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

2. Introduction l Feedback Linearization has formed the basis for most engineering applications of nonlinear control techniques l Basic Techniques - Static State Feedback Linearization 1) Hunt, Su and Meyer u Lie Algebraic approach 2) Gardner and Shadwick u Exterior calculus approach u GS Algorithm l 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 l Motivation l Background l Pfaffian Systems and Feedback Linearization l Conditions for Dynamic Feedback Linearization l Linearizability of Non-isothermal Chemical Reactors –Reactors with 2 chemical species –Reactors with 3 chemical species l Conclusions

4. Motivation l Linearizability of nonlinear control systems –CONTROLLER DESIGN –TRAJECTORY GENERATION Linear System Nonlinear Controller Linear Controller Nonlinear System

5. Motivation l 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 –T p M is the tangent space to M at a point p with basis –T p * M is cotangent space to M at a point p with basis –Elements of T p * M, called one-forms, are linear maps

7. 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

8. 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

9. Pfaffian Systems l 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 i th derived system of I defined by The number k for which is called the derived length of I.

10. Control Systems l 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 l 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)(S)

11. 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 k th derived system is trivial 2.  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  x) such that the combined system {S,P} is equivalent to a linear controllable form. l 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, . l The degree of precompensation is summarized by

14. Dynamic Precompensators l General Form l Precompensator Structure (i) Structure of precompensator determined by indices and (ii) Alternatively, with multiplicities

15. Dynamic Feedback Linearization l Linearization problem is summarized by l General problem reduces to special interconnection of nonlinear systems with precompensators of appropriate dimensions subject to DAE constraints S P 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  =  (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

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  i 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  =  (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: l 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 l 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 u 1 Tank Volumetric Flowrate u 2 Jacket Volumetric Flowrate c I Concentration of species I c Iin Inlet Concentration of species I TTank Temperature T in Tank Inlet Temperature T J Jacket Temperature T jin Jacket Inlet Temperature u 1, T, c A, c B, c c u 2, T Jin u 2, T J u 1, T in, c Ain, c Bin, c cin Cooling Jacket

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 r I Rate of production of species I Enthalpy of Reaction  Reactor-side heat transfer coefficient  Jacket-side heat transfer coefficient VTank 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 {c A, c B }. 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 u 1 to u 0 »let u 1 = 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 l Using a generalization of GS algorithm, a large class of linearizable chemical reactors was identified that is invariant to chemical kinetics. l Class can be increased considerably by considering more general precompensators and simple re-design l Some applicable commercial reactor systems: –Ammonia reactor –Nylon 6,6 and Nylon 6 polymerization reactors –Synchronous growth bioreactor –Multiproduct batch reactors l Primary applications –Feedback stabilization –Trajectory tracking –Improvement of MPC schemes l Challenge is to provide a “measurement” or estimate of the linearizing outputs