Fractional calculus approach to modeling and control of (bio)mechanical systems

Recently, an increasing attention has been paid to fractional calculus (FC) and its application in control and modeling of fractional order (bio)mechanical system. Fractional derivatives and integrals may have a wide application in describing complex properties of materials including long-term memory, non-locality of power law type and fractality [1]. In this presentation we applied the concept of fractional order for biomehanical modeling of human arm dynamics as well as soft tissues, specially human skin as well as human blood. Besides, it is also presented the connection between fractional order differintegral operators and behavior of the memsystems which can be used for modeling dynamics of (bio)mechanical systems. Further, we present robust feedback-(feedforward) loop fractional-order iterative learning control [2] for regular and singular fractional order system. Particularly, a feedback-(feedforward) PDalpfa / PIbetaDalpfa type iterative learning control (ILC) of fractional order system- (regular and degenerate type) which includes time delay are considered [3]. Sufficient conditions for the convergence of a proposed PD alpha type of learning control algorithm for a class of fractional state space time delay system are given in time domain. Finally, a simulation results show the feasibility and effectiveness of the suggested approach

Advanced fractional order modeling and control of dynamics of complex systems: recent results

In this presentation, we provide some applications of memristors and mem-systems with a particular focus on electromechanical systems and analogies that holds great promise for advanced modeling and control of complex objects and processes. In science and engineering, the ideas and concepts developed in one branch of science and engineering are often transferred to other branches. In addition to the analogy between mechanical and electrical systems, it was observed that phenomena from other physical domains exhibit similar properties, [1]. Representative example is nonlinear element -memristor which was postulated by Chua in 1971 [1] by analyzing mathematical relations between pairs of fundamental circuit variables. Besides, the relation between current and voltage which defines a memristive system, the relation between charge and voltage also specifies a memcapacitive system, and the flux-current relation gives rise to a meminductive system [2]. Here, we give a short review of available mem-systems integer order. In addition, important property of fractional operators is that they capture the history of all past events which means that fractional order systems [3] have intrinsically a memory of the previous dynamical evolution. Particularly, we present the connection between fractional order differintegral operators and behavior of the mem-systems which can be used for modeling dynamics of complex systems. Several potential applications of electromechanical analogies of integer and fractional order are discussed. Further, we investigate and suggest an open-closed-loop P/PDalpha type iterative learning control (ILC) [4] of fractional order singular complex system [5]. Particularly, we discuss fractional order linear singular systems in pseudo state space form. Sufficient conditions for the convergence in time domain of the proposed fractional order ILC for a class of fractional order singular system are given by the corresponding theorem together with its proof. Finally, numerical example is presented to illustrate the effectiveness of the proposed open-closed ILC scheme of fractional order for a class of fractional order singular complex system

Recent results on advanced control and stability issues of fractional-order dynamical systems

Recently, fractional calculus has attracted the increased attention of scientific society where fractional operators are often used for complex dynamical systems. Iterative learning control (ILC) is one of the recent topics in control theories and it is a powerful intelligent control concept that iteratively improves the behavior of processes that are repetitive in nature. Here, we present recently obtained results as well as new results on open-closed loop type ILC, for a given class of integer order and fractional order regular systems. We discuss PIDD2/ PID,PD2Dα,PDα/PD types ILC, particularly ILC schemes with Dα type which is more flexible for practical implementation. Sufficient conditions for the convergence in the time domain of the proposed ILC for a class of fractional and integer order systems are given by the corresponding theorems together with its proof. Finally, the simulation results, including an application to the suitable robot system and Neuro-Arm robot, are presented to illustrate the performance of the proposed ILC schemes. Also, some attention will be devoted to the finite-time stability/stabilization problem of fractional-order (uncertain) neutral time-delay systems. By use of the generalized Gronwall inequality and its extended form, new sufficient conditions for finite-time stability of such systems are obtained. Finally, numerical examples are given to illustrate the effectiveness and applicability of the proposed theoretical results.[http://www.mi.sanu.ac.rs/~icme2022/

Design of generalized fractional order gradient descent method

This paper focuses on the convergence problem of the emerging fractional order gradient descent method, and proposes three solutions to overcome the problem. In fact, the general fractional gradient method cannot converge to the real extreme point of the target function, which critically hampers the application of this method. Because of the long memory characteristics of fractional derivative, fixed memory principle is a prior choice. Apart from the truncation of memory length, two new methods are developed to reach the convergence. The one is the truncation of the infinite series, and the other is the modification of the constant fractional order. Finally, six illustrative examples are performed to illustrate the effectiveness and practicability of proposed methods.Comment: 8 pages, 16 figure

Further results on advanced modeling and control of complex mechanical systems

The investigation into the dynamics and control of robotic and complex (bio)mechanical systems has been an active topic of research for many years. The science of robotics/adaptronics has grown tremendously over the past twenty years, fueled by rapid advances in computer and sensor technology, as well as theoretical advances in control theory. Recently, calculus of general order R has attracted an increased attention of scientific society where fractional operators are often used for control issues and for modelling dynamic of complex systems,[1].The modelling complex rigid multibody systems using symbolic equations can provide many advantages over the more widely-used numerical methods of modelling these systems. In this presentation, we propose using procedure for symbolic form computation of the complete dynamics of (exoskeleton) robotic systems with kinematic chain structures using the Rodriquez approach, [2]. Dynamic equations are given as Lagrange equations of the second kind in the covariant form with external generalized forces of the gravity, motor-torque, viscous and spring. Mathematical model of the proposed NeuroArm robotic system due to a high gear ratio between the actuators and robot joints, can be reduced to a linear model. Robust control of general order with no overshoot can be obtained using fractional order compensator which is designed according to the symmetrical optimum principle, [3].The effectiveness of the proposed method will be illustrated through the control simulation of three degrees of freedom robot manipulator. Also, some attention will be devoted to problem the viscous friction in robotic joints. The calculus of general order and the calculus of variations are utilized to modelling the viscous friction which is extended to the fractional derivative of the angular displacement. This model is introduced into dynamic equations via generalized forces which are derived by using the principle of virtual work. Also, it is presented the tracking problem of exoskeleton robotic system for rehabilitation with three DOFs with revolute joints via intelligent control which includes advanced iterative learning control (ILC), [4]. First, a feedback linearization control technique based on computed torque method is applied on a given robotic system. Then, the proposed intelligent ILC algorithm takes the advantages offered by closed-loop control PD type and open-loop control sgnPDD2 type of ILC. Suggested robust ILC algorithm is applied to the linearized system to further enhance tracking performance for repetitive tasks and deal with the model uncertainties. Finally, a simulation example is presented to illustrate the effectiveness of the proposed robust ILC scheme for a proposed exoskeleton robot arm