Abstract
A new formulation of the minimum energy control problem for the positive 2D continuous-discrete linear systems is proposed. Necessary and sufficient conditions for the reachability of the systems are established. Conditions for the existence of the solution to the minimum energy control problem and procedures for computation of an input minimizing the given performance index are given. Effectiveness of the procedure is demonstrated on numerical example.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bistritz, Y.: A stability test for continuous-discrete bivariate polynomials. In: Proc. Int. Symp. on Circuits and Systems, vol. 3, pp. 682–685 (2003)
Busłowicz, M.: Stability and robust stability conditions for a general model of scalar continuous-discrete linear systems. Measurement Automation and Monitoring 56(2), 133–135 (2010)
Busłowicz, M.: Robust stability of the new general 2D model of a class of continuous-discrete linear systems. Bull. Pol. Acad. Sci. Techn. 57(4) (2010)
Dymkov, M., Gaishun, I., Rogers, E., Gałkowski, K., Owens, D.H.: Control theory for a class of 2D continuous-discrete linear systems. Int. J. Control 77(9), 847–860 (2004)
Farina, L., Rinaldi, S.: Positive Linear Systems; Theory and Applications. J. Wiley, New York (2000)
Gałkowski, K., Rogers, E., Paszke, W., Owens, D.H.: Linear repetitive process control theory applied to a physical example. Int. J. Appl. Math. Comput. Sci. 13(1), 87–99 (2003)
Kaczorek, T.: Minimum energy control of fractional positive continuous-time linear systems. In: Proc. of Conf. MMAR Międzyzdroje, Poland, August 26-29 (2013)
Kaczorek, T.: Minimum energy control of descriptor positive discrete-time linear systems. Compel 3(4) (2013) (in press)
Kaczorek, T.: New stability tests of positive standard and fractional linear systems. Circuit and Systems 2(4), 261–268 (2011)
Kaczorek, T.: Positive 1D and 2D systems. Springer, London (2001)
Kaczorek, T.: Positive 2D hybrid linear systems. Bull. Pol. Acad. Sci. Tech. 55(4), 351–358 (2007)
Kaczorek, T.: Positive fractional 2D continuous-discrete linear systems. Bull. Pol. Acad. Tech. 59(4), 575–579 (2011)
Kaczorek, T.: Positive fractional 2D hybrid linear systems. Bull. Pol. Acad. Tech. 56(3), 273–277 (2008)
Kaczorek, T.: Reachability and minimum energy control of positive 2D continuous-discrete systems. Bull. Pol. Acad. Sci. Tech. 46(1), 85–93 (1998)
Kaczorek, T.: Realization problem for positive 2D hybrid systems. Compel 27(3), 613–623 (2008)
Kaczorek, T.: Selected Problems of Fractional Systems Theory. LNCIS, vol. 411. Springer, Heidelberg (2011)
Kaczorek, T.: Stability of continuous-discrete linear systems described by general model. Bull. Pol. Acad. Sci. Tech. 59(2), 189–193 (2011)
Kaczorek, T.: Minimum energy control of positive continuous-time linear systems with bounded inputs. Int. J. Appl. Math. Comput. Sci. (2013) (in press)
Kaczorek, T.: Necessary and sufficient conditions for the minimum energy control of positive discrete-time linear systems with bounded inputs. Bull. Pol. Acad. Sci. Tech. (2013) (in press)
Kaczorek, T., Klamka, J.: Minimum energy control of 2D linear systems with variable coefficients. Int. J. of Control 44(3), 645–650 (1986)
Klamka, J.: Controllability and minimum energy control problem of fractional discrete-time systems. In: Baleanu, D., Guvenc, Z.B., Tenreiro Machado, J.A. (eds.) New Trends in Nanotechnology and Fractional Calculus, pp. 503–509. Springer, New York (2010)
Klamka, J.: Controllability of Dynamical Systems. Kluwer Academic Press, Dordrecht (1991)
Klamka, J.: Minimum energy control of 2D systems in Hilbert spaces. System Sciences 9(1-2), 33–42 (1983)
Klamka, J.: Relative controllability and minimum energy control of linear systems with distributed delays in control. IEEE Trans. Autom. Contr. 21(4), 594–595 (1976)
Kaczorek, T., Marchenko, V., Sajewski, Ł.: Solvability of 2D hybrid linear systems - comparison of three different methods. Acta Mechanica et Automatica 2(2), 59–66 (2008)
Narendra, K.S., Shorten, R.: Hurwitz stability of Metzler matrices. IEEE Trans. Autom. Contr. 55(6), 1484–1487 (2010)
Sajewski, Ł.: Solution of 2D singular hybrid linear systems. Kybernetes 38(7/8), 1079–1092 (2009)
Xiao, Y.: Stability test for 2-D continuous-discrete systems. In: Proc. 40th IEEE Conf. on Decision and Control, vol. 4, pp. 3649–3654 (2001)
Xiao, Y.: Stability, controllability and observability of 2-D continuous-discrete systems. In: Proc. Int. Symp. on Circuits and Systems, vol. 4, pp. 468–471 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Kaczorek, T. (2014). A New Formulation and Solution of the Minimum Energy Control Problem of Positive 2D Continuous-Discrete Linear Systems. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds) Recent Advances in Automation, Robotics and Measuring Techniques. Advances in Intelligent Systems and Computing, vol 267. Springer, Cham. https://doi.org/10.1007/978-3-319-05353-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-05353-0_11
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05352-3
Online ISBN: 978-3-319-05353-0
eBook Packages: EngineeringEngineering (R0)