Theodore Kottas

Adaptive Estimation of Fuzzy Cognitive Maps With Proven Stability and Parameter Convergence

Fuzzy cognitive maps (FCMs) have been introduced by Kosko to model complex behavioral systems in various scientific areas. One issue that has not been adequately studied so far is the conditions under which they reach a certain equilibrium point after an initial perturbation. This is equivalent to studying the existence and uniqueness of solutions for their concept values. In this paper, we study the existence of solutions of FCMs equipped with continuous differentiable sigmoid functions having contractive or, at least, non-expansive properties. This is done by using an appropriately defined contraction mapping theorem and the non-expansive mapping theorem. It is proved that when the weight interconnections fulfill certain conditions, the concept values will converge to a unique solution, regardless of the exact values of the initial concept values perturbations, or in some cases, a solution exists that may not necessarily be unique; otherwise, the existence or the uniqueness of equilibrium cannot be assured. Based on these results, an adaptive weight-estimation algorithm is proposed that employs appropriate weight projection criteria to assure that the uniqueness of FCM solution is not compromised. In view of these results, recently proposed extensions of FCM, which are the fuzzy cognitive networks (FCN), are invoked.

On the existence and uniqueness of solutions for the concept values in Fuzzy Cognitive Maps

Fuzzy cognitive maps (FCM) have been introduced by Kosko to model complex behavioral systems in various scientific areas. One issue that has not been adequately studied so far is the conditions under which they reach a certain equilibrium point after an initial perturbation. This is equivalent to studying the existence and uniqueness of solutions for their concept values. In this paper, we study the existence of solutions by using an appropriately defined contraction mapping theorem. It is proved that when the weight interconnections fulfill certain conditions the concept values will converge to a unique solution regardless the exact values of the initial concept values perturbations. Otherwise the existence or the uniqueness of equilibrium can not be assured. The results are considered very significant because set the basis for the development of reliable system identification and control schemes based on the concept of FCM. In view of these results recently proposed extensions of FCM, the fuzzy cognitive networks are invoked.

Bilinear Adaptive Parameter Estimation in Fuzzy Cognitive Networks

Fuzzy Cognitive Networks (FCN) have been introduced by the authors recently as an extension of Fuzzy Cognitive Maps (FCM). One important issue of their operation is the conditions under which they reach a certain equilibrium point after an initial perturbation. This is equivalent to studying the existence and uniqueness of solutions for their concept values. In this paper, we study the existence of solutions of FCNs equipped with continuous differentiable sigmoid functions. This is done by using an appropriately defined contraction mapping theorem. It is proved that when the weight interconnections and the chosen sigmoid function fulfill certain conditions the concept values will converge to a unique solution regardless the exact values of the initial concept values perturbations. Otherwise the existence or the uniqueness of equilibrium can not be assured. Assuming that these conditions are met, an adaptive bilinear weight estimation algorithm is proposed.

Indirect Adaptive Control Based on the Recurrent Neurofuzzy Model

The indirect adaptive regulation of unknown nonlinear dynamical systems with multiple inputs and states (MIMS) using F-RHONNs under the presence of parameter and dynamic uncertainties, is considered in this chapter. The method is based on the new NF dynamical systems definition introduced in Chap. 2, which uses the concept of adaptive fuzzy systems (AFS) operating in conjunction with recurrent high order neural networks. Since the plant is considered unknown, we first propose the calculation of fuzzy output centers by systems data or linguistic information and in the sequel the fuzzy rules are approximated by appropriate HONNs. Thus, the identification scheme leads up to fuzzy subsystems approximated by recurrent high order neural networks, which however takes into account the centers of the fuzzy output partitions (F-RHONNs). Every high order neural network approximates a group of fuzzy rules associated with each center. The indirect regulation is achieved by first identifying the system around the current operation point, and then using its parameters to device the control law. Weight updating laws for the involved HONNs are provided, which guarantee that, under the presence of both parameter and dynamic uncertainties, both the identification error and the system states reach zero, or at least uniform ultimate boundedness of all signals in the closed-loop. The control signal is constructed to be valid for both square and nonsquare systems by using a pseudo-inversion. The existence of the control signal is always assured by employing the method of parameter hopping instead of the conventional projection method.

Framework of Operation and Selected Applications

In this chapter, we present the framework of operation of FCN, which is based on the adaptive estimation algorithms developed in the previous chapter and on the proposal of a fuzzy rule-based mechanism for storing acquired knowledge during its operation and training. Moreover, selected applications are presented, which demonstrate the applicability of the FCN framework both in conventional benchmark control problems and in real life applications. Both parameter adaptation algorithms, the linear and the bilinear one, are tested, emphasizing their distinct characteristics and advantages. The applications include the control of an inverted pendulum, the control of a hydroelectric power plant, and the coordination of different renewable power sources in order to yield an overall optimal power production and consumption in a smart grid application.

Existence and Uniqueness of Solutions in FCN

In this chapter, we present a study for the existence of equilibrium points of FCNs equipped with continuous differentiable sigmoid functions that have contractive or at least nonexpansive properties. The study is done by using an appropriately defined contraction mapping theorem and the nonexpansive mapping theorem. It is proved that, when the weight interconnections fulfill certain conditions, related to the size of the FCN and the inclination of the sigmoid functions, the concept values will converge to a unique solution regardless of their initial states, or in some cases a solution exists that may not necessarily be unique. Otherwise the existence or the uniqueness of equilibria may or may not exist, it may depend on the initial states, but it cannot be assured. In case the FCN has also input nodes (that is nodes that influence but are not influenced by other nodes), the unique equilibrium does not depend solely on the weight set, as in the case of FCNs with no input nodes; it depends also on the values of the input nodes. Numerical examples explore the results and a thorough discussion interprets them.

Identification of Dynamical Systems Using Recurrent Neurofuzzy Modeling

In this chapter, we analyze the identification problem, which consists of choosing an appropriate identification model and adjusting its parameters according to some adaptive law, such that the response of the model to an input signal (or a class of input signals), approximates the response of the real system for the same input. As identification models, we use fuzzy-recurrent high-order neural networks (F-RHONNs). This model exploits the use of high-order networks (HONN), which are expansions of the first-order Hopfield and Cohen-Grossberg models that allow higher order interactions between neurons. It combines HONN with an underlying fuzzy model of Mamdani type assuming a standard defuzzification procedure such as centroid of area or weighted average. . Learning laws are proposed which ensure that the identification error converges to zero exponentially fast or to a residual set when a modeling error is applied. In the proposed method, there are two core ideas : (1) Several high-order neural networks are specialized to work around fuzzy centers, separating in this way the system in simpler (NF) subsystems with better approximation abilities and (2) the use of a novel method called switching parameter hopping to replace the commonly used \(\sigma \)-modification for the robustness of our system in order to restrict the weights and avoid drifting of their values to infinity.

Adaptive Estimation Algorithms of FCN Parameters

In this chapter, adaptive estimation algorithms are proposed, which estimate the FCN parameters based on sampled data that correspond to FCN equilibrium points. First, we assume that the only parameters that have to be estimated are the FCN weights. This requires the development of estimation algorithms that are based on a linear parametric model of the FCN equilibrium equation. Discrete time repetitive weight estimation laws are derived based on Lyapunov stability analysis and appropriate projection methods are employed to guarantee that the weight updating procedure does not compromise the conditions of existence and uniqueness of solutions, derived in the previous chapter. Next, we assume that apart from the FCN weights, the sigmoid inclination parameter of each node has to be appropriately estimated. This leads to bilinear parametric modeling of the FCN equilibrium equation and the derivation of respective adaptation algorithm. Similar to the linear case, appropriate projection methods are derived and it is proved that they do not compromise the stability results of the estimation error dynamics. Simulations and comparisons between the two approaches are given, which highlight the benefit of each of them.

Direct Adaptive Neurofuzzy Control of MIMO Systems

In this chapter we present the direct adaptive regulation and tracking of affine in control nonlinear MIMO dynamical systems possessing unknown nonlinearities. The method is based on the neurofuzzy modeling presented in Chaps. 3 and 4, which combines the definition of fuzzy dynamical systems with the recurrent high-order neural networks. When the neurofuzzy model matches the unknown plant, we provide a comprehensive and rigorous analysis based on Lyapunov stability of the closed-loop system. Convergence of the states to zero or to a residual set or to a reference signal, plus boundedness of all other signals in the closed-loop is guaranteed without the need of parameter (weights) convergence, which is ensured only if a sufficiency-of-excitation condition is satisfied. The existence of the control signal and the boundedness of the weight convergence is always ensured by introducing the method of parameter modified hopping which is incorporated in the weight updating laws. Furthermore, we present direct adaptive regulation results of unknown nonlinear dynamical systems, paying special attention to the analysis of modeling errors and the model order problem. In these results, the model error is expressed by adding in the NF model a new disturbance term. The disturbance depends both on input and system states and on a nonzero term that is not necessarily known. A robustifying analysis of the developed method is also presented for the several disturbance cases. Moreover, the proposed approximation method is analyzed for its robustness when a smaller number of states is assumed. The omission of states, referred to as model order problem, is modeled by introducing in the approximation equations a disturbance term, which depends on the unknown (omitted) states. The applicability of the method is also tested on a well-known simulated nonlinear system where it is shown that by following the proposed procedure one can obtain asymptotic regulation or trajectory tracking. Comparison is also made with simple RHONN controllers, showing that the NF approach works better in any case.

Introduction and Scope of Part I

In our world, there are two principal objectives in the scientific study of the environment: we want to understand (identification) and to control. These two goals are in continuous interaction with each other, since deeper understanding allows firmer control, while, on the other hand, systematic application of scientific theories inevitably generates new problems which require further investigation, and so on.