Abstract
This paper is concerned with the optimal model reduction for linear discrete periodic time-varying systems and digital filters. Specifically, for a given stable periodic time-varying model, we shall seek a lower order periodic time-varying model to approximate the original model in an optimal H 2 norm sense. By orthogonal projections of the original model, we convert the optimal periodic model reduction problem into an unconstrained optimization problem. Two effective algorithms are then developed to solve the optimization problem. The algorithms ensure that the H 2 cost decreases monotonically and converges to an optimal (local) solution. Numerical examples are given to demonstrate the computational efficiency of the proposed method. The present paper extends the optimal model reduction for linear time invariant systems to linear periodic discrete time-varying systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B.A. Bamieh and J.B. Pearson, 'The H 2 problem for sampled data systems,' Systems and Control Letters 19, 1–12, 1992.
L. Baratchart, M. Cardelli, and M. Olivi, 'Identification and rational L 2 approximation: A gradient algorithm,' Automatica, 413–417, 1991.
D. Enns, 'Model reduction with balanced realizations: An error bound and a frequency weighted generalization,' Proc. 23rd IEEE Conf. Decision and Control, Las Vegas, 1984.
W.A. Gardner (Ed.), Cyclostationary in Communications and Signal Processing, IEEE Press, 1994.
W.A. Gardner and L.E. Franks, 'Characterization of cyclostationary random signal processes,' IEEE Trans. Inform. Theory21, 4–14, 1975.
K.M. Grigoriadis, 'Optimal H ∞ model reduction via linear matrix inequalities: continuous and discrete-time cases,' Systems & Control Letters 26, 321–333, 1995.
K. Glover, 'All optimal Hankel-norm approximations of linear multivariable systems and their L ∞-error bounds,' Int. J. Control 39(6) 1115–1195, 1984.
M. Green, 'A relative error bound for balanced stochastic truncation,' IEEE Trans. Automat. Contr. 33(10) 961–965, 1988.
U. Helmke and J.B. Moore, Optimization and Dynamical Systems, London: Springer-Verlag, 1994.
A. Helmersson, 'Model reduction using LMIs,' Proc. 33th IEEE Conf. Decision and Control 3217–3222, 1994.
D.C. Hyland and D.S. Bernstein, 'The optimal projection equations for fixed-order dynamic compensation,' IEEE Trans. Automat. Contr. 29(11) 1034–1037, 1984.
D.C. Hyland and D.S. Bernstein, 'The optimal projection equations fro model reduction and the relationships among the methods of Wilson, Skelton amd Moore,' IEEE Trans. Automat. Contr. 30, 1201–1211, 1985.
R. Ishii and M. Kakishita, 'A design method for a periodically time-varying digital filter for spectrum scrambling,' IEEE Trans. on Acoust. Speech Signal Processing, ASSP-38, 1219–1222, 1990.
P.P. Khargonekar, K. Poolla and A. Tannenbaum, 'Robust control of linear time invariant plants using periodic compensation,' IEEE Trans. on Automatic Control, 30, 1088–1096, 1985.
F.L. Kitson and L.J. Griffiths, 'Design and analysis of recursive periodically time varying digital filters with highly quantized coefficients,' IEEE Trans. Acoust., Speech, Signal Processing, 36, 674–685, 1988.
C.W. King and C.A. Lin, 'A unified approach to scrambling filter design,' IEEE Trans. on Signal Processing, 43, 1753–1765, 1995.
C.A. Lin and C.W. King, 'Minimal periodic realizations of transfer matrices,' IEEE Trans. Automat. Contr. 38, 462–466, 1993.
Meyer, R.A. and C.S. Burrus, A unified analysis of multirate and periodically time varying filters. IEEE Trans. on Circuits and Systems 22, 162–168, 1975.
T. Chen and B. Francis, Optimal Sampled Data Control Systems, Springer, 1995.
B.C. Moore, 'Principal component analysis in linear systems,' IEEE Trans. Automat. Contr. 26, 2, 17–32, 1981.
L. Pernebo and L.M. Silverman, 'Model reduction via balanced state space representation,' IEEE Trans. Automat. Contr.27,(4) 382–387, 1982.
J.S. Prater and C.M. Loeffler, Analysis and design of periodically time varying IIR filters, with applications to Transmultiplexing, IEEE Trans. on Signal Processing, 40, 2715–2725, 1992.
J.T. Spanos, M.H. Milman and D.L. Mingori, 'A new algorithm for L 2 optimal model reduction,' Automatica, (5) 897–909, 1992.
Vaidyanathan, P.P., Multirate Systems and Filter Banks, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1993.
W. Wang and M.G. Safonov, 'Multiplicative-error bound for balanced stochastic truncation and model reduction,' IEEE Trans. Automat. Contr. 37(8) 1265–1267, 1992.
D.A. Wilson, 'Optimum solution of model-reduction problem,' Proc. IEE-D 1161–1165, 1970.
D.A. Wilson, 'Model reduction for multivariable systems,' Int. J. Control 20(1) 57–64, 1974.
B. Xie, R. Aripirala and V. L. Syrmos, 'Model reduction of linear discrete-time periodic systems using Hankel-norm approximations,' Proc. 13th IFAC World Congress, San Francisco, June 1996.
L. Xie, W. Yan and Y.C. Soh, 'L 2 optimal reduced order filter design', Proc. the 35th IEEE Conf. Decision and Control, Kobe, Japan, Dec. 1996, 4270–4275.
W.-Y. Yan and J. Lam, 'An approximation approach to H 2 Optimal Model Reduction,' IEEE Trans. Automat. Contr. 44(7) 1341–1358, 1999.
C. Zhang, J. Zhang and K. Furuta, Analysis of H 2 and H ∞ performance of discrete periodically time-varying controllers, Automatica 33(4) 619–634, 1997.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Xie, L., Zhang, C. & Li, L. A Gradient Flow Approach to Optimal Model Reduction of Discrete-Time Periodic Systems. Journal of Global Optimization 23, 373–399 (2002). https://doi.org/10.1023/A:1016543116069
Issue Date:
DOI: https://doi.org/10.1023/A:1016543116069