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