Fractal Geometry Meets Computational Intelligence: Future Perspectives

Characterizations in terms of fractals are typically employed for systems with complex and multiscale descriptions. A prominent example of such systems is provided by the human brain, which can be idealized as a complex dynamical system made of many interacting subunits. The human brain can be modeled in terms of observable variables together with their spatio-temporal-functional relations. Computational intelligence is a research field bridging many nature-inspired computational methods, such as artificial neural networks, fuzzy systems, and evolutionary and swarm intelligence optimization techniques. Typical problems faced by means of computational intelligence methods include those of recognition, such as classification and prediction. Although historically conceived to operate in some vector space, such methods have been recently extended to the so-called nongeometric spaces, considering labeled graphs as the most general example of such patterns. Here, we suggest that fractal analysis and computational intelligence methods can be exploited together in neuroscience research. Fractal characterizations can be used to (i) assess scale-invariant properties and (ii) offer numeric, feature-based representations to complement the usually more complex pattern structures encountered in neurosciences. Computational intelligence methods could be used to exploit such fractal characterizations, considering also the possibility to perform data-driven analysis of nongeometric input spaces, therby overcoming the intrinsic limits related to Euclidean geometry.

Entropy and Semi-Entropies of Regular Symmetrical Triangular Interval Type-2 Fuzzy Variables

Fuzzy entropy has a wide range of applications in uncertainty problems. Due to the dual-complexity of its characteristics and calculation, the study on type-2 fuzzy entropy is rare, let alone the semi-ones. Given this, the paper takes the lead in proposing the credibility-based type-2 entropy and semi-entropies delivered around a specific symmetric type-2 fuzzy variable. After presenting the relevant theorems and definitions, we give the corresponding examples of linear entropy and semi-entropies to illustrate and verify the favorable property of our study. This series of formulas on type-2 entropy proposed has a strong advantage in reducing computational complexity. It can be commonly applied to measure fuzziness and solve return-oriented and cost-oriented problems in the business field. A sequence of measures on type-2 fuzzy entropy developed in this paper delivers fresh insights into this field. It also provides a new reasonable tool for the decision-making on cost and investment control in companies.

Arithmetic Operations and Expected Values of Regular Interval Type-2 Fuzzy Variables

High computation complexity restricts the application prospects of the interval type-2 fuzzy variable (IT2-FV), despite its high degree of freedom in representing uncertainty. Thus, this paper studies the fuzzy operations for the regular symmetric triangular IT2-FVs (RSTIT2-FVs)—the simplest IT2-FVs having the greatest membership degrees of 1. Firstly, by defining the medium of an RSTIT2-FV, its membership function, credibility distribution, and inverse distribution are analytically and explicitly expressed. Secondly, an operational law for fuzzy arithmetic operations regarding mutually independent RSTIT2-FVs is proposed, which can simplify the calculations and directly output the inverse credibility of the functions. Afterwards, the operational law is applied to define the expected value operator of the IT2-FV and prove the linearity of the operator. Finally, some comparative examples are provided to verify the efficiency of the proposed operational law.

Cascade Adaptive MPC with Type 2 Fuzzy System for Safety and Energy Management in Autonomous Vehicles: A Sustainable Approach for Future of Transportation

A sustainable circular economy involves designing and promoting new products with the least environmental impact through increasing efficiency. The emergence of autonomous vehicles (AVs) has been a revolution in the automobile industry and a breakthrough opportunity to create more sustainable transportation in the future. Autonomous vehicles are supposed to provide a safe, easy-to-use and environmentally friendly means of transport. To this end, improving AVs’ safety and energy efficiency by using advanced control and optimization algorithms has become an active research topic to deliver on new commitments: carbon reduction and responsible innovation. The focus of this study is to improve the energy consumption of an AV in a vehicle-following process while safe driving is satisfied. We propose a cascade control system in which an autonomous cruise controller (ACC) is integrated with an energy management system (EMS) to reduce energy consumption. An adaptive model predictive control (AMPC) is proposed as the ACC to control the acceleration of the ego vehicle (the following vehicle) in a vehicle-following scenario, such that it can safely follow the lead vehicle in the same lane on a highway. The proposed ACC appropriately switches between speed and distance control systems to follow the lead vehicle safely and precisely. The computed acceleration is then used in the EMS component to find the optimal engine torque that minimizes the fuel consumption of the ego vehicle. EMS is designed based on two methods: type 1 fuzzy logic system (T1FLS) and interval type 2 fuzzy logic system (IT2FLS). Results show that the combination of AMPC and IT2FLS significantly reduces fuel consumption while the ego vehicle follows the lead vehicle safely and with a minimum spacing error. The proposed controller facilitates smarter energy use in AVs and supports safer transportation.

Bypassing the Natural Visual-Motor Pathway to Execute Complex Movement Related Tasks Using Interval Type-2 Fuzzy Sets

In visual-motor coordination, the human brain processes visual stimuli representative of complex motion-related tasks at the occipital lobe to generate the necessary neuronal signals for the parietal and pre-frontal lobes, which in turn generates movement related plans to excite the motor cortex to execute the actual tasks. The paper introduces a novel approach to provide rehabilitative support to patients suffering from neurological damage in their pre-frontal, parietal and/or motor cortex regions. An attempt to bypass the natural visual-motor pathway is undertaken using interval type-2 fuzzy sets to generate the approximate EEG response of the damaged pre-frontal/parietal/motor cortex from the occipital EEG signals. The approximate EEG response is used to trigger a pre-trained joint coordinate generator to obtain the desired joint coordinates of the link end-points of a robot imitating the human subject. The robot arm is here employed as a rehabilitative aid in order to move each link end-points to the desired locations in the reference coordinate system by appropriately activating its links using the well-known inverse kinematics approach. The mean-square positional errors obtained for each link end-points is found within acceptable limits for all experimental subjects including subjects with partial parietal damage, indicating a possible impact of the proposed approach in rehabilitative robotics. Subjective variation in EEG features over different sessions of experimental trials is modeled here using interval type-2 fuzzy sets for its inherent power to handle uncertainty. Experiments undertaken confirm that interval type-2 fuzzy realization outperforms its classical type-1 counterpart and back-propagation neural approaches in all experimental cases, considering link positional error as a metric. The proposed research offers a new opening for the development of possible rehabilitative aids for people with partial impairment in visual-motor coordination.

Performance analysis of extracted rule-base multivariable type-2 self-organizing fuzzy logic controller applied to anesthesia

We compare type-1 and type-2 self-organizing fuzzy logic controller (SOFLC) using expert initialized and pretrained extracted rule-bases applied to automatic control of anaesthesia during surgery. We perform experimental simulations using a nonfixed patient model and signal noise to account for environmental and patient drug interaction uncertainties. The simulations evaluate the performance of the SOFLCs in their ability to control anesthetic delivery rates for maintaining desired physiological set points for muscle relaxation and blood pressure during a multistage surgical procedure. The performances of the SOFLCs are evaluated by measuring the steady state errors and control stabilities which indicate the accuracy and precision of control task. Two sets of comparisons based on using expert derived and extracted rule-bases are implemented as Wilcoxon signed-rank tests. Results indicate that type-2 SOFLCs outperform type-1 SOFLC while handling the various sources of uncertainties. SOFLCs using the extracted rules are also shown to outperform those using expert derived rules in terms of improved control stability.

Teaching–learning-based optimal interval type-2 fuzzy PID controller design: a nonholonomic wheeled mobile robots

This paper introduces an optimal interval type-2 fuzzy proportional–integral–derivative (PID) controller to achieve the best trajectory tracking for nonholonomic wheeled mobile robots (WMRs). In the core of the proposed method, a novel population-based optimization algorithm, called teaching–learning-based optimization (TLBO), is employed for evolving the parameters of the controller as well as the parameters of the input and output membership functions. Two PID controllers are designed for each of two wheels separately whereas each controller has two inputs and one output that are logically connected by nine rules. The controller can handle the problem of integrated kinematic and dynamic tracking in the presence of uncertainties. Simulation results demonstrate the superiority of the proposed control scheme.

Overview of Type-2 Fuzzy Logic Systems

Fuzzy set theory has been proposed as a means for modeling the vagueness in complex systems. Fuzzy systems usually employ type-1 fuzzy sets, representing uncertainty by numbers in the range [0, 1]. Despite commercial success of fuzzy logic, a type-1 fuzzy set (T1FS) does not capture uncertainty in its manifestations when it arises from vagueness in the shape of the membership function. Such uncertainties need to be depicted by fuzzy sets that have blur boundaries. The imprecise boundaries of a type-2 fuzzy set (T2FS) give rise to truth/membership values that are fuzzy sets in [0], [1], instead of a crisp number. Type-2 fuzzy logic systems (T2FLSs) offer opportunity to model levels of uncertainty which traditional fuzzy logic type1 struggles. This extra dimension gives more degrees of freedom for better representation of uncertainty compared to type-1 fuzzy sets. A type-1 fuzzy logic system (T1FLSs) inference produces a T1FS and the result of defuzzification of the T1FS, a crisp number, whereas a T2FLS inference produces a type-2 fuzzy set, its type-reduced fuzzy set which is a T1FS and the defuzzification of the type-1 fuzzy set. The type-reduced fuzzy set output gives decision-making flexibilities. Thus, FLSs using T2FS provide the capability of handling a higher level of uncertainty and provide a number of missing components that have held back successful deployment of fuzzy systems in decision making.

Comments on "Dynamical Optimal Training for Interval Type-2 Fuzzy Neural Network (T2FNN)

In this comment, it will be shown that the backpropagation (BP) equations by Wang et al. are not correct. These BP equations were used to tune the parameters of the antecedent type-2 membership functions as well as the consequent part of the interval type-2 fuzzy neural networks (T2FNNs). These incorrect equations would have led to erroneous results, and hence this might affect the comparisons and findings presented by Wang et al. This comment will highlight the correct BP tuning equations for the T2FNN.

A simplified type-2 fuzzy logic controller for real-time control

Increasingly, genetic algorithms (GAs) are used to optimize the parameters of fuzzy logic controllers (FLCs). Although GAs provide a systematic design approach, the optimization process is generally performed off-line using a plant model. Differences between the model and physical plant may result in unsatisfactory control performance when the FLCs are deployed in practice. Type-2 FLCs are an attractive alternative because they can better cope with modeling uncertainties. Unfortunately, type-2 FLCs are computationally intensive. This paper presents a simplified type-2 FLC that is suitable for real-time applications. The key idea is to only replace some critical type-1 fuzzy sets by type-2 sets. Experimental results indicate that the proposed simplified type-2 FLC is as robust as a conventional type-2 FLC, while lowering the computational cost.