INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control 2002; 12:57}69 (DOI: 10.1002/rnc.625)
Application of nonlinear time}scaling for robust controller design of reaction systems P. Moya, R. Ortega * R, M. S. Netto, L. Praly, J. PicoH Dept. Ingeniern& a de Sistemas y AutomaH tica, Universidad Polite& cnica de Valencia, P.O. Box 22012, E-46071 Valencia, Italy Lab des Signaux et Syste% mes, CNRS, SupeH lec, Univ. Paris Sud., Plateau de Moulon, 91192 Gif sur Yvette, France E! cole Nationale Supe& rieure des Mines de Paris, Centre Automatique et Syste% mes, 35 Rue Saint Honore& , 77305 Fontainebleau Cedex, France
SUMMARY Even though the basic mechanisms of operation of reaction systems are relatively simple the dynamical models obtained from "rst principles are complex and contain highly uncertain terms. To develop reliable model-based controllers it is therefore necessary to simplify the system dynamics preserving the features which are essential for control. Towards this end, co-ordinate transformations identifying the states which are dependent/independent of the reactions and #ows have been reported in the literature. This has allowed, for instance, the design of observers which are insensitive to the (usually unknown) reaction functions. The main contribution of this paper is to show the utility of nonlinear state-dependent time-scaling to simplify the system dynamics, and consequently the controller design. In particular, we show that with time-scaling and an input transformation we can reveal the existence of attractive invariant manifolds, which allows us to reduce the dimension of the system. As an application we study the well-known fourth order baker's yeast fed-batch fermentation process model, whose essential dynamics is captured by a planar system perturbed by an exponentially decaying term. We then exploit this particular structure to design, with reduced control authority, a nonlinear asymptotically stabilizing control law which is robust with respect to the reaction function. Copyright 2001 John Wiley & Sons, Ltd. KEY WORDS:
reactions systems; time-scaling; fed-batch fermentation processes; nonlinear control
1. INTRODUCTION The concept of reaction systems [1] refers to a wide class of nonlinear dynamical systems that appears in "elds such as chemical engineering, biotechnology, ecology, etc. Even though the basic *Correspondence to: R. Ortega, Lab des Signaux et Syste% mes, CNRS, SupeH lec, Univ. Paris Sud., Plateau de Moulon, 91192 Gif sur Yvette, France. RE-mail:
[email protected] Contract/grant Contract/grant Contract/grant Contract/grant
sponsor: sponsor: sponsor: sponsor:
Spanish Ministry of Science; contract/grant number: AP97-29170993 Valencian Government; contract/grant number: GV97-TI-04-59 European Nonlinear Control Network (NCN); contract/grant number: CON99P023DR04 Brazilian Foundation CAPES
Published online 16 November 2001 Copyright 2001 John Wiley & Sons, Ltd.
Received 31 October 2000 Accepted 20 March 2001
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mechanisms of operation are relatively simple the models are usually quite complex and uncertain, therefore various transformations that simplify their dynamics have been proposed. In Reference [2] a linear change of co-ordinates is used to separate*in reaction systems without inlet and outlet streams*the reaction-variant states from the reaction-invariant ones. This decomposition is very useful*for instance, for observer design*since the reaction functions are poorly known. In References [Srinivasan, 3, 4] a nonlinear change of co-ordinates is introduced to extend the concept of reaction invariants of Reference [2] to include -ow invariants for reaction systems with inlet and outlet streams, thus leading to a decomposition of the state evolution into reaction and #ow variants/invariants. The inclusion of a nonlinear transformation, besides adding a signi"cant degree of complexity, might destroy the structure of the system, stymieing the physical interpretation of the transformed system states. As an alternative to the nonlinear change of co-ordinates of Reference [4], we propose in this paper a nonlinear state-dependent time-scaling which achieves the same objectives. Two important features of our time-scaling are (1) it has a clear physical interpretation in terms of the residence time; (2) it is explicitly computable, allowing its application not just for analysis purposes, but also for observer and controller design. Nonlinear time-scaling has already been used, among other applications, for feedback linearization in References [5, 6]. The main contributions of the paper are as follows. First, we combine the well-known linear change of co-ordinates of Reference [2] with the proposed time-scaling to derive two normal forms*applicable to continuous and fed-batch reactors, respectively*that clearly reveal the reactions and #ows variants/invariants and the existence of attractive invariant manifolds. The latter allows us to reduce the dimension of the system and simplify the controller design. We also show that using these normal forms we can easily design reaction-independent observers, which are simpler than the existing ones and have a guaranteed computable convergence rate in all operating regimes. Second, to illustrate the application of time-scaling for controller design we consider the problem of regulation of the well-known fourth order baker's yeast fed-batch fermentation process model. It is shown that in the new time-scale, besides the unaccessible subspace, an attractive manifold is revealed, which reduces the essential dynamics to a planar system perturbed by an exponentially decaying term. We then exploit this particular structure to design, with reduced control authority, a nonlinear asymptotically stabilizing control law which is robust with respect to the reaction function. Simulation results that illustrate the performance of our controller are also presented.
2. REACTION NETWORK AND KINETICS Consider a reaction system with R independent reactions involving S components References [1, 3]. Two di!erent mechanisms, reaction kinetics and exchange dynamics, are involved. The overall dynamics result from the S material balance equations for the S species and the continuity equation for the reactor volume as
n "N2r (n,
(2)
where r (c, )2 (I !C M2)n Q z "M2n z "Q2n
(3)
with M, Q constant matrices of suitable dimensions and ( ) )> the pseudoinverse, which transforms (1) into q zR "! z #r (z, )2n #A z #A z (8) R R Q Q where A O C !N2(N>)2C and A O Q !N2(N>)2Q . R Q Q R R Q Q We are in position to present the following proposition. The proof, which is carried out in the time-scale, follows immediately from (8), the normal forms (5), (7), and the observation that z ()"exp\Oz (0)P0. The latter property*which is revealed by the time-scale change*obvi ates the need to include an estimate of z . Henceforth, the resulting estimator will be simpler than, e.g. the one reported in Reference [7]. Proposition 3.2 Consider the reaction system (1). Assume "R#p, where O rank [N2, C ]; n 3R1Q , as de"ned in (8), is measurable; Q S *R and N2 has a unique left pseudoinverse. Q Q Under these conditions, the estimator (de"ned in the time-scale t) 1. 2. 3.
n( "N2(N>)2n #A z( q z( "! z( #q <
(9) (10)
ensures the estimation error converges to zero as 1 k k M" (c) O ! K c #c # K c "u* k W where c O [c , c , c ]. This stems from the (easily veri"able) fact that "! #u*. The existence of M and the structure of (14) motivates the linear change of co-ordinates z "c , z "c #k c , and z " , which yields W z "!z # (z)z u z "!z #k u*!k k z u W W K z "!z #u* where (z) O (1/k (z !z ), z !( /k k )z #( /k ) z ).A Finally, if we de"ne the input X W K W K transformation v "z u we obtain the following feedforward form: z "!z # (z) v X z "!z #k u*!k k v (15) W W K z "!z #u* We will now study the equilibria of (15). In view of Assumption (A5), and the fact that time scaling does not a!ect the equilibria, we have that (15), with v "v* O c*/Q*, has an equilibrium at the desired operating point. (Assumption (A5) is then tantamount to say that the reaction function is such that the algebraic equation
(c*, z*, u*) (z*!k u*)#k k c*"0 X W W K admits a solution z*"k (u*!k (c*/Q*)).) We can thus translate the equilibrium to the origin W K to get z "!z #[ (z #z*)! (z*)]v*# (z #z*)v X X X z "!z !k k v W K z "!z where we have introduced the notation ( ) ) O ( ) )!( ) )*. The control law (13) in these new coordinates reduces to v "v*, thus we obtain in closed loop the cascade system z "!z #[ (z #z*)! (z*)]v* X X z "!z (16) z "!z It is well known [11] that the cascade system (16) is (globally) asymptotically stable if (a) All the solutions of (16) are bounded. (b) The autonomous subsystem z "!z # I (zJ )v* (17) X where we de"ned I (zJ ) O (z #z*, z*, z*)! (z*, z*, z*), is (globally) asymptotically stable. X X X A
Notice that depends on the whole vector z O [z , z , z ] X
Copyright 2001 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2002; 12:57}69
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Boundedness of z , z is, of course, obvious from (16). On the other hand, boundedness of z is established considering the quadratic function < " z , invoking assumption (A3), and bound ing the derivative of < as follows:
