance is often called dead time

Dead time is different from lag Lag occurs because of the combination of

y and its derivative on the left-hand side of the equation Dead time

occurs because of a time delay in processing a disturbance on the

right-hand side

2.6 conclusion

Please become comfortable with handling ODEs View them as systems;

identify their inputs and outputs, their gains and time parameters

3.0 context and direction

A particularly simple process is a tank used for blending Just as promised

in Section 1.1, we will first represent the process as a dynamic system and

explore its response to disturbances Then we will pose a feedback control

scheme We will briefly consider the equipment required to realize this

control Finally we will explore its behavior under control

DYNAMIC SYSTEM BEHAVIOR

3.1 math model of a simple continuous holding tank

Imagine a process stream comprising an important chemical species A in

dilute liquid solution It might be the effluent of some process, and we

might wish to use it to feed another process Suppose that the solution

composition varies unacceptably with time We might moderate these

swings by holding up a volume in a stirred tank: intuitively we expect the

changes in the outlet composition to be more moderate than those of the

Our concern is the time-varying behavior of the process, so we should

treat our process as a dynamic system To describe the system, we begin

by writing a component material balance over the solute

Ao Ai

Ao FC FCVC

dt

In writing (3.1-1) we have recognized that the tank operates in overflow:

the volume is constant, so that changes in the inlet flow are quickly

duplicated in the outlet flow Hence both streams are written in terms of a

single volumetric flow F Furthermore, for now we will regard the flow as

constant in time

Balance (3.1-1) also represents the concentration of the outlet stream, CAo,

as the same as the average concentration in the tank That is, the tank is a

perfect mixer: the inlet stream is quickly dispersed throughout the tank

volume Putting (3.1-1) into standard form,

Ai Ao

we identify a first-order dynamic system describing the response of the

outlet concentration CAo to disturbances in the inlet concentration CAi

The speed of response depends on the time constant, which is equal to the

ratio of tank volume and volumetric flow Although both of these

quantities influence the dynamic behavior of the system, they do so as a

ratio Hence a small tank and large tank may respond at the same rate, if

their flow rates are suitably scaled

System (3.1-2) has a gain equal to 1 This means that a sustained

disturbance in the inlet concentration is ultimately communicated fully to

the outlet

Before solving (3.1-2) we specify a reference condition: we prefer that CAo

be at a particular value CAo,r For steady operation in the desired state,

there is no accumulation of solute in the tank

r , Ao r , Ai r

Thus, as expected, steady outlet conditions require a steady inlet at the

same concentration; call it CA,r Let us take this reference condition as an

initial condition in solving (3.1-2) The solution is

dt)t(Ce

eeC)

− τ

−

τ+

Equation (3.1-4) describes how outlet concentration CAo varies as CAi

changes in time In the next few sections we explore the transient

behavior predicted by (3.1-4)

3.2 response of system to steady input

Suppose inlet concentration remains steady at CA,r Then from (3.1-4)

r , A t

t r , A

t r , A

t

0

t r , A

t t r , A Ao

C1eeCe

C

eC

eeCC

+

=

τ τ

− τ

−

τ τ

− τ

−

(3.2-1)

Equation (3.2-1) merely confirms that the system remains steady if not

disturbed

3.3 leaning on the system - response to step disturbance

Step functions typify disturbances in which an input variable moves

relatively rapidly to some new value and remains there Suppose that

input CAi is initially at the reference value CA,r and changes at time t1 to

value CA1 Until t1 the outlet concentration is given by (3.2-1) From the

step at t1, the outlet concentration begins to respond

=

τ

−

− τ

−

−

τ τ τ

− τ

−

−

τ τ

− τ

−

−

) t 1

A

) t r A

t t t 1 A

) t r A

1 t

t

t 1 A

t ) t r A Ao

1 1

1 1

1 1

e1Ce

C

eeeCe

C

tte

C

ee

CC

(3.3-1)

In Figure 3.3-1, CA,r = 1 and CA1 = 0.8 in arbitrary units; t1 has been set

equal to τ At sufficiently long time, the initial condition has no influence

and the outlet concentration becomes equal to the new inlet concentration

After time equal to three time constants has elapsed, the response is about

95% complete – this is typical of first-order systems

In Section 3.1, we suggested that the tank would mitigate the effect of

changes in the inlet composition Here we see that the tank will not

eliminate a step disturbance, but it does soften its arrival

Figure 3.3-1 first-order response to step disturbance

3.4 kicking the system - response to pulse disturbance

Pulse functions typify disturbances in which an input variable moves

relatively rapidly to some new value and subsequently returns to normal

Suppose that CAi changes to CA1 at time t1 and returns to CA,r at t2 Then,

−

− τ

−

− τ

−

−

tte

1Ce

e1Ce

C

ttte

1Ce

C

tt0C

C

2

) t t r

A

) t ) t 1

A

) t r A

2 1

) t 1

A

) t r A

1 r

A

Ao

2 2

1 2 1

2

1 1

(3.4-1)

In Figure 3.4-1, CA,r = 0.6 and CA1 = 1 in arbitrary units; t1 has been set

equal to τ and t2 to 2.5τ We see that the tank has softened the pulse and

reduced its peak value A pulse is a sequence of two counteracting step

changes If the pulse duration is long (compared to the time constant τ),

the system can complete the first step response before being disturbed by

Figure 3.4-1 first-order response to pulse disturbance

3.5 shaking the system - response to sine disturbance

Sine functions typify disturbances that oscillate Suppose the inlet

concentration varies around the reference value with amplitude A and

frequency ω, which has dimensions of radians per time

( )tsinAC

+τ

ω+

ωτ

−

2 2

t 2 2 r

, A

1

Ae

1

AC

Solution (3.5-2) comprises the mean value CA,r, a term that decays with

time, and a continuing oscillation term Thus, the long-term system

response to the sine input is to oscillate at the same frequency ω Notice,

however, that the amplitude of the output oscillation is diminished by the

square-root term in the denominator Notice further that the outlet

oscillation lags the input by a phase angle tan-1(-ωτ)

In Figure 3.5-1, CA,r = 0.8 and A = 0.5 in arbitrary units; ωτ has been set

equal to 2.5 radians, and τ to 1 in arbitrary units The decaying portion of

the solution makes a negligible contribution after the first cycle The

phase lag and reduced amplitude of the solution are evident; our tank has

mitigated the inlet disturbance

Figure 3.5-1 first-order response to sine disturbance

3.6 frequency response and the Bode plot

The long-term response to a sine input is the most important part of the

solution; we call it the frequency response of the system We will

examine the frequency response for an abstract first order system

(Because we wish to focus on the oscillatory response, we will write

(3.6-1) so that x and y vary about zero The effect of a non-zero bias term can

be seen in (3.5-1) and (3.5-2).)

2 2

fr A

1

2 2 fr

fr fr

1

Kx

yR:ratioamplitude

)(tan:

anglephase

1

KAy

:amplitude

)tsin(

yy:resp

freq

)tsin(

Ax:

input

Kxydt

dyτ:system

τω+

τω+

=

φ+ω

=

ω

=

=+

−

(3.6-1)

The frequency response is a sine function, characterized by an amplitude,

frequency, and phase angle The amplitude and phase angle depend on

system properties (τ and K) and characteristics of the disturbance input (ω

and A) It is convenient to show the frequency dependence on a Bode

plot, Figure 3.6-1

The Bode plot abscissa is ω in radians per time unit; the scale is

logarithmic The frequency may be normalized by multiplying by the

system time constant Thus plotting ω is good for a particular system;

plotting ωτ is good for systems in general

The upper ordinate is the amplitude ratio, also on logarithmic scale RA is

often normalized by dividing by the system gain K The lower ordinate is

the phase angle, in degrees on a linear scale

In Figure 3.6-1, the coordinates have been normalized to depict first-order

systems in general; the particular point represents conditions in the

example of Section 3.5

For a first order system, the normalized amplitude ratio decreases from 1

to 0 as frequency increases Similarly, the phase lag decreases from 0 to

-90º Both these measures indicate that the system can follow slow inputs

faithfully, but cannot keep up at high frequencies

Another way to think about it is to view the system as a low-pass filter:

variations in the input signal are softened in the output, particularly for

high frequencies

The slope of the amplitude ratio plot approaches zero at low frequency;

the high frequency slope approaches -1 These two asymptotes intersect at

the corner frequency, the reciprocal of the system time constant At the

corner frequency, the phase lag is -45º

If we disturb our system, will it return to good operation, or will it get out

of hand? This is asking whether the system is stable We define stability

as "bounded output for a bounded input" That means that

• a ramp disturbance is not fair – even stable systems can get into

trouble if the input keeps rising

• a stable system should handle a step change in input, ultimately

coming to some new steady state (We must be realistic, however

If the system is so sensitive that a small input step leads to an

unacceptably high, though steady, output, we might declare it

unstable for practical purposes.)

• it should also handle a sine input; here the result is in general not

steady state, because the output may oscillate (Thus we

distinguish between 'steady state' and 'long-term stability'.)

The solutions for the typical bounded step, pulse, and sine disturbances,

given in Sections 3.3 through 3.5, show no terms that grow with time, so

long as the time constant τ is a positive value For these categories of

bounded input, at least, a first-order system appears to be stable We will

need to examine stability again when we introduce automatic control to

our process

3.8 concentration control in a blending tank

In Section 3.1 we described how variations in stream composition could

be moderated by passing the stream through a larger volume - a holding

tank Let us be more ambitious and seek to control the outlet composition:

we add a small inlet stream Fc of concentrated solution to the tank This

will allow us to adjust the composition in response to disturbances

Ao FC FC F F CVC

Ac

Ai c Ao

Ao c

FF

F1F

CCF

F1

1C

dtdCF

=

+

Notice that our equation coefficients each contain the input variable Fc

Notice, as well, that for dilute CAo and concentrated CAc stream Fc

(however it may vary) will not be very large in comparison to the main

flow F If this is the case, we may be justified in making an engineering

approximation: neglecting the ratio Fc/F in comparison to 1 Thus

c

Ac Ai Ao

F

CCCdt

dC

F

Now we have a linear first-order system Comparison with (3.1-2) shows

the same time constant V/F and the same unity gain for inlet concentration

disturbances There is a new input Fc, whose influence on CAo (i.e., gain)

increases with high concentration CAc and decreases with large

throughflow F

3.9 use of deviation variables in solving equations

In process control applications, we usually have some desired operating

condition We now write system model (3.8-3) at the target steady state

All variables are at reference values, denoted by subscript r

r c

Ac r Ai r

F

CC

We recognize that deviations from these reference conditions represent

errors to be corrected Hence we recast our system description (3.8-3) in

terms of deviation variables; we do this by subtracting (3.9-1) from

(3.8-3)

' c Ac '

Ai

' Ao

'

Ao

r , c c

Ac r , Ai Ai r

, Ao Ao r

, Ao Ao

FF

CCCdt

dC

F

V

FFF

CC

CC

Cdt

CC

−+

−

=

−+

−

(3.9-2)

where we indicate a deviation variable by a prime superscript The target

condition of a deviation variable is zero, indicating that the process is

operating at desired conditions Using deviation variables

• makes conceptual sense for process control because they indicate

deviations from desired states

• makes the mathematical descriptions simpler

Thus we shall use deviation variables for derivations and modeling For

doing process control (computing valve positions, e.g.) we will return to

the physical variables We can recover the physical variable by adding its

deviation variable to its reference value For example,

)t(CC)

t

(

Ao r , Ao

where we emphasize the variables that are time-varying

3.10 integration from zero initial conditions

As a rule, we will presume that our systems are initially at the reference

condition That is, the initial conditions for our differential equations are

zero Integrating (3.9-2) we find

dt)t(FeF

Cedt)t(Ce

e

c t

0

t Ac

t '

Ai t

0

t t '

− τ

τ

−

τ

+τ

Equation (3.10-1) shows how the outlet composition deviates from its

desired value CAo,r under disturbances to inlet composition CAi and the

flow rate of the concentrated makeup stream Fc, where both of these are

also expressed as deviations from reference values Equation (3.10-1) is

analogous to (3.1-4) for the simpler holding tank

3.11 response to step changes

Proceeding as in Section 3.3, we presume a step in inlet composition of

ΔCAi at time t1 and of ΔFc in makeup flow at time t2

−+

−

=

Δ

−τ

+Δ

−τ

=

τ

−

− τ

−

−

τ τ

− τ τ

−

∫

∫

) t t Ac

c 2

) t Ai

1

t

t

t Ac c 2

t t

t

t Ai 1

t '

Ao

2 1

2 1

e1F

CF)tt(Ue

1C)tt(U

dteF

CF)tt(U

edteC)tt(U

eC

(3.11-1)

CAo′ exhibits a first-order response to each of these step inputs

Example: try these numbers:

3 m

3 3

)0001.0(m

kg)400(m

kg)8(m

kg

)

10

(Notice that the exact steady-state balance, derived from (3.8-2), is

satisfied to within 1%, so that our approximation in deriving (3.8-3)

appears to be reasonable.) The time constant for our process is

s300

m)02.0(

)6(F

V

s

3 m

300 t 3

300 ) 120 t 5

3 300

) 0 t 3

'

Ao

e1m

kg)1)(

120t(Ue

1m

kg)1(

e1)02.0(

)105(m

kg)400)(

120t(Ue

1m

kg)1)(

0t(UC

(3.11-4)

where t must be computed with units of seconds In Figure 3.11-1, we can

see that the reduction in make-up flow at 120 s compensates for the earlier

increase in inlet composition Now we are ready to consider control

CONTROL SCHEME

3.12 developing a control scheme for the blending tank

A control scheme is the plan by which we intend to control a process A

control scheme requires:

1) specifying control objectives, consistent with the overall objectives

of safety for people and equipment, environmental protection,

product quality, and economy

2) specifying the control architecture, in which various of the system

variables are assigned to roles of controlled, disturbance, and

manipulated variables, and their relationships specified

3) choosing a controller algorithm

4) specifying set points and limits

3.13 step 1 - specify a control objective for the process

Our control objective is to maintain the outlet composition at a constant

value Insofar as the process has been described, this seems consistent

with the overall objectives

3.14 step 2 - assign variables in the dynamic system

The controlled variable is clearly the outlet composition The inlet

composition is a disturbance variable: we have no influence over it, but

must react to its effects on the controlled variable We do have available a

variable that we can manipulate: the make-up flow rate

We specify feedback control as our control architecture: departure of the

controlled variable from the set point will trigger corrective action in the

manipulated variable Said another way, we manipulate make-up flow to

control outlet composition

Figure 3.11-1 outlet composition response to opposing step inputs

3.15 step 3 - introduce proportional control for our process

The controller algorithm dictates how the manipulated variable is to be

adjusted in response to deviations between the controlled variable and the

set point We will introduce a simple and plausible algorithm, called

proportional control This algorithm specifies that the magnitude of the

manipulation is directly proportional to the magnitude of the deviation

( Ao , setpt Ao)

gain bias

In algorithm (3.15-1) the controlled variable CAo is subtracted from the set

point (Subtracting from the set point, rather than the reverse, is a

convention.) Any non-zero result is an error The error is multiplied by

the controller gain Kgain Their product determines the degree to which

manipulated variable Fc differs from Fbias, its value when there is no error

The gain may be adjusted in magnitude to vary the aggressiveness of the

controller Large errors and high gain lead to large changes in Fc

We must consider the direction of the controller, as well as its strength:

should the outlet composition exceed the set point, the make-up flow must

be reduced Algorithm (3.15-1) satisfies this requirement if controller gain

Kc is positive

3.16 step 4 - choose set points and limits

The set point is the target operating value For many continuous processes

this target rarely varies In our blending tank example, we may always

desire a particular outlet concentration In other cases, such as a process

that makes several grades of product, the set point might be varied from

time to time In batch processes, moreover, the set point can show

frequent variation because it provides the desired trajectory for the

time-varying process conditions

Several sorts of limits must be considered in control engineering:

safety limits: if a variable exceeds these limits, a hazard exists Examples

are explosive composition limits on mixtures, bursting pressure in a

vessel, temperatures that trigger runaway reactions

These limits are determined by the process, and the control scheme must

be designed to abide by them

expected variation: it is necessary to estimate how much variation might

be expected in a disturbance variable This estimate is the basis for

specifying the strength of the manipulated variable response In Section

3.11, our system model (based on the material balance) showed us how

much variation in make-up flow, at specified make-up composition, was

required to compensate for a particular change in the inlet composition

These limits are determined by the process and its environment No

amount of controller design can compensate for a manipulated variable

that is unequal to the disturbance task

tolerable variation: ideally the controlled variable would never deviate

from the set point This, of course, is unrealistic; in practice some

variation must be tolerated, because

• obtaining enough information on the process and disturbance is

usually impossible, and in any case too expensive

• exerting sufficient manipulative strength to suppress variation in the

control variable might be expected to require large variations in the

manipulated variable, which can cause problems elsewhere in the

process

Tolerable limits are determined by the safety limits, above, and then an

economic analysis that considers the cost of variation and the cost of

control We do not expect to achieve perfect control, but good control is

usually worth spending some money

For the blending tank example, then, we select:

• set point: CAo,setpt = 10 kg m-3 This would be determined by the user

of the stream

• safety limits: none apparent from problem statement

• expected variation: ±1 kg m-3; such a specification might come from

historical data or engineering calculations The steady-state material

balance (e.g., (3.11-1) applied at long times) shows that the make-up

flow must vary at least ±5×10-5 m3 s-1 to compensate such

disturbances However, might we need more capability during the

course of a transient??

• tolerable variation: ±0.1 kg m-3 This specification depends on the

user of the stream

EQUIPMENT

3.17 type of equipment needed for process control

Figure 3.17-1 shows our process and control scheme as two

communicating systems The system representing the process has two

inputs and one output Of these only one is a material stream; however,

we recall that systems communicate with their environment (and other

systems) through signals, and in the blending process the outlet

composition responds to the inlet composition and make-up flow rate

The system representing feedback control describes the needed operations,

but we have not described the nature of the equipment – could there be a

single device that takes in a composition measurement and puts out a

flow? Can we find a vendor to make such a device to execute controller

algorithm (3.15-1)? Can we have the gain knob calibrated in units

consistent with those we want to use for flow and composition?

-tank to hold liquid -agitator to mix contents -inlet and outlet piping

multiply gain and add bias

subtract from set point

measure signal

set point

adjust make-up flow

system representing process

system representing controller and other equipment

make-up flow

-tank to hold liquid -agitator to mix contents -inlet and outlet piping

multiply gain and add bias

subtract from set point

measure signal

set point

adjust make-up flow

system representing process

system representing controller and other equipment make-up flow

Figure 3.17-1: Closed loop feedback control of process

We will address these questions in later lessons For now, we assume that

there will be several distinct pieces of equipment involved, and that they

work together so that

( Ao , setpt Ao)

c bias

where we use the conventional symbol Kc for controller gain In the case

of (3.17-1), we notice that the dimensions of Kc are volume2 mass-1 time-1

In good time we will improve our description of both equipment and

controller algorithms When we do, however, we will find that the overall

concept of feedback control is the same as presented in Figure 3.17-1: the

controlled variable is measured, decisions are made, and the manipulated

variable is adjusted to improve the controlled variable

CLOSED LOOP BEHAVIOR

3.18 closing the loop - feedback control of the blending process

Our next task will be to combine our controller algorithm with our system

model to describe how the process behaves under control We begin by

expressing algorithm (3.17-1) in deviation variables At the reference

condition, all variables are at steady values, indicated by subscript r

( Ao,setptr Aor)

c bias r

Presumably the reference condition has no error, so that the set point is

simply the target outlet composition CAo,r Thus we learn that Fbias, the

zero-error manipulated variable value, is simply Fc,r Subtracting (3.18-1)

'

c

r Ao Ao r setpt , Ao setpt , Ao c bias bias r c

c

CC

K

F

CCC

CKFFF

We replace the manipulated variable in system model (3.9-2) with

controller algorithm (3.18-2) to find

Ao

' setpt , Ao c Ac '

Ai

' Ao

'

F

CCC

t

On expressing (3.18-3) in standard form, we arrive at a first-order

dynamic system model representing the process under proportional-mode

feedback control, as shown in Figure 3.17-1

' setpt , Ao c Ac

c Ac '

Ai c Ac

' Ao

' Ao c Ac

CF

KC1F

KCC

F

KC1

1C

dtdCF

KC

++

=

++

τ

(3.18-4)

Equation (3.18-4) describes a dynamic system (process and controller in

closed loop) in which the outlet composition varies with two inputs: the

inlet composition and the set point Figure 3.18-1 compares (3.18-4) with

the process model (3.9-2) alone; we see that

• the closed loop responds more quickly because the closed loop time

constant is less than process time constant τ

• the closed loop has a smaller dependence on disturbance C′Ai because

the gain is less than unity Both time constant and gain are reduced by

increasing the controller gain Kc

3.19 integration from zero initial conditions

In Section 3.10, we integrated our open-loop system model to find how

C′Ao responded to inputs C′Ai and F′c Now we integrate closed-loop

system model (3.18-4) in a similar manner

dtC

eK

edtCeK

e

setpt , Ao t

0

t SP CL

t '

Ai t

0

t CL CL

t '

CL CL

CL

∫

− τ

τ

−

τ

+τ

where

F

KC1F

KCK

F

KC1

1K

F

KC

c Ac

SP c

Ac

CL c

Ac CL

+

=+

=+

τ

=

' c Ac '

Ai

' Ao

'

F

CC

Cdt

τ

' setpt , Ao c Ac

c Ac '

Ai c Ac

' Ao

' Ao c Ac

CF

KC1F

KCC

F

KC1

1C

dtdCF

KC

(the process under control)

variable gain

other input

' c Ac '

Ai

' Ao

'

F

CC

Cdt

τ

' setpt , Ao c Ac

c Ac '

Ai c Ac

' Ao

' Ao c Ac

CF

KC1F

KCC

F

KC1

1C

dtdCF

KC

(the process under control)

variable gain

other input

Figure 3.18-1: Comparing open- and closed-loop system descriptions

3.20 closed-loop response to pulse disturbance

We test our controlled process by a pulse ΔC in the inlet composition that

begins at time t1 and ends at t2 We find

e1KC

ttte

1KC

tt00

C

2

) t t )

t CL

Ai

2 1

) t CL

1 2

CL 1

(3.20-1)

Figure 3.20-1 shows both uncontrolled (open-loop) and controlled

(closed-loop) process responses for the same operating conditions used in Section

3.11 We see the faster response and smaller error that we expected when

we examined (3.18-4) in Section 3.18 These characteristics improve as

gain increases Increasing gain also elicits stronger manipulated variable

action Thus automatic control appears to have improved matters

Kc= 0 m 6 kg -1 s -1

Figure 3.20-1: Response to pulse input under proportional control

3.21 closed-loop response to step disturbance - the offset phenomenon

Integrating (3.19-1) for a step of ΔC, we obtain

Figure 3.21-1 shows open- and closed-loop step responses Notice that for

no case does the controlled variable return to the set point! This is the

phenomenon of offset, which is a characteristic of the proportional control

algorithm responding to step inputs

CL

' Ao

setpt , Ao Ao

KC

)(C

C)(C

pointset -responselongterm

Recalling (3.19-2), increasing the controller gain decreases the closed-loop

disturbance gain KCL, and thus decreases the offset

We find that offset is implicit in the proportional control definition

(3.15-1) An off-normal disturbance variable requires the manipulated

variable to change to compensate For the manipulated variable to differ

from its bias value, (3.15-1) shows that the error must be non-zero Hence

some error must persist so that the manipulated variable can persist in

compensating for a persistent disturbance

3.22 response to set point changes

We apply (3.19-1) to a change in set point

SP setpt , Ao

'

We recall from (3.19-2) that KSP is less than 1 Thus, the outlet

composition follows the change, but cannot reach the new set point This

is again offset due to proportional-mode control Increasing controller

gain increases KSP and reduces the offset

3.23 tuning the controller

Choosing values of the adjustable controller parameters, such as gain, for

good control is called tuning the controller So far, our experience has

been that increasing the gain decreases offset - then should we not set the

gain as high as possible?

We should not jump to that conclusion In general, tuning positions the

closed-loop response between two extremes At one extreme is no control

at all, gain set at zero (open-loop) At the other is too much attempted

control, driving the system to instability In the former case, the

controlled variable wanders where it will; in the latter case,

over-aggressive manipulation produces severe variations in the controlled

variable, worse than no control at all Tuning seeks a middle ground in

which control reduces variability in the controlled variable This means

both rejection of disturbances and fidelity to set point changes