Robust fuzzy filter design for uncertain nonlinear singularly perturbed systems with Markovian jumps: An LMI approach
Introduction
Singularly perturbed systems are dynamical systems with multiple time-scales, which often occur naturally due to the presence of small “parasitic” parameter, typically small time constants, masses, etc. Example of singularly perturbed systems can be found in every discipline. In power system models small “parasitic” parameters can represent machine reactance or transients in voltage regulators. In industrial control systems, it may represent time constants of drives and actuators. In biochemical models, a small “parasitic” parameter can indicate a small quantity of an enzyme, while in a flexible booster model, a small “parasitic” parameter is due to bending modes and in nuclear reactor models it is due to fast neutrons. It is well established that a direct application of standard control methods to multiple time- scale processes, without accounting for the presence of time-scale multiplicity, may lead to controller ill-conditioning and/or closed-loop instability. Therefore, one of the main aims of singular perturbation analysis is to alleviate the ill-conditioning resulting from the interaction of slow and fast dynamic modes. Linear singularly perturbed systems have been studied by many researchers; see for example, [10], [18], [9], [20], [13], [12], [16], [17], [22], [19] and the references therein. For nonlinear singularly perturbed systems, in general, they can not be decomposed into slow and fast subsystems. Recently, there have been some attempts, in [6], [7], [24], [14], control for a class of singularly perturbed systems with nonlinear only in the slow variable have been examined. In [7], a local state feedback control problem for an affine nonlinear singularly perturbed has also been addressed. Global results for a class of nonlinear singularly perturbed systems described by a Takagi–Sugeno fuzzy model have been obtained in [1], [2], [3], [30]. The results derived in [1], [2], [3], [30] do not involve the separation of states into slow and fast ones.
In practice, many singularly perturbed systems may experience abrupt changes in their structure and parameters, caused by phenomena such as component and interconnection failures, parameters shifting, tracking, and the time required to measure some of the variables at different stages. Component and interconnection failures are very common in power systems and biochemical processes. This class of systems can be modelled by Markovian jump singularly perturbed systems (MJSPS) [21], [11], [5], [23], [4]. During the past decades, many researches have been done on linear MJSPS. In [21] a recursive algorithm for the regulator design has been given, and in [4] a parallel algorithm for the optimal controller design, which can yield arbitrary orders of accuracy has been proposed. In [5] the authors discussed control of linear MJSPS using the bounded real property, but the results are in the form of a set of coupled algebraic Riccati equations that is difficult to solve. Although the linear MJSPS has been widely considered, to the best of the authors’ knowledge, the nonlinear MJSPS has not been investigated yet.
Recently, fuzzy logic control has been proposed as an alternative approach to conventional control techniques for complex nonlinear systems. The TS fuzzy system is one of the most popular fuzzy systems in the model-based fuzzy control. In the TS fuzzy system, local dynamics in different state space regions are represented by local linear systems. The overall model of the system is obtained by blending these linear models through nonlinear fuzzy membership functions. Unlike conventional modelling where a single model is used to describe the global behaviour of a system, the fuzzy modelling is essentially a multi-model approach in which simple submodels (linear models) are combined to describe the global behaviour of the system. In [25], [31], [29], the authors have shown that the TS fuzzy system is an universal approximator of smooth nonlinear dynamics systems. There are two major ways in fuzzy modeling. One is fuzzy model identification [26], [28] using input–output data, and the other is fuzzy model construction, by the idea of sector nonlinearity [27]. In [27], a systematic procedure of fuzzy control system design that consists of fuzzy model construction, rule reduction, and robust compensation for nonlinear systems has been proposed.
The aim of a filter is to estimate the values of internal system variables that are not measured from the available output. Estimation problems arise in diverse fields such as communication, control and signal processing. To the best of the authors’ knowledge, the problem of filtering design for nonlinear MJSPS has not been investigated yet. The aim of this paper is to study the problem of designing a filter for a class of uncertain nonlinear MJSPS described by TS fuzzy systems. LMI-based sufficient conditions for a filter to have an performance are derived in terms of a family of linear matrix inequalities. The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard nonlinear singularly perturbed systems. The paper can be viewed as an extension our early work on filtering design for nonlinear singularly perturbed systems [1] to nonlinear singularly perturbed systems with Markovian jumps.
This paper is organized as follows. In Section 2, system description is presented. In Section 3, based on a LMI approach, we develop a technique for designing a robust fuzzy filter such that the -gain of the mapping from the exogenous input noise to the estimated error output is less than a prescribed value for the system described in Section 2. The validity of this approach is demonstrated by an example from a literature in Section 4. Finally, conclusions are given in Section 5.
Section snippets
System description
The class of uncertain nonlinear singularly perturbed system with Markovian jumps under consideration is described by the following TS fuzzy system:where , ε > 0 is the singular perturbation parameter, ν(t) = [ν1(t)…νϑ(t)] is the premise variable that may depend on states in many cases, μi(ν(t))
Robust fuzzy filter design
The structure of this section is as follows. In Section 3.1, we investigate the case when the premise variable of the fuzzy model ν(t) is available for feedback. Then, the case when the premise variable of the fuzzy model ν(t) is unavailable for feedback is examined in Section 3.2.
Illustrative example
Consider a tunnel diode circuit shown in Fig. 1 where the tunnel diode is characterized bywhere α is the characteristic parameter. Assuming that the inductance, L, is the parasitic parameter and letting x1(t) = vC(t) and x2(t) = iL(t) as the state variables, the circuit is governed by the following state equations:where w(t) is the disturbance noise input, y(t) is the
Conclusion
In this paper, we have investigated the problem of designing a robust fuzzy filter for a singularly perturbed TS fuzzy system with Markovian jumps. The proposed fuzzy filter guarantees the -gain from an exogenous input to an estimated error output to be less or equal to the prescribed value. Sufficient conditions for the existence of the proposed fuzzy filter have been given in terms of a family of LMIs. The proposed approach does not involve the separation of states into slow and fast
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