Exam II Flashcards
What does SISO stand for and what type of classification is it?
single input, single output; control
What does MIMO stand for and what type of classification is it?
multiple input, multiple output; control
What type of controller has remedial control?
feedback
What type of controller has predictive control?
Feedforward
Because it requires some time, control is considered what kind of process?
dynamic
What kind of control logic helps control non measurable outputs?
Inferential
What is an example of inferential control?
you can measure the variable B to control variable A knowing that A=f(B)
An open looped system is not being what?
controlled
What hardware is used for a FB control loop?
Process
Sensor or measuring device
Transmission line (pneumatic or electrical)
Controller
Final control element (or ACTUATOR, ie, valve)
What piece of hardware is neglected in dynamics?
Transmission line
In what general way do FB controllers differ?
they differ in the way they correlate epsilon(t) (error) with c(t) (actuating signal)
What are the three control laws?
Proportional action
Proportional-Integral action
Proportional-Integral-Derivative action
What is the control action equation for Proportional control?
c(t)=Kc*epsilon(t)+cs
constitutive equation
For proportional control, what does Kc equal?
Proportional gain
For proportional control, what does cs equal?
Bias signal
For proportional control, if epsilon(t)=0, what does cs represent?
actuating signal
What is sometimes used in place of the proportional gain and what is the equation for it?
proportional band
PB=100/Kc
For proportional control, the higher the Kc the higher this is to the error signal.
sensitivity of the controller
For proportional control, what is the constitutive equation in the deviation form?
C(s) = Kc * E(s)
Y G F
The P controller can reduce this but it cannot cancel it unless Kc is very very high
the error
What is the constitutive equation for PI control?
c(t) = Kc * epsilon(t) + (Kc/tau I) * integral of (epsilon(t) dt) + cs
In integral control, c(t) is proportional to this of the error.
time integral
What memory does the integral controller have?
past values of error
For PI what does tau I represent?
integral time constant
What is an adjustable parameter that falls between 0.1 and 50 minutes?
tau I
PI can eliminate even small what?
errors
The PI action lasts as long as there is a what?
non-zero error (can make the action very slow)
Because the integral action can make the action very slow it is never what?
used alone
What is the transfer function of a P controller?
Gc(s) = Kc
What is the transfer function of a PI controller?
Gc(s) = Kc {1 + [1 / (tauI * s)]}
The transfer function of a PI controller is not what, and is a function of what?
constant; s
What is the constitutive equation for a PID controller?
c(t) = Kc * epsilon(t) + (Kc/tau I) * integral of (epsilon(t) dt) + (Kc * tau D * (depsilon/dt)) + cs
In the derivative control, c(t) is proportional to what?
the derivative of the error
For derivative action, as the error increases what happens?
The D action compensates it without waiting for it to become to high
Derivative control is what kind of control?
anticipatory
What does tau D represent?
derivative time constant
In derivative control, if the error changes suddenly, what could happen?
the actuating signal can be very strong and the system would become too sensitive (or it would yield a large control action, although it is not needed).
What is the transfer function for a PID controller?
Gc(s) = Kc ( 1 + (1 /( tau I * s)) + tau D * s
What do you multiply the transfer function of a PID controller by to get it into standard form?
s
What does the final control action do? (Actuators)
receives c(t) and adjusts the manipulated variable
The final control element is usually a what?
a valve
What can valves control?
flow rate, temperature, level, composition, …..
What is the equation of a linear actuator?
f(x) = x
What is the equation of a square root actuator?
f(x) = sqrt (x)
What is the equation of an equal percentage actuator?
f(x) = alpha ^ (x-1)
What is the equation of a hyperbolic actuator?
f(x) = 1 / (alpha - (alpha - 1)x)
What is the equation for volumetric flow?
Q = Cv * f(x) * sqrt( delta p / rho) where Cv is the flux constant delta p is the pressure density rho is the density x is the degree of opening (position of the stem) 0 <= x <= 1 f(x) = the valve characteristic curve 0 <= f(x) <= 1
What kind of valve is commonly used and controlled by compressed air?
pneumatic
If air supply is lost, a pneumatic valve will do either this or that?
remain open (fail open) or will close (fail closed)
A valve being fail open or fail closed depends on what?
safety requirements
What is the force balance?
F = m (d^2 * x / dt^2)
mechanical action on the valve - elastic force - viscous resistance is equal to what?
mass times acceleration
mechanical action is defined how?
pressure * area
elastic force is defined how?
k * x
viscous force is defined how?
viscosity coefficient * (dx/dt)
What is the transfer function of a valve?
(A/k) / ( (m/k) s^2 + (viscosity coefficient / k) s + 1)
Valves behave as a what order system?
second order
If the mass of the stem is small the mass term can be eliminated making the system what order?
first order
G(s) = (A/k) / ((viscosity coefficient/k) s + 1)
What can be pneumatic or electrical?
Transmission lines
Unless the transmission line is very long, the dynamic behavior of a pneumatic transmission line can be what?
neglected
Electrical signals are very what?
fast
Globally, a FB- controlled process has how many inputs and how many outputs?
2; 1
set point + distrubance); (y
For the dynamic behavior of FB controlled processes, what has inputs and outputs?
each component (process, sensor, controller, actuator)
What does each component have?
its own transfer function
What is the transfer function of the process?
Y(s) = Gp(s) * M(s) + Gd(s)*D(s)
What is the transfer function of the sensor?
Ym(S) = Gm(s) * Y(s)
What is the transfer function of the controller?
C(s) = Gc(s) * E(s)
What is the transfer function of the actuator?
M(s) = Gf(s) * C(s)
What is the final value of Y(s) for a closed loop?
Y(s) = ((Gp(s) * Gf(s) * Gc(s))/(Gp(s) * Gf(s) * Gc(s) * Gm(s) + 1)) * Ysp(s)
+
(Gd(s) / (Gp(s) * Gf(s) * Gc(s) * Gm(s) + 1)) * D(s)
The closed-loop function gives the closed-loop what?
response of the process
In closed loop transfer functions, what is the same?
the denominators
What is the denominator?
the product of the transfer functions in the loop + 1 (everything but Gd(s))
In Servo control problems what is changed and what is omitted?
set point is changed but there is no distrubance
In Regulator control problems what does not change?
the set point but a disturbance acts on the system
Servo problems drive the process along a what?
trajectory
Regulator problems suppress what?
the effect of disturbances
What equation gives the poles for a FB controlled process?
Gp(s) * Gf(s) * Gc(s) * Gm(s) + 1 = 0
How is the dynamic behavior described if the poles are real and negative?
over-damped
How is the dynamic behavior described if the poles are complex?
oscillatory
How is the dynamic behavior described if the poles have a positive real part?
control makes the process unstable
To determine the effect of a controller what two terms are typically set equal to 1?
Gm and Gf
For P control, what does Gc equal?
Kc
For P control what does Y(s) equal?
Y(s) = ((Gp(s) * Kc) / (Gp(s) * Kc + 1)) Ysp(s) + (Gd(s) / (Gp(s) * Kc + 1)) D(s)
What does Gp equal for first order systems?
Gp = Kp / (taup * s + 1)
What does Gd equal for first order systems?
Gd = Kd / (taup * s + 1)
Does a P controller change the order of a system?
no
For a first order system under P control the time constant is (reduced/increased)?
reduced
For a first order system under P control the response is (faster/slower)?
faster
For a first order system under P control the static gains have been (decreased/increased)?
decreased
What does a P controller add?
offset
What does offset equal?
setpoint - actual final response
for P control if you take the limit has time approaches infinity of y(t) what does it equal?
Kd’ which is not zero (here is the offset)
What is the closed loop response of a P-FB controlled second order system?
Y(s) = (Kp’ / (tau’^2 * s^2 + 2 * zeta’ * tau’ * s + 1)) Ysp (s)
What does tau ‘ equal for a second order P-FB controlled system?
tau / sqrt(Kc * Kp + 1)
What does zeta’ equal for a second order P-FB controlled system?
zeta / sqrt (Kc * Kp + 1)
What does K’p equal for a second order P-FB controlled system?
(Kc * Kp) / (Kc * Kp + 1)
What does tau ‘ equal for a first order P-FB controlled system?
taup / (Kc * Kp + 1)
What does K’p equal for a first order P-FB controlled system?
(Kc * Kp) / (Kc * Kp + 1)
What does K’d equal for a first order P-FB controlled system?
Kd / (Kc * Kp +1)
the closed loop response of a 2nd order system under P control remains what?
second order
The static gain (increases/decrease) for a 2nd order system under P control?
decreases
The natural period of oscillation (tau) (increases/decreases for a 2nd order system under P control?
decreases
The damping factor (increases/decrease) for a 2nd order system under P control?
decreases (which can be undesirable)
We can find the final response using what?
final value theorem
What is the final value theorem?
y (t = infinity) = limit as s approaches 0 of sY = K’p = (Kc * Kp) / (KcKp + 1)
for a unit step change what is the offset?
(new set point which is the unit step change) - (actual response of the response) = 1 - ((Kc * Kp) / (Kc * Kp + 1)) = 1 / (Kc * Kp + 1)
What does the offset go to if Kc goes to infinity?
zero
zeta ‘ is (>) zeta?
less than
an overdamped process can become what when it is P controlled?
underdamped
generally tau’ and zeta’ do what with increasing Kc under P control?
decrease
If the proportional gain increases what happens to the system?
the system becomes faster, the offset decreases, but it may start to oscillate
The effect of I control on the system does what to the order of the system?
increases it by one
For a servo problem with I control where the system is first order, what does Y(s) equal?
Y(s) = (1 / (tau^2 * s^2 + 2 * zeta * tau * s + 1)) Ysp(s)
For a servo problem with I control where the system is first order, what does tau equal?
tau = sqrt((taup * tauI) / (Kp * Kc))
For a servo problem with I control where the system is first order, what does zeta equal?
zeta = (1/2) * sqrt (tauI / (taup * Kp * Kc))
The closed loop response of a 1st order system under I control (increases/decreases) its order?
increases
1st order system under I control increases the order which makes the response more what?
sluggish
a 1st order system under I control eliminates what?
offset
We can find the the final response of a 1st order system under I control using the final value theorem. For a unit step change, what does y(t=infinity) =?
= limit as s approaches zero of s * Y = 1
A rule of thumb is that if the gain of a controlled process is 1 there is no what?
offset
The form of a closed loop response for I controlled systems depends on what three variables?
zeta, tauI, and Kc
As tauI decreases for an I controlled system the response becomes (faster/slower) and the behavior can be (nonoscillating/oscillating)
faster; oscillating
Increasing the integral action (increasing Kc and decreasing tauI) makes the closed-loop response more what?
sensitive
As the integral action increases the overshoot of the closed loop response does what?
increases
What is one of the most common controllers?
PI
What kind of problem do we used to test the effect a control will have?
servo
The effect of PI control (increases/decreases) the order of the system?
increases
For a first order PI controlled system what happens to the offset?
eliminated
For a first order PI controlled system the integral action does what to the closed-loop response?
slows down due to increasing the order
For a first order PI controlled system as Kc increases and tauI decreases, the response becomes what?
faster but more oscillatory?
Why do we use a PI controller?
To eliminate the offset
Does D control stabilize the system?
yes
Does the order stay the same when affected by D control?
yes
The time constant increases with the addition of D control which causes what?
a slower closed-loop response
The D action does not have any effect on what?
the set point
If there is a step change in the set point of a D controlled system, the system reacts at t = 0 and then does what?
returns to the original steady state
With D control, when a regulator problem is considered, what does y(t) do?
goes to the same final value of the system if there was no control but slowly
What effect does D control have on the oscillations of a second order system?
stabilizes against oscillations (reduces oscillations)
what does D control do to the damping factor?
increases it
What effects does a PID control have on the response of a controlled system?
order increases (integral action) instability due to increasing Kc is stabilized offset is eliminated (due to integral action)
If a process parameter can vary into a range, which controller can you use?
P-FB can be used alone
If the process must assume a well defined value which controller can you use?
PI or PID FB (offset is eliminated)
If the system is complex which controller should you use? (higher order)
PID stabilizes oscillations (does not increase the order relative to the PI-controlled system)
What controller would be used for liquid level control at the bottom of a distillation column?
P control alone
What controller would be used to control flow rate that must take on a prescribed value and the control action should take place in a reasonable time (immediate action not needed)
PI
What controller would be used if it needs to act quickly?
PID
What units typically have a PID?
oven, reactors
temperature and composition control
What are the three tunable controller parameters?
Kc, tau I, and tau D
What is the Lyapunov stability criterion?
A linear system is defined stable if, for any given bounded input, it produces a bounded output
Bounded input/output remain confined between what?
an upper and lower limit
Poles that exhibit positive real parts give rise to what terms which produce what?
ci * e^(pi*t); unstable behaviors
A FB controlled system is stable if all the roots of Gp(s) * Gf(s) * Gc(s) * Gm(s) + 1 = 0 have what?
a negative real part
If any of the coefficients of the denominator set equal to zero are negative there is at least one what?
root with a positive real part
If all of the coefficients of the denominator set equal to zero are positive we can use what to determine stability?
the Routh array
How do you tune the controller parameters?
using either local/simple or integral performance criteria
What are simple/local performance criteria?
criteria that use only a few points of the response
What are integral performance criteria?
criteria that use the entire closed-loop response
What do we minimize to design a robust control system?
overshoot
rise time
decay ratio
Warning! the satisfaction of a single condition does not guarantee what?
that the closed loop response is the desired one
We can pick parameters that minimize which errors?
Integral of the square error (ISE)
Integral of the absolute value of the error (IAE)
Integral of the time-weighted absolute error (ITAE)
Which integral performance criteria should you use to suppress large errors?
ISE because the errors are squared and thus contribute more to the value of the integral
Which integral performance criteria should you use to suppress small errors?
IAE because the square of small numbers is an even smaller quantity
Which integral performance criteria should you use to suppress errors that persist over time?
ITAE because the presence of t in the integral amplifies the effect of even small errors
Who devised a semi-empirical method which relies on both simple/local performance and integral performance criteria?
Cohen and Coon
When a step change of amplitude A in the actuating signal c(t) takes place, the system’s response, ym (also called the reaction curve) is sigmoidal and can be approximated by the response of what?
A first order system with dead time
In the Cohen-Coon method, what three parameters do you work with?
K, td, tau
In the Cohen-Coon method, what does K equal?
B/A where A is the amplitude of the step
In the Cohen-Coon method, what does tau equal?
B/sigma where sigma is the slope of the tangent line of the sigmoidal curve section best fit by a line
What best values of the controller parameters did Cohen and Coon use?
One quarter decay ratio
minimum offset
minimum ISE
What parameters are present for P control using the Cohen Coon method?
Kc = (tau/(Ktd))(1 + (td/(3tau)))
What parameters are present for PI control using the Cohen Coon method?
Kc =(tau/(Ktd))(0.9 + (td/(12tau)))
tau I = td ((30 + 3(td/tau))/(9 + 20(td/tau))
What parameters are present for PID control using the Cohen Coon method?
Kc = (tau/(Ktd))((4/3) + (td/(4tau)))
tau I = td ((32 + 6(td/tau))/(13 + 8(td/tau))
tau D = td (4/ (11 + 2*(td/tau)))
What does cascade control do?
takes multiple measurements before acting on the manipulated variable
example (temperature control for a CSTR)
Cascade control is only good if the disturbance effects the measured control how?
directly
When K = B/A what is B? what is A?
B = the asymptotic response of the system found by plotting y(t) A = the amplitude of the disturbance
Using experimental data, what should you treat the Cohen-Coon model as?
a first order system with dead time
Fitting procedures are only what?
approximated
Using a few points is more rigorous and precise than what?
fitting
When you have to valves in parallel what are they called?
1 is called master and the other the slave
Be aware that welding points are what?
critical points
