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Discrete-time MPC for switched systems with applications to biomedical problems. (English) Zbl 1457.90096
Summary: This paper studies switched systems in which the manipulated control action is the time-depending switching signal. To control the switched systems means to select an autonomous system – at each time step – among a given finite family. Even when this selection can be done by solving a Dynamic Programming (DP) problem, such a solution is often difficult to apply, and state/control constraints cannot be explicitly considered. In this work a new set-based Model Predictive Control (MPC) strategy is proposed to handle switched systems in a tractable form. The optimization problem at the core of the MPC formulation consists in an easy-to-solve mixed-integer optimization problem, whose solution is applied in a receding horizon way. Applications to schedule therapies in viral infection and cancer treatments are studied. The numerical results suggest that the proposed strategy outperforms the schedule for available treatments.
90C11 Mixed integer programming
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
Gurobi; Matlab; YALMIP
Full Text: DOI
[1] Liberzon, D., Switching in system and control (2003)
[2] Chapman, M.; Mazumdar, E.; Langer, E.; Sears, R.; Tomlin, C., On the analysis of cyclic drug schedules for cancer treatment using switched dynamical systems, 3503-3509 (2018)
[3] Hernandez-Vargas, E. A., Modeling and control of infectious diseases in the host: with MATLAB and r, academic press - ELSEVIER (2019)
[4] Clavel, F.; J Hance, A., Hiv drug resistance, N Engl J Med, 350, 1023-1035 (2004)
[5] Martinez-Cajas, J.; Wainberg, M., Antiretroviral therapy - optimal sequencing of therapy to avoid resistance, Drugs, 68, 43-72 (2008)
[6] Molla et. al., A., Ordered accumulation of mutations in HIV protease confers resistance to ritonavir, Nat Med, 2 (1996)
[7] Risom, T.; Langer, E.; Chapman, M.; et. al., Differentiation-state plasticity is a targetable resistance mechanism in basal-like breast cancer, Nat Commun, 9, 1-17 (2018)
[8] Migliardi, G.; Sassi, F.; Torti, D.; Galimi, F.; Zanella, E.; Buscarino, M., Inhibition of MEK and PI3k/mTOR suppresses tumor growth but does not cause tumor regression in patient-derived xenografts of RAS-mutant colorectal carcinomas, Clin Cancer Res, 18, 2515-2525 (2012)
[9] Jokinen, E.; Koivunen, J., MEK and PI3k inhibition in solid tumors: rationale and evidence to date, Ther Adv Med Oncol, 7, 170-180 (2015)
[10] Grilley-Olson, J.; Bedard, P.; Fasolo, A.; et. al., A phase ib dose-escalation study of the mek inhibitor trametinib in combination with the PI3k/mTOR inhibitor GSK2126458 in patients with advanced solid tumors, Invest New Drugs, 34, 740-749 (2016)
[11] Bertsekas, D. P.; Bertsekas, D. P.; Bertsekas, D. P.; Bertsekas, D. P., Dynamic programming and optimal control, 1 (1995), Athena scientific Belmont, MA · Zbl 0935.90037
[12] Rawlings, J. B.; Mayne, D. Q.; Diehl, M., Model predictive control: theory, computation, and design, 2 (2017), Nob Hill Publishing Madison, WI
[13] Mayne, D. Q.; Rawlings, J. B.; Rao, C. V.; Scokaert, P. O.M., Constrained model predictive control: stability and optimality, Automatica, 36, 789-814 (2000) · Zbl 0949.93003
[14] Rawlings, J. B.; Mayne, D. Q., Model predictive control: theory and design (2009), Nob-Hill Publishing
[15] Anderson, A.; González, A. H.; Ferramosca, A.; Kofman, E., Finite-time convergence results in robust model predictive control, Optimal Control Applications & Methods (2018) · Zbl 1402.93095
[16] Blanchini, F.; Miani, S., Set-Theoretic methods in control, Systems & control: foundations & applications (2015), Springer International Publishing · Zbl 1417.93008
[17] Anderson, A.; González, A. H.; Ferramosca, A.; D’Jorge, A.; Kofman, E., Robust MPC suitable for closed-loop re-identification, based on probabilistic invariant sets, Systems & Control Letters, 118, 84-93 (2018) · Zbl 1402.93094
[18] Sun, Z.; Ge, S., Stability Theory of Switched Dynamical Systems, 1-253 (2011) · Zbl 1298.93006
[19] C. Geromel, J.; Colaneri, P., Stability and stabilization of discrete time switched systems, International Journal of Control - INT J CONTR, 79, 719-728 (2006) · Zbl 1330.93190
[20] Fiacchini, M.; Jungers, M., Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: a set-theory approach, Automatica, 50, 75-83 (2014) · Zbl 1298.93254
[21] Sun, Z.; Sam Ge, S., Switched linear systems: control and design (2005)
[22] Locatelli, A., Optimal control: an introduction, 55 (2001) · Zbl 1096.49500
[23] Hernandez-Vargas, E.; Colaneri, P.; Middleton, R.; Blanchini, F., Discrete-time control for switched positive systems with application to mitigating viral escape, Int J Robust Nonlinear Control, 21, 10, 1093-1111 (2011) · Zbl 1225.93072
[24] González, A.; Ferramosca, A.; Bustos, G.; Marchetti, J.; Fiacchini, M.; Odloak, D., Model predictive control suitable for closed-loop re-identification, Systems and Control Letters, 69, 23-33 (2014) · Zbl 1288.93035
[25] Löfberg, J., YALMIP : A Toolbox for modeling and optimization in MATLAB, Proceedings of the CACSD conference. Taipei, Taiwan (2004)
[26] The MathWorks Inc.. Matlab R2017b. 2017https://www.mathworks.com.
[27] Gurobi Optimization LLC. Gurobi optimizer reference manual. 2019. http://www.gurobi.com.
[28] Land, A. H.; Doig, A. G., An automatic method of solving discrete programming problems, Econometrica, 28, 3, 497-520 (1960) · Zbl 0101.37004
[29] AIDSInfo, Panel of antiretroviral guidelines for adults and adolescents, Department of Health and Human Services, Washington, Guidelines for the use of Antiretroviral Agents in HIV-1 Infected Adults and Adolescents (2013)
[30] Kumar N., Cramer G., Zamani Dahaj S.A., Sundaram B., Celli J., Kulkarni R.. Stochastic modeling of phenotypic switching and chemoresistance in cancer cell populations. 2019.
[31] Chapman, M.; Risom, T.; Aswani, A.; Dobbe, R.; Sears, R.; Tomlin, C., A model of phenotypic state dynamics initiates a promising approach to control heterogeneous malignant cell populations, 2481-2487 (2016)
[32] Goldman, A.; Majumder, B.; Dhawan, A.; Ravi, S.; Goldman, D.; Kohandel, M., Temporally sequenced anticancer drugs overcome adaptive resistance by targeting a vulnerable chemotherapy-induced phenotypic transition, Nat Commun, 6, 6139 (2015)
[33] Hernandez-Vargas, E. A.; Velasco-Hernandez, J. X., In-host modelling of COVID-19 kinetics in humans, Annu Rev Control (2020)
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