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Linearizability ofChemical Reactors
By
M. Guay
Department of Chemical and Materials Engineering
University of Alberta
Edmonton, Alberta, Canada
Work Supported by NSERC
Introduction
Feedback Linearization has formed the basis for most engineering applications of nonlinear control techniques
Basic Techniques - Static State Feedback Linearization
1) Hunt, Su and Meyer Lie Algebraic approach
2) Gardner and Shadwick Exterior calculus approach GS Algorithm
Application to Chemical Reactors
– Static state-feedback linearizability of chemical reactors has been exploited in a number of studies (Hoo and Kantor, Henson and Seborg, Chung and Kravaris, etc…)
– DFL observed by Rouchon and Rothfuss et al.
)(+=))(,( tBvAzztuxfx
Outline
Motivation
Background
Pfaffian Systems and Feedback Linearization
Conditions for Dynamic Feedback Linearization
Linearizability of Non-isothermal Chemical Reactors
– Reactors with 2 chemical species
– Reactors with 3 chemical species
Conclusions
Motivation
Linearizability of nonlinear control systems
– CONTROLLER DESIGN
– TRAJECTORY GENERATION
LinearSystem
NonlinearController
LinearController
NonlinearSystem
Motivation
Exterior Calculus Setting
– Provides systematic framework for the study of feedback equivalence (Cartan)
– Leads to general solution of linearization problem (beyond Lie algebraic and Diff. Algebraic approaches)
– Ease of symbolic computation
– Unified treatment of ODE, DAE (implicit) and PDE systems
Background
Let M be a n-dimensional manifold
– TpM is the tangent space to M at a point p with basis
– Tp*M is cotangent space to M at a point p with
basis
– Elements of Tp*M, called one-forms, are linear
maps
e en1
, ,
de den1 , ,
: Tp M R
Background
Associated with differential forms is an algebra called the Exterior Algebra,
– Defined by the (anti-commutative) exterior product
e.g. product of two one-forms
gives a (degree) two-form.
– Addition of forms of same degree
v w w v
Pfaffian Systems
Let be a submodule of M)
is called a Pfaffian system defined locally as
where is a set of one-forms.
defines an exterior differential system I
i ii
i C M( )
1 , , n
I d , Id ea l G en e ra ted b y , d
Pfaffian Systems
Important structure associated with a Pfaffian system is its derived flag
Definition 1:
The derived flag of a Pfaffian system , I, is a filtering resulting in a sequence of Pfaffian system such that
The system I(i) is called the ith derived system of I defined by
The number k for which
is called the derived length of I.
I I ( ) I (k). 1
I I d Ii i i( ) ( ) ( )m o d . 1 10
I Ik k( ) ( ) 1
Control Systems
A control affine nonlinear system is given by
where
S defines a Pfaffian system on the manifold with local
coordinates (x, u, t) generated by
The integral curves c(s) in M* of the control system are the solutions of
where is the velocity vector tangent to c(s).
( ) ( ) ( )x f x g x u t
x M R u Rn p , .
(S)
M M R Rp*
dx f x g x u d t dx f x g x u d tn n n
n
1 1 1
1
( ( ) ( ) ) , , ( ( ) ( ) )
, , .
( ( )) ( ) ,c s c s 0
( )c s
Feedback Linearization
Definition 2:
A control system is said to be feedback linearizable if there exist a static state feedback (x)+x)u and a coordinate transformation x) that transforms the nonlinear to a linear controllable one.
Using the derived flag of , linearizability by static state feedback is stated as
Theorem 1 (Gardner and Shadwick)
A control system S is static state feedback linearizable if and only if
1. The kth derived system is trivial
2. is generated by one-forms
that satisfy the congruences
ji
ii p j, ( , , , , , ) 1 1
d d t du
d d t
i
ii
ji
ji j
m o d
m o d ( )
1
Dynamic Feedback Linearization
Definition 3
A control system S is said to be feedback linearizable by dynamic state feedback if there exists a precompensator
with and a coordinate transformation x) such that the combined system {S,P} is equivalent to a linear controllable form.
Dynamic feedback linearizability implies that the combined system is generated by one-forms that fulfill Theorem 1
( , ) ( , )
( , ) ( , )
a x b x
u c x d x
R x u uq ( , , , )( )
y
y p pp
1 11( )
( )
Dynamic Precompensators
Precompensation can be achieved from differentiation of the process inputs, u, or of a static state feedback transformation of them, .
The degree of precompensation is summarized by
( , ) ,
( )
( ) ( )
( )
i i
i i
i ii
x u i p
1
1 2
1
1 , , . p ii q w ith
Dynamic Precompensators
General Form
Precompensator Structure
(i) Structure of precompensator determined by indices and
(ii) Alternatively, with multiplicities
( , , , , , , , )
, .
( )
( ) ( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
u u
u u
u u i p
x u u u u
j p
i i
i i
i i
p p
j j
j j
j j
i i
p
j
1
1 2
1
1 11 1
1
1 2
1
1
1
fo r
fo r
1 , , p ii q w ith 1 p .
k k m1 , , s sm1 , , .
Dynamic Feedback Linearization
Linearization problem is summarized by
General problem reduces to special interconnection of nonlinear systems with precompensators of appropriate dimensions subject to DAE constraints
SS PP
DifferentialAlgebraic
Constraints
Feedback LinearizableSystem {S,P}
Conditions for DFL
Definition 3:
Consider the control system S and a precompensator P based on the feedback v= (x,u) and indices with multiplicities
The first derived system, (1), associated with P is given by the set of forms, which satisfy
The second derived system associated with P is defined as the set of forms, which satisfy
or
By induction
k k m1 , , s sm1 , , .
d dv dv k kp s ( ) ( )m o d , , , ,2 11 2 2 10 1
1 if
d dv dv p s ( ) m o d , , , .110
1
d dv dv k kp s s ( ) ( )m o d , , , .2 11 2 10 1
1 2 if
d dv
dv
j k k i m
k k j k k j
p s s
i i
i i
i
( ) ( )m o d , , ,
, .
1 1
1
0
1 1 1
11
1
fo r
Conditions for DFL
Lemma 1:
For control system S and a precompensator P defined by the indices with multiplicities dynamic feedback linearization requires that
Lemma 2:
If a control system S is DFL with precompensator P, there exists p integers i such that
defined by
k k m1 , , s sm1 , , ,
n k k s k k sm m m m ( ) ( ) .1 1 1 1 0
ii n
i m
i m m m m m
i q q j jj qm
j qm
i q q j jj qm
j qm
i
k i s
k k k s i s s
k k k s i s
k k k s i s
k k k p s i p
1 1
1 1
1 1
1
1
1 1
1 1
1 2 21
2 1 1
,
( ) ,
( ) ,
( ) ,
( ) ,
Conditions for DFL
Theorem 2
A control system S is dynamic feedback linearizable by dynamic extension of a state feedback transformation v= (x,u) if and only if
i) P belongs to the set stated by Lemma 1
ii) the bottom derived system associated with P is trivial
iii) there exists generators that fulfill the congruences
where for
with
d d t
dv dv
d d t
q q
m j m j
jq
jq
jq
jp
jp
11
1
11
1
m o d , , ,( )
mij j d im ( / )( ) ( ) 1
1 11 j k k qm
q p s sl 1
k k j k k l m
k k j k k k l m
l m
m m m
1 1 1
1 1
1 1
1 1
w h en
an d
w h en .
Conditions for DFL
Some Comments on Theorem 2:
It provides a generalization of GS algorithm and can be used to compute linearizing outputs
For more general precompensators, extend original inputs to generate required derivatives u() to compute DAE constraints
and apply the theorem with precompensator
DAE constraints are not know a priori but theorem gives explicit equations (PDEs) for the required expressions
( , , , , , , , )( ) ( )x u u u up p
p1 1
1 11
, .
( )
( ) ( )
( )
j j
j j
j jj j p
1
1 2
1
fo r
Chemical Reactors
Consider Non-isothermal CSTRs
where
u1 Tank Volumetric Flowrate
u2 Jacket Volumetric Flowrate
cI Concentration of species I
cIin Inlet Concentration of species I
T Tank Temperature
Tin Tank Inlet Temperature
TJ Jacket Temperature
Tjin Jacket Inlet Temperature
u1, T, cA, cB, cc
u2, TJin
u2, TJ
u1, Tin, cAin, cBin, ccin
CoolingJacket
Chemical Reactors
Applying mass balances and energy balances assuming that
» constant hold-up
» incompressible flow, constant heat capacities and heat transfer coefficients
» negligible jacket heat transfer dynamics
general model form is obtained
where
rI Rate of production of species I
Enthalpy of Reaction
Reactor-side heat transfer coefficient
Jacket-side heat transfer coefficient
V Tank volume
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) / ( ) / ( ) / (
,
,
,
c r c c c T c c u V
c r c c c T c c u V
c r c c c T c c u V
T c c c T T T V T T u V
T T T V T T
A A A B C A IN A
B B A B C B IN B
C C A B C C IN C
A B C J IN
J J J Jin
1
1
1
1 J Ju V) /2
Linearizability
Case 1: Constant hold-up reactor with two chemical species
Result:
The chemical reactor model is dynamic feedback linearizable with precompensator
and linearizing outputs {cA, cB}.
Applying the precompensator
yields a dynamic feedback linearizable system with outputs
( , , ) ( ) /
( , , ) ( ) /
( , , ) ( ) / ( ) / ( ) / ( ) /
,
,
c r c c T c c u V
c r c c T c c u V
T c c T T T V T T u V
T T T V T T u V
A A A B A IN A
B B A B B IN B
A B J IN
J J J Jin J J
1
1
1
2
( )
( ) ( )
u u
u u
1 11
11
12
( )u u1 11
c c c c
c cTB A in A B in
B B in
,
Linearizability
Case 2: Two chemical species with variable hold-up
Result:
Applying the precompensator
yields a dynamic feedback linearizable system with outputs
( , , ) ( ) /
( , , ) ( ) /
( , , ) ( ) / ( ) / ( ) / ( ) /
,
,
c r c c T c c u V
c r c c T c c u V
T c c T T T V T T u V
T T T V T T u V
V u u
A A A B A IN A
B B A B B IN B
A B J IN
J J J Jin J J
1
1
1
2
1 0
( )u u1 11
c c c c
c c
V
cc cB A in A B in
B B in A inA A in
, .
Linearizability
Case 3: Two Chemical Species with heat transfer dynamics
Result:
The chemical reactor model is dynamic feedback linearizable with the precompensator
and linearizing outputs
( , , ) ( ) /
( , , ) ( ) /
( , , ) ( ) / ( ) / ( ) / ( ) / ( ) / ( ) /
,
,
c r c c T c c u V
c r c c T c c u V
T c c T T T V T T u V
T T T V T T V
T T T V T T u V
V u
A A A B A IN A
B B A B B IN B
A B W IN
W W W i J W W
J W J J Jin J J
1
1
1
0
2
1
u0
( )
( ) ( )
u u
u u
1 11
11
12
c c c c
c c
V
cc cB A in A B in
B B in B inB B in
, .
Linearizability
Case 4: Three chemical species and constant hold-up
Result:
The chemical reactor is dynamic feedback linearizable with precompensator
and linearizing outputs
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) / ( ) / ( ) / (
,
,
,
c r c c c T c c u V
c r c c c T c c u V
c r c c c T c c u V
T c c c T T T V T T u V
T T T V T T
A A A B C A IN A
B B A B C B IN B
C C A B C C IN C
A B C J IN
J J J Jin
1
1
1
1 J Ju V) /2
c c c c
c c
c c c c
c cB A in A B in
B B in
B C in C B in
B B in
, .
( )u u1 11
Linearizability
Case 5: Three chemical species, constant hold-up and heat transfer dynamics
Result:
The chemical reactor model is dynamic feedback linearizable with precompensator
and linearizing outputs
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) / ( ) / ( ) / (
,
,
,
c r c c c T c c u V
c r c c c T c c u V
c r c c c T c c u V
T c c c T T T V T T u V
T T T V T
A A A B C A IN A
B B A B C B IN B
C C A B C C IN C
A B C W IN
W W W i
1
1
1
1
0
J W W
J W J J Jin J J
T V
T T T V T T u V
) / ( ) / ( ) / 2
( )
( ) ( )
u u
u u
1 11
11
12
c c c c
c c
c c c c
c cA B in B A in
A A in
A C in C A in
A A in
, .
Linearizability
Case 6: Three chemical species with variable hold-up and heat-transfer dynamics
Result:
Cannot find a simple “linear” precompensator to linearize this process.
Consider design change
» switch control from u1 to u0
» let u1 = p(V)
» not endogenous feedback
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) / ( ) / ( ) / (
,
,
,
c r c c c T c c u V
c r c c c T c c u V
c r c c c T c c u V
T c c c T T T V T T u V
T T T V T
A A A B C A IN A
B B A B C B IN B
C C A B C C IN C
A B C W IN
W W W i
1
1
1
1
0
J W W
J W J J Jin J J
T V
T T T V T T u V
V u u
) / ( ) / ( ) /
2
1 0
Linearizability
Case 7: Three chemical species with design change
Result:
Applying the precompensator
yields a dynamic feedback linearizable system with outputs
( , , , ) ( ) ( ) /
( , , , ) ( ) ( ) /
( , , , ) ( ) ( ) /
( , , , ) ( ) / ( ) ( ) / ( )
,
,
,
c r c c c T c c p V V
c r c c c T c c p V V
c r c c c T c c p V V
T c c c T T T V T T p V V
T T T
A A A B C A IN A
B B A B C B IN B
C C A B C C IN C
A B C W IN
W W
0 / ( ) /
( ) / ( ) / ( )
V T T V
T T T V T T u V
V p V u
W i J W W
J W J J Jin J J
2
0
( )u u0 01
c c c c
c c
c c c c
c cA B in B A in
A A in
A C in C A in
A A in
, .
Linearizability
Case 8: Three chemical species
» Allow for control of inlet and outlet flow
Result:
The chemical reactor model is dynamic feedback linearizable with precompensator
and linearizing outputs
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) /
( , , , ) ( ) / ( ) / ( ) / (
,
,
,
c r c c c T c c u V
c r c c c T c c u V
c r c c c T c c u V
T c c c T T T V T T u V
T T T V T
A A A B C A IN A
B B A B C B IN B
C C A B C C IN C
A B C W IN
W W W i
1
1
1
1
0
J W W
J W J J Jin J J
T V
T T T V T T u V
V u u
) / ( ) / ( ) /
2
1 3
( )u u1 11
c c c c
c c
c c c c
c cTC B in B C in
C C in
C A in A C in
C C inW
, , .
Conclusions
Using a generalization of GS algorithm, a large class of linearizable chemical reactors was identified that is invariant to chemical kinetics.
Class can be increased considerably by considering more general precompensators and simple re-design
Some applicable commercial reactor systems:
– Ammonia reactor
– Nylon 6,6 and Nylon 6 polymerization reactors
– Synchronous growth bioreactor
– Multiproduct batch reactors
Primary applications
– Feedback stabilization
– Trajectory tracking
– Improvement of MPC schemes
Challenge is to provide a “measurement” or estimate of the linearizing outputs