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A decomposed sliding mode control of the drive with an interior permanent magnet synchronous motor and flexible coupling is presented. Decomposition exploits principles of vector control to divide motor into channel for control of magnetic flux and channel for control of torque separately. Sliding mode control principles are exploited to keep demanded value of magnetic flux and to control load angle in the presence of vibration modes and external disturbances. To obtain continues voltage as a control variable a smoothing integrator follows signum function in both channels. As a modification the switching governed by signum function is replaced by the high gain including rearrangement of the control system block diagram. The simulations indicate that the control system yields the desired robustness and further investigations are recommended.

Fulfillment of the increasing quality requirements to the performance of many industrial drives nowadays is not possible in the presence of the elastic features of the drive’s mechanical parts. Such features can cause torsion vibrations, which decrease the drive dynamic accuracy and in some specific cases can even lead to instability of the overall control system. Product degradation is usually a result of them. The problem of torsional vibration elimination is prevalent not only in high-power application such as rolling-mill drives but also conveyer and cage-host drives. Today due to the progress in power electronic and microprocessor systems that allow controlling the electromagnetic torque of motor almost without any delay and expand a pass-band up to the frequencies comparable to a resonance frequencies of the controlled plant mechanical parts, the problem of torsional vibrations appears also in many medium and low-power applications such as servo-drives, throttle drives, and robot arm drives including space applications [

It is well known that the classical PI controller is not effective in controlling the plant with elastic joints. Many different control concepts that effectively damped torsional vibrations have been proposed and developed, for example, [

In the case, if the control plant parameters can change, the problem is getting more complicated and many advanced control concepts have been proposed as solution. For example, the solution based on the robust control theory exploiting the

Different approaches depend on application of a sliding-mode control, robustness of which is capable to eliminate influence of control plant parameters changes and also to eliminate external disturbances if these are presented in the control system dynamics [

The main goal of this paper is to present the novel sliding mode control based control system for plant with elastics connections that combines robustness with chattering free control while exploiting voltage source inverter (VSI).

The control plant has three important parts:

VSI that feeds IPMSM;

Interior Permanent Magnet Synchronous Motor;

Elastic mechanical transmission, for example, a gearbox, with external load.

Mathematical models of these plant parts are necessary for control system design; therefore, their description is presented in the following

As it is known feed voltages corresponding to the demanded operation mode of the synchronous motor should be generated on its windings. A supplyline with three-phase voltages

The phase output voltage of the power converter represents in this case a sequence of voltage impulses. This high frequency sequence is averaged in the synchronous motor winding owing to its filtering property. The theoretical background is based on Shannon-Kotelnikov or the sampling theorem [

pulse—by nature;

continuous average—for electric motor control.

To realize the specified transformation of the three-phase voltages widely the scheme of power transformation with a direct current link (Figure

Power converter with a direct current link.

The voltage source inverter [

One of the mostly used VSI is the three-phase bridge with the isolated neutral, shown in Figure

The simplified schema of the three-phase VSI bridge.

Thus, depending on a control signal of each switch the output phase voltage

Vectors of VSI in complex plane.

The module of the above-mentioned six nonzero vectors depends on the load connection and on the value of the input dc line voltage

The VSI switch control, providing the needed output voltage, has two independent parts [

the modulation law that defines the part of the modulation period, in which the power switch is connected either to the positive potential (

the switching law that defines a sequence of switching of phase switches on the modulation period.

In this case, each phase output voltage represents a sequence of various duration squared impulses, magnitude of which is equal to

The classical mathematical model that considers the basic physical features of the processes inside of the motor, shown in Figure

equations of electric balance in its windings;

equation of the electromagnetic torque developed by the electric motor;

equation of mechanical movement (Newton’s second law for a rotary motion).

IPMSM.

The third equation, based on formula of mechanical movement, is general for all electric motors and has the following form:

The equations of electric balance based on Kirchhoff’s second law and the equations of electromagnetic torque are defined by the electric and magnetic circuits of the IPMSM and the physical processes inside of it. In the case of IPMSM the magnetic flux is created by permanent magnets located inside of the rotor. Such placement of permanent magnets gives to the rotor feature of the salient poles and the IPMSM has a nonuniform air gap. Due to nonuniformity of the air gap the magnetic flux creates an additional (

It is well known that mathematical description of the electromechanical transformation of power in the IPMSM using the actual phase currents and voltages as independent variables is capable to describe the physical processes inside of the motor. On the other hand such description is relatively complicated for analysis of the dynamical processes and results in a system of nonlinear differential equations with periodically changing coefficients due to change of rotor position.

The successful choice of transformation into axes

Second, orientation of a datum line 0 on the axis of rotation makes the differential equation independent regarding to a zero sequence current. It must be emphasized that the zero sequence current does not participate in the creation of air gap magnetic field and it has no influence on the electromechanical processes in the motor. It creates only an additional loading of windings and semiconductor devices. The traditional connection of motor phase windings “star” or “delta” automatically provides cancellation of a zero sequence current; that is, it eliminates additional thermal losses. It leads to simplification of mathematical description of synchronous motor. Elimination of differential equation for the zero sequence current reduces the system order. In this case the considered three-phase synchronous motor is replaced by a two-phase “idealized” one (

The transformation from the three-phase coordinates system to the rotating two-phase system (

The electric variables in a rotating system

The equations of electric balance and the electromagnetic moment for IPMSM in the rotating frame are as follows:

The behavior of the drive’s mechanical part (Figure

Elastics mechanical transmission.

Based on torque balance the behavior of the elastics mechanical transmission is described by following difference equations:

Sliding mode is a special type of behavior of a relay system [

The main feature of this mode is that none of the used general switched structures can realize such behavior. The sliding mode occurs in the intersection of all m surfaces
_{.} The vector function _{.} Usually it is an error function that must be led to zero by using the vector switching function

Formally, the aim of control is to force state system,

In this case the control design procedure is decoupled into the two tasks:

sliding mode design in the space of the vector function

motion design on the intersection of all m surfaces in the state space with order

Solution of the first task is based as it is considered in sliding mode analysis on the Lyapunov stability of the control plant in the space of the vector function

Motion on the sliding manifold is described by the equivalent control

In this case, the equivalent control,

An electrical motor in the drive structure carries out a transformation of the electric energy into a mechanical one that moves the mechanisms participating in the working process. Technology requirements for this process define the necessity and expediency of maintenance at the reference level of those or other mechanical variables, for example, position, speed, acceleration, torque, and so forth, of the mechanism tip.

In our case the main adjustable variable is a load position

stabilization of the load position at the reference constant level;

the load angular position change under the reference law;

restriction of the load angular position by an admissible value.

The number of the control variables depends on the number of the control plant independent controls. In the case of IPMSM there are two independent controls (see (

The main control goal is to maintain the reference value of the load angular position

Several basic approaches to the synthesis of IPMSM drives control can be defined:

single-loop control;

single-loop decomposition value reference;

cascade (subordinated) control.

In the first case the drive is considered a unit. The control synthesis interfaces to the solution of the nonlinear problem. The switching frequency and duration of power switches on states are generated automatically in the closed loop, as an auxiliary element in the solution of the primary drive control goal. Such systems possess high dynamics and small sensitivity to drive parameter changes and to external disturbances. Unfortunately, the automatically generated switching frequency of the power switches is not constant because it depends on the initial conditions. This leads to the power switches losses increase and to the drive mechanical noise.

In the second case two independent problems are considered: a problem of IPMSM control design and a problem of power converter control design. After transformation of IPMSM into system rotating at rotor speed the control law synthesis is based on the assumption that the power converter generates DC input currents and voltages responding to the solution of the primary control goal. Inverse Clark and Park transformations change the constant input currents and voltages into voltages and currents with variable frequency and magnitude. The problem of the power converter control is to select suitable switching of eight available switching vectors to follow required voltages with variable magnitude and frequency as close as possible providing the complete solution of the control problem.

Power converter control deals with the use of the modulation techniques based on a high frequency to generate the input voltages of the motor. DC output currents and voltages of control algorithm represent for the power converter a sequence of impulses of various durations. This sequence of impulses, being averaged due to the filtering effect of the load, that is, the IPMSM, forms a continuous load voltage, which is subject to control. In this case the indicators characterizing the discontinuous control, that is, its modulation such as switching frequency and pulse ratio are formed outside of power converter; that is, feed forward (

The control problem, in the third case, exploits the principles of the cascade (

As it was explained above the basic controlled variable in an electrical drive is the load angular position

Using such references, that is, the load angular position

The order of the error function depends on the model of control plant. Its derivation must be linear function of the control variables. To solve this control problem, for example, to make the error functions equal to zero, it means to synthesize switching control actions of power converter output voltages, which feeds the IPMSM stator windings. From the point of view of output voltages the above error function (

In this case, current,

One of the possible variants to maintain the simultaneous equality for functions (

As it was mentioned the possibility to organize sliding movement and to design the necessary control variables can be solved using the equation for initial dynamic system motion projection, for example, the IPMSM, at an error subspace of the controlled variables

Motor and load can be modeled by the system of differential equations of seven order (

After inspection of matrix

The control algorithm

By solving the control design problem for the second component of a control vector

It is important to notice that on “reasonable” operating modes the magnetic flux

As a first stage result, decomposition of an initial design problem was carried out and two one-dimensional problems about the occurrence of sliding modes were considered independently. The received equations (

It is a transition from the fictitiously entered control vector,

Each of these vectors has a fixed direction, and none of them coincides with the formally entered control vector,

For the output phase voltages of the power converter design it is possible to use (

The alternative approach to the design of power converter output phase voltages is based on a sufficient existence of a sliding mode condition in the systems with redundant control [

For the design of such control variables it is necessary to transform the domain of admissible controls received in the rotating coordinate system using an inverse linear Park transformation:

In this fixed coordinate system the relative position of this area and the output vectors of the power converter (

Sliding mode areas for admissible control

VSI output voltage vector can accept six nonzero values, separated an angular distance of

It is obvious that in this case there is a possibility to allocate ranges of angle values

The obtained conditions of sliding mode existence are sufficient for any instantaneous values

As it was written above the main disadvantage of the single-loop sliding mode control is that the automatically generated switching frequency of the power switches is not controllable. It depends on the initial conditions and could vary in a wide range. This leads to the negative effects as increase of power switches losses and to higher drive’s mechanical noise.

One of the possible ways to eliminate this disadvantage is exploitation of the single-loop decomposition control considering separately two independent problems: a problem of IPMSM control design and a problem of power converter control design. The power converter control problem deals with the exploitation of the modulation based on a high-frequency connection to the input voltages of motor phase windings independently.

As the first step of a single-loop decomposition control fictitiously entered control voltage vector,

In this case the IPMSM with the elastics joints is modeled by a nine-order system of differential equation (

In this case, the error between the reference value

For the design of the sliding mode on the intersection of the surfaces

For the second step, which is power converter real control design fictitiously entered control vector

Due to choice of switching frequency there is a possibility to minimize drive’s switching losses. However, it must respect the fact that to obtain good results the PWM frequency must be a few times higher than the band pass highest frequency of the controlled plant.

One disadvantage of this approach if compared with classical one is the need of the additional derivation of the control variables. It is well known that the derivations of high order have larger influence on the control system dynamics. Another approach that allows using the integral of the control variable error instead of the 5th order derivation was investigated. In this case all advantages of the above decomposed sliding mode control design are saved.

The behavior of the IPMSM with the elastics connection is described by the same seventh-order differential equation system of (

For the design of the sliding mode on the intersection of the surfaces

Proposed decomposed SMC strategy was verified in two steps. As the first step control of the plant, which consists of PMSM and flexible load, was verified by simulations. Secondly the strategy was applied to the PMSM only for control of its shaft position. In spite of some problem with generation of the observed rotor acceleration, which is necessary as one derivative feedback, this control structure was verified experimentally.

Overall block diagram of decomposed sliding mode control of the drive with IPMSM and flexible coupling is shown in Figure

Overall block diagram of decomposed SMC system with IPMSM and flexible coupling.

The feedback derivatives gains as well as ideal position trajectory,

As possible modification of the proposed control system the signum function is replaced by a transfer characteristic consisting of a proportional high gain with saturation, which changes the switching boundary to a boundary layer.

Operation in the modified sliding mode implies operation on the linear, high gain characteristic enabling rearrangement of the block diagrams (valid for both control channels) to avoid the need of the highest output derivative. After this rearrangement the flux control channel operation is based on the integration of error between demanded and real current,

Block diagram of modified decomposed SMC of torque channel.

The parameters of the IPMSM and load are listed in the Appendix. For whole simulation interval shaft of the motor is loaded with constant friction torque,

All the simulations are carried out with zero initial state variables and a step load position demand,

Decomposed SMC system of the drive with PMSM and flexible coupling.

Demanded voltages

Demanded voltages

Demanded currents

Load and rotor velocity

Load and rotor position

Ideal, load and error position

Modified control system of the drive with PMSM and flexible coupling.

Demanded voltages

Demanded voltages

Demanded currents

Load and rotor velocity

Load and rotor position

Ideal, load and error position

Subplots (a) show demanded voltages

Demanded current

The experimental results for decomposed SMC of rotor angle are shown in Figure

Finally, subplots (f) of both figures show the ideal response and response of the control system to the step load position demand,

Presented simulation results confirm possibility of decomposed sliding mode control of the load angle in spite of flexible coupling between motor and load. As can be seen from Figures

To avoid problem with high order derivatives the decomposed SMC strategy was applied to PMSM for rotor position control [

Overall block diagram of decomposed SMC of IPMSM rotor position.

Designed and measured switching surface for decomposed SMC of PMSM rotor angle.

Preliminary experimental results for decomposed SMC of PMSM rotor angle.

Applied stator

Stator

Stator

Rotor angular acceleration

Rotor angular velocity

Rotor and ideal position response

The experimental results for decomposed SMC of rotor angle (for the idle running PMSM) are shown in Figure

Comparison of PMSM with flexible coupling simulations and preliminary experiments with decomposed SMC of PMSM rotor position show a good agreement with the theoretical predictions made during the control system development. If suitable observer of control variables is developed it is highly desirable to investigate proposed decomposed SMC strategy experimentally.

The simulations and preliminary experiment presented predict that the proposed decomposed sliding mode control system and its modification can be made to follow the ideal closed loop dynamics of the load angle with moderate accuracy in spite of flexible coupling between motor and load. As a result, this is acceptable considering that no motor or driven load parameters were needed for control system design.

Some variables of modified control system show significant overshoots during transients, which were limited in subplots due to graphic purposes. In spite of very close tracking of ideal position modified control system shows also small overshoots of load position, which is not acceptable for position control systems requiring high precision.

The results obtained are sufficiently promising to warrant further experimental trials, which were not carried out due to problems with output derivatives of higher order. For the future work derivatives estimation methods such as observer with filtering properties or high gain multiple integrator observer together with measurement of load position with higher accuracy are recommended.

Parameters of PMSM are winding resistance, ^{2}.

Load parameters are load moment of inertia, ^{2}, and shaft spring constant, ^{−1}.

The authors wish to thank Slovak Grant Agency VEGA for funding Project no. 1/0355/11 “Optimal Control Techniques for Decreasing Losses of A.C. Drives.”