Robust Nonlinear Model Predictive Control using Volterra
Models and the Structured Singular Value ()
Rosendo Díaz-Mendoza and Hector Budman
ADCHEM 2009
July 12–15 2009
Background and Motivation
Background and Motivation
► 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
Introduction
Model Predictive Control
Model Predictive Control MPC
J  min Y
wrt U


y1sp k   yˆ1 k 






ynspy k   yˆ n y k  



Y

 y sp k  p   yˆ k  p  
1
 1




 y sp k  p   yˆ k  p 
ny
 ny

Parameters:
► p, prediction horizon
► m, control horizon
► p≥m
► ny, number of outputs
► nu, number of inputs
Diaz-Mendoza R. and Budman H
 u1 k  





 unu k  



U

 u1 k  m  





u k  m 
 nu

A model is required to
calculate ŷ
Robust NMPC using Volterra Models and the SSV
Introduction
Volterra Models
Volterra Models
Why Volterra Models?
►Represent a wide variety of nonlinear behavior
►Model structure: nominal model + uncertain model
M 1
u1 k0  n  
h 
x ,1
h 
x ,nu
yˆ x k0   Mn 01
n 0
n
n
M 1 M 1
 h
u k0  i u1 k0  j   
i , j  x ,1  1
i  0 j i
M 1 M 1
unu k0  n     hi , j x ,nu  unu k0  i unu k0  j 
i  0 j i
►M,
system memory
►nu, number of inputs
►x є [1,,ny]; ny, number of outputs
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
Volterra Models
CA
Comparison Real Process vs Nominal Model
0.4
CA+CB
Real process
Nominal Model
0.3
CSTR
0.2
A→B
cooling
fluid
Control Variable
cooling
fluid
0.1
0
-0.1
-0.2
 x2 
dx1

  x1  Da 1  x1  exp 
1 
dt
 1  x2  

dx2
x2
  x2  B Da 1  x1  exp 
1
dt
 1  x2 
Diaz-Mendoza R. and Budman H
-0.3
-0.4
0

  x2  xc 

10
20
30
40
50
Sampling Instant
►Truncation
error (M = 3)
►High order dynamics
Robust NMPC using Volterra Models and the SSV
60
Introduction
Volterra Models
Volterra Models
Identification
►Multilevel
►Nominal
pseudo random binary sequence (PRBS)
value = mean (parameters)
►Uncertainty
= 2  (parameters)
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.
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
Introduction
Volterra Models
Volterra Models
Output equation with parameter uncertainty
SISO System
M 1
M 1 M 1
n 0
i  0 j i
yˆ k0    hn  hn  u k0  n     hi , j  hi , j  u k0  i  u k0  j 
►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
Comparison Real Process vs Nominal Model
Comparison Real Process vs Nominal Model
0.4
0.4
Real process
Nominal Model
0.3
0.3
Model + 2
Model - 2
0.2
Control Variable
Control Variable
0.2
0.1
0
-0.1
0.1
0
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
Real process
Nominal Model
0
10
20
30
Sampling Instant
Diaz-Mendoza R. and Budman H
40
50
60
-0.4
0
10
20
30
40
Sampling Instant
Robust NMPC using Volterra Models and the SSV
50
60
Introduction
Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
J  min Y

wrt U
Worst Case Interpretation
6
wc(3) 
4
SISO System
wc(2) 
wc(1) 


y k   yˆ k 


Y


 y sp k  p   yˆ k  p 


sp
Manipulated Variable
2
0
 wc(5)
-2
-4
wc(4) 
-6
0
1
2
3
4
Sampling Instant
How to consider parameter
uncertainty?
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
5
6
Introduction
Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
Worst Case Interpretation for the
whole prediction horizon
6
wc( Y ) 
Manipulated Variable
5
 y1sp k   yˆ1 k 

max 

wrt hn ,hi , j 
 ynspy k   yˆ n y k 

4
3
2


  max Y
 wrt hn ,hi , j


1
0
0
1
2
3
4
5
6
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
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)
Worst Case Interpretation
6
wc(3) 
5
 wc(4)
Manipulated Variable
4
wc(2) 
3
wc(1) 
2
 wc(5)
1
0
0
1
2
3
Sampling Instant
4
5
6
 M 1

  h  h  u k  n  

n
n
 n 0

 M 1 M 1

max    hi , j  hi , j  u k  i  u k  j   
wrt hn ,hi , j
 i  0 j i

 wk  





d
k


Doyle, J., (1982). Analysis of feedback systems with structured uncertainties, IEE Proceedings D Control
Theory & Applications, 129 (6), 242–250
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  (SSV)
SSV Theorem
max
 y1sp k   yˆ1 k  


  max Y
max 

wrt hn ,hi , j 
 wrt hn ,hi , j
sp
 yn y k   yˆ n y k  


wrt hn ,hi , j

Y   kssv   M  kssv
Skew  problem (convex)
max
wrt hn ,hi , j
Y


max
wrt k ssv
k ssv
st   M  k ssv
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.
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
Introduction
Nonlinear Model Predictive Control
Nonlinear Model Predictive Control

max
wrt k ssv
st   M  k ssv
k ssv
M
M, interconnection matrix
Δ, uncertainty block structure
 M11 M12 
M

M 21 M 22 
M12  f k ssv , U 
M 21  f k ssv , Y 
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
Introduction
Nonlinear Model Predictive Control
Interconnection Matrix Example
k ssv
0
k ssv
0
k ssvu 1
0
2
k ssvu 1
0
0
0
0
0
0
0
0
0
k ssv
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
c1 0 ic 1
0
h1
0 c2  0 ic 2  0
Feedback
0
0
k ssv
0
0
0
k ssv
0
0
0
h2
0
0
0
k ssv
0
0
0
0
0
h1,1
0
0
0
0
0
0
k ssv
0
k ssv
0
0
h2, 2
0
k ssv
0
0
0
0
0
0
0
0
h1
Nominal
Diaz-Mendoza R. and Budman H
0
0
0
0
k ssv
0
0
0
0
0
h1,1
0
0
0
0
0
0
0
0
k ssv
0
h1, 2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
h1 0
0 h1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
h1,1
0
hy1 h1
0 h1,1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
hy1 h1,1 h2
0
0
0
0
0
0
0
0
0
0
h2, 2
0
0
0
0
0
0
0
0
0
0
h1, 2
Uncertain
Robust NMPC using Volterra Models and the SSV
0
k ssv
0
k ssv
0
k ssvu 1
0
2
k ssvu 1
k ssvu 2 
2
k ssvu 2 
k ssvu 1u 2 
0
Introduction
Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
NMPC Cost Function
max
wrt hn ,hi , j
 Y 
 u 


ulimits 


 tc 

max
wrt k ssv
st   M  k ssv

Diaz-Mendoza R. and Budman H
k ssv
 M11 M12 
M

M
M
22 
 21
M12  f k ssv , U 
M 21  f k ssv , Y, u, ulimits , tc
Robust NMPC using Volterra Models and the SSV
Introduction
Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
Additional terms
Manipulated variables movement penalization


W u k   u k  1


u  


Wmu u k  m   u k  m  1


u
1
u


max W1 u k   u k  1  k ssv





max W u u k  m   u k  m  1  k 
m
ssv 

Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
Introduction
Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
Additional terms
Manipulated variables constraints
ulimits
u k 


k
 ssv u



k
bound







u
k

m
k

ssv

ubound k  m  








u
k
  k ssv 
 max  k ssv

max  u k    ubound k 
ubound k  

 














u
k

m
  k ssv  max  u k  m    ubound k  m 
max  k ssv

ubound k  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 terms
Terminal Condition
Terminal Condition Representation
1.4
1.2
1
Control Variable
0.8
0.6
0.4

0.2
0

-0.2
-0.4
1
2
3
4
5
6
7
8
9
10
Sampling Instant

max 


M
M M
 sp

 y k    hi  hi u k  m    hi , j  hi , j u 2 k  m   wk   d k 


i 1
i 1 j i


Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV




Introduction
Nonlinear Model Predictive Control
Nonlinear Model Predictive Control
NMPC Cost Function
max
wrt hn ,hi , j
 Y 
 u 


ulimits 


 tc 

max
wrt k ssv
st   M  k ssv

k ssv
 M11 M12 
M

M
M
22 
 21
M12  f k ssv , U 
M 21  f k ssv , Y, u, ulimits , tc
NMPC Algorithm at each sampling instant



min  max
wrt U wrt hn ,hi , j



Diaz-Mendoza R. and Budman H






u 
 max k ssv 

min
 wrt U 

ulimits 
wrt k ssv


 st   M  k ssv 
tc  
Y
Robust NMPC using Volterra Models and the SSV
Case Studies
SISO
Case Study SISO System
CSTR with first order exothermic reaction
Control Specifications
CA
CA+CB
cooling
fluid
CSTR
A→B
►
►
cooling
fluid
►
 x2 
dx1

  x1  Da 1  x1  exp 
1 
dt
 1  x2  

dx2
x2
  x2  B Da 1  x1  exp 
1
dt
 1  x2 

  x2  xc 

CV: x1 (dimensionless reactant
concentration)
MV: xc (cooling jacket dimensionless 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.
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
Case Studies
SISO
Disturbance Characteristics
Disturbance Value
 dimensionless cooling rate
0.32
0.3

0.28
0.26
0.24
0.22
0.2
0
5
10
15
20
25
Sampling Instant
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
Case Studies
SISO
Control Variable
p=10 , m=2
0.05
0
Control Variable
-0.05
-0.1
-0.15
-0.2

Wi u = 2.5
-0.25

Wi u = 1.5

Wi u = 2.0

Wi u = 1.0
-0.3
0
5
10
15
20
Manipulated Variable
p=10 , m=2
25
0.7
Sampling Instant
Manipulated Variable
0.6
0.5
0.4
0.3
0.2

Wi u = 2.5
0.1

Wi u = 1.5

Wi u = 2.0

Wi u = 1.0
0
0
5
10
15
Sampling Instant
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
20
25
Case Studies
SISO
Controlled Variable
Case with constraints, p=10 , m=2 , W u = 0.5
i
0.05
Controlled Variable
0
-0.05
-0.1
-0.15
-0.2
-0.25
Manipulated Variable
Case with constraints, p=10 , m=2 , W u = 0.5
0
5
10
15
20
25
30
35
40
45
i
50
0.25
Sampling Instant
Constraint set at 0.2

Manipulated Variable
0.2
0.15
0.1
0.05
0
0
5
10
15
20
25
30
35
Sampling Instant
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
40
45
50
Case Studies
SISO
Control Variable
p=10 , m=2 , Wu = 2.5
i
0.05
0
Control Variable
-0.05
Sum absolute error Robust = 1.46
Sum absolute error Non-Robust = 1.55
6% improvement
-0.1
-0.15
-0.2
-0.25
-0.3
Robust Controller
Non-Robust Controller
0
5
10
15
20
Manipulated Variable
p=10 , m=2 , Wu = 2.5
i
25
0.7
Sampling Instant
Manipulated Variable
0.6
0.5
0.4
0.3
0.2
0.1
0
Robust Controller
Non-Robust Controller
0
5
10
15
Sampling Instant
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
20
25
Case Studies
WiΔu
SISO
1.5
Robust Controller is better
than Non-Robust
37%
1
41%
0.75
50%
0.50
66%
25 different disturbances for each weight
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
Case Studies
MIMO
Case Study MIMO System
D
Sf
►
X
S
P
Fermenter
►
►
►
►
X, biomass concentration
S, substrate concentration
P, product concentration
D, dilution rate
Sf, feed substrate concentration
dX
  dX   k X
dt
dS
1
 D S f  S  
k X
dt
YX / S
Control Specifications
► CV: X and P
► MV: D and Sf
► YX/S: process disturbance
dP
  DP  k   X
dt
 m 1  P / Pm S
k 
K m  S  S 2 / Ki
Parameter calculation
► Multilevel PRBS
► Parameter uncertainty
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
Manipulated Variable (Dilution Rate)
u
u
p=10 , m=2 , W = 0.2 , W = 0.5
1
2
0.14
0.12
0.1
Manipulated Variable
0.08
Control Variables
p=10 , m=2 , Wu = 0.2 , Wu = 0.5
1
2
0.06
Control Variables
0.04
0.06
0.04
0.02
0
0.02
-0.02
0
-0.04
-0.06
-0.02
0
5
10
15
20
25
20
25
Sampling Instant
-0.04
Subustrate Feed Concentration
u
u
p=10 , m=2 , W = 0.2 , W = 0.5
1
-0.06
2
0.4
-0.08
0.35
-0.1
-0.12
-0.14
0
5
10
15
Sampling Instant
20
25
Manipulated Variable
0.3
Biomass
Product
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
Sampling Instant
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV
Preliminary Conclusions
Preliminary Conclusions
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
Challenges
Current challenges
►
Computational demand
►
Multivariable control
Diaz-Mendoza R. and Budman H
Robust NMPC using Volterra Models and the SSV