Linearizability of Chemical Reactors

1
Linearizability ofChemical Reactors

• By
• M. Guay
• Department of Chemical and Materials Engineering
• University of Alberta
• Edmonton, Alberta, Canada
• Work Supported by NSERC

2
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.

3
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

4
Motivation

• Linearizability of nonlinear control systems
• CONTROLLER DESIGN
• TRAJECTORY GENERATION

Linear System
Nonlinear System
Nonlinear Controller
Linear Controller
5
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

6
Background

• Let M be a n-dimensional manifold
• TpM is the tangent space to M at a point p with
  basis
• TpM is cotangent space to M at a point p with
  basis
• Elements of TpM, called one-forms, are linear
  maps

7
Background

• Associated with differential forms is an algebra
  called the Exterior Algebra, W(M)
• 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

8
Pfaffian Systems

• Let S be a submodule of W(M)
• S is called a Pfaffian system defined locally as
• where is a set of one-forms.
• S defines an exterior differential system I

9
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.

10
Control Systems

• A control affine nonlinear system is given by
• where
• S defines a Pfaffian system S 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).

(S)
11
Feedback Linearization

• Definition 2
• A control system is said to be feedback
  linearizable if there exist a static state
  feedback u a (x)b (x)u and a coordinate
  transformation x f (x) that transforms the
  nonlinear to a linear controllable one.
• Using the derived flag of S, 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. S is generated by one-forms
• that satisfy the congruences

12
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 f (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

13
Dynamic Precompensators

• Precompensation can be achieved from
  differentiation of the process inputs, u, or of a
  static state feedback transformation of them, x.
• The degree of precompensation is summarized by

14
Dynamic Precompensators

• General Form
• Precompensator Structure
• (i) Structure of precompensator determined by
  indices and
• (ii) Alternatively, with
  multiplicities

15
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

P
S
Differential Algebraic Constraints
Feedback Linearizable System S,P
16
Conditions for DFL

• Definition 3
• Consider the control system S and a
  precompensator P based on the feedback v j(x,u)
  and indices with
  multiplicities
• The first derived system, S(1), associated with
  P is given by the set of forms, w(1), which
  satisfy
• The second derived system associated with P is
  defined as the set of forms, w(2), which satisfy
• or
• By induction

17
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 ri such that
• defined by

18
Conditions for DFL

• Theorem 2
• A control system S is dynamic feedback
  linearizable by dynamic extension of a state
  feedback transformation v j(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

19
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(b) 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

20
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

u1, Tin, cAin, cBin, ccin
u2, TJ
u2, TJin
Cooling Jacket
u1, T, cA, cB, cc
21
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
• l Enthalpy of Reaction
• a Reactor-side heat transfer coefficient
• b Jacket-side heat transfer coefficient
• V Tank volume

22
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

23
Linearizability

• Case 2 Two chemical species with variable
  hold-up
• Result
• Applying the precompensator
• yields a dynamic feedback linearizable system
  with outputs

24
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

25
Linearizability

• Case 4 Three chemical species and constant
  hold-up
• Result
• The chemical reactor is dynamic feedback
  linearizable with precompensator
• and linearizing outputs

26
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

27
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

28
Linearizability

• Case 7 Three chemical species with design change
• Result
• Applying the precompensator
• yields a dynamic feedback linearizable system
  with outputs

29
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

30
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