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On the assignment of eigenvalues in LTI systems
| Content Provider | Semantic Scholar |
|---|---|
| Author | Antoulas, Athanasios C. Gugercin |
| Copyright Year | 1999 |
| Abstract | In this note we discuss the assignability of natural frequencies of LTI systems, given their Hankel singular values. For details we refer to [l]. Preliminaries. Consider the LTI system I=: k( t ) = Az( t ) + Bu(t), y ( t ) = Cz( t ) , A E PnX", B,CT E P", We will assume that C is controllable, observable, and stable. There are two important invariants associated with C: the natural frequencies or poles A;@), and the Hankel singular values o;(C). The former quantities are defined as the eigenvalues of A: X;(C) = X;(A), i = l , . , n . The latter are defined as the singular values of the Hankel operator Xc, associated with E: Xx: Ut+ y+ = X ( u ) , where y+(t):=J_= h(t-T)u(T)dT, t > 0, and h(t) = CeAtB , t 2 0, is the impulse response of C. It turns out that 3Cc has a finite number of non-zero singular values: u;(c) = Jm, i = 1, ,n, where a1 1-.1 an > 0, and P > 0, Q > 0 are the controllability, observability grammians of E , respectively, satisfying the Lyapunov equations: 0 A P + PA^ + B B ~ = 0, A ~ & + Q A + C ~ C = o The eigenvalues X;(C) describe the dynamics of the system. The Hankel singular values .;(E) on the other hand just as the singular values in the case of constant matrices describe how well a linear dynamical system can be approximated by a similar system of lower dimension. The problem which arises is the relationship between &(E) and u;(C). More specifically, given the former, to what extent can one influence the latter, and vice versa. We will actually address a more specific question, namely: given A;@), to what extent is it possible to reduce the condition number of the singular values, and given ai@), the condition number of A. The former problem can be solved by noting that one can always construct an all-pass system with given poles (an all-pass system is perfectly conditioned, since all its Hankel singular values are equal). Concerning the distribution of the eigenvalues of A for pre-assigned Hankel singular values, we first note that 0-7803-5250-5/99/$10.00 |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.math.vt.edu/people/gugercin/ttpapers/GugercinC1.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Approximation algorithm Arabic numeral 0 Condition number Dynamical system Invariant (computer science) Language Technologies Institute Linear time-invariant theory Lyapunov fractal Observable PersonNameUse - assigned Quantity Singular |
| Content Type | Text |
| Resource Type | Article |
