Abstract
This paper addresses the issue of the regulation of the dehydration air temperature in a dryer. The classical optimization approach is used to solve the optimal control problem for this kind of system because the linearized mathematical model involves the delay state. However, the presence in the model of nonlinear uncertainties, which satisfy the matching condition, is a reason to apply the Lyapunov redesign approach. It gives robustness to the optimal linear control, improving the closed-loop performance. Furthermore, the optimal–robust control algorithm was tested in a dehydration process. This is the first time that the optimal and the proposed optimal–robust controller has been applied in a real-time process.
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References
Krasovskii, N.N.: On the analytic construction of an optimal control in a system with time lags. J. Appl. Math. Mech. 26(1), 50–67 (1962)
Ross, D.W., Flügge-Lotz, I.: An optimal control problem for systems with differential-difference equation dynamics. SIAM J. Control 7(4), 609–623 (1969)
Ross, D.W.: Controller design for time lag systems via a quadratic criterion. IEEE Trans. Autom. Control 16(6), 664–672 (1971)
Uchida, K., Shimemura, E.: Closed-loop properties of the infinite-time linear-quadratic optimal regulator for systems with delays. Int. J. Control 43(3), 773–779 (1986)
Uchida, K., Shimemura, E., Abe, N.: Circle condition and stability margin of the optimal regulator for systems with delays. Int. J. Control 46(4), 1203–1212 (1987)
Kim, A.V., Lozhnikov, A.B.: A linear-quadratic control problem for state-delay systems. Exact solutions for the Riccati equations. Autom. Rem. Control C/C Avtomatika i Telemekhanika 61(7;1), 1076–1090 (2000)
Santos, O., Mondié, S., Kharitonov, V.L.: Linear quadratic suboptimal control for time delays systems. Int. J. Control 82(1), 147–154 (2009)
Kharitonov, V.L., Zhabko, A.P.: Lyapunov Krasovskii approach to the robust stability analysis of time-delay systems. Automatica 39(1), 15–20 (2003)
Kirk, D.E.: Optimal control theory: an introduction. Courier Corporation, Chelmsford (2012)
Wu, H.S.: Sufficient conditions for robust stability of LQG optimal control systems including delayed perturbations. J Optim. Theory Appl. 96(2), 437–451 (1998)
Kosmidou, O.I., Boutalis, Y.S.: A linear matrix inequality approach for guaranteed cost control of systems with state and input delays. IEEE Trans. Syst. Man Cybern. Part A: Syst. Hum. 36(5), 936–942 (2006)
Lien, C.H.: Guaranteed cost output control for uncertain neutral systems with time-varying delays via LMI. Int. J. Syst. Sci. 37(10), 723–734 (2006)
Lin, Y.C., Lin, C.L.: Optimal control approach for robust control design of neutral systems. Optim. Control Appl. Methods 30(1), 87–102 (2009)
Chuang, C.H., Lin, C.L., Lin, Y.C.: Robust optimal control of time delay systems based on Razumikhin theorem. Asian J. Control 14(5), 1431–1438 (2012)
Ariola, M., Pironti, A.: H\(\infty \) optimal terminal control for linear systems with delayed states and controls. Automatica 44(10), 2676–2679 (2008)
Niamsupa, P., Phatb, V.N.: H\(\infty \) optimal control of linear time-varying systems with time-varying delay via a controllability approach. Sci. Asia 35, 284–289 (2009)
Glizer, V.Y., Turetsky, V.: Optimal quadratic control of linear time delay systems: one approach to numerical solution. In: 11th IEEE International Conference on Control and Automation (ICCA), pp. 797–802. IEEE (2014)
Erfani, A., Rezaei, S., Pourseifi, M., Derili, H.: Optimal control in teleoperation systems with time delay: a singular perturbation approach. J. Comput. Appl. Math. 338, 168–184 (2018)
Marzban, H.R., Hoseini, S.M.: Optimal control of linear multi-delay systems with piecewise constant delays. IMA J. Math. Control Inf. 35(1), 183–212 (2018)
Rodríguez-Guerrero, L., Santos-Sánchez, O.J., Cervantes-Escorcia, N., Romero, H.: Real-time discrete suboptimal control for systems with input and state delays: experimental tests on a dehydration process. ISA Trans. 71, 448–457 (2017)
Khalil, H.K.: Nonlinear Systems. Prentice-Hall, New Jersey (1996)
Rodríguez-Guerrero, L., Mondié, S., Santos-Sánchez, O.: Guaranteed cost control using Lyapunov redesign for uncertain linear time delay systems. IFAC-PapersOnLine 48(12), 392–397 (2015)
Rodríguez-Guerrero, L., Santos-Sánchez, O., Mondié, S.: A constructive approach for an optimal control applied to a class of nonlinear time delay systems. J. Process Control 40, 35–49 (2016)
Santos-Sánchez, N.F., Salas-Coronado, R., Santos-Sánchez, O.J., Romero, H., Garrido-Aranda, E.: On the effects of the temperature control at the performance of a dehydration process: energy optimization and nutrients retention. Int. J. Adv. Manuf. Technol. 86(9–12), 3157–3171 (2016)
Di Ferdinando, M., Pepe, P.: Robustification of sample-and-hold stabilizers for control-affine time-delay systems. Automatica 83, 141–154 (2017)
Li, Z., Wang, Q., Gao, H.: Control of friction oscillator by Lyapunov redesign based on delayed state feedback. Acta Mechanica Sinica 25(2), 257–264 (2009)
Pepe, P.: On Sontag’s formula for the input-to-state practical stabilization of retarded control-affine systems. Syst. Control Lett. 62(11), 1018–1025 (2013)
Acknowledgements
This work was supported by Conacyt-Mexico Project: 239371, and PRODEP-Mexico UAEH-PTC 776, 511-617-8021. The authors thank English Professor Carol Hayenga for her English revision.
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Communicated by Lars Grüne.
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López-Labra, HA., Santos-Sánchez, OJ., Rodríguez-Guerrero, L. et al. Experimental Results of Optimal and Robust Control for Uncertain Linear Time-Delay Systems. J Optim Theory Appl 181, 1076–1089 (2019). https://doi.org/10.1007/s10957-018-01457-9
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DOI: https://doi.org/10.1007/s10957-018-01457-9