Abstract
Based on the combinatorial Routhα-β andγ-δ expansions of a stable transfer function, a new energy decomposition tree for linear systems is developed. The pertinent properties to the energy decomposition tree are investigated, and an algorithm is derived for synthesizing transfer functions from the tree. The synthesis process naturally leads to a new family of Routh approximants to the system. It is indicated that the selection of Routh approximants based on the values of impulse-response energy is often inadequate because there may be a number of different Routh approximants with the same order and the same impulse-response energy. In such cases, an additional performance criterion, such as the integral of squared error of impulse response or unit-step response, has to be used to select a suitable Routh approximant.
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References
Bistritz, Y., 1983, A direct Routh stability method for discrete system modeling,Syst. Control Lett. 2, 83–87.
Guo, T. Y.; Hwang, C.; Shieh, L. S.; and Chen, C. H., 1993, Reduced-order of a 2-D linear discrete separable-denominator system using bilinear Routh approximations,IEE Proc., Part G 139, 45–56.
Hsieh, C. S., and Hwang, C., 1989, Model reduction of continuous-time systems using a modified Routh approximation method,IEE Proc., Part D 136, 151–156.
Hsieh, C. S., and Hwang, C., 1989, Order reduction of discrete-time system via bilinear Routh approximants,J. Chinese Inst. Engrs. 12, 529–538.
Hutton, M. F., and Friedland, B., 1975, Routh approximations for reducing order of linear time-invariant systems,IEEE Trans. Automat. Contr. 20, 329–337.
Hutton, M. F., and Rabins, M. J., 1975, Simplification of high-order mechanical system using Routh approximation,Trans. ASME, J. Dynamic Systems, Measurement, and Control 96, 383–392.
Hwang, C., 1984, Mixed method of Routh and ISE criterion approaches for reduced-order modeling of continuous-time systems,Trans. ASME, J. Dynamic Systems, Measurement, and Control 106, 353–356.
Hwang, C., 1986, A combined method using modified Routh stability array and MSE criterion for the reduction of discrete-time systems,J. Chinese Inst. Engrs. 9, 647–656.
Hwang, C., and Guo, T. Y., 1985, Matrix Routh-approximant reduced-order modeling for multivariable systems,Int. J. Systems Sci. 16, 697–712.
Hwang, C.; Guo, T. Y.; and Shieh, L. S., 1993, Model reduction using new optimal Routh approximant technique,Int. J. Control 55, 989–1007.
Hwang, C., and Shih, Y. P., 1990, Reduced-order modeling of discrete-time systems via bilinear Routh approximation,Trans. ASME, J. Dynamic Systems, Measurement, and Control 112, 292–297.
Hwang, C., and Shih, Y. P., 1983, Routh approximation for reducing order of discrete systems,Trans. ASME, J. Dynamic Systems, Measurement, and Control 104, 107–109.
Hwang, C., and Wang, K. Y., 1984, Optimal Routh approximations for continuous-time systems,Int. J. Systems Sci. 15, 249–259.
Krishnamurthy, V., and Seshadri, V., 1976, A simple and direct method of reducing order of linear systems using Routh approximants in frequency domain,IEEE Trans. Automat. Contr. 21, 797–799.
Krishnamurthy, V., and Seshadri, V., 1978, Model reduction using Routh stability criterion,IEEE Trans. Automat. Contr. 23, 729–731.
Lamba, S. S., and Bandyopadhyay, B., 1986, An improvement on the Routh approximation technique,IEEE Trans. Automat. Contr. 31, 1047–1050.
Langholz, G., and Feinmesser, G., 1978, Model reduction by Routh approximations,Int. J. Control 9, 493–496.
Lepschy, A., and Viaro, U., 1983, An improvement in the Routh-Padé approximant technique,Int. J. Control 36, 643–661.
Lucas, T. N., 1985, Linear system reduction by impulse energy approximations,IEEE Trans. Automat Contr. 30, 784–786.
Lucas, T. N., 1987, Further discussion on impulse energy approximation,IEEE Trans. Automat. Contr. 32, 189–190.
Lucas, T. N., 1988, Scaled impulse energy approximation for model reduction,IEEE Trans. Automat. Contr. 33, 791–793.
Ramakrishnan, J. V.; Rao S. V.; and Koval, L. R., 1990, Multivariable Routh-approximant model reduction method in the time domain,Optimal Control Methods and Applications 11, 233–247.
Rao, A. S., 1981, Routh approximant state space reduced order models for a system with uncontrollable modes,IEEE Trans. Automat Contr. 26, 1286–1288.
Rao, A. S., 1983, On Routh approximation,Proc. IEEE 71, 272–274.
Rao, A. S., 1986, On linear system reduction by impulse energy approximation,IEEE Trans. Automat. Contr. 31, 551–552.
Rao, A. S.; Lamba, S. S.; and Rao, S. V., 1978, On simplification of unstable systems using Routh approximant technique,IEEE Trans. Automat. Contr. 23, 943–944.
Rao, A. S.; Lamba, S. S.; and Rao, S. V., 1978, Routh-approximant time-domain reduced-order modeling for single-input, single-output systems,Proc. IEEE 125, 1059–1063.
Rao, A. S.; Lamba, S. S.; and Rao, S. V., 1980, Application of Routh approximant method for reducing the order of a class of time-varying systems,IEEE Trans. Automat. Contr. 25, 110–113.
Sarasu, J., and Parthasarathy, R., 1979, System reduction by Routh approximation and modified Cauer continued fraction,Electron. Lett. 15, 691–792.
Shamash, Y., 1978, Routh approximations using the Schwarz canonical form,IEEE Trans. Automat. Contr. 23, 940–941.
Shamash, Y., 1979, Note on the equivalence of the Routh and Hurwitz methods of approximation,Int. J. Control 30, 899–900.
Shamash, Y., 1980, Stable biased reduced order model using the Routh method of reduction,Int. J. Systems Sci. 11, 641–654.
Singh, V., 1979, Nonuniqueness of model reduction using the Routh approach,IEEE Trans. Automat. Contr. 24, 650–651.
Singh, V., 1981, Stable approximants for stable systems: A new approach,Proc. IEEE 69, 1155–1156.
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This work was supported by the National Science Council of the Republic of China under Grant NSC80-0402-E006-12.
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Hwang, C., Lee, Y.C. A new family of Routh approximants. Circuits Systems and Signal Process 16, 1–25 (1997). https://doi.org/10.1007/BF01183172
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DOI: https://doi.org/10.1007/BF01183172