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Realizations of the Attasi state space model for 2D filters. (English) Zbl 0539.93013
Given a sequence of matrix Markov parameters $$W_{ij},i,j=1,2,...,N$$, a matrix quadruple $$\{F_ 1,F_ 2,G,H\}$$ is sought so that $$W_{ij}=HF_ 2^{j-1}\quad F_ 1^{i-1}G. \{F_ 1,F_ 2,G,H\}$$ are the matrices in the local state-space realization, $x(i+1,\quad j+1)=F_ 1x(i,j+1)+F_ 2x(i+1,j)+F_ 0x(i,j)+Gu(i,j),y(i,j)=Hx(i,j),$ with i,$$j\geq 0$$, $$F_ 0=-F_ 1F_ 2$$ or $$F_ 0=-F_ 2F_ 1,u(i,j)$$ is the input, y(i,j) is the output and x(i,j) represents the local state vector. The requirement of commutativity, $$F_ 1F_ 2=F_ 2F_ 1$$, in the original Attasi model is dispensed with here. The existece of a realization from given $$\{W_{ij}\}$$ is established, conditions concerning uniqueness of the realization are given and a necessary and sufficient condition for the state-space model to be minimal is proved.
Reviewer: N.K.Bose
##### MSC:
 93B20 Minimal systems representations 70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics 93B15 Realizations from input-output data 93B10 Canonical structure 93C05 Linear systems in control theory 93C35 Multivariable systems, multidimensional control systems 93C55 Discrete-time control/observation systems 93E11 Filtering in stochastic control theory
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##### References:
 [1] ABRAMATIC J. F., I.E.E.E. Trans. Acoustics Speech Signal Processing 27 pp 445– (1979) · Zbl 0432.94019 [2] ATTASI S., Systems Identification Advances and Case Studies pp 289– (1973) [3] EISING R., I.E.E.E. Trans, autom. Control 23 pp 793– (1978) · Zbl 0397.93022 [4] FORNASINI E., I.E.E.E. Trans, autom. Control 12 pp 484– (1976) · Zbl 0332.93072 [5] GANTMACHER F. R., The Theory of Matrices (1959) · Zbl 0085.01001 [6] GOLOMB M., Bull. Ane. Math. Soc 49 pp 581– (1943) · Zbl 0061.26703 [7] HADAMARD J., J. Math. Pures. Appl. 8 pp 101– (1892) [8] HINAMOTO T., I.E.E.E. Trans. Circuits Syst. 27 pp 36– (1980) · Zbl 0443.93015 [9] Ho B. L., Proc. 3rd Allerton Conf. Circuits Syst (1965) [10] KALMAN R. E., Aspects of Network and System Analysis (1971) [11] Koo C. S., Proc. I.E.E.E. Int. Symp. Circuits Syst. (1978) [12] LUENBERGER D. G., I.E.E.E. Trans, autom. Control 12 pp 290– (1967)
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