Abstract

This study provides a novel analysis and design of the state-dependent Riccati equation (SDRE) control in cancer treatment application. The key assumption to ensure continuous SDRE controllers—in terms of the solvability of pointwise Riccati equations—is replaced by a simplified equivalent condition, which largely alleviates the computational burden. At the discontinuities, an alternative solution is novelly suggested, because the conventional/empirical αparameterization technique to seek a continuous SDRE implementation without breakdowns is analyzed to be ineffective, which is the first counterexample in literature. Representatively, among discontinuities, an objective conflict against tumor eradication is discovered. Another value of the proposed analysis is supported by the generality demonstrations, in various fields beyond biomedical systems. Finally, the robustness of SDRE scheme to parameter variations is established via simulations, which more promotes the alternative solution as applied throughout the treatment course.

References

1.
Shi
,
J.
,
Alagoz
,
O.
,
Erenay
,
F. S.
, and
Su
,
Q.
,
2014
, “
A Survey of Optimization Models on Cancer Chemotherapy Treatment Planning
,”
Ann. Oper. Res.
,
221
(
1
), pp.
331
356
.10.1007/s10479-011-0869-4
2.
Cacace
,
F.
,
Cusimano
,
V.
, and
Palumbo
,
P.
,
2020
, “
Optimal Impulsive Control With Application to Antiangiogenic Tumor Therapy
,”
IEEE Trans. Control Syst. Technol.,
28(1), pp.
106
117
.10.1109/TCST.2018.2861410
3.
Bahrami
,
K.
, and
Kim
,
M.
,
1975
, “
Optimal Control of Multiplicative Control Systems Arising From Cancer Therapy
,”
IEEE Trans. Autom. Control
,
20
(
4
), pp.
537
542
.10.1109/TAC.1975.1101019
4.
Çimen
,
T.
,
2010
, “
Systematic and Effective Design of Nonlinear Feedback Controllers Via the State-Dependent Riccati Equation (SDRE) Method
,”
Annu. Rev. Control
,
34
(
1
), pp.
32
51
.10.1016/j.arcontrol.2010.03.001
5.
Mellal
,
L.
,
Folio
,
D.
,
Belharet
,
K.
, and
Ferreira
,
A.
,
2018
, “
Modeling Approach of Transcatheter Arterial Delivery of Drug-Loaded Magnetic Nanoparticles
,”
The Encyclopedia of Medical Robotics
(Micro and Nano Robotics in Medicinech), Vol.
2
,
J. P.
Desai
,
R.
Patel
,
A.
Ferreira
, and
S.
Agrawal
, eds.,
World Scientific
,
Singapore
, pp.
207
229
.
6.
Babaei
,
N.
, and
Salamci
,
M. U.
,
2015
, “
Personalized Drug Administration for Cancer Treatment Using Model Reference Adaptive Control
,”
J. Theor. Biol.
,
371
, pp.
24
44
.10.1016/j.jtbi.2015.01.038
7.
Wang
,
X.
,
Yaz
,
E. E.
, and
Schneider
,
S. C.
,
2018
, “
Coupled State-Dependent Riccati Equation Control for Continuous Time Nonlinear Mechatronics Systems
,”
ASME J. Dyn. Sys., Meas., Control
,
140
(
11
), p.
111013
.10.1115/1.4040295
8.
Cloutier
,
J. R.
,
D'Souza
,
C. N.
, and
Mracek
,
C. P.
,
1996
, “
Nonlinear Regulation and Nonlinear H Control Via the State-Dependent Riccati Equation Technique: Part 1—Theory; Part 2—Examples
,”
Proceedings of the International Conference on Nonlinear Problems in Aviation and Aerospace
(With Supplementary Materials), Daytona Beach, FL, pp.
117
141
.
9.
Ghadami
,
S. M.
,
Amjadifard
,
R.
, and
Khaloozadeh
,
H.
,
2014
, “
Optimizing a Class of Nonlinear Singularly Perturbed Systems Using SDRE Technique
,”
ASME J. Dyn. Sys., Meas., Control
,
136
(
1
), p.
011003
.10.1115/1.4024602
10.
Padmanabhan
,
R.
,
Meskin
,
N.
, and
Haddad
,
W. M.
,
2017
, “
Reinforcement Learning-Based Control of Drug Dosing for Cancer Chemotherapy Treatment
,”
Math. Biosci.
,
293
, pp.
11
20
.10.1016/j.mbs.2017.08.004
11.
Qin
,
B.
,
Sun
,
H.
,
Ma
,
J.
,
Li
,
W.
,
Ding
,
T.
, and
Zomaya
,
A.
,
2019
, “
Robust H Control of Doubly Fed Wind Generator Via State-Dependent Riccati Equation Technique
,”
IEEE Trans. Power Syst.
,
34
(
3
), pp.
2390
2400
.10.1109/TPWRS.2018.2881687
12.
Unal
,
C.
, and
Salamci
,
M. U.
,
2017
, “
Drug Administration in Cancer Treatment Via Optimal Nonlinear State Feedback Gain Matrix Design
,”
Proceedings of the 20th IFAC Triennial World Congress
, Toulouse, France, pp.
9979
9984
.
13.
Ghane
,
H.
,
Sterk
,
A. E.
, and
Waalkens
,
H.
, “
Chaotic Dynamics From a Pseudo-Linear System
,”
IMA J. Math. Control Inf.
, epub.10.1093/imamci/dnz005
14.
Zhang
,
P.
,
Yuan
,
Y.
,
Guo
,
L.
, and
Liu
,
H.
,
2019
, “
Near-Optimal Control for Time-Varying Linear Discrete Systems With Additive Nonlinearities and Random Gains
,”
IEEE Trans. Autom. Control
,
64
(
7
), pp.
2968
2975
.10.1109/TAC.2018.2874707
15.
Çimen
,
T.
,
2012
, “
Survey of State-Dependent Riccati Equation in Nonlinear Optimal Feedback Control Synthesis
,”
AIAA J. Guid. Control Dyn.
,
35
(
4
), pp.
1025
1047
.10.2514/1.55821
16.
Topputo
,
F.
,
Miani
,
M.
, and
Bernelli-Zazzera
,
F.
,
2015
, “
Optimal Selection of the Coefficient Matrix in State-Dependent Control Methods
,”
AIAA J. Guid. Control Dyn.
,
38
(
5
), pp.
861
873
.10.2514/1.G000136
17.
Yu
,
M.-J.
, and
Bernstein
,
D. S.
,
2018
, “
Retrospective Cost Subsystem Estimation and Smoothing for Linear Systems With Structured Uncertainty
,”
AIAA J. Aerosp. Inf. Syst.
,
15
(
10
), pp.
566
584
.10.2514/1.I010581
18.
Chandra
,
K. P. B.
, and
Gu
,
D.-W.
,
2019
,
Nonlinear Filtering
,
Springer
,
Cham, Switzerland
, pp.
7156
161
.
19.
Lin
,
L.-G.
,
Vandewalle
,
J.
, and
Liang
,
Y.-W.
,
2015
, “
Analytical Representation of the State-Dependent Coefficients in the SDRE/SDDRE Scheme for Multivariable Systems
,”
Automatica
,
59
, pp.
106
111
.10.1016/j.automatica.2015.06.015
20.
Zhou
,
K.
,
Doyle
,
J. C.
, and
Glover
,
K.
,
1996
,
Robust and Optimal Control
,
Prentice Hall
,
NJ
.
21.
de Pillis
,
L. G.
,
Gu
,
W.
,
Fister
,
K. R.
,
Head
,
T.
,
Maples
,
K.
,
Murugan
,
A.
,
Neal
,
T.
, and
Yoshida
,
K.
,
2007
, “
Chemotherapy for Tumors: An Analysis of the Dynamics and a Study of Quadratic and Linear Optimal Controls
,”
Math. Biosci.
,
209
(
1
), pp.
292
315
.10.1016/j.mbs.2006.05.003
22.
Matveev
,
A. S.
, and
Savkin
,
A. V.
,
2002
, “
Application of Optimal Control Theory to Analysis of Cancer Chemotherapy Regimens
,”
Syst. Control Lett.
,
46
(
5
), pp.
311
321
.10.1016/S0167-6911(02)00134-2
23.
Kim
,
S.
, and
Kwon
,
S. J.
,
2017
, “
Nonlinear Optimal Control Design for Underactuated Two-Wheeled Inverted Pendulum Mobile Platform
,”
IEEE/ASME Trans. Mechatronics
,
22
(
6
), pp.
2803
2808
.10.1109/TMECH.2017.2767085
24.
Korayem
,
A. H.
,
Nekoo
,
S. R.
, and
Korayem
,
M. H.
,
2019
, “
Optimal Sliding Mode Control Design Based on the State-Dependent Riccati Equation for Cooperative Manipulators to Increase Dynamic Load Carrying Capacity
,”
Robotica
,
37
(
2
), pp.
321
337
.10.1017/S0263574718001030
25.
Safi
,
M.
,
Mortazavi
,
M.
, and
Dibaji
,
S. M.
,
2018
, “
Global Stabilization of Attitude Dynamics: SDRE-Based Control Designs
,”
AUT J. Mod. Simul.
,
50
(
2
), pp.
203
210
.10.22060/MISCJ.2018.13961.5087
26.
Wang
,
X.
,
Reitz
,
M.
, and
Yaz
,
E. E.
,
2018
, “
Field Oriented Sliding Mode Control of Surface-Mounted Permanent Magnet AC Motors: Theory and Applications to Electrified Vehicles
,”
IEEE Trans. Veh. Technol.
,
67
(
11
), pp.
10343
10356
.10.1109/TVT.2018.2865905
27.
Pittner
,
J.
, and
Simaan
,
M. A.
,
2018
, “
Streamlining the Tandem Hot-Metal-Strip Mill: Threading Progress Stems From the Use of Advanced Control With Virtual Rolling
,”
IEEE Ind. Appl. Mag.
,
24
(
2
), pp.
35
44
.10.1109/MIAS.2017.2740473
28.
Shochat
,
E.
,
Hart
,
D.
, and
Agur
,
Z.
,
1999
, “
Using Computer Simulations for Evaluating the Efficacy of Breast Cancer Chemotherapy Protocols
,”
Math. Models Meth. Appl. Sci.
,
09
(
04
), pp.
599
615
.10.1142/S0218202599000312
29.
de Pillis
,
L. G.
, and
Radunskaya
,
A.
,
2003
, “
The Dynamics of an Optimally Controlled Tumor Model: A Case Study
,”
Math. Comput. Model.
,
37
(
11
), pp.
1221
1244
.10.1016/S0895-7177(03)00133-X
30.
Benner
,
P.
, and
Saak
,
J.
,
2013
, “
Numerical Solution of Large and Sparse Continuous Time Algebraic Matrix Riccati and Lyapunov Equations: A State of the Art Survey
,”
GAMM-Mitteilungen
,
36
(
1
), pp.
32
52
.10.1002/gamm.201310003
31.
Laub
,
A. J.
,
1979
, “
A Schur Method for Solving Algebraic Riccati Equations
,”
IEEE Trans. Autom. Control
,
24
(
6
), pp.
913
921
.10.1109/TAC.1979.1102178
32.
Benner
,
P.
, and
Werner
,
S. W. R.
,
2019
, “
MORLAB—Model Order Reduction LABoratory (Version 5.0, Aug. 2019)
,” Max Planck Institute for Dynamics of Complex Technical Systems (“ml care nwt fac” function), Magdeburg, Germany, accessed Jan. 20, 2020, http://www.mpi-magdeburg.mpg.de/projects/morlab
33.
Van Dooren
,
P.
,
1981
, “
The Generalized Eigenstructure Problem in Linear System Theory
,”
IEEE Trans. Autom. Control
,
26
(
1
), pp.
111
129
.10.1109/TAC.1981.1102559
34.
Do
,
T.
,
Kwak
,
S.
,
Choi
,
H.
, and
Jung
,
J.
,
2014
, “
Suboptimal Control Scheme Design for Interior Permanent Magnet Synchronous Motors: An SDRE-Based Approach
,”
IEEE Trans. Power Electron.
,
29
(
6
), pp.
3020
3031
.10.1109/TPEL.2013.2272582
35.
Golub
,
G. H.
, and
Van Loan
,
C. F.
,
1996
,
Matrix Computations
, 3rd ed.,
Johns Hopkins University Press
,
Baltimore, MD
.
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