control theory - iit madrasche.iitm.ac.in/~nkaisare/courses/mpcshort.pdf · control advanced...
TRANSCRIPT
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Model Predictive Control1. Theory
Niket S. KaisareDepartment of Chemical Engineering
Indian Institute of Technology – Madras
Summer School on Process Identification and Control NIT-Tiruchirappalli, June 25th 2008
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Hierarchy of Process Control
Regulatory Control
Advanced Process Control
Real-TimeOptimizer
PlanningScheduling
StrategicDecisions
seconds
min ~ hours
hours ~ day
week ~ month
quarterly ~ annual
Time-scale
$$$
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Example: Shell Heavy Oil Fractionator
� Hierarchical control strategy� PID loops for level &
inventory control� Quality control loops� Minimise energy usage
� Several operating constraints
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What is Model Predictive Control?
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Exemplary MPC Algorithm
Plant
input output
disturbances
observerstate / information
objectivemax productivity∑
model( )duxfx ,,=�
constraints
maxmin xxx ≤≤
controller
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Dynamic Matrix Control:Industrial Origin of MPC
� Initiated at Shell Oil and other refineries (1970s)
� Various commercial software� DMCplus – Aspen Tech� RMPCT –Honeywell� Dozen+ other players (e.g., 3DMPC-ABB)
� >4500 worldwide installations
� Predominant in oil and petrochemical industries
� The range of applications is expanding
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A Brief History of MPCAlgorithm Model Objective Pred. Horizon Constraints
LQG (1960) L SS min ISE IO infinity -
IDCOM (1976) L conv min ISE O p IO
DMC (1979) L conv min ISE IOM p IO
QDMC (1983) L conv min ISE IOM p IO
GPC (1987) L ARMA min ISE IO p -
IDCOM-M L conv min ISE O p IO(1988) min ISE I
SMOC (1988) L SS min ISE IO p IO
Rawlings and L SS min ISE IO infinity IO Muske (1993)
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Multivariable Process ControlConventional Structure
Global SSOptimization
Plant-Wide Optimization
Local SSOptimization
Unit 1 Local Optimization Unit 2 Local Optimization
DynamicSupervisory/ConstraintControl
High/Low Select Logic
PID Lead/Lag PID
SUM SUM
ModelPredictive
Control (MPC)
MPC Structure
Basic DynamicControl
FCPC
TCLC
FCPC
TCLC
Unit 1 Distributed ControlSystem (PID)
Unit 2 Distributed ControlSystem (PID)
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Design Effort
Almost all models used in MPC are typically empirical models “identified” through plant tests rather than first-principles models.
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MPC Definition
Model Predictive Control (MPC) refers toa class of algorithms that utilize
an explicit process model to computea manipulated variable profile that will
optimize an open-loop performance objectiveover a future time interval.
The performance objective typicallypenalizes predicted future errors and
manipulated variable movementsubject to constraints.
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MPC Definition
Model Predictive Control (MPC) refers toa class of algorithms that utilize
an explicit process model to computea manipulated variable profile that will
optimize an open-loop performance objectiveover a future time interval.
The performance objective typicallypenalizes predicted future errors and
manipulated variable movementsubject to constraints.
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Receding Horizon Control Concept
k k+1 k+m-1 k+p-1
Past Future
setpoint
input
output
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Analogy to Chess
hgfedcba
1
2
3
4
5
6
7
8
Me
Opponent (Plant)
Model Predictive Control
Opponent’s Move
New Piece Position
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Major Components of MPC
1. Dynamic state (stored in memory)
2. Multi-step prediction equation
3. Objectives and Constraints
4. Optimization solver (Linear / Quadratic Program)
5. State update scheme
Predicted future outputs = Effect of current “state” (memory) +Future input adjustments +Feedforward measurement +Feedback correction
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General Setup
Current State
Prediction Modelfor Future Outputs
Future Input Moves(To Be Determined)
To Optimization
Feedback / FeedforwardMeasurements
Prediction Measurement
Correction
Dynamic Model State:Compact representationOf the past input record
?
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General Setup
Previous State(in Memory)
Current State
Prediction Modelfor Future Outputs
New Input Move(Just Implemented)
Future Input Moves(To Be Determined)
To Optimization
Feedback / FeedforwardMeasurements
State Update
Prediction Measurement
Correction ?
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MPC Options
� Model Types� Finite Impulse Response Model or Step Response Model� State-Space Model� Linear or Nonlinear
� Measurement Correction� To the prediction (based on open-loop state calculation)� To the state (through state estimation)
� Objective Function� Linear or Quadratic� Constrained or Unconstrained
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Step Response Model
Assumptions:• No immediate effect: S0 = 0• Stable process settles after time n: for 1j nS S j n= ≥ +
1. Dynamic Model
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Step Response Model:Superposition Principle
0( ) ( )
k
iv k v i
=
= ∆∑
1. Dynamic Model
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Step Response Model:Superposition Principle
1 2( ) ( 1) ( 2) ( ) ( 1) (0)( )
n n ny k v k S v k S v k n S v k n S v Sv k n Sn
= ∆ − + ∆ − + + ∆ − + ∆ − − + +
−
" "�����������������
1
1( ) ( )
n
i k ni
y k v k i S v S−
=
= ∆ − +∑
1. Dynamic Model
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Extension to Multivariate Systems
1. Dynamic Model
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State Definition
Possible choices of state:Past n input moves, orFuture n output response assuming v is held constant at v(k)
Effect of past inputs is kept in memory as the system state
1b. Definition of “State”
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State Definition
( )0 1 1
( ) ( ) ( )n
T T T Tx k y k y k y k−
⎡ ⎤= ⎣ ⎦"
assuming ∆v(j)=0 for( ) ( )iy k y k i= +
Define:
2. Effect of past inputs on n-step future predictions is the current state:
1.
1b. Definition of “State”
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1. Dynamic State for Step Response Model
� Step response model
� Definition of the state� Stores n future output response in the memory
assuming the input is kept constant at u(k-1)
( ) ( )1
( )n
i ni
y k S v k i S v k n=
= ∆ − + −∑
( )
( )( )
( )⎥⎥⎥⎥
⎦
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⎢⎢⎢⎢
⎣
⎡
=
− ky
kyky
kY
n 1
1
0
~#
( ) ( )( ) ( ) 01
with==+∆=∆
+="kuku
ikykyi
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Multi-Step Prediction
Main interest for control: predicting future output behavior.
Æ Future output = function of (past input, hypothesized future input)
Æ Dynamic state: memory about the effect of past input
Æ Future output = function of (dynamic state, future input)
2. Multi-step Prediction
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Multi-Step Prediction
Effect of future input moves (to be determined)
x(k)
2. Multi-step Prediction
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2. Multi-Step Prediction( )( )
( )=
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⎣
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#21 ( )
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dp
d
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+
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kpke
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#21
Effect of past inputs
Effect of current and future inputs
Effect of measured disturbances Unmeasured disturbances
+
+
+
2. Multi-step Prediction
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2. Multi-Step PredictionAssume measured disturbance remains constant: d(k) = d(k+1) = d(k+2) = …
∆d(k+1) = ∆d(k+2) = … = 0
Previous prediction error is added as a constant bias terme(k+1) = e(k+2) = … e(k+p) = ymeasured(k) – y(k)
( )( )
( )( )
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S
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3. Objective Function and Constraints
Constraint violation may not be acceptable
Shaded region is the setpoint tracking error
ySP
yMax
( ) ( )
( )[ ] ( )[ ]
( )[ ] ( )[ ] ⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
+∆+∆
++−+−
∑
∑−
=
=
−+∆∆ 1
0
1
1,,min
m
l
p
iSPSP
mkukulkuRTlku
kikyyQTkikyy
"
3. Objective function and constraints
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Output Trajectories
quadratic penalty
past future
Reference trajectory
quadratic penalty
past future
Setpoint
quadratic penalty
past future
Zonequadratic penalty
past future
Funnel
3. Objective function
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Constraints
� Constraints include� Input magnitude constraints:
� Input rate constraints:
� Output magnitude constraints:
maxmin )( uikuu ≤+≤
max)( uiku ∆≤+∆
maxmin )( yikyy ≤+≤
3. Constraints
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Constraint Formulations
Hard constraint
past future
Soft constraint
past future
quadratic penalty
past future
Setpoint approximation of soft constraint
quadratic penalty
3. Constraints
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Optimisation in Receding Horizon Control
k k+1 k+m-1 k+p-1
Past Future
setpoint
input
output Which input move to choose?
3&4. Objective function and Optimisation
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4. Optimization: Quadratic Program
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3&4. Objective Function for QP
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3&4. Constraint Formulation for QP
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3&4. Constraint Formulation in QP
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3&4. Constraint Formulation in QP
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3&4. Constraint Formulation in QP
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Receding Horizon Control Concept
k k+1 k+m-1 k+p-1
Past Future
setpoint
input
output
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5. State Update Scheme
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−
−
)()(
)()(
1
2
1
0
kyky
kyky
n
n
#
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
++
++
−
−
)1()1(
)1()1(
1
2
1
0
kyky
kyky
n
n
#
N N )()()1( 1 kvSkxMkxresponsestepshift
∆+=+
drop
S1 S2
Sn-1
Sn
Recall from previous lecture: State Update Equation
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MPC Definition
Model Predictive Control (MPC) refers toa class of algorithms that utilize
an explicit process model to computea manipulated variable profile that will
optimize an open-loop performance objectiveover a future time interval.
The performance objective typicallypenalizes predicted future errors and
manipulated variable movementsubject to constraints.
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Model Predictive Control2. Applications
Niket S. KaisareDepartment of Chemical Engineering
Indian Institute of Technology – Madras
Summer School on Process Identification and Control NIT-Tiruchirappalli, June 25th 2008
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Example 1: Blending System Control
� Control rA and rB� Control blend flow q� Constraints on MV
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Example 1: Blending System Control
� Control rA and rB� Control blend flow q� Constraints on MV
Classical Solution MPC Solution
Minimize deviation from the set-point for p steps into the future.
( ) ( )( )∑
= ⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+
−+−
−++
p
i sp
spBB
spAA
uuu qq
rrrr
mkkk 1
2
22
...,,,1
1
minγ
maxmin uuu i ≤≤
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Example 2: Heavy Oil Fractionator
� Objective� Keep temperature y7 above
a minimum value� Keep y1 and y2 at setpoint� Minimize u3 to maximize
heat recovery
� Several constraints
� Non-square system
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Example 3: Ethylene Plant
Furnaces
PrimaryFractionator
QuenchTower
Charge GasCompressor
Chilling
DemethanizerDeethanizer
EthyleneFractionator
DebutanizerPropyleneFractionator
DepropanizerFuel OilFuel Oil
HydrogenHydrogen
MethaneMethane
EthyleneEthylene
EthaneEthane
PropylenePropylene
PropanePropane
B - BB - B
GasolineGasolineLight H-CLight H-CNaphthaNaphtha
FeedstockFeedstock
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Typical Scale of Modeling and Control123456789
1011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
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Popularity of MPCSystematic and Integrated Solution
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Popularity of MPCSystematic and Integrated Solution
Advantages over traditional APC� Integrated solution
� automatic constraint handling� Feedforward / feedback� No need for decoupling or delay compensation
� Efficient Utilization of degrees of freedom� Can handle nonsquare systems (e.g., more MVs and CVs)� Assignable priorities, ideal settling values for MVs
� Consistent, systematic methodology� Realized benefits
� Higher on-line times� Cheaper implementation� Easier maintenance
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Popularity of MPCEmerging Popularity of On-line Optimization
� Process optimization and control are often conflicting objectives� Optimization pushes the process to the boundary of constraints.� Quality of control determines how close one can push the
process to the boundary.
� Implications for process control� High performance control is needed to realize on-line
optimization.� Constraint handling is a must.� The appropriate tradeoff between optimization and control is
time-varying and is best handled within a single framework
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Popularity of MPCEmerging Popularity of On-line Optimization
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Popularity of MPCEmerging Popularity of On-line Optimization
Bi-level Optimization in MPC
Steady-State Optimization(LP)
Dynamic Optimization(QP)
Optimal setpoints
Regulate low-level loops Measurements(feedback update)
S.S. prediction model
Economic optimization
Minimization oftracking error
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Example: Model Predictive Control of Mammalian Cell Cultures
Niket S. Kaisare, Jay H. LeeA. A. Namjoshi, D. Ramkrishna
Presented at AIChE Meeting, November 2000
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Overview of the Talk
� Introduction� Hybridoma cell reactor
� Cybernetic Modeling� Case Study
� Simpler microbial cell system� Full hybridoma cell system
� Conclusions
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Introduction and Motivation
� Hybridoma Cells� Hybrid mammalian cells� Produce monoclonal
antibodies
� Applications of antibodies� Protein
detection/purification� Immunotherapy� Tumor imaging
B-lymphocytes
Myeloma
Hybridoma
• Tumor cells• Rapid in-vitro growth
• Produce antibodies
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Hybridoma Cell Reactor
Air (O2)
• Glucose (s1f)• Glutamine (s2f)• Amino acids
• Unconsumed substrates• Biomass w/ antibodies• Waste metabolites
(Lactate, Alanine etc.)
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Steady State Multiplicity
0
2
4
6
0 100 200 300Time (hrs)
Cel
l con
cn (x
106
cel
ls/m
l)
Three types of steady states observed for same input conditions
Differ in the metabolic states of the cells
•
•
Start up Experiments, Hu et al. (1998)
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Hybridoma Reactor Control
� Previous Work� Detremblay et al. (1993): Estimation and PI control� Lenas et al. (1997): Adaptive fuzzy control� Dhir et al. (2000)
� Fuzzy update of model parameters� Heuristic random optimizer (HRO)
� Europa et al.(2000): Heuristic scheme
� Our focus: Model based optimal control
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Bioreactor Modeling
� Monod type model
� Phenomenological modifications possible � Does not capture cellular metabolic regulations
� Detailed description of extremely large number of metabolic reactions� Unsuitable for on-line optimization and control
sKsr PS +
=→maxµ
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Cybernetic Modeling (1)
� Characteristics� Captures internal cellular regulatory mechanisms� Relatively fewer metabolic reactions
� Accurate description of average cell in a batch, fed-batch or continuous culture� Diauxie growth phenomenon� Mammalian cultures� PHB storage intermediate synthesis
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Cybernetic Modeling (2)
� Postulates� Kuberne’tes (Greek): “Steersman”
� A Metabolic Network has physiological objectives� Limited pool of resources� A cell directs the synthesis and activity of enzymes
in a metabolic competition� Mathematically, through cybernetic regulation variables� u modifies enzyme synthesis
� v modifies enzyme activity
� Cells are optimal strategists
^
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Analysis of a Simpler Model
� Single cybernetic competition
� Bacterial cell growth� Diauxie phenomenon
(Kompala et al., 1986)� To understand cybernetic
competitions
� Model Equations
( ) ( ) 2,1=−−= icYvrSSDdtdS
iiiiifi
( ) 2,1* =+−−= irereurdtde
ii eigiiei β
( )cDrdtdc
g −=
⎥⎦
⎤⎢⎣
⎡+
= maxmax
i
i
ii
ii e
eSK
Sr µ ),max( 21
11 rr
rvsc =
S1 S2
C
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The Steady State Multiplicity
� Two stable steady states for certain range of substrate feeds
� Objectives are� Maintain the reactor
at the steady state� Drive the reactor to
other steady state
Bifurcation diagram for the system(Namjoshi et al., unpublished)
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Step Variations in s2f
� At steady state, we apply a series of step changes in s2f
� Measurement noise
� The system drifts to the other steady state in open loop
0 50 100 150 2000
0.02
0.04
0.06
0.08
Time (hrs)
Bio
mas
s c(
g/lit
re)
Reactor Drifts in Open Loop
0 50 100 150 200
0.16
0.18
0.2
Time (hrs)
s2f (
g/lit
re)
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Model Predictive Control
� Dynamic Model� Cybernetic model
� Linear Approximation� NLP – difficult to solve� Linearize in the horizon
� Optimizer� Measurement Correction
� Extended K.F.� Glucose and biomass
feedback PREDICTION HORIZON
k k+m-1PAST FUTURE
TARGET
PROJECTEDOUTPUT
INPUTMOVES
ymax
( ) ( ) ⎥⎦⎤
⎢⎣⎡ ∆Λ+−Λ ++∆
2
2
2
2|11min ku
kkky
uUYR
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Manipulated and Controlled Variables(case A: Two M.V.s and two C.V.s)
� C.V.: biomass, s2
� M.V.: D, s1f
� Controller is able to control the reactor outputs at the given set points
� However, the cells are in physiologically different state
0 20 40 60 80 100 120 140 160 180 2000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Time (hrs)
Bio
mas
s c(
g/lit
re)
:
0 20 40 60 80 100 120 140 160 180 200
0.7
0.75
0.8
0.85
0.9
0.95
Time (hrs)
Dilu
tion
Rat
e D
(1/h
our)
0 20 40 60 80 100 120 140 160 180 2000.04
0.05
0.06
0.07
0.08
0.09
time (hrs)
s1f (
Man
ipul
ated
)
Cell Mass Concentration
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Case B: One M.V. and two C.V.s
� Changes implemented� Decreased weight on exit s2
concentration� M.V.: Dilution rate D
� Controlled at the desired steady state
� Behavior is better than case 1
0 50 100 150 2000.01
0.02
0.03
Time (hrs)
Bio
mas
s c(
g/lit
re)
Case 2(b)
0 50 100 150 2000.75
0.8
0.85
0.9
Time (hrs)
Dilu
tion
Rat
e D
(1/h
our)
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Varying the Steady State
� At t=50 hours, step change in reference trajectory to another steady state
� White measurement noise
0 50 100 150
0.02
0.04
0.06
Time (hrs)
Bio
mas
s c(
g/lit
re)
Driving the reactor to other S.S.
MeasurementsEstimate
0 50 100 150
0.7
0.75
0.8
0.85
0.9
Time (hrs)D
ilutio
n R
ate
D(1
/hou
r)
Reference Trajectory
The controller is able to reject disturbances and track the reference trajectory
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CELL
Model Equations
Hybridoma CellsMetabolic Network
C’
M2 M4
S2
S1
M1M3L
( ) ( )crrvrvvrssDdtds mmsdsdsc
f 2122111111 +++−−=
Namjoshi et al.,Session #271b
Set of 16 ODEs
4 sets of cybernetic regulation variables
Result from 4 substitutable cybernetic competitions
•
•
•
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Step Response at the LOW State
� We give a 10% step down change in D in our simulated reactor
� Highly nonlinear and stiff nature of the system
� Biomass increases with decrease in D
1000 1500 2000 25000
0.1
0.210% step down in D at steady state
s1 (g
/litre
)
1000 1500 2000 25000.4
0.6
0.8
C (g
/litre
)
1000 1500 2000 25000
0.5
1
time (hrs)
C'n
etic
Var
v7s
d
LOW
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Control Actions: LOW to HIGH State
� Starting the reactor at the low biomass state
� Step change in the reference trajectory
� Controller drives reactor to the high biomass steady state
0 100 200 300 400 500 6000
0.5
1
1.5
Time (hrs )
Bio
mas
s c(
g/lit
re)
Driving the reactor from to state
0 100 200 300 400 500 6000.01
0.02
0.03
0.04
Time (hrs)
Dilu
tion
Rat
e D
(1/h
our)
LOW HIGH
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Step Response at the HIGH State
� We give a 10% step up change in D in our simulated reactor
� Biomass first increases with increase in D
� When metabolic change is triggered, there is a sudden decrease
� S1 behavior is relatively monotonous
1000 1500 2000 25000
0.005
0.01
0.01510% step up in D at steady state
s1 (g
/litre
)
1000 1500 2000 25000.75
1
1.25
C (g
/litre
)
1000 1500 2000 25000
0.5
1
time (hrs)
C'n
etic
Var
. v1s
d
HIGH
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Control Actions: HIGH State
� Drive the reactor from high to intermediate biomass steady state
� Controlled Variable: c
� Prediction horizon is not long enough
0 100 200 300 400 500 600
0.8
0.9
1
Controlled Variable: biomass
biom
ass
C(g
/litre
)
0 100 200 300 400 500 6000
2
4
6 x 10-3
gluc
ose
s1(g
/litre
)
0 100 200 300 400 500 6000.01
0.02
0.03
0.04
time (hrs)
M.V
. - D
(1/h
our)
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Control Actions: HIGH State
� Same as previous case
� Controlled var.: c and s1
� Controller takes the reactor to the intermediate state
� We use monotonicity of s1 advantageously
0 100 200 300 400 500 6000.8
1
1.2
Controlled variables: biomass and glucose
biom
ass
C(g
/litre
)
0 100 200 300 400 500 6000
0.005
0.01
0.015
gluc
ose
s1(g
/litre
)
0 100 200 300 400 500 6000.03
0.035
0.04
0.045
time (hrs)
M.V
. - D
(1/h
our)
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Conclusions
� MPC algorithms were used to drive biological reactors from one state to another “multiple” state
� Successive linear approximation based controller was applied effectively to the cybernetic models displaying multiplicity features in biological systems
� Necessary to look at the entire state matrix rather than just the biomass or nutrient conc.
� Non-differentiability of the cybernetic v variable was overcome by using relevant approximations in different metabolic states