Orthogonality Measures and Applications in Systems Theory in One and More Variables
The representation or order reduction of a rational transfer function by a linear combination of orthogonal rational functions offers several advantages, among which the possibility to work with prescribed poles and hence the guarantee of system stability
- PDF / 836,323 Bytes
- 8 Pages / 430 x 660 pts Page_size
- 39 Downloads / 186 Views
2
Dept of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B3001 Heverlee, Belgium [email protected] Dept of Mathematics and Computer Science, University of Antwerp Middelheimlaan 1, B2020 Antwerp, Belgium {annie.cuyt,brigitte.verdonk}@ua.ac.be
Abstract. The representation or order reduction of a rational transfer function by a linear combination of orthogonal rational functions offers several advantages, among which the possibility to work with prescribed poles and hence the guarantee of system stability. Also for multidimensional linear shift-invariant systems with infinite-extent impulse response, stability can be guaranteed a priori by the use of a multivariate Pad´e-type approximation technique, which is again a rational approximation technique. In both the one- and multidimensional case the choice of the moment functional with respect to which the orthogonality of the functions in use is imposed, plays a crucial role.
1
Introduction
Let {ci }i∈N be a sequence of complex numbers and let c be a linear functional defined on the space of polynomials C[t] with complex coefficients c(ti ) = ci , i = 0, 1, . . . Then c is called the moment functional determined by the moment sequence {ci }i∈N . By means of c a formal series development of h(z) with coefficients ci (for instance a transfer function h(z) with impulse response ci for i = 0, 1, 2, . . .) can be viewed as ∞ 1 i 2 2 h(z) = ci z = c(1) + c(t)z + c(t )z + . . . = c . (1) 1 − tz i=0 Let Ln = span{1, . . . , tn } denote the space of polynomials of degree n and let ∂V denote the exact degree of a polynomial V (t) ∈ C[t]. A sequence of polynomials {Vm (z)}m∈N is called orthogonal with respect to the moment functional c provided that Vm ∈ Lm \ Lm−1 and c ti Vm (t) = 0, i = 0, . . . , m − 1, c Vm2 (t) = 0. (2) For an arbitrary polynomial Vm (z) ∈ Lm with coefficients bi , we can also construct the associated polynomial Wm−1 (z) by Vm (t) − Vm (z) Wm−1 (z) = c . (3) t−z I. Lirkov, S. Margenov, and J. Wa´ sniewski (Eds.): LSSC 2007, LNCS 4818, pp. 243–250, 2008. c Springer-Verlag Berlin Heidelberg 2008
244
A. Bultheel, A. Cuyt, and B. Verdonk
m−1−i Then Wm−1 ∈ Lm−1 with coefficients ai = j=0 cj bi+j+1 [3, p. 10]. It can ˜ m−1 (z) = z m−1 Wm−1 (1/z) and be proven [3, p. 11] that the polynomials W V˜m (z) = z m Vm (1/z) satisfy [3, p. 11] ∞ ˜ ˜ h − Wm−1 /Vm (z) = di z i (4) i=q
˜ m−1 /V˜m with q = m. In this way it is easy to obtain rational approximants W for a given transfer function h(z). Choosing Vm (z) in (4) allows to control the poles of the rational approximant. If however Vm (z) is fixed by the orthogonality conditions (2), then q = 2m in (4) and many more moments are matched, but the control over the poles is lost. We recall that a system is called bounded-input bounded-output (BIBO) stable if the output signal is bounded whenever the input signal is bounded. Since stability is guaranteed when the rational approximant has all its poles inside the unit disk or polydisk, respectively, the aim is to
Data Loading...
