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Voltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion). The dynamic model of VSI-PMSMs is multivariable and exhibits complicated nonlinear dynamics. The inverters’ currents, which are generated through a pulsewidth modulation process, are used to control the stator currents of the PMSM, which in turn control the rotational speed of this electric machine. So far, several nonlinear control schemes for VSI-PMSMs have been developed, having as primary objectives the precise tracking of setpoints by the system’s state variables and robustness to parametric changes or external perturbations. However, little has been done for the solution of the associated nonlinear optimal control problem. The purpose of this study/paper is to provide a novel nonlinear optimal control method for VSI-fed three-phase PMSMs.

The present article proposes a nonlinear optimal control approach for VSI-PMSMs. The nonlinear dynamic model of VSI-PMSMs undergoes approximate linearization around a temporary operating point, which is recomputed at each iteration of the control method. This temporary operating point is defined by the present value of the voltage source inverter-fed PMSM state vector and by the last sampled value of the motor’s control input vector. The linearization relies on Taylor series expansion and the calculation of the system’s Jacobian matrices. For the approximately linearized model of the voltage source inverter-fed PMSM, an H-infinity feedback controller is designed. For the computation of the controller’s feedback gains, an algebraic Riccati equation is iteratively solved at each time-step of the control method. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control for this system, the H-infinity Kalman filter is proposed as a state observer. The proposed control method achieves fast and accurate tracking of the reference setpoints of the VSI-fed PMSM under moderate variations of the control inputs.

The proposed H-infinity controller provides the solution to the optimal control problem for the VSI-PMSM system under model uncertainty and external perturbations. Actually, this controller represents a min–max differential game taking place between the control inputs, which try to minimize a cost function that contains a quadratic term of the state vector’s tracking error, the model uncertainty, and exogenous disturbance terms, which try to maximize this cost function. To select the feedback gains of the stabilizing feedback controller, an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. To analyze the stability properties of the control scheme, the Lyapunov method is used. It is proven that the VSI-PMSM loop has the H-infinity tracking performance property, which signifies robustness against model uncertainty and disturbances. Moreover, under moderate conditions, the global asymptotic stability properties of this control scheme are proven. The proposed control method achieves fast tracking of reference setpoints by the VSI-PMSM state variables, while keeping also moderate the variations of the control inputs. The latter property indicates that energy consumption by the VSI-PMSM control loop can be minimized.

The proposed nonlinear optimal control method for the VSI-PMSM system exhibits several advantages: Comparing to global linearization-based control methods, such as Lie algebra-based control or differential flatness theory-based control, the nonlinear optimal control scheme avoids complicated state variable transformations (diffeomorphisms). Besides, its control inputs are applied directly to the initial nonlinear model of the VSI-PMSM system, and thus inverse transformations and the related singularity problems are also avoided. Compared with backstepping control, the nonlinear optimal control scheme does not require the state-space description of the controlled system to be found in the triangular (backstepping integral) form. Compared with sliding-mode control, there is no need to define in an often intuitive manner the sliding surfaces of the controlled system. Finally, compared with local model-based control, the article’s nonlinear optimal control method avoids linearization around multiple operating points and does not need the solution of multiple Riccati equations or LMIs. As a result of this, the nonlinear optimal control method requires less computational effort.

Voltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion), The solution of the associated nonlinear control problem enables reliable and precise functioning of VSI-fd PMSMs. This in turn has a positive impact in all related industrial applications and in tasks of electric traction and propulsion where VSI-fed PMSMs are used. It is particularly important for electric transportation systems and for the wide use of electric vehicles as expected by green policies which aim at deploying electromotion and at achieving the Net Zero objective.

Unlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is defined by the present value of the system’s state vector and by the last sampled value of the control input vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation, which is used for computing the feedback gains of the controller, is new, as is the global stability proof for this control method. Comparing with nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (H-infinity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of state-dependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems that can be transformed to the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions.

The efficient speed controller is found to be an important requirement to run the motor for the brushless direct current (BLDC) motor. This requirement is considered as superior, as it may increase the operating speed and system efficiency. In the existing methods, proportional plus integral (PI) controller has been included because of its simple architecture. But the PI controller produces load disturbance, control complexity and some parametric (Proportional plus integral) variations. The purpose of this proposed controller is to overcome the problems produced by PI controller in BLDC motor.

The proposed BLDC motor is developed with fixed order H-infinity controller. In this architecture, both the weight functions and transfer functions were included to design the controller. This controller has been included in this BLDC to detect the rotor position. The optimal position of rotor is identified by introducing particle swarm optimization algorithm.

The torque that obtained in the motor is highly reduced by this proposed controller and also enhances the speed. The BLDC motor is modelled in a MATLAB environment.

The performance of the torque, speed and back electro-motive force is analysed and compared with the existing controllers such as fuzzy proportional plus integral plus derivative, sensing algorithm and fuzzy proportional plus derivative controller.

Simulation results show that the proposed technique gives better results than the other existing controllers.

The paper's aim is to reconsider the feasibility at both the heights of mathematics and philosophy of the statement that each predicate determines a unique set.

A conceptual approach is taken.

In the naive and the modern axiomatic set theories, it is a well‐known fact that each predicate determines precisely one set. That is to say, for any precisely defined predicate P, there is always A={x|P(x)} or x∈A↔P(x). However, when the authors are influenced by the thinking logic of allowing both kinds of infinities and compare these two kinds of infinities, and potentially infinite and actually infinite intervals and number sets, it is found that the expressions of these number sets are not completely reasonable.

Detailed analyses are given for the introduction of new symbols and representations for either potential or actual infinite sets.

The paper's aim is to reveal the return of the Berkeley paradox of the eighteenth century.

This is a conceptual discussion.

Since, long time ago, the common belief has been that the establishment and development of the theory of limits had provided an explanation for the Berkeley paradox. However, when the authors revisit some of the age‐old problems using the thinking logic of allowing both the concepts of potential and actual infinities, they find surprisingly that the shadow of the Berkeley paradox does not truly disappear in the foundation of mathematical analysis.

Show the incompleteness of the theory of limits, which is not the same as what has been believed in the history of mathematics.

The distinction between essence and existence cannot be a distinction in God: in the actual infinite, essence and existence coincide and are one. In it, maximum and minimum coincide. Coincidentia oppositorum is a Latin phrase meaning coincidence of opposites. It is a neo-Platonic term, attributed to the fifteenth-century German scholar Nicholas of Cusa in his essay, Docta Ignorantia. God (coincidentia oppositorum) is the synthesis of opposites in a unique and absolutely infinite being. God transcends all distinctions and oppositions that are found in creatures. The purpose of this paper is to study Cusanus’s thought in respect to infinity (actual and potential), Spinoza’s relationship with Cusanus, and present a mathematical theory of coincidentia oppositorum based on complex numbers.

Mathematical development of a dialectical logic is carried out with truth values in a complex field.

The conclusion is the same as has been made by thinkers and mystics throughout time: the inability to know and understand the idea of God.

The history of the Infinite thus reveals in both mathematics and philosophy a development of increasingly subtle thought in the form of a dialectical dance around the ineffable and incomprehensible Infinite. First, the authors step toward it, reaching with their intuition beyond the limits of rationality and thought into the realm of the paradoxical. Then, they step back, struggling to express their insight within the limited scope of reason. But the Absolute Infinite remains, at the border of comprehensibility, inviting them with its paradoxes, to once again step forward and transcend the apparent division between finite and Infinite.

The purpose of this article is to treat the nonlinear optimal control problem in EV traction systems which are based on 5-phase induction motors. Five-phase permanent magnet synchronous motors and five-phase asynchronous induction motors (IMs) are among the types of multiphase motors one can consider for the traction system of electric vehicles (EVs). By distributing the required power in a large number of phases, the power load of each individual phase is reduced. The cumulative rates of power in multiphase machines can be raised without stressing the connected converters. Multiphase motors are also fault tolerant because such machines remain functional even if failures affect certain phases.

A novel nonlinear optimal control approach has been developed for five-phase IMs. The dynamic model of the five-phase IM undergoes approximate linearization using Taylor series expansion and the computation of the associated Jacobian matrices. The linearization takes place at each sampling instance. For the linearized model of the motor, an H-infinity feedback controller is designed. This controller achieves the solution of the optimal control problem under model uncertainty and disturbances.

To select the feedback gains of the nonlinear optimal (H-infinity) controller, an algebraic Riccati equation has to be solved repetitively at each time-step of the control method. The global stability properties of the control loop are demonstrated through Lyapunov analysis. Under moderate conditions, the global asymptotic stability properties of the control scheme are proven. The proposed nonlinear optimal control method achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

Comparing to other nonlinear control methods that one could have considered for five-phase IMs, the presented nonlinear optimal (H-infinity) control approach avoids complicated state-space model transformations, is of proven global stability and its use does not require the model of the motor to be brought into a specific state-space form. The nonlinear optimal control method has clear implementation stages and moderate computational effort.

In the transportation sector, there is progressive transition to EVs. The use of five-phase IMs in EVs exhibits specific advantages, by achieving a more balanced distribution of power in the multiple phases of the motor and by providing fault tolerance. The study’s nonlinear optimal control method for five-phase IMs enables high performance for such motors and their efficient use in the traction system of EVs.

Nonlinear optimal control for five-phase IMs supports the deployment of their use in EVs. Therefore, it contributes to the net-zero objective that aims at eliminating the emission of harmful exhaust gases coming from human activities. Most known manufacturers of vehicles have shifted to the production of all-electric cars. The study’s findings can optimize the traction system of EVs thus also contributing to the growth of the EV industry.

The proposed nonlinear optimal control method is novel comparing to past attempts for solving the optimal control problem for nonlinear dynamical systems. It uses a novel approach for selecting the linearization points and a new Riccati equation for computing the feedback gains of the controller. The nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of state-dependent Riccati equations.

This paper is the fourth part of the effort to resolve the following two problems that urgently need an answer: how can an appropriate theoretical foundation be chosen for modern mathematics and computer science? And, under what interpretations can modern mathematics and the theory of computer science be kept as completely as possible?

The paper is a conceptual discussion.

The paper lays out the set theoretical foundation for the mathematical system of potential infinities.

This work is the non‐logical axiomatic part of the mathematical system of potential infinities: the axiomatic set theoretic system. At the end, the problem of consistency of this axiomatic set theory is discussed.

Permanent magnet synchronous spherical motors can have wide use in robotics and industrial automation. They enable three-DOF omnidirectional motion of their rotor. They are suitable for several applications, such as actuation in robotics, traction in electric vehicles and use in several automation systems. Unlike conventional synchronous motors, permanent magnet synchronous spherical motors consist of a fixed inner shell, which is the stator, and a rotating outer shell, which is the rotor. Their dynamic model is multivariable and strongly nonlinear. The treatment of the associated control problem is important.

In this paper, the multivariable dynamic model of permanent magnet synchronous spherical motors is analysed, and a nonlinear optimal (H-infinity) control method is developed for it. Differential flatness properties are proven for the spherical motors’ state-space model. Next, the motors’ state-space description undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a time-varying operating point, which is defined by the present value of the motors’ state vector and by the last sampled value of the control input vector. For the approximately linearized model of the permanent magnet synchronous spherical motors, a stabilizing H-infinity feedback controller is designed. To compute the controller’s gains, an algebraic Riccati equation has to be repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. Finally, the performance of the nonlinear optimal control method is compared against a flatness-based control approach implemented in successive loops.

Due to the nonlinear and multivariable structure of the state-space model of spherical motors, the solution of the associated nonlinear control problem is a nontrivial task. In this paper, a novel nonlinear optimal (H-infinity) control approach is proposed for the dynamic model of permanent magnet synchronous spherical motors. The method is based on approximate linearization of the motor’s state-space model with the use of first-order Taylor series expansion and the computation of the associated Jacobian matrices. Furthermore, the paper has introduced a different solution to the nonlinear control problem of the permanent magnet synchronous spherical motor, which is based on flatness-based control implemented in successive loops.

The presented control approaches do not exhibit any limitations, but on the contrary, they have specific advantages. In comparison to global linearization-based control schemes (such as Lie-algebra-based control), they do not make use of complicated changes of state variables (diffeomorphisms) and transformations of the system's state-space description. The computed control inputs are applied directly to the initial nonlinear state-space model of the permanent magnet spherical motor without the intervention of inverse transformations and thus without coming against the risk of singularities.

The motion control problem of spherical motors is nontrivial because of the complicated nonlinear and multivariable dynamics of these electric machines. So far, there have been several attempts to apply nonlinear feedback control to permanent magnet-synchronous spherical motors. However, due to the model’s complexity, few results exist about the associated nonlinear optimal control problem. The proposed nonlinear control methods for permanent magnet synchronous spherical motors make more efficient, precise and reliable the use of such motors in robotics, electric traction and several automation systems.

The treated research topic is central for robotic and industrial automation. Permanent magnet synchronous spherical motors are suitable for several applications, such as actuation in robotics, traction in electric vehicles and use in several automation systems. The solution of the control problem for the nonlinear dynamic model of permanent magnet synchronous spherical motors has many industrial applications and therefore contributes to economic growth and development.

The proposed nonlinear optimal control method is novel compared to past attempts to solve the optimal control problem for nonlinear dynamical systems. Unlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation which is used for computing the feedback gains of the controller is new, and so is the global stability proof for this control method. Compared to nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (H-infinity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of state dependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems which can be transformed into the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes, which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions. Furthermore, the second control method proposed in this paper, which is flatness-based control in successive loops, is also novel and demonstrates substantial contribution to nonlinear control for robotics and industrial automation.

The purpose of this paper is to compare three cultural approaches from anthropology and business literature: National Culture Approach (NCA), Corporate Culture Approach (CCA), and Transactional Culture Approach (TCA). The author grounds these approaches in different epistemological standpoints and locate them at different positions on the unity-infinity continuum. The author outlines their strengths and weaknesses, and offer the Douglasian Cultural Framework (DCF) as a transactional tool for cultural sense-making.

Reviewing conventional NCA/CCA frameworks reveals that while their simplicity renders them attractive to users, their assumption of stable, internally homogenous and coherent cultures has its limitations. Conversely, reviewing anthropology-based TCA literature reveals that while TCA overcomes some limitations of NCA/CCA frameworks, it also has its weaknesses – it overemphasizes “self-interest” as the preferred form of rationality, and some TCA scholars render cultural comparisons impossible by supporting cultural infinity. Finally, examining DCF reveals that it overcomes some limitations of NCA/CCA frameworks, while simultaneously advancing TCA. Nevertheless, DCF too has limitations which are also exposed.

Most NCA/CCA scholars support the “unity” argument of culture, while some transactional scholars support the “infinity” argument. DCF finds a perfect balance between the two through “constrained relativism”. Also, since DCF focuses on human transactions, it is not limited in its applications to specific levels and scales. It can therefore be applied to scenarios spanning across levels and scales. Finally, it offers a compromise between the differentiation and fragmentation perspectives of corporate culture, and brings out the best of the interpretivist and post-modernistic traditions.

The exposition of DCF opens up new avenues for research which have hitherto remained unexplored for want of appropriate frameworks, for instance the UN Peace Corps., NATO, Medecins Sans Frontiers, etc.

By focusing on human transactions, the paper allows for a much more dynamic conceptualization of culture as compared to static NCA/CCA frameworks.

On the basis of previous work, the authors aim to further study the problem of infinity existing in between predicates and sets.

A conceptual approach is taken in the paper.

The authors modify the conventional rule of thinking that each predicate determines a unique set, and establish a principle regarding the relationship between predicates and sets. Then, the authors study the structures of actually infinite, rigid sets.

The structure of actually infinite sets is detailed.