From Multi-Parametric
Programming Theory to
MPC-on-a-chip Multi-scale
Systems Applications
Stratos Pistikopoulos
FOCAPO 2012 / CPC VIII
Acknowledgements

Funding
 EPSRC - GR/T02560/01, EP/E047017, EP/E054285/1
 EU - MOBILE, PRISM, PROMATCH, DIAMANTE, HY2SEPS
 CPSE Industrial Consortium, KAUST
 Air

Products
People




J. Acevedo, V. Dua, V. Sakizlis, P. Dua, N. Bozinis, N. Faisca
Kostas Kouramas, Christos Panos, Luis Dominguez, Anna Vöelker,
Harish Khajuria, Pedro Rivotti, Alexandra Krieger, Romain Lambert,
Eleni Pefani, Matina Zavitsanou, Martina Wittmann-Hoghlbein
John Perkins, Manfred Morari, Frank Doyle, Berc Rustem, Michael
Georgiadis
Imperial & ParOS R&D Teams
Outline

Key concepts & historical overview

Recent developments in multi-parametric
programming and mp-MPC

MPC-on-a-chip applications

Concluding remarks & future outlook
Outline

Key concepts & historical overview

Recent developments in multi-parametric
programming and mp-MPC

MPC-on-a-chip applications

Concluding remarks & future outlook
What is On-line Optimization?
MODEL/OPTIMIZER
Control
Actions
Data Measurements
SYSTEM
What is Multi-parametric Programming?

Given:



a performance criterion to minimize/maximize
a vector of constraints
a vector of parameters
z ( x)  min f (u, x)
u
s.t. g (u, x)  0
xR
n
u R
s
What is Multi-parametric Programming?

Given:




u
a performance criterion to minimize/maximize
s.t. g (u, x)  0
a vector of constraints
a vector of parameters
x Rn
Obtain:

z ( x)  min f (u, x)
u R
s
the performance criterion and the optimization
variables as a function of the parameters
 the regions in the space of parameters where these
functions remain valid
Multi-parametric programming
(1) Optimal look-up function
z ( x)  min f (u, x)
u
s.t. g (u, x)  0
x Rn
u R
(2) Critical Regions
s
u (x)
Obtain optimal solution u(x) as a
function of the parameters x
Multi-parametric programming
Problem Formulation
min
 3 u1  8 u2 
u1 ,u2
st .
1
5

 8

 4
1
 1
 4   u1   0





22  u 2   0


 1
0
 10  x1  10
0
  13  0
0   x1    20  0


 



 1 x 2   121 0


  
0
 8   0
 100  x 2  100
Multi-parametric programming
Critical Regions
4 Feasible Region Fragments
100
CR001
CR002
CR003
CR004
80
60
40
x2x2
20
0
-20
-40
-60
-80
-100
-10
-8
-6
-4
-2
0
x1
x1
2
4
6
8
10
Multi-parametric programming















U  















Multi-parametric Solution
 1  0.031 
  6.71 
 1

 5 
0

  x1 



0
10
 1



  0.33 0  x 1 
  1.67 

 x 2 


 1.33 0   x    14.67  if  0
1

  2




 100 



1
 0

 100 

 1  0.115
  1 0.031

  1 0.045
 0.73  0.03  x 1 
5.5

0
0.26 0.03   x   7.5 if  1

  2


 0
1

1
 0

8.65 

 6.71 



  x1 
 7.5 

   10 
x
  2



 100 




 100 
 1  0.045
 1
0
if 
1

 0

  7.5
x


1

  5 

 


x 2 
100






 0 0  x 1 
0


1 0   x 
13

  2
 
0 0.05  x 1 
11.8
0 0.06    x    9.8 

  2


  1 0.11 
  8.65
x


if  1
0    1    10 

 x 2 


0

1
100








Multi-parametric programming
min  3u1  8u2 
u
st .
1
1
 1 0 
  13  0
 5  4 u
0 0 x
  20  0

   1  
   1  
 
  8 22  u2   0  1  x2   121 0





  
  4  1
0 0
 8  0
 10  x1  10,  100  x2  100
4 Feasible Region Fragments
100
CR001
CR002
CR003
CR004
80
60
40
x2
20
0
-20
-40
-60
-80
-100
-10
-8
-6
-4
-2
0
x1
2
4
6
8
10















U  















  0.333
 1.333

 0.7333
0.26667

0  x1 
  1.6667 

   14.6667 
0
  x2 


 0.0333  x1 
5.5

  7.5
0.03333 
  x2 


if
if
0
1

0  x1 
0

  13
0
  x2 


if
0
0

0.05128   x1 
11.8462 

  9.80769 
0.0641 
  x2 


if
 1
 1

 1

 0

 0
 0.03125 
  6.71875 



0
5
  x1 



0
10




  x2 


1
100






1
100



 1
 1

 1

 1
 0

 0
 0.115385 
8.65385 
6.71875 
0.03125 



0.0454545   x1 
7.5


 



x
0
10
2








1
100



1

 100

 1
 1

 0

 0.0454545 
  7.5
   x1    5 
0

 


x
1
  2



 100 
 1
 1

 0

0.115385 
  8.65385 
   x1   

0
10

 


x
1
100
  2





Only 4 optimization problems solved!
On-line Optimization via off-line
Optimization
POP
PARAMETRIC PROFILE
OPTIMIZER
Control
Actions
System
State
SYSTEM
Control
Actions
System
State
SYSTEM
Function Evaluation!
Multi-parametric/Explicit Model
Predictive Control

Compute the optimal sequence of manipulated inputs which minimizes
tracking error = output – reference
subject to constraints on inputs and outputs

On-line re-planning: Receding Horizon Control
Multi-parametric/Explicit Model
Predictive Control

Compute the optimal sequence of manipulated inputs which minimizes
Solve a QP at each time interval

On-line re-planning: Receding Horizon Control
Multi-parametric Programming Approach
State variables  Parameters
 Control variables  Optimization variables

MPC  Multi-Parametric Programming
problem
 Control variables  F(State variables)

Multi-parametric Quadratic Program
Explicit Control Law
2
CR0
CR1
CR2
1.5
J ( x(t ))  min
ut |t , ut 1|t
 x
1
j 0
T
t  j |t
x t  j |t  0.01 ut2 j |t  xTt 2 |t P x t 2 |t
 0.7326  0.0861 
0.0609 
s.t x t  j 1|t  
 x t  j |t  0.0064  ut  j |t
0
.
1722
0
.
9909




 2  ut  j |t  2 j  1,2 x t |t  x(t )
1
0.5
x2
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
x1

 0.7059 0.7083 
0.2065







6
.
8355

6
.
8585
x
t
if
x
t


 0.7059  0.7083 
0.2065






ut   
2
if  0.7059  0.7083  x t    0.2065



0.7059 0.7083  xt    0.2065
2
if

0.5
1
1.5
2
Multi-parametric Controllers
(1) Optimal look-up function
Optimization Model
(2) Critical Regions
Parametric Controller
Measurements
Control Action
SYSTEM
System Outputs
Input Disturbances
 Explicit Control Law
MPC-on-a-chip!
 Eliminate expensive, on-line computations
 Valuable insights !
Key milestones-Historical Overview
AIChE J.,Perspective (2009)

Number of publications
Multi-Parametric
Programming
Multi-Parametric
MPC &
applications
Pre-1999
>100
0
Post-1999
~70
250+
2002 Automatica paper ~ 580 citations
 Multi-parametric programming – until 1992 mostly

analysis & linear models

Multi-parametric/explicit MPC – post-2002 much
wider attention
Patented Technology

Improved Process Control
European Patent No EP1399784, 2004

Process Control Using Co-ordinate Space
United States Patent No US7433743, 2008
Outline
Key concepts & historical overview
 Recent developments in multi-parametric
programming and mp-MPC

 Model
reduction/approximation
 mp-NLP & explicit nonlinear mp-MPC
 mp-MILP
 Robust explicit mp-MPC
 State estimation and mp-MPC
 Framework for mp-MPC
A framework for multi-parametric
programming & MPC (Pistikopoulos 2008, 2009)
Modelling/
Simulation
Identification/
Approximation
‘High-Fidelity’
Dynamic Model
System Identification
Model Reduction
Techniques
‘Approximate Model’
Model-Based Control
& Validation
Multi-Parametric
Programming (POP)
Extraction of
Parametric Controllers
u = u ( x (θ) )
Closed-Loop
Control System Validation
Model Reduction/Approximation
Replace discrete
dynamical
System with a set
of affine algebraic
models
N-step ahead
predictionenables use of
Linear MPC
routines
Model Reduction/Approximation


N-step-ahead approximation based on initial conditions
(measurements) and sequence of controls (constant
control vector parameterization). Set of affine algebraic
models
For all j point over the time horizon - approximations are
constructed as follows
y
t
mp-NLP Algorithms for Explicit NMPC
Strategy:
Direct Approach
Multi-parametric Nonlinear Dynamic Optimization Problem
Discretize state and controls via Orthogonal Collocation Techniques
Approximate
Multi-parametric Nonlinear Programming Problem (mp-NLP)
Quadratic Approximation Based
Solve sequence of mp-QP‟s
Nonlinear Sensitivity based
Solve sequence of NLP‟s
Partition state space recursively
mp-NLP Algorithms for Explicit NMPC
Key features: Two implementations for the characterization of the Parameter space
Quadratic Approximation based (General mp-NLP)
• Characterizes the parameter space by sub-partitioning CRs where the QA approximation
provides “poor” solutions.
NLP Sensitivity Based (NMPC mp-NLP)
• Characterizes the parameter space using NLP
sensitivity information and linearization of the
constraints.
v(x)
v0
(x) = 0
(x)
0
 (M 0 ) 1 N 0 + (x  x 0 )+(||x||)
Validity of approximation:
(x) = O(||x||)  (x)/||x|| → 0 as x → 0.
x0 v*
x0 v*
mp-NLP Algorithms for Explicit NMPC
Key Advantage: Fast implementation of the control laws
• State-of-the art multi-parametric solvers (e.g. mp-QP)
• Straightforward characterization of critical regions
• Complexity reduction through region merging
• Extension to address hybrid systems
Multiparametric Mixed-Integer Nonlinear Programming
Strategy: Decompose mp-MINLP into two sub-problems
Pre-processing
Step 1
Characterize feasible region
Simplicial
Approximation
Primal sub-problem (mp-NLP)
Approximate
via mp-QPs
x = f()
Step 2
y = y*
Master sub-problem (MINLP)
Iterate until master sub-problem is infeasible
mp-MILP
Explicit Solution of the general mpMILP Problem
Applications
• Pro-active Scheduling under price, demand and processing time
uncertainty (seee poster & paper)
• Explicit Model Predictive Control of Hybrid Systems:
Control actions as optimization variables, states as parameters,
input and model disturbances as parameters
• Integration of scheduling & MPC
Hybrid Approach - Two-Stage Method for
mp-MILP1
Stage 1 – Reformulation
Partially robust RIM-mp-MILP* model;
Solutions are immunized against all
immeasurable parameters and complicating
constraint matrix uncertainty
Stage 2 – Solution
Suitable multi-parametric programming
algorithms (e.g. Faisca et al. (2009))
Optimal partially robust solution; Upper bound
on optimal objective function value
*objective function coefficient and
1 Wittmann-Hohlbein,
Pistikopoulos (2011)
right hand side vector uncertainty
Global Optimization of mp-MILP1
Constraint matrix uncertainty poses major challenge
mp-MINLP
Multi-Parametric Global Optimization:
• Adaptation of strategies from the deterministic case to
multi-parametric framework: Parametric B&B procedure
• Globally optimal solution is a piecewise affine function over
polyhedral convex critical regions
Challenges in Global Optimization of mp-MILP Problems:
• Comparison of parametric profiles, not scalar values
• High computational requirements
Can we find “good solutions” of an mp-MILP problem with less effort?
1 Wittmann-Hohlbein, Pistikopoulos; JOGO, submitted , 2011
Robust Explicit mp-MPC

Famous control problem: Dynamic Systems with Model Uncertainties
(Mayne, Rawlings, Rao & Scokaert, 2000)
 N 1

V ( x )  min 
( x k Qx k  u k Ru k )  x N Px N 

U 
 k 0
x k 1  Ax k  Bu k + Wθk
Cx k  Du k  d

Mx N  
u min  u  u max
Parametric Uncertain System
 

 b
aij  aij : aij  aij , 0  aij , 0
bij
: bij  bij , 0  bij , 0
 min     max

x : systemstates
Uncertainty due to modelling, identification errors,
measurement errors etc.
u : control inputs
 u N 1 
ij


Exogenous Disturbance
x0  x
U  u0
 
A  aij  R nn , B  bij  R nm

Constraints represent safety, operational constraints

It is very critical that the system does not violate them

Immunize against uncertainty
33
Robust Explicit mp-MPC

Robustification – robust reformulation step (Ben-Tal &
Nemirovski, 2000; Floudas& Co-workers, 2004-2007)

Dynamic Programming framework to Robust MPC

Novel Multi-parametric Programming algorithm to
constrained Dynamic Programming (Faísca, Kouramas,
Saraiva, Rustem & Pistikopoulos, 2008)


Small mp-QP at each stage
No need for global optimization
MHE & mp-MPC
min xN MPC
x0 ,uk
2
PMPC

N MPC

k 0
2
xk Q
MPC

N MPC 1

k 0
uk
2
RMPC
s.t. xk 1  Axk  Buk  Gwk (actual system),
xk 1  Axk  Buk (nominal system),
xˆk 1  Axˆk  Buk  t (estimated system step 1.3),


u0  u 0*  K xˆ0*  x 0* , u  U
xk  X= X
K  S, xN MPC  X=f ,
S, k  1...N MPC  1, S  ExX S, xˆ0  x0  S,
S is mRPI of xˆk 1  xk 1   A  BK   xˆk  xk   t.
Main idea:
Step 1. Formulate the dynamics that govern the estimation error eT  f  eT 1,wT 1 
Step 2. Use these dynamics to find the set that bounds the estimation error eT  S
Step 3. Incorporate the bounding set into the controller to „robustify‟ against the
estimation error
Moving Horizon Estimation (MHE)
min
xˆT  N |T ,WˆT  N |T
s.t.
xˆT  N |T  xT  N |T
2
P
xˆk 1  Axˆk  Buk  Gwˆ k ,
1

YTTN1
2
 O xˆT  N |T  cbU TT N2 1
W
yˆ k  Cxˆk  vˆk ,
xˆk  X,
T 1


k T  N
wˆ k  W,
2
wˆ k Q 1

T

k T  N
vˆk  W
xT  N |T  Axˆ *T  N 1|T 1  BuT  N 1|T 1  Gwˆ T*  N 1|T 1 (smoothed update of arrival cost)

Model-based state estimator

Obtains current state estimate xT
Main advantage: incorporates system constraints
MHE is dual to MPC: backwards MPC


vˆk
2
1
R
A framework for multi-parametric
programming & MPC (Pistikopoulos 2008, 2009)
Modelling/
Simulation
Identification/
Approximation
‘High-Fidelity’
Dynamic Model
System Identification
Model Reduction
Techniques
‘Approximate Model’
Model-Based Control
& Validation
Multi-Parametric
Programming (POP)
Extraction of
Parametric Controllers
u = u ( x (θ) )
Closed-Loop
Control System Validation
A framework for multi-parametric
programming and MPC (Pistikopoulos 2010)
On-line Embedded
Control:
Off-line Robust
Explicit Control
Design:
Modelling/
Simulation
Identification/
Approximation
EMBEDDED
CONTROLLER
REAL SYSTEM
‘High-Fidelity’
Dynamic Model
Model Reduction
Techniques
System
Identification
‘Approximate
Model’
Model-Based
Control &
Validation
Multi-Parametric
Programming
(POP)
Extraction of
Parametric
Controllers
u = u ( x(θ) )
Closed-Loop
Control System
Validation
Outline
Key concepts & historical overview
 Recent developments in multi-parametric
programming and mp-MPC
 MPC-on-a-chip applications

 PSA
system
 Fuel Cell system
 Biomedical systems
PSA system and the cycle
Pure product
REPRES
DEP 1
DEP 2
DEP 3
Impurities
Feed
BED 1
Off gas
FEED
DEP 1
BED 2
PE 2
REPRES
BED 3
Bd
Pu
PE 1
PE 2
BED 4
DEP 1
DEP 2
DEP 3
Bd
DEP 2
DEP 3
FEED
Time
Bd
Pu
PE 1
PE 2
DEP 1
DEP 2
DEP 3
Bd
Pu
PE 1
DEP 1
DEP 2
DEP 3
REPRES
Pu
PE 1
FEED
PE 2
REPRES
REPRES
FEED
A framework for multi-parametric
programming and mp-MPC for PSA
‘High Fidelity’ PSA
Model (PDAE)
Modeling &
Simulation
System
Identification
‘Approximate’ Model
Multi-Parametric
Programming
Model Based Control &
Validation
MATLAB
POP Toolbox
Extraction of explicit MPC
controllers u = u(x(θ))
In-silico closed
loop controller
validation
Modelling - internal Bed
Mass balance
Species Accumulation
Mass transfer with adsorbent

2
Ci UCi
 Qi
Ci

(  (1   ) )

  p (1   )
 D
b
b p t
b
b Zi Z 2
Z
t
Bulk fluid convection



Dispersion in axial direction
Radial effects neglected
Transport properties independent of state variables
Axial mass dispersion (Wakao and Funazkri, 1978), velocity dependent
neglected
Energy balance
Energy accumulation in gas phase
NCOMP
(  (1   ) )
b
b p
NCOMP
U
C
i 1
pi
Ci
C C
i 1
vi
i
T
T
NCOMP
 (1   ) 
b p
t
 (  (1   ) ) RT
b
b p
Z
Energy convection




Energy accumulation in solid phase
Energy accumulation in adsorbed phase
NCOMP

i 1

i 1
Cvi
 T
T
Qi
 (1   ) C p 

p
b
s
t
t
NCOMP
C
i  (1   )
b  p i 1
t


2
Q

i (  H )   T
2
i
t
Z
Heat of adsorption
Heat dispersion
Lumped energy balance on gas and solid phase
Radial effects neglected
Specific heat, transport properties independent of state variables
Axial mass dispersion (Wakao et.al., 1978), velocity dependent
neglected
Momentum balance & adsorption
characteristics
1.751     C MW
2
i
i
150  1   
i

1


U
UU
3 2
3
Z
 dp
 dp
NCOMP
P
Ergun‟s equation, steady
state pressure drop


 Qi
 *

 K LDFi  Qi  Qi 
t


ai
*


*
NCOMP Q
Qi
i 

max  ai K i Ci RT 1  
Qi
i  1 Q max 
i


  H i 
K i  K i exp 

 RT 
LDF Rate expression
Nitta et.al. (1984), Ribeiro et.al. (2008), multisite
Langmuir adsorption isotherm (multi-component
mixture)
Valve Equation (for boundary
conditions)
Chou and Huang (1994), Nilchan and Pantelides (1998)
2

P

P
CV 1   High Low 



P


U 
 PHigh
Otherwise
CV
P

Pcritical
 2
 
1 




1

if
Cp
Cv
P = PHigh if gas leaving the bed
= PLow if gas entering the bed
PLow
PHigh

 Pcritical 





•Prictical constant since Cp
and Cv are assumed constant
•For REPRES and DEP Cp
and Cv calculated
at yH2 = 0.7, yCH4 = 0.3
•For blowdown and purge
(off gas) Cp and Cv calculated
at yH2 = 0.5, yCH4 = 0.5
Constraints - Boundary conditions
Z=0
Z=L
Z=L
UA  Q SLPM ( PFEED , TFEED ) P  PPRODUCT
Ci 
PYFeed
RT
T  TFeed
i
C i
Z
T
Z
Feed
Step
 0
 0
Z=0
Z=L

U  fValve PPURGE , PCODEP , CV
PURGE
C i
Z
Z=0
A boundary
condition for each
process step
 0

U 
Ci 
U i C i CODEP
Z=L
i C i
PC CODEP
i
RT i C CODEP
i
Purge
Step
Z=0
T
Z
 0
T  TCODEP
Base case system
99.99 % H2
Number of Beds
4
Activated
Carbon
Adsorbent
REPRES
DEP 1
Feed pressure
7 bars
Bed length
1m
Blowdown
pressure
1.01325
bars
Bed diameter
0.12 m
Bed Porosity
0.4
Feed
temperature
303.15 K
Feed
Composition
70 % H2,
30 %CH4
Feed flow rate
8.0 SLPM
DEP 2
DEP 3
BED 1
FEED
Feed
DEP 1
BED 2
PE 2
REPRES
BED 3
Bd
Pu
PE 1
PE 2
BED 4
DEP 1
DEP 2
DEP 3
Bd
DEP 2
DEP 3
FEED
Bd
Pu
PE 1
PE 2
DEP 1
DEP 2
DEP 3
Bd
Pu
PE 1
DEP 1
DEP 2
DEP 3
REPRES
Pu
PE 1
Off gas
FEED
PE 2
REPRES
t FEED  tDEP1  t DEP2  tDEP3  t Bd  tPu  t PE1  t PE2  t REPRES  
REPRES
FEED
Adsorption time
Objective and process variables

Changes in adsorption time effects purity the
most
 Adsorption time – Manipulated variable

Purity – Controlled variable

Fast tracking of H2 purity to the set point
99.99%

Regulate changes in adsorption time
 Avoid bed saturation
 Avoid high fluid inlet velocities as it causes
mechanical damage

Hard constraints on adsorption time has to be
satisfied for safe and economical operation
A framework for multi-parametric
programming and MPC
‘High Fidelity’ PSA
Model (PDAE)
Modeling &
Simulation
System
Identification
‘Approximate’ Model
Multi-Parametric
Programming
Model Based Control &
Validation
MATLAB
POP Toolbox
Extraction of explicit MPC
controllers u = u(x(θ))
In-silico closed
loop controller
validation
System Identification - Approximation

PDAE model not suitable for current model
based control approaches




Process model approximations are
needed
Input – Adsorption time
Output – H2 purity
Sampling time – 1 PSA cycle

Input signal design for system perturbation

Random pulse employed for persistent
excitation


Maximum amplitude decided by hit and
trial studies
Pulse duration (constant) calculation
based on closed loop response
System identification
Model fit to the input
output data above by an
8th order state space system
x k 1  Axk  Bu k
y k  Cx k
A framework for multi-parametric
programming and MPC
‘High Fidelity’ PSA
Model (PDAE)
Modeling &
Simulation
System
Identification
‘Approximate’ Model
Multi-Parametric
Programming
Model Based Control &
Validation
MATLAB
POP Toolbox
Extraction of explicit MPC
controllers u = u(x(θ))
In-silico closed
loop controller
validation
MPC Formulation for PSA

 
N 1

M 1
min Z   yk  y Q yk  y   uk Ruk
u
k 1
R '
k
s.t.
xk 1  Axk  Buk
yk  Cx k  ymismatch
ulow  uk  uhigh
yk  1

k 0
'


y = hydrogen purity at the end of
adsorption stage
u = adsorption time, sec

N = 4, M = 2 Q = 1

2 optimization variables u0, u1

Optimal R based on the closed
loop response
Constraints on u
Low u: low adsorption time/cycle time, fast PSA cycles
 More ON/OFFs of the switch valves per unit time



R
k
Extra wear and tear of manipulative variable hardware
Fast loading-unloading of adsorbent leading to its degradation
High u: high adsorption time/cycle time, long PSA cycles
 Risk of over saturation, or irreversible adsorption of adsorbent
mp-MPC for PSA control
EXPLICIT/MULTI-PARAMETRIC MPC CONTROLLER
(2) Optimal Look-up Function
(1) Critical Regions
Measurements
Control Action
System Outputs
Input Disturbances
MPC on a chip
 Explicit Control Law Eliminate expensive, on-line computations
 Valuable insights!
Explicit/Multi-Parametric MPC Design
Critical Regions from POP software
Solve the mp-optimization problem for all values of the parameters to obtain the
explicit control laws (u = D1x + u0) and the corresponding critical region
maps (D2x.≤ q).
A framework for multi-parametric
programming and MPC
‘High Fidelity’ PSA
Model (PDAE)
Modeling &
Simulation
System
Identification
‘Approximate’ Model
Multi-Parametric
Programming
Model Based Control &
Validation
MATLAB
POP Toolbox
Extraction of explicit MPC
controllers u = u(x(θ))
In-silico closed
loop controller
validation
MPC Vs PID
Step Disturbance in PSA feed rate – 10 % of Design
Controller
Response time
(Cycles)
Average ∆U
(Seconds)
Maximum ∆U (Seconds)
mp-MPC
13
0.74
1.8
PID
25
0.84
5.09
Impulse Disturbance in PSA feed rate – 35 % of Design
mp-MPC
7
0.75
1.6
PID
5
4.72
12.12
Open Loop
9
Impulse Disturbance in PSA feed rate – 54 % of Design
mp-MPC
7
1.77
4.18
PID1
4
17.11
32.29
PID2
5
9.44
21.16
Open Loop
10
MPC Vs PID
Outline
Key concepts & historical overview
 Recent developments in multi-parametric
programming and mp-MPC
 MPC-on-a-chip applications

 PSA
system
 Fuel Cell system
 Biomedical systems
PEM Fuel Cell System
PI
MassFlow
H2
PI
N2
TE
MassFlow
Electronic
Load
PT
TE
VENT
Hydrator
PI
TE
PT
TE
PT
A
VENT
PDT
K
Air
MassFlow
TE
TE
Water
PT
TE
H2O
PT
TE
M
Hydrator
Filter
Radiator
PEM Fuel Cell System
Develop 1kW PEM fuel cell system
Collect data for the PEM fuel cell, fan, hydrogen storage
Design controller for the integrated system
Tamb
Ist
u: mair,Vfan, mcool
d: Tamb,Ist
y: Tst ,λO2
θ: xt , Tamb,Ist , Tst ,Tst,sp
mair
Vfan
mcool
PEM Fuel
Cell System
Tst
λO2
PEM Fuel Cell System - Controller Design

Optimized PID Controller

Nominal MPC Controller
N 1
M 1
min J   ( y k  y k ) QRk ( y k  y k )  ( y N  y N ) P( y N  y N )   (u k  u R ) T Rk (u k  u R )
x , y ,u
R T
R
k 0
xt 1  Axt  But
yt 1  Cxt
Robust MPC Controller

R
k 1
Subject to:

R
Include in the controller design the model error
u: mair,Vfan, mcool
d: Tamb,Ist
y: Tst ,λO2
θ: xt , Tamb,Ist , Tst ,Tst,sp
PEM fuel cell system
 Dynamic model
 Ideal and uniformly distributed gases
 The fuel and the oxidant are humidified
 No liquid can go into the membrane because it is waterproof
 Uniform temperature in the fuel cell stack
 Simplified mathematical models for humidifier, radiator and pump
Controller evaluation (closed-loop simulation)
Incorporate
controller into high fidelity model and perform computational
studies
yt
yt
ut+1
ut+1
Incorporate
MATLAB
+
POP Software
controller into the PEM Fuel Cell System - perform experiments
Tamb
Ist
mair
mcool
PEM Fuel
Cell System
Tst
λO2
PEM Fuel Cell System
Unit Specifications
 Fuel Cell : 1.2kW
 Anode Flow : 5..10 lt/min
 Cathode Flow : 8..16 lt/min
 Operating Temperature : 65 – 75 °C
 Ambient Pressure
Control Strategy
Start-up Operation
Heat-up Stage : Control of coolant loop
Nominal Operation
Control Variables :
 Mass Flow Rate of Hydrogen & Air
 Humidity via Hydrators temperature
 Cooling system via pump regulation
 Known Disturbance : Current
PEM Fuel
Cell System
mH2
mAir
mcool
TYHydrators
Vfan
Tst
HTst
(1) Optimal look-up
function
(2) Critical Regions
Outline
Key concepts & historical overview
 Recent developments in multi-parametric
programming and mp-MPC
 MPC-on-a-chip applications

 PSA
system
 Fuel Cell system
 Biomedical systems
ERC MOBILE
Development of models and model based control and optimisation algorithms
for biomedical systems
Anaesthesia
Provide hypnosis, analgesia and muscle relaxation while
maintaining the vital functions
Multiple input multiple output model predictive control
Type 1 diabetes
Maintain blood glucose concentration within the normal range
by optimising insulin delivery
Model predictive control problem
Acute Myeloid Leukaemia
Provide optimal chemotherapy dose to minimise the cancer cells
While keeping normal cells above a minimum level
Scheduling Problem
Muscle Relaxation
Hyper
Normal
Hypo
Framework towards optimal drug delivery systems
Model Development
Model Reduction
Individual Patient
High Fidelity Model
Model Predictive Control/ Optimisation
Measurement
device/ State
Estimation
mp-MPC
Optimal Scheduling
16.5
16
Optimal control
law/ trajectory
x2
15.5
15
14.5
MEASUREMENTS/
STATES
14
13.5
-37
-36
-35
-34
Cancer
cells
-33
x1
mp-MPC on a Chip
Normal
cells
1st Cycle
Patient
OPTIMAL DRUG DELIVERY/DOSAGE
Disturbances
Optimal dose
drug 1
drug 2
2nd Cycle
3rd Cycle
4th Cycle
Set points
Individual constraints
Diabetes Type I
Mathematical Modelling
240
220
Glucose
Profile
200
glucose(mg/dl)
Model Development
Individual Patient
High Fidelity Model
180
160
140
120
100
80
35g
100g
35g
40g
90g
50g
50g
80g
20g
60
40
0
day2
day1
10
20
30
day3
40
time(hr)
50
60
70
Anaesthesia
Pharmacokinetics
Anaesthetic
concentrations
Lung
Brain
Heart
Gut
Liver
Spleen
Kidney
Pancreas
Adipose
Skin
Pharmacodynamics
Emax
Effect
Skeleton
Artery
Vein
Muscle
Efficacy
Individual
variability
E50
Potency
E0
Cell Cycle
C50
Leukaemia
Concentration
Cell population profiles
ERC MOBILE
Step 1: The sensor measures
the glucose concentration from
the patient
Step 2: The sensor then inputs
the data to the controller which
analyses it and implements the
algorithm
2
Sensor
Controller
1
Patient
Insulin Pump
3
Step 3: After analyzing the
data the controller then signals
the pump to carry out the
required action
4
Step 4: The Insulin Pump
delivers the required dose to
the patient intravenously
Outline

Key concepts & historical overview

Recent developments in multi-parametric
programming and mp-MPC

MPC-on-a-chip applications

Concluding remarks & future outlook
MPC-on-a-chip technology –
Reflections
(10 years since 2002 Automatica paper appeared .. )

Scientific/academic impact ?

Application/industrial impact ?
MPC-on-a-chip technology –
Reflections
(10 years since 2002 Automatica paper appeared .. )

Scientific/academic impact ? HIGH – many
un-resolved issues ..

Application/industrial impact ? Limited –
not panacea to all MPC solutions ..
MPC-on-a-chip – Perspectives

Application types for Multi-parametric
Programming & MPC
 Type
1 - Large scale and expensive industrial
processes with slow/medium dynamics
 Type
2 - Medium scale and cost industrial
processes with medium/fast dynamics
 Type
3 - Small scale and inexpensive
processes/equipment with medium/fast dynamics
MPC-on-a-chip – Perspectives

Type 1 – Large scale and expensive
industrial processes with slow/medium
dynamics
MPC-on-a-chip – Perspectives

Type 1 - Large scale and expensive
industrial processes with slow/medium
dynamics
 Control
hardware/software availability
 MPC implementation mainly via online
optimization
 Explicit MPC can play a role for low level
process control
 Hybrid (on-line + off-line) approach possible –
accelerate on-line dynamic optimization step
MPC-on-a-chip – Perspectives

Type 2 – medium scale and cost
industrial processes with medium/fast
dynamics
Reboiler/condenser
Product - GAN
Waste
LIN
Column
Air
HEX
MPC-on-a-chip – Perspectives

Type 2 – medium scale and cost
industrial processes with medium/fast
dynamics
 Limited
Control hardware/software availability
 Online optimization/MPC usually prohibitive
 Multi-parametric MPC ideal – proved in
previous applications (Air Separation,
Automotive)
MPC-on-a-chip – Perspectives

Type 3 – small scale and inexpensive
processes/equipment with medium/fast
dynamics
Patient
MPC-on-a-Chip
Mechanical
Pump
Glucose
Sensor
MPC-on-a-chip – Perspectives

Type 3 – small scale and inexpensive
processes/equipment with medium/fast
dynamics
 Available
control hardware/software limited not suitable for online MPC
 Multi-parametric MPC technology ideal/
essential
 MPC-on-a-Chip part of embedded (all-in-one)
system
 Suitable for new technologies (FPGA, wireless)
From Multi-Parametric
Programming Theory to
MPC-on-a-chip Multi-scale
Systems Applications
Stratos Pistikopoulos
FOCAPO 2012 / CPC VIII