Balanced reduction of linear periodic systems. (English) Zbl 1274.93112

Summary: For linear periodic discrete-time systems, the analysis of the model error introduced by a truncation on the balanced minimal realization is performed, and a bound for the infinity norm of the model error is introduced. The results represent an extension to the periodic systems of the well known results on the balanced truncation for time-invariant systems. The general case of periodically time-varying state-space dimension has been considered.


93C05 Linear systems in control theory
93D15 Stabilization of systems by feedback
93C55 Discrete-time control/observation systems
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[1] Al-Saggaf U. M., Franklin G. F.: An error bound for a discrete reduced order model of a linear multivariable system. IEEE Trans. Automat. Control AC-32 (1987), 9, 815-819 · Zbl 0622.93021 · doi:10.1109/TAC.1987.1104712
[2] Bittanti S.: Deterministic and stochastic linear periodic systems. Time Series and Linear Systems (S. Bittanti, Springer-Verlag, Berlin 1986 · Zbl 0611.62107
[3] Bolzern P., Colaneri P.: Existence and uniqueness conditions for the periodic solutions of the discrete-time periodic Lyapunov equations. Proc. of 25th Conference on Decision and Control, Athens 1986, pp. 1439-1443
[4] Bolzern P., Colaneri P., Scattolini R.: Zeros of discrete-time linear periodic systems. IEEE Trans. Automat. Control AC-31 (1986), 1057-1059 · Zbl 0606.93036 · doi:10.1109/TAC.1986.1104172
[5] Colaneri P., Longhi S.: The lifted and cyclic reformulations in the minimal realization of linear discrete-time periodic systems. Proc. of the 1st IFAC Workshop on New Trends in Design of Control Systems, Smolenice 1994, pp. 329-334
[6] Colaneri P., Longhi S.: The realization problem for linear periodic systems. Automatica 31 (1995), 775-779 · Zbl 0822.93019 · doi:10.1016/0005-1098(94)00155-C
[7] Evans D. S.: Finite-dimensional realizations of discrete-time weighting patterns. SIAM J. Appl. Math. 22 (1972), 45-67 · Zbl 0242.93024 · doi:10.1137/0122006
[8] Flamm D. S.: A new shift-invariant representation for periodic linear systems. Systems Control Lett. 17 (1991), 9-14 · Zbl 0729.93034 · doi:10.1016/0167-6911(91)90093-T
[9] Fortuna L., Nunnari G., Gallo A.: Model Order Reduction Techniques with Applications in Electrical Engineering. Springer-Verlag, Berlin 1992
[10] Glover K.: All optimal Hankel-norm approximation of linear multivariable systems and their \(L^\infty \)-errors bounds. Internat. J. Control 39 (1984), 6, 1115-1193 · Zbl 0543.93036 · doi:10.1080/00207178408933239
[11] Gohberg I., Kaashoek M. A., Lerer L.: Minimality and realization of discrete time-varying systems. Oper. Theory: Adv. Appl. 56 (1992), 261-296 · Zbl 0242.93024 · doi:10.1137/0122006
[12] Grasselli O. M., Longhi S.: Disturbance localization by measurements feedback for linear periodic discrete-time systems. Automatica 24 (1988), 375-385 · Zbl 0653.93033 · doi:10.1016/0005-1098(88)90078-7
[13] Grasselli O. M., Longhi S.: Zeros and poles of linear periodic discrete-time systems. Circuits Systems Signal Process. 7 (1988), 361-380 · Zbl 0662.93015 · doi:10.1007/BF01599976
[14] Grasselli O. M., Longhi S.: Robust tracking and regulation of linear periodic discrete-time systems. Internat. J. Control 54 (1991), 613-633 · Zbl 0728.93065 · doi:10.1080/00207179108934179
[15] Grasselli O. M., Longhi S.: The geometric approach for linear periodic discrete-time systems. Linear Algebra Appl. 158 (1991), 27-60 · Zbl 0758.93044 · doi:10.1016/0024-3795(91)90049-3
[16] Gree M., Limebeer D. J. N.: Linear Robust Control. Prentice Hall, Englewood Cliffs, N. J. 1995
[17] Mayer R. A., Burrus C. S.: A unified analysis of multirate and periodically time-varying digital filters. Trans. Ccts Syst. CSA-22 (1975), 162-168
[18] Moore B. C.: Principal component analysis in linear systems: controllability, observability and modal reduction. Trans. Automat. Control AC-26 (1981), 17-32 · Zbl 0464.93022 · doi:10.1109/TAC.1981.1102568
[19] Park B., Verriest E. I.: Canonical forms on discrete linear periodically time-varying systems and a control application. Proc. of the 28th IEEE Conference on Decision and Control, Tampa 1989, pp. 1220-1225
[20] Pernebo L., Silverman L. M.: Model reduction via balanced state space representations. IEEE Trans. Automat. Control AC-27 (1982), 2, 382-387 · Zbl 0482.93024 · doi:10.1109/TAC.1982.1102945
[21] Salomon G., Zhou K., Wu E.: A new balanced realization and model reduction method for unstable systems. Proc. of 14th IFAC World Congress, Beijing 1999, vol. D, pp. 123-128
[22] Shokoohi S., Silverman L. M., Dooren P. Van: Linear time-variable systems: balancing and model reduction. Trans. Automat. Control AC-28 (1983), 8, 810-822 · Zbl 0517.93014 · doi:10.1109/TAC.1983.1103331
[23] Shokoohi S., Silverman L. M., Dooren P. Van: Linear time-variable systems: stability of reduced models. Automatica 20 (1984), 1, 59-67 · Zbl 0534.93037 · doi:10.1016/0005-1098(84)90065-7
[24] Varga A.: Periodic Lyapunov equations: some applications and new algorithms. Internat. J. Control 67 (1997), 69-87 · Zbl 0873.93057 · doi:10.1080/002071797224360
[25] Xie B., Aripirala R. K. A. V., Syrmos V. L.: Model reduction of linear discrete-time periodic systems using Hankel-norm approximation. Proc. of IFAC 13th World Congress, San Francisco, 1996, pp. 245-250
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