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The purpose of this paper is to give reviews on the platform modeling and design of a controller for autonomous vertical take-off and landing (VTOL) tilt rotor hybrid unmanned aerial vehicles (UAVs). Nowadays, UAVs have experienced remarkable progress and can be classified into two main types, i.e. fixed-wing UAVs and VTOL UAVs. The mathematical model of tilt rotor UAV is time variant, multivariable and non-linear in nature. Solving and understanding these plant models is very complex. Developing a control algorithm to improve the performance and stability of a UAV is a challenging task.

This paper gives a thorough description on modeling of VTOL tilt rotor UAV from first principle theory. The review of the design of both linear and non-linear control algorithms are explained in detail. The robust flight controller for the six degrees of freedom UAV has been designed using H-infinity optimization with loop shaping under external wind and aerodynamic disturbances.

This review will act as a basis for the future work on modeling and control of VTOL tilt rotor UAV by the researchers. The development of self-guided and fully autonomous UAVs would result in reducing the risk to human life. Civil applications include inspection of rescue teams, terrain, coasts, border patrol buildings, police and pipelines. The simulation results show that the controller achieves robust stability, good adaptability and robust performance.

The review articles on quadrotors/quadcopters, hybrid UAVs can be found in many literature, but there are comparatively a lesser amount of review articles on the detailed description of VTOL Tilt rotor UAV. In this paper modeling, platform design and control algorithms for the tilt rotor are presented. A robust H-infinity loop shaping controller in the presence of disturbances is designed for VTOL UAV.

The purpose of this paper is to present a novel optimization approach to design a robust H-infinity controller.

To use a modified particle swarm optimization (PSO) algorithm and to search for the optimal parameters of the weighting functions under the circumstance of the given structures of three weighting matrices in the H-infinity mixed sensitivity design.

This constrained multi-objective optimization is a non-convex, non-smooth problem which is solved by a modified PSO algorithm. An adaptive mutation-based PSO (AMBPSO) algorithm is proposed to improve the search accuracy and convergence of the standard PSO algorithm. In the AMBPSO algorithm, the inertia weights are modified as a function with the gradient descent and the velocities and positions of the particles.

The AMBPSO algorithm can efficiently solve such an optimization problem that a satisfactory robust H-infinity control performance can be obtained.

To provide high torques needed to move a robot’s links, electric actuators are followed by a transmission system with a high transmission rate. For instance, gear ratios of 100:1 are often used in the joints of a robotic manipulator. This results into an actuator with large mechanical impedance (also known as nonback-drivable actuator). This in turn generates high contact forces when collision of the robotic mechanism occur and can cause humans’ injury. Another disadvantage of electric actuators is that they can exhibit overheating when constant torques have to be provided. Comparing to electric actuators, pneumatic actuators have promising properties for robotic applications, due to their low weight, simple mechanical design, low cost and good power-to-weight ratio. Electropneumatically actuated robots usually have better friction properties. Moreover, because of low mechanical impedance, pneumatic robots can provide moderate interaction forces which is important for robotic surgery and rehabilitation tasks. Pneumatic actuators are also well suited for exoskeleton robots. Actuation in exoskeletons should have a fast and accurate response. While electric motors come against high mechanical impedance and the risk of causing injuries, pneumatic actuators exhibit forces and torques which stay within moderate variation ranges. Besides, unlike direct current electric motors, pneumatic actuators have an improved weight-to-power ratio and avoid overheating problems.

The aim of this paper is to analyze a nonlinear optimal control method for electropneumatically actuated robots. A two-link robotic exoskeleton with electropneumatic actuators is considered as a case study. The associated nonlinear and multivariable state-space model is formulated and its differential flatness properties are proven. The dynamic model of the electropneumatic robot is linearized at each sampling instance with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. Within each sampling period, the time-varying linearization point is defined by the present value of the robot’s state vector and by the last sampled value of the control inputs vector. An H-infinity controller is designed for the linearized model of the robot aiming at solving the related optimal control problem under model uncertainties and external perturbations. An algebraic Riccati equation is solved at each time-step of the control method to obtain the stabilizing feedback gains of the H-infinity controller. Through Lyapunov stability analysis, it is proven that the robot’s control scheme satisfies the H-infinity tracking performance conditions which indicate the robustness properties of the control method. Moreover, global asymptotic stability is proven for the control loop. The method achieves fast convergence of the robot’s state variables to the associated reference trajectories, and despite strong nonlinearities in the robot’s dynamics, it keeps moderate the variations of the control inputs.

In this paper, a novel solution has been proposed for the nonlinear optimal control problem of robotic exoskeletons with electropneumatic actuators. As a case study, the dynamic model of a two-link lower-limb robotic exoskeleton with electropneumatic actuators has been considered. The dynamic model of this robotic system undergoes first approximate linearization at each iteration of the control algorithm around a temporary operating point. Within each sampling period, this linearization point is defined by the present value of the robot’s state vector and by the last sampled value of the control inputs vector. The linearization process relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. The modeling error which is due to the truncation of higher-order terms from the Taylor series is considered to be a perturbation which is asymptotically compensated by the robustness of the control algorithm. To stabilize the dynamics of the electropneumatically actuated robot and to achieve precise tracking of reference setpoints, an H-infinity (optimal) feedback controller is designed. Actually, the proposed H-infinity controller for the model of the two-link electropneumatically actuated exoskeleton achieves the solution of the associated optimal control problem under model uncertainty and external disturbances. This controller implements a min-max differential game taking place between: (i) the control inputs which try to minimize a cost function which comprises a quadratic term of the state vector’s tracking error and (ii) the model uncertainty and perturbation inputs which try to maximize this cost function. To select the stabilizing feedback gains of this H-infinity controller, an algebraic Riccati equation is being repetitively solved at each time-step of the control method. The global stability properties of the H-infinity control scheme are proven through Lyapunov analysis.

Pneumatic actuators are characterized by high nonlinearities which are due to air compressibility, thermodynamics and valves behavior and thus pneumatic robots require elaborated nonlinear control schemes to ensure their fast and precise positioning. Among the control methods which have been applied to pneumatic robots, one can distinguish differential geometric approaches (Lie algebra-based control, differential flatness theory-based control, nonlinear model predictive control [NMPC], sliding-mode control, backstepping control and multiple models-based fuzzy control). Treating nonlinearities and fault tolerance issues in the control problem of robotic manipulators with electropneumatic actuators has been a nontrivial task.

The novelty of the proposed control method is outlined as follows: preceding results on the use of H-infinity control to nonlinear dynamical systems were limited to the case of affine-in-the-input systems with drift-only dynamics. These results considered that the control inputs gain matrix is not dependent on the values of the system’s state vector. Moreover, in these approaches the linearization was performed around points of the desirable trajectory, whereas in the present paper’s control method the linearization points are related with the value of the state vector at each sampling instance as well as with the last sampled value of the control inputs vector. The Riccati equation which has been proposed for computing the feedback gains of the controller is novel, so is the presented global stability proof through Lyapunov analysis. This paper’s scientific contribution is summarized as follows: (i) the presented nonlinear optimal control method has improved or equally satisfactory performance when compared against other nonlinear control schemes that one can consider for the dynamic model of robots with electropneumatic actuators (such as Lie algebra-based control, differential flatness theory-based control, nonlinear model-based predictive control, sliding-mode control and backstepping control), (ii) it achieves fast and accurate tracking of all reference setpoints, (iii) despite strong nonlinearities in the dynamic model of the robot, it keeps moderate the variations of the control inputs and (iv) unlike the aforementioned alternative control approaches, this paper’s method is the only one that achieves solution of the optimal control problem for electropneumatic robots.

The use of electropneumatic actuation in robots exhibits certain advantages. These can be the improved weight-to-power ratio, the lower mechanical impedance and the avoidance of overheating. At the same time, precise positioning and accurate execution of tasks by electropneumatic robots requires the application of elaborated nonlinear control methods. In this paper, a new nonlinear optimal control method has been developed for electropneumatically actuated robots and has been specifically applied to the dynamic model of a two-link robotic exoskeleton. The benefit from using this paper’s results in industrial and biomedical applications is apparent.

A comparison of the proposed nonlinear optimal (H-infinity) control method against other linear and nonlinear control schemes for electropneumatically actuated robots shows the following: (1) Unlike global linearization-based control approaches, such as Lie algebra-based control and differential flatness theory-based control, the optimal control approach does not rely on complicated transformations (diffeomorphisms) of the system’s state variables. Besides, the computed control inputs are applied directly on the initial nonlinear model of the electropneumatic robot and not on its linearized equivalent. The inverse transformations which are met in global linearization-based control are avoided and consequently one does not come against the related singularity problems. (2) Unlike model predictive control (MPC) and NMPC, the proposed control method is of proven global stability. It is known that MPC is a linear control approach that if applied to the nonlinear dynamics of the electropneumatic robot, the stability of the control loop will be lost. Besides, in NMPC the convergence of its iterative search for an optimum depends on initialization and parameter values selection and consequently the global stability of this control method cannot be always assured. (3) Unlike sliding-mode control and backstepping control, the proposed optimal control method does not require the state-space description of the system to be found in a specific form. About sliding-mode control, it is known that when the controlled system is not found in the input-output linearized form the definition of the sliding surface can be an intuitive procedure. About backstepping control, it is known that it cannot be directly applied to a dynamical system if the related state-space model is not found in the triangular (backstepping integral) form. (4) Unlike PID control, the proposed nonlinear optimal control method is of proven global stability, the selection of the controller’s parameters does not rely on a heuristic tuning procedure, and the stability of the control loop is assured in the case of changes of operating points. (5) Unlike multiple local models-based control, the nonlinear optimal control method uses only one linearization point and needs the solution of only one Riccati equation so as to compute the stabilizing feedback gains of the controller. Consequently, in terms of computation load the proposed control method for the electropneumatic actuator’s dynamics is much more efficient.

This paper aims to provide a mathematical modeling and design of H-infinity controller for an autonomous vertical take-off and landing (VTOL) Quad Tiltrotor hybrid unmanned aerial vehicles (UAVs). The variation in the aerodynamics and model dynamics of these aerial vehicles due to its tilting rotors are the key issues and challenges, which attracts the attention of many researchers. They carry parametric uncertainties (such as non-linear friction force, backlash, etc.), which drives the designed controller based on the nominal model to instability or performance degradation. The controller needs to take these factors into consideration and still give good stability and performance. Hence, a robust H-infinity controller is proposed that can handle these uncertainties.

A unique VTOL Quad Tiltrotor hybrid UAV, which operates in three flight modes, is mathematically modeled using Newton–Euler equations of motion. The contribution of the model is its ability to combine high-speed level flight, VTOL and transition between these two phases. The transition involves the tilting of the proprotors from 90° to 0° and vice-versa in 15° intervals. A robust H-infinity control strategy is proposed, evaluated and analyzed through simulation to control the flight dynamics for different modes of operation.

The main contribution of this research is the mathematical modeling of three flight modes (vertical takeoff–forward, transition–cruise-back, transition-vertical landing) of operation by controlling the revolutions per minute and tilt angles, which are independent of each other. An autonomous flight control system using a robust H-infinity controller to stabilize the mode of transition is designed for the Quad Tiltrotor UAV in the presence of uncertainties, noise and disturbances using MATLAB/SIMULINK. This paper focused on improving the disturbance rejection properties of the proposed UAV by designing a robust H-infinity controller for position and orientation trajectory regulation in the presence of uncertainty. The simulation results show that the Tiltrotor achieves transition successfully with disturbances, noise and uncertainties being present.

A novel VTOL Quad Tiltrotor UAV mathematical model is developed with a special tilting rotor mechanism, which combines both aircraft and helicopter flight modes with the transition taking place in between phases using robust H-infinity controller for attitude, altitude and trajectory regulation in the presence of uncertainty.

The purpose of this paper is to develop a multiobjective differential evolution (MODE)-based extended H-infinity controller for autonomous helicopter control.

Development of a MATLAB-based MODE suitable for controller synthesis. Formulate the H-infinity control scheme as an extended H-infinity loop shaping design procedure (H _∞ -LSDP) with incorporation of v-gap metric for robustness to parametric variation. Then apply the MODE-based algorithm to optimize the weighting function of the control problem formulation for optimal performance.

The proposed optimized H-infinity control was able to yield set of Pareto-controller candidates with optimal compromise between conflicting stability and time-domain performances required in autonomous helicopter deployment. The result of performance evaluation shows robustness to parameter variation of up to 20 per cent variation in nominal values, and in addition provides satisfactory disturbance rejection to wind disturbance in all the three axes.

The formulated H-infinity controller is limited to hovering and low speed flight envelope. The optimization is focused on weighting function parameters for a given fixed weighting function structure. This thus requires a priori selection of weighting structures.

The proposed MODE-infinity controller algorithm is expected to ease the design and deployment of the robust controller in autonomous helicopter application especially for practicing engineer with little experience in advance control parameters tuning. Also, it is expected to reduce the design cycle involved in autonomous helicopter development. In addition, the synthesized robust controller will provide effective hovering/low speed autonomous helicopter flight control required in many civilian unmanned aerial vehicle (UAV) applications.

The research will facilitate the deployment of low-cost, small-scale autonomous helicopter in various civilian applications.

The research addresses the challenges involved in selection of weighting function parameters for H-infinity control synthesis to satisfy conflicting stability and time-domain objectives. The problem of population initialization and objectives function computation in the conventional MODE algorithm are addressed to ensure suitability of the optimization algorithm in the formulated H-infinity controller synthesis.

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.

A distinctive feature of tilt-rotor UAVs is that they can be fully actuated, whereas in fixed-angle rotor UAVs (e.g. common-type quadrotors, octorotors, etc.), the associated dynamic model is characterized by underactuation. Because of the existence of more control inputs, in tilt-rotor UAVs, there is more flexibility in the solution of the associated nonlinear control problem. On the other side, the dynamic model of the tilt-rotor UAVs remains nonlinear and multivariable and this imposes difficulty in the drone's controller design. This paper aims to achieve simultaneously precise tracking of trajectories and minimization of energy dissipation by the UAV's rotors. To this end elaborated control methods have to be developed.

A solution of the nonlinear control problem of tilt-rotor UAVs is attempted using a novel nonlinear optimal control method. This method is characterized by computational simplicity, clear implementation stages and proven global stability properties. At the first stage, approximate linearization is performed on the dynamic model of the tilt-rotor UAV with the use of first-order Taylor series expansion and through the computation of the system's Jacobian matrices. This linearization process is carried out at each sampling instance, around a temporary operating point which is defined by the present value of the tilt-rotor UAV's state vector and by the last sampled value of the control inputs vector. At the second stage, an H-infinity stabilizing controller is designed for the approximately linearized model of the tilt-rotor UAV. To find the feedback gains of the controller, an algebraic Riccati equation is repetitively solved, at each time-step of the control method. Lyapunov stability analysis is used to prove the global stability properties of the control scheme. Moreover, the H-infinity Kalman filter is used as a robust observer so as to enable state estimation-based control. The paper's nonlinear optimal control approach achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs. Finally, the nonlinear optimal control approach for UAVs with tilting rotors is compared against flatness-based control in successive loops, with the latter method to be also exhibiting satisfactory performance.

So far, nonlinear model predictive control (NMPC) methods have been of questionable performance in treating the nonlinear optimal control problem for tilt-rotor UAVs because NMPC's convergence to optimum depends often on the empirical selection of parameters while also lacking a global stability proof. In the present paper, a novel nonlinear optimal control method is proposed for solving the nonlinear optimal control problem of tilt rotor UAVs. Firstly, by following the assumption of small tilting angles, the state-space model of the UAV is formulated and conditions of differential flatness are given about it. Next, to implement the nonlinear optimal control method, the dynamic model of the tilt-rotor UAV undergoes approximate linearization at each sampling instance 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. The linearization process is based on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. The modelling error, which is due to the truncation of higher-order terms from the Taylor series, is considered to be a perturbation that is asymptotically compensated by the robustness of the control scheme. For the linearized model of the UAV, an H-infinity stabilizing feedback controller is designed. To select the feedback gains of the H-infinity controller, an algebraic Riccati equation has to be repetitively solved at each time-step of the control method. The stability properties of the control scheme are analysed with the Lyapunov method.

There are no research limitations in the nonlinear optimal control method for tilt-rotor UAVs. The proposed nonlinear optimal control method achieves fast and accurate tracking of setpoints by all state variables of the tilt-rotor UAV under moderate variations of the control inputs. Compared to past approaches for treating the nonlinear optimal (H-infinity) control problem, the paper's approach is applicable also to dynamical systems which have a non-constant control inputs gain matrix. Furthermore, it uses a new Riccati equation to compute the controller's gains and follows a novel Lyapunov analysis to prove global stability for the control loop.

There are no practical implications in the application of the nonlinear optimal control method for tilt-rotor UAVs. On the contrary, 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 to the linear parameter varying (LPV) 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 stability properties of the Galerkin series expansion-based optimal control approaches are still unproven.

The proposed nonlinear optimal control method is suitable for using in various types of robots, including robotic manipulators and autonomous vehicles. By treating nonlinear control problems for complicated robotic systems, the proposed nonlinear optimal control method can have a positive impact towards economic development. So far the method has been used successfully in (1) industrial robotics: robotic manipulators and networked robotic systems. One can note applications to fully actuated robotic manipulators, redundant manipulators, underactuated manipulators, cranes and load handling systems, time-delayed robotic systems, closed kinematic chain manipulators, flexible-link manipulators and micromanipulators and (2) transportation systems: autonomous vehicles and mobile robots. Besides, one can note applications to two-wheel and unicycle-type vehicles, four-wheel drive vehicles, four-wheel steering vehicles, articulated vehicles, truck and trailer systems, unmanned aerial vehicles, unmanned surface vessels, autonomous underwater vessels and underactuated vessels.

The proposed nonlinear optimal control method is a novel and genuine result and is used for the first time in the dynamic model of tilt-rotor UAVs. The nonlinear optimal control approach exhibits advantages against other control schemes one could have considered for the tilt-rotor UAV dynamics. For instance, (1) compared to the global linearization-based control schemes (such as Lie algebra-based control or flatness-based control), it does not require complicated changes of state variables (diffeomorphisms) and transformation of the system's state-space description. Consequently, it also avoids inverse transformations which may come against singularity problems, (2) compared to NMPC, the proposed nonlinear optimal control method is of proven global stability and the convergence of its iterative search for an optimum does not depend on initialization and controller's parametrization, (3) compared to sliding-mode control and backstepping control the application of the nonlinear optimal control method is not constrained into dynamical systems of a specific state-space form. It is known that unless the controlled system is found in the input–output linearized form, the definition of the associated sliding surfaces is an empirical procedure. Besides, unless the controlled system is found in the backstepping integral (triangular) form, the application of backstepping control is not possible, (4) compared to PID control, the nonlinear optimal control method is of proven global stability and its performance is not dependent on heuristics-based selection of parameters of the controller and (5) compared to multiple-model-based optimal control, the nonlinear optimal control method requires the computation of only one linearization point and the solution of only one Riccati equation.

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 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.

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.