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Top1. Introduction
The gain Scheduling (GS) design has become a popular method for non-linear system design, especially during the last decade (Leith and Leithead 2000; Rugh and Shamma 2000). It has special features that make it easy to apply compared with other design methods for non-linear systems. Among those features, the most attractive is that GS employs linear design tools in the design stage. Over the past several decades, the principal ingredients of GS theory emerged and applied to wide spectrum of fields. The theory has received further boost with the introduction of Liner Fractional Transformation approach, see (Packard 1994; Lu et al. 1991; Johansen 1994) and their references. Recently, GS techniques have become one of the popular approaches to ‘linear parameter-varying’ (LPV) systems following the work of Shamma (1988). LPV systems take the following structure:
(1) where the parameter
h(t) is an exogenous time-varying quantity (independent of the state
x of the system) and takes values in some allowable set. In turn, an LPV system becomes simply a particular form of linear time-varying system.
On another research front, one of the main causes of instability and poor performance of dynamical systems is time-delays. Of interest is the class of LPV systems with time delay, which we will henceforth call LPVTD systems. The stability and the performance studying of the LPVTD systems are both theoretically and practically important. Recently, the problem of gain-scheduled H∞-control for LPVTD systems was investigated in (Wu and Grigoriadis 2001). The gain-scheduled
H∞-filtering problem has not been fully treated. On one hand, the purpose of H∞ filtering is to design a filter so as to guarantee that the resulting filtering error system is stable and the L2- induced gain from the noise to the estimation error is less than a prescribed level. It is known that H∞ problems are often solved by Linear Matrix Inequality (LMI) tools.
To evaluate the published results on filtering of LPV systems, Tables 1 and 2 contain some useful information.
Table 1. | Reference | Model | Nature | Delay pattern |
| Bokor (2000) | Continuous | Non-delayed | - |
Mahmoud
and Boujarwah (2001) | Continuous | Non-delayed | - |
|
Barbosa et al. (2005)
| Continuous | Non-delayed | - |
|
Daafouz et al. (2005)
| Discrete | Non-delayed | - |
|
Velni and Grigoriadis (2005)
| Continuous | Delayed | Constant |
|
Sato (2006)
| Continuous | Non-delayed | - |
|
Wang et al. (2007)
| Discrete | Delayed | Time-varying |
|
Zhou et al. (2007)
| Discrete | Non-delayed | - |
|
Velni and Grigoriadis (2008)
| Continuous | Delayed | Parametric delay |