State Feedback of Complex Systems Using Fuzzy Cognitive Maps

Abstract

Complex systems have become a research area with increasing interest over the last years. The emergence of new technologies, the increase in computational power with reduced resources and cost, the integration of the physical world with computer based systems has created the possibility of significantly improving the quality of life of humans. While a significant degree of automation within these systems exists and has been provided in the past decade with examples of the smart homes and energy efficient buildings, a paradigm shift towards autonomy has been noted. The need for autonomy requires the extraction of a model; while a strict mathematical formulation usually exists for the individual subsystems, finding a complete mathematical formulation for the complex systems is a near impossible task to accomplish. For this reason, methods such as the Fuzzy Cognitive Maps (FCM) have emerged that are able to provide with a description of the complex system. The system description results from empirical observations made from experts in the related subject – integration of expert’s knowledge – that provide the required cause-effect relations between the interacting components that the FCM needs in order to be formulated. Learning methods are employed that are able to improve the formulated model based on measurements from the actual system. The FCM method, that is able to inherently integrate uncertainties, is able to provide an adequate model for the study of a complex system. With the required system model, the next step towards the development of a autonomous systems is the creation of a control scheme. While FCM can provide with a system model, the system representation proves inadequate to be utilized to design classic model based controllers that require a state space or frequency domain representation. In state space representation, the state vector contains the variables of the system that can describe enough about the system to determine its future behavior in absence of external variables. Thus, within the components – the nodes of the FCM, ideally those can be identified that constitute the state vector of the system. In this work the authors propose the creation of a state feedback control law of complex systems via Fuzzy Cognitive Maps. Given the FCM representation of a system, initially the components-states of the system are identified. Given the identified states, a FCM representation of the controller occurs where the controller parameters are the weights of the cause-effect relations of the system. The FCM of the system then is augmented with the FCM of the controller. An example of the proposed methodology is given via the use of the cart-pendulum system, a common benchmark system for testing the efficiency of control systems.

Modeling the spread of dangerous pandemics with the utilization of a hybrid-statistical–Advanced-Fuzzy-Cognitive-Map algorithm: the example of COVID-19

“…However, despite the many theoretical developments of FCMs, based on the material presented in the previous section and their success to many medical applications (and not only), FCMs still have a number of limitations and drawbacks (Groumpos 2018 ; Mpelogianni et al 2018 ). All theories are essentially based on the material of the “ Methods” section, which will be referred from now on as the “classical FCM theories ”.…”

“…Our perceptions, understanding, and ability to model medical problems enable us to develop policies, processes, and products that are invented and required in solving them. This requires new advanced interdisciplinary theories and technologies (Groumpos 2018 ; Mpelogianni et al 2018 ).…”

“…While running several simulations, we have observed that, due to the way weights are being calculated, if the number of iterations of the algorithm is increased, in order to reach a steady-state, the causality sometimes reverses all or some of the Wij values. This is a very serious drawback as it changes the causality between concepts (Mpelogianni et al 2018 ) and in several occasions, instead of having a lower, we are going to have a larger result causing serious problems not only in the interpretation of the obtained results but also on stability issues to a number of real-life systems. This is very crucial to medical problems and especially for COVID-19.…”

“…The procedure of constructing the AFCM is remaining the same (Mpelogianni et al 2018 ; Apostolopoulos et al 2021 ). First, experts should define all the basic system’s variables-concepts.…”

“…The mathematical description of the system and the combination of initial states and inputs are sufficient to provide information about both the future states and outputs. Mathematically, the standard form of the new model is described by the system general equations (Mpelogianni et al 2018 ), …”