Normalized coprime representations for time-varying linear systems

Abstract

Abstract-By considering the behaviour of stabilizable and detectable, linear time-varying state-space models over doublyinfinite continuous time, we establish the existence of so-called normalized coprime representations for the system graphs; that is, stable and stably left (resp. right) invertible, image (resp. kernel) representations that are normalized with respect to the inner product on L 2 (−∞, ∞); this is consistent with the notion of normalization used in the time-invariant setting. The approach is constructive, involving the solution of timevarying differential Riccati equations with single-point boundary conditions at either +∞ or −∞. The contribution lies in accommodating state-space models that may not define an exponential dichotomy.

Robust stability analysis for feedback interconnections of unstable time-varying systems

“…Now, without assuming that A defines an exponentially dichotomous evolution, if the pairs (A, B) and (A, C) are stabilizable and detectable, respectively, then from results in [19] there exists normalized coprime representations of the graph satisfying the properties identified in Section III; see also [1], [20] where the same result is obtained under stronger assumptions; …”

“…we now assume that the graphs of H and ∆ admit normalized right and left coprime representations; this is known to be the case for various classes of linear systems, including time-invariant systems with constantly proper transfer functions in the Callier-Desoer algebra [6] and time-varying systems with stabilizable and detectable finite-dimensional state-space realisations [19], for example. In particular, we assume the existence of bounded causal operators V, U, V, U, X H , Y H , X H , Y H and N, M, N, M, X ∆ , Y ∆ , X ∆ , Y ∆ , defined on the whole of L 2 (T), such that the following properties hold: (A1) the double Bezout identity …”

“…In the development, the normalized representations of the graphs are assumed to exist on the basis that this is true for important classes of linear system. These include: time-invariant mappings corresponding to multiplication by constantly proper Callier-Desoer transfer functions (see also [6]); and linear time-varying systems that admit stabilizable and detectable finite-dimensional state-space realizations with coefficients of bounded and continuous variation across time, as recently shown in [19].…”

Strong Detectability and Observers for Linear Time Varying Systems

“…In the following, the time dependency is omitted for a better readability. Differentiating (10b) gives ẋ = ż + Ṡy + S ẏ (12) and by using relation (10a) one obtains…”

“…The pair (A(t), C(t)) is called (uniformly exponentially) stabilizable, if its corresponding dual system is detectable. Details regarding duality between stabilizability and detectability can be found, e.g., in [12].…”