Abstract

Through chemostat reactors, organisms can be observed under laboratory conditions. Hereby, the reactor contains the biomass, whose growth can be controlled via the dilution rate, respectively, the speed of a pump. Due to physical limitations, input constraints need to be considered. The population density in the reactor can be described by a hyperbolic nonlinear integro partial differential equation (IPDE) of first order. The steady-states and generalized eigenvalues and eigenmodes of these IPDE are determined. In order to track a desired reference trajectory, an optimal and an inversion-based feedforward control are designed. For the optimal feedforward control, the singular arc of the control is calculated and a switching strategy is stated, which explicitly considers the input constraints. For the inversion-based feedforward control, the IPDE is first linearized around the steady-state. To comply with the input constraints, a control system simulator is designed. For the simulation model, the IPDE is approximated using Galerkin's method. Simulations show the functionality of the designed controls and provide the basis for comparison. The inversion-based feedforward control operates well near the steady-state, whereas the performance of the optimal feedforward control is not bounded to the proximity to the steady-state.

References

References
1.
Smith
,
H. L.
, and
Waltman
,
P.
,
1995
,
The Theory of the Chemostat-Dynamics of Microbial Competition
, Cambridge Studies in Mathematical Biology, 13,
Cambridge University Press
,
Cambridge, UK
.
2.
Bayen
,
T.
, and
Mairet
,
F.
,
2013
, “
Minimal Time Control of Fed-Batch Bioreactor With Product Inhibition
,”
Bioprocess Biosyst. Eng.
,
36
, pp.
1485
1496
.10.1007/s00449-013-0911-9
3.
Toth
,
D.
, and
Kot
,
M.
,
2006
, “
Limit Cycles in a Chemostat Model for a Single Species With Age Structure
,”
Math. Biosci.
,
202
(
1
), pp.
194
217
.10.1016/j.mbs.2006.03.008
4.
Boucekkine
,
R.
,
Hritonenkoand
,
N.
, and
Yatsenko
,
Y.
,
2013
,
Optimal Control for Age-Structured Populations in Economy, Demography, and the Environment
,
Routledge
,
New York
.
5.
Brauer
,
F.
, and
Castillo-Chavez
,
C.
,
2001
,
Mathematical Models in Population Biology and Epidemiology
, Vol.
40
,
Springer
,
New York
.
6.
Bittanti
,
S.
,
Fronza
,
G.
, and
Guardabassi
,
G.
,
1973
, “
Periodic Control: A Frequency Domain Approach
,”
IEEE Trans. Autom. Control
,
18
(
1
), pp.
33
38
.10.1109/TAC.1973.1100225
7.
Mazenc
,
F.
,
Malisoff
,
M.
, and
Harmand
,
J.
,
2008
, “
Further Results on Stabilization of Periodic Trajectories for a Chemostat With Two Species
,”
IEEE Trans. Autom. Control
,
53
, pp.
66
74
.10.1109/TAC.2007.911315
8.
Karafyllis
,
I.
,
Malisoff
,
M.
, and
Krstic
,
M.
,
2015
, “
Sampled-Data Feedback Stabilization of Age-Structured Chemostat Models
,” American Control Conference (
ACC
), Chicago, IL, July 1–3, pp.
4549
4554
.10.1109/ACC.2015.7172045
9.
Schmidt
,
K.
,
Karafyllis
,
I.
, and
Krstic
,
M.
,
2018
, “
Yield Trajectory Tracking for Hyperbolic Age-Structured Population Systems
,”
Automatica
,
90
, pp.
138
146
.10.1016/j.automatica.2017.12.050
10.
Karafyllis
,
I.
, and
Krstic
,
M.
,
2014
, “
On the Relation of Delay Equations to First-Order Hyperbolic Partial Differential Equations
,”
ESAIM: Control, Optim. Calc. Var.
,
20
(
3
), pp.
894
923
.10.1051/cocv/2014001
11.
Karafyllis
,
I.
,
Malisoff
,
M.
, and
Krstic
,
M.
,
2015
, “
Ergodic Theorem for Stabilization of a Hyperbolic PDE Inspired by Age-Structured Chemostat
,” arXiv:1501.04321.
12.
Karafyllis
,
I.
, and
Krstic
,
M.
,
2017
, “
Stability of Integral Delay Equations and Stabilization of Age-Structured Models
,”
ESAIM: Control, Optim. Calc. Var.
,
23
(
4
), pp.
1667
1714
.10.1051/cocv/2016069
13.
Brokate
,
M.
,
1985
, “
Pontryagin's Principle for Control Problems in Age-Dependent Population Dynamics
,”
J. Math. Biol.
,
23
(
1
), pp.
75
101
.10.1007/BF00276559
14.
Feichtinger
,
G.
,
Tragler
,
G.
, and
Veliov
,
V. M.
,
2003
, “
Optimality Conditions for Age-Structured Control Systems
,”
J. Math. Anal. Appl.
,
288
(
1
), pp.
47
68
.10.1016/j.jmaa.2003.07.001
15.
Chen
,
J.-J.
, and
He
,
Z.-R.
,
2009
, “
Optimal Control for a Class of Nonlinear Age-Distributed Population Systems
,”
Appl. Math. Comput.
,
214
(
2
), pp.
574
580
.10.1016/j.amc.2009.04.018
16.
Skritek
,
B.
, and
Veliov
,
V. M.
,
2015
, “
On the Infinite-Horizon Optimal Control of Age-Structured Systems
,”
J. Optim. Theory Appl.
,
167
(
1
), pp.
243
271
.10.1007/s10957-014-0680-x
17.
Barbu
,
V.
, and
Iannelli
,
M.
,
1999
, “
Optimal Control of Population Dynamics
,”
J. Optim. Theory Appl.
,
102
(
1
), pp.
1
14
.10.1023/A:1021865709529
18.
Arguchintsev
,
A. V.
,
2002
, “
On Optimization of Hyperbolic Systems With Smooth Controls and Integral Constraints
,”
IFAC Proc. Volumes
,
35
(
1
), pp.
323
327
.10.3182/20020721-6-ES-1901.00303
19.
Zhao
,
C.
,
Zhao
,
P.
, and
Wang
,
M.-S.
,
2006
, “
Optimal Harvesting for Nonlinear Population Dynamics
,”
Math. Comput. Model.
,
43
(
3–4
), pp.
310
319
.10.1016/j.mcm.2005.06.008
20.
Hritonenko
,
N.
, and
Yatsenko
,
Y.
,
2007
, “
The Structure of Optimal Time- and Age-Dependent Harvesting in the Lotka-Mckendrik Population Model
,”
Math. Biosci.
,
208
(
1
), pp.
48
62
.10.1016/j.mbs.2006.09.008
21.
Alt
,
S.
,
Schmidt
,
K.
,
Arnold
,
E.
,
Zeitz
,
M.
, and
Sawodny
,
O.
,
2017
, “
Inversion-Based Feedforward Control Design for Linear Transport Systems With Spatially Acting Input
,”
Int. J. Syst. Sci.
,
48
(
5
), pp.
930
940
.10.1080/00207721.2016.1221481
22.
Bastin
,
G.
, and
Coron
,
J.-M.
,
2011
, “
On Boundary Feedback Stabilization of Non-Uniform Linear 2 × 2 Hyperbolic Systems Over a Bounded Interval
,”
Syst. Control Lett.
,
60
(
11
), pp.
900
906
.10.1016/j.sysconle.2011.07.008
23.
Coron
,
J.-M.
,
Vazquez
,
R.
,
Krstic
,
M.
, and
Bastin
,
G.
,
2013
, “
Local Exponential h2 Stabilization of a 2 × 2 Quasilinear Hyperbolic System Using Backstepping
,”
SIAM J. Control Optim.
,
51
(
3
), pp.
2005
2035
.10.1137/120875739
24.
Schmidt
,
K.
, and
Sawodny
,
O.
,
2017
, “
Efficient Simulation of Semilinear Populations Models for Age-Structured Bio Reactors
,” IEEE Conference on Control Technology and Applications (
CCTA
), Kohala Coast, HI, Aug. 27–30, pp.
1716
1721
.10.1109/CCTA.2017.8062704
25.
Finlayson
,
B. A.
,
2013
,
The Method of Weighted Residuals and Variational Principles
,
SIAM
,
Philadelphia, PA
.
26.
Pelovska
,
G.
, and
Iannelli
,
M.
,
2006
, “
Numerical Methods for Lotka-Mckendrick's Equation
,”
J. Comput. Appl. Math.
,
197
(
2
), pp.
534
557
.10.1016/j.cam.2005.11.033
27.
Kurth
,
A.-C.
,
Schmidt
,
K.
, and
Sawodny
,
O.
,
2020
, “
Optimal Control for Population Dynamics With Input Constraints in Chemostat Reactor Applications
,” European Control Conference (
ECC
), Saint Petersburg, Russia, May 12–15, pp.
271
276
.10.23919/ECC51009.2020.9143791
28.
Kurth
,
A.-C.
,
Schmidt
,
K.
, and
Sawodny
,
O.
,
2020
, “
Optimal Trajectory Tracking for Population Dynamics With Input Constraints in Chemostat Reactor Applications
,” American Control Conference (
ACC
), Denver, CO, July 1–3, pp.
5022
5027
.10.23919/ACC45564.2020.9147788
29.
Gurtin
,
M.
, and
MacCamy
,
R.
,
1974
, “
Non-Linear Age-Dependent Population Dynamics
,”
Arch. Ration. Mech. Anal.
,
54
(
3
), pp.
281
300
.10.1007/BF00250793
30.
Kelley
,
C. T.
,
2003
,
Solving Nonlinear Equations With Newton's Method
,
Siam
,
Philadelphia, PA
.
You do not currently have access to this content.