FUNDAMENTALS OF ELECTRICAL DRIVE CONTROLS Joško Deur and Danijel Pavković University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, I. Lučića 5, HR-10002 Zagreb, Croatia Keywords: Electrical drives, control, modeling, DC motor, permanent-magnet synchronous motor, cascade control, chopper, sensors, speed control, position control, pointing, tracking, friction, compliance, backlash, state control, nonlinear compensation. Contents 1. Introduction 2. Elements of controlled electrical drive

2.1. Separately-excited DC motor 2.1.1. Dynamic model 2.1.2. Steady-state curve

2.2. Electronic power converters 2.3. Sensors 2.4. Electronic control unit and control algorithms

3. Adjustment of DC motor speed 3.1. Speed adjustment by armature resistance control 3.2. Speed adjustment by armature voltage and field control

4. Design of DC drive cascade control system 4.1. Cascade control structure 4.2. Damping optimum criterion 4.3. Armature current control 4.4. Speed control 4.5. Position control

4.5.1. Small-signal operating mode 4.5.2. Large-signal operating mode

5. Design of tracking system 5.1. Tracking of a-priori known reference 5.2. Tracking of a-priori unknown reference

6. Control of permanent-magnet synchronous motor 6.1. Modeling of motor 6.2. Control

7. Compensation of transmission compliance, friction, and backlash effects 7.1. Model of two-mass elastic system with friction and backlash 7.2. Compliance compensation 7.3. Friction compensation 7.4. Backlash compensation

8. Conclusion Summary Controlled electrical drives can be regarded as the most flexible and efficient source of controlled mechanical power. Understanding and developing the controlled electrical drive systems require a multi-disciplinary knowledge, starting from electrical machine theory, through electronic power converter technology to control system design techniques. This article gives a systematic overview of elements of a controlled electrical drive with emphasis on the control system design. The basic procedure of feedback and feedforward cascade control system design is presented for the

separately-excited DC motor. It is then demonstrated that the basic principle of current/torque control can be applied to AC machines modeled in the rotational field coordinate frame, while the superimposed speed and position controller structure remains the same as with the DC motor. Finally, a notable attention is paid to analysis of transmission compliance, friction, and backlash effects, and their compensation by means of advanced control algorithms. 1. Introduction Electrical drives represent a dominant source of mechanical power in various applications in production, material handling, and process industries. Applying the feedback control techniques to electrical drives substantially improves their performance in terms of achieving precise and fast motion control (servo-control) with a high efficiency. Traditionally, the controlled electrical drives were based on direct-current (DC) motors and analog controllers. However, the rapid development of power electronics and microprocessor technology in the last three decades has propelled application of servo-control to brush-less, alternating-current (AC) drives, and provided implementation of advanced motion control algorithms including compensation of transmission compliance, friction, and backlash effects. The overall control performance, efficiency, reliability, and availability of the controlled electrical drives have been substantially improved, thus accelerating their penetration into various engineering applications. This article presents an overview of controlled electrical drive technology with emphasis on control system design. The presentation is based on the separately-excited DC motor, since control of this motor can be easily understood and readily extended to AC motors. First, the elements of a controlled electrical drive are described (Section 2), which include DC motor and its mathematical model, electronic power converters, sensors, and electronic control units including the basic control algorithms. Next, the steady-state form of DC motor model is used to describe the motor speed adjustment (or open-loop control) in the regions below and above the rated speed, as well as the controlled starting and regenerative braking of the motor (Section 3). This serves as a basis for presenting a cascade structure of motor feedback control, including optimal tuning of current, speed, and position controllers (Section 4). For tracking applications, the feedback system is extended by feedforward paths or a feedforward compensator, in order to reduce the dynamic tracking error (Section 5). Section 6 shows, on an example of permanent-magnet synchronous motor (PMSM), how the naturally decoupled armature and field control of DC motor can be applied to the coupled dynamics of three-phase AC motors. Finally, Section 7 analyzes influences of transmission insufficiencies related to compliance, friction and backlash effects on the static and dynamic behavior of a servodrive, and presents control algorithms for compensating these effects. The theoretical discussions are illustrated by a number of computer simulation results. 2. Elements of Controlled Electrical Drive Figure 1 shows the structural block diagram of a controlled electrical drive. An electrical motor is coupled to a working mechanism in order to provide a transfer of mechanical power. The main additional features of controlled electrical drives compared to their conventional counterparts are: (i) the power transfer is made time variant/controllable using an electronic power converter , and (ii) the drive motion can be controlled in a precise manner based on the use of feedback paths containing sensors and electronic control unit. The control tasks can be different, starting from current control (corresponding to open-loop torque/force control), through speed and position control, and towards force control. Normally, the controlled power flows from the electrical grid to the working mechanism. However, during transients or occasional continuous braking intervals, the motor switches to a generator mode and the power flows back to the grid. If the power converter does not support the regenerative braking feature (typically in low-power drives), the braking power is dissipated on a braking resistor.

Figure 1. Structural block diagram of controlled electrical drive. 2.1. Separately-Excited DC Motor Direct-current (DC) motor (see cross-section schematic in Fig. 2a) consists of a magnetic field flux (excitation) circuit (placed on the stator), armature circuit (placed on the rotor), and a commutator which inverts the current in an armature coil whenever it passes through the neutral zone that is perpendicular to the stator field axis. The power is transferred to the armature through brushes that are fixed in the neutral zone and leaned to the commutator. The excitation and armature circuits can be connected separately from each other, or a series or parallel connection can be utilized instead. The separately-excited DC motors are mostly used in controlled drives, owing to the possibility of independent field and armature current control and related superior control features in a wide speed range.

Figure 2. Simplified cross-section schematic (a) and equivalent scheme of separately-excited DC motor.

2.1.1. Dynamic Model Figure 2b shows an equivalent scheme of the separately-excited DC motor. The stator magnetic flux acts upon the armature current ai , thus producing the motor torque. On the other hand, when the

rotor rotates, the voltage e (back electromotive force, EMF) is induced in the armature winding. The motor dynamics are described by the following set of differential equations (see Nomenclature), given in both time ( t ) and Laplace ( s ) domain:

aa a a a a a a a a

( )( ) ( ) ( ) ( ) ( ) ( ) ( )

di tu t R i t L e t u s R i s L si s e s

dt (1a)

e( ) ( ) ( )e t K t t (1b)

m l m l( )

( ) ( ) ( ) ( ) ( )d t

J m t m t Js s m s m sdt

(1c)

m m a( ) ( ) ( )m t K t i t (1d)

MM M M M M M M M

( ( ))( ) ( ) ( ) ( ) ( )

d i tu t R i t N u s R i s N s s

dt

(1e)

( )

( ) ( ) ( )d t

t s s sdt

(1f)

where M( )i is the nonlinear static magnetizing curve. The armature circuit, Eq. (1a), can be

described by the following transfer function

a a

a a a a

( ) 1

( ) e( ) 1

i s K

u s s L s R T s

, (2)

where a a1K R and a a aT L R are the armature gain and armature time constant, respectively.

Based on Eqs. (1) and (2), a block diagram of the motor model can be created, as shown in Figure 3a. In the basic case of constant excitation circuit voltage M( const. const.)u or permanent-

magnet excitation, the block diagram reduces to the one shown in Figure 3b based on the following substitutions: t mK K and v eK K .

Figure 3. Block diagram of DC motor: (a) general case and (b) constant-flux case. 2.1.2. Steady-State Curve Under the steady-state conditions, the time-derivatives of motor dynamic variables vanish (e.g.

a 0; 0di dt s ). After rearranging the steady-state forms of motor equations (1), the following

expression for the motor steady-state curve is obtained:

a am2

e e m

0 m( )

u Rm

K K K

m

. (3)

The steady-state curve is shown in Figure 4. Since the armature resistance aR is relatively small

(particularly for high-power machines), the steady-state curve is rather stiff, i.e. the motor speed drop due to the increase of load l mm m is small compared with the idle speed 0 . The drive

operating point is determined as the cross-section point of the motor and load static curves (Fig. 4; note that m lm m is valid for the steady-state conditions according to Eq. (1c)). If the motor speed

is lower than the idle speed: 0 a a 0e u i , the motor operates in the driving mode (1st

quadrant of the coordinate system in Fig. 4). Otherwise, for the case when 0 ( ae u and

a 0i ), the machine operates in the generator braking mode, thereby producing the electric energy

and transmitting it to the grid (2nd quadrant in Fig. 4). For the reverse motion ( 0 ), the driving and braking modes relate to the 3rd and 4th quadrants, respectively (see also Section 3).

Figure 4. Steady-state curve of DC motor and construction of operating point. 2.2. Electronic Power Converters The electronic power converters transform the grid power system with constant parameters (e.g. voltage, frequency) into the motor power system with variable parameters and potentially different voltage waveform. They can be divided into several characteristic groups, as shown in Figure 5. A rectifier is connected to AC grid and it provides variable-voltage power supply for DC motors. Inverters are utilized when variable-frequency AC motors are supplied from a DC grid (e.g. in transport applications). When a power converter allows for both power directions to cover the driving and generator braking modes, it is called regenerative converter. Other power converter types include DC/DC converters and AC/AC converters.

Figure 5. Basic types of electronic power converters. Unless very low-power drives are considered (e.g. up to 50W), power converters are designed as switching devices. This is because of a great efficiency of electronic valves when used in switching (on/off) mode. Historically, thyristors were the first semiconductor switching components. Their

main advantage is that they are able to withstand extremely large currents (typically up to 10,000 A), while their main disadvantage is that they are actually semi-controllable semiconductor switches (i.e. they cannot be turned off actively). Thyristors have conveniently been used in line-commutated power converters (rectifiers and regenerative converters, see left part of Fig. 5) aimed at supplying the DC motors directly from AC grid. However, the response time of these converters is relatively slow, as it closely relates to the inverse of AC grid frequency, and the produced voltage waveform is characterized by a significant ripple. In order to overcome these disadvantages, which become more relevant when supplying AC machines, pulse width-modulated (PWM) voltage-source transistor converters can conveniently be used. In the low power range (e.g. up to several kW) fast and efficient MOSFET components are used, while power converters with larger power ratings (up to 1MW) rely on insulated gate bipolar transistor (IGBT) components. The switching frequency is typically significantly larger than 1 kHz (more than 20 kHz for MOSFET converters), so that the response time is a fraction of millisecond. Figure 6 shows the voltage-source transistor converter used in DC drives, which is usually called four-quadrant (4Q) chopper or H-bridge converter. The corresponding "idealized" voltage and current waveforms for the 1st-quadrant operation are shown in Figure 7. The full-bridge three-phase rectifier converts the AC grid voltage to DC link voltage. One of the diagonal pairs of chopper transistors is switched on at any time, thus bringing the positive or negative DC link voltage UD to the motor. Since the (constant) switching frequency c 1 21f T T is very high (typically 5 kHz),

the motor speed relates to the average motor voltage, which reads (Fig. 7):

1 2

1 2a a a D 1 c D D

1 2 1 20

1( ) ( ) 2 1 (2 1)

T TT T

u u t u t dt U T f U UT T T T

, (4)

where 1 c 1 1 2 /T f T T T is the output square-wave voltage duty cycle.

Moreover, since c a 1/f T is usually satisfied, the armature current ai , and thus the torque mm , is

effectively smoothed by the lag (low-pass) nature of the armature dynamic model (2). Thus, for the square-wave (AC) armature voltage signal au t the armature current ai has a DC form, while

opposite is valid for the DC link voltage DU and current Di (Fig. 7). According to Eq. (4), the

armature voltage can be arbitrarily varied in the range D D[ , ]U U by changing the duty cycle

(pulse width modulation, PWM). During the interval 1T of the 1st-quadrant converter operation, the current is conducted through the

transistors 1 1Q Q , while for the interval 2T the still positive motor current flows through the

freewheeling diodes 2 2D D . Since 1 2T T , more power flows from the DC link to the motor than

vice versa, and the energy spent is covered by the power grid through the rectifier. This could be expected since the motor operates in the 1st quadrant.

Figure 6. Electrical scheme of H-bridge transistor converter.

Figure 7. "Idealized" waveforms of H-bridge converter. Table 1 outlines the converter operation in all four quadrants of the speed vs. torque coordinate system in Figure 4. For instance, in the 2nd quadrant, 1 2T T is still valid since the motor speed

and the armature voltage au are larger than zero (cf. Fig. 4). However, the armature current ia is

negative (for the negative motor torque mm ), which means that the current flows through

freewheeling diodes 1 1D D during the interval 1T and through transistors 2 2Q Q during the

interval 2T . Thus, since 1 2T T the average motor power is negative, i.e. it flows to the DC link and

increases the DC link (capacitor) voltage. Since the simple rectifier topology cannot provide the power flow to the grid, the braking resistor is switched on by the transistor bQ (Fig. 7) when the

capacitor voltage exceeds the upper threshold, thus dissipating the motor braking power. When the DC link voltage drops below the lower threshold (indicating the driving operation and the power flow from the grid), the transistor bQ is again turned off.

Table 1. Description of four-quadrant chopper operation.

Quadrant:

1st (motor) 2nd (generator) 3rd (motor) 4th (generator)

ua and > 0 > 0 < 0 < 0

ia and mm > 0 < 0 < 0 > 0

T1 > T2 Yes Yes No No

T1 '11 ,QQ '

11 , DD '11 , DD '

11 ,QQ

T2 '22 , DD '

22 ,QQ '22 ,QQ '

22 , DD

The H-bridge transistors can be directly controlled by a current controller of a "bang-bang" type, as shown in Figure 6. Depending on the sign of armature current control error aR a–i i , the current

controller switches on one of the diagonal transistor pairs. In general, a lower level of current ripple can be achieved by means of open-loop PWM voltage control. The principle of conventional PWM control logic is illustrated in Figure 8. The reference armature voltage is compared with a carrier signal, whose frequency corresponds to the converter switching frequency, and the resulting logical signal and its complement are fed to the H-bridge transistor gates.

Figure 8. Illustration of PWM functionality.

The topology of voltage-source power converter in Figure 6 can be directly applied for supplying three-phase AC motors. The main differences are: (i) there are three pairs of inverter transistors for supplying the three-phase motor winding, and (ii) under the steady-state conditions the PWM reference voltages are not constant (they have a sinusoidal shape). Each power converter is characterized by a dynamic behavior in terms of a pure delay occurring in the process of settling the demanded armature voltage aRu . If the controller sampling time is

synchronized with the converter switching frequency, the pure delay is, statistically-speaking, equal to half of the switching period c1/ f . Thus, the converter dynamics may be approximately described

as

a chchch ch

aR ch

( ) 1( ) , with

( ) 1 2T s

c

u s KG s e T

u s T s f

. (5)

2.3. Sensors The main motor/drive variables need to be measured for the purpose of feedback control (Fig. 1). The main requirements on the electrical drive sensors are high static accuracy (e.g. in the range from 0.1%-1%), high bandwidth, galvanic insulation in the case of measurement of drive electrical quantities, small volume/mass, high reliability, and acceptable cost. The motor current can be inexpensively measured by a shunt resistor placed in the motor power line. However, due to the obvious weaknesses in terms of limited accuracy and a lack of galvanic insulation, this solution is limited to low power/low cost drives. A standard solution for modern electrical drives is use of a Hall effect-based current sensor. The Hall sensor of magnetic flux is placed in the air gap of a magnetic core, which is excited by two coils conducting the motor current and the measured current signal. The measured current, which is to be proportional to the actual current, is generated in a closed-loop circuit that regulates the Hall sensor flux to zero. The motor speed was traditionally measured by a DC or AC tacho-generator. However, since the modern AC drives require the motor angular position signal even for current control (Section 6), it has been convenient to use a single sensor for both motor speed and position measurements. Here, it should be mentioned that the speed signal can be reconstructed from the position signal by time-differentiation, while the position reconstruction based on speed integration is not viable due to inherent drift error in the presence of speed measurement offset. A potentiometer-based position sensor can be used in low-cost (typically low-power) applications. Standard solutions for industrial electrical drives include encoders, either incremental or absolute, and resolvers. Encoder sensors are generally more accurate than resolvers: the position measurement accuracy of standard encoder sensors used in electrical drives can be up to 100,000 pulses per revolution after electronic interpolation. However, they are generally more expensive and less reliable in harsh environments. The position is measured in a digital way by counting the encoder pulses. The speed signal is reconstructed by measuring either the encoder pulse frequency or pulse width, or by combining these two approaches for a favorable accuracy over a wide speed range. A superior speed measurement accuracy can be achieved by flash A/D converter-based electronic interpolation of the originally analog encoder signals, where the typical interpolation factor is 1024. The absolute encoders are more expensive than the incremental ones, but they are immediately ready for operation (no drive initialization procedure is required). Resolver is a two-phase brush-less AC generator whose rotor winding is fed by a high-frequency signal. The induced sine and cosine armature signals are fed into a resolver/digital (R/D) converter,

which provides demodulation of these signals, and their processing by a hybrid (analog-digital) servo-circuit that gives an analog speed signal and digital absolute-position signal. The resolver is an affordable and a very robust sensor. The sensors including the associated signal processing/filtering circuits are also characterized by some dynamic behavior. Since the sensor delay is relatively small, the sensor dynamics is usually approximately described by the simple, first-order lag term (cf. Eq. (5)). The sensor equivalent time constant is typically related to an inverse of the filter bandwidth. Exceptionally, for the digital speed measurement by encoders, the delay is equal to a half of the encoder counter sampling time. 2.4. Electronic Control Unit and Control Algorithms The electrical drive analog control devices and later hybrid (analog-digital) control devices are nowadays replaced by fully digital, microprocessor-based electronic control units (ECU). The basic structure and main features of the electrical drive ECU are similar to those encountered in many other control applications. However, it should be noted that the electrical drive ECUs are rather demanding in terms of high sampling frequency (larger than 1 kHz), complex control and estimation algorithms (especially for AC drives), high requirements on electromagnetic compatibility etc. The input signals are acquired by counter circuits (e.g. encoder position signal) or analog/digital (A/D) converters (e.g. current signal). These signals are processed by a microprocessor based on the control code, and the PWM timer circuits generate the digital output signals that are fed to the power converter. The core control algorithms, such as speed and current controllers, are derived from the earlier-generation analog controllers. Figure 9a shows the block diagram of an analog proportional-integral (PI) armature current controller (see Section 4). The integral (I) term integrates the current control error signal ia aR a–e i i and varies the output armature voltage reference aRu until it reduces the

control error to zero (elimination of steady-state error). The proportional (P) term stabilizes and speeds up the response.

Figure 9. Block diagram of analog PI controller (a) and its discrete-time digital counterpart (b), derivation of PI controller difference equation (c), and its implementation into controller code (d).

3. Adjustment of DC Motor Speed According to Eq. (3), the DC motor speed can be adjusted (i.e. controlled in an open-loop manner) by varying the total armature resistance aR , armature voltage au , or magnetic field flux .

Similar methods can be employed to provide electrical braking of the drive. 3.1. Speed Adjustment by Armature Resistance Control A variable resistor (rheostat) can be added in series with the armature (Fig. 10a) in order to change the motor static curve. This inexpensive method can be used to achieve appropriate motor starting, speed adjustment, and braking. Increase of the overall resistance in the armature circuit results in a proportional increase of the speed drop , with no influence on the idle speed 0 (Eq. (3), Fig.

4). Thus, the motor static curve become softer, as illustrated in Figure 10b. This allows speed adjustment below the rated static curve a, as shown by the curve b in Figure 10b. Further, by adding a proper value of the armature resistance aaR (e.g. 5 to 10 times larger than aR ,

curve c in Fig. 10b), the motor starting torque msM , and thus the starting current as ms~I M , can be

reduced from the largely excessive short-circuit current a a/u R , which would damage the motor, to

regular values that provide a safe, yet fast starting. The added resistance would have to be gradually decreased during starting in order to keep the motor torque high (not shown in Fig. 10b).

Figure 10. Illustration of use of additional armature resistance for DC motor speed adjustment, starting, and reversal braking: (a) electrical scheme, (b) steady-state curves.

Finally, armature resistance control can also be used in reversal braking of the motor. For the active (e.g. gravitational) character of the load torque curve ( lsgn( ) const.m ), it is enough to increase the

resistance aaR beyond the motor starting value, in order to allow the load to reverse the motor into

the braking 4th quadrant (curve d in Fig. 10b). For instance, for a crane system, varying aaR results

in varying load descending speed. On the other hand, if the armature voltage polarity is changed and the resistance aaR added, the transient reversal braking in the 2nd quadrant results in an effective

stopping of the motor for the case of passive (e.g. frictional) load (curve e in Fig. 10b). The braking torque can be adjusted by changing the resistance aaR .

The main disadvantage of using the armature resistor for speed adjusting, and also for frequent starting and braking operation, is low efficiency, i.e. large thermal losses on the additional resistor,

particularly at low speeds where a large resistance aaR should be added into the armature circuit.

Also, the control transient performance is significantly slower when compared with electronic controls, which are discussed in the next section. 3.2. Speed Adjustment by Armature Voltage and Field Control From the standpoint of drive efficiency, it is more desirable to reduce the motor speed (i.e. the mechanical power mm ) by reducing the armature voltage au (i.e. the electrical power a au i )

than by adding the armature resistance that dissipates the electrical power. Armature voltage is adjusted by a high-efficiency electronic power converter (Section 2). As the idle speed 0 is

proportional to the armature voltage au and the speed drop is independent of au (see Eq. (3)

and Fig. 4), the armature control static curves are parallel to the nominal curve (curves b in Fig. 11). Since the armature voltage must be lower than the rated voltage in order to protect the motor/commutator insulation, the armature control can be used below the rated speed only. The speed can be increased beyond the rated speed by weakening the magnetic field flux , because the idle speed 0 is inversely proportional to the flux (Eq. (3)). Since the speed drop

is inversely proportional to 2 , the static curves become somewhat softer in the field weakening region (curves c in Fig. 11). The field control is achieved by varying the excitation circuit voltage Mu by means of another electronic power converter. The power of this converter is

only a fraction of the armature converter power, because the motor magnetizing power is much smaller than the armature power. For the continuous-duty drive operating mode, the armature current ai must be lower than the rated

current anI to protect the motor from overheating. This means that in the armature control region,

where n and m a~m i , the continuous-duty motor torque is limited to the rated torque mnM

(Fig. 11). On the other hand, in the field control region, n is valid, so that the motor torque

m m am K i must be reduced below the rated torque mnM . More specifically, since the flux is

controlled to be inversely proportional to the speed, the continuous-duty motor torque is inversely proportional to the motor speed, as well, i.e. the motor power is limited to the rated power

n mn nP M in the field control region (Fig. 11). Hence, under the light-load (e.g. idling)

conditions the motor speed can be arbitrarily increased above the rated speed (provided that adequate bearings are used), thus reducing the “machining” time, i.e. increasing the productivity of different machines (e.g. machine tools). It should be mentioned that at very high motor speed the DC motor power should be reduced below the rated power due to the commutation constraints (Fig. 11). For the intermittent-duty modes, the current can be increased over the rated value (typically up to twice the rated current value for the DC motors, m,max mn2M M ), which results in a significant

extension of the torque/power operating range, as illustrated in Figure 11.

Figure 11. Steady-state curves illustrating DC motor speed adjustment by combined armature and field control.

After an abrupt armature voltage reduction, the motor (transient) operating point moves to a new static curve. Since the electrical transients are usually much faster than the mechanical transients, during the initial period of armature voltage change the motor speed may be assumed to remain the same, while the motor torque may abruptly change. This results in an abrupt transition in the braking 2nd quadrant (see curve d in Fig. 11). Because the actual motor speed is greater than the new idle speed 0' , the regenerative braking occurs. Energy losses during this braking phase are much

smaller than those associated with reversal braking since no resistance is added. Moreover, the overall efficiency is superior due to the fact that the electrical energy is recuperated into the grid. If the motor needs to be stopped, the armature voltage would be gradually decreased, so that the braking torque is kept at a constant value (fast deceleration without jerk; see curves e in Fig. 11). After the voltage drops to zero, it can be further gradually decreased below zero to provide constant-torque starting of the motor in the reverse direction (3rd quadrant, curves f). In the case of passive torque ( m lsgn( ) sgn( )m m ), the transient will end up in a driving operating point (Point 1

in Fig. 11). Otherwise, for the active (e.g. gravitational) load, the operating point lies in the 4th quadrant (Point 2 in Fig. 11), where a crane load can be lowered with an arbitrary speed and a great regenerative-braking efficiency.

The above-described, electronically-controlled or manual speed adjustment cannot be very fast and precise. For instance, the steady-state accuracy is affected by the load torque disturbance, i.e. for the constant armature voltage the speed varies with the load torque (Fig. 4). Also, an adequate profile of armature voltage change for the constant-torque braking and starting is difficult to be realized in an open-loop manner. These are the motivations for using a closed-loop speed control system (Section 4), which can reach a superior behavior of the electrical drive under steady-state and transient conditions. 4. Design of DC Drive Cascade Control System The electrical drive control systems are usually implemented in a cascade structure and designed by applying practical optima such as damping optimum criterion. 4.1. Cascade Control Structure According to Eqs. (1) and Figure 3b, the constant-flux DC drive model includes the following measurable state variables: armature current ai , speed , and position . Control theory suggests

that high performance of such a high-order system can be achieved by using either a state variable controller or a compact high-order controller. The cascade control structure is a special form of state variable control, where a superimposed controller commands the reference of an inner controller. Figure 12 shows the block diagram of a basic cascade control system for the DC motor operating in the constant-flux region. The innermost control loop corresponds to the innermost, and at the same time, the fastest state variable of the motor model in Figure 3b, and this is the armature current ia. Based on the same reasoning, the speed controller is superimposed to the current controller, while the outermost control loop includes the position controller. The main advantages of the cascade control structure are:

1. Design and verification of each control loop is preformed sequentially (step-by-step), from the innermost current control loop to the outermost position control loop.

2. The disturbance suppression is effectively performed at the local level (e.g. the current control loop deals with the electromotive force disturbance, while the speed control loop suppresses the load torque disturbance).

3. The drive state variables can be effectively limited by saturating the output of the immediately superimposed controller: e.g. saturation of the speed controller output (the current reference iaR) effectively limits the armature current, thus protecting the motor and power converter.

4. Off-line or even on-line switching between different modes of operation (position/ speed/current control) is straightforward due to the modular structure of the control system.

A potential weakness of the hierarchical/cascade control structure is that it may generally be slower compared to the compact control. Nevertheless, due to the significance of the aforementioned advantages (particularly that in Point 3), the cascade structure is used in a great majority of electrical drive control applications. However, in some applications the compact control concept can be advantageous. For example, for low-power and low-cost DC drives, which are characterized by a soft steady-state curve and a non-excessive short-circuit current asI (see curve c in Fig. 10b),

there is no need for direct over-current protection of the drive. Moreover, it is beneficial to omit the current sensor for the cost reasons and provide as fast position response as possible. Therefore, the inner current control loop, and also the speed loop, can be omitted, and a compact higher-order position controller can designed instead. An example of such a DC drive control system is the automotive electronic throttle DC drive.

Figure 12. Block diagram of cascade structure of DC drive control. While the innermost armature current controller is used along with the outer speed or position controller, it may also be utilized as a stand-alone controller (case b in Fig. 12). This is because the closed-loop current control is closely related to the open-loop torque control (since m a~m i ), and

the torque control (or force control for linear output motion) has many applications such as those in electric vehicle power trains, winding (spooling) systems, robot grippers, and clutch clamping systems. Closed-loop speed control (case c) provides a high static and dynamic accuracy of speed adjustment when compared to the open-loop speed control discussed in Section 3 (case a). Some of typical applications are machine tools main drives, rolling mills, and operator-guided manipulators. Probably the most important servodrive applications are those related to position-control systems (case d). They can be divided in two main groups: (i) pointing mechanisms and (ii) tracking systems. Typical applications include robotics, machine tools, and moving-target tracking applications. When the open-loop torque control (case b) cannot provide a favorable accuracy due to, for instance, transmission friction disturbance, it is desirable to design a closed loop torque/force control system. The force controller is either superimposed to the position controller (not shown in Fig. 12), or used instead of it if the controlled force is directly proportional to the motor position. The control system structure shown in Figure 12, and the related design procedure given in the following subsections relate to the basic constant-flux operating region (motor speed is below the rated speed). According to the discussion in Section 3, the motor speed can be increased above the rated speed by weakening the motor flux. In that case, the control structure needs to be extended by an additional loop that controls the excitation current Mi . Here, the excitation current reference MRi

can be determined from the motor speed according to the flux weakening open-loop control strategy (Section 3) and the known magnetizing curve M( )f i (cf. Fig. 3a). The excitation current

controller is designed in the same manner as will be described below for the armature current controller. Another, generally more accurate, field-weakening control method relates to establishing a closed-loop control of the electromotive force e . The EMF reference Re is set slightly below the

rated motor voltage to provide some voltage reserve (e.g. 10%) for armature current control. The actual EMF variable e is reconstructed from the armature voltage and current.

4.2. Damping Optimum Criterion Let us consider a single-input/single-output feedback control system with the block diagram shown in Figure 13. The controller acts upon the control error signal R1ye y y and calculates the

commanded signal ( )u t (the process input signal), which should provide that the process output

( )y t follows the reference R ( )y t with a minimum transient error ( )ye t , and achieves steady-state

accuracy. The response ( )y t should also be well damped for at least two reasons: (i) any oscillatory behavior is generally undesirable in servodrives since it is directly associated with a high level of vibrations, and (ii) because of inevitable presence of process parameter variations, the control system tuned for a weakly-damped behavior may even tend to become unstable.

Figure 13. Block diagram of feedback system. Typically, the controller zeros (roots of the numerator polynomial of the transfer function c ( )G s )

should not cancel the process poles (roots of the denominator polynomial of the transfer function

p ( )G s ), because the system may, otherwise, be largely suboptimal with respect to disturbance

variable . Since the non-cancelled controller zeros can often cause a large step response overshoot, they are typically cancelled by the prefilter placed in the reference path (Fig. 13). Assuming that the process transfer function p ( )G s does not include zeros or that they are (if stable)

eventually cancelled by the prefilter, as well, the control system closed-loop transfer function reads:

c p c ccl

R c c p c c p c c

c

c p c c

( ) ( ) ( )( ) 1 1( )

( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( )

1,

( ) ( ) ( ) ( )

G s G s K B sy sG s

y s B s G s G s B s A s A s K B s

K

A s A s K B s A s

(6)

where ( )A s is the closed-loop characteristic polynomial given by

1 3 2c c p c 3 2 1( ) ( ) ( ) ( ) ... 1n

nA s K A s A s B s a s a s a s a s , (7)

and n is the closed-loop system order. If ( )A s is indeed such that cl0 1 0 1A G , the closed-

loop system is assumed to be accurate under the steady-state conditions. This is typically achieved by including the integral term 1/ s in the controller transfer function, so that c 0 0A , which for

c 0 1B gives 0 1A .

The nominal characteristic polynomial of the damping optimum design method is defined as:

2 1 2 3 3 2 21 2 e 3 2 e 2 e e( ) ... 1n n n

n nA s D D D T s D D T s D T s T s (8)

where , 2,...,iD i n , are the characteristic dimensionless ratios, and eT is the closed-loop equivalent

time constant. Analytical expressions for the controller parameters are obtained by equating the coefficients of the actual polynomial (7) and the nominal polynomial (8). The main characteristics of the damping optimum design method are as follows:

1. For a full-order controller, it is possible to set all characteristic ratios 2 3, , ..., nD D D to the

optimal value of 0.5. This gives a quasi-aperiodic closed-loop step response with the overshoot of around 6% and the approximate settling time of 1.8Te for any system order n (Fig. 14a).

2. For a reduced-order controller, with the number of free parameters equal to r , the dominant characteristic ratios 2 , ..., rD D should be set to the optimal value of 0.5. If the response largely

differs from the nominal one described in Point 1 (e.g. if high-frequency oscillations occur), this indicates that a higher-order controller should be applied to adequately tune the non-dominant characteristic ratios 1, ...,r nD D .

3. For the higher-order controllers and processes with dominant lag dynamics, setting the non-dominant characteristic ratios (e.g. 4 5, ,...D D ) to 0.5 may result in a response that is too slow.

In that case, it is worthwhile to investigate the possibility of increasing some of these characteristic ratios above 0.5.

4. The response damping is primarily influenced by the most dominant characteristic ratio 2D .

Reducing this ratio to approximately 0.35 gives the fastest (boundary) aperiodic step response (no overshoot). On the other hand, if 2D is increased above 0.5 the response damping

decreases. These features are illustrated in Figure 14b.

5. For the purpose of design of slower superimposed controller, the inner closed-loop system, which is tuned according to the damping optimum criterion, may be approximately described by the equivalent transfer function

eqe

e

( )1

KG s

T s

. (9)

where eqK is the inner control loop gain.

Figure 14. Step response of system tuned according to damping optimum (a) and illustration of damping adjustment by varying dominant characteristic ratio 2D (b).

4.3. Armature Current Control Figure 15 shows the block diagram of the innermost, current control loop from Figure 12, which contains the DC motor model from Figure 3b, the PI controller (Fig. 9a), and the first-order lag terms describing the chopper and sensor dynamics (Section 2). The armature current control loop in Figure 15 is “interconnected” with the internal process loop passing through the back electromotive force ve k . However, in most cases the drive rotational

dynamics are much slower than the dynamics of the inner current control loop. Thus, the back electromotive force e may be treated as a slow (quasi-steady-state) disturbance which is compensated for by the controller integral action. If this is not the case (e.g. in low inertia/low power DC drives), the disturbance action of the fast electromotive feedback path can be compensated for by means of an EMF compensator based on the motor speed measurement m

(see dash-dotted path in Fig. 15).

Figure 15. Block diagram of armature current control loop. The controller integral time constant ciT is equated with the dominant process time constant aT

(zero-pole canceling approach), in order to substantially speed up the current response with respect to reference variable aRi .

ci aT T (10)

It should be noted that the pole a1/ T is not cancelled in the transfer function with respect to

disturbance variable e , so that the disturbance compensation is rather weak/suboptimal. However, this is not an issue for the particular control loop, because the disturbance is characterized by the mild/quasi-stationary behavior or it is already directly compensated for as explained above. In order to further simplify the design procedure, the small (“parasitical”) time constants Tch and Ti are lumped into the equivalent parasitic time constant of the current control loop:

i chiT T T (11)

This open-loop equivalent time constant should be increased by si / 2T ( siT = sampling time) if the

current controller is implemented in digital form. Taking into account Eq. (10) and the approximation

(11), the closed-loop current control system in Figure 15 is described by the following transfer function:

amci

2ci ciaR

ci ch i a ci ch i a

( ) 1( )

( ) 1i

i sG s

T T Ti s s sK K K K K K K K

. (12)

The steady-state accuracy ( ci 0 1G ) is achieved by the controller integral action. The optimal

damping is obtained by equating the coefficients of the characteristic polynomial in the transfer function (12) and the damping optimum characteristic polynomial (8) of the second order ( 2n ). After rearranging and inserting the optimal value of characteristic ratio 2i 0.5D , this yields:

ei2i

2ii

TT T

D

, (13)

ci ci 2 cici

ei ch i a ch i a ch i a

1

2i

i i

T T D TK

T K K K T K K K T K K K

(14)

According to Eq. (13), the equivalent closed-loop time constant eiT is two times larger than the

equivalent "parasitic" open-loop time constant iT . Since iT is typically around 1 ms (for transistor

choppers and fast Hall-effect sensors; and also for fast-sampling digital controllers), the equivalent time constant eiT equals approximately to 2 ms, thus giving the response settling time of ei2 4msT .

Such a fast response for a relatively slow armature system ( a eiT T ) can only be achieved by a large

control effort, i.e. high transient peaks of the armature voltage reference aRu . Fortunately, the

electrical motors allow for large control efforts, because the steady-state armature voltage is typically much smaller than the rated voltage (the armature gain a a1/K R Ka = 1/Ra is relatively large).

The simulation results shown in Figure 16 (see Appendix for the control system parameter values) illustrate the above current response features related to short settling time, good damping, and large control effort.

Figure 16. Simulation response of armature current control loop.

4.4. Speed Control By approximating the inner current control loop transfer function a aR/i s i s by the equivalent lag

term with the time constant eiT and the steady-state gain a am i/ 1/I I K (see Section 4.2 and cf. Eq.

(12)), the speed control loop from Figure 12 can be described by the block diagram shown in Figure 17. Although the process includes an integral term1/ Js , the controller needs to include an additional integrator in order to achieve steady-state accuracy in the presence of (load torque) disturbance variable acting upstream of the process integrator. The controller proportional action is used to stabilize the double-integrator open-loop system. The prefilter pole cancels the controller zero, thus avoiding a large step response overshoot (Section 4.2).

Figure 17. Block diagram of speed control loop.

The “parasitic” time constants of the inner current loop ( eiT ) and the speed sensor (T ) are lumped

into the equivalent first-order lag term open-loop time constant

eiT T T . (15)

Using this approximation, the closed-loop system transfer function is given by

mc

3 2c i c iRc

c t c t

( ) 1( )

( ) 1

sG s

T JT K T JKs s s T sK K K K K K

. (16)

Equating the characteristic polynomial of Eq. (16) with the damping optimum characteristic polynomial given by Eq. (8) for 3n , and rearranging yields the following expressions for controller parameters:

c e2 3

4T

T T TD D

, (17)

3 i ic

t t2

D JK KJK

T K K T K K

. (18)

This result is also known under the name “symmetrical optimum”, which has its frequency-domain interpretation (i.e. it maximizes the phase margin of the open-loop system), and may be considered as a distinctive design method and a special case of the damping optimum.

Eq. (17) suggests that in the presence of double integrator in the open loop (Fig. 17), the equivalent closed-loop time constant eT must be four times larger than the equivalent open-loop time

constant in order to obtain the well-damped closed-loop response. For the realistic example

ei 2T ms and 1T ms (e.g. incremental encoder sensor with the sampling time of 2 ms), the

design yields 3wT ms and e 12T ms, which gives the speed response settling time of

approximately 25 ms. The step response features are illustrated by the simulation results shown in Figure 18a. The same figure also shows the responses obtained in the case of absent prefilter. In that case the control effort is higher and the speed response rise time is shorter, but this is paid for by the speed response overshoot of 45% which is unacceptable in many applications. The filter is often omitted in tracking systems, in order to reach the steady-state accuracy of ramp following (see Section 5). Figure 18a also shows that the response (with or without prefilter) with respect to load torque step is well-damped and has the same settling time as the reference response when prefilter is used. In a majority of electrical drives, it is unrealistic to expect that the settling time could be only 25 ms in the case of large step change of speed reference (e.g. from zero speed to rated speed). This is because the required large control effort would not be feasible due to the armature current limit (see the limiter placed at the speed controller output in Fig. 12). Immediately after the large-magnitude reference step, the speed controller enters the limit (large-signal operating mode, Fig. 18b). This effectively opens the speed control loop and the drive behaves in accordance with Eq. (1c). Assuming that the motor torque limit is much larger than the load torque peak, mmax lmaxM m ,

Eq. (1c) is simplified, the speed response takes on a ramp form, and the large-signal operating mode response time s et T is given by

Rs

mmax

| |Jt

M

. (19)

Obviously, the response time st is shorter for smaller values of the moment of inertia J and a larger

value of the maximum motor torque mmaxM . Hence, in servodrives it is of interest to utilize motors

with small inertia (e.g. motors with light and slender permanent-magnet rotors, Section 5) and large torque overload ability (AC motors).

Figure 18. Simulation responses of speed control system in small- (a) and large-signal operating mode (b). 4.5. Position Control 4.5.1. Small-Signal Operating Mode The outermost position control loop in Figure 12 can be represented by the simplified linear system shown in Figure 19, where the inner speed control loop is approximated by the equivalent lag term. Since the process model includes an integral term and does not include a disturbance variable, the integral term is omitted from the controller in order to avoid unnecessary response slowdown. For the negligible position sensor delay ( 0T ), the equivalent open-loop time constant is equal to:

eT T . (20)

Based on the above position control loop block diagram, the closed-loop transfer function reads:

mc

2R

c c

( ) 1( )

( ) 1

sG s

T K Ks s sK K K K

. (21)

Applying the damping optimum criterion, based on Eq. (8) with 2n , and rearranging yields

e2

TT

D

, (22)

2c

D KK

T K

. (23)

Figure 19. Block diagram of position control loop. For the optimal setting 2 0.5D and the realistic numerical example from the previous subsection

( e 12msT ), it follows that e 24msT , which gives the step response settling time around 50 ms

for the small-signal operating mode. However, it is usually requested that a pointing system has an aperiodic response, in order to avoid damaging the drive mechanism while overshooting mechanical limiters. The fastest aperiodic tuning is obtained by the following setting (Section 4.2):

2 0.35D . (24) However, according to Eq. (22) the response becomes somewhat slower in this case. The response features of the position control loop in the small-signal operating mode are illustrated by the simulation results shown in Figure 20a.

Figure 20. Simulation responses of position control loop in small- (a) and large-signal operating mode (b) ( l 0m ).

4.5.2. Large-Signal Operating Mode The proportional position controller, with the gain cK , can be represented by the straight line in

the R e coordinate system, as shown in Figure 21 ( R me ). The controller saturation

effect (speed reference limit Rmax ) is represented by the limit-speed horizontal line in the same

coordinate system. If the controller is saturated during acceleration and deceleration portion of the response (large-signal operating mode), the pointing response would simply be slower than in the small-signal operating mode (similar effect as explained with speed control in Section 4.4). However, in some cases the response may further deteriorate due to the fact that the saturated motor braking torque may be insufficient to stop the motor, controlled by the proportional controller , to the target point 0e (origin of the coordinate system in Fig. 21) without an overshoot.

By assuming that the maximum motor torque is much larger than the load torque ( mmax lmaxM m , the maximum angular deceleration in the final (braking) stage of the pointing

response is defined as

max m max /M J (25)

Therefore, the maximum speed value, which the drive should posses at the remaining distance

/e K without the risk of position overshoot, is given by max2 /e K , i.e. the speed

reference should be limited to

R lim max2K

eK

(26)

Figure 21. Construction of nonlinear position controller static curve. If this square-root curve intercepts the proportional controller straight line below the speed limit line

R R,max (Fig. 21), this means that the proportional controller requests unfeasible speed

references at higher position errors, thus resulting in a dangerous, relatively large, position response overshoot (Fig. 20b). From Figure 21, the condition for the safe linear controller-only operation reads:

2 2

R max max m maxR lim R max c

c max R max

2 2arg ( )

a R a

K K Me K

K K K J

. (27)

Obviously, the linear controller is less likely to be adequate for drives with higher inertia, lower torque limit, and higher speed limit. On the other hand, the square-root static curve itself cannot be used for pointing, because its gradient (gain) at the target position (i.e. at 0e ) tends to infinity,

which would result in an unstable control system response in the small-signal operating mode. The problem may obviously be resolved by combining the limited proportional controller with the square-root curve, so that the resulting minimum-level curve is used as a nonlinear position controller. However, the discontinuous transition from the nonlinear to linear mode in Point C (Fig. 21) requires a jump of the requested braking torque deeper in the saturation region, thus again resulting in the pointing overshoot. In order to provide a smooth transition between the two modes, the square-root curve may be shifted horizontally to right, so that the linear controller curve becomes a tangent to the shifted curve (Point A in Fig. 21 with A 2e e ). The resulting

controller is shown in Figure 21 by bold line, where it can be readily shown that the characteristic, control mode switching points are defined by the following expressions:

RAA

c

eK

(28)

2

A RBB

RA

12

ee

(29)

where 2 1 1RA c maxK K K and RB Rmax . For the sake of simplicity of implementation, the

nonlinear controller may be approximated by a piecewise-linear controller, shown by the dashed bold line in Figure 21. It should be noted that the controller static curve has the same shape but negative sign for the reverse motion ( 0e , 3rd quadrant in Fig. 21).

Figure 20b illustrates the effectiveness of the piecewise-linear implementation of nonlinear position controller in terms of avoiding the overshoot, and having actually a smaller settling time compared to the case of linear controller in the large signal operating mode. Note that the response is characterized by an idealized (time-optimal) triangular speed profile and a quasi-rectangular (“bang-bang”) torque profile, where the first part of the response corresponds to the driving mode, while the second part corresponds to the regenerative braking mode (cf. Section 3). It should be mentioned that, if the load torque has a passive characteristic (see Fig. 11), neglecting the load torque in Eq. (25) represents a safe approximation in terms of a conservative estimate of the total braking torque potential and, consequently, avoiding the overshoot. Still, one may correct (boost) max , given by Eq. (25), by an average load torque contribution, in order to avoid an

increase of the response time in the case of a non-negligible load torque. In the case of an active load torque (e.g. the gravitational torque in Fig. 11), the approximation (25) is unsafe and it needs to be corrected (reduced) by the load torque contribution in the case lsgn sgn( )m .

5. Design of Tracking System This section deals with the tracking problem applied to the position control system. The position reference trajectory can be known in advance (pre-planned), such as in machine tools, CNC manufacturing stations, and industrial robots. On the other hand, a-priori unknown position reference is typically encountered in drives that are remotely controlled by means of a human

operator (e.g. x-by-wire systems in vehicles), or when the position trajectory is generated as a function of target position measurement (i.e. moving target tracking applications). 5.1. Tracking of A-Priori Known Reference In order to accurately track gradual changes of the a-priori known position reference, the conventional cascade structure of feedback control system (Fig. 12) can be extended with additional feedforward reference paths to the fast inner speed and current control loops (Fig. 22). The main task of feedback loops is to reject the disturbances (e.g. EMF, load torque, and friction), while the feedforward control (FFC) paths provide the direct/immediate reference signals to the fast inner control loops. Hence, the combined feedforward/feedback system should result in fast and accurate tracking and good disturbance rejection.

Figure 22. Block diagram of tracking system based on direct feedforward paths from a-priori known reference trajectory.

The feedforward reference signals RFF and RFFi can be calculated in an exact, off-line manner

based on the following expressions:

RRFF

K d

K dt

(30)

2

i RFF i RaRFF 2

m m

K d K dJ Ji

K K dt K K dt

(31)

Note that due to the off-line calculation, the position reference differentiation-related noise issues are effectively avoided. However, the pre-planned position reference should be smooth, so that it has continuous first and second time derivatives. The simulation results shown in Figure 23a illustrate the effectiveness of the feedforward control on an example of sinusoidal pulse reference tracking. The purely feedback system with the prefilter included into the speed reference path cannot provide accurate tracking. By omitting the prefilter, the tracking ability of the position control system is improved due to the faster speed response.

However, a substantial improvement of the tracking accuracy can only be achieved by using the feedforward control.

Figure 23. Simulation response of position tracking systems with a-priori known reference (a) and a-priori unknown reference (b).

5.2. Tracking of A-Priori Unknown Reference The feedforward control structure from Figure 22 can be extended by a state-variable filter, in order to provide an on-line reconstruction of the feedforward references RFF and aRFFi from a-

priori unknown position reference R . A state-variable filter is convenient for this task, because it

can provide a good trade-off of reconstruction accuracy and noise suppression, as well as a straightforward implementation. For the particular case of a second-order feedforward compensator and the damping optimum tuning, the state variable filter structure is shown in Figure 24a. The characteristic ratio is set to the optimal value ( 2f 0.5D ; corresponds to the case of Butterworth

filter), while the tuning of filter equivalent time constant efT is a trade-off between the tracking

accuracy and the noise suppression ability. It can be shown that the feedforward paths in Figure 22 introduce zeros in the overall transfer function between the position reference R and the actual position . The derivative action of the

feedforward zeros and the related increase of closed-loop bandwidth represents another explanation of the effect of improved tracking accuracy. Similar effect may be achieved by implementing a compact feedforward compensator placed in the position reference path (Fig. 24b). Since the feedforward paths are absent, this solution can conveniently be used as an external intervention that does not need an access to the internal control loops.

Figure 24. State variable filter (a) and feedforward compensator (b) used in position tracking systems with a-priori unknown reference.

The denominator coefficients of the feedforward compensator in Figure 24b can be tuned according to the damping optimum, where the filtering time constant efT is chosen in the range from e0.1T

to e0.3T , where eT is given by Eq. (22). The numerator coefficients can conveniently be tuned

according to the magnitude optimum, which provides a maximum closed-loop bandwidth without resonant peaks. Equation for the first-order and second-order compensator parameters are given in Table 2, where 2D is given by Eq. (24), while 3 2D D and 4 3D D represent the less-

dominant characteristic ratios of the closed-loop position control system, which correspond to the inner speed control loop characteristic ratios 2 3 0.5D D .

Figure 23b shows that by applying the state variable filter and feedforward compensator gives a similar level of (improved) tracking performance. However, their performance is still less effective compared to the direct FFC from Figure 22 (cf. Figs. 23b and 23a), because of the use of filtering action needed to provide on-line implementation of feedforward compensator and noise suppression.

Table 2. Expressions for first-order and second-order feedforward compensator parameters.

Feedforward compensator:

First-order Second-order

b2 - 2 22 e 3 4 3 21 2 2D T D D D D

b1 e 21 2T D 2

e 2 3 4 3 21 2 1 1 2 2T D D D D D

6. Control of Permanent-Magnet Synchronous Motor The permanent-magnet synchronous motor (PMSM) has a conventional three-phase winding stator (armature) and a permanent-magnet excitation placed on the rotor (Fig. 25a). The motor has a substantially higher power and torque density and a lower moment of inertia than the DC motor. Since this is a brush-less machine, it is essentially free of maintenance.

Figure 25. Cross-section of PMSM (a) and equivalent scheme with coordinate frame definition (b). 6.1. Modeling of Motor The basic motor model, which is usually established in the stator coordinate frame (frame a-b in Fig. 25b), is highly nonlinear and inconvenient for the control system design. To avoid the strong nonlinearities, the model is transformed into the moving d-q coordinate frame by using the Park transform, where the d-axis relates to the rotor field direction and the q-axis is perpendicular to it. Under the assumption of exterior permanent magnets (Fig. 25a), the d-q motor model with the parameters denoted in Figure 25b reads:

dd d q

diu Ri L p Li

dt , (32a)

qq q d r

diu Ri L p Li p

dt , (32b)

m l r q l3

2

dJ m m p i m

dt

, (32c)

where p is the number of pole pairs. Noting that the terms rp and r1.5p correspond to the voltage constant vK and the torque

constant tK , it is evident that the structure of the model (32) is very similar to the DC model

structure (1). The main difference is that instead of real armature and excitation currents ai and Mi

(and voltages au and Mu ), the model (32) includes the "virtual" currents di and qi (and voltages du

and qu ). The relations between the actual and "virtual" variables are given by the following direct-

and inverse-Park transform equations, respectively:

1d e e

2q e e

3

1 0 0cos sin

sin cos 0 3 / 3 3 / 3

ii

ii

i

(33)

1de e

2qe e

3

1 0cos sin

1/ 2 3 / 2sin cos

1/ 2 3 / 2

uu

uu

u

(34)

where e mp is the rotor “electrical” angular position. Another difference between the two

models is that the PMSM model (32) includes additional nonlinear cross-coupling terms

d qu p Li and q du p Li . However, these measurable additive nonlinear terms are easily

compensated for by appropriate feedforward actions (Section 6.2). The overall motor model is shown in the lower portion of the block diagram in Figure 26. 6.2. Control Since the PMSM model in the d-q coordinate frame (see the lower portion of Figure 26 and note that a 1/K R and a /T L R ) is very similar to the DC motor model (Fig. 3a), the PMSM can be

controlled using the same principle of decoupled field and torque control, as applied for the DC motor (Section 4). This principle is usually called field-oriented control or vector control. The PMSM field-oriented control system is shown in Figure 26, where the upper part of the figure relates to the controller itself. The field and torque currents di and qi , respectively, are calculated

by using the Park transform (33). It should be noted here that it is enough to measure two motor currents (e.g. 1i and 2i ), because they define the third current ( 3 2 3i i i ) for the common case of

no neutral line connected. The currents di and qi are controlled by PI controllers, which are tuned

in the same way as explained for the DC motor (Section 4). To provide this possibility, the aforementioned nonlinear cross-coupling terms du and qu are compensated for based on the

available speed and current measurements (feedforward d-q axis decoupling). The current controller outputs, i.e. the voltage references dRu and qRu , are used to calculate the phase voltage references

1R 2R 3R, , and u u u using the inverse Park transform (34). These voltage references are then used as

inputs to the PWM-control unit of the voltage-source power converter (Section 2). For the motor operation in the basic region below the rated speed, the motor flux d r dLi

should be kept at the rated value (see Sections 3 and 4). Since the rated flux equals r (permanent-

magnet flux), the field current di should be kept at zero. Another explanation for this approach is

that, with d 0i , the motor has minimum (steady-state) current 2 2 1/2d q( )I I I for the given torque

m t qM K I . Thus, for the basic operating region, the current references are simply calculated as

R 0di and qR mR ti m K , and the motor torque limit mmaxm is set to the maximum motor torque.

The motor torque reference mRm is generated by the PI speed controller, which is designed

(together with the position controller, as needed) in the same way as for DC drives (Section 4). In the operating region above the rated speed, the field d r dLi needs to be weakened below

the rated and constant permanent-magnet field r (Section 3). That can only be achieved by

imposing a relatively large negative field current ( d 0i ). This represents a disadvantage of the

PMSM motor, because the overall current and related winding resistive losses (copper losses) would significantly increase for the same torque when compared to the DC and induction motors. The field-current reference dRi can generally be generated from the reference torque mRm in two

ways: (i) model-based open-loop generation, and (ii) closed-loop generation based on the goal to keep the motor voltage constant in the field-weakening region. The torque-current reference Rqi is

again calculated from the reference motor torque mRm , but taking into account the permissible

current capacity of the power converter ( 2 2 1/2d q max ( )I I I I ).

The PMSM has comparable small-signal operating mode control features as the DC drives (Section 4). The large-signal operating mode performance is better due to smaller inertia and higher maximum motor current/torque (up to five times larger than the rated current/torque). The induction motor is cheaper than the PMSM (no need for expensive rare-earth permanent magnets). However, it is weightier and has lower power and torque density. The control principle is again based on field-orientation approach. However, the field-oriented control system is more complex and somewhat less effective when compared with the PMSM control, because the variable rotor flux value and orientation should be estimated on-line using a relatively uncertain, time-varying motor model.

Figure 26. Block diagram of field-oriented control of permanent-magnet synchronous motor (PMSM).

7. Compensation of Transmission Compliance, Friction, and Backlash Effects The effects of transmission compliance, friction, and backlash, which were neglected in the previous sections, can significantly affect the static and dynamic accuracy of servodrive control. Since mechanical hardware design methods for avoiding or mitigating these effects may be expensive and/or cumbersome, it is of interest to consider their compensation by means of proper extensions of basic control algorithms. 7.1. Model of Two-Mass Elastic System with Friction and Backlash The mechanical subsystem of an electrical drive with compliant transmission, friction, and backlash can be represented by a two-mass elastic system model, whose block diagram is shown in Figure 27 (see Nomenclature). The transmission torque m depends on the torsional angle , and it is transferred only when the transmission is outside the backlash zone ( b ). The friction is

usually dominant on the load side ( fl fmm m ). Its static characteristic (Fig. 28a) is highly

nonlinear, particularly in the zero-speed (stiction) region, which makes the position and low-speed control more difficult.

Figure 27. Block diagram of two-mass elastic system model including friction and backlash effects.

Figure 28. Static friction curve (a) and block diagram of Karnopp friction model (b).

The stiction static curve in Figure 28a is not uniquely defined, i.e. zero-speed friction can assume any value between positive and negative maximum static friction SM . This problem may be

overcome by approximating the stiction curve by a steep straight line crossing the origin. However, this simple modification suffers from numerical/simulation inefficiencies due to the stiff stiction approximation. Numerically efficient and physically justified friction model has been proposed by Karnopp (Fig. 28b). This model defines a narrow, zero-speed region, where the friction torque (stiction) is equated with the applied torque app m lm m m saturated to the maximum static friction

SM . Outside the zero-speed region, the friction torque is calculated according to the generalized

Stribeck model shown in Figure 28a. Further improvements in friction modeling relate to capturing the stiction dynamics effects. In particular, the presliding motion effect may be of relevance for various drives (e.g. for transmissions with emphasized rolling bearing friction). Instead of being stuck under the stiction regime, the friction element shows some presliding motion explained by asperity contact compliance. More about dynamic friction effects and models can be found in the comprehensive survey paper by Armstrong-Hélouvry, Dupont & Canudas de Wit (1994). The two-mass elastic system, shown in Figure 27, is in the linear operating mode when b and

l fl C| | m M . The system linear model can be transformed into the input-output form

shown in Figure 29, where the natural frequency and damping ratio parameters are given by

0 0l0 0l 0 l

m l l

1 1, , ,

2 2

d c dc

J J c J c

. (35)

It should be noted that the system is usually weakly damped, i.e. l 1 . The two-mass elastic

system behavior is characterized by two dimensionless parameters: the inertia ratio M l m/r J J and

the frequency ratio 1EM 0 ei/r T , where 1

eiT [rad/s] is the current (torque) control loop

bandwidth (Section 4). Table 3 defines the characteristic cases related to values of these ratios.

Figure 29. Block diagram of input/output model of linear two-mass elastic system.

Table 3. Characteristic dimensionless parameters of two-mass elastic system.

Parameter value Inertia ratio rM = Jl/Jm Frequency ratio rEM = 0/Tei-1

<< 1 Light load Soft coupling

1 Balanced load Median coupling

>> 1 Heavy load Stiff coupling

7.2. Compliance Compensation Obviously, the mechanical compliance cannot affect the current control loop behavior. Therefore, when analyzing and compensating for the compliance effect, the main emphasis is on the speed control loop. The bandwidth of speed controller tuned according to the damping/symmetrical optimum is determined by the current loop equivalent time constant eiT (see Eqs. (17) and (15)).

Clearly, if the mechanical resonance frequency 0 is low, or, more precisely, if the

electromechanical ratio EM 0 eir T is small (soft coupling), the compliance effect will affect the

speed control loop in terms of occurrence of weakly-damped torsional vibrations with the frequency 2

0 0(1 ) (see simulation results in Fig. 30a). A similar effect is present in the case of

median coupling ( EM 1r , e.g. from 0.5 to 2) and light or heavy load. In this regard, it is convenient

to design the mechanical part of the drive for a balanced load ( EM 1r ). Finally, in the case of stiff

coupling ( EM 1r ), the compliance does not have a notable influence, except in appearance of

high-frequency and low-amplitude torsional vibrations. The above results are valid for the "default" case of motor speed feedback. However, if the load speed sensor is used to establish the feedback, the system tuned according to the symmetrical optimum will become unstable for soft and median coupling. This is because the transfer function with respect to load speed l (unlike that with respect to m , Fig. 29) does not include the

conjugate-complex zeros at the frequency 0l , which have the stabilizing effect over the resonance

poles at the frequency 0 .

It should also be mentioned that the rapid development of power electronics and also sensors (see Section 2) has historically resulted in reduction of the equivalent time constant eiT , thus

emphasizing the effect of compliant transmission ( EMr was decreasing). In other words, the

transmission compliance has become a stronger obstacle in obtaining an extended speed/position control loop bandwidth based on the increasingly faster current control loop. In order to mitigate the compliance effect, the PI controller (with motor speed feedback) should be tuned by taking into account the full mechanical system model in Figure 29. Applying the straightforward damping optimum design method, described in Section 4, results in the PI controller parameter expressions given in Table 4 for the case of soft coupling (the more general sets of parameter equations for other characteristic EMr cases and the case on non-negligible damping

ratios l, 0 can be found in (Deur, Koledić & Perić, 1998). Here (and throughout Section 7), for

the sake of simplicity, the sensors gains ( i ,K K , etc.) are assumed to have unit values. It is

important to note that the equivalent time constant eT is not anymore linked to the "electrical"

time constant eiT , but rather to the inverse of mechanical parameter 0l. In other words, since

0l 0 ei1/ T , the speed control loop should be significantly slowed down (larger eT ) when

compared to the symmetrical optimum tuning. This is particularly emphasized in the case of heavy load ( M 1r ), because the natural frequency 0l decreases with the increase of l M mJ r J . Unlike

the cases of heavy and balanced load, where the slowed-down response can be damped by the properly tuned PI controller, the response is still undamped in the case of light load ( M 1r ). This

is because it is difficult to control the light load, vibrating on the elastic transmission coupled to a heavy motor, by means of motor speed feedback only. The above results are illustrated by the simulation responses shown in Figure 30b.

Figure 30. Comparative simulation step responses for two-mass system with "soft" transmission

( EM 0.25r ) and different controllers and inertia ratios M l m/r J J .

The torsional vibration problem for light load can be resolved by using a state controller. The reduced-order controller (Fig. 31a) adds the feedback path of the transmission torque m to the PI controller. The advantage of this controller is that the transmission torque can readily be estimated,

based on the known motor model structure and parameters. The generally more effective, full-order controller in Figure 31b can conveniently be used when the position sensors are installed on both motor and load side. This gives the information of motor speed m , load speed l , and torsional

angle m l , which are used for establishing a full-state feedback path. The damping

optimum design results for the state-controller parameters are included in Table 4, again for the case of soft coupling. Figure 30 illustrates that the state controllers, in particular that with the full state feedback, provide a favorable system behavior for a wide range of inertia ratio Mr values.

Figure 31. Block diagram of reduced-order (a) and full-order state controller (b). Table 4. Damping optimum results of PI and state controller design for two-mass system with soft

coupling ( EM 1r ) and negligible damping ratio values ( l 0 ).

PI controller Reduced-order state controller

TI = Te 2 3 0l

1

D D

2 3 0l

1

D D

Km 2

I m l 0l2 2

2 I 0l

( )

1

T J J

D T

m l2 3 3 2

4 3 2 I 0

J J

D D D T

Kmm - 2 2

2 I 0lm2 3 4 2 2

l 4 3 2 I 0 0l

11 1

D TJ

J D D D T

Full-order state controller

TI = Te 2345 DDDD

T

Km 20l

m l 00 4 3 2 I 0

1( )J J T

D D D T

K 2 2

2 I 0l m l m l 0l( ) ( )D T K K J J

Kl m l

m2 3 3 24 3 2 I 0

J JK

D D D T

7.3. Friction Compensation For the sake of simplicity, the analysis and compensation of friction effect is considered for the single-mass system (no transmission compliance included). Namely, the friction model in Figure 28 is included into the stiff-transmission drive model in Figure 3b. The friction effects are illustrated on the example of a pointing system. Similar effects can occur in the speed-controlled drive in the zero-speed region, and in the tracking system. Zero-speed friction or stiction causes the drive to be stuck ( 0 ) until the controller integral action builds up enough torque to overcome the maximum static friction SM (Fig. 32a – dashed line). Clearly, this standstill

effect makes the response slower, and it is more emphasized for lower reference position steps (lower speed magnitudes). In a tracking system, with a low rate of position reference change, the standstill effect would be manifested in a large tracking error. Finally, in the case of emphasized Stribeck effect of a falling static curve at low speeds (Fig. 28a), the positioning response can exhibit a relatively large overshoot at small position reference steps (Fig. 32b). The overshoot is caused by the friction drop during the transition from static to kinetic friction (breakaway friction drop

S C–M M ), which abruptly unloads the motor and provokes a sudden increase of speed. The

subsequent breakaway transitions with speed overshoots result in the so-called stick-slip motion (Fig. 32b), which is undesirable from the standpoint of vibrations, wear, and motor losses.

Figure 32. Illustration of friction effects and their compensation in position-controlled drive with (a) Coulomb friction ( S CM M ) and (b) static+Coulomb friction ( S CM M ).

Figure 33. Friction compensation and stabilization algorithm. The above friction effects can be effectively compensated for by extending the speed controller by a nonlinear friction compensation algorithm (Brandenburg & Schäfer, 1991). The upper part of Figure 33 shows such an algorithm, which reflects the structure of Karnopp friction model (cf. Fig. 28b). If the drive is stuck ( m ) and the drive motion is commanded ( R ), static

friction is compensated for by adding the scaled static friction torque fc S Rsgn( )m M to the

current command aRi . Otherwise, the drive is supposed to be in motion, or it is stuck with zero

speed command. In that case, the friction torque compensation signal is equal to kinetic (Coulomb) friction torque and its sign depends on the actual speed sign ( fc Csgn( )m M ). The zero-speed

hysteresis is introduced to avoid chattering of compensation signal under the stuck condition. Note that during the transition from static to kinetic friction compensating mode, the compensation signal abruptly reduces from SM to CM , thus compensating for the friction drop effect and related

position control overshoot. In order to avoid the stick-slip motion, the overall speed controller is extended by the stabilization action (lower part of Fig. 33). This action sets the integral term input to zero when the position control error is below a threshold for which the position response overshoot ceases. This holds the controller integral action, which caused the stick-slip motion. The simulation results in Figure 34 illustrate the effectiveness of the presented friction compensation method. The standstill interval and the response overshoot are reduced, and the stick-slip motion is avoided.

If the presliding displacement effect is emphasized, the simple and robust compensation algorithm shown in Figure 34 may be more appropriate. Depending on the sign of position control error signal e, the friction compensation signal assumes the value of SM . This signal is filtered by a first-

order lag term to account for the compliant/presliding friction dynamics and to avoid chattering around zero control error e. The lag-term time constant can conveniently be made time-variant, where larger values are used at lower control errors. In the presence of emphasized presliding displacement effect, there is no need for stabilization action. This is because the motor position will creep to the reference value during the presliding motion, without causing an overshoot and stick-slip motion. The experimental results shown in Figure 35 illustrate the benefits of applying the friction compensation algorithm on the example of automotive electronic throttle control. Regardless of position reference step magnitude, the compensated system response time is always similar (around 70 ms). This points out that the compensator can force a highly-nonlinear drive system to behave in a desired linear-like manner.

Figure 34. Robust friction compensation algorithm based on position control error signal input.

Figure 35. Experimental results of friction compensation in electronic throttle positioning system.

7.4. Backlash Compensation The transmission backlash, in conjunction with compliance effects, can notably limit the servodrive control performance, especially in position control applications. This is due to the discontinuous nature of transmitted torque from the motor to the load side, caused by the backlash dead-zone nonlinearity () in Figure 27. Figure 36 illustrates that the pointing servosystem with backlash is prone to the so-called limit cycle oscillations, even though the inner state-variable speed controller and the outer proportional position controller are tuned for a desirable aperiodic response. The limit cycle oscillations are avoided when the elastic transmission is preloaded, as illustrated in Figure 36 for the case of load torque engagement. In this case the controller needs to increase the torque (current) command in order to match the load torque and maintain the load position. Consequently, the system is outside the backlash zone (of the width 2b), and it largely behaves as a linear system (one without backlash). A servosystem with Coulomb-type load-side friction is also less prone to limit cycle oscillations, since friction acts as a passive load torque. It should be noted that the “true” and most effective transmission preloading measures are hardware oriented, which may increase the servosystem cost. Hence, different control measures for servosystems with backlash have been proposed in the literature.

Figure 36: Comparative simulation responses of pointing servosystems with soft coupling ( EM 0.25r ), and balanced load ( M 1r ).

The limit cycle oscillations can be avoided if controller integral action is turned off (the so-called stabilization intervention). Since this may result in a notable steady-state pointing error, active backlash compensation measures should be considered as well. Depending on the availability of the load position (speed) measurement, two distinctive approaches have been considered in (Schäfer, 1992):

In the case when only motor-side measurements are available, the linear position controller is supplemented with a nonlinear observer (estimator), which detects whether the servosystem has entered the backlash zone. If backlash operation tends to occur, appropriate relay logic augments the motor position reference in order to achieve alignment with the backlash deadzone boundary and, thus, to avoid the operation inside the backlash zone.

When both the motor-side and load side measurements are available, the position control system is based on a dual controller structure. Here a linear controller is active outside the backlash

zone, while in the case of operation within the nonlinear backlash region the control mode is switched to a dedicated nonlinear controller which facilitates smooth transition from the backlash region to the desired linear operating regime.

An alternative to the above switching-type backlash compensation strategies would be to utilize a more systematic nonlinear control approach, such as the exact input-output linearization of the servosystem with backlash (see e.g. Schöling & Orlik, 1999). The linearizing effect is achieved by means of an appropriate control action, as shown in Figure 37a, where the dynamic term R x is a

nonlinear feedback controller which linearizes the control system, while the nonlinear term M x represents a variable gain in the path of the position reference signal R (a reference scaling

factor).

Figure 37. Block diagram of servosystem with nonlinear controller (a), and smooth approximation of dead-zone static curve (b).

The nonlinear control law, which linearizes the servosystem with backlash, is derived based on the simplified representation of the process model in Figure 27, which assumes immediate inner control action ( ei 0T ), negligible torsional damping ( 0d ) and negligible load-side friction ( fl 0m mfl =

0). Exact linearization defines the nonlinear control law which results in closed-loop dynamic behavior equivalent to the linear control system (without backlash), represented by the following transfer function characteristic polynomial (e.g. a damping optimum characteristic polynomial):

* 1 * *1 1 0( ) n n

nA s s a s a s a . (36)

The final nonlinear control law reads:

*0 m l

* * **0 m l 1 m l 2 m

2 2 3 m 1 2

2

22m l

m 1 2l

( ) ,( )

( ) ( ) ( )( ) ( ) ( )

( )( )

( ) .( )

a J J

c

a J J a J J a Ja J

c c

J Jc J

J

M x

R x (37)

In order to implement the above control law, the discontinuous dead-zone nonlinearity ( ) is replaced in the above expressions with the following smooth approximation (see Figure 37b):

bb

( ) tanh

. (38)

The comparative responses in Figure 36 indicate that the nonlinear control strategy based on exact linearization results in a reasonable pointing accuracy (pointing error is approximately half the backlash dead-zone width), while avoiding limit cycle oscillations. However, its application is characterized by a notable control effort (abrupt changes of armature current reference aRi ), which

is required in order to achieve quick transition of the servosystem through the backlash zone. This is primarily due to the large magnitudes of the inverse value of the backlash-approximation first

partial derivative 1( ( ) / ) inside the backlash deadzone (Fig. 37b). Since this represents a

singularity in the above control law, the position difference used in the control strategy is held at a relatively small value when its actual value approaches zero. 8. Conclusion For practical reasons (e.g. simplicity of tuning and over-current protection), the electrical drives are usually controlled in the cascade structure, where the outermost force and position controllers are superimposed to the inner speed and current controllers. In the case of low-power/low-cost drives, characterized by a non-excessive short-circuit current, a compact position controller can be used instead, in order to avoid the use of current sensor and to speed up the response. For tracking systems, the feedback-control cascade structure should be extended with feedforward actions that are fed directly to the fast inner loops, thus significantly reducing the trajectory tracking error. The performance is generally better for tracking systems with a-priori known reference trajectory, because the inner loop feedforward references can be calculated in an exact manner and there is no noise amplification issue. In applications, where the access to the inner control loops is not possible, a compact, typically first-order or second-order feedforward compensator can be placed into the position reference path to speed up the feedback system and reduce the tracking error. AC motors, particularly the permanent-magnet synchronous motor (PMSM), have many advantages over the DC motors, such as larger power and torque density, lower mass, larger maximum torque, lower moment of inertia, and they are essentially maintenance free. This is the reason why they have been aggressively displacing the DC motors since the modern power electronics and microelectronics were developed to support the implementation of a more complex nonlinear structure of current control loop. The structure of the superimposed speed, position, and force control loops remains the same regardless of the motor type. As the innermost current control loop was getting faster due to the use of faster and more precise power converters and sensors, the electrical drive transmission insufficiencies such as compliance, friction, and backlash played a bigger role in limiting the performance of control of mechanical variables (speed, position, and force). Since it is sometimes not simple and/or cost effective to remove these insufficiencies by mechanical design measures, many efforts have been devoted to developing “smart” algorithms for compensating these harmful transmission effects. In the case of transmission compliance, the PI speed controller should be retuned to take into account the increased complexity of the mechanical system, or, if possible, extended with additional state feedback paths. In the case of friction and backlash effects, specific model-based or robust nonlinear compensation algorithms are needed.

The damping optimum criterion has been proven to be a powerful analytical method of designing the electrical drive control systems. It can be applied to systems with conventional PID controllers and more advanced reduced-order of full-order state controllers. The feedforward compensator zeros can conveniently be tuned according to another practical analytical optimum criterion - the magnitude optimum. Appendix: Control system parameter values

DC motor:

N 500 WP

N 1500 rpm

aN 3.4 AI

aN 220 VU

a 16.35R

a 0.3 HL 2

m 0.0157 kgmJ J

2l 0.0031, 0.0157, 0.0785 kgmJ

a 0.0612 A / VK

a 18.3 msT

t 0.936 Nm / AK

v 1.047 Vs / radK

18.17, 54.51, 90.86 Nm / radc

0.005 Nms / radd d = 0.005 Nms/rad Friction:

C 0.8 NmM

S 1.6 NmM

Backlash zone width:

b2 2 dega

H-bridge power converter:

ch

ch

K 44 V / V

T 0.25 ms

Sensors:

i

i

a

1.57 V / A

0.75 ms

0.065 Vs / rad

1 ms

1304 pulses / rad

K

T

K

T

K

Basic controllers:

2

ci

ci

2 3

0.5

2.12

18.3 ms

0.5

iD

K

T

D D

c

c

2

3c

67.5

12 ms

0.35

1.45 10

K

T

D

K

Glossary Alternating current (AC): Electrical current which periodically changes its direction.

Armature: The power-producing winding of an electrical machine, which, depending on the machine type, can be situated either on the stator (e.g. synchronous machine), or on the rotor (e.g. DC machine).

Back electromotive force (EMF): Refers to voltage generated in electrical machine armature winding by means of electromagnetic induction.

Backlash: Clearance between transmission components, which may result in the loss of torque transfer.

Bandwidth: Width of the frequency range of a signal which can be transferred by the dynamic system without noticeable loss of signal magnitude.

Cascade control system: A hierarchical structure of control system where a superimposed controller commands the reference (target) value to the inner control loop.

Compliance: Propensity of mechanical component to deform when external force/torque is applied along a given degree of freedom.

Control: Means or ability to manage or adjust a physical quantity to a desired (target) value, either based on a previously known characteristic of the controlled object (open-loop control), or by means of feedback (closed-loop control).

Damping optimum: An analytical method for tuning linear control systems which provides easy and straightforward adjustment of the closed-loop system response damping based on the prototype form of transfer function characteristic polynomial of any order.

Electrical drive: A mechatronic system which provides controlled motion of the working machine (load) by means of an electrical machine.

Electronic power converter: Power electronics device which transforms the grid power system characterized by fixed parameters (voltage, frequency) into the AC or DC motor power system with variable parameters.

Direct current (DC): Electrical current which keeps constant direction.

DC motor: An electrical motor that utilizes direct current (DC) power supply.

Feedback controller: Device or algorithm which adjusts the controlled object (process) input based on the measurement of controlled variable or variables in order to reach a desired behavior.

Feedforward compensator: Device or algorithm which acts upon the controlled object (plant) in an open-loop manner in order to improve the transient behavior of the feedback control system.

Field flux (excitation) circuit: Part of electrical machine that generates the magnetic field flux which interacts with the armature current thereby producing the electrical machine torque. It can be realized by means of excitation windings or permanent magnets.

Field-oriented control (vector control): Method of controlling the torque (current) of AC electric machines similar to naturally decoupled torque and field control of DC motor.

Friction: A dissipative force or torque resisting the relative motion of contacting surfaces.

Laplace transform: A signal transformation from the time (t) domain to the Laplace (complex frequency s) domain.

Load: Torque or force which needs to be overcome by the electrical drive.

Modeling: A procedure for obtaining a precise mathematical model of the controlled object.

Permanent-magnet synchronous motor (PMSM): An AC machine equipped with conventional three-phase winding stator (armature) and permanent-magnet excitation placed on the rotor.

Pointing: Precise control of electrical drive position for a constant position reference (target) value.

Process: A dynamic system representing the object of open-loop or closed-loop control.

Pulse-width modulation (PWM): A method of generating an arbitrary square-wave voltage signal by using switching action with variable duration of time intervals during which the electronic power switches are alternatively turned on and off.

Sensor: A device that measures a physical quantity and converts it into an electrical signal which is suitable for further electronic processing.

Simulation: A computer code which runs a dynamic model of a system (e.g. a closed-loop control system), thus yielding a prediction of the system behavior for a predefined set of parameters, initial conditions and inputs.

State-variable controller: A controller based on the measurement or estimation of some or all process state variables, in contrast to traditional single-loop controller which utilizes a single feedback signal.

State-variable filter: A dynamic model with internal feedback loops which reconstructs the time-derivative state variables of the given input signal.

Tracking: Accurate following of a time-varying reference (target) variable.

Transfer function: A mathematical representation of the relation between the input and output of a linear time-invariant dynamic system in the Laplace (s) domain.

Nomenclature c : Stiffness coefficient of compliant shaft d : Damping coefficient of compliant shaft

2() 3(),D D : Characteristic ratios of damping optimum

e : Induced voltage (back electromotive force, EMF) e : Position control error

a M,i i : Armature and magnetic-excitation current

aR am,i i : Armature current reference and measured armature current

d q,i i : Field and torque current

J : Total drive moment of inertia referred to motor shaft

l m,J J : Load and motor moment of inertia

aK Ka: Armature gain

c(i, , )w aK : Proportional gain of (current, speed, and position) controllers

chK : Steady-state gain of power converter (chopper)

e v,K K : Motor voltage constants

m t,K K : Motor torque constants

a ,L L : Armature inductance

m mR,m m : Motor torque and motor torque reference

lm : Load torque

f lm : Friction torque (at load side)

fcm : Friction compensation signal

S C,M M : Maximum static friction torque and kinetic (Coulomb) friction

MN : Number of turns of excitation winding

a M,u u : Armature and field-circuit voltage

d q,u u : Field and torque voltage

p : Number of pole pairs

EMr : Frequency ratio ( 10 ei/ T )

Mr : Inertia ratio ( l m/J J )

a ,R R : Armature resistance

MR : Field (excitation) winding resistance

s : Operator of Laplace transform t : Time

aT : Armature time constant

c(i, )T : Integral time constant of (current and speed) controllers

chT : Equivalent time constant of power converter (chopper)

e(i, , )T : Equivalent time constant of (current, speed and position) closed loops

i ,T T : Equivalent time constant of current and speed sensors

S( , , )iT : Equivalent time constant of (current, speed and position) open loops

x : State variable vector z : Operator of discrete-time Laplace ( z ) transform : Motor angular position

b : Half of backlash angle

R m, : - Position reference and measured position signal

: Torsional angle in linear two-mass elastic system

b : Torsional angle in two-mass elastic system with backlash

: Chopper voltage square-wave duty cycle : Angular acceleration

r, : DC motor magnetic field flux, and PMSM rotor field flux

( ) : Static function of backlash deadzone element

m, : Motor speed

l : Load speed

R m, : Speed reference and measured speed signal

Rlim : Speed reference limit for achievable braking during positioning

Rmax : Maximum speed reference

0 l, : Natural frequency of two-mass elastic system and load

l, : Damping ratio of two-mass elastic system and load

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Pavković D., Deur J., Lisac A. (2011). A Torque Estimator-based Control Strategy for Oil-Well Drill-string Torsional Vibrations Active Damping Including an Auto-tuning Algorithm, Control Engineering Practice, 19 (8), 836-850. [Presents an adaptive speed control system for an oil-well drill-string electrical drive based on a reduced-order state controller and compliant-drill-string torque feedback.]

Pavković D., Deur, J. (2011). Modeling and Control of Electronic Throttle Drive: A practical approach – from experimental characterization to adaptive control and application, 178 pp., Lambert Academic Publishing, Saarbrücken, Germany. [Presents a comprehensive and experimentally verified approach to modeling, identification, and position control of an automotive electronic throttle DC drive by means of a compact, PID-type, position controller extended with robust compensators of nonlinear friction and return spring effects.]

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Biographical Sketches Joško Deur (http://www.fsb.hr/acg/jdeur) received his B.Sc., M.Sc., and Ph.D. degrees in Electrical Engineering from the University of Zagreb, Croatia, in 1989, 1993, and 1999, respectively. He is an Associate Professor at the Faculty of Mechanical Engineering and Naval Architecture of the University of Zagreb, where he teaches courses in electrical machines and drives, servodrive controls, digital control systems, and automotive mechatronics. The topic of his M.Sc. thesis was on vector control of permanent magnet synchronous motors, while the Ph.D. thesis dealt with compensation of compliance and friction effects in electrical servodrive transmissions. In 2000, he spent a year with Ford Research

Laboratory in Dearborn, MI, working on different aspects of modeling and control of automotive power train systems. Prof. Deur has led about 20 projects supported by the Ministry of Science of the Republic of Croatia, the Ford Motor Company, the Jaguar Cars Ltd., and the CROSCO Integrated Drilling & Well Services Company. His main research interests include modeling, estimation, and control of automotive systems and servosystems. He has published about 20 journal papers and 70 conference papers. He received the Best Paper Award at the 19th IAVSD symposium in Milan, Italy, 2005, and the National Science Award in 2006. Danijel Pavković received his B.Sc. and M.Sc. degrees in Electrical Engineering in 1998 and 2003, respectively, and his Ph.D. degree in Mechanical Engineering in 2007, all from the University of Zagreb, Croatia. Since 2007, he has been a Senior Assistant at the Faculty of Mechanical Engineering and Naval Architecture of the University of Zagreb, teaching subjects in the field of electrical machines, electrical servodrive controls, and digital control systems. He has participated on 13 research and technology projects supported by the Ministry of Science of the Republic of Croatia, the Ford Motor Company, and the CROSCO Integrated Drilling & Well Services Company. His research interests include estimation and control of electrical servodrive systems, including automotive applications. His research efforts were acknowledged by the National Science Award for young researchers in 2005.