359
0.4
voltage
358
FC controller output
0.3
357
0.2
356
0.1
0
355
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
time t [s]
time t [s]
(c) FC controller output u 1
(d) capacitor voltage v 2
1.6
1
current i 2
reference i 2r
0.9
1.4
source voltage | v
|/ 100
s
0.8
1.2
2 0.7
u
1
[A]
0.6
i 2 0.8
0.5
current
0.4
0.6
0.3
PFC controller output
0.4
0.2
0.2
0.1
0
0
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.3
0.35
0.4
0.45
0.5
0.55
0.6
time t [s]
time t [s]
(e) source current i 2
(f) PFC controller output u 2
Fig. 12. (C1) Behaviors when a load resistance R changes from 1000 to 0.25 ohms in steady
state where a controller is designed for a different nominal load resistance from that in Fig.11
Entries of the matrix Y f c in (21) are clearly different from those in (19). On the other hand,
difference of the Yp f c between the matrix (22) and (20) is much small. Those differences give
the different behaviors as mentioned above.
Fig.12 demonstrates the discussion in Section 4.2 that controllers needs be designed in
consideration of undesirable behavior of system. For a large load resistance the output voltage
of the FC oscillates and for small load resistance the source current and the capacitor voltage
of the PFC respond to variations larger than those in Fig.11.
Lyapunov-Based Robust and Nonlinear Control
for Two-Stage Power Factor Correction Converter
35
Lyapunov-Based Robust and Nonlinear Control for Two-Stage Power Factor Correction Converter
15
7
6
voltage v 1
reference v 1r
6
5
5
4
[V]
4
[A]
v 1
i 1 3
3
voltage
current
2
2
1
1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time t [s]
time t [s]
(a) output voltage v 1
(b) output current i 1
1
363
voltage v 2
reference v
0.9
2r
362
0.8
361
0.7
1
u
360
0.6
[V]
v 2
0.5
359
0.4
voltage
358
FC controller output
0.3
357
0.2
356
0.1
0
355
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
time t [s]
time t [s]
(c) FC controller output u 1
(d) capacitor voltage v 2
1
current i 2
reference i 2r
0.9
source voltage | v
|/ 280
s
0.5
0.8
2 0.7
u
0.4
[A]
0.6
i 2 0.3
0.5
current
0.4
0.2
0.3
PFC controller output
0.2
0.1
0.1
0
0
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.3
0.35
0.4
0.45
0.5
0.55
0.6
time t [s]
time t [s]
(e) source current i 2
(f) PFC controller output u 2
Fig. 13. (C2) Behaviors when an efficient value of source voltage Ve changes from 100 to 85
volts for 2 seconds in steady state
5. Conclusion
For a two-stage power factor correction converter, a nonlinear controlled converter system
was systematically designed on the basis of its nonlinear model. The systematic controller
design clearly analyzed the behavior of nonlinear system to improve the performance. The
synthesis step were clearly shown. It was clarified that a nominal load resistance characterizes
the controller performance.
Finally, computer simulations demonstrated efficiencies of
the approach. The nonlinear controller synthesis was shown as a natural extension of a
well-known Lyapunov-based linear controller synthesis.
36
Applications of Nonlinear Control
16
Nonlinear Control
6. Acknowledgment
The author would like to thank Hideho Yamamura of Hitachi,Ltd. and Kazuhiko Masamoto
of NEC Networks for their valuable discussions and comments in the viewpoint of industrial
practice.
7. References
Brockett, R.W. & Wood, J.R. (1974). Electrical Networks Containing Controlled Switches. IEEE
Symposium on Circuit Theory, pp.1-11
Kassakian, J.G.; Schlecht, M.F. & Verghese, G.C. (1991). Principles of Power Electronics,
Addison-Wesley, ISBN:0-201-09689-7
Banerjee, S. & Verghese, G.C. (2001). Nonlinear Phenomena in Power Electronics, IEEE Press,
Piscataway, NJ, ISBN:0-7803-5383-8
Orabi, M. & Ninomiya, T. (2003). Nonlinear Dynamics of Power-Factor-Correction Converter.
IEEE Transactions on Industrial Electronics, pp.1116-1125
Dranga, O.;Tse, C.K. & Siu Chung Wong (2005). Stability Analysis of Complete Two-Stage
Power-Factor-Correction Power Supplies Proc. European Conference on Circuit Theory
and Design, pp.I/177-I/180
Mohler, R.R. (1991). Nonlinear Systems, Vol.2, Applications to Bilinear Control. Prentice Hall,
Englewood Cliffs, NJ, ISBN:0-13-623521-2
Escobar, G.; Chevreau, D.; Ortega, R. & Mendes, E. (1999). An Adaptive Passivity-Based
Controller for a Power Factor Precompensator. Proc. 5th European Control Conference,
BP4-1
Escobar, G.; Chevreau, D.; Ortega,R. & Mendes, E. (2001). An Adaptive Passivity-Based
Controller for a Unity Power Factor Rectifier. IEEE Trans. Control Systems Technology,
Vol.9, No.4, pp.637-644
Sasaki, S. & Uchida, K. (1998). Nonlinear H∞ Control System Design via Extended Quadratic
Lyapunov Function. Proc. IFAC Nonlinear Control Systems Design Symposium,
pp.163-168
Sasaki, S. (2002). System Clarification through Systematic Controller Design for a Forward
Converter with Power Factor Corrector. IEEE 33rd Annual Conf. on Power Electronics
Specialists Conference, Vol.3, pp.1083-1088
Sasaki,S. (2009). Systematic Nonlinear Control Approach to a Power Factor Corrector Design.
European Transactions on Electrical Power, Vol.19, No.3, pp.460-473
Redl, R. (1994). Power-Factor Correction in Single-Phase Switching-Mode Power Supplies –
An Overview. International Journal of Electronics, pp.555-582
3
Nonlinear Control Applied to the Rheology of
Drops in Elongational Flows with Vorticity
Israel Y. Rosas1, Marco A. H. Reyes2, A. A. Minzoni3 and E. Geffroy1
1Instituto de Investigaciones en Materiales,
2Facultad de Ingeniería,
3Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas
Universidad Nacional Autónoma de México, Mexico City
Mexico
1. Introduction
Fluid systems where a liquid phase is dispersed in other liquid, as emulsions, are present in
many industrial processes, technological applications and in natural systems. The flow of
these substances shows a rheological behavior that depends on the viscosities ratio, the
surface tension, surfactants, flow-type parameter and the coupled effects of the fluid
structure and the kinematics properties of the flow, mainly in the non linear regimen, which
is the reason because the dynamics of the fluid particles is an area of current research. As an
approach to understand the physical properties of these fluids, studies on the deformation,
break-up and coalescence of drops have been performed since the pioneering work of
Taylor (1932).
The deformation and breakup of liquid drops is dependent on the kinematics properties of
the imposed flow, in particular of the second invariant of the rate of deformation tensor II2D
and the flow-type parameter, α (see Astarita, 1979). In experimental studies of the dynamics,
break-up and coalescence of drops, then it is necessary to be able to modify on demand the
external flow field parameters causing the drop deformation. For example, Taylor (1934) use
two flow devices, a Parallel Band Apparatus (PBA) and a Four-Roll Mill (FRM), which
manually manipulated each of the rollers speeds to position a drop. Once the drop was in
place, Taylor was able to track changes that occur on the drop shape as a function of the
imposed flow, although for a short time, until the drop was ejected from its unstable
position.
Four-Roll Mills and Parallel Band cover a wide interval in the flow-type parameter;
however, there is a gap between them that the flow fields generated by co-rotating Two-Roll
Mill (TRM) geometries fill. PBA works for simple shear flow, corresponding to a flow-type
parameter α = 0; FRM setup is more effective in the interval 0.4 ≤ α ≤ 1 [Yang et al. (2001)],
whereas TRM is effective in the interval 0.03 ≤ α ≤ 0.3 [Reyes and Geffroy (2000b)].
In the case of the Four-Roll Mill, Bentley and Leal (1986a) have shown how to control -for
long times- the position of drops within the flow field generated by a FRM by adjusting with
a computer and in real-time the speed of rotation of each cylinder, and maintaining
38
Applications of Nonlinear Control
simultaneously constant values for the elongational flow field parameters. This is achieved
by modifying the angular velocity of the cylinders, which displaces the stagnation point to a
new position, in order to drive the motion of the drop along the stable flow direction
towards the stagnation point. Leal’s research group at the Chemical Engineering
Department at UC Santa Barbara had worked since that with the FRM apparatus, mainly in
the pure elongational flow regimen. In the case of the Parallel Band Apparatus, Birkhoffer
(2005) proposes computer-controlled flow cell based on the PBA using a digital
proportional-integral-differential controller.
We present a nonlinear control applied to study the rheology of a drop in an elongational
flow field with vorticity. Large deformations on fluid particles such as drops occur typically
in regions containing saddle points. However, the kinematics of a saddle-point flow does
not allow for long observation times of the deformed drop due to the outgoing streamlines,
which advect the drop away from the flow region of interest. Thus, it is necessary to
accurately control the centroid of the drop to study its desired rheology. A suitable control
mechanism is essential to maintain the drop position under known flow conditions for times
that are long compared to the intrinsic time scales of the dynamics.
In this work, the control mechanism of the position of a drop around the stagnation point of
the flow generated by a co-rotating Two-Roll Mill is presented. This control is based on the
Poincaré-Bendixson theorem for two-dimensional ordinary differential equations. Namely,
when a particle moves within a closed region containing a saddle point inside and the
vector field of the equation points inwards at the boundary of the region, the particle
undergoes a stable attractive periodic motion. We show that given a prescribed tolerance
region, around the unstable stagnation point, an incoming flow can always be generated
when the center of mass of the drop reaches the boundary of the tolerance region. This
perturbed flow is produced by adjusting the angular velocity of the cylinders, calculated
using an analytical solution for the flow, without significant change in the flow parameters.
This gives the time dependent analogue of the Poincaré-Bendixson situation just described.
The drop is controlled in a perturbed attracting periodic trajectory around the saddle point,
while being confined to a prescribed tolerance area. This mechanism is very different from
the one used for the proportional control which modifies the unstable nature of the saddle
point by adjusting the angular velocities of the cylinder to project the motion along the
stable direction only. In the proposed control the effect of the unstable direction combined
with the flow readjustment produces the periodic motion.
A nonlinear control strategy, based on the Poincaré-Bendixson theory, is proposed and
studied. In essence, the control generates the planar motion of a particle (the centroid in this
case) to ensure a periodic motion of the drop centroid inside a prescribed area around the
saddle point. In addition, a numerical study and an experimental situation are presented to
illustrate the effect of the control on the drop motion. The implementation of the nonlinear
control is within a closed region containing the saddle point, with the velocity field pointing
inward at every point on the boundary, while the particle undergoes a stable attractive
periodic motion. Thus, given a prescribed tolerance region around the stagnation point, it is
always possible to generate a controlled incoming flow whenever the center of mass of the
liquid drop reaches the boundary of the tolerance region to force the centroid back into the
prescribed tolerance region.
Nonlinear Control Applied to the Rheology of Drops in Elongational Flows with Vorticity
39
The proposed control scheme is capable of keeping the values of the global flow parameters
within a small tolerance, and redirecting the liquid-drop centroid toward the stagnation
point on a time scale that is much shorter than that of the evolution, with minimal impact
upon the liquid-drop dynamics.
We also performed detailed studies of the sensitivity of the control to imperfections in the
shape of the geometry that generates the flow, and the variation of the velocity in the
servomotors which control the position of the stagnation point. Also a detailed comparison
between numeric and experimental data is also presented. This provides a complete study of
the two dimensional situation. Finally, we present some open questions both in the modeling
and in the theory. In particular in the possibility of extending the current results to drops of
complex fluids such as viscoelastic drops, vesicles, capsules, and other immersed objects.
Given that there are no detailed studies available on the influence of the control scheme
upon the drop's forms, we study numerically the influence of this control on the motion of a
two dimensional drop by solving the Stokes equations in a container subjected to the
appropriate boundary conditions on the cylinders and the free surface of the drop. These
equations are solved with the Boundary Element Method for a variety of flows and drop
parameters in order to study the perturbation effects introduced by the application of the
control scheme and to provide the appropriate parameters for future experimental studies.
In particular, an easy to implement control scheme is an essential tool prior to undertakings
any experimental studies of the drop's dynamics -under elongational flows with vorticity at
small Capillary numbers-, for large deformation of drops capsules and other objects, as well
as for break up and coalescence of embedded objects.
The proposed method is simple. As the drop evolves under flow conditions, its centroid is
tracked. When the drop drifts away and its centroid overtakes the prescribed domain about
the nominal stagnation point, the flow is modified by adjusting the angular velocity of the
cylinders according to the values obtained from the approximate solution for flows
generated by TRMs. Essentially, by adjustment the angular velocities of the cylinders, the
outgoing streamline environment is change into an incoming one, reversing the direction of
motion of the drop, which is now towards the nominal stagnation point along a stable
direction. The reversal of direction does not alter significantly the deformation rates applied
upon the drop; thus, the drop's dynamics is essentially undisturbed. The process is repeated
as needed and the drop is confined for long times under steady and known conditions.
This study is complementary to the previous one (Bentley 1986a, Bentley 1986b) because it
allows detailed studies of the time evolution of the drop's parameters -mainly its
deformation and the orientation. In contrast to FRMs, the numerical results presented allow
us to study the dynamics of embedded objects under the nominal flow conditions when the
drop remains at the stagnation point, as well as the drop in the “controlled flow” with the
corresponding parameters. The study was carried out both under several viscosity ratios
and various geometric setups, assessing the robustness of the proposed method and the
very small influence it has on the drop behavior. The influence of the control scheme on the
drop's parameters is small with respect to the nominal flow (around 1%). As well, the
proposed control scheme is capable of relocating the stagnation point on a time scale much
shorter than the time scale of the drop's evolution. Moreover, this control would remain
effective during times much longer than the internal time scales for the drop evolution.
40
Applications of Nonlinear Control
The fact that a two-dimensional numerical model for the drop is used does not compromise
the results here presented, mainly because the control scheme should be applicable for small
deformation of drops or for drops with very small trajectories about the stagnation point.
For highly elongated drops, capsules and other complex objects, the simulated values of the
drop's shape may differ from the experimental values, but the applicability of the control
scheme should be equally robust.
2. Formulation of the control problem
As already demonstrated by Bentley (1986a), the only way to maintain fixed the position of
the drop with respect to the flow field is by changing the location of the stagnation point via
adjustments of the angular velocities of the cylinders, with the constraint that these changes
must avoid significant modifications of the flow field. Consequently, a useful control
scheme for flows by TRMs or FRMs has to maintain the drop as close as possible to the
stagnation point for a sufficiently long time, making possible studies of the drop dynamics.
From now on, the selected flow field conditions of a TRM are called nominal, and its
properties such as the shear rate, flow-type parameter and the position of the stagnation
point will be denoted by the subscript Nom.
Figure 1 shows the streamlines around the stagnation point of the unperturbed flow field
generated by a co-rotating TRM. When a liquid or rigid particle is placed around the
unstable stagnation point, the particle drifts in the direction of the outgoing streamlines. The
objective of the control is to maintain a drop about the stagnation point of the nominal flow
conditions. To construct the control scheme, the trajectory of the centroid of the drop is
analyzed as the solution of a two dimensional dynamical system. In this case we assume
that the centroid is away but near the unstable saddle point. Now if the vector field is
oncoming on the boundary of a box surrounding the unstable stagnation point the system
ω1
R 1
g
R 2
ω2
Fig. 1 . Streamlines generated by a co-rotating Two-Roll Mill with different radii, showing
the stagnation point in the gap between the rollers, g. The position of the stagnation point
along the vertical can be moved changing the angular velocities of the rollers. The value of
the flow-type parameter is a function of the position of the stagnation point.
Nonlinear Control Applied to the Rheology of Drops in Elongational Flows with Vorticity
41
will settle into a periodic orbit inside the box provided the vector field is time independent.
When there exists time dependence, a bounded motion (which can be periodic or quasi-
periodic) is produced inside the box. This motion is equally robust as the time independent
case, confining the centroid of the drop to any prescribed region provided the appropriate
incoming flow can be produced at the boundary