Fourier series expansion for synchronization of permanent magnet electric motors

Learning control techniques, based on Fourier approximation theory, are used to solve the tracking control problem via state (rotor position and speed and stator currents) feedback for permanent magnet synchronous motors performing repetitive tasks: smooth rotor position and speed reference signals, whose foreknowledge is not assumed and which are periodic of uncertain period, are required to be tracked for any motor initial condition. The effectiveness of the proposed solution is illustrated by its successfully application to a master-slave synchronization problem. The presented result illustrates the high potentiality of merging together functional approximation theory and advanced nonlinear control techniques.     

Application of least-squares techniques for induction motor parameters estimation

This paper deals with a new approach to the parameters estimation of three-phase induction motors with least-squares techniques. The identification method is based on the steady-state electrical equations of induction motor and allows the use of linear identification algorithms. Measurements of the stator currents and voltages and rotor angular speed are required for the identification procedure based on a simple algorithm. The main feature of the proposed procedure is the possibility of estimating the rotor resistance, self-inductance of the rotor winding and the stator leakage inductance parameters without measuring the rotor flux magnitude. The estimated parameters are compared with similar available data obtained with a 3 kW induction motor to evaluate the consistence and performance of the proposed method.     

Design,modelling and simulation of a new nonlinear and full adaptive backstepping speed tracking controller for uncertain PMSM

In this study, a new nonlinear and full adaptive backstepping speed tracking control scheme is developed for an uncertain permanent magnet synchronous motor (PMSM). Except for the number of pole pairs, all the other parameters in both PMSM and load dynamics are assumed unknown. Three phase currents and rotor speed are supposed to be measurable and available for feedback in the controller design. By designing virtual control inputs and choosing appropriate Lyapunov functions, the final control and parameter estimation laws are derived. The overall control system possesses global asymptotic stability; all the signals in the closed loop system remain bounded, according to stability analysis results based on Lyapunov stability theory. Further, the proposed controller does not require computation of regression matrices, with the result that take the nonlinearities in quite general. Simulation results clearly exhibit that the controller guarantees tracking of a time varying desired reference speed trajectory under all the uncertainties in both PMSM and load dynamics without singularity and overparameterization. The results also show that all the parameter estimates converge to their true values on account of the fact that reference speed signal chosen to be sufficiently rich ensures persistency of excitation condition. Consequently, the proposed controller ensures strong robustness against all the parameter uncertainties and unknown bounded load torque disturbance in the PMSM drive system. Numerical simulations demonstrate the performance and feasibility of the proposed controller.     

Dynamic properties of an individual wheelset drive of a railway vehicle

Spatial vibrations of an individual wheelset drive supported by rubber silent blocks in a two–axle bogie's frame of a railway vehicle are studied. The method, used for a creation of a mathematical model, is based on a decomposition of the wheelset drive into subsystems — a drive assembled from an engine and a gear transmission, a flexible composite hollow shaft and a flexible wheelset. Subsystems are modeled in local configuration spaces and they are mutually connected by deformable couplings. An adhesive characteristic in a wheel–rail contact is described using the Kalker's theory. The nonlinear mathematical model respects spatial vibrations of the drive's components and the flexible wheelset, a deformable rail ballast and it is used for investigating transient dynamic responses on different types of excitations. Experimentally investigated spatial bogie's frame displacements or displacements derived from a total model of the bogie caused by spatial irregularities of the rail track are main sources of excitations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical methods of stabilizer construction via guidance control

This work concerns guidance stabilization of non‐autonomous control systems. Global stabilization problem is usually quite complex and difficult to solve. To overcome this difficulty, guidance control is used. A guidance stabilizer uses a trajectory of a globally asymptotically stable auxiliary system as a guide. A local stabilizer keeps the trajectory of the system in a neighborhood of a solution of the auxiliary system. In this way, the trajectory of the system tends to the equilibrium position. The main idea of this method is to solve the global stabilization problem by applying local stabilization methods. The presented procedure also yields additional possibilities for the design of a stabilizer that eliminates the peak effect, that is, the large deviation of the solutions from the equilibrium position at the beginning of the stabilization process. This effect represents a serious obstacle to the design of cascade control systems and to guidance stabilization. The optimal control problem used in this paper eliminates this effect that we have when we apply known construction methods of local stabilizers to obtain a high speed of damping of the control systems trajectories. According to this approach and using ε‐strategies introduced by Pontryagin in the frame of differential games theory, the stabilizing control is constructed as a function of time defined in a small time interval and not as a feedback. An application to a mechanical stabilization problem is provided here. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical solutions for optimal control of monodomain equations

We present the numerical solutions of optimality systems corresponding to optimal control problems governed by the mono-domain equations which are widely used for describing the electrical activity of the cardiac tissue. This mono-domain model is based on a parabolic equation and a system of stiff ordinary differential equations. The space discretization of the state variables and dual variables is done using piecewise linear finite elements and the time discretization is based on linearly implicit Runge-Kutta methods. The main goal of this work is to study the optimal control behavior of the electrical potentials under the influence of extracellular ionic currents as control variables. The numerical results presented are based on first and second order optimization methods. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of a rotating ball

Influece of form errors of a chamber filled with a liquid on the movement and stability of a ball, rotating in the chamber, is studied. Two cases of the influence of the chamber form errors on the forces, acting on the ball, are defined. The first case describes the situation when limitations on the rotor shift are not imposed and disturbances of the chamber form are set by spherical harmonics not above the first order. In the second case disturbance of a chamber form is arbitrary and the rotor shift is supposed small. A rising here diflective moment tends to direct the angular speed vector along the small semiaxis of the ellipsoid, i.e., a stable position of the rotor appears.     

Synchronization of permanent magnet electric motors: New nonlinear advanced results

The nonlinear learning control techniques, based on Fourier approximation theory and used by Verrelli (2011) [2] to solve the synchronization problem for uncertain permanent magnet synchronous motors (performing repetitive tasks of uncertain repetition period), are considered in this paper. We show that, if the exogenous rotor position reference signal (which is to be globally tracked without assuming its foreknowledge) is restricted to the class of sinusoidal signals with uncertain bias, amplitude, frequency and phase, a stronger result can be derived by resorting to nonlinear advanced identification techniques. In contrast to Verrelli (2011) [2], neither availability of the rotor speed reference signal is required nor infinite memory identification schemes are used. The application to the problem of synchronizing a drumming robotic arm with a drumming human arm is presented: simulation results show satisfactory closed loop performances and confirm the effectiveness of the proposed solution.     

Goal-oriented error estimates including algebraic errors in discontinuous Galerkin discretizations of linear boundary value problems

We deal with a posteriori error control of discontinuous Galerkin approximations for linear boundary value problems. The computational error is estimated in the framework of the Dual Weighted Residual method (DWR) for goal-oriented error estimation which requires to solve an additional (adjoint) problem. We focus on the control of the algebraic errors arising from iterative solutions of algebraic systems corresponding to both the primal and adjoint problems. Moreover, we present two different reconstruction techniques allowing an efficient evaluation of the error estimators. Finally, we propose a complex algorithm which controls discretization and algebraic errors and drives the adaptation of the mesh in the close to optimal manner with respect to the given quantity of interest.     

On the importance and uses of feedback information in FEA

Procedures for verification and validation through adaptive control of the errors of discretization and idealization in finite element analysis (FEA), guided by feedback information, are discussed. A hierarchic framework for the construction of sequences of finite element spaces and working models is outlined and an example is presented.     

Mathematical Modelling of Diesel-Electric Propulsion Systems for Marine Vessels

This article presents a mathematical model of a complete diesel-electric propulsion system, including components as diesel generators, distribution network, variable speed thruster-drives, and conventional motor loads. The model is split into two parts: One power generating part where the load is specified with an aggregated active and reactive power load demand. Secondly, a power consumption part where the effects of the different load types as thruster drives, motors and other loads are modelled. The model is written in a state-space form suitable for the purpose of simulation and control design. PID-controllers represent speed governors and automatic voltage regulators.     

Speed and current regulation of a permanent magnet synchronous motor via nonlinear and adaptive backstepping control

This paper proposes a new speed and current control scheme for a Permanent Magnet Synchronous Motor (PMSM) by means of a nonlinear and adaptive backstepping design. All the parameters in both PMSM and load dynamics are considered unknown. It is assumed that all state variables are measurable and available for feedback in the controller design. The final control and parameter estimation laws are derived by the design of the virtual control inputs and the Lyapunov function candidate. The overall control system is asymptotically stable according to stability analysis results based on Lyapunov stability theory. Simulation results clearly show that the controller guarantees tracking of a time varying reference speed owing to the fact that the speed and current tracking errors asymptotically converge to zero despite all the parameter uncertainties/perturbations and load torque disturbance variation. Numerical simulations reveal the performance and feasibility of the proposed controller.     

Control-oriented Rotordynamics

The rotor vibrations are coupled by many parameters: angular speed couples two radial directions, unbalance couples torsional and lateral vibrations, rotor support couples rigid body and flexible modes. Small changes of these parameters strongly influence the rotor dynamics. The full analysis of the rotor coupled vibrations is much simpler when we can divide the system into smaller subsystems. In this case the calculations are simplified and we have deep insight into mechanisms leading to good or bed behaviour of the rotor motion. It is particularly important in the case of the rotor working in the wide range of the angular speeds. For the vibration analysis the methods known from control theory can be applied. The proposed approach was testified in the paper on the simple 3-mode rotor model. The torsional vibrations were separated from the lateral vibrations and a feedback among the subsystems was established. The subsystems are coupled by rotor unbalance and root locus method allows to show the critical values of the unbalance which destabilize rotor motion for different angular speeds. The lateral vibrations are stabilized by angular speed and again it is possible to find how big value of the rotor speed is sufficient to stabilize the rotor motion. Such analysis of the rotor vibrations appeared very useful for the choice of the control strategy. It indicated such control laws which amplify the stabilizing mechanisms in the rotor dynamics. Such procedures can also lead to the energy saving control laws. In the case of the lateral vibrations there were considered four control strategies. And these strategies were compared to indicate optimal one. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study on chaos control of second-order non-autonomous phase-locked loop based on state observer

With system parameters falling into a certain area, the second-order non-autonomous phase locked loop (PLL) is experiencing chaotic behavior which is undesirable in system, where it is necessary to estimate the phase of a received signal. In order to control chaos in PLL and drive it to the locked state, dynamical equation for phase error model of PLL is firstly derived. Then, the state values of phase and transient frequency errors were estimated by a state observer. Moreover, by exploiting these state estimations, a non-linear feedback controller is designed. Since the presented controller does not need to change the controlled system structure and not to use any information of system except the system state variables, the designed controller is simple and desirable. Simulation results show that the presented control law is very effective.     

Method of estimating transients in induction machines

The present paper is concerned with induction motors with wound and squirrel-cage rotors. It is assumed that the magnetic field produced by the stator windings is constant in magnitude and rotates with constant angular velocity. Differential equations that describe relations between the electromagnetic torque and main electric and mechanical quantities of the induction motors under consideration are derived in detail, the geometry of the rotors of motors being fully taken into account. A special nonsingular transformation is applied to the initial systems of equations (with angular coordinates) to split them up and reduce to a lower-order (in fact, third-order) system. A local stability analysis of the resulting equations is carried out. The stable equilibrium states that correspond to the operating modes of induction motors are determined. Methods of speed control of induction motors with wound and squirrel-cage rotors are considered. The limit load problem on motors and the control problem of the speed of motors are discussed; these problems lead to the necessity to estimate the transients in induction motors. The method of estimating transients that occur due to changes of the operating parameters of motors is developed based on a modification of the nonlocal reduction method. Using this method in combination with Barbashin and Tabueva’s methods to apply to the obtained systems made it possible to find analytical estimates of the ultimate permissible loads on induction motors and the control ranges of the parameters of the system that correspond to additional external active and inductive resistances. Moreover, estimates for the region of the attraction of stable equilibrium states of systems that describe the dynamics of induction motors are obtained.     

Identification of harmful time harmonic interactions in a high power squirrel-cage traction machine

The paper offers a methodology for the identification the time harmonic interactions with a strong negative impact on the rotor of a high power induction machine. These potentially dangerous pulsating torques may be effectively reduced by carefully setting the machine’s drive control. A novel approach based on a complex finite element model and further post-processing is used. Results obtained from electromagnetic calculations in the form of pulsating torques are used as mechanical loading of a structural dynamic model. This complex model intended for rotor vibration analysis is described in brief in the mechanical part of the paper. The proposed methodology is applied to the machine used as a propulsion unit for multisystem locomotives of 1600 kW power.     

A control reduced primal interior point method for a class of control constrained optimal control problems

A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparatively coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples. Supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. This paper appeared as ZIB Report 04-38.     

Numerical Solution of Hamilton-Jacobi-Bellman Equations by an Upwind Finite Volume Method

In this paper we present a finite volume method for solving Hamilton-Jacobi-Bellman(HJB) equations governing a class of optimal feedback control problems. This method is based on a finite volume discretization in state space coupled with an upwind finite difference technique, and on an implicit backward Euler finite differencing in time, which is absolutely stable. It is shown that the system matrix of the resulting discrete equation is an M-matrix. To show the effectiveness of this approach, numerical experiments on test problems with up to three states and two control variables were performed. The numerical results show that the method yields accurate approximate solutions to both the control and the state variables.     

On the piezoelectric shear effect in ultrasonic motors

A new actuator for piezoelectric ultrasonic motors (USM) using the d₁₅ effect was conceived. Whereas the piezoelectric d₃₃ and d₃₁ effects are normally used in commercial motors, there exist hardly any USM based on shear actuation. The actuator is a piezoelectric block polarized in axial direction and electroded circumferentially with four electrodes. The suitable superposition of two standing waves generates ultrasonic traveling waves in the actuator, which drives the rotor. The dimensions of the actuator are optimized with respect to the dynamic piezoelectric coupling factor. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)