Module 3: Feedback Control

1
Module 3 Feedback Control

• The Feedback Loop

2
Recall Module 1 - pieces of a feedback loop

• sensor
• control calculation - algorithm
• actuator - final control element
• setpoint - operating target

3
Heated Duct - Process Schematic

T
air
heater
4-20 mA
fan
TC
4
The Heated Duct Experiment - Components

• sensor RTD probe to measure temperature
• final control element - electric heater
• controller- PID controller running on PC.
• setpoint desired air temperature in the duct.

5
If we focus on sensor/final element, we have...

• RTD (resistance thermometer device) converts
  temperature to a voltage. Voltage is converted
  to an electric current signal, which is sent to
  the controller.
• Controller compares current signal to the
  setpoint, performs PID control calculation and
  sends a current signal to the power source.
  Power source provides current to the heater.
• standard language for communication between
  elements is typically 4-20 mA electrical signal

6
If we have a pneumatic valve...

• need an I/P conversion
• convert from electrical signal to pneumatic
• pneumatic signal
• typical range -gt 3-15 psig
• we can have different valve configurations
• e.g., air-to-open, air-to-close
• based on fail-safe considerations

7
Block Diagram Representation
Disturbance

• note - typically include signal conversions with
  final control element and sensors

Final Element
Process
Controller
SP(s)
Y(s)

-
Sensor
8
Block Diagram Representation
D(s)
Gd

Final Element
Process
Controller

SP(s)
Gv
Gp
CV(s)
Gc


F(s)
U(s)
-
Gs
M(s)
Sensor
9
Where are the significant dynamics?

• process
• final element
• sensor
• signal conversion
• usually we can neglect time lags associated with
  signal conversion and assume conversion is a pure
  gain process

10
Insignificant Dynamics

• transmission
• sometimes signal conversion
• When dealing with gains, watch your units!

11
Two Types of Control Problems

• Servo - move the process to new operating points
  - setpoint changes
• Load or Disturbance Rejection - eliminate the
  effects of disturbances on the controlled variable

12
Servo Transfer Function

• From block diagram algebra

13

Disturbance Rejection Transfer Function

From block diagram algebra
14
What are the ideal servo and load transfer
functions?

• Servo -gt 1
• Load -gt 0
• we can never achieve these in reality
• Deadtime
• Inertia of process

15
Servo Examples
IAE 322.7769ISE 1714.5888ITAE
36796.563
IAE 84.0372ISE 513.414ITAE 9580.2361
10
1
15
1

8
0.5
0.5
10
6
CV
0
D
CV
0
D
4
5
-0.5
-0.5
2
0
-1
0
-1
0
50
100
150
0
50
100
150
0
50
100
150
0
50
100
150
time
time

30
30
20
20
MV
MV
10
10
0
0
0
50
100
150
0
50
100
150
time
time
16
Assessing Servo Controller Performance - Criteria

• number of different measures for comparing
  response of controlled variable
• offset - sustained deviation from target
• rise time - time to reach new target
• decay ratio
• period of oscillation
• settling time
• manipulated variable overshoot

17
Assessing Controller Performance

• integral measures - applicable to both servo and
  load problems
• represent accumulated error over time frame of
  interest
• integral absolute error (IAE)
• integral squared error (ISE)
• integral time absolute error (ITAE)
• When would it be appropriate to use each
  particular measure?

18
Types of Disturbances

• step - shift in operation
• e.g., change to a new feed tank
• sinusoidal
• e.g., due to other poorly tuned controllers or to
  rotating equipment
• stochastic - random fluctuations
• uncorrelated (white noise)
• correlated (e.g., noise that drifts with time)

19
Specific Measures of Disturbance Rejection

• maximum deviation in controlled variable (from
  target)
• ISE - related to variance of controlled variable
  about target
• other integral measures (IAE, ITAE)

20
A First Pass at Design - Choosing CV/MV Pairs

• CV
• related to process objectives
• e.g., quality --gt composition
• reliable measurements must be available
• MV
• External variable, can be adjusted independently
• causal relationship with CV
• sufficient range
• can tolerate increased variability

21
Fundamental Action of Process Control

• transfer variability from the controlled variable
  to the manipulated variable
• variability is not completely eliminated
• rather, it is moved to a part of the process that
  can tolerate variability

22
Desired Process Relationships in CV/MV Pairs

• causal
• fast dynamics
• fast
• small dead time
• no inverse response
• gain
• consistent (i.e., always positive or negative)

23
Types of Control

• manual
• no automatic adjustment
• human operator provides feed back
• Wood stove example
• automatic controls - using a variety of
  algorithms
• on/off - will result in continuous cycling
  (think of your house heating, or your oven)
• PID and other control algorithms
• emergency control
• alarms, pressure venting
• sequence of shutdown steps

24
Proportional-Integral-Derivative (PID) Control

• most widely-used control algorithm
• robust - insensitive to errors
• widely applicable
• simple to calculate
• eliminates offset
• enhancements available

25
How would you design a control calculation?

• start with a correction proportional to the
  error
• what happens when we make a setpoint change?
• initial jump in manipulated variable
• do we reach our new target?

26
The Deficiency of Proportional Control

• making an adjustment to the MV that is
  proportional to the error is reasonable BUT, if
  we still have an error when we get to steady
  state, the controller wont take any further
  action.
• leads to sustained deviation - offset
• proportional controllers make adjustments based
  on what is happening in the present

27
Eliminating Offset - Integral Action

• keep a running total of our error
• cumulative error
• make adjustment based on this cumulative error -
  integral action
• focus on past

28
Integral Action

• as long as there is non-zero error, the integral
  term keeps accumulating
• MV adjustment based on this term will keep
  changing until there is no more sustained error.
• once we reach the new setpoint, integral stops
  changing
• integral action eliminates offset

29
Derivative Action

• try to anticipate where process is going
• look at rate of change of error
• if error is growing rapidly, we should take
  preventive action
• (think about driving a car or temperature in
  a reactor).
• derivative action
• Improves control by taking account of the future
• useful for sluggish processes with substantial
  inertia

30
Putting it all together - the PID algorithm

• in the time domain
• What is the bias for?
• Conceptually
• MV change f (present, past, future)

31
PID Terms and their Jobs

• Proportional
• provides initial adjustment - kick in the right
  direction
• can cause instability if tuned improperly
• Integral
• eliminates offset
• takes care of persistent errors
• can cause instability if tuned improperly
• Derivative
• accounts for process inertia
• useful when dead time is large
• does not affect or eliminate offset
• can cause undesirable high-frequency variations -
  amplification of measurement noise.

32
Modification to Derivative Term

• if we will be making many setpoint changes, use
  the following form for derivative action
• instead of
• Why?
• drop dependence on SP to avoid infinite rate of
  change associated with step changes in setpoint

33
PID Controller in the Laplace Domain

• working in deviation variables eliminates the
  bias term

34
Assessment of Closed-Loop PID Behaviour

• use the servo or load closed-loop transfer
  function
• examine poles, zeros
• example - first-order mixing tank with Kp1
• proportional-only
• PI
• PID

35
Tank with Proportional Only

• Start with process transfer function
• Controller transfer function with only
  Proportional control --
• Servo transfer function

36
Tank with PI and PID Control

• Proportional-integral and
• Proportional-integral-derivative
• and

37
Points from the Tank Example

• integral action increases the order of the
  closed-loop process by 1
• from 1 to 2 - potential for underdamped behaviour
• proportional gain cant make 1st order process
  unstable (if there is no dead time)
• excessive integral action can produce sustained
  oscillations, but not unstable behaviour (without
  deadtime)
• marginally stable for very small TI
• We can find the poles of Gservo and determine the
  stability of the control system for specific
  values of Kc, TI and Td

38
Tuning PID Controllers

• many approaches -
• correlations
• based on desired shape of setpoint response
• e.g., 1/4 decay ratio (Ziegler-Nichols)
• direct optimization
• specify performance objectives
• find optimal tuning parameters by doing lots of
  optimization
• e.g., Ciancone and Marlin graphs

39
Correlations

• typically based on FOPDT characterization from
  process reaction curve
• 1/4 decay - can be too underdamped for some
  applications, and may exhibit excessive MV action
• can give insight into dependence of tuning on
  process characteristics
• e.g., impact of increasing deadtime

40
Direct Optimization

• focus on -
• controlled variable performance
• use integral measure (e.g., IAE, ISE, ITAE)
• robustness
• examine performance over a range of models
• manipulated variable behaviour
• avoid excessive manipulated variable action
• enforced by placing limits on allowable variation
  and penalizing violations of limits

41
Optimization Criterion

• minimize subject to MV constraints
• minimize the CV deviations over a range of
  possible models while enforcing limits on the
  manipulated variable action

42
Generalizing the Results - Graphical Correlations

• for FOPDT process models
• e.g., from process reaction curves
• introduce fractional deadtime
• obtain values for dimensionless controller gain,
  integral time, and derivative time

43
Example 9.2 - Three Tank Process

• estimate FOPDT from third-order response
• time delay 5.5 min, gain0.039 A/valve, time
  constant10.5 min
• tune for disturbance, setpoint changes
• Note - optimal tuning constants will depend on
  whether disturbance rejection or setpoint changes
  are the primary goal

44
Fine-Tuning PID Performance

• examine CV, MV plots
• use setpoint change
• we can impose this
• we can see P, I effects
• proportional kick - initial change
• 40 - 150 of final MV change
• watch for excessive oscillation, long rise time
  (or settling time)
• If control is too aggressive oscillatory,
  usually Kc is too large or TI is too short.

45
Stability

• bounded-input bounded-output stability
• when we enter a bounded input signal, will the
  process response always be bounded, or can it
  grow to infinity?
• assess by examining behaviour of linear model
• use linearization if necessary - local results
  may not apply over the whole operating region of
  the plant.
• examine poles of transfer function

46
Closed-Loop Stability

• examine closed-loop transfer fn. poles
• common eqn. between servo, load cases
• characteristic equation
• denominator of servo, load transfer fns.
• roots are closed-loop poles

47
What happens if we have deadtime?

• analysis of poles valid only for systems without
  time delays
• transfer functions are polynomials
• solution - use frequency response methods
• Bode Stability Criterion
• the closed-loop experiment...

48
The Closed-Loop Experiment

• break the feedback loop
• adjust setpoint sinusoid frequency until phase
  angle is -180 o
• close loop and let the loop ring

SP
CV

Gc
Gp
-
49
Bode Stability Criterion

• lagged sinusoid becomes in phase when multiplied
  by -1 at setpoint junction
• process is stable if the AR (for controller and
  process together) at the critical frequency is
  less than 1.
• What would happen if AR gt1?
• critical frequency - point at which phase angle
  is -180 o. This is the worst-case bounded input
  that the system could encounter.
• apply this test to GOL - product of elements in
  feedback loop

50
Example - First-Order Plus Deadtime Process

• three-tank example
• gain 0.039
• time constant 10.5
• deadtime 5.5
• time delay causes significant phase lag
• confirm stability with PID constants
• delay-free 1st-order process never has a phase
  lag of 180

51
Failure Mode of a Valve (Section 12.2)

• based on safety considerations
• safe position if pneumatic air/power is lost
• air-to-open (fail closed)
• increasing signal increases valve opening
• air-to-close (fail open)
• increasing signal decreases valve opening

52
Sensors (Section 12.2)

• range - operating values over which the sensor
  provides a (reasonable) measurement
• must include intended range of process operation
• in some instances, multiple sensors with
  different ranges may be required
• start-up
• normal operation

53
Characteristics of Sensors (Section 12.2)

• accuracy
• degree of conformity to true value under specific
  conditions
• can think of as maximum bias in sensor reading
• reproducibility
• consistency of sensor for repeated measurements
  at the same conditions

54
Sensors - Statistical Interpretation

• accuracy
• errors in mean reading
• reproducibility
• variance in readings
• possible to have good reproducibility with poor
  accuracy
• consistent measurements, but biased