Controlling BLDC Motor Drives

Control Using PID Regulator

As it’s indicated by static characteristics and transients, BLDC motor drives are controlled in a similar manner to other DC motors, i.e. by changing armature volt age. In a system with an electronic commutator this occurs as a result of the change of the pulse width factor ku. This may happen at every particular pulse as a result of modifying the signal controlling the series of the pulses. In most cases we have to do with discreet control in which the value of the pulse width factor ku is relative to the values of variables in the drive for the duration of the pulse that precedes. Control is relative to the angle of rotation theta_r, just as presented earlier on during the discussion of the operation of the electronic commutator. The other values that are responsible for the control include rotational speed theta_r and armature current i_a.

Since for the purposes of the control it’s necessary that the angle of rotation is familiar, one of the common solutions involves the application of an encoder for determination of the position of the rotor and a differential system in order to indirectly obtain speed theta_r. It’s sometimes the case that the armature current is applied for regulation as the value that is an auxiliary one and can be used for the purposes of limiting control, just as presented in the example devoted to start-up of a motor.

Sometimes the control of BLDC drive does not apply position sensors and the necessary rotor angle is obtained indirectly on the basis of measurements of voltages and currents in armature windings by use of state observers. This method will be presented in the further part of the current section. A typical control system for a BLDC motor drive with a position sensor.

This section will be concerned with the presentation of examples of BLDC drive control systems with speed stabilization for changing loads and stepwise change of the input value for a constant motor load. Such a basic control task using PID regulator reveals the ability of BLDC motors to operate in the drive control based systems on changing the value of the supply voltage. +-+-+- results of simulations conducted using an untransformed model of the motor for stabilizing rotational speed for a stepwise change of the load torque from the rated value of Tl = 3 [Nm] initially to Tl = 9 [Nm], i.e. three times overloading the motor followed by a stepwise change to reach the value of Tl = 1[Nm], which is one third of the rated load.

+-+-+- Control voltage components and armature voltage during stabilization of speed: a) P component b) I component c) D component d) armature voltage e) ku factor.

+-+-+- Stabilization of rotor speed by PID regulator but for different control values: kP = 10; kI = 50; kD = 0.1: a) armature currents b) rotor speed c) speed error.

---a correct stabilization of the rotor speed for very dynamic changes of the load. For high gain values of PID regulator the error of speed regulation is in the range |e?| = 0.4%, while for a lower gains this error ranges around |e?| = 2 %, which denote values that are approximately proportional to the gains applied for kI, kP, kD factors. Stability state is achieved after around 0.5 [s]. Beneficial conditions for regulation are secured by a surplus of the regulation that involves input of the supply voltage U = 300 [V] to a motor while the rated value of the voltage is U = 120 [V]. As one can see for such an intensive regulation the system of the supply, commutator and motor itself have to be designed to withstand the maximum value of the voltage (300 [V]). The following waveforms, i.e. the ones presented illustrate PID regulation for the stepwise change of input speed ?d : 500 ? 1500 [rev/min] for both untransformed and transformed models of the motor.

+-+-+- PID regulation of rotor speed of the 6.6 [kW] motor, after stepwise change of required value of speed omega_d : 500 >> 1500 [rev/min]. Results for untransformed model and: kI = 1000, kP = 500, kD = 10: a) stator currents b) electromagnetic torque c) rotor speed d) armature voltage.

+-+-+- PID regulation of rotor speed - transients, but resulting from transformed d,q model: a) d,q currents b) electromagnetic torque c) rotor speed d) armature voltage.

+-+-+- PID regulation of rotor speed of the 6.6 [kW] motor, with armature current limitation to ir = 110 [A], after stepwise change of required value of speed omega_d: 500 -> 1500 [rev/min]. Results for untransformed model and kI= 1000, kP = 500, kD = 10: a) stator currents; b) electromagnetic torque c) rotor speed d) armature voltage.

The waveforms indicate that the transients resulting from the application of transformed and untransformed models are very similar. The regulation is realized quickly and effectively despite large difference in terms of the target speed. However, this happens under the assumption of accessibility of the higher value of supply voltage U = 1.5Un and an additional condition that the motor is capable of the generation of a surge torque of about Te = 500 [Nm], i.e., 20 times the value of Tn. Such regulation properties can be gained as a result of using a motor and supply system that is oversized in relation to the rated value during operation. For smaller requirements regarding the regulation speed it’s possible to apply a more economical supply system and introduce current delimiter in the motor design. The two figures present the regulation of the motor for a stepwise change of the input speed and simultaneous application of a current delimiter which wouldn't allow armature currents surges that exceed set multiple of the rated value, as in the previous examples. --- the results for the stepwise increase of the input value of the rotational speed while the curve in presents the example involving the reduction of the input value. For both cases the armature current is limited to the value of ir = 110 [A], which is about five times the value of the rated current.

+-+-+- PID regulation of rotor speed of the 6.6 [kW] motor, with armature current limitation to ir = 110 [A], after stepwise change of required value of speed omega_d: 1500 >> 500 [rev/min]. Results for untransformed model and kI= 1000, kP = 500, kD = 10: a) stator currents b) electromagnetic torque c) rotor speed d) armature voltage

Control with a Given Speed Profile

BLDC motor drives are capable of performing complex control tasks including the ones involving a given profile of drive speed and a given profile of angular position of the rotor. This type of control is named tracking control since the role of the control system is to follow the given trajectory of the motion while the role of the drive regulator (for instance, PID type) is to correct the error of the performed trajectory. This type of practical issues are well worth designing with the aid of signal processors that have been developed specifically to collect signals from measuring converters, perform numerical calculations associated with the control and send command signals to be executed in order to influence the operation of an electronic commutator.

Signal processors are equipped with internal components that serve for the purposes of control for instance several generators of PWM signal thus executing the tasks of the control of an converter. An example of a control system consisting of a BLDC motor and a signal processor.

The results of a computer simulation of a sample control task for a BLDC drive with a given speed profile with trapezoidal shape. In this task the rotor speed increases from 0 to 2.500 [rev/min] within 0.8 [s] and remains at this level for another 1.2 [s], after which within 1.0 [s] it decreases to 500 [rev/min]. The calculations for the case of this simulation have been conducted with the aid of untransformed mathematical model of a motor and with preserving d,q transformation. -- contains a comparison of the results of control for various values of gains of the PID regulator.

+-+-+- Control system of a BLDC motor, without a rotor position gauge with a signal processor

+-+-+- Rotor speed control of the 6.6 [kW] PMDC motor, according to trapezoidal shape of speed trajectory, under the nominal load of T_l = 25 [Nm]. PID regulator settings are: kI = 1000, kP = 500, kD = 10. Untransformed model employed: a) armature currents b) rotor speed c) electromagnetic torque d) motor voltage e) speed error [rev/min]

+-+-+- The same control problem, but transformed d,q model of BLDC is employed: a) d,q transformed armature currents b) armature currents c) motor voltage d) rotor speed e) electromagnetic torque f) speed error.

On the basis of the results presented one can conclude that the error of the drive control is roughly proportional to the values of regulator's gains.

Control for a Given Position Profile

The application of a signal processor as an electronic system that combines the properties of a converter of measured quantities, a digital calculation processor and a generator of control signals offers the possibility of performing complex is sues regarding a BLDC motor drive control with an alternative of eliminating a position sensor. --- presents a system including a signal processor that collects voltage signals from a EMF sampler and with the aid of such a system that is capable of determining the periods during which the transistors are in the ON state without direct measurement of the angle of rotation. The question of the sensorless control forms a complex task since it requires the de termination of the position of the rotor even before start-up when the induced EMF is too small in order to determine rotor position on its basis. Subsequently, after the required threshold is exceeded, on the basis of the induced EMF in the unsupplied phase of the armature winding at a desired instant the control signal for the electronic commutator occurs. --- presents EMF induced between clamps 1,2 during the start-up of the motor.

+-+-+- EMF induced in armature winding of a BLDC motor shortly after start-up For the operation of the motor with higher speed the determination of the instant of the commutation on the basis of the measured currents and EMF in the windings is easier since there is a considerable angular correlation between the two values that are relative to the position of the rotor. +-+-+- the correlation between SEM and armature currents contain the correlation of the voltages between the clamps and currents.

As it has already been mentioned, during the start-up the situation is more complex and in order to determine the rotor position it’s possible to apply the pulse method. The application of this method, however, requires that the rotor has a variable reluctance during the rotation, i.e. a situation which involves the salient pole rotor. --- electromotive forces (EMFs) and currents for the case of the start-up, for which the waveforms confirm the lack of simple correlation between the curves.

An example of the control of the reversible BLDC drive including the regulation of the position and the speed is presented in the following example.

Example 2 ---a diagram of the drive of a large, massive pendulum (swing) with a controlled amplitude and period of oscillations diverging from a natural period, which anyhow is dependent on damping of the motion. The drive applies a motor with a rated power of 0.95 [kW].

+-+-+- View of a massive pendulum driven by BLDC motor.

The data for the mechanical part of the drive are as follows: mass of the swing; length of the pendulum

- moment of inertia of the motor and the fast side of the gear

- moment of inertia of the slow side of the gear and the pendulum relative to the center of mass

- coefficient of pendulum damping

- coefficient of motor damping

- transmission ratio

Application of Lagrange's method requires the calculation the kinetic energy of the mechanical part of the system:

The virtual work of the system is equal to:

... where: Te - denotes the electromagnetic torque of the motor.

In its mechanical part the system has a single degree of freedom. Since the calculations are to be performed from the point of view of the motion of the pendulum, in this case q1 = f should be adopted as generalized coordinate. The constraints resulting from the transmission take the form:

After the introduction of the constraints , Lagrange's function for this sys tem takes the form:

After the application of the Lagrange's equation for the generalized coordinate f we obtain the equation of motion for the mechanical variable in the form:

The assumed pulsation of the motion of the pendulum is equal to the pulsation of the mathematical pendulum with moment of the inertia ...

The assumed motion of the pendulum involves the maintenance of the undamped fluctuations with the amplitude f0 and pulsation ?. Hence, the required trajectory is determined as follows:

The other type of the given motion is the exponential start-up of the pendulum from the initial angle f0 to angle fa with the time constant of Ta.

The mathematical model of the system also in the mechanical part is nonlinear and this is so to the larger degree the greater is amplitude of the motion of the pendulum f0, fa. This could be explained by the linearizing approximation of the pendulum swing... that is fulfilled with much increasing error for higher values of f. The regulation of the drive follows as a result of the application of PIDD regulator, i.e. the one that is responsible for the control of 4 types of error:

The figures that follow, i.e. the results of the regulation, which denotes swinging of the pendulum with a given amplitude. But the first of the figures shows the free motion of the pendulum with damping and without drive regulation. This gives the basis for the comparisons for examples analyzed further in which the regulation system and the actuator, i.e. BLDC motor and the transmission are engaged to perform the motion in accordance with the given trajectory.

The figures that come below present the motion of the pendulum and regulation curves resulting in the achievement of the designed trajectory.

+-+-+- Curves for electromechanical variables for the PIDD controlled pendulum swinging, f0 = -70º: a) i1(f) current b) armature currents c) electromagnetic torque Te = f(t) d) electromagnetic torque Te = f(f) e) ku factor f) Ua/U motor voltage curve

+-+-+- Mechanical variables' curves for the PIDD controlled pendulum swinging, f0 = -70º: a) position angle f1 b) angular velocity ?1 c) angular acceleration a1 d) position error ef e) speed error e?.

The presented results of computer simulation of the pendulum drive indicate that such complex regulation issues regarding tracking control of drive reversal can be realized with the aid of a PID regulator; however, considerable limitations are imposed on it. The motion of the pendulum has to be limited to the degree that only slightly exceeds the linear approximation of the model and an adequate selection of the gains of the regulator must be accounted for. In addition, motion needs to be sufficiently slow and in this case the period of the motion of the pendulum is Tp > 3 [s]. For the angles |fa |> 70º it’s very difficult to obtain a stable operation of the pendulum and the errors of the regulations are becoming greater.

However, we can completely confirm the applicability of BLDC motor for this type of drive.

BLDC Motor Drive--Formal Linearization

The transformed model of the motor for variables: ud, uq. Under the assumption that we aim to reduce reluctance torque of the motor that interferes with transients and concurrently we need to perform a trajectory of the motion, the course of action can be the following. Let us assume that approximately id ˜ 0 and simultaneously on the basis of under standard control conditions the relation ud ˜ 0 is fulfilled, it’s possible to establish the following equality on the basis of:

.... after the application of we will obtain the function for desired control uqd(t). The control system developed in this manner has to ensure the execution of the planned regulation of uqd(t) and additionally has an adjustment system with feed back, which has been designed to ensure the desired precision of the control de spite the occurrence of interference. Moreover, one can note that usually the value of A is small in comparison to B?, ?q since it’s related to the variable term Ms of inductance and that could be premises for disregarding the second term in the parentheses of the expression. Hence, in most simplified form, the examined control can be reduced to:

A problem is associated with the fact that the calculated controls, have to be performed in an untransformed system, which means that there is a necessity to apply the inverse transformation to obtain a system of phase voltages of the armature. The proposed transients for ud = 0 are relatively easy in execution with the aid of pulse width factor ku. Another type of control designed to match the planned trajectory of motion without transformation of the variables will be presented in the section that follows.

BLDC Motor with Inverse Dynamics--Regulation

This type of regulation is effective with regard to a system in which case it’s possible to plan the trajectory of motion, which involves the determination of the position, speed and acceleration of the mechanical variables. In a brushless DC motor, this means that the desired transients ...

... Subscript d denotes the desired value, i.e. the one that marks an ideal trajectory of motion. The actual trajectory usually does and will diverge from the desired one as a result of the effect of a number of factors which are disregarded when stating initial assumptions that tend to simplify either the mathematical model of the drive, constant parameters of the supply or other factors that get in the way of the process of regulation. The difference between the transients forms the error of the regulation. It’s further applied for improving the control signal in the additional component of the regulation system, i.e. the corrector. ..

+-+-+- BLDC drive control by the inverse dynamics method.

... width factor ku, into the system of the electronic commutator. The diagram of the control system that applies the inverse dynamics method.

This method finds a common application in a number of drive systems whose task involves tracking of a given trajectory. The regulation system determines the basic control procedures on the basis of the computed required state while the PID corrector offer an input of an additional signal thus minimizing the control error.

The benefits resulting from the application of this method include increase of the regulation speed, considerable reduction of errors since the predominant part of the control signal is given in advance and regardless of the error. Other advantages of this type of regulation are associated with effective control in non-linear systems since in this case the limitations regarding PID regulation are no longer in force. This method finds application in a number of industrial drives including the control of manipulators and robots. Below is a demonstration of the operating principle in practice with reference to an object from control of a pendulum.

____ This task involves the control of a large, massive pendulum to match a given trajectory, which is considerably diverges from its natural motion.

The details are identical; besides, serves for the analysis. The control of the motion, which forms an example of tracking control can be based on the method of inverse dynamics. The trajectories are given by the functions, for the variables of the pendulum in motion and not by the actual rotor movement. The determination of the desired torque in accordance with offers the following result:

On the basis of the comparison of the desired torque T_ed and the torque actually generated by the motor Te it’s possible to determine the regulation ku, which is additionally adjusted by the signal from PID corrector in the function of the regulation error. This is performed in accordance with the diagram presented, however, the system is simulated. This practically means that the values theta_r, theta_r, ar as well as Te are determined on the basis of calculations on the mathematical model and are not measured. Similarly as in the case, the examination will involve two versions of the desired trajectory: motion of the pendulum with a constant amplitude - starting from the maximum deflection of the pendulum and the other version in which the trajectory of the pendulum begins with start-up from a small angle of deflection, equal to 10º, and achieving maximum displacement (80º, 100º) at a time constant of T = 5 [s]. --- presents the results of the task in the first version and the motion of the pendulum with an amplitudes of 80º and the following an amplitude of 100º. +-+-+-76a presents the given torque Ted kg calculated in accordance with for the planned trajectory.

+-+-+- Electromechanical variables for inverse dynamics control of swinging massive pendulum, for f0 = -80º: a) pre-computed desired torque Ted kg b) position angle f1 c) angular speed, d) angular acceleration e) position error f) speed error e? g) ku factor.

+-+-+- Electromechanical variables for inverse dynamics control of swinging massive pendulum, for f0 = -100º: a) desired torque Ted kg b) position angle f1 c) angular speed d) angular acceleration e) position error ef; f) speed error e? g) motor voltage h) electromagnetic torque Te kg...(f0 = -80º and f0 = -100º) leads to the conclusion that there is a fundamental difference between the two. One can clearly conclude that the latter group i.e. results that are obtained by applying regulation with inverse dynamics, has considerable advantages over the ones presented earlier. The differences involve the stabilization of the transients to obtain large values of amplitudes of the swing of the pendulum and also very relevant improvement of the accuracy of the regulation as a result of applying inverse dynamics. The reduction of the positional error for stabilized transients is over 5 times, not to mention the possibility of steady operation for larger displacements. A set of figures that follows presents the results of the regulation for the alternative version of the task, i.e. for the start-up of the pendulum. Some samples of transients gained are presented for a rising amplitude of the motion to reach the values fa = 80º and fa = 100º in the stable state of the pendulum.

In the commentary of the results that were obtained one can compare the results presented with the ones since both of them refer to the same given trajectory, i.e. the start-up of a pendulum to reach fa = 80º for various control procedures. For the case of PIDD regulation the start-up directly leads to non-stability and the error of the angular position increases and even exceeds already 10°. In contrast, in the case of the application of control using in verse dynamics, the error ef during the start - up tends to stabilize and reaches the range of ±2º for a minimum value of the speed error. The results indicate that increasing the amplitude to fa = 100º offers the possibility of effective regulation of the position in the tracking motion while the position error stabilizes in the range of ±5º. The examples presented here indicate that the method is very effective despite the fact that it has been applied without the precise selection of adequate regulation parameters.