# zbMATH — the first resource for mathematics

All optimal Hankel-norm approximations of linear multivariable systems and their $$L^{\infty}$$-error bounds. (English) Zbl 0543.93036
The problem is considered of approximating a linear system, with transfer function matrix $$G(s)$$ of McMillan degree n, by a linear system with transfer function matrix $$\hat G(s)$$ of McMillan degree $$k<n.$$ A complete characterization is derived of all approximations that minimize the Hankel norm of the error system: $$\| G-\hat G\|_ H.$$ The Hankel norm is an induced operator norm obtained by considering a linear system as a mapping between square integrable ”past” inputs and square integrable ”future” outputs. The key to the solution to this problem is the characterization of all matrices, with rational entries, in the form $$\hat G(s)+F(s)$$ which minimize $\| G-\hat G-F\| =\sup_{\omega}{\bar \sigma}(G(j\omega)-\hat G(j\omega)-F(j\omega)),$ where $$\hat G(s)$$ has McMillan degree k, $$F(s)$$ is anti-causal and $${\bar \sigma}$$(.) denotes the maximum singular value of a matrix. The solution to this problem is obtaind using results on balanced realizations, all- pass functions and the inertia of matrices. An algorithm is presented for finding Hankel norm approximations, and various error bounds on the approximation error are given. For one class of approximants it is shown that $$\| G-\hat G\| \leq \sum^{n}_{i=k+1}\sigma_ i(G),$$ where $$\sigma_ i(G)$$ is the $$i^{th}$$ largest Hankel singular value of G. Bounds such as this are important for predicting the performance of control schemes designed for $$\hat G$$ but applied to G.
Although the impetus for obtaining the results contained in this paper owes much to the pioneering work of V. M. Adamyan, D. Z. Arov and M. G. Krejn [Mat. Sb., Nov. Ser. 86(128), 34-75 (1971; Zbl 0243.47023); Izv. Akad. Nauk Arm. SSR, Mat. 6, 87-112 (1971; Zbl 0311.15012)] on the approximation of Hankel matrices, the author has developed his own rather ingenious methods of obtaining closed form solutions to the finite dimensional case. The paper is selfcontained, and although rather long, is extremely well written and will reward its reader with many interesting results.
Reviewer: D.A.Wilson

##### MSC:
 93C35 Multivariable systems, multidimensional control systems 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 41A20 Approximation by rational functions 15B57 Hermitian, skew-Hermitian, and related matrices 93C05 Linear systems in control theory 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 93A15 Large-scale systems
Full Text:
##### References:
 [1] DOI: 10.1070/SM1971v015n01ABEH001531 · Zbl 0248.47019 [2] ANDERSON , B. D. O. , and LATHAM , G. A. , 1984 , report under preparation . [3] ANDERSON B. D. O., Network Analysis and Synthesis A Modem Systems Theory Approach (1973) [4] BARNETT S., Matrices in Control Theory with Applications to Linear Programming (1971) · Zbl 0245.93002 [5] BARTELS R. H., Communs. Ass. comput. Mach 415 pp 820– (1972) [6] BULTHEEL A., I.E.E.E. Trans. Circuits Syst [7] BETTAYEB , M. , SILVERMAN , L. M. , and SAFONOV , M. G. , 1980 ,Proc. I.E.E.E. Conj. on Decision and Control, Albuquerque , New Mexico , p. 195 . [8] DOI: 10.1007/BFb0065310 [9] CURTAIN , R. F. , and GLOVER , K. , 1984 , report under preparation . [10] DUREN P. L., Theory of HP Spaces (1970) · Zbl 0215.20203 [11] DOI: 10.1007/BF02288367 · JFM 62.1075.02 [12] GANTMACHER F. R., The Theory of Matrices 11 (1966) · Zbl 0136.00410 [13] GLOVER , K. , 1984 , report under preparation . [14] GLOVER K., I.E.E.E. Trans.autom. Control (1983) [15] HAMMARLINO S. J., I.M.A.J. of Numer. Anal 2 pp 303– (1982) [16] HILDEBRAND K., Introduction to Numerical Analysis (1956) · Zbl 0070.12401 [17] HOFFMAN K., Banach Spaces of Analytic Functions (1962) · Zbl 0117.34001 [18] JONCKHEERE , E. A. , SAFONOV , M. G. , and SILVERMAN , L. M. , 1981 ,Proc. 20thI.E.E.E. Conf. on Decision and Control, p. 118 . [19] KAILATH T., Linear Systems (1980) [20] DOI: 10.1109/TAC.1981.1102736 · Zbl 0553.93038 [21] LAUB A. J., Joint Automatic Control Conf (1980) [22] DOI: 10.1007/BF01599021 · Zbl 0512.93008 [23] LUENBERGER D. G., Optimization by Vector Space Methods (1969) · Zbl 0176.12701 [24] DOI: 10.1093/qmath/11.1.50 · Zbl 0105.01101 [25] DOI: 10.1109/TAC.1981.1102568 · Zbl 0464.93022 [26] DOI: 10.2307/1969670 · Zbl 0077.10605 [27] DOI: 10.1016/0022-247X(62)90030-6 · Zbl 0112.01401 [28] DOI: 10.1109/TAC.1982.1102945 · Zbl 0482.93024 [29] DE PRONY R., J. Ec polytech. (Paris) 1 pp 24– (1795) [30] RUDIN W., Real and Complex Analysis (1966) · Zbl 0142.01701 [31] SAFONOV M. G., Proc. American Control Conf (1983) [32] SARASON D., Am. Math. Soc. Transl 127 pp 180– (1967) [33] SILVERMAN L. M., Proc. Joint Automatic Control Conf (1980) [34] STEWART G. W., Introduction to Matrix Computation (1973) · Zbl 0302.65021 [35] DOI: 10.1016/0024-3795(74)90060-3 · Zbl 0288.15015 [36] DOI: 10.1109/TIT.1961.1057636
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.