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
Hierarchy of Process Control
Regulatory Control
Advanced Process Control
Real-TimeOptimizer
PlanningScheduling
StrategicDecisions
seconds
min ~ hours
hours ~ day
week ~ month
quarterly ~ annual
Time-scale
$$$
Example: Shell Heavy Oil Fractionator
� Hierarchical control strategy� PID loops for level &
inventory control� Quality control loops� Minimise energy usage
� Several operating constraints
What is Model Predictive Control?
Exemplary MPC Algorithm
Plant
input output
disturbances
observerstate / information
objectivemax productivity∑
model( )duxfx ,,=�
constraints
maxmin xxx ≤≤
controller
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
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)
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)
Design Effort
Almost all models used in MPC are typically empirical models “identified” through plant tests rather than first-principles models.
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.
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.
Receding Horizon Control Concept
k k+1 k+m-1 k+p-1
Past Future
setpoint
input
output
Analogy to Chess
hgfedcba
1
2
3
4
5
6
7
8
Me
Opponent (Plant)
Model Predictive Control
Opponent’s Move
New Piece Position
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
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
?
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 ?
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
Step Response Model
Assumptions:• No immediate effect: S0 = 0• Stable process settles after time n: for 1j nS S j n= ≥ +
1. Dynamic Model
Step Response Model:Superposition Principle
0( ) ( )
k
iv k v i
=
= ∆∑
1. Dynamic Model
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
Extension to Multivariate Systems
1. Dynamic Model
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”
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”
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=
= ∆ − + −∑
( )
( )( )
( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
− ky
kyky
kY
n 1
1
0
~#
( ) ( )( ) ( ) 01
with==+∆=∆
+="kuku
ikykyi
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
Multi-Step Prediction
Effect of future input moves (to be determined)
x(k)
2. Multi-step Prediction
2. Multi-Step Prediction( )( )
( )=
⎥⎥⎥⎥⎥
⎦
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⎢⎢⎢⎢⎢
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#21 ( )
( )
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#2
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( ) ( ) ( )100
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0
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−+∆
⎥⎥⎥⎥⎥
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⎣
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+++∆
⎥⎥⎥⎥⎥
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⎤
⎢⎢⎢⎢⎢
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⎡
+∆
⎥⎥⎥⎥⎥
⎦
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⎢⎢⎢⎢⎢
⎣
⎡
+−−
mku
S
ku
S
Sku
S
SS
mppp
#"
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( ) ( ) "##
++∆
⎥⎥⎥⎥⎥
⎦
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⎢⎢⎢⎢⎢
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⎡
+∆
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−
1
0
1
12
1
kd
S
Skd
S
SS
dp
d
dp
d
d ( )( )
( )⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+
++
kpke
kkekke
#21
Effect of past inputs
Effect of current and future inputs
Effect of measured disturbances Unmeasured disturbances
+
+
+
2. Multi-step Prediction
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)
( )( )
( )( )
( )( )
( )
( ) ( ) ( )( )kykykd
S
SS
mku
kuku
SSS
SSS
kY
kpky
kkykky
dp
d
d
u
mppp
02
1
11
12
1
~
1
11
1
1000
~21
−
⎥⎥⎥⎥
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3. Objective Function and Constraints
Constraint violation may not be acceptable
Shaded region is the setpoint tracking error
ySP
yMax
( ) ( )
( )[ ] ( )[ ]
( )[ ] ( )[ ] ⎪⎪⎭
⎪⎪⎬
⎫
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⎪⎪⎨
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0
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1,,min
m
l
p
iSPSP
mkukulkuRTlku
kikyyQTkikyy
"
3. Objective function and constraints
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
Constraints
� Constraints include� Input magnitude constraints:
� Input rate constraints:
� Output magnitude constraints:
maxmin )( uikuu ≤+≤
max)( uiku ∆≤+∆
maxmin )( yikyy ≤+≤
3. Constraints
Constraint Formulations
Hard constraint
past future
Soft constraint
past future
quadratic penalty
past future
Setpoint approximation of soft constraint
quadratic penalty
3. Constraints
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
4. Optimization: Quadratic Program
3&4. Objective Function for QP
3&4. Constraint Formulation for QP
3&4. Constraint Formulation in QP
3&4. Constraint Formulation in QP
3&4. Constraint Formulation in QP
Receding Horizon Control Concept
k k+1 k+m-1 k+p-1
Past Future
setpoint
input
output
5. State Update Scheme
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−
−
)()(
)()(
1
2
1
0
kyky
kyky
n
n
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⎥⎥⎥⎥⎥
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)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
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.
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
Example 1: Blending System Control
� Control rA and rB� Control blend flow q� Constraints on MV
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 ≤≤
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
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
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
Popularity of MPCSystematic and Integrated Solution
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
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
Popularity of MPCEmerging Popularity of On-line Optimization
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
Example: Model Predictive Control of Mammalian Cell Cultures
Niket S. Kaisare, Jay H. LeeA. A. Namjoshi, D. Ramkrishna
Presented at AIChE Meeting, November 2000
Overview of the Talk
� Introduction� Hybridoma cell reactor
� Cybernetic Modeling� Case Study
� Simpler microbial cell system� Full hybridoma cell system
� Conclusions
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
Hybridoma Cell Reactor
Air (O2)
• Glucose (s1f)• Glutamine (s2f)• Amino acids
• Unconsumed substrates• Biomass w/ antibodies• Waste metabolites
(Lactate, Alanine etc.)
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)
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
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
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
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
^
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
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)
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)
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
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
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)
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
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
•
•
•
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
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
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
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)
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)
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
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