A flatness-based control approach which is implemented in successive loops is used to solve the control problem for the multivariable and nonlinear dynamics of industrial robotic manipulators and autonomous vehicles. The state-space model of these robotic systems is separated into two subsystems, which are connected between them in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system and control about it can be performed with inversion of its dynamics as in the case of input-output linearized flat systems. The state variables of the second subsystem become virtual control inputs for the first subsystem. In turn exogenous control inputs are applied to the first subsystem. The whole control method is implemented in two successive loops and its global stability properties are also proven through Lyapunov stability analysis. The validity of the control method is confirmed in two case studies: (a) control of a 3-DOF industrial rigidlink robotic manipulator, (ii) control of a 3-DOF autonomous underwater vessel.

A nonlinear optimal (H-infinity) control method is developed for the Lotka-Volterra dynamical system. First, differential flatness properties are proven. The state-space description undergoes linearization, at each sampling instance, with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. Next, for the approximately linearized model of the system 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. Global stability properties are proven through Lyapunov analysis. Finally, the nonlinear optimal control method is compared against a flatness-based control approach implemented in successive loops.

In this article, the problem of nonlinear optimal (H-infinity) control for a translational oscillator with rotating actuator (TORA) system is treated. The dynamic model of a translational oscillator with rotational actuator is generated through Euler-Lagrange analysis. This model undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearized state-space model of the system, a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller’s feedback gains, an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control without the need to measure the entire state vector of the TORA the H-infinity Kalman Filter is used as a robust state estimator.

Voltage-source inverter-fed Permanent Magnet Synchronous Machines are widely used in industry (for instance for the actuation of robotic and mechatronic systems, of cranes, in water pumping stations) as well as in transportation systems (for the traction of trains and electric vehicles). The present article proposes a nonlinear optimal control approach for voltage source inverter-fed Permanent Magnet Synchronous Machines (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 machine’s control inputs vector. The linearization relies on Taylor series expansion and on 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. This controller stands for the solution of the nonlinear optimal control problem for the voltage source inverter-fed PMSM under model uncertainty and external perturbations. For the computation of the controller’s feedback gain an algebraic Riccati equation is iteratively solved at each time-step the control method. The global asymptotic stability properties of the control method are proven through Lyapunov analysis.

Fault diagnosis in energy conversion systems is performed with the use of neural networks and statistical decision making. An energy conversion system comprising a solar power unit, a DC-DC converter and a DC motor is considered and the related condition monitoring problem is solved. A neural network is used to model the dynamics of this energy conversion system after processing its input and output measurements, being accumulated at different operating conditions. The considered neural model is trained with the use of first-order gradient algorithms and consists of a hidden layer of Gauss-Hermite polynomial activation functions and of an output layer with linear weights. The neural network and the resulting model represents the fault-free functioning of the energy conversion system. At a next stage, the measurements of the real output of the energy conversion system are compared against the estimated outputs which are provided by the neural model. This provides, the residuals sequence. It holds that the sum of the squares of the residuals' vectors, multiplied with the inverse of the associated covariance matrix, stands for a stochastic variable (statistical test) which follows the χ2distribution. One can have a precise and almost infallible decision making tool about the appearance of faults in the energy conversion system, by selecting the 96% or the 98% confidence intervals of this distribution. When the upper or lower bound of the confidence interval are persistently exceeded one can conclude that the system has been subject to a fault. Finally, fault isolation can be also accomplished, by applying the statistical test into subspaces of the energy conversion system's state-space model.

1A nonlinear H-infinity (optimal) control approach is developed for the problem of the control of the spherical rolling robot. The solution of such a control problem is a nontrivial case due to underactuation and strong nonlinearities in the system's state-space description. The dynamic model of the robot undergoes approximate linearization around a temporary operating point which is recomputed at each timestep of the control method. The linearization relies on Taylor series expansion and on the computation of the system's Jacobian matrices. For the linearized dynamics of the spherical robot an H-infinity controller is designed. To compute the controller's feedback gains an algebraic Riccati equation in solved at each iteration of the control algorithm. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, for the implementation of sensorless control for the spherical rolling robot, the H-infinity Kalman Filter is used as a robust state estimator.

A nonlinear optimal control approach is developed for the UAV and suspended load system. The dynamic model of the UAV and payload system undergoes approximate linearization. This makes use of Taylor series expansion around a temporary operating point which recomputed at each iteration of the control method. The linearization procedure relies on the computation of the Jacobian matrices of the state-space model of the system. Next, an H-infinity feedback controller is designed for the approximately linearized model. The proposed control method stands for the solution of the optimal control problem for the nonlinear and multivariable dynamics of the UAV and payload system, under model uncertainties and external perturbations. To compute the controller's feedback gains an algebraic Riccati equation is solved at each time-step of the control algorithm. The new nonlinear optimal control approach achieves fast and accurate tracking for all state variables of the UAV and payload system, under moderate variations of the control inputs. Finally, Lyapunov analysis is used to prove the global stability properties of the control scheme.

Control of the milling process of mining products (ore milling) is a non-trivial problem due to being related with a strongly nonlinear and multivariable state-space model. To provide an efficient solution to this problem, in this article a nonlinear optimal (H-infinity) control method is developed. In the considered nonlinear optimal control method, the dynamic model of the mining products' mill undergoes first approximate linearization with the use of Taylor series expansion and with the computation of the associated Jacobian matrices. The linearization point (temporary equilibrium) is recomputed at each time step of the control method and comprises the present value of the system's state vector and the last value of the control inputs' vector that was exerted on it. For the linearized description of the mill's functioning the optimal control problem is solved by applying an H-infinity controller. The feedback gain is computed again at each iteration of the control algorithm through the solution of an algebraic Riccati equation. The stability of the control scheme is confirmed through Lyapunov analysis. First, it is shown that the control method satisfies the H-infinity tracking performance, and this signifies elevated robustness against model uncertainty and external perturbations. Next, under moderate conditions, it is proven that the control loop is globally asymptotically stable.

Permanent Magnet Linear Synchronous Motors are of wide use in industry in applications where actuation through rotational motors and a gears-based transmission system can be costly and prone to failures. In this article, a nonlinear optimal (H-infinity) control method is proposed for Permanent Magnet Linear Synchronous Motors (PMLSM). The dynamic model of the Permanent Magnet Linear Synchronous Motor undergoes approximate linearization around a temporary operating point (equilibrium) which is recomputed at each iteration of the control method. The linearization procedure is based on first-order Taylor-series expansion and on the computation of the Jacobian matrices of the motor’s model. For the approximately linearized model of the motor an H-infinity feedback controller is designed. This controller stands for the solution of the motor’s optimal control problem under model uncertainty and external disturbances. The computation of the controller’s feedback gain requires the solution of an algebraic Riccati equation, which is performed again at each time-step of the control algorithm. The stability properties of the control scheme are proven trough Lyapunov analysis. First, it is confirmed that the controller satisfies the H-infinity tracking performance criterion which ascertains its robustness. Moreover, it is proven that the control loop is globally asymptotically stable. Finally, to implement sensorless control of the motor the H-infinity Kalman Filter is used as a robust state estimator.

The article proposes a method for diagnosing faults and cyberattacks in electric power generation units that consist of a wind-turbine and of an asynchronous (DFIG) generator. The method relies on a differential flatness theory-based implementation of the nonlinear Kalman Filter, known as Derivative-free nonlinear Kalman Filter. The estimated outputs provided by the Kalman filter are subtracted from the real outputs measured from the power unit, thus generating the residuals sequence. It is proven that the sum of the squares of the residuals vectors, weighted by the inverse of the residuals covariance matrix, stands for a stochastic variable that follows the χ2distribution. By exploiting the statistical properties of the χ2distribution one can define confidence intervals which allow for deciding at a high certainty level about the appearance of a fault or cyberattack in the wind-power system.

A nonlinear optimal H-infinity control approach is proposed for bioreactors aiming at improved biofuels production. The dynamic model of the bioprocess taking place in the bioreactor undergoes approximate linearization round temporary equilibria which are recomputed at each iteration of the control method. The linearization makes use of Taylor series expansion and of the computation of the system's Jacobian matrices. For the approximately linearized model of the bioprocess an H-infinity feedback controller is designed. The feedback gain of the controller is found from the repetitive solution of an algebraic Riccati equation, taking place at each iteration of the control method. The stability of the proposed control scheme is evaluated through Lyapunov analysis. First, it is demonstrated that the control system satisfies the H-infinity tracking performance criterion, which signifies robustness against modelling uncertainty and external perturbations. Moreover, under moderate conditions it is proven that the control loop is globally asymptotically stable. The proposed control method solves finally the nonlinear optimal control problem for bioreactors in a computational efficient and of proven convergence manner.

A novel adaptive neurofuzzy H-infinity control approach to feedback control and stabilization of the nonlinear dynamical model of bioreactors used in biofuels production is developed. The form and the parameters of the differential equations that constitute the dynamic model of the bioreactor are considered to be unknown, while there is only knowledge about the order of the system. The model of the controlled system undergoes approximate linearization round a temporary equilibrium which is recomputed at each iteration of the control algorithm. The linearization procedure makes use of Taylor series expansion and the computation of Jacobian matrices. For the approximately linearized model of the bioreactor it is possible to design a stabilizing H-infinity feedback controller, provided that knowledge about the matrices of the linearized state-space description is available. Neurofuzy networks are used to estimate the unknown dynamics of the system and its Jacobians. The computation of the feedback controller's gain comes from the solution of an algebraic Riccati equation taking place at each iteration of the control method, and this allows the implementation of the H-infinity feedback controller. The learning rate of the neurofuzzy approximators is chosen from the requirement the first derivative of the system's Lyapunov function to be always a negative one, thus assuring the stability of the control loop. The global asymptotic stability and the robustness properties of the control method are proven through Lyapunov stability analysis.

The PDE model of the commodities price dynamics is shown to be equivalent to a multi-asset Black-Scholes PDE. Actually it is a diffusion process evolving in a 2D assets space, where the first asset is the commodity's spot price and the second asset is the convenience yield. By applying semi-discretization and a finite differences scheme this multi-asset PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For the local subsystems, into which the commodities PDE is decomposed, it becomes possible to apply boundary-based feedback control. The controller design proceeds by showing that the state-space model of the commodities PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem's dynamics and can eliminate the subsystem's tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the multi-factor commodities' PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the commodities PDE system so as to assure that all its state variables will converge to the desirable setpoints.

A new method of adaptive control for active power filters is developed in this article. By proving that the active power filter is a differentially flat system, its transformation to the linear canonical (Brunovsky) form becomes possible. In this new description the control input of the active power filter comprises unknown nonlinear terms which are identified by neurofuzzy networks and through an adaptation / learning procedure. These estimated parts of the system's dynamics are used in an indirect adaptive control scheme, which finally makes the outputs of the active power filter converge to the desirable setpoints. The learning rate in the aforementioned adaptation procedure is given a value which assures that a suitably chosen Lyapunov function will remain negative definite. Under the proposed control method, the closed loop of the active power filter is shown to satisfy the H-infinity tracking criterion, which implies a maximum capability for rejection of external perturbations as well as of modelling errors. flatness-based adaptive fuzzy control based on differential flatness theory is a completely model-free control method. When designing the controller, there is no need for prior knowledge of the system's parameters and state-space equations.

To assure power quality and stability of the electricity network it is important to perform control and synchronization of the inverters which are used in the connection of distributed DC power generation units to the grid. It is proven that the model of the inverters, is a differentially flat one. By exploiting differential flatness properties it is shown that the multiple inverters model can be transformed into a set of local inverter models which are decoupled and linearized. For each local inverter the design of a state feedback controller becomes possible, e.g. using pole placement methods. Such a controller processes measurements not only coming from the individual inverter but also coming from other inverters which are connected to the grid. Moreover, to estimate the non-measurable state variables of each local inverter, the Derivative-free nonlinear Kalman Filter is used. This consists of the Kalman Filter recursion applied to the local linearized model of the inverter and of an inverse transformation that is based on differential flatness theory, which enables to compute estimates of the state variables of the initial nonlinear model of the inverter. Furthermore, by redesigning the aforementioned filter as a disturbance observer it becomes also possible to estimate and compensate for disturbance terms that affect each local inverter.

The paper proposes a systematic method for forecasting default probabilities for financial firms with particular interest in electric power corporations. According to credit risk theory a company's proximity to default is determined by the distance of its assets' value from its debts. The assets' value depends primarily on the company's market (option) value through a complex nonlinear relation. Therefore, by forecasting with accuracy the enterprize's option value it becomes also possible to estimate the future value of the enterprize's asset value and the associated probability of default. This paper proposes a systematic method for forecasting the probability to default for companies (option / asset value forecasting methods) using a new nonlinear Kalman Filtering method under the name Derivative-free nonlinear Kalman Filter. The company's option value is considered to be described by the Black-Scholes nonlinear partial differential equation. Using differential flatness theory the partial differential equation is transformed into an equivalent state-space model in the so-called canonical form. Using the latter model and by redesigning the Derivative-free nonlinear Kalman Filter as a m-step ahead predictor, estimates are obtained of the company's future option values. Thus, by forecasting the company's market (option) values, it becomes also possible to forecast the associated asset value and volatility and finally to estimate the company's future default risk.

The paper proposes a new method for the control of nonlinear dynamical systems which is based on differential flatness theory. The method assumes that the system is already found or can be transformed to the so-called triangular form. The controller design proceeds by showing that each row of the statespace model of the nonlinear system stands for a differentially flat system, where the flat output is chosen to be the associated state variable. Next, for each subsystem which is linked with a row of the state-space model a virtual control input is computed, that can invert the subsystem's dynamics and can eliminate the subsystem's tracking error. From the last row of the state-space description, the control input that is actually applied to the nonlinear system is found. This control input contains recursively all virtual control inputs which were computed for the individual subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the nonlinear system so as to assure that all its state vector elements will converge to the desirable setpoints. The proposed flatness-based control method can solve efficiently several nonlinear control problems. Indicative evaluation results are presented in the manuscript in the form of simulation experiments. These confirm also the potential application of the proposed control method to electric power generators and to renewable power generation systems.

The paper considers financial derivatives and option pricing models which are described with the use of diffusiontype partial differential equations (e.g. Black-Scholes models). Using this approach a new filtering method for distributed parameter systems is developed, for estimating option prices variations without knowledge of initial conditions. The proposed filtering method is the so-called Derivative-free nonlinear Kalman Filter and is based on a decomposition of the nonlinear partial-differential equation of the financial system into a set of ordinary differential equations with respect to time. Next, each one of the local models associated with the ordinary differential equations is written in the linear canonical form through a transformation which is based on differential flatness theory. This transformation provides a model of the nonlinear dynamics of the option pricing model for which state estimation is possible by applying the standard Kalman Filter recursion. Based on the obtained state estimate, validation of the Black-Scholes PDE model can be performed and the existence of inconsistent parameters in the Black-Scholes PDE model can be concluded.

This research work introduces a new method for feedback control of nonlinear dynamical systems and considers as application example the problem of trajectory tracking for autonomous robotic vehicles. The control method consists of a repetitive solution of an H-infinity control problem for the mobile robot, that makes use of a locally linearized model of the robot and takes place at each iteration of the control algorithm. The vehicle's model is locally linearized round its current position through the computation of the associated Jacobian matrices. Using the linearized model of the vehicle an H-infinity feedback control law is computed. The known robustness features of H-infinity control enable to compensate for the errors of the approximative linearization, as well as to eliminate the effects of external perturbations. The efficiency of the proposed control scheme is shown analytically and is confirmed through simulation experiments. The method can be applied to a wide class of nonlinear dynamical systems.

This research work introduces a new control method for feedback control of nonlinear power electronics systems with application example the problem of three-phase voltage source converters. The control method consists of a repetitive solution of an H-infinity control problem for the voltage source converter, that makes use of a locally linearized model of the converter and takes place at each iteration of the control algorithm. The converter's model is locally linearized round its current operating point through the computation of the associated Jacobian matrices. Using the linearized model of the converter an H-infinity feedback control law is computed. The known robustness features of H-infinity control enable to compensate for the errors of the approximative linearization, as well as to eliminate the effects of external perturbations. The performance of the proposed control scheme is validated analytically and is confirmed through simulation experiments.

A chaos-based communication system is designed in which extraction of the information signal at the receiver is performed with the use of a nonlinear filtering method of improved efficiency (Derivative-free nonlinear Kalman Filter). In the transmitter's side the source of information undergoes modulation (encryption) using as carrier a chaotic signal generated by the Duffing oscillator. The modulated signal is transmitted through a communication channel and at the receiver's side demodulation takes place, by exploiting the estimation provided for the state vector of the chaotic oscillator by the Derivative-free nonlinear Kalman Filter. The proposed filtering method has improved performance over the Extended Kalman Filter and reduces significantly transmission errors.

The paper studies differential flatness properties and an input-output linearization procedure for doubly-fed induction generators (DFIGs). By defining flat outputs which are associated with the rotor's angle and the magnetic flux of the stator an equivalent DFIG description in the Brunovksy (canonical) form is obtained. For the linearized canonical model of the generator a feedback controller is designed. Moreover, a comparison of the differential flatness theory-based control method against Lie algebra-based control is provided. At a second stage, a novel Kalman Filtering method (derivative-free nonlinear Kalman Filtering) is introduced. The proposed Kalman Filter is redesigned as disturbance observer for estimating additive input disturbances to the DFIG model. These estimated disturbance terms are finally used by a feedback controller that enables the generator's state variables to track desirable setpoints. The efficiency of the proposed state estimation-based control scheme is tested through simulation experiments.

The paper is concerned with proving differential flatness of the three-phase voltage source converter (VSC) model and its resulting description in the Brunovksy (canonical) form. For the linearized canonical model of the converter a feedback controller is designed. At a second stage, a novel Kalman Filtering method (derivative-free nonlinear Kalman Filtering) is introduced. The proposed Kalman Filter is redesigned as disturbance observer for estimating perturbations in the VSC model. These estimated disturbance terms are finally used by a feedback controller that enables the DC output voltage to track desirable setpoints. The efficiency of the proposed state estimation-based control scheme is tested through simulation experiments.

The existing passive distribution networks, characterized by unidirectional power flows and a partial and centralized control, limit optimal management of Renewable Energy Sources (RES), therefore restraining their exploitation. A deep revision of the planning and management methodologies for distribution electrical networks is therefore required due to the increasing penetration of RES in distribution electrical networks In order to overcome the integration problems of Distributed Generation (DG) and RES, existing electrical distribution networks will, therefore, evolve from passive to active networks and smart grids, managed through systems based on Information and Communication Technology (ICT). In this paper a hybrid optimization method able to maximize the Net Present Value related to the investment made by Wind Turbines developers in an active distribution network and smart grids is proposed. The method, that combines Genetic Algorithms with a multi-period optimal power flow, is validated on a 69-bus 11 kV radial distribution network.

Sensorless control for induction motors using Unscented Kalman Filtering is studied. The complete 6-th order dynamic model of the induction motor is analyzed and a nonlinear controller based on differential flatness theory is developed. The Unscented Kalman Filter is proposed to estimate the state vector of the nonlinear electric motor using a limited number of sensors, such as the ones measuring stator currents. Next, control of the induction motor is implemented through feedback of the estimated state vector. The efficiency of the Unscented Kalman Filter-based control scheme, is tested through simulation experiments.

The paper provides technical analysis and implementation cost assessment of Sigma-Point Kalman Filtering and Particle Filtering in autonomous navigation systems. As a case study, the sensor fusion-based navigation of an unmanned aerial vehicle (UAV) is examined. The UAV tracks a desirable flight trajectory by fusing measurements coming from its Inertial Measurement Unit (IMU) and measurements which are received from a satellite or ground-based positioning system (e.g. GPS or radar). The estimation of the UAV's state vector is performed with the use of (i) Sigma-Point Kalman Filtering (SPKF), (ii) Particle Filtering (PF). Trajectory tracking is succeeded by a nonlinear controller which is derived according to flatness-based control theory and which uses the UAV's state vector estimated through filtering. The performance of the remote sensing navigation system which is based on the aforementioned state estimation methods is evaluated through simulation tests.

The paper studies the problem of autonomous navigation of a multi-UAV system with the use of the Unscented Information Filter (UIF) and the Distributed Particle Filter (DPF). It is considered that m UAV (helicopter) models are monitored by n different ground stations. At each monitoring station a filter is used to track each UAV by fusing measurements which are provided by various UAV sensors, while by fusing the state estimates from the distributed local filters an aggregate state estimate for each UAV is obtained. The UIF and DPF estimated state vector is in turn used by a flatness-based controller that makes the UAV follow the desirable trajectories.

This paper studies a model of cooperative behavior in a multi-robot system that consists of N mobile robots. It is assumed that the robots correspond to diffusing particles, and interact to each other as the theory of Brownian motion predicts. Brownian motion is the analogous of the quantum harmonic oscillator (Q.H.O.), i.e. of Schrodinger's equation under harmonic (parabolic) potential. It is shown that the motion of the robots can be described by Langevin's equation which is a stochastic linear differential equation. It is proved that Langevin's equation is a generalization of conventional gradient algorithms. Therefore the kinematic models of mobile robots which follow conventional gradient algorithms can be considered as a subcase of the kinematic models which are derived from the diffusion analogous of the Q.H.O model.

Observer-based adaptive fuzzy H(infinity) control is proposed to achieve H(infinity) tracking performance for a class of nonlinear systems, which are subject to model uncertainty and external disturbances and in which only a measurement of the output is available. The key ideas in the design of the proposed controller are (i) to transform the nonlinear control problem into a regulation problem through suitable output feedback, (ii) to design a state observer for the estimation of the non-measurable elements of the system's state vector, (iii) to design neuro-fuzzy approximators that receive as inputs the parameters of the reconstructed state vector and give as output an estimation of the system's unknown dynamics, (iv) to use an H(infinity) control term for the compensation of external disturbances and modelling errors, (v) to use Lyapunov stability analysis in order to find the learning law for the neuro-fuzzy approximators, and a supervisory control term for disturbance and modelling error rejection. The control scheme is tested in the cart-pole balancing problem and in a DC-motor model.

Quantum associative memories are derived from the Hopfleld memory model under the assumption that the elements of the correlation weight matrix W are stochastic variables. The probability density function of each weight is given as a solution of Schrodinger's diffusion equation. Spectral analysis of quantum associative memories follows previous studies on the wavelets' energy spectrum. Spectral analysis shows that (i) the basis functions of the stochastic weights express the distribution of energy with respect to the weights' values, (ii) the stochastic weights satisfy the principle of uncertainty.

Quantum computation is proposed for the parallelization of a fuzzy logic control (FLC) algorithm. Quantum computation speeds up the fuzzy inference since serial operations between matrices of large dimensionality are now replaced by a one-step quantum addition or a quantum subtraction. The unitarity properties of the algorithm prove that the FLC stands for a simulator of a quantum computing machine.

A fuzzy reinforcement learning (FRL) scheme which is based on the principles of sliding-mode control and fuzzy logic is proposed. The FRL uses only immediate reward. Sufficient conditions for the convergence of the FRL to the optimal task performance are studied. The validity of the method is tested through simulation examples of a robot which deburrs a metal surface.