Showing papers in "International Journal of Dynamics and Control in 2017"

A delay fractional order model for the co-infection of malaria and HIV/AIDS

Abstract: We propose a delay fractional order model for the co-infection of malaria and the human immunodeficiency virus, where personal protection and vaccination against malaria are considered. We compute the reproduction number of the model and study the stability of the disease free equilibrium. The numerical simulations of the model are performed for distinct values of the order of the fractional derivative, $$\alpha \in (0,1]$$ . We have also varied relevant parameters, such as the susceptibility to malaria of individuals showing symptoms of acquired immunodeficiency syndrome, $$ u _2$$ , the degree of sexual activity due to malaria, $$\epsilon _2$$ , the human immunodeficiency virus related mortality due to co-infection, $$\psi $$ , and the level of personal protection against malaria, b. The outputs of the model are biologically meaningful.

Sliding mode control design for synchronization of fractional order chaotic systems and its application to a new cryptosystem

Abstract: In this paper, the synchronization method of two identical fractional order chaotic systems is developed with lower order than the existing fractional order 3 by designing suitable sliding mode control. Further, a new cryptosystem is derived for an image encryption and decryption based on the synchronized lowest fractional order 2.01 chaotic systems. To validate the theoretical results, numerical simulations are presented for the proposed cryptosystem.

On the analysis of a multi-regions discrete SIR epidemic model: an optimal control approach.

Abstract: In this paper, we devise a discrete time SIR model depicting the spread of infectious diseases in various geographical regions that are connected by any kind of anthropological movement, which suggests disease-affected people can propagate the disease from one region to another via travel. In fact, health policy-makers could manage the problem of the regional spread of an epidemic, by organizing many vaccination campaigns, or by suggesting other defensive strategies such as blocking movement of people coming from borders of regions at high-risk of infection and entering very controlled regions or with insignificant infection rate. Further, we introduce in the discrete SIR systems, two control variables which represent the effectiveness rates of vaccination and travel-blocking operation. We focus in our study to control the outbreaks of an epidemic that affects a hypothetical population belonging to a specific region. Firstly, we analyze the epidemic model when the control strategy is based on the vaccination control only, and secondly, when the travel-blocking control is added. The multi-points boundary value problems, associated to the optimal control problems studied here, are obtained based on a discrete version of Pontryagin's maximum principle, and resolved numerically using a progressive-regressive discrete scheme that converges following an appropriate test related to the Forward-Backward Sweep Method on optimal control.

Uncertainty and disturbance estimator based sliding mode control of an autonomous underwater vehicle

Abstract: In this paper, a highly non-linear model of an autonomous underwater vehicle (AUV) with six degrees-of-freedom is linearized to yaw (horizontal) and pitch (vertical) planes under several working conditions. For controlling steering and diving planes, an uncertainty disturbance estimator based sliding mode control (UDE-SMC) scheme is proposed and designed separately as single-input single-output controllers for horizontal and vertical plane dynamics of an AUV system. The proposed UDE-SMC scheme is effective in compensating the uncertainties in the hydrodynamic parameters of the vehicle and rejecting unpredictable disturbances due to ocean currents. The UDE-SMC consists of an equivalent and estimated lumped uncertain terms to suppress the effect of external disturbances and parametric uncertainties acting on the vehicle dynamics. Numerical simulations were performed to validate the UDE-SMC.

Fractional calculus in economic growth modelling: the Spanish and Portuguese cases

Abstract: This work presents a fractional order approach to model the growth of national economies, namely, their gross domestic products (GDPs). Land area, arable land, population, school attendance, gross capital formation, exports of goods and services, general government final consumption expenditure and money and quasi money are taken as variables to describe GDP. The particular cases of the national economies of Spain and Portugal are studied along the last five decades. Results show that fractional models have a better performance than the other alternatives considered in the literature.

Synchronization between fractional order complex chaotic systems

Abstract: In this article, the authors have studied synchronization between a pair of fractional order complex systems viz., Lorenz and Lu systems, Lu and T systems, Lorenz and T systems using active control method. The numerical results and simulation show that this method is effective to synchronize the fractional order complex dynamical systems. The main feature of the article is the comparison of time of synchronization when pair of systems approach from integer order to fractional order. The numerical results are carried out using MATLAB.

Hybrid function projective synchronization of chaotic systems via adaptive control

Abstract: In this manuscript, hybrid function projective synchronization of Bhalekar–Gejji and Pehlivan chaotic systems is established by applying adaptive control technique where the system parameters are unknown. In this manuscript both the master and slave system are chosen in such a way that none of them can be derived from the member of the unified chaotic system. We construct an adaptive controller in such a manner that master and slave system attain global chaos synchronization. The results derived for the synchronization have been established using adaptive control theory and Lyapunov stability theory. Fundamental dynamical properties of both the chaotic systems are also described. The results are validated by numerical simulation which are performed by using Matlab.

Synchronization of uncertain fractional-order hyper-chaotic systems via a novel adaptive interval type-2 fuzzy active sliding mode controller

Abstract: In this paper, a novel adaptive interval type-2 fuzzy active sliding mode control (AIT2FASMC) approach is proposed for synchronization of fractional-order hyper-chaotic systems. The synchronization is achieved in front of uncertainties facing the fuzzy logic controller (FLC) in fractional–order hyper-chaotic systems such as uncertainties in control outputs, uncertainties in inputs to the fuzzy logic controller, and linguistic uncertainties. Two fractional-order hyper-chaotic systems can be synchronized based on Lyapunov stability theorem by using direct AIT2FASMC approach. Also, the proposed method reduces the chattering phenomena in the control signal, significantly. This novel fractional-order sliding mode controller is proposed for robust stabilization/synchronization problem of a class of fractional-order hyper-chaotic systems in the presence of external noise. Type-2 fuzzy active sliding mode control $$\left( {FASMC}\right) $$ have the ability to overcome the limitations of type-1 FASMC when system is corrupted by high levels of uncertainty. Finally, the proposed approach is applied to two identical and non-identical fractional-order hyper-chaotic systems when the slave system is perturbed by external noise. Simulation results show applicability and feasibility of the proposed finite-time control strategy.

Synchronization of fractional-order hyper-chaotic systems based on a new adaptive sliding mode control

Abstract: In this paper, a novel adaptive sliding mode approach for synchronization of fractional-order identical and non-identical chaotic and hyper-chaotic systems is proposed. The effects of unknown parameters and model uncertainties are also fully taken into account. The upper bound of the uncertainties is used to obtain the controller parameters. Here in this paper a novel sliding surface is proposed and its finite-time convergence to origin is analytically proved. Appropriate adaptation laws are obtained to undertake the unknown parameters of the controller. The stability of the resulting synchronization error system in a given finite-time is proved. Finally, particle swarm optimization algorithm is used for optimizing the controller parameters. Illustrative examples for a recently revealed 4-D fractional-order chaotic system are presented. This new 4-D fractional-order chaotic system has non chaotic behavior for its integer-order system and is different from the other previous fractional-order chaotic systems. Simulation results show the applicability and feasibility of the proposed finite-time control strategy.

Analysis and hyper-chaos control of a new 4-D hyper-chaotic system by using optimal and adaptive control design

Abstract: In this manuscript,a new 4-D hyper-chaotic system is proposed. Followed by optimal and adaptive control theory. Firstly, we analyze the chaotic properties of new 4-D hyper-chaotic system such as dissipation, equilibrium, stability, time series, phase portraits, Lyapunov exponents, bifurcation and Poincare maps. Next, we study the optimal control for the new 4-D hyper-chaotic system which is based on the Pontryagin minimum principle. Further, Lyapunov stability theory is used for adaptive control approach and a parameter estimation update law is given for the new 4-D hyper-chaotic system with completely unknown parameters. Finally, to demonstrate the effectiveness of the proposed method we use MATLAB bvp4c and ode45 for numerical simulation which illustrate the stabilized behaviour of states and control functions for different equilibrium points. The plots displaying the time history of states functions and the parameters estimates have been drawn for the different values of equilibrium points.

Hyper-chaotic analysis and adaptive multi-switching synchronization of a novel asymmetric non-linear dynamical system

Abstract: In this paper, a new 4D autonomous asymmetric hyperchaotic system obtained from a three dimensional autonomous chaotic system is introduced. We analyse the hyperchaotic properties of the new system such as dissipation, equilibrium, Lyapunov exponent, stability, time series, phase portraits, Poincare map and bifurcation diagram. Furthermore, the multi-switching synchronization of the new asymmetric hyperchaotic system is analysed by using the adaptive control strategy. Based on the theory of Lyapunov stability, we sketch the input controllers and updating laws of distinct switching, and it is extended to examine the synchronization problems of response and drive systems with distinct combinations. Brief theoretical analysis and numerical results are presented to show the dynamical behaviour of the new 4D hyperchaotic system.

A novel fractional adaptive active sliding mode controller for synchronization of non-identical chaotic systems with disturbance and uncertainty

Abstract: In this paper, a class of integer and fractional-order chaotic systems, which undergoes external disturbances and system uncertainties, are considered. A robust synchronization scheme that incorporates a sliding mode controller established on a new fractional-order surface. A fractional order derivative provides an additional degree of freedom in the sliding surface. After that, the stability analysis for closed loop system is studied. Then, based on a Lyapunov function candidate an adaptive switching gain is derived which make the controller capable to bring the synchronizing error to zero without any disturbance exerted upon the stability. The proposed method is designed for a wide class of chaotic systems. Furthermore, the results are extended for fractional-order version of chaotic systems. The proposed controller can be used to both integer and fractional order chaotic systems. The design is simple with rigorous stability analysis. Several numerical simulations are provided to verify the effectiveness of the theoretical results.

Fractional derivatives and periodic functions

Abstract: This paper studies the periodic functions in the perspective of fractional calculus application. It is shown that the fractional derivative of a periodic signal is periodic if it is defined on the whole real line. Several common fractional derivative formulations are considered, namely the Grunwald–Letnikov, Liouville and Caputo on $$\mathbb {R}$$ , and the two-sided fractional derivatives. It is verified that the fractional derivative of a causal periodic signal is never causal periodic. The periodic behaviour of the fractional linear systems is also studied. If such systems are defined with suitable derivatives the output corresponding to periodic input is also periodic. Is is concluded that only the integer order linear systems can have a sinusoidal impulse response.

Emergence of complex dynamical behaviors in improved Colpitts oscillators: antimonotonicity, coexisting attractors, and metastable chaos

Abstract: Emerging applications of chaotic oscillators require a classification and characterization of their complex behaviour with respect to their numerous parameters. In the present work, the dynamics of improved Colpitts oscillator is revisited based on a smooth (i.e. exponential) mathematical model of the system. In contrast to previous literature related to the improved Colpitts oscillator, the numerical simulations reveal very rich and striking phenomena including antimonotonicity, coexistence of attractors, and metastable chaos. Various complex dynamics regimes are characterized in terms of the system parameters by using bifurcation diagrams, Lyapunov exponents and phase space trajectory plots. Some PSpice simulations of the nonlinear dynamics of the oscillator are carried out to validate the theoretical analysis. Owing to the richness of bifurcation modes observed in this work, the improved Colpitts oscillator may be useful in theoretical, numerical and experimental studies of various aspects of nonlinear dynamics in autonomous systems and related topics. It also has promising applications in the fields of High Frequency chaos based communications, sonar sensors as well as radar systems.

Explicit criteria for stability of fractional h-difference two-dimensional systems

Abstract: In the paper the explicit conditions for stability of linear fractional order h-difference systems with the Grunwald–Letnikov-type operator are presented. The state variables of the considered systems are taken from the plane. As the tool the \({\mathcal {Z}}\)-transform, which can be considered as an effective method for the stability analysis of linear systems, is used. The main result gives the sufficient and necessary condition for the asymptotic stability of the considered system according to the entries of the given matrix associated with the system.

Passenger seat vibration control of a semi-active quarter car system with hybrid Fuzzy–PID approach

Abstract: In this paper, semi-active quarter car system with three degrees of freedom is considered for modeling and evaluation of passenger ride comfort. Experimental results of magneto-rheological shock absorber are modeled using polynomial model. The considered algorithms in semi-active quarter car suspension system include PID controller, fuzzy logic controller, hybrid fuzzy–PID controller and hybrid fuzzy–PID controller with coupled rules. Simulation responses of the controlled semi-active and uncontrolled quarter car systems are compared under bump type of road excitation in time domain. Simulation results demonstrate that the semi-active suspension system having hybrid fuzzy–PID controller with coupled rules provide best performance in controlling the passenger seat acceleration and displacement response compared to uncontrolled and other controlled cases.

Routes of periodic motions to chaos in a periodically forced pendulum

Abstract: In this paper, with varying excitation amplitude, bifurcation trees of periodic motions to chaos in a periodically driven pendulum are obtained through a semi-analytical method. This method is based on the implicit discrete maps obtained from the midpoint scheme of the corresponding differential equation. Using the discrete maps, mapping structures are developed for specific periodic motions, and the corresponding nonlinear algebraic equations of such mapping structures are solved. Further, semi-analytical bifurcation trees of periodic motions to chaos are also obtained, and the corresponding eigenvalue analysis is carried out for the stability and bifurcation of the periodic motions. Finally, numerical illustrations of periodic motions on the bifurcation trees are presented in verification of the analytical prediction. Harmonic amplitude spectra are also presented for demonstrating harmonic effects on the periodic motions. The bifurcation trees of period-1 motions to chaos possess a double spiral structure. The two sets of solutions of period- $$2^{l}$$ motions $$(l=0,1,2,\ldots )$$ to chaos are based on the center around $$2m\pi $$ and $$(2m-1)\pi (m=1,2,3,\ldots )$$ in phase space. Other independent bifurcation trees of period-m motions to chaos are presented. Through this investigation, the motion complexity and nonlinearity of the periodically forced pendulum can be further understood.

Nonlinear model-based parameter estimation and stability analysis of an aero-pendulum subject to digital delayed control

Abstract: Digitization and time delay are known to modify the stability properties of feedback controlled systems. Although their effects have been widely investigated and they occur in most of the systems equipped with digital processors, they are usually neglected in industrial approaches, by virtue of the high sampling frequencies of modern processors. However, these approaches are not conservative with respect to stability. In this work, we investigate, first analytically, then numerically and experimentally, the stability properties of the so-called Aeropendulum. The Aeropendulum is a mechanical pendulum with a propeller at its free end. A motor, activating the propeller, allows an active control of the pendulum in a feedback loop. The system exhibits most of the difficulties encountered in more involved industrial robotic systems. The estimation of the parameter values is performed through a model-based estimation, which allows to successfully define damping coefficients of order zero, one and two. Stability charts obtained with different controllers are compared, showing the larger stability region obtainable with the act-and-wait controller under proper conditions, as predicted by the theory.

Global qualitative analysis of a phytoplankton–zooplankton model in the presence of toxicity

Abstract: In this paper, we investigate the dynamical behavior of toxic-phytoplankton–zooplankton model with delay. Here, we describe a phytoplankton–zooplankton system that exhibits a Holling type II functional response in the presence of toxicity. Combined effort (E) is used to harvest the population. It is assumed that the phytoplankton is affected directly by an external toxic substance and the feeding of zooplankton on the affected phytoplankton is influenced indirectly by the toxic substance. Firstly we consider the elementary dynamical properties of the toxic-phytoplankton–zooplankton interacting model system in absence of time delay. Then we establish the existence of local Hopfbifurcation as the time delay crosses a threshold value and also prove the existence of stability switching phenomena. Theoretical results of this paper justified with numerical simulations.

Impact of generalist type sexually reproductive top predator interference on the dynamics of a food chain model

Abstract: Food uptake ability of higher trophic level species are more complicated and interesting due to their choice and availability of food and consequently their growth. In the present study, the effect of top predator interference on the dynamics of a tritrophic food chain model involving an intermediate and a generalist type top predator are considered. It is assumed that the interaction between the logistically growing prey and intermediate predator follows the Volterra scheme, while that between the top predator and its favorite food i.e. the intermediate predator follows Crowley–Martin type functional response. Also, it is considered that the growth of top predator is by sexual reproduction. The boundedness of the system, existence of an attracting set, local and global stability of non-negative equilibrium points and uniform persistence are established by very complicated and highly nonlinear biological considerations. With the help of empirical results from different field and laboratory experiment, it is clarified that Crowley–Martin type functional response gives better result than other predator dependent functional responses for the issue of interference among own population. Also it is clarified from literature survey that high trophic species are mostly generalist type i.e. they can survive rather than their favorite food depending upon the present environmental situation and their growth rate depends upon the mating success. Moreover, increasing the top predator interference stabilizes the system, while increasing the normalization of the residual reduction in the top predator population destabilizes the system. Different numerical examples support these ecological phenomenon.

Optimal control of a delayed hepatitis B viral infection model with cytotoxic T-lymphocyte and antibody responses

Abstract: The dynamics of a model describing the hepatitis B viral infection model with cytotoxic T-lymphocyte and antibody responses is studied in this paper. The model consists of a system of differential equations describing the interaction between hepatocytes, the free virus and the immune responses. Both the treatment and the intracellular delay are incorporated into the model. Existence, positivity and boundedness of solutions are investigated. Also, the existence of the optimal control pair is established and the Pontryagin’s minimum principle is used to characterize these optimal controls. The optimal controls represent the efficiency of drug treatment in inhibiting viral production and preventing new infections. The optimality system is derived and solved numerically using the forward and backward difference approximation. The obtained results show that the optimal treatment strategies reduce the viral load and then increase the uninfected hepatocytes, this improves the patient life quality.

Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete implicit mappings

Abstract: In this paper, periodic motions of a periodically forced, damped, Duffing oscillator with double-well potential are analytically predicted through discrete implicit mappings. The discrete implicit maps are obtained from the differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions of the Duffing oscillator are predicted analytically, and the corresponding stability and bifurcation analysis of periodic motions are carried out through the eigenvalue analysis. Finally, from the analytical prediction, numerical results of periodic motions are performed by the numerical method of the differential equation to verify the semi-analytical prediction, and the corresponding harmonic amplitudes are computed through discrete Fourier series of the analytically predicted node points of periodic motions, and the complexity of periodic motions can be measured by the harmonic amplitude. The frame work presented in this paper can provide a semi-analytical method to find periodic motions and to determine routes of periodic motions to chaos rather than numerical simulation only in nonlinear dynamical systems, and the stable and unstable periodic motions and even chaos can be predicted analytically.

Stationary patterns induced by self- and cross-diffusion in a Beddington–DeAngelis predator–prey model

Abstract: The study of spatial pattern formation through diffusion-driven instability of reaction–diffusion models of interacting species has long been one of the fundamental problems in mathematical ecology. The present article is concerned with interacting predator–prey reaction–diffusion model with Beddington–DeAngelis type functional response. The essential conditions for Hopf and Turing bifurcations are derived on the spatial domain. The parameter space for Turing spatial structure is established. Based on the bifurcation analysis, the spatial pattern formation in Turing space through numerical simulations is carried out in order to study the evolution procedure of the proposed model system in the vicinity of coexistence equilibrium point. The consequences of the results obtained reveal that the effects of self- and cross-diffusion play significant role on the steady state spatiotemporal pattern formation of the reaction–diffusion predator–prey model system which concerns the influence of intra-species competition among predators. Finally, ecological implications of the present results obtained are discussed at length towards the end in order to validate the applicability of the model under consideration.

Phase and anti-phase synchronizations of fractional order hyperchaotic systems with uncertainties and external disturbances using nonlinear active control method

Abstract: In this paper, the phase and anti-phase synchronizations between fractional order hyperchaotic Lu and 4D integral order systems with parametric uncertainties and disturbances are studied using nonlinear active control method. A new lemma is used to design the controller. Numerical simulations are presented to demonstrate the effectiveness of the method to synchronize and anti-synchronize the fractional order hyperchaotic systems. The striking feature of the article is the comparison of time of synchronization and anti-synchronization with and without the presence of uncertainties and external disturbances through graphical presentations for different particular cases.

A hybrid method of evolutionary algorithm and simple cell mapping for multi-objective optimization problems

Abstract: A hybrid method is proposed to take advantages of evolutionary algorithms (EAs) and the simple cell mapping (SCM) for multi-objective optimization problems (MOPs). The hybrid method starts with a random search for Pareto optimal solutions with an EA, and follows up with a neighborhood based search and recovery algorithm using the SCM. The non-dominated sorting genetic algorithm-II (NSGA-II) is used as an example of EAs. It is found that the SCM based search and recovery algorithm can reconstruct the branches of the Pareto set even when only one point in the vicinity of the set is available from the random search by the EA. We have chosen several benchmark MOPs to compare NSGA-II and SCM separately with the EA $$+$$ SCM hybrid method while using the Hausdorff distance as a performance metric, and applied the method to develop multi-objective optimal designs of PID controls for a nonlinear oscillator with time delay. The results show that the EA $$+$$ SCM hybrid method is very promising.

When behaviour turns contagious: the use of deterministic epidemiological models in modeling social contagion phenomena

Abstract: Mathematical models offer crucial insights into the transmission dynamics and control of infectious diseases. These models have also been applied to investigate a variety of ‘contagious’ social phenomena like crime, opinions, addiction and fanaticism. We review the use and adaptation of models from epidemiology (compartmental models) to investigate the transmission dynamics of different social contagion processes- all of which are spread by contact only.

A new analysis of infection dynamics: multi-regions discrete epidemic model with an extended optimal control approach

Abstract: We propose in this paper an extended optimal control approach applied to multi-regions discrete model for the study of infection dynamics and control of an epidemic when it emerges in many regions which are supposed to be accessible for health authorities. We apply optimal control theory for investigating the effectiveness of an optimal control approach for the prevention of disease outbreaks, and which is based on both vaccination and travel-blocking strategies. First, the vaccination method taken into account in this work, aims to highlight the importance of mass vaccination campaigns that health policy-makers could lead in all regions affected by an epidemic, and important to control, for reducing the number of their infected people and increasing the number of their removed people. Second, the travel-blocking optimal control approach introduced here, is presented as a defensive strategy to limit the number of people traveling from regions with high-risk of infection towards regions with a risk relatively smaller. These regions are also controlled by vaccinations, and movements of their people intending to reach other regions, are restricted, in order to help contain the spread of the epidemic by following convenient vaccination for each region.

Analysis of a mathematical model of periodically pulsed chemotherapy treatment

Abstract: In this paper, we have considered competition models describing tumor–normal–immune cell interaction with the added effects of periodically pulsed chemotherapy The parametric conditions needed to prevent relapse following attempts to remove the tumor or tumor metastasis are obtained The effects of resistant tumor sub-populations are also investigated and recurrence prevention strategies are discussed Our analytical findings are explained through numerical simulation which show the reliability of our models from the epidemiological point of view

Global dynamics of a prey-predator model with Allee effect and additional food for the predators

Abstract: Provision of additional/alternative food to the predators for controlling predator-prey dynamics has been receiving considerable attention from theoretical as well as experimental biologists. This is due to environment friendly role played by the additional food in controlling and managing the interacting population. Theoretical investigations done on additional food provided predator-prey models reveal that provision of additional food to predators has a significant role to play in enhancement of commercially important predator species and also in reduction of prey population. The quality and quantity of the additional food provided to predators play vital role in shaping the dynamics of the interacting system. So far as our knowledge goes, all theoretical investigations carried out in this direction assume logistic growth for the prey species. In reality, the per capita growth rate of the prey population is often an increasing function at low prey density. Incorporation of this realistic growth rate induces Allee effect into the dynamics of the prey species. In this paper, we consider an additional food provided predator-prey model wherein the prey population is subjected to Allee effect. The model includes both strong and weak Allee effects. This article presents a comprehensive analysis of the considered model that highlights the influence of Allee effect in prey and additional food for the predators on the system dynamics.

Design of fractional-order PI controllers and comparative analysis of these controllers with linearized, nonlinear integer-order and nonlinear fractional-order representations of PMSM

Abstract: Mathematical model of permanent magnet synchronous motor (PMSM), which is also popularly known as sinusoidal BLDC motor, is highly nonlinear. Speed control of this motor is a cascade control structure which has two distinct loops: an inner current control loop and outer speed control loop. In this paper, we present the design of fractional proportional-integral (PI) controller using small-signal model of PMSM. The basic scheme used for speed control is field-oriented control wherein constant torque angle control is obtained by zero direct-axis current. This requires three PI controllers: one for direct-axis current to keep at zero; another for quadrature axis current to maintain at maximum; and the third for the speed to keep at the reference level. In this paper, we have replaced all three PI controllers with fractional-order (FO) PI controllers, where order of integrator is fractional. These three FOPI controllers are tested on integer-order nonlinear model as well as fractional-order nonlinear model of PMSM using MATLAB simulink and FOMCON toolbox. The comparative analysis shows superior performance of the FOPI controllers for FO PMSM.