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Finite settling time stabilisation of a family of discrete time systems. (English) Zbl 0768.93061
Summary: The problem of finding a compensator $$C$$, that stabilises in the Finite Settling Time (FST) sense, a family of $$k$$ discrete time plants $$\{P_ i,\;i=1,\dots,k\}$$, is referred to as Simultaneous FST Stabilisation Problem (S-FSTSP) and it is examined here. The general case of many input, many output plants is considered first and algebraic conditions for the existence of a S-FSTS controller are derived in terms of the properties of the plants family matrix. For the special case of single input many output (SIMO), and many input single output (MISO) plant families, testable necessary and sufficient conditions for solvability of S-SFTSP are derived and whenever a solution exists, the family of solution is given. The nature of the results is algebraic, since they depend on the properties of a rational vector space associated with the family; however, the final conditions are expressed as standard linear algebra tests.
##### MSC:
 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93C55 Discrete-time control/observation systems
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##### References:
 [1] G. D. Forney: Minimal bases of rational vector spaces with applications to multivariable linear systems. SIAM J. Control Optim. 13 (1975), 493 - 520. · Zbl 0269.93011 · doi:10.1137/0313029 [2] B. K. Ghosh, C. I. Byrnes: Simultaneous stabilisation and simultaneous pole placement by non-switching dynamic compensation. IEEE Trans. Automat. Control AC-28 (1983), 735-741. [3] R. E. Kalman: On the general theory of control systems. Proc. 1st IFAC Congress Automat. Control, Moscow 1960, 4, pp. 481-492. [4] N. Karcanias: Matrix Equations over Principal Ideal Domains. City Univ., Control Engin. Centre Research Report, London 1987. [5] N. Karcanias, E. Milonidis: Total finite settling time stabilisation for discrete time SISO systems. IMA Control Theory Conf. Univ. of Strathclyde, Glasgow 1988. · Zbl 1125.93386 [6] V. Kučera: The structure and properties of time-optimal discrete linear control. IEEE Trans. Automat. Control AC-16 (1971), 375-377. [7] V. Kučera: Discrete Linear Control: The Polynomial Equation Approach. J. Wiley, New York 1979. [8] V. Kučera: Polynomial design of dead-beat control laws. Kybernetika 16 (1980), 198-203. [9] E. Milonidis, N. Karcanias: Total Finite Settling Time Stabilisation for Discrete Time MIMO systems. City University, Control Engineering Centre, Research Report CEC/ EM-NK/101, London 1990. · Zbl 1125.93386 [10] R. Saeks, J. Murray: Fractional representation, algebraic geometry, and the simultaneous stabilisation problem. IEEE Trans. Automat. Control AC-27 (1982), 4, 895 - 903. · Zbl 0495.93045 · doi:10.1109/TAC.1982.1103005 [11] M. Vidyasagar: Control System Synthesis: A Factorization Approach. MIT Press, Boston, Mass. 1985. · Zbl 0655.93001 [12] Y. Zhao, H. Kimura: Multivariate dead-beat control with robustness. Internat. J. Control 47 (1988), 229-255. · Zbl 0638.93026 · doi:10.1080/00207178808906009
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