Continuing the library of dynamic components

1
Look Ahead to Remainder of CHEE 319

• Continuing the library of dynamic components
• Empirical modeling
• Feedback control - elements
• PID control algorithm
• definition
• tuning
• Controller performance measures

2
Look Ahead continued...

• Stability - importance and assessment
• Controller Enhancements
• cascade control
• feedforward control

3
Building a Library of Process Dynamic Components

• Chapters 5 and 6, Marlin

4
Outline

• 1st order, 1st order plus dead time
• second order
• integrators
• estimating First Order Plus Deadtime (FOPDT)
  models

5
First Order Processes

• transfer function
• gain, time constant

6
First Order Process - Step Response

2
1.8
1.6
1.4
y
1.2
1
0.8
0.6
0.4
greatest slope at time 0
0.2
0
0
5
10
15
20
25
30
time
7
First-Order Process - Frequency Response
At low frequencies, amplitude ratio is
the process gain.

PROCESS (GP) BODE PLOT
0
10
-1
10
Amplitude Ratio
maximum phase lag of -90o or ?/2 radians
-2
10
-2
-1
0
1
10
10
10
10
Frequency, w (rad/time)
0
-50
Phase Angle (degrees)
-100
-2
-1
0
1
10
10
10
10
Frequency, w (rad/time)
Notice log scales for AR and ?
8
Time Delay y(t) u(t-?)

• transfer function
• one parameter length of time delay
• frequency response
• already in polar form
• amplitude ratio 1 for all frequencies
• phase angle is a steadily increasing lag
• - causes trouble for controllers

9
Time Delay - Frequency Response

PROCESS (GP) BODE PLOT
1
amplitude ratio of 1
10
Amplitude Ratio
0
continuously increasing phase lag - doesnt
approach any limit
10
-2
-1
0
1
10
10
10
10
Frequency, w (rad/time)
0
-100
Phase Angle (degrees)
-200
-2
-1
0
1
10
10
10
10
Frequency, w (rad/time)
10
First-Order Plus Deadtime Model

• product of first-order dead time models
• three parameters
• step response
• frequency response
• combination of plots for first-order process,
  dead time
• Why?

11
First-Order Plus Dead time Model - Step response
for step at time zero.
2

1.8
1.6
1.4
y
1.2
1
0.8
0.6
0.4
0.2
0
initial delay
0
5
10
15
20
25
30
35
time
12
Bode Plots for Products of Transfer Functions

• consider
• polar form

13
Bode Plots for Products

• Overall AR is product of individual ARs
• - add AR plots because they are on a log scale
• phase angle
• sum of the individual phase angles

14
1st Order Plus Dead time - Frequency Response

PROCESS (GP) BODE PLOT
PROCESS (GP) BODE PLOT
0
0
10
10
-1
-1
10
10
Amplitude Ratio
Amplitude Ratio
-2
-2
10
10
-2
-1
0
1
-2
-1
0
1
10
10
10
10
10
10
10
10
Frequency, w (rad/time)
Frequency, w (rad/time)
0
0
-100
-50
Phase Angle (degrees)
Phase Angle (degrees)
-200
-300
-100
-2
-1
0
1
-2
-1
0
1
10
10
10
10
10
10
10
10
Frequency, w (rad/time)
Frequency, w (rad/time)
first order
first order plus dead time
15
1st Order Plus Deadtime - Frequency Response

• amplitude ratio - remains the same as for first
  order process
• phase angle - lag is now increased, and keeps
  increasing due to time delay

16
Outline

• 1st order, 1st order plus deadtime
• second order
• integrators
• estimating First Order Plus Deadtime (FOPDT)
  models

17
Second-Order Processes

• arise from processes modeled by two first-order
  ODEs in series, two interacting ODEs or by a
  second order ODE.
• Recall that our non-isothermal CSTR example had
  second order transfer functions.
• parameterized by gain, time constant and damping
  coefficient
• transfer function

18
Second-Order Processes

• damping coefficient, ?, can be determined by
  placing the transfer function in this standard
  form and then finding ? and ?
• roots of denominator are poles of the transfer
  function
• ? is called the damping coefficient.

19
Second-Order Processes - Qualitative Behaviour

• poles are
• look at influence of damping coefficient, ?
• ? gt1 - two distinct real poles
• overdamped response (no oscillations).
• ? 1 repeated, real poles
• critically damped - on the verge of oscillatory
  step response

20
Second-Order Processes - Qualitative Behaviour

• ?lt1 - underdamped
• corresponds to complex roots
• step response exhibits oscillations
• our nonisothermal CSTR example was underdamped.
• Maple animation

21
Second-Order Processes - Frequency Response

• amplitude ratio is
• amplitude ratio can be bigger than Kp over a
  range of frequencies.
• AR plot can exhibit resonance
• at some frequencies, 2nd order systems can
  amplify oscillations.

22
Second-Order Processes - Frequency Response

• phase angle
• lag tends to -180 o at high frequencies

23
Second Order Process - Frequency Response

PROCESS (GP) BODE PLOT
resonant peak
1
10
0
10
Amplitude Ratio
-1
10
-2
10
-2
-1
0
1
10
10
10
10
Frequency, w (rad/time)
0
max phase lag of 180o
-100
Phase Angle (degrees)
-200
-2
-1
0
1
10
10
10
10
Frequency, w (rad/time)
24
Outline

• 1st order, 1st order plus deadtime
• second order
• integrators
• estimating First Order Plus Deadtime (FOPDT)
  models

25
Integrator

• transfer function G(s)1/s
• how? - think of tank with constant outflow
• level accumulates or decreases depending on
  inflow to tank
• step response level increases constantly (ramp)
• unstable system - not self-regulating
• pole - at the origin (0)

26
Integrator - Frequency Response

• amplitude ratio
• phase angle --gt constant at -90 o

27
Self-Regulation

• does rate of process change depend on current
  state?
• concentration mixing in tank - YES
• level accumulation - integrator NO
• autocatalytic reaction in CSTR YES !!
• self-regulation - stable - process response is
  limited
• non-self regulating - process response changes
  without bound - unstable
• positive feedback response increases w/o bound

28
Outline

• 1st order, 1st order plus deadtime
• second order
• integrators
• estimating First Order Plus Deadtime (FOPDT)
  models empirically

29
Chapter 6

• Empirical Models for Process Dynamics

30
Empirical Models of Process Dynamics

• empirical - estimated from data
• cf. mechanistic models considered so far
• why use empirical models?
• less development time less
• complex process - development of mechanistic
  model will be difficult (Less skill and knowledge
  are required to develop empirical models.)
• reduced computational requirements for use
• first-order model vs. detailed PDE model

31
Fundamental Concept

• Perturb process in a known way and under known
• conditions, collect data, choose model structure
  and estimate model parameters.

32
Marlins Six Step Procedure

• Experimental Design
• Plant Experiment(s)
• Model Structure Determination
• Parameter Estimation
• Diagnostic Evaluation
• Model Verification

33
Experimental Design - An Essential Step

• what is to be modeled?
• base conditions - reference point
• type, size of input perturbations
• duration of the experiment
• involves collection of background knowledge of
  process
• contact engineers/operators/design engineers

34
Plant Experiment

• ensure process is operating smoothly, near
  desired reference point
• watch secondary variables
• did disturbances enter during the experiment?
• allow output to get to steady-state after input
  perturbation
• want to ensure cause and effect

35
Model Structure

• do we use first-order, 2nd order, ...?
• structural determination can be difficult
• use knowledge of response characteristics
• e.g., over vs. underdamped, pure dead-time,
  first-order response
• there are limited quantitative methods
• in this course we will use first-order plus
  dead-time models

36
Parameter Estimation

• model parameters can be estimated statistically
  via regression or using other simpler
  techniques.
• What parameters will we have to estimate for a
  first-order plus dead time model?

37
Diagnostics and Verification

• examine predicted response and compare with
  observed response
• include several perturbations in experiment to
  check for process changes (disturbances during
  the experiment).
• verification
• examine predictions vs. new data

38
Process Reaction Curve

• implement step change in process input
• allow to reach steady-state
• estimate parameters using a graphical analysis
• use Method II in Marlin (not Method I)

39
Estimating 1st-order Plus Deadtime Models -
Method II

• process gain
• 63.2 time corresponds to
• 28 time - corresponds to

40
Marlin Example

• series of heated stirred tanks
• 5 change in valve to steam line
• ultimate change -gt 13.1 C
• gain 2.6 C/ open
• 28, 63 times -gt 10.7 min, 14.7 min corresponding
  to 3.7 C, 8.3 C
• time constant -gt 6.0 min
• time delay -gt 3.7 min

41
Empirical Modeling Example

T I
INPUT
OUTPUT
42
Issues

• size of step input - signal to noise
• guideline (input step/3 std. devns)gt 5
• trade-off - need for signal vs. large disruptions
  in process operation
• diagnostics
• use a step up and step down to see if process has
  changed during experiment
• gain, time constant may depend on size of step
  if process is highly nonlinear