This paper presents an algorithm for determining the minimum dwell time constraint for switched linear fault detection filters. When applying switched systems, ensuring the stability is a crucial part, which can be guaranteed, when we switch slowly enough between the subsystems, more precisely the intervals between two consecutive switching, called dwell time, are large enough. The problem formulation is based on multiple Lyapunov functions and is expressed through a special form of linear matrix inequities (LMIs), which include a nonlinear term with the dwell time. This represents a multivariable time dependent optimization problem. To solve this special formulated LMIs, we propose an algorithm, called -iteration, which is a combination of the procedure of interval halving with an LMI solver. The results of the illustrative example suggest further benefits.
Słowa kluczowe: Switched linear system, Dwell time constraint, Switched linear fault detection filter, MFARE
W artykule zaprezentowano algorytm do określania minium czasowego miedzy kolejnymi przełączeniami liniowego filtru do detekcji błędu . Sformułowanie problemu bazuje na wielokrotnej funkcji Lapunva. Do rozwiązania tego problmu zaproponowano algorytm -iteration który jest kombinacją procedury połówkowania intwerwału z wykorzystanie solvera LMI.
Keywords: filtr przełączający, funkcja dwell, minimum czasu miedzy przełączeniami
Switched systems for purpose of nonlinear control have been studied extensively in the two past decades and useful results are now available, see e.g. , , ,  and . As it was stated by several authors e.g. (Liberzon and Morse in 1999, Hespana in 2004, Chen and Saif in 2004, Colaneri in 2008,) the asymptotic stability can be ensured when we switch slow enough between the subsystems, more precisely the intervals between two consecutive switchings -called dwell time-, are large enough. This problem has been specially addressed in the synthesis of switched state estimator of Luenberger type, e.g. (Prandini in 2003, Chen and Saif in 2004) and it is also a crucial part in our objective of the designing a switched linear fault detection filter. In earlier researches different methods have been proposed for determining the minimum dwell time, see , , , ,  and . The most commonly used and powerful algorithms, like e.g. the representation based on Kronecker products (Geromel and Colaneri, 2006) or Logic- Based Switching Algorithms (Hespana, 1998), are based on multiple Lyapunov functions and expressed in form of linear matrix inequalities (LMIs), see in , , ,  and . Since we deal with filtering, the basic Lyapunov theorem needs to be extended to cope with performance requirements such as the root mean square (RMS) property of a switched system, which corresponds normally to determining an upper bound of the minimum dwell time. To this aim, in our research we consider a method used by (Geromel and Colaneri, 2008) for nonlinear control and we have adopted it to the classical detection filtering problem, see in , , ,  and . More exactly, the concept of the switched control in  can be associated to the switched filtering problem by duality and [...]
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