تجزیه و تحلیل سیستم بازخورد فازی با استفاده از ماتریس های انتقال

An analytical characterization of fuzzy feedback systems based on transition matrices is carried out in this paper. The analysis faces both systems which use linear operators (sum and product) and those based on the max–min operators. We focus on the asymptotic trend of the system when the external input is held constant; such study becomes a matrix convergence problem by means of the transition matrices that define the system behavior. For the non-linear case a sufficient condition of convergence of the system (that, in particular, avoids oscillations) is demonstrated.

Fuzzy feedback systems (FFSs) have been traditionally used as controllers, where they have demonstrated their effectiveness in a myriad of applications. The stability of these controllers is usually studied by non-linear analysis techniques, such as Lyapunov’s Methods [10,11,15–18,21], which give a confidence of the reliability of the system, but such techniques do not characterize its recursive behavior. There have been several interesting attempts to rigorously formalize FFSs; one of them is due to Tong [19] in 1980, where the author obtained a theoretical description of the closed loop response as a function of the initial state. Some interesting further works are the ones by Chen and Tsao [8] (where the authors stated that the main cause of the failure of FFSs is due to the use of the max–min operator) and Kang [12]. In the former, the key was to map the fuzzy system (FS) into a non-FS so that its behavior could be better understood. In the latter, Kang proposed a systematic design method of linguistic fuzzy controllers. More recently, Adamy and Kempf [1,13] have introduced a particular case of FFS, the recurrent FSs, the study of which is based on the similarity with automata and recurrent neural networks. If the FFSs are appropriately designed they have an automaton-like behavior, but in other cases they may exhibit chaotic behavior. In a significant subclass of these systems the dynamic behavior can deduced directly from the rule base. Figs. 1–3 show the system configurations used in [19,8,13,1], respectively. In this paper, we will focus on the analysis of FFSs where, unlike the cases depicted in Figs. 1 and 2, the (fuzzy) output is directly fed back into one of the inputs (see Fig. 4) without an intermediate defuzzification process. In this case, the output is a combination (linear or not) of the fuzzy sets defined over the output space, so the feedback input will also be. According to [3], a FS for which this requirement holds can be efficiently dealt with using transition matrices. By doing so, the input–output relation becomes a matrix relation and the analysis of the steady state of the system when the external input (the one not from the feedback) is held constant becomes a problem of matrix convergence. Taking advantage of this property, in this paper, we will use the fast inference using transition matrices (FITM) methodology described in [3] to perform such a matrix analysis. This methodology has been recently proposed to perform inferences in standard additive model (SAM) [15] FSs efficiently; it is based on representing each input to the FS as a vector, the coordinates of which are the contribution to the input of each of the elements of the input linguistic variable (LV). With this assumption, a considerable reduction of the overall computational complexity is achieved and storage needs turn out to be moderately low. This procedure has been extended to other types of t-norms and t-conorms that satisfy a requirement of distributivity [5]. The idea of studying a FFS with transition matrices was presented in a previous paper [6] two years before theFITMprocedurewas designed. The system described in [6] aims at finding tracks in a multitarget environment when descriptions about the targets are reported in text-based form (as, for instance, a policeman would do). In order to find the long-term behavior of such a system, the external input was held constant (meaning that the same target was reported successively) and both the output set and its centroid were calculated. A methodology as such gives insight into what the system output is according to the rule base, so it is the ground for rule base tuning. However, the analysis in [6] is restricted to the external input being equal to one of the fuzzy sets on the input LV (not an arbitrary input) and it is only valid for linear operators. In this paper, we will show that transition matrices constitute a general framework to characterize FFSs, irrespective of their linear or non-linear character. We will describe precise guidelines to derive the transition matrices, with generality about the inputs. Additionally, we will analytically demonstrate that FFSs that use the max–min t-conorm and t-norm, respectively, may or may not converge—when the external input is held constant—depending on the structure of the transition matrix. Specifically, we will demonstrate that a sufficient condition for convergence is that the elements in the principal diagonal of the transition matrix are greater than the remaining elements in the matrix. Whenever this happens, the system achieves convergence in a number of iterations that depends on the size of the transition matrix. The paper is organized as follows: in Section 2 some basics on theFITMprocedure are given. Afterwards the main analysis of the FFS is done: the long-term behavior of a FFS when the external inputs are held constant is studied, first for linear (Section 3) and then for non-linear operators (Section 4). Finally, a comparison between a FFS using linear and non-linear operators is carried out. Appendix A briefly elaborates on normalization in FITM. In Appendix B the matrix and vector operations used in the paper, and defined for general t-norms and t-conorms, are summarized. Appendices C and D contain a theorem and its proof needed for the non-linear analysis.

The analytical characterization of a FFS as a function of time step has been carried out. Writing the fuzzy inference with transition matrices (using the FITM procedure) allows us to study the asymptotic behavior of the system when the external inputs are held constant. From this study we can draw the following conclusions: (1) If linear operators (such as the sum–product) are chosen to perform the inference, the system asymptotically reaches a steady state provided that the transition matrix is diagonalizable, eigenvalues are different and the largest eigenvalue is real. This final state can be predicted a priori and it is totally independent of the initial state of the system. It only depends on the transition matrix, i.e., on the rule base, the activation matrices of the inputs and the constant input value. (2) The convergence value of the linear FFS is the centroid of the eigenvector associated to the maximum eigenvalue of the transition matrix. (3) If non-linear operators are chosen as t-norms and t-conorms, the convergence of the output value is not guaranteed. If the system arrives at a steady state, its convergence value will depend on the initial state. (4) If the max–min operator is used, the convergence is guaranteed provided that the elements in the principal diagonal of the transition matrix are greater than or equal to the elements outside the diagonal. To conclude, we would like to remark that the use of transition matrices in the FFSs analysis gives an important insight into the behavior of the system under different conditions, i.e., for different inputs. This information may be essential for FFS design or tuning.