The realization problem for linear periodic systems. (English) Zbl 0822.93019

Summary: The realization problem for discrete-time linear periodic systems is considered. Deep insight on the various complicated aspects of minimal, quasi-minimal and uniform realization is provided based on the knowlege of the structural properties of discrete-time periodic systems. The results hinge on the lifting isomorphism between such systems and time- invariant ones.


93B15 Realizations from input-output data
93B20 Minimal systems representations
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems


Full Text: DOI


[1] Bittanti, S., Deterministic and stochastic linear periodic systems, (Bittanti, S., Time Series and Linear Systems (1986), Springer-Verlag: Springer-Verlag Berlin), 141-182 · Zbl 0611.62107
[2] Bittanti, S.; Colaneri, P.; De Nicolao, G., The difference periodic Riccati equation for the periodic prediction problem, IEEE Trans. Autom. Control, AC-33, 706-712 (1988) · Zbl 0656.93067
[3] Bittanti, S.; Colaneri, P.; De Nicolao, G., An algebraic Riccati equation for the discrete-time periodic prediction problem, Syst. Control Lett., 14, 71-78 (1990) · Zbl 0692.93070
[4] Bolzern, P.; Colaneri, P.; Scattolini, R., Zeros of discrete-time linear periodic systems, IEEE Trans. Autom. Control, AC-31, 1057-1058 (1986) · Zbl 0606.93036
[5] Colaneri, P., Zero-error regulation for discrete-time linear periodic systems, Syst. Control Lett., 15, 377-380 (1990) · Zbl 0712.93046
[6] Colaneri, P., Output stabilization via pole-placement of discrete-time linear periodic systems, IEEE Trans. Autom. Control, AC-36, 739-742 (1991)
[7] Colaneri, P., Hamiltonian matrices for lifted systems and periodic Riccati equations in \(H_2/H_∞\) analysis and control, (Proc. 29th IEEE Conference on Decision and Control. Proc. 29th IEEE Conference on Decision and Control, Brighton, U.K. (1991)), 1914-1919
[8] Colaneri, P.; Longhi, S., Discrete-time linear periodic systems: the minimal realization problem, (Proc. 12th IFAC World Congress, Sydney, Australia, Vol. 5 (1993)), 69-74
[9] Dahleh, M. A.; Voulgaris, P. G.; Valavani, L. S., Optimal and robust controllers for periodic and multirate systems, IEEE Trans. Autom. Control, AC-37, 90-99 (1992) · Zbl 0747.93028
[10] Flamm, D. S., A new shift-invariant representation for periodic linear systems, (Proc. American Control Conf.. Proc. American Control Conf., San Diego, CA (1990)), 1510-1515 · Zbl 1354.70035
[11] Gohberg, I.; Kaashoek, M. A.; Lerer, L., Minimality and realization of discrete time-varying systems, Operator Theory: Adv. Applics, 56, 261-296 (1992) · Zbl 0747.93054
[12] Grasselli, O. M.; Longhi, S., Zeros and poles of linear periodic discrete-time systems, Circ. Syst. Signal Process., 7, 361-380 (1988) · Zbl 0662.93015
[13] Grasselli, O. M.; Longhi, S., Disturbance localization by measurements feedback for linear periodic discrete-time systems, Automatica, 24, 375-385 (1988) · Zbl 0653.93033
[14] Grasselli, O. M.; Longhi, S., Robust tracking and regulation of linear periodic discrete-time systems, Int. J. Control, 54, 613-633 (1991) · Zbl 0728.93065
[15] Grasselli, O. M.; Longhi, S., Pole-placement for nonreachable periodic discrete-time systems, Math. Control Signals Syst., 4, 439-455 (1991) · Zbl 0739.93034
[16] Grasselli, O. M.; Longhi, S., The geometric approach for linear periodic discrete-time systems, Linear Algebra Applics, 158, 27-60 (1991) · Zbl 0758.93044
[17] Grasselli, O. M.; Longhi, S., Block decoupling with stability of linear periodic systems, J. Math. Syst. Estim. Control, 3, 427-458 (1993) · Zbl 0785.93062
[18] Ho, B. L.; Kaiman, R. E., Effective construction of linear state-variable models from input-output functions, Regelungstechnik, 14, 545-548 (1966) · Zbl 0145.12701
[19] Kabamba, P. T., Control of linear systems using generalized sampled-data hold functions, IEEE Trans. Autom. Control, AC-32, 772-783 (1987) · Zbl 0627.93049
[20] Kalman, R. E.; Falb, P. L.; Arbib, M. A., (Topics in Mathematical System Theory (1969), McGraw-Hill: McGraw-Hill New York), Section 10 · Zbl 0231.49001
[21] Khargonekar, P. P.; Poolla, K.; Tannenbaum, A., Robust control of linear time-invariant plants using periodic compensators, IEEE Trans. Autom. Control, AC-30, 1088-1092 (1985) · Zbl 0573.93013
[22] Lin, C.-A.; King, C.-W., Minimal periodic realizations of transfer matrices, IEEE Trans. Autom. Control, AC-38, 462-466 (1993) · Zbl 0789.93089
[23] Mayer, P. G., A new class of shift-varying operators, their shift-invariant equivalents, and multi-rate digital schemes, IEEE Trans. Autom. Control, AC-35, 429-433 (1990) · Zbl 0705.93013
[24] Mayer, R. A.; Burrus, C. S., A unified analysis of multirate and periodically time-varying digital filters, IEEE Trans. Circuits Syst., CAS-22, 162-168 (1975)
[25] Sànchez, E.; Hernàndez, V.; Bru, R., Minimal realization for discrete-time periodic systems, Linear Algebra Applics, 162-164, 685-708 (1992) · Zbl 0759.93053
[26] Willems, J. L.; Kuc̆era, V.; Brunovsky, P., On the assignment of invariant factors by time-varying feedback strategies, Syst. Control Lett., 5, 75-80 (1984) · Zbl 0553.93031
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