Abstract
A spatially discrete control law is proposed for a class of systems described by scalar linear differential equations of parabolic and hyperbolic types with unknown parameters and disturbances. A finite set of discrete measurements (with respect to the spatial variable) of the plant state is available. The control law depends on a function that depends on the spatial variable and on a finite set of measurements of the plant state. Examples of this function, which allows realizing the control signal only at certain intervals in the spatial variable and providing lower control costs than some other analogs, are given. The exponential stability of the closed-loop system and robustness with respect to interval uncertain parameters of the plant and exogenous bounded disturbances are proved. Numerical modeling examples confirm the results of calculations and show the efficiency of the algorithm compared with some existing analogs.
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REFERENCES
Candogan, U.O., Ozbay, H., and Ozaktas, H.M., Controller implementation for a class of spatially-varying distributed parameter systems, IFAC Proc. Vols. (Proc. 17th IFAC World Congr.), 2008, vol. 41, no. 2, pp. 7755–7760.
Hagen, G. and Mezic, I., Spillover stabilization in finite-dimensional control observer design for dissipative evolution equations, SIAM J. Control Optim., 2003, vol. 42, no. 2, pp. 746–768.
Smagina, E. and Sheintuch, M., Using Lyapunov’s direct method for wave suppression in reactive systems, Syst. Control Lett., 2006, vol. 55, no. 7, pp. 566–572.
Demetriou, M.A., Guidance of mobile actuator-plus-sensor networks for improved control and estimation of distributed parameter systems, IEEE Trans. Autom. Control, 2010, vol. 55, pp. 1570–1584.
Smyshlyaev, A. and Krstic, M., On control design for PDEs with space-dependent diffusivity or time-dependent reactivity, Automatica, 2005, vol. 41, pp. 1601–1608.
Krstic, M. and Smyshlyaev, A., Adaptive boundary control for unstable parabolic PDEs—part I: Lyapunov design, IEEE Trans. Autom. Control, 2008, vol. 53, pp. 1575–1591.
Delchamps, D.F., Extracting state information from a quantized output record, Syst. Control Lett., 1989, vol. 13, pp. 365–372.
Brockett, R.W. and Liberzon, D., Quantized feedback stabilization of linear systems, IEEE Trans. Autom. Control, 2000, vol. 45, pp. 1279–1289.
Baillieul, J., Feedback coding for information-based control: operating near the data rate limit, Proc. 41st IEEE Conf. Decis. Control (Las Vegas, Nevada, USA, 2002), pp. 3229–3236.
Zheng, B.-C. and Yang, G.-H., Quantized output feedback stabilization of uncertain systems with input nonlinearities via sliding mode control, Int. J. Robust Nonlinear Control, 2012, vol. 24, no. 2, pp. 228–246.
Khapalov, A.Y., Continuous observability for parabolic system under observations of discrete type, IEEE Trans. Autom. Control, 1993, vol. 38, no. 9, pp. 1388–1391.
Cheng, M.B., Radisavljevic, V., Chang, C.C., Lin, C.F., and Su, W.C., A sampled data singularly perturbed boundary control for a diffusion conduction system with noncollocated observation, IEEE Trans. Autom. Control, 2009, vol. 54, no. 6, pp. 1305–1310.
Logemann, H., Rebarber, R., and Townley, S., Generalized sampled-data stabilization of well-posed linear infinite-dimensional systems, SIAM J. Control Optim., 2005, vol. 44, no. 4, pp. 1345–1369.
Fridman, E. and Blighovsky, A., Robust sampled-data control of a class of semilinear parabolic systems, Automatica, 2012, vol. 48, pp. 826–836.
Liu, K., Fridman, E., and **a, Y., Networked Control under Communication Constraints: A Time-Delay Approach. Advances in Delays and Dynamics, New York: Springer Int. Publ., 2020.
Hardy, G.H., Littlewood, J.E., and Polya, G., Inequalities, Cambridge: Cambridge Univ. Press, 1988.
Fridman, E., Introduction to Time-Delay Systems. Analysis and Control, Basel: Birkhäuser, 2014.
Henry, D., Geometric Theory of Semilinear Parabolic Equations, New York: Springer-Verlag, 1993.
Curtain, R. and Zwart, H., An Introduction to Infinite-Dimensional Linear Systems Theory, New York: Springer-Verlag, 1995.
Funding
The results in Sec. 3 were produced with support from the Russian Science Foundation, project no. 18-79-10104, at the Institute for Problems in Mechanical Engineering, Russian Academy of Sciences (IPME RAS). The results in Secs. 4 and 5 were produced with support from the Russian Foundation for Basic Research, project no. 19-08-00246, at IPME RAS. The results in Secs. 6 and 7 were supported by a grant from the President of the Russian Federation, project no. MD-1054.2020.8, at IPME RAS.
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Translated by V. Potapchouck
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Furtat, I.B., Gushchin, P.A. Spatially Discrete Control of Scalar Linear Distributed Plants of Parabolic and Hyperbolic Types. Autom Remote Control 82, 433–448 (2021). https://doi.org/10.1134/S0005117921030048
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DOI: https://doi.org/10.1134/S0005117921030048