(245l) Nonlinearity Analysis of Periodically Forced Bioreactors

Chi Zhai^a, Ahmet Palazoglu^b,
Wei Sun^a*, Zengzhi Du^a

^a
China Beijing Key Lab of Membrane Science and Technology, College of Chemical
Engineering, Beijing University of Chemical Technology, 100029 Beijing, CHINA

^b
Department of Chemical Engineering, University of California, Davis,

 CA 95616, USA

Abstract

Nonlinearity is the underlying
characteristic in process systems that allows for time average performance
enhancement by periodic forcing of external parameters^{1, 2}. The well-known pi-criterion provides a sufficient condition for
optimal periodic operation³, and is based on the
linearization of the system and second-order truncation of the performance
index, which is valid only for weakly nonlinear systems with infinitesimal
forcing amplitudes. On the other hand, bioreactors often exhibit highly
nonlinear dynamics which poses difficulties on
the systematical analysis of
periodic operation. Hence, it becomes desirable to explore how inherent nonlinearities would contribute to performance
improvement if periodic forcing is applied on the process. In this paper, a two-compartment structured
model for ethanol production is studied⁴. By judiciously setting the
scope of the design/operation parameters and implementing bifurcation analysis
on the unforced model, Hopf bifurcation points are detected which
separate the parameter space into a hyperbolically
stable region and a self-oscillatory
region.

  For a hyperbolically
stable point, analysis of the periodic operation can
be accomplished by applying the
center
manifold theory to the forced nonlinear system. With a
multivariable expansion of the center manifold equations, higher-order terms of
the nonlinear system are evaluated⁵. This method is a higher-order correction of the pi-criterion,
which will reduce to the pi-criterion by truncation up to 2^nd order
terms. But this method is only suitable for sinusoidal-like forcing
inputs. Typically, when pulsed periodic inputs are applied on the system,
Fourier transform of the inputs would complicate this approach significantly⁶.
Previous studies⁷ used Carleman linearization as another method for
pulsed periodic optimization problems. This method is carried out by Carleman
linearization on the nonlinear model and solving it analytically, and then the
state terms are substituted into the Taylor expansion of the performance index.
Since Carleman linearization is cumbersome and suffers from lack of parsimony,
a more compact functional expansion method using the Laplace-Borel transform is
applied⁸. This method is found to be comparatively easy for dealing
with both pulsed inputs and sinusoidal-like inputs⁹.
These three methods are compared with a case study on the two-compartment
bioreactor model, and the superiority of the functional expansion method is
addressed.

  This
bioreactor may exhibit self-oscillatory behavior by adjusting the parameters
properly. We show that for a self-oscillatory point, external periodic forcing may bifurcate
the system to more complex dynamic behaviors. Bifurcation analysis can be
implemented on the forcing waveform that is continuously differentiable, i.e.,
when the sinusoidal input is exerted on the self-oscillator, complex dynamics
such as invariant torus or chaos would emerge. In this regard, the route to
chaos by period-doubling cascades is discussed in this case study. While,
intermittency¹⁰ may also lead the system to chaos, and one may only
concentrate on the onset of chaos in industry as chaotic behavior is viewed as an
operational state to be avoided. An algebraic criterion on the onset of chaos
is adopted in this case study, which is the Laplace-Borel transformation of the
autocorrelation function. Unlike other criterion for the formation of chaos,
which is usually simulation-based, this criterion is analysis-based and offers
better insight.

Keywords: Carleman
linearization; functional expansion; Laplace-Borel transform; central manifold;
the onset of chaos.

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