Fundamentals of electrical drive controls_Deur_Pavković_2012

Lučića 5, HR-10002 Zagreb, Croatia Keywords: Electrical drives, control, modeling, DC motor, permanent-magnet synchronous motor, cascade control, chopper, sensors, speed control, posi

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

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

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 Deur, J., Pavković, D., “Fundamentals of Electrical Drive Controls”, UNESCO Encyclopedia of Life Support Systems, Chap 6.39.21, 2012

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 i , thus producing the motor torque On the other hand, when the a

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:

Figure 3 Block diagram of DC motor: (a) general case and (b) constant-flux case

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

(particularly for high-power machines), the steady-state curve is rather stiff, i.e the motor speed drop  due to the increase of load ml mm 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 mm ml is valid for the steady-state conditions according to Eq (1c)) If the motor speed

is lower than the idle speed:   0 e ua   , the motor operates in the driving mode (1ia 0 st

quadrant of the coordinate system in Fig 4) Otherwise, for the case when   0 (e u a and

i  ), 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 fc 1T T1 2 is very high (typically 5 kHz), the motor speed relates to the average motor voltage, which reads (Fig 7):

where T f1 c T1/T T1 2 is the output square-wave voltage duty cycle

Moreover, since 1/fc  Ta is usually satisfied, the armature current i , and thus the torque a m , is m

effectively smoothed by the lag (low-pass) nature of the armature dynamic model (2) Thus, for the square-wave (AC) armature voltage signal u ta  the armature current i has a DC form, while a

opposite is valid for the DC link voltage U and current D i (Fig 7) According to Eq (4), the D

armature voltage can be arbitrarily varied in the range [U UD, D] by changing the duty cycle  (pulse width modulation, PWM)

During the interval T of the 11 st-quadrant converter operation, the current is conducted through the transistors Q1Q1, while for the interval T the still positive motor current flows through the 2

freewheeling diodes D2D2 Since T1 , more power flows from the DC link to the motor than T2

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, T1 is still valid since the motor speed T2 and the armature voltage u are larger than zero (cf Fig 4) However, the armature current ia a is

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

freewheeling diodes D1D1 during the interval T and through transistors 1 Q2 Q2 during the interval T Thus, since 2 T1 the average motor power is negative, i.e it flows to the DC link and T2

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 Q (Fig 7) when the b

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 Q is again turned off b

Table 1 Description of four-quadrant chopper operation

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 u If the controller sampling time is aR

synchronized with the converter switching frequency, the pure delay is, statistically-speaking, equal

to half of the switching period 1/ f Thus, the converter dynamics may be approximately described c

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 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 eia iaR –iaand varies the output armature voltage reference u until it reduces the aR

earlier-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 R , armature voltage a u , or magnetic field fluxa  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 R (e.g 5 to 10 times larger than aa R , a

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

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

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 (sgn( ) const.ml  ), it is enough to increase the resistance R beyond the motor starting value, in order to allow the load to reverse the motor into aa

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

in varying load descending speed On the other hand, if the armature voltage polarity is changed and the resistance R added, the transient reversal braking in the 2aa nd 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 resistanceR aa

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 R should be added into the armature circuit aa

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 u (i.e the electrical power a ua ) ia

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 u and the speed drop a  is independent of u (see Eq (3) a

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  , the static curves become somewhat softer in the field 2

weakening region (curves c in Fig 11) The field control is achieved by varying the excitation

circuit voltage u by means of another electronic power converter The power of this converter is M

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 i must be lower than the rated a

current I to protect the motor from overheating This means that in the armature control region, an

where    and n mm ~i , the continuous-duty motor torque is limited to the rated torque a Mmn

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

m K  must be reduced below the rated torque i Mmn 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 n

P 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, Mm,max 2Mmn), 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 (sgn(mm) sgn( ) ml ), 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 4thquadrant (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 i , speeda  , 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 i a 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 i aR) 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 I (see curve c in Fig 10b), as

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 mm ~i ), and a

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 i Here, the excitation current reference M iMR

can be determined from the motor speed according to the flux weakening open-loop control strategy (Section 3) and the known magnetizing curve   f i( )M (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 e is set slightly below the R

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 e y yR1 and calculates the y

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

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

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 functionG s ) c( )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 G s does not include zeros or that they are (if stable) p( )eventually cancelled by the prefilter, as well, the control system closed-loop transfer function reads:

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 D D2, 3, .,D to the n

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 D2, .,D should be set to the optimal value of 0.5 If the response largely r

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 D r1, .,D n

3 For the higher-order controllers and processes with dominant lag dynamics, setting the dominant characteristic ratios (e.g.D D4, 5, ) to 0.5 may result in a response that is too slow

non-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 D 2

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

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

where K is the inner control loop gain eq

Figure 14 Step response of system tuned according to damping optimum (a) and illustration of

damping adjustment by varying dominant characteristic ratio D (b) 2

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 forcee k v 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 T is equated with the dominant process time constant ci T a

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

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 ch

i

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

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:

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

equivalent "parasitic" open-loop time constant Ti Since Ti is typically around 1ms (for transistor choppers and fast Hall-effect sensors; and also for fast-sampling digital controllers), the equivalent time constant T equals approximately to 2 ms, thus giving the response settling time ofei 2Tei4ms Such a fast response for a relatively slow armature system (Ta Tei) can only be achieved by a large control effort, i.e high transient peaks of the armature voltage reference u Fortunately, the aR

electrical motors allow for large control efforts, because the steady-state armature voltage is typically much smaller than the rated voltage (the armature gain Ka 1/RaKa = 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 i s ia / aR s by the equivalent lag

term with the time constant T and the steady-state gain ei Ia /Iam1/Ki (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 (T ) and the speed sensor (Tei ) are lumped into the equivalent first-order lag term open-loop time constant

Eq (17) suggests that in the presence of double integrator in the open loop (Fig 17), the equivalent closed-loop time constant Te 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 2

T  ms and T  ms (e.g incremental encoder sensor with the sampling time of 2 ms), the 1design yields Tw  ms and 3 Te 12ms, 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, Mmmax maxml ,

Eq (1c) is simplified, the speed response takes on a ramp form, and the large-signal operating mode response time ts Te is given by

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 (T  ), the equivalent open-loop time constant is equal to: 0

Figure 19 Block diagram of position control loop

For the optimal setting D2 0.5 and the realistic numerical example from the previous subsection (Te 12ms), it follows thatTe 24ms, 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 0.35

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) (ml  ) 0

4.5.2 Large-Signal Operating Mode

The proportional position controller, with the gain Kc, can be represented by the straight line in the R  coordinate system, as shown in Figure 21 (e e R m) 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 e  (origin of the coordinate system in Fig 21) without an overshoot 0

By assuming that the maximum motor torque is much larger than the load torque (Mmmax max ml , the maximum angular deceleration in the final (braking) stage of the pointing response is defined as

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: