On The Similarity State Transformation for Linear Parameter-Varying Systems

Abstract

Similarity state transformations between equivalent State-Space (SS) representations of discrete-time Linear Parameter-Varying (LPV) systems are investigated. Based on previous results, it is known that to characterize all equivalent LPV-SS representations, the statetransformation matrix must depend dynamically on the scheduling variable. However, preserving static dependence of a LPV-SS representation, i.e. characterizing all equivalent SS representation with static dependence, has primary importance both in control and identification. Therefore, first, the state transformation problem is investigated from an algebraic (behavior) point of view, then conditions are developed to guarantee preservation of the static dependence in similarity state transformations. Additional geometric interpretation of the obtained results, together with computational approach to synthesize state-transformations are also developed. An illustrative example is provided to demonstrate the validity of the obtained results.

From structurally independent local LTI models to LPV model

“…It does not exclude other possible definitions, notably those based on p-dependent state transformation matrices T (p). Such considerations, related to LPV system equivalent state-space representations as investigated in [3], would be out of the scope of this technical communiqué. In practice, local LTI models are estimated from a finite data sample subject to random uncertainties, thus the definition of coherent local models is understood in an approximative sense.…”

On Local LTI Model Coherence for LPV Interpolation

“…However, as pointed out in [28], by applying a p-dependent linear transformation z(t) = T (p(t))x(t), the LPV system (1) becomes…”

“…Nevertheless, it has been reported in [28] that, in some particular case, there do exist p-dependent transformations preserving the static p-dependence of LPV models. How can this be possible?…”

Comparing global input-output behavior of frozen-equivalent LPV state-space models

“…It is important to note that TΣ ,Σ is not an LPV isomorphism between the two LPV models. In order for TΣ ,Σ to be an LPV isomorphism, A(p(t)) = TΣ ,Σ (p(t + 1))Â(p(t))T −1 Σ,Σ (p(t)), should hold for any scheduling signal p ∈ P, see Kulcsár and Tóth (2011); Tóth (2010) for more details.…”

“…As shown, e.g., in Tóth (2010); Lopes dos Santos et al (2011), existing techniques dedicated to the identification of Linear Parameter-Varying (LPV) systems can be split up into two main families: the local and the global approach. On the one hand, the global approach focuses on a global procedure for which it is assumed that one global experiment can be performed in which the control inputs as well as the scheduling variables can be both excited (see, e.g., Bamieh and Giarré (2002); Felici et al (2007)).…”