Acta Technica 84. (1977)
1-2. szám - Csáki, F.: The Role of Truncated Polynomials in Some Discrete-Time State-Space Techniques
Acta Technica Acadcmiac Scientiarum Hungaricae, Tomus 84 (1 -2), pp. 1 16 (1977) THE ROLE OF TRUNCATED POLYNOMIALS IN SOME DISCRETE-TIME STATE-SPACE TECHNIQUES F. CSÁKI* ORD. MEMBER OF THE HUNG. AC. OF SCI. (Manuscript received 1 May, 1976) After defining truncated polynomials on the base of the characteristic equation, some application possibilities are shown. The resolvent matrix, the state-transition matrix, the transfer function and the transfer-function matrix, the inverse of the Vandermonde and confluent Vandermonde matrices, the inverse of canonical transformation matrices are expressed in terms of truncated polynomials for discrete-time systems demonstrating their importance in some state-space techniques. Introduction Some classical г-transform methods and recent state-space techniques, concerned with time-invariant, lumped parameter, linear, discrete-time systems, do tacitly apply to so-called truncated polynomials, which may he defined for l = 0, 1, 2, . . . , n as IV,(*) â -1 + ... +fl+1Z + f, (1) or briefly ВД - J/У-'; (/,,= 1), (1*) i=t where / (i — 0, 1, 2, . . .,n — 1) and fn = 1 are the constant coefficients of the characteristic polynomial l)(z) ^ Zn -+- /„-I*"-1 + • • • + fiz + fa 1 (2) or (/,,= 1). (2*) Note that IV,(z) is a polynomial of power n — l in which the last coefficient is/,. Thus Ni(z) can be obtained by Horner’s scheme, see Eq. (77), or by breaking off the characteristic polynomial at fzl and dividing it by z‘. Accordingly: IV0(z) = D(z); Nn(z) = 1. (3) * Prof. Dr. F. Csáki, Váczi u. 8, 11-1052 Budapest, Hungary Acta Technica Academiae Scientiarum Hungaricae 84, 1977