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Variable Structure - SRM Sliding Mode Control
Sliding mode control is widely used in a up-to-date electric drives. This results from the fact that switching of constant supply voltage to electric motor windings formally constitutes of one of the possible control modes with variable structure.
This, in turn, was made possible as a result of the major development in the field of power electronics. Sliding mode control involves the control of the motion along a sliding surface s(q,t) =0. This group of methods involves control applying PWM technique (Pulse Width Modulation), as well as other more advanced control methods, including DTC (Direct Torque Control), with regard to induction motor drives for instance. The natural stability of the system together with high frequency of the switching make it possible in majority of cases for the trajectory of the drive's motion to follow in a close vicinity of a sliding edge even without application of special efforts and precise selection of the parameters. Sometimes engineers involved in its practical application don’t see it necessary to bother themselves with proving stability of a drive. In such cases the experience resulting from laboratory tests and a narrow range of requirements regarding the control of the drive make it possible to design control on the basis of one's experience. However, in a wide range of other cases and, in particular, in actuators realizing complex and variable trajectories of motion the selection of control parameters tends to be more formal and most often it’s based on the direct Lyapunov method for the analysis of the stability of the system [25,4/1,16/1]. It finds application in servomechanisms with stepper motors as well as BLDC drives [53/3]. Similarly, in SRM drives it’s possible to realize the given trajectory of the motion by a proper switching the supply voltage in the range ±u, or ±u, 0 with an adequately high frequency. The sliding control of SRM motor that is applied here has certain limitations resulting from the nature of the motor as well as the adopted assumptions. One of them is that the control occurs simultaneously only for a single phase winding of the motor, as a result of which it involves just one dimension. The second restriction concerns the fact that the control occurs only for the conduction of transistors, i.e. for the flow of energy from the source. In contrast, during the diode conduction and return of the energy into the source the motor's current and torque are uncontrolled for the duration of this stage of operation. Obviously, there is a possibility of application of transistor-diode control mode, i.e. -u, 0 during the periodic switching of the winding and some kind of the effect of current control is obtained in this way; however, it works only in the direction of increasing the time needed for the decay of the current and it won’t be applied here. The presentation further on will concentrate on the application of sliding control with regard to SRM machine realizing given trajectory of motion and limiting torque ripple, i.e. overcoming one of the drawbacks of this drive in terms of vibration and noise production. The presentation will include DTC control and current control of the motor in accordance with the given sliding curve realizing these targets. It’s possible to plan other control techniques involving sliding, all of which realize the given trajectories of the motion of the drive. The presentation of practical examples of such systems will be the subject of the following sections.
Current Control of SRM Drive
The current control used here involves the sliding control of SRM drive in which the sliding surface is defined by the given function of the currents of the phase windings of a machine ... and the control itself is defined in the standard way:
The sliding control described, which is further called current control due to the fact that the sliding surface is designed on the basis of the values of the phase currents, is not a typical one. This comes as a consequence of the fact that the analytical expression of sliding surface does not involve time t in an explicit way but the control is relative to other variable, i.e. the angle of the rotation of the rotor theta_r, thus, it’s a phase surface. Secondly, as it will be presented later, the sliding surface is determined in the function of the sum of phase currents. In the practice of SRM motor control this means the dependence of the sliding surface on a single phase current that is drawn from the source, which is controllable- since it’s supplied through the transistors, and possibly on a single or more phase currents that are in the phase of decay. This, in turn, means that the control is single-dimensional dependent on the sum of the currents in windings, while only one of the currents is controllable, i.e. the one that is energized from the source. The selection of the sliding surface has two roles to play: it minimizes the pulsations of the torque and executes the given curve of the electromagnetic torque of the motor that is defined to adequately reflect the required trajectory of the motion of the drive. As one can see, the task set in this way requires that the problem that is inverse to the motor torque function has to be solved, i.e. ...
Since this problem is non-linear and concurrently the current is normally con ducted through more than a single phase winding, the above should be restated as follows
In practice, this problem is associated with the determination of the sum of two currents being conducted through adjacent windings, i.e. one that is supplied and one that returns the energy, in a way that enables generation of a desired instantaneous value of electromagnetic torque. The solution of this task is the reverse to the relation presented respectively for the SRM motor with the teeth number Zs/Zr = 6/4 and Zs/Zr = 8/6.
+-+-+- Starting of motor B (Zs/Zr = 6/4) with current controller set for segmented constant torque values: a) required Sik current b) phase currents c) electromagnetic torque d) rotor speed e) partial and resultant torque for a final speed .
+-+-+- Starting of motor A (Zs/Zr = 8/6) with current controller set for segmented constant torque values: a) required Sik current b) phase currents c) electromagnetic torque d) rotor speed e) partial and resultant torque for a final speed
+-+-+-Trajectory formed in the B SRM machine under a current controller: a) required current b) phase currents c) electromagnetic torque d) rotational speed e) partial and resultant torques for theta_r = 180 [rad/min]
+-+-+-Trajectory formed in A SRM machine under a current control rule: a) required current b) phase currents c) electromagnetic torque d) rotational speed e) partial and resultant torques for theta_r = 200 [rad/min]
From the results it stems that gaining high value of the torque for motor B (Zs/Zr = 6/4) poses a much more difficult task than for the motor with 4 phase windings, i.e. motor A (Zs/Zr = 8/6) as a result of the larger distances between the teeth for the first of them. However, in both cases it’s possible to per form the current control by the presented method, which will be demonstrated on the basis of several examples. First, we will discuss the start-up of motors with the application of current control.
Current control, as presented can be effectively used to form the trajectory of the motion of a drive as a result of applying a given waveform of torque produced by a motor. This torque for the application of the current control needs to be subsequently transformed this gaining the required current waveform necessary to perform the task given by Sik. This is presented on the illustrations of the operation of the drive for both motors A and B. The presented examples of the application of current control prove that the presented method is effective with regard to the both motors considered as exemplary ones, i.e. for the motors with 3 and 4 phase windings. This allows one to form the waveforms of rotational speed, torque and cur rent, the latter of which is in this method the quantity that is directly regulated, as well as enables one to limit the pulsations of the torque. However, this method can be effective only within the range in which there is an adequate surplus of the regulation, which in this case means a sufficient surplus of the supply voltage, that will enable one to perform the planned current control. This limited surplus of the control is the reason that in the start-up of motors is carried out with a torque decreasing by stepwise sections along with increasing speed.
The forming of the trajectory of the motion occurs regarding the rotational speed that is permitted by the supply voltage in order to ensure that the given current shape resulting from assumed trajectory of the motion were possible to perform by the control system.
SRM Drive--Direct Torque Control (DTC)
DTC control with regard to SRM motor also forms an application of sliding method for drive regulation since it occurs as a result of rapid switching of the voltage applied to the windings of a machine's stator in a way that ensures that the given waveform of electromagnetic torque is realized. This method in its practical application is similar to current control, which has already been the focus of presentation earlier in the section. The specific characteristics of DTC control involve the fact that the sliding surface is constructed on the basis of the desired waveform of the torque:
This type of control, as it has been mentioned earlier, translates into the control of the voltage of machine phases as a result of rapid switching of the supply ...
It appears that this method should offer more advantages than current control since it’s more direct with regard to realizing a given trajectory of the drive motion.
However, one has to bear in mind that we don’t have the measurement of the mo tor torque whereas currents are measured at each phase and this is done with a high degree of precision. For the case of the DTC control instead of the measurement of the torque it’s necessary that we apply an estimation of the torque, i.e. usually a torque observer that offers the actual value of the machine torque on the basis of the accessible measurements and computations based on mathematical model. In conclusion, DTC is also an indirect method for the control of the trajectory of the motion. The issue thereof will be subsequently transferred into the stage of determining the error of the executed trajectory. Thus, the effectiveness of this control is relative to the precision of the observer and its ability to reduce the error of observation. The DTC method has, however, an advantage that the trajectory can be given on-line, which is more difficult to execute using current control method. The figures that follow illustrate the results of DTC control with regard to SRM motor.
By looking at the application of DTC method for the control of SRM motor drive one has to recognize high efficiency of this type of control. However, one can also note that the results presented here refer to the ones gained on the basis of computer simulation employing the previously developed mathematical model instead of results of measurements on a real system. In consequence the results are idealized in the sense of not being charged with the error of the method associated with the application of the torque observer in the control. In this case the electromagnetic torque calculated on the basis of the mathematical model is equal to the measured torque and in this way one of the sources of the significant error is absent.
SRM Drive: Sensor- and Sensorless Control
For the control of SRM motor it’s indispensable that we are familiar with the position of the rotor in the sense of the precise knowledge of the of rotation angle theta_r.
This is due to the switching of transistors, which is used to control the supply of the phase winding for the angles of rotation equal to aon and aoff, respectively. For this reason the most typical solution involves the application of the quadrature en coder in the control system, whose signals are transformed into information regarding the position of the rotor, its rotational speed and direction of the rotational motion. A block diagram of the control using the signal from position sensor.
+-+-+- Block diagram of control of SRM drive applying encoder sensor signal
An alternative to the control system using encoder involves sensorless control, which in brief means that the sensor is absent from the system.
Such a solution is made possible as a result of applying a position estimator or ob server of motor state, thus, leads to savings in terms of the investment, reduction of mass and space occupied by the system and increase in efficiency. This improvement in terms of reliability is connected with the lack of an additional mechanical device on the shaft that is also common to the rotor and the connection leading from it to the control system. It’s not always possible to apply sensorless control, and in particular, it may not be possible to use it in systems in which it’s necessary to have a very high degree of precision of regulation. The observer itself will be the subject of discussion later and now we will focus on the earlier concept regarding position estimator. It’s formed by a complex measurement and calculation unit of the control system with the previously prepared characteristics of magnetization or characteristics of windings' inductance. Here we apply the relations that are reverse to the magnetization characteristics, that is:
where subscript k denotes the number of a phase winding, ?k - magnetic flux coupled with this winding, and _k - angle of rotation reduced to the pitch of the teeth for the k-th winding. In order to use the relation it’s necessary to measure phase currents and voltages supplying phase windings so that the instantaneous value of the flux linkage associated with the k-th winding is familiar:
While we have the value of the flux linkage and current available, it’s possible to precisely determine the value of the angle of rotation in terms of the rotor's tooth position in respect to the axes of the given winding on the basis of look-up tables based on the reverse characteristics of magnetization. It’s self-evident that a useful device for such a control is a signal processor (DSP) and the particular manufacturers offer publications and exemplary solutions using their equipment beside part catalogues. Another method for the estimation of rotor position theta_r applies the technique of test pulses injected to the unsupplied phase of SRM ma chine, which is quite similar to the one used in the determination of the start-up sequence. An example of such sensorless control.
+-+-+- General block diagram of a sensorless control of SRM with a position estimator based on flux linkage
SRM Sensorless Control: State Observer Application
Sensorless control denotes here, just as in the previous examples, the lack of a position/speed sensor, such as encoder or resolver in a system. Concurrently, the system has to contain sensors of phase current, which are applied commonly and don’t pose any technical problem. Their use allows for the application of adequate emergency devices and various diagnostic methods regarding the state the drive.
In this manner, they lead to an increase of reliability of the system at a low cost and can be applied in the operation of the state observer. The role of the observer consists in on-line determination of estimates q ˆ of variables q and reduction of the observation error:
... to zero within a given time. Taking after the non-linear model of the dynamic state in the form
... the vector of the observer in the form:
... and the difference from the estimation of the state vector:
The equations of the observer form a direct repetition of the dynamic equations to which undetermined correction functions ?1, ?2 are supplemented and applied with regard to the equations whose variables are not observed. A problem that is widely discussed in the literature involves a method of finding correction functions for a non-linear system. Thus, the lack of a general method leads to a number of specific solutions, which are applied on the basis of analogy to similar systems. In these circumstances it’s important to select an appropriate method of testing whether the estimation error decays in time for the experimentally selected correlation functions. This is possible with the aid of the generalized Lyapunov method after the selection of positively determined candidate function V in an given area. This function needs to be positively determined and relative to the estimation errors. In order to secure the asymptotic error decay the first derivative of the function has to be negative in that area, in accordance with the Laypunov theorem. In the examined case of estimation of the position and rotational speed of SRM motor, the candidate function can be assumed in the form:
This function is self-evidently positively determined in the entire area of the occurrence of the estimation error. Concurrently, the requirement of the asymptotic decay error comes down the inequality in the form ...
Hence, it’s necessary to study two inequalities ...for the presented model of the observer and selected correlation functions ?1, ?2. When it comes to the selection of these functions, the most extreme solution involves the use of a sliding mode observer, which switches a constant function depending on the sign of the observation error. Thus, it imposes the function to remain in the vicinity of the observed value. In the examined case the application of the sliding observer with regard to means that:
... an assume positive or negative values depending on the sign of estimation error. By looking at the conditions of estimation error decay one can imagine how the sign changes of correlation functions
lead to the negative value required in these conditions. After testing and selecting adequate gain factors K1, K2 this practically enables one to apply such an observer in the control of SRM motor without the application of the position sensor.