Sub-optimal controller design for time-delay nonlinear partial differential equation systems: an extended state-dependent differential Riccati equation approach

References

• Akkouche, A., Maidi, A., & Aidene, M. (2014). Optimal control of partial differential equations based on the variational iteration method. Computers & Mathematics with Applications, 68(5), 622–631. https://doi.org/10.1016/j.camwa.2014.07.007
   Web of Science ®Google Scholar
• Aksikas, I., Winkin, J. J., & Dochain, D. (2007). Optimal LQ-feedback regulation of a nonisothermal plug flow reactor model by spectral factorisation. IEEE Transactions on Automatic Control, 52(7), 1179–1193. https://doi.org/10.1109/TAC.2007.900823
   Web of Science ®Google Scholar
• Apkarian, P., & Noll, D. (2020). Boundary control of partial differential equations using frequency domain optimization techniques. Systems & Control Letters, 135, 104577. https://doi.org/10.1016/j.sysconle.2019.104577
   Web of Science ®Google Scholar
• Arias, E., Hernández, V., Ibáñez, J. J., & Peinado, J. (2007). A fixed point-based BDF method for solving differential Riccati equations. Applied Mathematics and Computation, 188(2), 1319–1333. https://doi.org/10.1016/j.amc.2006.11.001
   Web of Science ®Google Scholar
• Balasubramanian, P., Kosuri, M., Pushpavanam, S., & Kienle, A. (2003). Effect of delay on the stability of a coupled reactor- separator system. Industrial & Engineering Chemistry Research, 42(16), 3758–3764. https://doi.org/10.1021/ie0208826
   Web of Science ®Google Scholar
• Batmani, Y., & Khaloozadeh, H. (2016). On the design of suboptimal sliding manifold for a class of nonlinear uncertain time-delay systems. International Journal of Systems Science, 47(11), 2543–2552. https://doi.org/10.1080/00207721.2014.999263
   Web of Science ®Google Scholar
• Batmani, Y., Takhtabnus, M., & Mirzaei, R. (2022). DC microgrid fault-tolerant control using state-dependent Riccati equation techniques. Optimal Control Applications and Methods, 43(1), 123–137. https://doi.org/10.1002/oca.v43.1
   Web of Science ®Google Scholar
• Bavarsad, A., Fakharian, A., & Menhaj, M. B. (2021). Nonlinear observer-based optimal control of an active transfemoral prosthesis. Journal of Central South University, 28(1), 140–152. https://doi.org/10.1007/s11771-021-4592-2
   Web of Science ®Google Scholar
• Breiten, T., Dolgov, S., & Stoll, M. (2019). Solving differential Riccati equations: A nonlinear space-time method using tensor trains. arXiv preprint arXiv:1912.06944.
   Google Scholar
• Cimen, T. (2012). Survey of state-dependent Riccati equation in nonlinear optimal feedback control synthesis. Journal of Guidance, Control, and Dynamics, 35(4), 1025–1047. https://doi.org/10.2514/1.55821
   Web of Science ®Google Scholar
• Cruz-Quintero, E., & Jurado, F. (2020). Boundary control for a certain class of reaction–advection–diffusion system. Mathematics, 8(11), 1854. https://doi.org/10.3390/math8111854
   Web of Science ®Google Scholar
• Ding, X. L., & Jiang, Y. L. (2018). Analytical solutions for multi-term time-space coupling fractional delay partial differential equations with mixed boundary conditions. Communications in Nonlinear Science and Numerical Simulation, 65, 231–247. https://doi.org/10.1016/j.cnsns.2018.05.022
   Web of Science ®Google Scholar
• Fang, Z., & Gao, C. (2019). Lyapunov function partial differential equations for chemical reaction networks: Some special cases. SIAM Journal on Applied Dynamical Systems, 18(2), 1163–1199. https://doi.org/10.1137/17M1145884
   Web of Science ®Google Scholar
• Geng, F., & Li, X. (2012). A new method for Riccati differential equations based on reproducing kernel and quasilinearization methods. Abstract and Applied Analysis, 2012(SI2), 1–8. https://doi.org/10.1155/2012/603748
   Google Scholar
• He, L., & Liu, A. (2010). Existence and uniqueness of solutions for nonlinear impulsive partial differential equations with delay. Nonlinear Analysis: Real World Applications, 11(2), 952–958. https://doi.org/10.1016/j.na.2008.01.024
   Web of Science ®Google Scholar
• John, V. (2013). Numerical methods for partial differential equations. Lecture notes.
   Google Scholar
• Kalamian, N., Khaloozadeh, H., & Ayati, M. (2019). Design of state-dependent impulsive observer for non-linear time-delay systems. IET Control Theory & Applications, 13(18), 3155–3163. https://doi.org/10.1049/cth2.v13.18
   Web of Science ®Google Scholar
• Kalamian, N., Khaloozadeh, H., & Ayati, S. M. (2020). Adaptive state-dependent impulsive observer design for nonlinear deterministic and stochastic dynamics with time-delays. ISA Transactions, 98, 87–100. https://doi.org/10.1016/j.isatra.2019.08.034
   PubMed Web of Science ®Google Scholar
• Kamalifar, A., Menhaj, M., Monfared, M. N., & Fakharian, A. (2022). Design of robust model reference adaptive controller for a wider class of nonlinear systems. Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 46(1), 127–139. https://doi.org/10.1007/s40998-021-00451-8
   Web of Science ®Google Scholar
• Khodakaramzadeh, S., Batmani, Y., & Meskin, N. (2019). Automatic blood glucose control for type 1 diabetes: A trade-off between postprandial hyperglycemia and hypoglycemia. Biomedical Signal Processing and Control, 54, 101603. https://doi.org/10.1016/j.bspc.2019.101603
   Web of Science ®Google Scholar
• Korayem, M. H., & Nekoo, S. (2015a). Finite-time state-dependent Riccati equation for time-varying nonaffine systems: Rigid and flexible joint manipulator control. ISA Transactions, 54, 125–144. https://doi.org/10.1016/j.isatra.2014.06.006
   PubMed Web of Science ®Google Scholar
• Korayem, M. H., & Nekoo, S. (2015b). State-dependent differential Riccati equation to track control of time-varying systems with state and control nonlinearities. ISA Transactions, 57, 117–135. https://doi.org/10.1016/j.isatra.2015.02.008
   PubMed Web of Science ®Google Scholar
• Lei, Y., Liu, X., & Xie, C. (2020). Stabilization of an ODE–PDE cascaded system by boundary control. Journal of the Franklin Institute, 357(14), 9248–9267. https://doi.org/10.1016/j.jfranklin.2020.07.007
   Web of Science ®Google Scholar
• Leugering, G., Imanuvilov, O., Zhang, B. Y., & Triggiani, R. (2005). Control theory of partial differential equations. CRC Press.
   Google Scholar
• Liavoli, F. B., & Fakharian, A. (2017). Nonlinear optimal control of air handling unit via state dependent Riccati equation approach. In 2017 5th International Conference on Control, Instrumentation, and Automation (ICCIA) (pp. 138–143). IEEE.
   Google Scholar
• Liavoli, F. B., & Fakharian, A. (2019). Sub-optimal observer-based controller design using the state dependent Riccati equation approach for air-handling unit. In 2019 27th Iranian Conference on Electrical Engineering (ICEE) (pp. 991–996). IEEE.
   Google Scholar
• Liu, Z., Liu, J., & He, W. (2017). Partial differential equation boundary control of a flexible manipulator with input saturation. International Journal of Systems Science, 48(1), 53–62. https://doi.org/10.1080/00207721.2016.1152416
   Web of Science ®Google Scholar
• Mazenc, F., & Prieur, C. (2011). Strict Lyapunov functionals for nonlinear parabolic partial differential equations. IFAC Proceedings Volumes, 44(1), 12550–12555. https://doi.org/10.3182/20110828-6-IT-1002.01572
   Google Scholar
• Monfared, M. N., Fakharian, A., Menhaj, M. B., & Abbasi, R. (2020). Optimal control of immunogenic tumor cells population growth. Journal of Control Engineering and Applied Informatics, 22(3), 5–12.
   Google Scholar
• Nasiri, N., Fakharian, A., & Menhaj, M. B. (2020). Observer-based robust control for flexible-joint robot manipulators: A state-dependent Riccati equation-based approach. Transactions of the Institute of Measurement and Control, 42(16), 3135–3155. https://doi.org/10.1177/0142331220941653
   Web of Science ®Google Scholar
• Nasiri, N., Fakharian, A., & Menhaj, M. B. (2021). A novel controller for nonlinear uncertain systems using a combination of SDRE and function approximation technique: Regulation and tracking of flexible-joint manipulators. Journal of the Franklin Institute, 358(10), 5185–5212. https://doi.org/10.1016/j.jfranklin.2021.04.037
   Web of Science ®Google Scholar
• Nekoo, S. R. (2013). Nonlinear closed loop optimal control: A modified state-dependent Riccati equation. ISA Transactions, 52(2), 285–290. https://doi.org/10.1016/j.isatra.2012.10.005
   PubMed Web of Science ®Google Scholar
• Nekoo, S. R., Acosta, J., & Ollero, Á. (2022). Quaternion-based state-dependent differential Riccati equation for quadrotor drones: Regulation control problem in aerobatic flight. Robotica, 40(9), 1–16. https://doi.org/10.1017/S0263574722000091
   Web of Science ®Google Scholar
• Nguyen, T., & Gajic, Z. (2009). Solving the matrix differential Riccati equation: A Lyapunov equation approach. IEEE Transactions on Automatic Control, 55(1), 191–194. https://doi.org/10.1109/TAC.2009.2033841
   Web of Science ®Google Scholar
• Pitarch, J., Rakhshan, M., Mardani, M., Sadeghi, M., & de Prada, C. (2016). Distributed nonlinear control of a plug-flow reactor under saturation. IFAC-PapersOnLine, 49(24), 87–92. https://doi.org/10.1016/j.ifacol.2016.10.760
   Google Scholar
• Polyanin, A. D., & Sorokin, V. G. (2021). A method for constructing exact solutions of nonlinear delay PDEs. Journal of Mathematical Analysis and Applications, 494(2), 124619. https://doi.org/10.1016/j.jmaa.2020.124619
   Web of Science ®Google Scholar
• Quarteroni, A., & Valli, A. (2008). Numerical approximation of partial differential equations. (Vol. 23). Springer Science & Business Media.
   Google Scholar
• Rafee Nekoo, S. (2020). A PDE breach to the SDRE. Asian Journal of Control, 22(2), 667–676. https://doi.org/10.1002/asjc.v22.2
   Web of Science ®Google Scholar
• Ruas, V. (2016). Numerical methods for partial differential equations: An introduction. John Wiley & Sons.
   Google Scholar
• Shadi, R., Liavoli, F. B., & Fakharian, A. (2021). Nonlinear sub-optimal controller for Ebola virus disease: State-dependent Riccati equation approach. In 2021 7th International Conference on Control, Instrumentation and Automation (ICCIA) (pp. 1–6). IEEE.
   Google Scholar
• Shahradfar, E., & Fakharian, A. (2021). Optimal controller design for DC microgrid based on state-dependent Riccati equation (SDRE) approach. Cyber-Physical Systems, 7(1), 41–72. https://doi.org/10.1080/23335777.2020.1811381
   Google Scholar
• Smyshlyaev, A., & Krstic, M. (2010). Adaptive control of parabolic PDEs. Princeton University Press.
   Google Scholar
• Solodushkin, S. I., Yumanova, I. F., & De Staelen, R. H. (2015). First order partial differential equations with time delay and retardation of a state variable. Journal of Computational and Applied Mathematics, 289, 322–330. https://doi.org/10.1016/j.cam.2014.12.032
   Web of Science ®Google Scholar