Some applications of fractional calculus on control problems in robotics and system stability

In recent years, there have been extensive research activities related to applications of fractional calculus (FC), [5] in nonlinear dynamics, mechatronics as well as control theory. In this paper, they are presented recently obtained results which are related to applications of fractional calculus in mechanics - specially stability and control issues. Some of these results [1-4] are presented at the Fifth symposium of fractional differentiation and its applications FDA2012, was held at the Hohai University, Nanjing, China in the period of May 14-May 17, 2012. Also, fractional order dynamic systems and controllers have been increasing in interest in many areas of science and engineering in the last few years.In that way, our objective of using fractional calculus is to apply the fractional order controller to enhance the system control performance as well as it has better disturbance rejection ratios and less sensitivity to plant parameter variations. First, they are introduced and obtained the new algorithms of fractional order PID control based on genetic algorithms in the position control of a 3 DOF’s robotic system driven by DC motors. Then, the main task is to find out the optimal settings for a fractional PI D controller in order to fulfill the proposed design specifications for the closed-loop system. In addition, this method allows the optimal design of all major parameters of a fractional PID controller and then enhances the flexibility and capability of the PID controller. Last,in simulations, they are compared step responses of these two optimal controllers where it will be shown that fractional order PID controller improves transient response as well as provides more robustness in than conventional PID. Second, we propose sufficient conditions for finite time stability for the (non)homogeneous fractional order systems with time delay. Specially, the problem of finite time stability with respect to some of the variables (partial stability) is considered. Namely, along with the formulation of the problem of stability to all variables, Lyapunov also formulated a more general problem on the stability to a given part of variables (but not all variables) determining the state of a system,[6].The problem of the stability of motion with respect to some of the variables also known as partial stability arises naturally in applications. So, in this presentation, it will be proposed finite time partial stability test procedure of perturbed (non) linear (non)autonomous time varying delay fractional order systems. Time-delay is assumed to be varying with time but its upper bound is assumed to be known over given time interval. New stability criteria for this class of fractional order systems will be derived using “classical” Bellman-Gronwall inequality,as well as another another suitable inequality, [7]. Last,a numerical example is provided to illustrate the application of the proposed stability procedure. Third, some attention is devoted to the problem of stability of linear discrete-time fractional order systems is addressed, [8]. It was shown that some stability criteria for discrete time-delay systems could be applied with small changes to discrete fractional order state-space systems. Accordingly, simple conditions for the stability and robust stability of a particular class of linear discrete time-delay systems are derived. These results are modified and used for checking the stability of discrete-time fractional order systems. The systems under consideration involve time delays in the state and parameter uncertainties. The parameter uncertainties are assumed to be time-varying and norm bounded

Some applications of fractional calculus on control problems in robotics and system stability

In recent years, there have been extensive research activities related to applications of fractional calculus (FC), [5] in nonlinear dynamics, mechatronics as well as control theory. In this paper, they are presented recently obtained results which are related to applications of fractional calculus in mechanics - specially stability and control issues. Some of these results [1-4] are presented at the Fifth symposium of fractional differentiation and its applications FDA2012, was held at the Hohai University, Nanjing, China in the period of May 14-May 17, 2012. Also, fractional order dynamic systems and controllers have been increasing in interest in many areas of science and engineering in the last few years.In that way, our objective of using fractional calculus is to apply the fractional order controller to enhance the system control performance as well as it has better disturbance rejection ratios and less sensitivity to plant parameter variations. First, they are introduced and obtained the new algorithms of fractional order PID control based on genetic algorithms in the position control of a 3 DOF’s robotic system driven by DC motors. Then, the main task is to find out the optimal settings for a fractional PI D controller in order to fulfill the proposed design specifications for the closed-loop system. In addition, this method allows the optimal design of all major parameters of a fractional PID controller and then enhances the flexibility and capability of the PID controller. Last,in simulations, they are compared step responses of these two optimal controllers where it will be shown that fractional order PID controller improves transient response as well as provides more robustness in than conventional PID. Second, we propose sufficient conditions for finite time stability for the (non)homogeneous fractional order systems with time delay. Specially, the problem of finite time stability with respect to some of the variables (partial stability) is considered. Namely, along with the formulation of the problem of stability to all variables, Lyapunov also formulated a more general problem on the stability to a given part of variables (but not all variables) determining the state of a system,[6].The problem of the stability of motion with respect to some of the variables also known as partial stability arises naturally in applications. So, in this presentation, it will be proposed finite time partial stability test procedure of perturbed (non) linear (non)autonomous time varying delay fractional order systems. Time-delay is assumed to be varying with time but its upper bound is assumed to be known over given time interval. New stability criteria for this class of fractional order systems will be derived using “classical” Bellman-Gronwall inequality,as well as another another suitable inequality, [7]. Last,a numerical example is provided to illustrate the application of the proposed stability procedure. Third, some attention is devoted to the problem of stability of linear discrete-time fractional order systems is addressed, [8]. It was shown that some stability criteria for discrete time-delay systems could be applied with small changes to discrete fractional order state-space systems. Accordingly, simple conditions for the stability and robust stability of a particular class of linear discrete time-delay systems are derived. These results are modified and used for checking the stability of discrete-time fractional order systems. The systems under consideration involve time delays in the state and parameter uncertainties. The parameter uncertainties are assumed to be time-varying and norm bounded

Optimal v-plane robust stabilization method for interval uncertain fractional order pid control systems

Robust stability is a major concern for real-world control applications. Realization of optimal robust stability requires a stabilization scheme, which ensures that the control system is stable and presents robust performance for a predefined range of system perturbations. This study presented an optimal robust stabilization approach for closed-loop fractional order proportional integral derivative (FOPID) control systems with interval parametric uncertainty and uncertain time delay. This stabilization approach, which is carried out in a v-plane, relies on the placement of the minimum angle system pole to a predefined target angle within the stability region of the first Riemann sheet. For this purpose, tuning of FOPID controller coefficients was performed to minimize a root angle error that is defined as the squared difference of minimum angle root of interval characteristic polynomials and the desired target angle within the stability region of the v-plane. To solve this optimization problem, a particle swarm optimization (PSO) algorithm was implemented. Findings of the study reveal that tuning of the target angle can also be used to improve the robust control performance of interval uncertain FOPID control systems. Illustrative examples demonstrated the effectiveness of the proposed v-domain, optimal, robust stabilization of FOPID control systems. © 2021 by the authors. Li-censee MDPI, Basel, Switzerland

On the Selection of Tuning Methodology of FOPID Controllers for the Control of Higher Order Processes

In this paper, a comparative study is done on the time and frequency domain tuning strategies for fractional order (FO) PID controllers to handle higher order processes. A new fractional order template for reduced parameter modeling of stable minimum/non-minimum phase higher order processes is introduced and its advantage in frequency domain tuning of FOPID controllers is also presented. The time domain optimal tuning of FOPID controllers have also been carried out to handle these higher order processes by performing optimization with various integral performance indices. The paper highlights on the practical control system implementation issues like flexibility of online autotuning, reduced control signal and actuator size, capability of measurement noise filtration, load disturbance suppression, robustness against parameter uncertainties etc. in light of the above tuning methodologies.Comment: 27 pages, 10 figure

Design and practical implementation of a fractional order proportional integral controller (FOPI) for a poorly damped fractional order process with time delay

One of the most popular tuning procedures for the development of fractional order controllers is by imposing frequency domain constraints such as gain crossover frequency, phase margin and iso-damping properties. The present study extends the frequency domain tuning methodology to a generalized range of fractional order processes based on second order plus time delay (SOPDT) models. A fractional order PI controller is tuned for a real process that exhibits poorly damped dynamics characterized in terms of a fractional order transfer function with time delay. The obtained controller is validated on the experimental platform by analyzing staircase reference tracking, input disturbance rejection and robustness to process uncertainties. The paper focuses around the tuning methodology as well as the fractional order modeling of the process' dynamics

Universal direct tuner for loop control in industry

This paper introduces a direct universal (automatic) tuner for basic loop control in industrial applications. The direct feature refers to the fact that a first-hand model, such as a step response first-order plus dead time approximation, is not required. Instead, a point in the frequency domain and the corresponding slope of the loop frequency response is identified by single test suitable for industrial applications. The proposed method has been shown to overcome pitfalls found in other (automatic) tuning methods and has been validated in a wide range of common and exotic processes in simulation and experimental conditions. The method is very robust to noise, an important feature for real life industrial applications. Comparison is performed with other well-known methods, such as approximate M-constrained integral gain optimization (AMIGO) and Skogestad internal model controller (SIMC), which are indirect methods, i.e., they are based on a first-hand approximation of step response data. The results indicate great similarity between the results, whereas the direct method has the advantage of skipping this intermediate step of identification. The control structure is the most commonly used in industry, i.e., proportional-integral-derivative (PID) type. As the derivative action is often not used in industry due to its difficult choice, in the proposed method, we use a direct relation between the integral and derivative gains. This enables the user to have in the tuning structure the advantages of the derivative action, therefore much improving the potential of good performance in real life control applications

Application of a Fractional Order Integral Resonant Control to increase the achievable bandwidth of a nanopositioner

The congress program will essentially include papers selected on the highest standard by the IPC, according to the IFAC guidelines www.ifac-control.org/publications/Publications-requirements-1.4.pdf, and published in open access in partnership with Elsevier in the IFAC-PapersOnline series, hosted on the ScienceDirect platform www.sciencedirect.com/science/journal/24058963. Survey papers overviewing a research topic are also most welcome. Contributed papers will have usual 6 pages length limitation. 12 pages limitation will apply to survey papers.Publisher PD