Modelling and Control of
Nonlinear Processes
Jianying (Meg) Gao and Hector Budman
Department of Chemical Engineering
University of Waterloo
1
Outline
Motivation

Modelling
•Volterra
series
•State-affine

PI
•RS&RP

Nonlinear process examples

Two major difficulties: modelling and control!
Empirical Modelling

Control
•G-S
Motivation


Volterra series, state-affine
Robust Control

Robust Stability (RS) and Robust Performance (RP)

Proportional-Integral (PI) control

Gain-scheduling PI (G-S PI)
Results and Conclusions

Continuous stirred tank reactor (CSTR)

Future application
2
Nonlinear Process Example 1
Motivation
Modelling
series
•State-affine

Fed-batch Bioreactor Mass Balance

Linear process: constant
dS
X
 D( S i  S ) 
dt
YX / S
•Volterra
Control
•G-S
PI
•RS&RP

Substrate
Si

Nonlinear process: Monod
Input
 ( S )   max
, X , S
S
Ks  S
Output
dS
S
X
 D( S i  S )   max
dt
K s  S YX / S
3
Nonlinear Process Example 2
Motivation

Continuous Stirred Tank Reactor: CSTR
Modelling
A
•Volterra
series
•State-affine
Control
•G-S
PI
•RS&RP
A  B
Heat
1st order
Q f , C f ,T f
A&B
Cooling
u  Tc
Input



Mass Balance:
Q f , C, T
V , ,T
Output
dC
V
 Q f (C f  C )  VkC
dt
k
Linear process: constant
Nonlinear process: Arrhenius
k (T )  e
E
RT
4
Nonlinear Control
Motivation

1st Difficulty: simple & accurate model
Modelling

•Volterra

series
•State-affine
Control
•G-S
PI
•RS&RP

Accurate: the model gives a good data fit
Simple: the model structure is simple to apply for
control purpose
2nd Difficulty: model is never perfect!



Uncertainty: model/plant mismatch
Controllers are desired to be ROBUST to model
uncertainty!
Robust control: takes into account uncertainty!
nonlinear
model
Linear
model
uncertainty
5
Empirical Modelling
Motivation
Type
First principles model
Empirical model
How?
Mass, energy balance
Input/output data
Choose?
Difficult, complex
Easy
Modelling
•Volterra
series
•State-affine
Control
•G-S
PI
•RS&RP
1g/l
model
y set e
-y
controller
u
process
(inlet
concentration)
v
y
measurement
1g/l
soft sensor
6
1st Order Volterra Series
Motivation
Modelling
series
•State-affine


No priori knowledge of the process is required!
Black box model between input and output
•Volterra
Control
•G-S
PI
•RS&RP
n
y (t )   hi u (t  i )
i 1
 h1u(t  1)  h2 u(t  2)  
S  y(t )
u (t  2)
Si  u
at different time
at current time
t-2 t-1 t
t
7
1st Order Volterra Series
Motivation

Tc
Modelling
•Volterra
series
•State-affine
Control
Impulse response
A
Q , C ,T
f
f
Cooling Tc
Input
0
1
•G-S
PI
•RS&RP
t
f
A&B
Q f , C, T
V , ,T
Output
C
t
u  [1,0,0,0]

1
n
y  [0, h1 , h2 ,hn ,0]
1st order Volterra kernels: h1 , h2 , hn
8
1st Order Volterra Series
C (1)
Tc (1)
Motivation
Modelling
•Volterra
series
•State-affine
=
Control
•Robust
control
•Gain-scheduling
•PI
•MPC
+
+
1
Tc ( 2)
2
1
C ( 2)
2
1
Tc (1)  Tc (2)
2
Cooling
Temperature
=
n
y (t )   hi u (t  i )
i 1
1
C (1)  C (2)
2
Reactor
concentration
9
Volterra Series Model
Motivation
Modelling


•Volterra
series
•State-affine
Control
•G-S
PI
•RS&RP
No priori knowledge of the process is required!
Black box model between input and output

More terms, better data fit
n
n
n
y (t )   hi u (t  i)  hij u (t  i )u (t  j )  
i 1
i 1 j 1
 h1u(t  1)  h2 u(t  2)  
 h11u(t  1)u(t  1)  h12u(t  1)u(t  2)  

2nd order Volterra kernels: h11 , h12 , hnn
10
Identify Volterra Kernels
Motivation

Modelling
Identification of Volterra kernels
u (t  i)  ui
•Volterra
series
•State-affine
y(t )  h1u1  h2 u 2  h11u1u1  h12u1u 2
Control
•G-S
PI
•RS&RP
w1  u1 , w2  u 2 , w3  u1u1 , w4  u1u 2
y(t )  h1 w1  h2 w2  h11w3  h12 w4

Linear least squares
11
Volterra Series Model
Motivation

Advantages
Modelling

•Volterra

series
•State-affine

Control
•G-S
PI
•RS&RP

From input/output data Tc  C
Straightforward generalization of the linear system
description
Linear least squares algorithm: hi , hij
Disadvantages

The output depends on past inputs raised to different
powers and in different product combinations, e.g.
u(t  1) 2 , u(t  2)u(t  3), u(t  2) 3

Not suitable for robust control approach
Linear
model
uncertainty
12
State-affine Model
Motivation
Modelling

State-affine Model (Sontag, 1978)

•Volterra
series
•State-affine

Control

•G-S

PI
•RS&RP
I/O
Data
State-affine system, i.e. systems that are affine in the
state variables but are nonlinear with respect to the
inputs
It can cover a wide range of nonlinear processes
Identified from Volterra series model kernels
Suitable for robust control
Least
squares
Volterra
series model
Intermediate
step
Sontag’s
algorithm
State-affine
model
13
State-affine Model
Motivation

Model structure
x(t  1)  F(u )x(t )  G (u )u (t )
Modelling
series
•State-affine
y (t )  h 0 x(t )
•Volterra

Where
Control
F(u )  F0  F1u (t )  F2 u (t ) 2  
G (u )  G 1  G 2 u (t )  G 3u (t ) 2  
•G-S
PI
•RS&RP

Identification of matrix coefficients Fi , G i , h 0


Iterative matrix manipulation of Volterra kernels
Sontag (1978), Budman and Knapp (2000,2001)
14
State-affine Model
Motivation

x(t  1)  (F0  F1u (t )) x(t )  (G 1  G 2 u (t ) 2 )u (t )
Modelling
•Volterra
series
•State-affine
Control
PI
•RS&RP
A simple example
y(t )  h 0 x(t )

How to treat the nonlinearity as Uncertainty?
•G-S
Nominal point
Nonlinearity
x(t  1)  F0 x(t )  G 1u (t )  F1u (t )x(t )  G 2 u (t ) 2 u (t )
y(t )  h 0 x(t )
 1,t
Linear
model
uncertainty
 2 ,t
15
Nonlinearity
Motivation

Modelling
Uncertainty is function of input: Key advantage!
 i ,t  u(t )i , 1,t  u(t ),  2,t  u(t ) 2
•Volterra
series
•State-affine
Control
Uncertainty

Uncertainty bounds
Tc  5 50( C )
•G-S
PI
•RS&RP
0
u(t )  5 50
 1,t  5 50
50
 1,t
5
t
16
Results 1 (modelling)
Motivation
State-affine model
True process output
Modelling
•Volterra
series
•State-affine
0.3
Control
0.1
•G-S
PI
•RS&RP
0.2
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
50
100
150
200
250
300
350
400
450
17
Results 1 (modelling)
Motivation
State-affine model
True process output
Modelling
•Volterra
series
•State-affine
0.3
Control
0.1
•G-S
PI
•RS&RP
0.2
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
50
100
150
200
250
300
350
400
450
18
Results 1 (modelling)
Motivation
Modelling
•Volterra
series
•State-affine
Control
•G-S
PI
•RS&RP

State-affine model for CSTR
x(t  1)  (F0  F1u (t )) x(t )  (G 1  G 2 u (t ))u (t )
y (t )  h 0 x(t )
 0.1188  0.0345
F0  


2
.
3416
0
.
0937


1
G1   
0 
0.1076 0
F1  

1
.
2289
0


0 
G2   
1
h 0  0.1755  0.0382
19
Conclusions 1 (modelling)
Motivation

A general modelling approach is proposed!
Modelling

•Volterra

series
•State-affine

For a Nonlinear Process:
Obtained an empirical model from I/O data!
No priori knowledge required! So it can be applied to
processes with unknown dynamics!
Control
•G-S
PI
•RS&RP

Nonlinearity is dealt with as uncertainty!

Methods for quantifying the model uncertainty
from experimental data are studied.
x(t  1)  (F0  F1u (t )) x(t )  (G 1  G 2 u (t ))u (t )
y (t )  h 0 x(t )
 1,t
 1,t
20
G-S PI Design
Motivation

PI controller:
Modelling

•Volterra
series
•State-affine

u (t )  K c [e(t ) 
Proportional gain: K  2
c
Integration time:   4 min
I
Control
e
•G-S
PI
•RS&RP
PI
1
0

t
1
I
t
 e(i)]
i 1
Slope=
u
Kc
I

2
4
Kc  2
0
t
Gain-Scheduling PI controller
K c  f (Tc )
 I  g (Tc )
u  Si  5 50( 0C )
21
Traditional Gain-Scheduling
A
Motivation
Modelling
•Volterra
series
•State-affine
Control
•G-S
PI
•RS&RP
Tc  5 50( C )
0
5
Q f , C f ,T f
A&B
Q f , C, T
Tc
V , ,T
50
5 10
20 25
Linear
Model 1
Kc  2
I  4
45 50
Linear
Model 2
?
switch
Kc  3
I  3
Output
Linear
Model 3
?
switch
Kc  4
 I  1.5
22
G-S PI Design
Motivation

Continuous G-S PI controller: state-space
1 t
u (t )  K c (u )[e(t ) 
e(i)]

 I (u ) i 1
Modelling
•Volterra
series
•State-affine
Math manipulation
Control
 (t  1)   (t )  e(t )
•G-S
PI
•RS&RP
u  Tc
e  Cset  C
u (t )  (C c  Wc u (t )) (t )  ( Dc  Wd u (t ))e(t )
Cc 

Kc
I
, Dc  K c 
Design parameters:
Kc
I
K c , I ,Wc ,Wd
23
Closed-loop System: APS
Motivation

Modelling
 η(t  1)  A(δ t ) B(δ t )   η(t) 
 e(t)   C(δ ) D(δ )  ν(t) 

 
t
t 

η(0)  η0
•Volterra
series
•State-affine
Control
PI
•RS&RP
Affine Parameter-dependent System: the
closed-loop
•G-S

Assumption1:Affine dependence on the
uncertain parameters
A(δ t )  A 0  A1δ1,t  A 2 δ2,t   A n δn,t
δ t  ( 1 ,  2 ,,  n )  R n
24
Uncertain Parameter
Motivation

Modelling
Assumption 2: Each uncertain parameter is
bounded   [ ,  ]
i ,t
•Volterra
series
•State-affine
i
i
100
Tc   1,t
Control
•G-S
PI
•RS&RP
5
1
t

Convexity: Parameter vector
δ t is valued in a hyper-rectangle
called the parameter box
2
W
4
3
W : {(1 ,  2 ,,  n ) : i  { i ,  i }}
25
Robust Stability
Motivation

Modelling
•Volterra
series
•State-affine
V
Lyapunov function
0
V 0

Energy

Stable position: zero energy

Path:
Control
•G-S
PI
•RS&RP
V 0
V 0
dV
0
dt
V 0
V 0
26
Robust Stability
Motivation

CSTR: avoid overheating, maintain target
Modelling
•Volterra
series
•State-affine
Max
S
Control
Min
•G-S
PI
•RS&RP
stable
1
W
4

2
3
unstable
General RS condition:
A( )T PA ( )  P  0, for all   W
27
Robust Performance
Motivation
Modelling
•Volterra
series
•State-affine

Disturbance Rejection; performance index


Smaller  , better performance
Larger  , worse performance
e

Control
•G-S
PI
•RS&RP
v
e
L2
  1
L2
 =Disturbance in A
A&B
Q f , C, T
1g/l
(Cset  C )
V , ,T
Output
28
Robust Performance
Motivation

Fed-batch bioreactor: product quality!
Modelling
•Volterra
series
•State-affine
Control
•G-S
PI
•RS&RP
Max
S
Min
Good

Bad
RP: Solve for  and controller parameters
 A( ) T PA ( )  P A( ) T PB CT 

T
T
2
T
B PA ( )
B PB   I D   0 for all   W


C
D
 I 

29
Robust Control Design
Motivation

Modelling

•Volterra
series
•State-affine
Control
Empirical model of the
nonlinear process

•G-S
PI
•RS&RP

State-affine model
Controller structure
K c , I

PI:

G-S PI:
K c , I ,Wc ,Wd
Closed-loop system

RS and RP conditions are
checked
linear
model
uncertainty
controller
closed-loop
RS ?
RP ?
30
Results 2 (Linear PI)
Motivation

Linear PI: RS and RP regions
Modelling
20
•Volterra
18
series
•State-affine
Control
•G-S
PI
•RS&RP
16
I
Wc  Wd  0
RS region
Good
14
12
RP region
10
8
6
4
2
0.5
1
1.5
2
2.5
3
3.5
Kc
31
Results 2 (G-S PI RS)
Motivation
Improve over linear PI Kc=2.42,taui=1.1545

Modelling
0.5
•Volterra
series
•State-affine
Good
Control
PI
•RS&RP
0
•G-S
Wc
-0.5
-1
-1
-0.5
0
Wd
0.5
1
1.5
32
Results 2 (G-S PI RP)
Motivation

Improve over linear PI Kc=2,taui=1.1545
Modelling
0.5
•Volterra
series
•State-affine
Good
Control
PI
•RS&RP
0
•G-S
Wc
-0.5
-1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Wd
33
Results 2 (PI)
Motivation

Simulation: G-S PI is much BETTER!
Modelling
•Volterra
series
•State-affine
Control
•G-S
PI
•RS&RP
1
Disturbance
0.5
0
-0.5
-1
0
2
4
6
8
10
12
14
16
18
20
1
Linear PI
0.5
0
G-S PI:better
-0.5
-1
2
4
6
8
10
12
14
16
18
20
34
Conclusions 2 (Control)
Motivation

Design and simulation results
Modelling
PI
•Volterra
series
•State-affine
Linear
Control
•G-S
PI
•RS&RP
G-S


I
Wc
Wd
2
1.15
0
0
0.96
0.38
1.37
2.95
-0.004
0.001
0.39
0.22
 optimal  simulation
is a good performance index


Kc
Consistence between analysis and simulation
A general robust design approach is proposed!


Based on empirical model from I/O data!
G-S PI! Much better performance for wide
operation range!
35
Conclusions
Motivation

Two difficulties are solved efficiently!
Modelling
•Volterra
series
•State-affine


Modelling: state-affine
Control: robust control
Control
•G-S
PI
•RS&RP

Our contributions!



Quantify uncertainty from I/O data!
Develop global RP conditions!
Propose Continuous G-S PI structure!
36
Application
Motivation

Modelling
Empirical Modelling

•Volterra
series
•State-affine
Control
•G-S
PI
•RS&RP

Models:
nonlinear
biochemical processes
chemical
and
Robust Control Design


Nonlinear processes when nonlinearity is
treated as uncertainty!
Uncertain processes with real and timevarying uncertainty!
37
Motivation
Modelling
•Volterra
series
•State-affine
Control
•G-S
PI
•RS&RP
38
Unconstrained MPC
K+m
p: prediction horizon;m: control horizon
k: sampling point, same as t
Unconstrained MPC
Quadratic Design Objective
2
1
2
ˆ
min { Γ[Y(k  1 / k )  R (k  1)]  ΛU }
[1]
u ( k ) 2
ˆ (k  1 / k )  M Y
ˆ (k )  W(k  1 / k )  S u U(k )
s.t. Y
p
p
Solution
0
0
 Γε ( k  1 / k ) 
ΓS u 
[2]

 U(k )  

0


 Λ 
ε(t  1 / k )  R(k  1 / k )  M p Y(k )  W(k  1 / k )
Unconstrained MPC
Least Squares Solution
x  (A T A) 1 A Tb
Ax  b
MPC Solution: Best Sequence of m Control Moves
uT
p
1
uT
p
U(k )  (S Γ ΓS  Λ Λ) S Γ Γε(k  1 / k )
T
u
p
T
T
Present Control Move is Implemented
u (k )  K MPCε(k  1 / k )
K MPC  1 0  01m (S Γ ΓS  Λ Λ) S ΓT Γ
uT
p
T
u
p
T
1
uT
p
MPC Design Parameters
No general guide on design parameters!
Control horizon: m
Small: a robust controller that is relatively insensitive to model
errors
Large: computational effort increases; excessive control action
Prediction horizon: p
Large: more conservative control action which has a stabilizing
effect but also increase the computational effort
weighting matrix for outputs :
Usually set
Γ  const
Γ
State-space MPC
Weighting matrix for inputs:
Λ  I
More important than other parameters
Small
Large
Λ
Λ
more aggressive control, less stable
less aggressive control, more stable
State-space MPC (Zanovello and Budman,1999)
U(k )  E2  T1K mpcN 2  U(k  1)

 u(k )   C


K
N
y
(
k
)
mpc 2  

  u1

Closed-loop System: APS
Robust G-S MPC
Operation Range
Step 1
1(u=-1)
Step 2
1   1 0
Step 3
1 {1,0}
Step 4
2(u=0)
1
u  [1,1]
2
State
Affine 1&2
State
Affine 2&3
MPC 1-2
MPC 2-3
Switching
3(u=1)
1  0 1
1 {0,1}
RS
&
RP
Outline
Traditional G-S Design
Design procedure and disadvantages
Robust G-S Design
Affine parameter-dependent systems (APS)
RS and its LMI formulation
RP and its LMI formulation
Robust G-S MPC design
State-affine model and uncertainty quantification
State-space formulation of MPC
Results and Conclusions
Case study: nonlinear CSTR
Nonlinear CSTR
Pure A
Q f , C f ,T f
Mix of A and B
V , ,T
u(t)=Tc
1st –order exothermal reaction
Q f , C ,T
Results (1)
Optimization Design Results: Table 1

# of ranges
u  [1,1]
k 5
k4
k 3
k2
k 1

[0.2033,0.7817,0.9618,1.4984,1.0681]
[0.1909,0.7518,1.0746,1.6372]
0.5044
0.4440
[0.3521,0.2013,1.2408]
0.4729
[0.1121,1.7121]
0.4210
[1.1650]
0.4461
Evenly Separated Ranges
Conclusions (1)
Efficient Robust G-S MPC Design
Simulation test with disturbances, e.g. IMA, and etc.
Global G-S MPC designed with guaranteed RS and RP
Analysis is the worst case which covers all the simulations
Observations of the Robust G-S MPC Design
Performance index close to each other
Even separations may not capture the process nonlinearity
Conservatism of the design
Robust G-S MPC Performance depends on
# of separations
Separation point locations: evenly or not
Nonlinear dynamics
Results (2)
Comparison of two controllers: Analysis & Simulation
G-S MPC 5-1:
G-S MPC 5-2:
Λ
Λ
designed based on optimization
chosen randomly
Table 2
optmization
Yes
No
 ( k  5 , u  [1,1] )
[0.2033,0.7817,0.9618,1.4984,1.0681]
[10,10,10,10,10]

 simulation
0.5044
0.7230
0.2115
0.3144
Results (2)
CSTR Simulation: Figure 1
G-S MPC output and disturbance
1
0.8
Disturbance
0.6
0.4
No optimization
0.2
0
-0.2
Optmization
-0.4
-0.6
-0.8
-1
0
5
10
15
20
25
30
35
40
45
50
Conclusions (2)

is a Good performance index
Consistence between analysis and simulation (Table 2)
Robust G-S MPC Design
Global G-S MPC designed with guaranteed RS and RP
Analysis is the worst case
Future Directions
Separation of operation range, # of separations
Reducing conservatism of the design
Process with more nonlinearity
Traditional G-S Design
A typical G-S design procedure for nonlinear plants
Step1: select n operating points which cover the range of the
plant’s dynamics
Step 2: Linearize a first principle model or identify linear models
around each operating point
Step 3: Design local linear controller for each local linear model
Step 4: In between operating points, the gains of the local
controllers are interpolated, or scheduled, resulting in a global
controller.