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Robust Nonlinear Model Predictive Control using Volterra Models and the Structured Singular Value ( ). Rosendo D í az-Mendoza and Hector Budman ADCHEM 2009 July 12–15 2009. Background and Motivation. Background and Motivation. Chemical processes are nonlinear - PowerPoint PPT Presentation
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Robust Nonlinear Model Predictive Control using Volterra Models and the Structured Singular Value ()
Rosendo Díaz-Mendoza and Hector Budman
ADCHEM 2009July 12–15 2009
► Chemical processes are nonlinear
► Nonlinear Model Predictive Control (NMPC)► First principles or empirical models► Robustness issues
► Robustness of NMPC► Simulation studies for different parameter
values
► Develop a Robust-NMPC methodology that considers parameter uncertainty
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Background and Motivation
Background and Motivation
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A model is required to calculate ŷ
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Model Predictive Control
Model Predictive Control MPC
Parameters:► p, prediction horizon► m, control horizon► p ≥ m► ny, number of outputs► nu, number of inputs
Volterra Models
Why Volterra Models?►Represent a wide variety of nonlinear behavior►Model structure: nominal model + uncertain model
1
0
1
0
1
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1
0
1
0
1
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,,
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M
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Schetzen, M., The Volterra and Wiener theories of nonlinear systems; Robert E. Krieger, 1989
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Volterra Models
CSTRA→B
ca
a
xxxxxDBx
dtdx
xxxDx
dtdx
212
212
2
12
211
1
1exp1
1exp1
CA
CA+CB
coolingfluid
coolingfluid
►Truncation error (M = 3)►High order dynamics
0 10 20 30 40 50 60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4Comparison Real Process vs Nominal Model
Sampling Instant
Cont
rol V
aria
ble
Real processNominal Model
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Volterra Models
Volterra Models
Nowak, R. D., and Van Veen, B. D. (1994). Random and pseudorandom inputs for Volterra filter identification, IEEE Transactions on Signal Processing, 42 (8), 2124–2135.
►Multilevel pseudo random binary sequence (PRBS)
►Nominal value = mean (parameters)
►Uncertainty = 2 (parameters)
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Volterra Models
Volterra ModelsIdentification
Output equation with parameter uncertaintySISO System
1
0
1
0
1
00,,00ˆM
n
M
i
M
ijjijinn jkuikuhhnkuhhky
►hn, hi,j, nominal value►hn, hi,j, parameter uncertainty
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Volterra Models
Volterra Models
0 10 20 30 40 50 60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4Comparison Real Process vs Nominal Model
Sampling Instant
Con
trol V
aria
ble
Real processNominal ModelModel + 2
Model - 2
0 10 20 30 40 50 60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4Comparison Real Process vs Nominal Model
Sampling Instant
Con
trol V
aria
ble
Real processNominal Model
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Volterra Models
Volterra Models
Nonlinear Model Predictive Control
pkypky
kyky
ˆ
ˆ
sp
sp
Y
U
Ywrt
min
J
SISO System
How to consider parameter uncertainty?
0 1 2 3 4 5 6-6
-4
-2
0
2
4
6
Sampling Instant
Man
ipul
ated
Var
iabl
e
Worst Case Interpretation
wc(1) wc(2)
wc(3)
wc(4)
wc(5)
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
0 1 2 3 4 5 60
1
2
3
4
5
6
Sampling Instant
Man
ipul
ated
Var
iabl
e
Worst Case Interpretation for thewhole prediction horizon
wc( Y )
Y
jin
yy
jin hh
nn
hhkyky
kyky
,, ,wrtsp
1sp1
,wrtmax
ˆ
ˆmax
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction
Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
Structured Singular Value ()
Calculation of the worst ŷ(k) to ŷ(k+p) when parameter uncertainty is taken in consideration, i. e., for ŷ(k)
kdkw
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nkuhhM
i
M
ijjiji
M
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hh jin
1
0
1
,,
1
0
,wrt ,
max
0 1 2 3 4 5 60
1
2
3
4
5
6
Sampling Instant
Man
ipul
ated
Var
iabl
e
Worst Case Interpretation
wc(1)
wc(2)
wc(3)
wc(4)
wc(5)
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
Doyle, J., (1982). Analysis of feedback systems with structured uncertainties, IEE Proceedings D Control Theory & Applications, 129 (6), 242–250
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jin
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jin hh
nn
hhkyky
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,, ,wrtsp
1sp1
,wrtmax
ˆ
ˆmax
Structured Singular Value (SSV)
SSV Theorem
ssvssvhhkk
jin
MY
,,wrtmax
ssv
k
khhk
ssv
ssvjin
M
Y
st
wrt ,wrt maxmax
,
Skew problem (convex)
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
Braatz, R. D., Young, P. M., Doyle, J. C., and Morari, M. (1994). Computational complexity of calculation, IEEE Transactions on Automatic Control, 39 (5), 1000–10002.
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UMMMMM
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k
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wrt max M
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
2,12,221,11,1112,11,112,22
1,111,11
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0
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
Interconnection Matrix Example
Nominal UncertainFeedback
NMPC Cost Function
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k
khhk
tcuu
ssv
ssvjin
M
Y
st
wrt limits
,wrt maxmax
, tcuukfkf
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,,,,,
limits21
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2221
1211
YMUM
MMMM
M
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
Additional termsManipulated variables movement penalization
1
11
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Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
mkumku
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max
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Additional termsManipulated variables constraints
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u
ssv
ssv
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Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
Additional termsTerminal Condition
1 2 3 4 5 6 7 8 9 10-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Terminal Condition Representation
Sampling Instant
Cont
rol V
aria
ble
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
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i
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1
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NMPC Cost Function
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st
wrt limits
,wrt maxmax
, tcuukfkf
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limits21
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MMMM
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ssv
ssvjin
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Y
st
wrt wrt limits
,wrt wrt maxminmaxmin
,
NMPC Algorithm at each sampling instant
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Introduction Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
CSTRA→B
ca
a
xxxxxDBx
dtdx
xxxDx
dtdx
212
212
2
12
211
1
1exp1
1exp1
Control Specifications
► CV: x1 (dimensionless reactant concentration)
► MV: xc (cooling jacket di-mensionless temperature)
► β: process disturbance
Parameter Calculation► Multilevel PRBS► Parameter uncertainty
Doyle III, F. J., Packard, A., and Morari, M. (1989). Robust controller design of a nonlinear CSTR, Chemical Engineering Science, 44 (9), 1929–1947.
CA
CA+CB
coolingfluid
coolingfluid
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Case Studies SISO
Case Study SISO System
CSTR with first order exothermic reaction
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Case Studies SISO
Disturbance Characteristics
0 5 10 15 20 250.2
0.22
0.24
0.26
0.28
0.3
0.32
Sampling Instant
Disturbance Value dimensionless cooling rate
0 5 10 15 20 25-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Control Variablep=10 , m=2
Sampling Instant
Con
trol V
aria
ble
Wiu = 2.5
Wiu = 2.0
Wiu = 1.5
Wiu = 1.0
0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Manipulated Variablep=10 , m=2
Sampling Instant
Man
ipul
ated
Var
iabl
e
Wiu = 2.5
Wiu = 2.0
Wiu = 1.5
Wiu = 1.0
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Case Studies SISO
0 5 10 15 20 25 30 35 40 45 50-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Controlled VariableCase with constraints, p=10 , m=2 , W
iu = 0.5
Sampling Instant
Con
trolle
d V
aria
ble
0 5 10 15 20 25 30 35 40 45 500
0.05
0.1
0.15
0.2
0.25
Manipulated VariableCase with constraints, p=10 , m=2 , W
iu = 0.5
Sampling Instant
Man
ipul
ated
Var
iabl
e
Constraint set at 0.2
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Case Studies SISO
Sum absolute error Robust = 1.46Sum absolute error Non-Robust = 1.556% improvement
0 5 10 15 20 25-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Control Variablep=10 , m=2 , W
iu = 2.5
Sampling Instant
Con
trol V
aria
ble
Robust ControllerNon-Robust Controller
0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Manipulated Variablep=10 , m=2 , W
iu = 2.5
Sampling Instant
Man
ipul
ated
Var
iabl
e
Robust ControllerNon-Robust Controller
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Case Studies SISO
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Case Studies SISO
WiΔu Robust Controller is better
than Non-Robust1.5 37%
1 41%
0.75 50%
0.50 66%
25 different disturbances for each weight
im
mmk
k
kSX
f
k
KSSKSPP
XDPtP
XY
SSDtS
XdXtX
//1
dd
1dddd
2
/
► X, biomass concentration► S, substrate concentration► P, product concentration► D, dilution rate► Sf, feed substrate concentration
Control Specifications► CV: X and P► MV: D and Sf
► YX/S: process disturbance
Parameter calculation► Multilevel PRBS► Parameter uncertainty
FermenterXSP
DSf
Saha, P., Hu, Q., and Rangaiah, G., P. (1999). Multi-input multi-output control of a continuous fermenter using nonlinear model based controllers, Bioprocess Engineering, 21, 533–542.
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Case Studies MIMO
Case Study MIMO System
0 5 10 15 20 25-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Control Variablesp=10 , m=2 , W
1
u = 0.2 , W2
u = 0.5
Sampling Instant
Con
trol V
aria
bles
BiomassProduct
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Case Studies MIMO
0 5 10 15 20 25-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Manipulated Variable (Dilution Rate)p=10 , m=2 , W
1
u = 0.2 , W2
u = 0.5
Sampling Instant
Man
ipul
ated
Var
iabl
e
0 5 10 15 20 250
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Subustrate Feed Concentrationp=10 , m=2 , W
1
u = 0.2 , W2
u = 0.5
Sampling Instant
Man
ipul
ated
Var
iabl
e
Conclusions
► A Robust-NMPC algorithm was developed
► The algorithm considers all the features of previous NMPC formulations
► In average the robust controller results in better performance as the input weight is decreased
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Preliminary Conclusions Preliminary Conclusions
► Computational demand
► Multivariable control
Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV
Current challenges
Preliminary Conclusions Challenges