access icon free Solvability conditions for the positive real lemma equations in the discrete time

Passivity theory and the positive real lemma have been recognised as two of the cornerstones of modern systems and control theory. As digital control is pervasive in virtually all control applications, developing a general theory on the discrete-time positive real lemma appears to be an important issue. While for minimal realisations the relations between passivity, positive-realness and existence of solutions of the positive real lemma equations is very well understood, it seems fair to say that this is not the case in the discrete-time case, especially when the realisation is non-minimal and no conditions are assumed on left- and/or right-invertibility of the transfer function. The purpose of this study is to present a necessary and sufficient condition for existence of solutions of the positive real equations under the only assumption that the state matrix A is asymptotically stable.

Inspec keywords: matrix algebra; computability; transfer functions; asymptotic stability; discrete time systems; digital control

Other keywords: positive-realness; solvability conditions; digital control; right-invertibility; discrete time; left-invertibility; transfer function; general theory; state matrix; positive real lemma equations; asymptotic stability; passivity theory

Subjects: Control system analysis and synthesis methods; Programming and algorithm theory; Discrete control systems; Algebra; Stability in control theory

References

    1. 1)
      • 8. Pandolfi, L.: ‘An observation on the positive real lemma’, J. Math. Anal. Appl., 2001, 255, pp. 480490.
    2. 2)
      • 2. Hitz, L., Anderson, B.D.O.: ‘Discrete positive-real functions and their application to system stability’, Proc. Inst. Electr. Eng., 1969, 116, pp. 153155.
    3. 3)
      • 1. Ferrante, A., Picci, G.: ‘Representation and factorization of discrete-time rational all-pass functions’, IEEE Trans. Autom. Control, 2017, 62, (7), pp. 32623276, DOI: 10.1109/TAC.2016.2628163.
    4. 4)
      • 5. Collado, J., Lozano, R., Johansson, R.: ‘On Kalman–Yakubovitch–Popov lemma for stabilizable systems’, IEEE Trans. Autom. Control, 2001, 46, (7), pp. 10891093.
    5. 5)
      • 4. Yiping, C.: ‘A proof of the discrete-time KYP lemma using semidefinite programming duality’. Proc. of the 26th Chinese Control Conf., Zhangjiajie, Hunan, China, 26–31 July 2007.
    6. 6)
      • 6. Iwasaki, T., Hara, S., Yamauchi, H.: ‘Structure/control design integration with finite frequency positive real property’. Proc. of the American Control Conf., Chicago, IL, 2000.
    7. 7)
      • 9. Ferrante, A., Pandolfi, L.: ‘On the solvability of the positive real lemma equations’, Syst. Control Lett., 2002, 47, pp. 211219.
    8. 8)
      • 14. Baggio, G., Ferrante, A.: ‘On minimal spectral factors with zeros and poles lying on prescribed regions’, IEEE Trans. Autom. Control, 2016, 61, (8), pp. 22512255.
    9. 9)
      • 7. Meinsma, G., Shrivastava, Y., Fu, M.: ‘A dual formulation of mixed μ and on the loss lessness of (D,G) scaling’, IEEE Trans. Autom. Control, 1997, 42, (7), pp. 10321036.
    10. 10)
      • 13. Baggio, G., Ferrante, A.: ‘On the factorization of rational discrete-time spectral densities’, IEEE Trans. Autom. Control, 2016, 61, (4), pp. 969981.
    11. 11)
      • 12. Youla, D.C.: ‘On the factorization of rational matrices’, IRE Trans. Inf. Theory, 1961, IT-7, pp. 172189.
    12. 12)
      • 11. Stefanovski, J.: ‘Structure-preserving numerical algorithm for solving discrete-time LMI and DARS’, Syst. Control Lett., 2011, 60, pp. 205210.
    13. 13)
      • 10. Ferrante, A.: ‘Positive real lemma: necessary and sufficient conditions for the existence of solutions under virtually no assumptions’, IEEE Trans. Autom. Control, 2005, 50, (5), pp. 720724.
    14. 14)
      • 3. Rantzer, A.: ‘On the Kalman–Yakubovich–Popov lemma’, Syst. Control Lett., 1996, 28, pp. 710.
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