Models of Continuous-Time Linear Time-Varying Systems with Fully Adaptable System Modes

Abstract

Linear time-varying (LTV) models have a niche of their own in the representation of systems whose behaviour may be fairly approximated as linear but whose parameters are not constant or when their behaviour shows some other properties not displayed by linear timeinvariant (LTI) models. LTV models have been applied successfully in a wide range of problems ranging from system modelling and control (see, for instance, Benton & Smith, 2005; Kim et al., 2005; Vaishya & Singh, 2001) or electronic circuit design (Darabi et al., 2004) to less common applications such as the modelling of soil carbon dynamics (Martin et al., 2007) or the analysis of the stability of oscillatory (bio)chemical systems (Zak et al., 2005). A common problem found in systems science and which is relevant for automation techniques and robotics is the proper identification of the parameters present in a given system. For this kind of problems, methods have been derived for the identification of LTV dynamics (see, for instance, Lortie & Kearney, 2001). However, in particular applications where it is required to estimate in real time the parameters of a given plant with LTV behaviour and there is limited computing power for this purpose, it would be desirable to resort to another less expensive scheme. A traditional scheme used to identify certain parameters of a given system is depicted in Figure 1. In this scheme, the output of the system under investigation x is compared to the output of the adaptive filter y and the difference e is used to tune the filter. Traditionally, the filter is based on an algorithm implemented on a suitable computing platform and adequate interfacing circuitry. However, nothing precludes the implementation of such a filter using analogue techniques which may lead to the synthesis of systems for analogue computing (Cowan et al., 2006). Nowadays circuit design techniques are advanced enough to synthesise in a rather convenient way all kinds of static and dynamic linear and nonlinear functions without relying in implementations comprising several operational amplifiers, resistors, capacitors and multipliers and working with low power consumption indexes and relatively high speed processing capabilities (Mulder et al., 1998). For instance, a second-order low-