Compre Flashcards
Advantages of automatic control of a process
- Enhanced process safety
- Satisfying environmental constraints
- Meeting ever-stricter product quality specifications
- More efficient use of raw materials and energy
- Increased profitability
Control systems
used to maintain process conditions at their desired values by manipulating certain process variables to adjust the variables of interest.
common attributes of control systems
- The ablity to maintain the process variable at its desired value in spite of disturbances that might be experienced (this is termed disturbance rejection)
- The ability to move the process variable from one setting to a new desired setting (this is termed set point tracking)
The concept of using information about the deviation of the system from its desired state to control the system
feedback control
type of control system where the controller automatically acts to return the controlled variable to its desired value
closed-loop feedback control system
type of control system where the measurement signal is disconnected from the controller, and the controller output has to be manually adjusted to change the value of the controlled variable
open-loop feedback control system
manual mode system
open-loop feedback control system
automatic mode system
closed-loop feedback control system
most common type of signal feedback
Negative feedback
the error signal is computed from the difference between the set point and the measured signal
Negative feedback
The negative value of the measured signal is “fed back” to the controller and added to the set point to compute the error.
type of control where the controller should change the heat input by an amount proportional to the error
proportional control
integral control: controller response
The controller is instructed to change the heat input by an additional amount proportional to the time integral of the error.
two adjustable parameters of integral control
a multiplier for the error and a multiplier for the integral of the error
disadvantage of integral control
the system has a tendency to be more oscillatory
apparent error
the controller receives measured values of the temperature, rather than the actual values
type of reponse where the control system has actually caused a deterioration in performance due to the increase in the controller gain (the proportionality constants), which makes the tank temperature oscillate with increasing amplitude until the physical limitations of the heating system are reached.
unstable response
Diagram that indicates the flow of information around the control system and the function of each part of the system.
Block diagram
The process variable that we want to maintain at a particular value.
Controlled variable
A device that outputs a signal to the process based on the magnitude of the error signal.
Controller
One goal of a control system, which is to enable the system to “reject” the effect of disturbance changes and maintain the controlled variable at the set point.
Disturbance rejection
Any process variables that can cause the controlled variable to change.
Disturbances
variables that we have no control over
Disturbances
Process variable that is adjusted to bring the controlled variable back to the set point.
Manipulated variable
the error is the difference between the set point and the measured variable
Negative feedback
The steady-state value of the error
Offset
the measured value of the controlled variable is not fed back to the controller
Open loop
the measured temperature is added to the set point
Positive feedback
The desired value of the controlled variable.
Set point
One goal of a control system, which is to force the system to follow or “track” requested set point changes.
Set point tracking
The Laplace transform of a function f (t)
F(s) = L{f(t)}
the time when the process is disturbed from steady state
t = 0
Mercury thermometer assumptions
- All the resistance to heat transfer resides in the film surrounding the bulb (i.e., the resistance offered by the glass and mercury is neglected).
- All the thermal capacity is in the mercury. Furthermore, at any instant the mercury assumes a uniform temperature throughout.
- The glass wall containing the mercury does not expand or contract during the transient response.
Mercury thermometer energy balance
hA(x-y) = mC(dy/dt)
the rate of flow of heat through the film resistance surrounding the bulb causes the internal energy of the mercury to increase at the same rate
The increase in internal energy of the mercury is manifested by what
The increase in internal energy is manifested by a change in temperature and a corresponding expansion of mercury, which causes the mercury column, or “reading” of the thermometer, to rise.
What are deviation variables
the differences between the variables and their steady-state values
X = x - xs
Y = y - ys
time constant symbol and units
tau; units of time
time constant of a mercury thermometer
tau = mC/hA
m - mass of mercury , kg
C - heat capacity , J/(kg K)
h - film coefficient, W/(K m^2)
A - surface area, m^2
(thermometer) transfer function definition
It is the ratio of the Laplace transform of the deviation in thermometer reading (output) to the Laplace transform of the deviation in the surrounding temperature (input).
Any physical system for which the relation between Laplace transforms of input and output deviation variables is of the form given by the transfer function 1/(Ts+1)
first-order system
Synonyms for first-order systems
first-order lag
single exponential stage
The naming of all these terms is motivated by the fact that the transfer function results from a first-order, linear differential equation,
X - Y = T(dY/dt)
standard first-order transfer function
Y(s)/X(s) = Kp / (Ts+1)
The important characteristics of the standard form of the transfer function
- The denominator must be of the form Ts+1.
- The coefficient of the s term in the denominator is the system time constant T.
- The numerator is the steady-state gain Kp .
steady-state value that the system attains after being disturbed by a unit-step input
steady-state gain Kp
PROPERTIES OF TRANSFER FUNCTIONS
In general, a transfer function relates two variables in a physical process; one of these is the cause (forcing function or input variable), and the other is the effect (response or output variable). In terms of the example of the mercury thermometer, the surrounding temperature is the cause or input, whereas the thermometer reading is the effect or output.
The transfer function completely describes the dynamic characteristics of the system.
input -> G(s) -> output
The transfer function results from a linear differential equation; therefore, the principle of superposition is applicable.
Forcing function
input, X(s)
Response
output, Y(s)
common forcing functions
step, impulse, ramp, and sinusoidal functions
This function increases linearly with time
RAMP FUNCTION
radian frequency w relation to the frequency f
The radian frequency w is related to the frequency f in cycles per unit time by
w = 2(pi)f
step response features
- The value of Y ( t) reaches 63.2 percent of its ultimate value when the time elapsed is equal to one time constant t. When the time elapsed is 2 t, 3 t, and 4 t, the percent response is 86.5, 95, and 98, respectively. From these facts, one can consider the response essentially completed in three to four time constants.
- The slope of the response curve at the origin is 1. This means that if the initial rate of change of Y(t) were maintained, the response would be complete in one time constant.
- A consequence of the principle of superposition is that the response to a step input of any magnitude A may be obtained directly from Fig. 4–7 by multiplying the ordinate by A. Figure 4–7 actually gives the response to a unit-step function input, from which all other step responses are derived by superposition.
A resistance that has a linear relationship between flow and head, q = h/R
linear resistance
When is a pipe a linear resistance
A pipe is a linear resistance if the flow is in the laminar range.
holding tank mass balance
(rho) q(t) - (rho) q0(t) = (rho) A dh/dt
what is the term R in holding tanks
conversion factor that relates h(t) to q(t) when the system is at steady state
dimensions of the steady-state gain for the transfer function
Q0(s)/Q(s) = 1/(Ts+1)
dimensionless
the input variable q (t) and the output variable qo (t) have the same units (volume/time)
a pulse of unit area as the duration of the pulse approaches zero
unit-impulse function
a system that grows without limit for a sustained change in input is said to have what
nonregulation
systems that have a limited change in output for a sustained change in input are said to have what
regulation
Self-regulating process example
An example of a system having regulation is the step response of a first-order system. If the inlet flow to the process is increased, the level will rise until the outlet flow becomes equal to the inlet flow, and then the level stops changing.
a transient mass balance around a holding tank
Rate of mass flow in - Rate of mass flow out = Rate of accumulation of mass in tank
transient mass balance for the salt in a mixing tank
Flow rate of salt in - Flow rate of salt out = Rate of accumulation of salt in tank
mixing tank mass balance in terms of symbols
qx - qy = V dy/dt
holding tank time constant
tau = AR
mixing tank time constant
tau = V/q
A transient energy balance on a heating tank
Rate of energy flow into tank - Rate of energy flow out of tank + Rate of energy flow in from heater = Rate of accumulation of energy in tank
heating tank energy balance in symbols
mC(Ti - Tref) - mC(T - Tref) + q = (rho)VC dT/dt
heating tank time constant
tau = (rho)V/m = V/v
how does a noninteracting system function
the outlet flow from tank 1 discharges directly into the atmosphere before spilling into tank 2, and the flow through R1 depends only on h1. The variation in h2 in tank 2 does not affect the transient response occurring in tank 1
interacting system relations
the flow through R1 depends on the difference between h1 and h2
liquid level assumptions
we shall assume the liquid to be of constant density, the tanks to have uniform cross-sectional area, and the flow resistances to be linear
delay that is observed when two or more first-order systems are connected in series
transfer lag
when is there no transfer lag
For a single first-order system, there is no transfer lag; i.e., the response begins immediately after the step change is applied, and the rate of change of the response (slope of response curve) is maximal at t = 0.
Generalization for Several Noninteracting Systems in Series
The overall transfer function for two or more noninteracting first-order systems connected in series is simply the product of the individual transfer functions.
The term interacting is often referred to as what
loading
The second tank is said to load the first tank.
effect of interaction on the response
interaction slows up the response
If the same size step change is introduced into a noninteracting and an interacting system, the flow from tank 1 (q1) for the noninteracting case will not be reduced by the increase in level in tank 2. However, for the interacting case, the flow q1 will be reduced by the buildup of level in tank 2. At any time t1 following the introduction of the step input, q1 for the interacting case will be less than for the noninteracting case with the result that h2 (or q2) will increase at a slower rate.
effect of interaction on a system containing two first-order lags
In general, the effect of interaction on a system containing two first-order lags is to change the ratio of effective time constants in the interacting system. In terms of the transient response, this means that the interacting system is more sluggish than the noninteracting system.
For two systems in series, if the output from system 1 is not affected by the output from system 2, the systems are said to be what
noninteracting
The output from system 1 is affected by the output from system 2. The overall transfer function for the process is not merely the product of the transfer functions in series.
Interacting systems
There is the presence of the cross-product term in the denominator.
type of control where the controller changes the heat input to the tank by an amount that is proportional to error
proportional control
source of heat input q
electricity or steam
In either case, the output signal from the controller should adjust q in such a way as to maintain control of the temperature in the tank.
If an electrical source for the heat input were used, what is the final control element?
The final control element might be a variable transformer that is used to adjust current to a resistance heating element.
If the heat input source were steam, what is the final control element?
The final control element would be a control valve that adjusts the flow of steam.
Components of a control system
- Process (stirred-tank heater).
- Measuring element (thermometer).
- Controller.
- Final control element (variable transformer or control valve).
What is the set point
the desired value of the controlled variable
a change in any variable that may cause the controlled variable of the process to change
load
Examples of load variables
inlet temperature
flow rate
heat loss from the tank
the measured value of the controlled variable is returned or “fed back” to a device
closed-loop system or a feedback system
device where the controlled variable is compared with the desired value or set point
comparator
adjusts the final control element to return the controlled variable to the set point
controller
principle that involves the use of the controlled variable T to maintain itself at a desired value TR
feedback principle
ensures that the difference between TR and Tm is used to adjust the control element so that the tendency is to reduce the error
Negative feedback
system where the signal to the comparator is obtained by adding TR and Tm
positive feedback system
This action would cause T to increase further. It should be clear that this situation would cause T to “run away” and control would not be achieved.
problem where we assume that there is no change in load Ti and that we are interested in changing the bath temperature according to some prescribed function of time
servomechanism-type (or servo) problem
For this problem, the set point TR would be changed in accordance with the desired variation in bath temperature.
servomechanism-type (or servo) problem
well-known examples of the servo-type problem
tracking of missiles and aircraft and the automatic machining of intricate parts from a master pattern
The servo problem can be viewed as trying to follow a moving target (i.e., the changing set point).
In this problem, the desired value TR is to remain fixed, and the purpose of the control system is to maintain the controlled variable at TR in spite of changes in load Ti.
regulator problem
This problem is very common in the chemical industry.
regulator problem
servo problem
the response of a linear control system to a change in set point
regulator problem
response to a change in load
senses the bath temperature T and transmits a signal Tm to the controller
temperature measuring element
may exhibit some dynamic lag
lag is first-order
commonly used temperature sensing devices in industry
Thermocouples
They have time constants on the order of 6 to 20 seconds. The size of the time constant depends on the mass (size) of the thermocouple.
what is the bias value
the steady-state heat output from the controller/heater
output from the controller when the error is zero (i.e., steady state)
bias value
A device that outputs a signal to the process or final control element based on the magnitude of the error signal.
Controller
a controller that outputs a signal proportional to the error
proportional controller
The difference between the actual value of a variable and its steady-state value. Block diagrams are always constructed using this.
Deviation variable
A device that provides a modulated input to the process in response to a signal from the controller.
Final control element
The change in any process variable that can cause the controlled variable to change.
Load
A sensor used to determine the value of the controlled variable and to send it to the comparator/controller.
Measuring element
Measuring element examples
a thermocouple (temperature), a strain gauge (pressure), a gas chromatograph (composition), and a pH electrode (acidity)
These sensors typically have some dynamic behavior associated with them and can affect the design of the control system.
Measuring element
The goal of a control system for this type of problem is to enable the system to compensate for load changes and maintain the controlled variable at the set point.
Regulator problem
The goal of a control system for this type of problem is to force the system to “track” the requested set point changes.
Servo problem
components of a control hardware
Transducer (temperature-to-current)
Computer/ Controller (current-to-current)
Converter (current-to-pressure)
Control valve (pressure-to-flow rate)
The external power needed for each component in a control system
120 V
valve where the plug moves downward and restricts the flow of fluid through the valve as the air pressure increases
air-to-close valve
this valve opens and allows greater flow as the valve-top air pressure increases
air-to-open valve
steady-state gain
Kc, constant of proportionality
The simplest type of controller
proportional controller
adjustable parameter of a proportional controller
controller gain, proportional gain, or sensitivity
actual behavior of a proportional controller
The controller output will saturate (level out) at pmax = 15 psig or 20 mA at the upper end and at pmin = 3 psig or 4 mA at the lower end of the output. The ideal transfer function does not predict this saturation phenomenon.
A special case of proportional control
on/off control
If the gain Kc is made very high, the valve will move from one extreme position to the other if the process deviates only slightly from the set point.
on/off action
the valve is either fully open (on) or fully closed (off); i.e., the valve acts as a switch
phenomenon where the controller will rapidly cycle on and off as the error fluctuates about zero
chattering
two adjustable parameters of integral control
the gain and the integral time
reciprocal of the integral time
reset rate
It acts upon the derivative of the error, so it is most active when the error is changing rapidly.
Derivative control or PROPORTIONAL-DERIVATIVE (PD) CONTROL
It serves to reduce process oscillations.
Other terms that are used to describe the derivative action
rate control and anticipatory control
Derivative action basis
Derivative action is based on how rapidly the error is changing, not the magnitude of the error or how long the error has persisted. It is based on the slope of the error versus time curve at any instant in time. Therefore, a rapidly changing error signal will induce a large derivative response.
difference between this new steady-state value and the original value
offset
disadvantage of PI controller
oscillatory behavior
advantage of PI controller
no offset
If excessive oscillations had to be eliminated, what may be added
derivative action might be added
overall transfer functions
apply to the entire system
The series of blocks between the comparator and the controlled variable
forward path
The block between the controlled variable and the comparator
feedback path
product of all transfer functions
open-loop transfer function
It relates the measured variable B to the set point R if the feedback loop is disconnected (i.e., opened) from the comparator.
solution to the servo problem
The response to a change in set point R, obtained by setting U = 0
solution to the regulator problem
The response to a change in load variable U, obtained by setting R = 0
“brute-force” technique
another approach to finding the closed-loop transfer functions from the block diagram
involves “breaking the loop” and working your way across the block diagram
Process in which the feedback loop is connected to the comparator.
Closed-loop process
Transfer functions relating two variables in the process when the feedback loop is connected to the comparator.
Closed-loop transfer function
The path that connects the controlled variable and the comparator.
Feedback path
The transfer functions that lie between two signals in the block diagram moving left to right.
Forward path
Process in which the feedback loop is disconnected from the comparator.
Open-loop process
Product of all transfer functions in the loop relating B and R when the feedback loop is disconnected from the comparator.
Open-loop transfer function
What is process control
It is the study and application of automatic control in chemical engineering which combines knowledge of chemical processes and of dynamic systems.
Define process
It is a collection of equipment and materials, marked by a boundary in space, exchanging energy and materials.
Define system
It is a collection of equipment and operations within a boundary communicating by a set of input and output signals.
stimuli = responses
in a ramp function, the steady-state difference between the input and output after the transient response is
b*tau
in a ramp function, the output lags by
tau
after an initial transient period in a ramp function, the response is
parallel with input
after an initial transient period for a sinusoidal input, the response is
periodic with the same frequency as the input
in interacting systems, the denominator has what
There is the presence of the cross-product term in the denominator.
Inserted to an on/off controller
In practice, a dead band is inserted into the controller. With a dead band, the error reaches some finite positive value before the controller “turns on.” Conversely, the error must fall to some finite negative value before the controller “turns off.”
