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4. Complex Control Systems
3. COMPLEX CONTROL SYSTEMS
1 Introduction
In practice, control engineers may encounter:
•
A process has significant disturbances.
•
A process with multiple outputs is controlled by a single input
•
A process has multiple inputs to control a single output.
•
The output of a process is not available.
As controllers designed by the techniques discussed before will not perform acceptably, more
complex control structures has to be used and will be discussed in this Section.
Processes with Significant Disturbances
Example 1: A conventional feedback control system is shown below
Temperature at bottom of distillation column is controlled by adjusting the steam flowrate of the
reboiler which is controlled by appropriate valve opening.
Problem: for a given valve opening, change in the steam supply pressure affect steam delivered
to the reboiler, causing alteration in column temperature. If such fluctuations in steam supply
pressure are frequent and substantial, the performance of the conventional scheme will
deteriorate. In such cases, a more effective control scheme is required for improved
performance. The block diagram of such problem is shown in following
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Example 2: shows a stirred heating tank under conventional feedback control.
The objective is to regulate T, the temperature within the tank, by adjusting the rate of steam
flow through the coil.
Problem: By conventional scheme, changes in Ti must first be registered as an upset in the value
of T before the controller can take corrective action. However, where the fluctuations in Ti are
frequent and substantial, an alternative scheme, which detects changes in the disturbance Ti and
implement preventive control action before an upset is registered in T, will be far more effective.
Same block diagram may also be represented for the stirred heating tank control problem.
These two examples are illustrative of processes with significant disturbances. There are two
common controller structures that are often used for this type of problem:
•
Cascade control: when significant disturbances affect the input
•
Feedforward control: when significant disturbances affect the output.
Example 3: Consider an Air Handling Unit (AHU) for Air-conditioning system.
•
Air enters the AHU through the outdoor air damper, depends on mixing box damper
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settings, may be mixed with air passing through the re-circulation air damper.
•
The temperatures and flow rates of the outdoor and re-circulation air streams determine
the conditions at the exit of the mixed air plenum.
•
Air exiting the mixed air plenum passes through the heating or cooling coils (only one of
the two coils will be active at any given time).
•
After being conditioned in the coils, the air is distributed to the zones through the supply
air ductwork. The supply air temperature is measured downstream of the supply fan.
•
Return air is drawn from the zones by the return fan and is either exhausted or recirculated, depending on the position of the mixing box dampers. The return air
temperature is measured downstream of the return fan.
Depending on control scheme, variable to be controlled are
•
OA% and RA%,
•
Supply Air temperature,
•
Supply Airflow rate,
•
Duct pressure, etc.
Manipulating variables are
•
Chilled Water pump
•
Chilled Water valve
•
Fan speed, etc.
2 Cascade Control
Example 1: Considering the distillation column temperature control problem and compare
In the second scheme, a flow controller (FC) is put in between the temperature controller (TC)
and the control valve.
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Purpose: irrespective of fluctuations in the steam supply pressure, the desired steam flow rate
can be delivered.
Configuration: the output of the TC (the primary controller) is the set-point for the FC (the
secondary controller).
The temperature controller needs only to adjust the set-point of the flow controller.
Characteristics: the output of one is the set-point of the other.
With proper cascade controller design, the effect of supply pressure disturbances on the steam
flow conditions can be significantly reduced.
Example 2: The cooling only AHU is shown in Figure below, it consists cooling coil, air
dampers, fans, chilled water pumps and valves.
Exhust air
Return air
Damp
Exhust fan
MA
Cooling coil
Supply fan
Fresh air
Supply air
Chilled water
TC
TT
MA= mixed air
TT=temperature transmitter
TC=temperature controller
Let Gp and Gc be the AHU process and temperature controller, respectively, The drawback of a
conventional PID controller is that it has a single degree of freedom in tuning, and this makes it
difficult for field engineers to tune PID parameters to meet different specifications. Considering
a central HVAC plant as shown in below, where the main chilled water distribution pipe may
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supply chilled water to many AHUs. Therefore, flow rate change in one AHU branch will affect
all other loops. When conventional single loop temperature control scheme is used with the
output of the temperature controller applied directly to the control valve, no correction will be
made until its effect reaches the temperature measuring elements. Thus, there is a considerable
lag in correcting flow rate disturbances.
To improve the response of the simple feedback control to changes in chilled water supply
pressure, the pressure drop across the control valve is measured to adjust the valve position
before the cooling exchanger reacting mixture has felt its effect. Thus, if pressure goes up,
decrease the valve opening of the chilled water to keep constant chilled water supply, and vice
versa. With a cascade control technique using a separate flow rate control loop, the flow rate
controller will correct any change in the chilled water flow rate. Fig. shows a schematic diagram
of the cascade control system for supply air temperature
Exhust air
Return air
Damp
Exhust fan
MA
Cooling coil
Supply fan
Fresh air
Supply air
FC
FT
Chilled water
TC
TT
MA= mixed air
TT=temperature transmitter
TC=temperature controller
FT=flow transmitter
FC=flow controller
Basic Principles of Cascade Control
General structure of block diagram of cascade control
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y
the bottom's temperature
u
the steam flowrate
g(s)
the main transfer function
gv(s)
the transfer function between the controller signal and the control valve stem
position
ud
the desired steam flowrate
d2
steam pressure header disturbance
d1
is other disturbances to the bottom's column temperature not related to steam
flowrate.
Primary control loop: The main process g(s) has a single output y to be controlled by
manipulating a single input U using controller gc1.
Secondary control loop: this controller is set up to regulate U by adjusting final control element
with transfer function gv(s).
The signal from the primary controller becomes the set-point for the secondary controller to
ensure that the main process receives the input U as originally intended by the primary
controller.
The two cascade controllers are also referred to as master (primary) and slave (secondary)
controllers.
Two requirements for cascade control:
1.
The disturbance to be regulated, e.g., d2 must be within the inner loop, so that it is
controllable by the secondary controller.
2.
The inner control loop must respond much more quickly that the outer control loop in
order to allow U to be regulated before it disturbs the entire process.
For this example, this means that a steam flow measurement device must be installed.
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Closed-Loop Characteristics
In cascade control, even it is a single-input single-output (SISO) system, it have two control
loops. The cascade control system is a multi-loop, SISO control system.
To analyze the overall behavior of the system, we deal first with the inner loop. The closed-loop
relationship for the inner loop elements is:
u=
gd 2
gc2 gv
d2
ud +
1 + g c 2 g v h2
1 + g c 2 g v h2
u = g1*ud + g 2* d 2
The block diagram is consolidated to
The overall closed loop transfer function representation is immediately obtained as:
gg1* gc1
g d1
gg2*
y=
yd +
d2 +
d1
*
*
1 + gg1 gc1h1
1 + gg1 gc1h1
1 + gg1* gc1h1
To compare conventional feedback control with cascade control, we obtain closed loop transfer
function of conventional feedback control from the block diagram
y=
ggv gc1
ggd 2
gd 1
d1
yd +
d2 +
1 + ggv gc1h1
1 + ggv gc1h1
1 + ggv gc1h1
In cascade control, gv is replaced by:
g1* =
g c2 g v
1 + g c 2 g v h2
g 2* =
gd 2
1 + g c 2 g v h2
and gd2 is replaced by:
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When h2 = 1:
•
As gc2 increases, g1* → 1 and g2* → 0
•
And, as a result U → Ud
With high-performance inner-loop control, the overall effect of the disturbance d2 on U (and
hence on the process system) is totally eliminated.
Conclusion: with a fast-acting inner loop (high magnitude for gc2), the overall cascade system
becomes much less vulnerable to the effects of the fluctuations in d2 than the equivalent system
under conventional feedback control.
Because the inner loop of a cascade controller faster than the outer loop, the steady-state values
of g1*, and g2* can often be used. This amounts to a "quasi-steady-state approximation" for the
inner loop dynamics. The above example illustrates the point. The inner loop (which needs only
to move the steam valve to cause an almost instantaneous adjustment in steam flowrate) has a
timescale of a few seconds, by contrast, the outer control loop (which includes the very slow
dynamics of the steam heating the entire liquid contents of the column bottoms) would have a
timescale of many minutes.
Cascade Controller Tuning
It is clear that the controller tuning exercise for a cascade control system must proceed in two
stages:
1.
The inner loop is usually tuned "very tightly" (i.e., as high a proportional gain value as
feasible) to give a fast inner-loop response required for effective cascade control.
2.
The outer loop is then tuned, with the inner loop in operation, using any of the methods
discussed before.
It is useful to note that proportional controllers can be used for the inner loop of a cascade
controller because any offset in y can be corrected by using integral action in the outer loop.
3 Feedforward Control
To illustrate the concepts of feedforward control, consider the stirred heating tank example.
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Conventional feedback control suffers from frequent disturbances in feed temperature, and the
controller must wait until these have upset the process before feedback action can be taken.
Cascade control is not a solution, as the disturbance is not associated with the manipulated
variable.
However, if we measure the feed temperature Ti, then configure a feedforward controller shown
as following. The feedforward controller then makes adjustments in the steam flowrate to the
heating coil in response to observed changes in Ti.
The difference between feedforward and conventional feedback controls:
•
Feedforward control: compensation for the effect of a measured disturbance is possible
before the process is affected
•
Conventional feedback control: corrective action can only be taken after the process has
registered the effect of the disturbance.
Perfect disturbance rejection – not possible with feedback control-is possible (theoretically) with
feedforward control.
The process output is never measured in feedforward control.
Design of Feedforward Controllers
Consider the following general transfer function model for a process relating manipulated
variable and disturbance to process output, with no controller yet attached:
y ( s ) = g ( s )u ( s ) + g d ( s ) d ( s )
The objective is for y(s) to equal its set-point value, yd(s), at all times, regardless of changes in
d(s).
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Derive the u(s) required to make y(s) = yd(s) for all time.
This translates to the problem of solving the above equation for u upon setting y = yd; doing this
gives:
u ( s) =
g ( s)
1
d ( s)
yd − d
g ( s)
g ( s)
or
u ( s ) = g st ( s ) yd + g ff ( s )d ( s)
gst(s) is the set-point tracking controller;
gff(s) is the disturbance rejection controller.
The block diagram of the feedforward control system is shown in the following
Note:
1.
The scheme can, in principle, be used both for set-point tracking (the servo problem), and
disturbance rejection (the regulatory problem). Observe that for servo problems (d =0),
the controller equation is:
u ( s ) = g st ( s ) yd
for the regulator problem (yd = 0), we have:
u ( s ) = g ff ( s ) d ( s )
However, it is almost exclusively associated with disturbance rejection; it is hardly ever
used for set-point tracking.
2.
In both set-point tracking and disturbance rejection, the controller has no access to
measurements of y, it is focused on the potential cause of the upset (changes in yd or d). It
relies on the accuracy of the process transfer function models to ensure that the desired
behavior is obtained, even though y itself is not monitored.
The feedforward control is possible when:
•
The disturbance d is measured, and
•
The effect of d on the output y is available.
Example: DESIGN OF A FEEDFORWARD CONTROLLER FOR STIRRED MIXING TANK.
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The first task is to monitor the inlet temperature Ti, so as to have a measurement of the
disturbance d.
Its transfer function model is given as:
⎛ βθ ⎞
g (s) = ⎜
⎟
⎝ 1 + θs ⎠
⎛ 1 ⎞
g d (s) = ⎜
⎟
⎝ 1 + θs ⎠
where the parameters β and θ are given by
θ=
V
F
and
β=
λ
ρVC P
respectively, where V: tank volume, ρ: liquid density, Cp: specific heat capacity, λ: latent heat of
vaporization of steam, F: volumetric flowrate.
Solution: According to the design procedure given above, the equation for the feedforward
controller therefore is:
u (s) =
1 + θs
βθ
yd −
1
βθ
d (s)
Now, we investigate the controller performance and the overall behavior of the stirred mixing
tank system in response to a step change of magnitude A in the set-point yd
The feedfdrward controller equation for set-point tracking is:
u (s) =
1 + θs
βθ
yd
and since, in this case, yd(s) = A/s, we have:
u ( s) =
A
A
+
βθs β
The overall system response is now obtained:
y ( s) =
βθ
u (s)
1 + θs
which gives:
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y ( s) =
A ⎛1
⎞
⎜ +θ ⎟
1 + θs ⎝ s
⎠
result
A
s
y( s) =
which, when inverted back to time, gives
y (t ) = A
The implication of this result is that the system output, y(t), tracks the desired set-point perfectly.
It means; so long as the process model matches the actual process behavior, this feedforward
scheme always gives the result that y = yd for all time regardless of the particular form of yd.
In practice, the process model will typically not match the process behavior exactly and such
perfect set-point tracking will no longer be possible and steady-state offsets will be inevitable.
Disturbance rejection: The feedforward controller equation for disturbance rejection is:
u (s) = −
1
βθ
d (s)
inverted back to time gives:
u (t ) = −
1
βθ
d (t )
Substitute β and θ into u(t)
u (t ) = −
FρC p
λ
d (t )
If d(t) increases from its initial value of 0 to a new value δ (i.e., Ti increases from Tis to Tis + δ);
then
u (t ) = −
F ρC p
λ
δ
control action calls for the energy input to the process to be reduced by that factor.
As FρCpδ is the amount of additional energy brought into the system as a result of the increase
in Ti, the control is an exact counterbalancing of the disturbance effect.
The tank temperature is unaffected by whatever fluctuations the inlet stream temperature Ti may
experience, result a situation of perfect disturbance rejection.
Practical Considerations
The system suffers from some significant drawbacks:
1.
It is not useful if d cannot be measured;
2.
Even when d is measurable, the scheme requires a perfect process model to achieve its
objectives;
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3.
Even a perfect process model exists, the derived feedforward controller transfer functions both gst and gff involve the reciprocal of the process transfer function g as
u (s) =
g (s)
1
d (s)
yd − d
g (s)
g (s)
may therefore present any of the following problems:
(a)
They may be too complicated to be realizable in time.
(b)
If g has a time delay element e-αs then its reciprocal will contain the term eαs,
implying that implementing gst in real time will require a prediction.
(c)
If both g and gd have time-delay elements, and the time delay associated with gd is
less than that associated with g, then gff will contain a term in eγs where γ > 0; again,
prediction is required to implement such a controller in real time.
Under the conditions specified in (b) and (c), the controllers are said to be physically
unrealizable since, at time t, each of these controllers will require knowledge of the value that
yd(t), or d(t) will take at some time in the future; in (b) yd(t + α) is required, in (c), d(t +γ) is
required. It is, of course, impossible to predict these quantities exactly.
Feedforward control has proved to be a very powerful process control scheme, especially for
disturbance rejection.
(1)
Many process disturbances are indeed measurable;
(2)
Even when the available process models are imperfect and/or the derived gff is too
complicated, the lead/lag unit provides reasonably good performance as a feedforward
controller.
Application of Lead/Lag Units in Feedforward Control
Recall the Cohen and Coon process reaction curve technique for approximate process dynamic
characterization, in which a general approximate representation for many g and gd elements is:
Ke −αs
g ( s) =
1 + τs
and
K d e − βs
g d ( s) =
1+τ d s
leading to a gff having the following generalized form:
K d e − βs 1 + τs
g ff ( s) = −
1 + τ d s Ke −αs
This simplifies to:
g ff ( s) = −
K ff (1 + τs )
1+τ d s
e −γs
which is a lead/lag unit with a time delay. In the event that α and β are approximately equal, the
above Eq. is a pure lead/lag element.
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This offers an objective justification for the effectiveness of lead/lag units in the implementation
of feedforward control schemes even in situations when the dynamic behavior of the processes in
question is poorly modeled.
Tuning the Lead/Lag Unit
1.
Obtain approximate characterizing process parameters Obtain process reaction curves for
(a) The response of the process output to a step change in the input U (with d =0), and
(b) The corresponding process response to a step change in the disturbance d (with U = 0).
Characterize each response by their respective steady-state gains, and effective time
constants, and time delays as given in above equations.
2.
Estimate lead/lag unit parameters
From the six parameters obtained from the two process reaction curves, we obtain the
lead/lag unit parameters as follows:
Gain term: Kff=Kd/K
Lead time constant:τ
Lag time constant: :τd
Associated delay: γ=β−α
4. Feedback/Feedforward Control
Weaknesses of feedforward control:
1.
Identification of all disturbances and their direct measurement, not possible for many
processes.
2.
Any changes in the parameters can not be compensated, because they can not be
detected.
3.
Requires a good model for the process, not possible for many systems.
Feedback control is insensitive to all three of these drawbacks but has other drawbacks:
Feedback
Feedforward
Advantages
Not require identification and measurement of
disturbance
Acts before a disturbance has been felt by the system.
Insensitive to modeling errors.
Good for slow systems or significant dead time.
Insensitive to parameter changes.
Not introduce instability in the closed-loop response.
Disadvantages
Control action is taken after disturbances have
been felt.
Requires all disturbances and their direct measurement.
Unsatisfactory for slow and significant dead time
process.
Cannot cope with unmeasured disturbances.
It may create instability in the closed-loop
response.
Sensitive to process parameter variations.
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In practice, it is common to augment a feedforward control scheme with feedback compensation.
Purpose: complement the disturbance rejection properties of the feedforward controller with the
"ruggedness" of the feedback controller.
In this scheme, feedforward control is used to contain the effects of disturbances while the
feedback is used to take care of the errors introduced as a result of the inevitable imperfections
of feedforward control and/or other unmeasured disturbances.
The block diagram is shown in the follwoing, the dual objective here is to choose gff for perfect
disturbance rejection and gc, so that the closed-loop system is stable.
Upon simple block diagram manipulations on the Figure, we obtain the equivalent block diagram
as following.
The closed loop transfer functions are easily obtained as:
y=
g + gg ff
ggc
d
yd + d
1 + ggc h
1 + ggc h
If we now select gff as:
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g ff ( s ) = −
g d (s)
g (s)
the effect of d on y is theoretically eliminated.
Important points about the feedforward/feedback scheme:
1. The overall system stability is determined by the roots of the closed-loop system’s
characteristic equation:
1 + gg c h = 0
Observe that this has nothing to do with gff. The stability of the feedback loop remains
unaltered by including feedforward compensation. This is a very significant point in
favor of the feedforward control scheme; in terms of closed loop stability, we have
nothing to lose by applying it.
2. When gd and g are perfectly known, perfect compensation is possible; in the more
realistic case where this is not the case, and/or the disturbances are (partially or
completely) unmeasurable, the feedback loop picks up the residual error and eliminates
it with time.
5
Ratio Control
In certain situations, the value observed for d is to be a certain percentage of another process
input variable.
In this case, the objective is to maintain a constant ratio between d and this other variable. Thus
as d changes, this other variable also changes in order to maintain the desired ratio.
The control scheme is called ratio control; it is a special form of feedforward control.
Process Flow Applications
Ratio control mostly applied in systems that involve mixing of two or more process streams. In
such process flow applications, there are typically two streams whose flowrates are to be
maintained at a certain prespecified proportion.
Both flowrates are measurable, usually the flowrate of one stream can be controlled while the
other flowrate cannot be controlled but can be measured.
The stream whose flowrate cannot be controlled is referred as: wild stream.
Consider the process as shown in the Figure, a salt solution mixer for which the salt concentrate
flowrate can be regulated but the water flowrate cannot. It makes the water stream the "wild
stream."
To maintain uniform salt concentration within the mixing tank, it is necessary to maintain a
constant ratio of water and salt concentrate flowrates.
A ratio control scheme to achieve this objective can be set up in one of the two ways shown in
the following Figures
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Under the strategy indicated in Figure (a),
•
Both flowrates are measured and their current ratio obtained by an electronic divider.
•
The observed ratio is transmitted to the ratio controller where it is compared to the desired
ratio set-point; the error is used to set the flowrate required for the salt concentrate stream.
The actual current ratio of the flowstreams is the information obtained from the process and
passed on to the controller; the controller will now have to use this information to calculate, and
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implement the change needed in the salt concentrate flowrate to attain to the desired ratio.
Another strategy indicated in Figure (b)
•
The "wild stream" flowrate is measured and multiplied at the "ratio station" by the desired
ratio; this produces the value of the flowrate at which the other stream must be set to
maintain the desired ratio.
•
The output of the ratio station is the set-point for the flow controller; it is compared with the
actual flowrate of the salt concentrate stream, and the controller acts on the basis of the
observed deviation.
In this case there is a flow controller that directly regulates the flowrate of the salt concentrate
stream; the information it receives from the ratio station is its set-point.
Two Loops Ratio Control
In the case it requires high control quality and no wild stream, It is nature to control both loops.
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Compared with normal ratio control scheme, this ratio control scheme is more robust as it can
reject disturbance of main control loop so that the response of secondary loop becomes
smoother.
Variable Ratio Control
The ratios are different at different set points.
6
Single Input Control Multiple Outputs
One manipulated variable and several outputs to be controlled is called selective control.
Depending on the application, there are several approaches to selective controller design.
Override Control
It is often used to handle the situation in which the single manipulated variable to regulate one
output under normal operation, and used to control another output variable in abnormal
operation.
This usually be applied when the process would be unsafe under abnormal operation without the
"override" to the second output variable.
Example 1: Consider the crude oil preheater furnace. The feedback control system is shown in
the following Figure. It adjusts the furnace fuel flowrate to maintain the desired exit temperature
of the heated crude oil stream.
Both feedforward and cascade control could be added to better compensate for fuel header
pressure fluctuations and crude oil feedrate disturbances.
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Problem: The outside metal surface temperature of the tubes containing the crude oil must
remain below the metallurgical limit of the tube material.
Override controllers can be used to maintain the tube surface temperature below this critical
value. The override control scheme is shown in the following
The optical pyrometers are used to measure the tube surface temperature at several locations
along the tube.
One of these tube temperature measurements are selected and send it to a temperature controller
and High Selector (HS) switch. When the measured tube temperature exceeds a preset critical
value, the HS switches from the "normal" loop to the "abnormal" loop in which the fuel flowrate
is adjusted to control tube skin temperature at a preset safe maximum.
When the "normal" loop calls for less fuel than required to maintain the maximum tube surface
temperature, the selector will switch back to the "normal" loop.
In this way the crude oil exit temperature from the furnace will be maintained at its set-point
except when the tube surface temperature is exceeded.
During "abnormal" operation, the crude oil exit temperature will be somewhat below its setpoint, but as close as possible while ensuring safe operation of the furnace.
Example 2: Protection of a boiler system:
1.
The steam pressure in a boiler is controlled through the use of a pressure control loop on the
discharge.
2.
Water level in the boiler should not fall below a lower limit which is necessary to keep the
heating coil immersed in water and thus prevent its burning out.
Figure shows the override control system using a low switch selector (LSS). Whenever the liquid
level falls below the allowable limit, the LSS switches the control action from pressure control to
level control (loop 2 in the Figure) and closes the valve on the discharge line.
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Auctioneering Control
A second type of selective control is so-called "auctioneering" control. This is where there is a
fixed control loop, but the output variable to be used may change. This may be illustrated by the
furnace example of the previous section.
The tube surface temperature was measured at several locations, for the "abnormal" loop we are
interested in the location with the highest tube surface temperature. A HS switch is added to
choose the highest temperature of those measured in regulating the tube surface temperature. It
ensures that the highest measured temperature is maintained at a safe level.
Example: Several highly exothermic reactions take place in tubular reactors filled with a catalyst
bed.
The Figure shows the temperature profile along the length of the tubular reactor.
Hot spot: The point of highest temperature along the length of the tubular reactor.
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Variation Factors:
•
Feed conditions (temperature, flow rate, concentration)
•
Catalyst activity
•
Temperature and flow rate of the coolant.
7 Multiple Inputs Control Single Output
In this configuration, there are two or more manipulated variables available for the control of a
single-output variable.
Split-Range Control
Split-range control is the usual solution when one requires multiple manipulated variables in
order to span the range of possible set-points.
Example: Suppose there is a nonisothermal batch reactor with a certain specified temperature
program required to produce the product.
•
The temperature ranging from 150C at the beginning of the batch to 1000C at the end.
•
The cooling water is at 50C
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4. Complex Control Systems
•
The low-pressure steam is at 1800C.
It requires both cooling water and steam in order to span the temperature range of interest.
Split-range control allows both cooling water and steam to be used as shown in the Figure:
Note that the schedule allows the appropriate fraction of cooling water and steam to be used for
each value of the controller output signal and reactor jacket inlet temperature required. Usually
the design uses overlap in the range of each valve for more precise control.
Split-range control is often employed for broad span temperature control of small plant, and for
year-round heating and cooling of office buildings.
Multiple Inputs improving Dynamics
Multiple inputs are also employed in order to speed up the dynamic response of a process during
serious upsets or transitions between set-points.
Example: Consider the stirred heating tank, but now assume that there is an additional auxiliary
electrical heater and cooler that can be used to control the tank temperature as shown in the
Figure:
The heating tank has several temperature set-points depending on the product being made in a
downstream reactor. The normal regulatory operation at each of these set-points can be readily
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4. Complex Control Systems
handled with the process steam going to the tank heating coil.
It is important for the downstream reactor that set-point changes be made rapidly but the stirred
heating tank temperature responds rather slowly even with the steam valve full open or full shut.
Thus for an increase in temperature set-point the auxiliary heater helps speed the move to the
new set-point. Similarly when the temperature set-point change is down, the auxiliary cooler
helps the tank quickly achieve the new set-point.
The drawback is that this auxiliary heater and cooler require expensive electrical energy while
the steam to the coil is very cheap because it is produced as a byproduct from another process
unit. Thus, the auxiliary heater and cooler be used only during set-point changes.
This could be accomplished with three controllers as shown in Figure above with a schedule
defining when each controller is active. For example, one could choose the following schedule to
achieve the control objective.
CONTROLLER SCHEDULE
Deviation from set-point
∆T < - 10 C
0
Controller active
TC3+TC1
- 10 C < ∆T < + 10 C
TC1
∆T > + 10 C
TC2+TC1
0
0
0
The controllers TC2 and TC3 could be on-off controllers or proportional controllers with
suitable gain. The controller for the steam coil would have to have integral action and be tuned to
reject disturbances.
8 Inferential control
Generic problem
There exist a number of processes in which the primary variable to be controlled is difficult to
measure or is a sampled measurement with a long delay in the sampling and analysis process.
Sometimes the quantity to be controlled is a calculated variable. In such cases, control of the
process is usually accomplished by measuring secondary variables (for which sensors are more
reliable, cheaper or more readily available and installed) and setting up a feedback control
system using these secondary variables. Such control strategies are referred to as inferential
control, the control scheme can be used when:
1. Measurement of the true controlled variable is not available in a timely manner because
•
An on-stream sensor is not possible.
•
An on-stream sensor is too costly.
•
Sensor has unfavorable dynamics (e.g., long dead time or analysis time) or is
located far downstream.
2. A measured inferential variable is available.
Example 1: In distillation the primary variables to be regulated are the product compositions
(bottoms and distillate purity) as shown in Figure below. Gas chromatographs typically used to
measure these are very expensive, difficult to maintain/calibrate and introduce significant
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25
measurement delays because of the time needed to purge the sample line and to heat the sample.
In this case, control is accomplished using temperature measurements on the intermediate trays.
Example 2: In polymerization reactors, primary variables of interest are the molecular weight
and viscosity of the product, as shown in Figure below. Control is accomplished using
secondary measurements such as temperature and pressure in the reactor. The Low Density
PolyEthylene (LDPE) can be detected from online measurements such as the temperature profile
down the reactor and the solvent flow rate, which are available on a more frequent rate than
fundamental polymer properties. The estimation of polymer viscosity in a polymerization
reactor using viscometer is subject to a significant time delay but the torque from a variable
speed drive provides an instantaneous indication of reactor viscosity.
Example 3: Figure below shows a third application of inferential control. This pertains to control
of industrial drying processes. The control of such drying processes is to determine solids
moisture, a variable not usually measurable, and using this to manipulate the temperature of
drying air until the desired target in solids moisture is obtained.
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4. Complex Control Systems
Figure below shows the schematic of the generic problem tackled by inferential control. The
control objective typically is to keep the primary variable on target in presence of unmeasured
disturbances. We first look at some classical techniques used, which while being simple to
implement can be costly because of poor controller performance.
Generic inferential control problem
For the simple case of a linear system with a single disturbance, single primary output and single
secondary measurement as in the Figure, the process can be modeled as:
y ( s) = guy ( s)u ( s ) + g dy ( s )d ( s )
Primary output
z ( s ) = guz ( s )u ( s ) + g dz ( s )d ( s )
Secondary measurements
(1)
Block diagram of a process with one primary and one secondary measurement
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4. Complex Control Systems
Classical Control
Consider direct feedback control of the secondary measurement z using the manipulated variable
u as shown in Figure. This strategy can be used if the primary variable and secondary variable
are very closely related. For example in distillation, it is well known that temperature is a very
good indicator of product composition. Hence, by maintaining one of the tray temperatures
constant, we can often maintain good control of the product quality.
A classical approach control secondary variable
Taking setpoint of z to be zero without loss of generality, the transfer function for disturbance
rejection under perfect control of z can be derived as:
⎛ guy ( s ) g dz ( s )
⎞
y( s) = ⎜ −
+ g dy ( s ) ⎟ d ( s )
guz ( s )
⎝
⎠
Note that y(s)=0 if g dz ≈ g dy and guz ≈ guy , i.e. when the disturbance d and manipulated variable
u affect both z and y in a similar manner. If this is not so (which is often the case) this strategy
may result in poor control.
Cascade Control
In the cascade control strategy, shown in Figure, the inner loop tries to maintain the secondary
variable at a set point which is adjusted by the outer loop to bring y back to its set point. This
strategy is usually employed when there are significant delays and lags associated with
measurement of y. To implement this strategy, we must have a measurement of the primary
variable. The disturbance rejection transfer function is given by:
y(s)
= guy ( s ) g1 ( s ) + g dy ( s )
d (s)
(3)
where
g1 = −
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g cz g cy g dy + g cz g dz
1 + g cz g cy guy + g cz guz
(4)
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4. Complex Control Systems
Structure of the cascade control system
This strategy has the advantage that steady state error in control of y will be reduced to zero if
we use integral action in the outer loop controller. But whether this strategy will work well in a
process depends on a number of factors. The inner loop should be able to react fast enough to
follow frequent set-point changes. If there are significant lags in the inner loop then the system
will not have enough time to settle down, and control system performance will be poor. If the
disturbances come in at a low frequency such that the outer loop has enough time to correct it.
this structure might be acceptable. However if the disturbances come at a frequency that keeps
the system from settling down then the controller on the outer loop will not have time to settle
down. Because of the large delays involved in the measurement this will usually mean poor
performance of the control system.
More importantly, both of the above strategies have no easy extension to the case of multiple
secondary measurements. Usually multiple secondary measurements contain more information
about the state of the system. Thus methods that use multiple measurements have an advantage
over these multi-loop strategies.
Estimator Based Control
In this strategy, an estimator for the unmeasured output y is built first which is then used in a
feedback control mode. Figure 5 shows the structure and block diagram of this control strategy.
Structure of the controller using state estimator
The disturbance rejection transfer function is given by
g dy ( s )
y(s)
=
d ( s ) 1 + guy ( s ) g cy ( s )
Note that this is the same as what we get if we had direct feedback control on the primary
measurement itself. The advantage with this approach is that an estimate of the unmeasured state
y is available through the secondary measurements, which is useful for the operator. If g dy
and gcy have large lags associated with them, then this can result in poor performance since the
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4. Complex Control Systems
optimum performance achievable using direct feedback control is limited by the time lags and
time delays present in the feedback loop. This controller may be worse than direct feedback
control on z in some cases if z responds faster to the disturbance and manipulated variable. In
addition, state estimators are never perfect and introduce additional errors in the feedback
control loop that necessitate de tuning the controller to some extent. The construction of state
estimators is not trivial.
Inferential Control
In the classical 2-degree of freedom IMC structure an estimate of the disturbance effect on y is
fed back as shown in Figure. The question inferential control tackles is how to generate
d y = g dy d ( s ) if no measurement of y is available but only z(s) is available.
In this case, we can first compute
d z ( s ) = g dz ( s)d ( s ) = z ( s ) − guz ( s )u ( s )
(6)
And then obtain an estimate of d(s) as:
d e ( s ) = g dz−1 ( s ) [ z ( s ) − guz ( s )u ( s ) ]
(7)
d y can then be estimated as follows:
d ye ( s ) = g dy ( s )d e ( s ) = g dy ( s ) g dz−1 ( s ) [ z ( s ) − guz ( s )u ( s ) ]
(8)
Using this equation we get the structure of the inferential control system as shown in Figure
The design of the disturbance controller g cd ( s ) is as in IMC. The output response is given by
y ( s) = guy ( s)u ( s ) + d y ( s )
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(9)
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4. Complex Control Systems
To keep y close to the set point (= 0) we choose the controller so that
u ( s ) = − guy−1 ( s )d y ( s )
(10)
Using the estimate of d y in place of d y we get:
u ( s ) = − guy−1 ( s ) g dy ( s ) g dz−1 ( s ) ( z ( s ) − guz ( s )u ( s ) )
(11)
The controller transfer function derived above may not be realizable since we have to invert the
process transfer function. If the process transfer functions contain RHP zeros or time delays then
we must add a filter f(s), designed as in the IMC controller, to make the controller realizable:
u ( s ) = − f ( s ) guy−1 ( s ) g dy ( s ) g dz−1 ( s ) ( z ( s ) − guz ( s )u ( s ) )
(12)
The disturbance rejection transfer function for the control scheme is given by:
y(s)
= g dy ( s ) (1 − f ( s ) )
d (s)
(13)
This transfer function is very similar to a feedforward controller. If the filter transfer function
used is close to 1 then we have nearly perfect rejection of disturbances as in feed forward
control. In general the filter must have lag terms to compensate for modeling errors and make
the system robust.
It is important to differentiate the above structure from state estimation techniques like Kalman
filter that uses z to predict y and then control y. If the secondary measurements respond faster to
the disturbances then one can take faster corrective action and hence get better control system
performance by using z in an inferential structure. If the output y responds to disturbances
slower than z, then the state estimator will add additional lag in the feedback loop and hence will
adversely affect the performance of the control system. Usually it is possible to identify
secondary measurements closer to the disturbance and hence in general the inferential control
strategy is preferred.
Both the inferential control scheme and the state-estimation schemes outlined above can be
extended to nonlinear and multivariable situations.
Summary
An inferential variable must satisfy the following criteria:
1.
The inferential variable must have a good relationship to the true controlled variable
for changes in the potential manipulated variable.
2.
The relationship in criterion 1 must be insensitive to changes in operating conditions
(i.e., unmeasured disturbances) over their expected ranges.
3.
Dynamics must be favorable for use in feedback control.
Correction of inferential variable
1.
By primary controller in automated cascade design
2.
By plant operator manually, based on periodic information
3.
When inferential variable is corrected frequently, the sensor for the inferential
variable must provide good reproducibility, not necessarily accuracy
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