Transcript ECE100: Intro to Engineering Design, Presentation No. 1

Engineering Process Control Perspectives on
Adaptive Time-Varying Interventions
Daniel E. Rivera and Michael Pew*
Control Systems Engineering Laboratory
Department of Chemical and Materials Engineering
Arizona State University
[email protected]
*undergraduate researcher and junior in Industrial Engineering
http://www.fulton.asu.edu/~csel
Presentation Outline
• Motivation: Adaptive, time-varying interventions
• A few control engineering basics
• An illustration: hypothetical intervention based on the Fast Track
program
• Some brief thoughts on Model Predictive Control, system
identification, and Model-on-Demand Estimation and Control
• Summary and conclusions
Motivation
• Explore the relationships between engineering process control
and the problem of adaptive, time-varying interventions in
prevention and treatment.
• In adaptive, time-varying interventions, different dosages of
prevention or treatment components are assigned to different
individuals across time, with dosage corresponding to the need
of the individual.
• Similar to process control, an effective adaptive time-varying
provides the following advantages:
–
–
–
–
minimizes negative effects,
increases compliance,
reducesw inefficiencies and waste,
enhances intervention potency
Control Engineering
• Control engineering is a broadly-applicable field that spans all
areas of engineering:
–
–
–
–
–
–
–
Chemical
Electrical
Mechanical and Aerospace
Civil / Construction
Industrial
Biomedical
Computer Science and Engineering
• Control engineering principles play a part in everyday life
activities.
Control Engineering (Continued)
Considers how to manipulate or adjust system variables
so that dynamic behavior is transformed from
undesirable from desirable
• Open-loop: refers to system behavior without a
controller or decision rules
• Closed-loop: refers to system behavior once a
controller or decision policy is implemented.
Signal Definitions
• Controlled Variables (y): system variables that we wish to keep
at a reference value (or goal), also known as the setpoint (r).
• Control Error (e=r-y): the difference between the controlled
variable and the setpoint; we wish to take this to zero.
• Manipulated Variables (u): system variables whose adjustment
influences the response of the controlled variable; their value is
determined by the controller/decision policy.
• Disturbance Variables (d): system variable that influences the
controlled variable response, but cannot be manipulated by the
controller; disturbance changes occur external to the sys tem
(hence sometimes referred to as exogeneous variables)
• Noise (n): used to describe the measurement error
(or unreliability) in the controlled variables
The “Shower” Control Problem
Controlled:
Temperature,
Total Water Flow
Disturbances:
Inlet Water Flows,
Temperatures
Manipulated: Hot and Cold
Water Valve Positions
Hot
Cold
Think about what may constitute
controlled, manipulated
and disturbance variables in this system
Operational Objectives of an Engineering
Control System
• Setpoint Tracking. Refers to the ability of the control
system to manipulate system variables such that
the controlled variable follows a reference (setpoint)
trajectory as closely as possible.
• Disturbance Rejection. Refers to the ability of the
control system to accomplish setpoint tracking,
despite substantial variations in the disturbance
variables.
Feedback and Feedforward
Control Strategies
• In feedback control strategies, a controlled variable
(y) is examined and compared to a reference value or
setpoint (r). The controller issues actions (decisions
on the values of a manipulated variable (u)) on the
basis of the discrepancy between y and r.
• In feedforward control, changes in a disturbance
variable (d) are monitored and the manipulated
variable (u) is chosen to counteract anticipated
changes in y as a result of d.
Closed-Loop Feedback Control “Block
Diagram”
Disturbances:
d
Manipulated:
Hot and Cold
Water Valve
Positions
Reference:
Desired Temperature,
Total Water Flow
r
ec = r - ym
-
C
Pd
u
+
P
+
Inlet Water Flows,
Temperatures
Controlled:
y Temperature,
Total Water Flow
+
ym
C = Controller
P = Plant Model/“Transfer Function”
Pd = Disturbance Model/“Transfer Function”
n
From Open-Loop Operation to
Closed-Loop Control
Measured Output
20
Temperature
Deviation
(Measured
Controlled
Variable)
10
0
Open-Loop
(Before Control)
-10
-20
0
500
1000
1500
2000
2500
3000
3500
4000
Time[Min]
Input
Hot Water
Valve
Adjustment
10
(Manipulated
Variable)
-10
Closed-Loop
Control
0
0
500
1000
1500
2000
2500
3000
3500
4000
Time[Min]
The transfer of variance from an expensive resource to a cheaper one is
one of the major benefits of engineering process control
Components of a Closed-Loop Control System
• Sensors: needed to measure the controlled and
(possibly) the disturbance variables.
• Actuators: needed to achieve desired settings for the
manipulated variables
• Controllers (i.e., decision rules). These relate control
errors, previous manipulations and disturbance
measurements to current settings of the manipulated
variable.
Open-Loop (Manual) vs. Closed-Loop
(Automatic) Control
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
O-L “Manual”
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
C-L “Automatic”
Fast Track Program:
Adaptive, Time Varying Intervention
• A multi-year program held in the Baltimore area
designed to prevent conduct disorders in at-risk
children
• Two adaptive components:
– Reading tutoring assigned to children demonstrating
academic difficulties
– Home-based counseling visits assigned to families based on
the level of parental functioning
Parental Function-Home Visits Adaptive
Intervention Control Problem
• Controlled Variable (Tailoring Variable):
Estimated Parental Function PF(k)
• Manipulated Variable (Intervention):
Frequency of Home Visits I(k)
• Disturbance Variable:
“Depletion” D(k)
Modeling Parental Function Dynamics
Basic Conservation Principle:
Accumulation
Within
The System
=
Input
Through the
System
Boundaries
-
Output
Through the
System
Boundaries
Parental Function “Open Loop” Dynamics
PF(k+1) = PF(k) + KI I(k – q) - D(k) + N(k)
k = integer reflecting review instance
KI = intervention gain
D(k) = depletion
N(k) = unreliability
I(k) = intervention dosage
q = delay
Estimated Parental Function (End of Review Instance)
= Parental Function (Start of Review Instance)
+ Parental Function Contributed by Intervention
- Parental Function Depletion
+ Measurement Unreliability
Parental Function - Home Visits Intervention as
an Inventory Control Problem - Dynamics
Intervention
CTL
Parental
Function
“Inventory”
LT
Depletion
Parental function is built up by providing an intervention (frequency of home
visits), that is potentially subject to delay, and is depleted by potentially
multiple disturbance factors (which could be measured or unmeasured)
Parental Function Dynamics
Modeling Challenges
• Quantitative models for supply mechanisms (how particular
magnitudes of an intervention contributes to specific levels of
parental function) can be stochastic, nonlinear, and
autoregressive in nature
• Depletion mechanisms can be nonlinear, stochastic, and
autoregressive as well; could be measured or unmeasured.
• Choice of review interval, measurement of outcomes, defining
tailoring variables, and deciding intervention dosage levels all
play significant, important roles.
Feedback-Only Parental Function
Control Problem
I(k) (Manipulated)
CTL
LT
PF(k)
(Controlled)
Depletion
D(k) (Disturbance)
In the feedback-only control problem, intervention dosages are calculated
based only on perceived changes to “inventory” (parental function PF(k)).
Parental Function Feedback Loop Block Diagram*
(to decide on home visits for families with at risk children)
Clinical Judgment
Goal
+
Decision
Rules
+
Disturbances
Intervention
Outcomes
Process
I(k)
Review
Interval
If PF(k) is “Low” then Weekly Home Visits
If PF(k) is “Medium” then Bi-Weekly Visits
If PF(k) is “High” then Monthly Home Visits
If PF(k) is “Acceptable” then No Visits
Tailoring Variable
Estimation
+
+
Reliability/
Measurement
Error
Estimated Parental Function PF(k)
*Based on material from Collins, Murphy, and Bierman, “A Conceptual Framework
for Adaptive Preventive Interventions,” to appear in Prevention Science.
Rule-based Controller
(i.e.,decision rules per Collins et al., in press)
•If parental function is “Low” (0 < PF(k) < PF low) the intervention
dosage should correspond to weekly home visits (I (k) = I weekly),
•If parental function is “Medium” (PF low < PF(k) < PF medium)
then intervention dosage should correspond to bi-weekly home
visits (I (k) = I biweekly),
•If parental function is “High” (PF medium < PF(k) < PF high)
then intervention dosage should correspond to monthly home
visits (I (k) = I monthly),
•If parental function is higher than goal (PF(k) > PF high)
then intervention dosage should correspond to no home
visits (I (k) = 0).
PF Simulation Details
• Chose KI = 0.1 and q=0; no measurement unreliability (N(k))
• Settings for PF levels, Intervention Strengths:
PF low = 33%, PFmedium = 67%, PF high = 90%
I weekly = 100%, I biweekly = 67% , I monthly = 33%
• Goal = 90% parental function
• Intervention executed for 36 months with monthly simulation of
the system dynamics and a quarterly (T = 3 months)
review interval.
• Written in Excel.
Rule-Based Controller, D(k) = 0
PF Level
Chart Title
100
90
80
70
60
50
40
30
20
10
0
PF
Factor
Goal
1
11
21
Months 31
Intervention Dosage
Recommended Intervention Dosage
3
2
1
0
1
11
21
Months
31
PF Level
Rule-Based Controller, D(k) = 2
PF Factor
100
90
80
70
60
50
40
30
20
10
0
PF
Factor
Goal
1
11
21
Months 31
Intervention Dosage
Recommended Intervention Dosage
3
2
1
0
1
11
21
Months
31
PF Level
Rule-Based Controller, D(k) = 4
PF Factor
100
90
80
70
60
50
40
30
20
10
0
PF
Factor
Goal
1
11
21
Months 31
Intervention Dosage
Recommended Intervention Dosage
3
2
1
0
1
11
21
Months
31
Model-Based Controller Tuning
• A model-based tuning rule simplifies the choice of controller tuning
parameters.
• We will consider a tuning rule for an “integrating” system which
relies on the concept of Internal Model Control (IMC).
• User supplies the intervention gain (KI) and delay (q and only one
adjustable parameter (), which is inversely proportional to the
closed-loop speed of response
– Increasing  makes the system sluggish; decreasing  speeds it up.
– Increasing the speed-of-response results in higher dosage levels and
makes the closed-loop system more sensitive to model variations (i.e.,
the control system is less robust to uncertainty).
Proportional-Integral-Derivative
(PID) with Filter Control
Kc t
de(t)
du(t)


u(t) = Kc e(t) 
 e(t )dt  Kc  D
 F
 0
dt
dt
I
• Choice of Proportional (Kc), Integral (I), Derivative
(D) and Filter (F) tuning parameters influence
how control error (e = r - y) determines the value
of the manipulated variable.
• Four “adjustable” tuning knobs can represent a
tuning challenge.
Internal Model Control
PID Controller Tuning (Rivera et al. 1986)
User supplies intervention gain (KI), delay (q and the
adjustable parameter ()
 = =
2(   )  
Kc =
2
2
K I (2   4   )
 I = 2(   )  
q
2
2 (   )
D =
2(   )  
2
F = 2
2
2  4  
Discrete-Time PID Controller
Implementation
I( k )
= I (k  1)  K e (k )  K e (k  1)  K e( k  2)  K I ( k  1)
1
2
3
4
K1
I( k )
TK c  T  D
1 
= I (k  1) 
 F T   I T

e (k )

K2
K3
K4
I (k1)
TK c  2 D 
K
F

1
e (k  1)  c D e (k  2) 
(I (k 1)I (k 2))
 F T 
T 
 F T
 F T
T is the “sampling time” or review period
Discrete-Time PID Controller Summary
I(k ) = I(k 1) K1e(k) K 2e(k 1) K 3e(k  2) K 4I(k 1)
Current Dosage = Previous Dosage
+ Scaled Corrections from Current and Prior Control Errors
+ Scaled Previous Dosage Change
K1, K2, K3, and K4 are tuning constants in the controller;
e(k) = (PF(k) - R(k)), where R(k) is the setpoint/goal, is the
control error
The dosage decision I(k) is normally a continuous value
between 0 and 100%, but for purposes of this example it is
quantized into the nearest of the four dosage levels
(I weekly , I biweekly , I monthly,0 )
“Quantized” PID controller Implementation
Clinical Judgment
+
Goal
PID
Controller
Actuation
+
Disturbances
Intervention
Outcomes
Process
-
Review
Interval
Control Error = Goal – Estimated Parental Function
Tailoring Variable
Estimation
+
+
Estimated Parental Function
Unreliability/
Measurement
Error
PF Level
PID Controller, D(k) = 2, Lambda = 3
PF Factor
100
90
80
70
60
50
40
30
20
10
0
PF
Factor
Goal
1
11
21
Months 31
Intervention Dosage
Recommended Intervention Dosage
3
2
1
0
1
11
21
Months
31
Controller Comparison, D(k) = 2
Rule-Based
PF Factor
100
90
80
70
60
50
40
30
20
10
0
PF
Factor
Goal
1
11
21
PF Level
PF Level
PID
PF Factor
100
90
80
70
60
50
40
30
20
10
0
Months 31
PF
Factor
Goal
1
3
2
1
21
Months 31
Recommended Intervention Dosage
Intervention Dosage
Intervention Dosage
Recommended Intervention Dosage
11
3
2
1
0
0
1
11
21
Months
31
1
11
21
Months
31
PID Controller, D(k) = 2, Various Lambdas
Lambda = 3
PF
Factor
Goal
1
11
21
PF Factor
100
90
80
70
60
50
40
30
20
10
0
Months 31
PF
Factor
Goal
1
0
21
3
2
1
11
21
Months
31
11
21
Months 31
Recommended Intervention Dosage
3
2
1
0
0
1
PF
Factor
Goal
1
Months 31
Intervention Dosage
3
1
11
PF Factor
100
90
80
70
60
50
40
30
20
10
0
Recommended Intervention Dosage
Intervention Dosage
Intervention Dosage
Recommended Intervention Dosage
2
Lambda = 10
PF Level
PF Factor
100
90
80
70
60
50
40
30
20
10
0
PF Level
PF Level
Lambda = 1
1
11
21
Months
31
1
11
21
Months
31
PID Controller, D(k) = 2, Various Lambdas
PF Analysis
18
14
14
12
10
8
10
Months
Months
12
8
6
6
4
4
2
2
0
0
Low
Medium
High
Months On Intervention Level
16
Low
Acceptable
12
12
10
10
Months
14
8
6
High
Acceptable
Months On Intervention Level
16
14
Medium
2
2
Weekly
Bi-Weekly
Monthly
No Visits
High
Acceptable
Months On Intervention Level
15
10
5
0
0
Medium
20
6
4
Low
25
8
4
PF Analysis
20
18
16
14
12
10
8
6
4
2
0
Months
Months
PF Analysis
16
16
Months
Lambda = 10
Lambda = 3
Lambda = 1
0
Weekly
Bi-Weekly
Monthly
No Visits
Weekly
Bi-Weekly
Monthly
No Visits
PF Level
PID Controller, D(k) = 4, Lambda = 3
PF Factor
100
90
80
70
60
50
40
30
20
10
0
PF
Factor
Goal
1
11
21
Months 31
Intervention Dosage
Recommended Intervention Dosage
3
2
1
0
1
11
21
Months
31
Controller Comparison, D(k) = 4
Rule-Based
PF Factor
100
90
80
70
60
50
40
30
20
10
0
PF
Factor
Goal
1
11
21
PF Level
PF Level
PID
PF Factor
100
90
80
70
60
50
40
30
20
10
0
PF
Factor
Goal
1
Months 31
Intervention Dosage
Intervention Dosage
2
1
21
Months 31
Recommended Intervention Dosage
Recommended Intervention Dosage
3
11
3
2
1
0
0
1
11
21
Months
31
1
11
21
Months
31
Controller Comparison, D(k) = 4
PID
Rule-Based
PF Analysis
14
14
12
12
Months
Months
10
8
6
10
8
6
4
4
2
2
0
0
Low
Medium
High
Low
Acceptable
Months On Intervention Level
16
12
12
10
10
Months
14
8
6
High
Acceptable
8
6
4
4
2
2
0
Medium
Months On Intervention Level
16
14
Months
PF Analysis
16
0
Weekly
Bi-Weekly
Monthly
No Visits
Weekly
Bi-Weekly
Monthly
No Visits
Additional Cases
• Consider measurement unreliability
• Cases with nonlinear and stochastic gain and delay
values
Fab/Test Node Dynamics
Load
Time
Control-Relevant Modeling Principles
(Adapted from Skelton, "Model Error Concepts in Control Design,"
International Journal of Control, 49, 1725, 1989)
• Errors that may appear to be “small” in the open-loop may lead
to bad closed-loop performance
• Errors that may appear to be “large” in the open-loop may not
necessarily lead to bad closed-loop performance
• Control requirements dictate model accuracy, not vice versa
• Very simple models can be adequate descriptions for control
design (even for highly nonlinear, stochastic dynamical systems)
when properly applied - Rivera addendum
Model Predictive Control
• Refers to a class of discrete-time control systems
which
– make explicit use of a model to predict the process
output at future time instants (the prediction
horizon)
– employ a receding or moving horizon strategy, so
that at each instant the horizon is displaced
towards the future.
– calculate a corresponding control sequence (the
move horizon) that minimizes a certain objective
function
MPC - Moving Horizon Representation
(Parental Function)
(Depletion)
(Anticipated Depletion)
(Intervention)
Model Predictive Control Objective Function
Model Predictive Control Advantages
• Ability to handle large multivariable systems
• Ability to enforce constraints on manipulated and
controlled variables
• Effective integration of feedback, feedforward
controller modes; ability to incorporate anticipation
• Novel formulations (such as hybrid MPC) enable the
application to systems involving both discrete-event
and continuous variables.
System Identification
“Identification is the determination, on the basis of input and
output, of a system within a specified class of systems, to
which the system under test is equivalent.”
- L. Zadeh, (1962)
Disturbances
Inputs
System
Outputs
System identification focuses on the modeling of
dynamical systems from experimental data
System Identification Procedure
Construct the
experiment and
collect data
Data
Should data
be preprocessed?
Polish and
present data
Processed data
Choice of model
structure
Data
not OK
Fit the model
to the data
Model
Validate
the model
Model structure
not OK
No
Can the model
be accepted?
Yes
• Statistically
• Physically
Data-centric Modeling and Control
• Provides the potential for addressing bias/variance
tradeoffs in a manner appealing to the practicing
process control engineer,
• Enables the process control engineer to perform
nonlinear modeling and control while retaining the
intuition associated with linear methods
• The increased sophistication (relative to linear
methods) does not have to be matched with equally
sophisticated training.
Model-on-Demand Estimation
• Applies a localized goodness-of-fit measure (e.g. localized AIC,
Akaike’s FPE) over an exponentially increasing bandwidth to
determine the size of the local neighborhood.
error
 0.3 
hi = 1 
hi 1
d 

d = dim( )
hmax
hmin
desired bandwidth
2
current
operating
point
1
Conclusions
• Adaptive, time-varying interventions are feedback control
systems, and therefore can benefit from a control-theoretic
perspective.
• A hypothetical adaptive, time-varying intervention has been
simulated using both a rule-based controller (explicit decision
rules) vs. an engineering based PID (Proportional-IntegralDerivative) controller.
• Model Predictive Control, Data-Centric System Identification are
concepts that can impact novel control algorithms geared to
this (and other problems) in the behavioral sciences.