Investigation of robust stability of fractional order multilinear affine systems: -convex parpolygon approach
Introduction
Although having a 300 years old history, fractional order systems (FOS) have gained much interest in the last decades. Especially, the studies that light up the advantages of fractional calculus against the classical control methods constitute the background of this interest [1]. As real life processes are mostly fractional, modeling of these kind of systems with the non-integer calculus has an inevitable place in the mathematical world [2]. On the other hand, real life processes include uncertainties that may complicate the controlling procedure and yield to undesirable results [3]. The field of uncertainty has been studied in the last half century and there have been done much work related to uncertain systems [4]. Uncertainties may be due to additive unknown internal or external noise, environmental influence, external disturbance and parameter perturbations, which are unavoidable in real world systems and may disturb the stability of the control process [5]. Not only the problem of controlling FOS has been an important phenomenon in the last few decades but also controlling FOS in the presence of uncertainties is a challenging task in recent studies [6]. One of the important uncertainty structures is affine uncertainty. Many physical systems, for example motor drives, robot manipulators, etc. can be appropriately modeled as affine system. Several studies in the literature related to such systems have been proposed [7]. One can consider an uncertain polynomial family consisting of multiples of independent uncertain polynomials in the form of affine uncertainty structure to constitute multi-linear affine uncertainty (MLAU) [8] that will be more complicated to analyze. In our literature survey, limited researches have been encountered related with MLAU structure. For example, MLAU representation is discussed in [9]. An approach is proposed to solve the Robust Fault Detection problem for systems with MLAU in [10]. Consequently, the investigation of the characteristics and stability problems of such systems will be important. Most of the studies related with this subject have been proposed for classical integer order systems. However, investigation of the characteristics of FOS with MLAU structure will be important.
This paper deals with the stability of a fractional order control system with the MLAU structure extending the classical-convex parpolygon (-CP) approach to fractional order uncertain systems. The applications of -CP for integer order systems are studied in [8], [11]. The successful application of the -CP approach to FOS comes from the fact that the frequency response analysis of FOS is parallel to integer order ones, where the value set is in a convex polygon gained by mapping theorem [12]. Frequency response analysis of classical control systems has been a main research subject for several decades. Frequency response analysis of classical uncertain systems [13], [14], Nyquist envelope of an interval plant family [15] and a simplification procedure on the computation of frequency response of uncertain systems [16] are some of the studies among the several others in the literature. Consequently, frequency response analysis and stability investigation of the FOS with MLAU structure using -CP approach will contribute the researches in fractional order control studies.
In this paper, the numerator and the denominator of the transfer function of the control system are considered to be multiplication of the fractional order polynomials with affine uncertainty structures. It is known from the complex plane geometry that the uncertain system can be defined with an uncertainty box in the complex plane [17]. When MLAU structure is considered, multiplication of the affine polynomials dramatically increases the number of vertex and edge polynomials of the uncertainty box of the system that makes the computation of the vertex and edge sets more complicated. In order to avoid this difficulty the -CP approach, which is used for integer order systems in [8], [11], [18], [19], has been extended to the fractional order multi-linear affine system (FOMLAS) to reduce the amount of vertex and edge polynomials. This method considers the outer vertices and the outer edges when computing the value set of the family of polynomials. As a result, the decrease in the number of vertex and the edge polynomials of the uncertainty box yields an easy way of computation of the value set.
In the literature, different methods for robust stability analysis of uncertain systems have been proposed. Some of the analysis methods are based on value set computation [3] while the others use root locations, eigenvalues and Linear Matrix Inequality approach [4], [20], [21], [22]. In this paper, an algorithm is proposed to obtain the outer edges of the value set using -CP approach for stability investigation of FOS with MLAU structure.
The paper is organized as follows: problem statement and preliminaries are given in Section 2. The procedure for constructing the -CP for fractional order affine polynomials is given in Section 3. Section 4 includes the construction of the value set of the FOMLAS and Section 5 introduces the investigation of its stability. A case study is presented in Section 6. Section 7 includes the concluding remarks.
Section snippets
Problem statement and preliminaries
Consider the fractional order polytopic polynomial family of the form
whose coefficients depend linearly on and uncertainty box is where and specify the lower and upper bounds of the -th perturbation respectively. are non-integer orders of the polynomial. In order to make a frequency response analysis, one can substitute in the Laplace transform of the fractional order polynomial in
Construction of -convex parpolygon for fractional order affine polynomial
The corresponding polytope of a family of fractional order polynomials in the form of Eq. (1) in the coefficient space has vertices and exposed edges of . This polynomial can be re-written in the following form which is proposed in [8], [11] for integer order affine polynomials, where, , , .
Then, the vertex polynomials of the polytope of fractional order can be
Construction of the value set of fractional order multilinear affine system
Consider a FOMLAS, whose numerator and denominator polynomials are fractional order multi-linear affine polynomial (FOMLAP) of the form, where, , is a fixed polynomial and are polynomial in the form of Eq. (1). Typical block diagram of a multi-linear affine system can be shown in Fig. 1 [8].
Consider the plants in Fig. 1, be fractional order affine uncertainty structure in its numerator and denominator, then one can
Stability of the fractional order multilinear affine system
Well known zero exclusion principle can be successfully combined with the -CP structure of the value set of the FOMLAS.
Theorem 6 Definethen the characteristic equation of fractional order uncertain system in Eq. (24) is stable if is stable.
Proof The proof of this theorem comes from the results of Theorem 4, Theorem 5. One can say using Eqs. (26), (27) that the boundary of the value sets of the polynomials and are contained in the convex polygons and
Case study
Consider the fractional order version of the system with affine uncertainty structure in [25] with the plant and the controller as follows, The open loop transfer function of the system is as follows, where the controller has fixed polynomials in its numerator and denominator,
Conclusion
This study is dedicated to the investigation of the value set of FOMLAS using the -CP approach. The multi-linear uncertainty and the affine uncertainty structures are combined in a fractional order system and a fractional order controller has been considered. The value set of the system has been found both using the classical method and the -CP approach to illustrate the advantages of the -CP approach for FOMLAS. The results are shown via an illustrative example. Then the stability of the
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